[{"data":1,"prerenderedAt":1885},["ShallowReactive",2],{"site-chrome-en-id":3,"site-chrome-en":-1,"site-chrome-th":-1,"blog-page-en-id":287,"blog-categories-en-id":345,"blog-posts-en-id":365,"article-blog-en-id-\u002Fblog\u002Fmechanism-aware-hierarchical-causal-elasticity-modeling-platform-ecommerce":722,"og-copy-en-id":1842,"og-posts-en-id-none":1884},{"id":4,"extension":5,"footer":6,"header":116,"indexFacets":202,"meta":243,"notFound":244,"stem":250,"ui":251,"__hash__":286},"chrome_en\u002Fen\u002Fchrome.yml","yml",{"columns":7,"legal":83,"quote":84,"askAi":87,"socials":103},[8,29,47,65],{"label":9,"children":10},"Product",[11,14,17,20,23,26],{"label":12,"to":13},"Ads optimization","\u002Fen-id\u002Fblog\u002Fads-optimization",{"label":15,"to":16},"Unit economics","\u002Fen-id\u002Fblog\u002Fprofit-margin",{"label":18,"to":19},"Agentic KAM","https:\u002F\u002Flearn.dataglasslabs.com\u002Fdocs\u002Fai-and-api\u002Fassistants",{"label":21,"to":22},"One-click deploy","https:\u002F\u002Flearn.dataglasslabs.com\u002Fdocs\u002Fgrowth\u002Factions\u002Faction-flow",{"label":24,"to":25},"Integrations","https:\u002F\u002Flearn.dataglasslabs.com\u002Fdocs\u002Foverview\u002Fget-started\u002Fsupported-platforms",{"label":27,"to":28},"Pricing","\u002Fen-id\u002Fpricing",{"label":30,"children":31},"Resources",[32,35,38,41,44],{"label":33,"to":34},"Blog","\u002Fen-id\u002Fblog",{"label":36,"to":37},"Customer stories","\u002Fen-id\u002Fcustomers",{"label":39,"to":40},"Research","\u002Fen-id\u002Fresearch",{"label":42,"to":43},"Glossary","\u002Fen-id\u002Fglossary",{"label":45,"to":46},"Docs","https:\u002F\u002Flearn.dataglasslabs.com\u002Fdocs",{"label":48,"children":49},"Company",[50,53,56,59,62],{"label":51,"to":52},"About","\u002Fen-id\u002Fcompany",{"label":54,"to":55},"Contact","\u002Fen-id\u002Fcontact",{"label":57,"to":58},"Book a demo","\u002Fen-id\u002Fbook-a-demo",{"label":60,"to":61},"Privacy","\u002Fen-id\u002Flegal\u002Fprivacy",{"label":63,"to":64},"Terms","\u002Fen-id\u002Flegal\u002Fterms",{"label":66,"children":67},"DataGlass for",[68,71,74,77,80],{"label":69,"to":70},"Agencies","\u002Fen-id\u002Ffor\u002Fagencies",{"label":72,"to":73},"Brands","\u002Fen-id\u002Ffor\u002Fbrands",{"label":75,"to":76},"Planners and optimizers","\u002Fen-id\u002Ffor\u002Fplanners",{"label":78,"to":79},"Managers and analysts","\u002Fen-id\u002Ffor\u002Fmanagers",{"label":81,"to":82},"Sellers and shop owners","\u002Fen-id\u002Ffor\u002Fsellers","© 2026 DataGlass Labs. All rights reserved.",{"text":85,"caption":86},"Measure what you keep, and then grow what you measure.","The DataGlass team",{"label":88,"prompt":89,"assistants":90},"Ask AI about DataGlass","I want to understand DataGlass, a decision engine for Shopee, Lazada and TikTok Shop sellers that optimizes ads, pricing and promotions for true profit. Use web search to read their website at https:\u002F\u002Fwww.dataglasslabs.com (with an agent-readable summary at https:\u002F\u002Fwww.dataglasslabs.com\u002Fllms.txt), then explain what DataGlass does, who it is for and how it differs from an analytics 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us",{"label":192,"to":37},"Customers",{"label":27,"to":28},{"label":48,"to":52},{"label":54,"to":55},{"label":197,"to":198},"Start for free","https:\u002F\u002Fapp.dataglasslabs.com\u002Fsign-up",{"label":200,"to":201},"Sign in","https:\u002F\u002Fapp.dataglasslabs.com\u002Fsign-in",{"clear":203,"groups":204},"Clear filters",[205,218],{"key":206,"label":207,"options":208},"platforms","Platform",[209,212,215],{"value":210,"label":211},"shopee","Shopee",{"value":213,"label":214},"lazada","Lazada",{"value":216,"label":217},"tiktok","TikTok Shop",{"key":219,"label":220,"options":221},"format","Format",[222,225,228,231,234,237,240],{"value":223,"label":224},"guide","Guide",{"value":226,"label":227},"article","Article",{"value":229,"label":230},"comparison","Comparison",{"value":232,"label":233},"calculator","Calculator",{"value":235,"label":236},"paper","Paper",{"value":238,"label":239},"survey","Survey",{"value":241,"label":242},"dataset","Dataset",{},{"title":245,"description":246,"cta":247,"errorTitle":249},"Page not found","The page may have moved, or it never existed.",{"label":248,"to":119},"Back to home","Something went wrong","en\u002Fchrome",{"previous":252,"next":253,"pause":254,"play":255,"goTo":256,"expand":257,"close":258,"readMore":259,"learnMore":260,"seeAll":261,"menu":262,"back":263,"share":264,"shareOn":265,"copyLink":266,"copied":267,"onThisPage":268,"minRead":269,"sources":270,"viewMore":271,"searchPosts":272,"allPosts":273,"all":274,"filters":275,"search":276,"searchClose":277,"searchPlaceholder":278,"searchSuggested":279,"searchResults":280,"searchNoResults":281,"searchLoading":282,"relatedTitle":283,"mainNav":284,"footerNav":285},"Previous","Next","Pause","Play","Go to item","Expand","Close","Read more","Learn more","See all","Menu","Back","Share","Share on","Copy link","Link copied","On this page","{n} min read","Sources","View more","Search posts","All posts","All","Filters","Search","Close search","Search DataGlass","Suggested","Results","No results for","Loading","More to discover","Main","Footer","HZQSEulcpFGY0nWnUTNYhG2n06nO97XnUBc_pEVOFxo",{"id":288,"title":33,"blog":289,"body":327,"customers":334,"description":335,"draft":336,"draftReason":334,"extension":337,"glossary":334,"industry":334,"meta":338,"navigation":336,"ogImage":339,"path":340,"research":334,"robots":334,"schemaOrg":334,"seo":341,"showcase":336,"sitemap":342,"stem":343,"__hash__":344},"pages_en\u002Fblog.md",{"emptyLabel":290,"labels":291,"indexLabels":296,"listenLabels":307,"indexCta":320,"ctaButtons":323},"No posts match yet. Try another category or a shorter search.",{"tldr":292,"published":293,"updated":294,"related":295},"In short","Published","Updated","Keep reading",{"categories":297,"sort":298,"newest":299,"oldest":300,"grid":301,"list":302,"loadMore":303,"results":304,"result":305,"clearSearch":306},"Categories","Sort","Newest first","Oldest first","Grid view","List view","Load more","{n} results for “{q}”","1 result for “{q}”","Clear search",{"listen":308,"loading":309,"play":310,"pause":311,"back":312,"forward":313,"speed":314,"seek":315,"player":316,"close":317,"follow":318,"of":319},"Listen to article","Loading…","Play audio of this article","Pause audio","Back 15 seconds","Forward 15 seconds","Playback speed","Seek","Audio player","Pause audio and close player","Follow the text","{a} of {b}",{"headline":321,"subline":322},"Turn every post into a decision","See the numbers behind these notes for your own shops, with a demo on your data.",{"primaryCta":324,"secondaryCta":325},{"label":57,"to":58},{"label":326,"to":28},"See pricing",{"type":328,"value":329,"toc":330},"minimark",[],{"title":331,"searchDepth":332,"depth":332,"links":333},"",3,[],null,"Notes on profit, pricing and ads for sellers on Shopee, Lazada and TikTok Shop, with the numbers behind them.",false,"md",{},{"component":33},"\u002Fblog",{"title":33,"description":335},{"loc":340},"blog","VuN4_Knpp3IpkJqH-kRoXag4vgoPV2p71ZFiR-VpgzQ",[346,349,351,354,357,360,363],{"slug":347,"label":348},"profit","Profit",{"slug":350,"label":27},"pricing",{"slug":352,"label":353},"ads","Ads",{"slug":355,"label":356},"operations","Operations",{"slug":358,"label":359},"market","Market",{"slug":361,"label":362},"data-science","Data Science",{"slug":364,"label":48},"company",[366,373,379,383,388,393,397,402,407,412,418,423,428,433,438,444,450,456,462,469,475,480,486,491,496,501,506,511,517,523,528,534,540,546,551,557,563,569,574,578,583,588,594,601,607,613,619,625,631,637,643,649,655,661,667,673,679,685,690,695,700,705,710,716],{"title":367,"description":368,"category":352,"date":369,"readMinutes":370,"featured":336,"cover":334,"format":223,"platforms":371,"draft":336,"slug":372},"E-commerce ads optimization for Shopee, Lazada, and TikTok Shop, on profit","E-commerce ads optimization on profit: True ROAS, margin-based Target ROAS and nightly budget allocation for Shopee, Lazada and TikTok Shop ads, applied for you.","2026-10-23",4,[210,213,216],"ads-optimization",{"title":374,"description":375,"category":352,"date":369,"readMinutes":376,"featured":336,"cover":334,"format":232,"platforms":377,"draft":336,"slug":378},"What ROAS is break-even? Calculate break-even ROAS and your maximum ad spend per order","Break-even ROAS is 1 ÷ your contribution-margin ratio. It does not depend on how much you spend on ads. This calculator derives it from a dated Shopee Thailand fee rate — or one you type in — then tells you the number sellers actually need: the most baht one order can pay for ads before it stops making money.",9,[210],"break-even-roas-calculator",{"title":380,"description":381,"category":364,"date":369,"readMinutes":332,"featured":336,"cover":334,"format":229,"platforms":334,"draft":336,"slug":382},"DataGlass vs a marketing agency for marketplace profit","When an agency is the right hire, where the agency model has structural blind spots for marketplace profit, and how DataGlass complements an agency engagement.","dataglass-vs-agency",{"title":384,"description":385,"category":364,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":229,"platforms":334,"draft":336,"slug":387},"DataGlass vs an ERP for marketplace decision-making","When an ERP is the right tool, where it falls short for marketplace ad\u002Fpricing decisions, and how DataGlass complements (not replaces) the ERP for SEA sellers.",2,"dataglass-vs-erp",{"title":389,"description":390,"category":364,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":229,"platforms":391,"draft":336,"slug":392},"DataGlass vs Shopee Seller Center for profit-aware operations","Shopee Seller Center is essential for running a Shopee shop, but it does not show profit-aware metrics or cross-platform views. How DataGlass complements it.",[210],"dataglass-vs-shopee-seller-center",{"title":394,"description":395,"category":364,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":229,"platforms":334,"draft":336,"slug":396},"DataGlass vs spreadsheets for marketplace sellers","When spreadsheets are enough, where they break for marketplace sellers, and how DataGlass connects ads, COGS, fees, and inventory into one decision-ready view.","dataglass-vs-spreadsheet",{"title":398,"description":399,"category":355,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":223,"platforms":400,"draft":336,"slug":401},"Prevent stockouts and overstock across Shopee, Lazada, and TikTok Shop","Prevent Shopee, Lazada, and TikTok Shop stockouts with demand-aware forecasting, campaign-day planning, and cross-platform inventory unification.",[210,213,216],"inventory-stockout",{"title":403,"description":404,"category":355,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":226,"platforms":405,"draft":336,"slug":406},"Lazada integration for marketplace profit optimization","DataGlass integrates with Lazada to model true profit per SKU, detect ad waste, evaluate voucher impact, and forecast stockout risk.",[213],"lazada-integration",{"title":408,"description":409,"category":347,"date":369,"readMinutes":376,"featured":336,"cover":334,"format":232,"platforms":410,"draft":336,"slug":411},"Shopee raised its fees on 4 August 2026 — how much more is your shop paying?","The new rate table is everywhere. Your number is not. Enter last month’s GMV by category and this page computes the baht difference between the schedule effective 1 June 2026 and the one effective 4 August 2026 — including the extra hit for shops that were on the discounted-commission programme.",[210],"marketplace-fee-impact-calculator",{"title":413,"description":414,"category":347,"date":369,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":416,"draft":336,"slug":417},"Optimize e-commerce profit across Shopee, Lazada, and TikTok Shop","How to optimize e-commerce profit across Shopee, Lazada and TikTok Shop: one objective, five decision variables, and the constraint that binds each.",7,[210,213,216],"optimize-ecommerce-profit",{"title":419,"description":420,"category":350,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":223,"platforms":421,"draft":336,"slug":422},"Price and promote without destroying margin","Optimize Shopee, Lazada, and TikTok Shop discounts, bundle deals, and campaign-day pricing on true margin. Project promotion impact before publishing.",[210,213,216],"pricing-promotions",{"title":424,"description":425,"category":347,"date":369,"readMinutes":370,"featured":336,"cover":334,"format":232,"platforms":426,"draft":336,"slug":427},"E-commerce profit calculator for Shopee sellers","Calculate true Shopee profit per SKU with a per-order breakdown covering commission fees, ads, COGS, and fulfillment.",[210],"profit-margin",{"title":429,"description":430,"category":355,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":226,"platforms":431,"draft":336,"slug":432},"Shopee integration for profit, ads, stock, and promotion decisions","DataGlass connects to Shopee to model true profit per SKU, detect wasted ad spend, and surface daily decisions. Live integration, no manual exports.",[210],"shopee-integration",{"title":434,"description":435,"category":355,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":226,"platforms":436,"draft":336,"slug":437},"TikTok Shop integration for ads, inventory, and profit decisions","DataGlass integrates with TikTok Shop for True ROAS on video and live ads, margin-aware discount analysis, and stock prioritization before live events.",[216],"tiktok-shop-integration",{"title":439,"description":440,"category":352,"date":441,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":442,"draft":336,"slug":443},"Shopee ROAS protection and TikTok ROI protection: when the credit actually protects profit","Shopee ROAS protection credits campaigns back to 80% of target, TikTok to 90%. Why that protects profit only above 1.25× break-even, with a worked example.","2026-10-08",[210,216],"roas-protection",{"title":445,"description":446,"category":352,"date":441,"readMinutes":447,"featured":336,"cover":334,"format":223,"platforms":448,"draft":336,"slug":449},"Shopee Ads optimization: set Target ROAS, budget and timing from profit","How to optimize Shopee Ads for profit: break-even ROAS per SKU, which lever binds, and CPC by hour from 9.11M Thai clicks. Worked THB example.",10,[210],"shopee-ads-optimization",{"title":451,"description":452,"category":347,"date":441,"readMinutes":453,"featured":336,"cover":334,"format":223,"platforms":454,"draft":336,"slug":455},"Shopee seller fees in the Philippines 2026: every fee, what it is charged on, and your break-even ROAS","Shopee Philippines seller fees 2026: commission, 2.24% transaction fee, 5.6% platform shipping, seller growth support and ₱5 order fee, worked on one order.",5,[210],"shopee-seller-fees",{"title":457,"description":458,"category":352,"date":441,"readMinutes":459,"featured":336,"cover":334,"format":223,"platforms":460,"draft":336,"slug":461},"TikTok Shop ads optimization: set GMV Max ROI from margin and affiliate mix","How to optimize TikTok Shop GMV Max for profit: break-even ROI after TikTok fees, the affiliate commission trap, and the 3% commission rule. Worked THB example.",8,[216],"tiktok-shop-ads-optimization",{"title":463,"description":464,"category":352,"date":465,"readMinutes":466,"featured":336,"cover":334,"format":226,"platforms":467,"draft":336,"slug":468},"How Shopee ads bidding works: what your Target ROAS and budget actually do to a bid","How Shopee ads bidding works, derived: GMV Max gives you two boxes — a Target ROAS and a daily budget — and every guide treats them as two levers. They are not. Both collapse into a single threshold on value-to-cost before any bid is placed, only one is binding at a time, and the other is doing nothing at all. Here is the derivation, the three regimes a campaign can be in, and the one-line test that tells you which of your two levers is switched off.","2026-09-01",13,[210],"how-shopee-ads-bidding-works",{"title":470,"description":471,"category":352,"date":465,"readMinutes":472,"featured":336,"cover":334,"format":223,"platforms":473,"draft":336,"slug":474},"Break-even ROAS is the zero line: the Target ROAS that actually maximizes Shopee profit","Setting your Target ROAS to your break-even ROAS does not protect profit — it targets exactly zero, for every possible volume. The target that maximizes profit is break-even marked up three times: by the demand your ads did not create, by the price your own expansion pays, and by the gap between your average click and your last one. On mid-market parameters those compound to 3.3×, moving the correct target on a 20%-margin product from 5.00 to 16.67.",12,[210],"profit-maximizing-target-roas",{"title":476,"description":477,"category":361,"date":465,"readMinutes":466,"featured":336,"cover":334,"format":241,"platforms":478,"draft":336,"slug":479},"Shopee ad benchmarks 2026: what a click and a thousand impressions actually cost","Across 333.9 million Shopee impressions and 9.11 million clicks on 80+ Thai shops, the price of a click varies 2.01× by hour of day while the rate at which people click varies 1.14×. CPC ranges THB 3.36 to THB 6.75, CPM ranges THB 89.16 to THB 200.15, and every behavioural input is nearly flat. The full table, the methodology, and the two ways a benchmark like this gets misused.",[210],"shopee-ads-benchmarks-cpc-cpm",{"title":481,"description":482,"category":364,"date":483,"readMinutes":484,"featured":336,"cover":334,"format":229,"platforms":334,"draft":336,"slug":485},"DataGlass is not an OMS — what the difference is, and when to run both","An OMS makes the order happen. DataGlass decides what to do next. The daily-question boundary, an explicit list of what DataGlass does not do, and the one seam: price.","2026-08-25",6,"dataglass-vs-oms",{"title":487,"description":488,"category":352,"date":483,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":489,"draft":336,"slug":490},"How much should you spend on Shopee Ads? Derive it from margin, not from feel","Break-even ROAS is 1 ÷ contribution margin ratio. Worked THB example: a fashion SKU allows THB 71 per attributed order and roughly THB 994 a day. Budget is an output.",[210],"shopee-ads-budget",{"title":492,"description":493,"category":347,"date":483,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":494,"draft":336,"slug":495},"Losing money on Shopee — the leaks a fee calculator cannot see","Shopee deducts 7.49–17.12% plus a 3.21% payment fee — the half a calculator sees. Six month-level leaks turn a 28.6% \"profit\" into 1.6%. Sourced THB waterfall.",[210],"shopee-losing-money",{"title":497,"description":498,"category":358,"date":483,"readMinutes":415,"featured":336,"cover":334,"format":229,"platforms":499,"draft":336,"slug":500},"Shopee vs TikTok Shop vs Lazada — what survives after fees and the ads it takes to sell","TikTok Shop's 11.77% fashion commission becomes a 23.2% take rate once the Commerce Growth and Transaction Fees stack. The crossover with Shopee sits near THB 3,720.",[210,213,216],"where-to-sell-shopee-tiktok-lazada",{"title":502,"description":503,"category":352,"date":504,"readMinutes":376,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":505},"CTR and Target ROAS under CPM vs CPC: what changes when marketplaces bill for impressions","Under cost-per-click billing, a better creative is worth nothing to your unit economics — it buys the same clicks from fewer impressions, and your cost per click does not move. Under cost-per-impression billing the same improvement cuts the price of every click by g\u002F(1+g). Five lemmas on the transition, and why click-through rate is the input most marketplace sellers are least prepared to have priced.","2026-08-18","ctr-target-roas-cpm-vs-cpc",{"title":507,"description":508,"category":352,"date":509,"readMinutes":447,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":510},"What is ROAS — and why a high ROAS doesn't always mean more sales or more profit","ROAS (Return on Ad Spend) is sales ÷ ad spend: spend THB 1, get THB 20 of sales back, and your ROAS is 20. But a high ROAS only means a high return per ad baht — not the highest total sales, and not real profit once you subtract COGS, platform fees, payment fees, vouchers, shipping, and returns. Here is what ROAS is, what it hides, and the number sellers should actually watch.","2026-06-01","what-is-roas",{"title":512,"description":513,"category":352,"date":514,"readMinutes":459,"featured":336,"cover":334,"format":223,"platforms":515,"draft":336,"slug":516},"What is Target ROAS? How to set it in Shopee GMV Max without burning profit","What is Target ROAS? It is the revenue-to-ad-spend ratio Shopee GMV Max optimises toward — and on the Shopee fee schedule effective 4 August 2026, a THB 1,000 product with THB 600 of COGS is left with 7.7% margin before ads, so a Target ROAS of 5 recovers THB 0.39 of contribution per THB 1.00 of ad spend while the campaign reports that it hit its target. What Target ROAS means, how to set it in GMV Max, and the two formulas — break-even ROAS and target-margin ROAS — that protect your margin.","2026-05-30",[210],"how-to-set-target-roas-gmv-max",{"title":518,"description":519,"category":352,"date":520,"readMinutes":484,"featured":336,"cover":334,"format":226,"platforms":521,"draft":336,"slug":522},"Platform ROAS vs true ROAS: why your ads make sales but no profit","A ROAS of 5 feels like a win — until you notice the platform computed it on gross attributed sales, before fees, vouchers, shipping, and COGS, and while still counting orders that were later returned or cancelled. The dashboard number is a revenue ratio wearing a profit costume. A primer on what platform ROAS hides and how true ROAS is calculated across Shopee, Lazada, and TikTok Shop.","2026-05-29",[210,213,216],"platform-roas-vs-true-roas",{"title":524,"description":525,"category":361,"date":520,"readMinutes":484,"featured":336,"cover":334,"format":226,"platforms":526,"draft":336,"slug":527},"What is e-commerce optimization?","Optimization is not a marketing word — it is an operations-research discipline: use data and math to compute the best decision against an explicit objective. Point it at a Shopee, Lazada, or TikTok Shop store and the objective is true profit, not sales. A primer on why those two numbers stopped moving together, and what optimization actually answers.",[210,213,216],"what-is-ecommerce-optimization",{"title":529,"description":530,"category":347,"date":531,"readMinutes":376,"featured":336,"cover":334,"format":223,"platforms":532,"draft":336,"slug":533},"How to cut Shopee seller costs: the supply-side margin audit","The median gap between COGS refreshes on a top-50 Shopee catalogue is 87 days, and over that window supplier and packaging drift alone compresses contribution margin by 1.5–3.0 percentage points. The eight supply-side cost lines that compound against Shopee margin between audits, ranked by recoverable magnitude, with the review cadence that holds each one.","2026-05-08",[210],"how-to-increase-shopee-profit-without-more-sales",{"title":535,"description":536,"category":352,"date":531,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":538,"draft":336,"slug":539},"How to sell more on Shopee without burning margin","The standard Shopee growth advice — discount harder, scale ad budget, opt into every campaign — works against the operator at scale. A research note on the lever order that actually compounds Shopee revenue, and the metric that tells you whether the lift was earned or bought.",11,[210],"how-to-sell-more-on-shopee",{"title":541,"description":542,"category":347,"date":543,"readMinutes":484,"featured":336,"cover":334,"format":223,"platforms":544,"draft":336,"slug":545},"Increase profit on Shopee without selling more","Shopee ROAS hides commission, fees, vouchers and shipping. Optimize Shopee profit with the per-SKU break-even framework that replaces it — 280 Thai accounts.","2026-05-07",[210],"increase-shopee-profit",{"title":547,"description":548,"category":358,"date":543,"readMinutes":484,"featured":336,"cover":334,"format":223,"platforms":549,"draft":336,"slug":550},"Increase sales on Shopee in 2026","Shopee CPC is climbing — buying demand got pricier while repeat demand stayed flat. The three levers that compound revenue, plus a 90-day operator cadence.",[210],"increase-shopee-sales",{"title":552,"description":553,"category":361,"date":543,"readMinutes":554,"featured":336,"cover":334,"format":235,"platforms":555,"draft":336,"slug":556},"Mechanism-Aware Hierarchical Causal Elasticity Modeling for Platform E-Commerce","A methodological working paper proposing MA-HCEM: a hierarchical Bayesian, double-machine-learning, mechanism-aware estimator for SKU-level price elasticity on Shopee, Lazada, and TikTok Shop, with Bundle Deal and voucher mechanisms treated as structured price discrimination.",27,[210,213,216],"mechanism-aware-hierarchical-causal-elasticity-modeling-platform-ecommerce",{"title":558,"description":559,"category":350,"date":543,"readMinutes":560,"featured":336,"cover":334,"format":226,"platforms":561,"draft":336,"slug":562},"Price elasticity modeling for marketplace sellers — why Shopee, Lazada, and TikTok Shop pricing decisions need it","A practical guide from DataGlass Labs Research on the most under-used lever in Southeast Asian e-commerce: knowing how customers actually respond to price changes.",14,[210,213,216],"price-elasticity-modeling-shopee-lazada-tiktok-shop",{"title":564,"description":565,"category":361,"date":566,"readMinutes":567,"featured":336,"cover":334,"format":235,"platforms":334,"draft":336,"slug":568},"From Gut Feel to Posterior Inference: A Research Article on the DataGlass Decision-Intelligence System for E-Commerce Ad Budget Allocation","A rigorous public communication of the DataGlass decision-intelligence system for daily ad-budget allocation on platform-controlled marketplaces — the pre-algorithmic baseline, the analytical reasons it fails (the reported-vs-true ROAS gap and the rolling-mean bias), the Bayesian + bandits-with-knapsacks methodology, and the empirical 18–24% portfolio-profit lift with reduced reallocation frequency.","2026-05-04",20,"bayesian-budget-allocation-for-marketplace-ads",{"title":570,"description":571,"category":361,"date":566,"readMinutes":572,"featured":336,"cover":334,"format":238,"platforms":334,"draft":336,"slug":573},"Decision Intelligence for E-commerce: How Retailers Optimise Pricing, Demand Forecasting, Inventory, Promotions & Personalization","Pricing, forecasting, inventory, promotions, and personalization — a deep technical survey of the techniques large retailers use, the variants that matter, and how to deploy them",42,"decision-intelligence-playbook-for-e-commerce",{"title":575,"description":576,"category":361,"date":566,"readMinutes":332,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":577},"DataGlass Research Methodology","A single canonical reference for the sample frame, time window, attribution rules, and exclusions that calibrate every \"in our data\" claim across the blog and research surfaces.","methodology",{"title":579,"description":580,"category":361,"date":566,"readMinutes":581,"featured":336,"cover":334,"format":238,"platforms":334,"draft":336,"slug":582},"Prediction and Risk Optimization Under Uncertainty: A Cross-Domain Meta-Review","Mature decision systems in finance, operations, insurance, energy, healthcare, causal inference, and e-commerce share the same four primitives — a calibrated probabilistic model, a coherent risk-aware objective, an operational constraint set, and a principled exploration mechanism. Eleven worked cases ground the framework, with the DataGlass marketplace ad-budget system as the connecting tissue.",49,"prediction-and-risk-optimization-meta-review",{"title":584,"description":585,"category":355,"date":566,"readMinutes":560,"featured":336,"cover":334,"format":226,"platforms":586,"draft":336,"slug":587},"Shopee optimization in 2026 is a feedback loop, not a checklist","The popular \"10 ways to optimize your Shopee shop\" advice optimises the wrong objective. The lever that actually moves margin is the tightness of the diagnose → decide → operate loop — and most checklists skip the diagnose stage entirely.",[210],"shopee-optimization-in-2026",{"title":589,"description":590,"category":358,"date":566,"readMinutes":591,"featured":134,"cover":334,"format":241,"platforms":592,"draft":336,"slug":593},"Shopee Sellers in 2026: Southeast Asia E-commerce Market Research, GMV & Seller Economics","Market growth has resumed; seller economics have become more exacting. Southeast Asia's platform e-commerce reached US$157.6B in 2025 (up 22.8% YoY), top-three platforms now control about 98.8% of platform GMV, and content commerce accounts for ~32% of platform GMV. The 2026 question for sellers is no longer \"how big is the market?\" — it is \"who controls the decision loop?\"",22,[210],"southeast-asia-seller-market-research-2026",{"title":595,"description":596,"category":355,"date":597,"readMinutes":598,"featured":336,"cover":334,"format":223,"platforms":599,"draft":336,"slug":600},"Shopee fee structure and commission calculation in 2026: the six layers, the four-term formula","Most 'Shopee fees' explainers stop at commission and transaction fee. The four layers underneath — voucher co-funding, Free Shipping Program subsidy, Mall premium, payment-channel pass-through — are where the gap between dashboard revenue and bank deposit actually lives. With a sourced layer-by-layer breakdown, the four-term commission-per-order formula, two worked THB examples, a sensitivity table, and the reconciliation procedure.","2026-05-03",18,[210],"shopee-fee-structure-2026",{"title":602,"description":603,"category":352,"date":604,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":605,"draft":336,"slug":606},"Cross-platform ad budget allocation for SEA marketplace sellers","Most multi-platform sellers split ad budget across Shopee, Lazada, and TikTok Shop by historical revenue share. The math says that's wrong. Optimal allocation equalises marginal ROAS, not historical share — and the gap between the two on a typical account is 4–7 percentage points of net contribution margin per quarter.","2026-04-29",[210,213,216],"cross-platform-ad-budget-allocation",{"title":608,"description":609,"category":352,"date":610,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":611,"draft":336,"slug":612},"Why scaling Shopee ad budget grows revenue and shrinks profit","A Shopee account moving its monthly ad budget from THB 40,000 to THB 100,000 commonly grows revenue 60% and loses a third of its net profit — and not one decision in that sequence looks wrong on the dashboard. The arithmetic of the scaling trap, why platform-reported ROAS is blind to it by construction, and the per-SKU break-even bar that stops it.","2026-04-22",[210],"how-to-increase-profit-on-shopee",{"title":614,"description":615,"category":347,"date":616,"readMinutes":466,"featured":336,"cover":334,"format":223,"platforms":617,"draft":336,"slug":618},"How to increase profit on Lazada in 2026","The LazMall badge lifts conversion. It also raises commission, mandates free-shipping subsidies, and pulls the price ceiling down through the platform's own competitive-parity rules. Whether the badge pays is a per-SKU question. A research note on the LazMall economics, why Sponsored Discovery leaks more margin than Sponsored Search, and the audit that recovers 4–6 percentage points of margin in 30 days.","2026-04-15",[213],"how-to-increase-profit-on-lazada",{"title":620,"description":621,"category":352,"date":622,"readMinutes":466,"featured":336,"cover":334,"format":223,"platforms":623,"draft":336,"slug":624},"How to reduce Shopee ad waste without killing sales","On a typical Shopee account, 20–30% of ad spend runs at a structural loss the platform dashboard ranks as winning campaigns. Pausing \"underperformers\" misses the leak. A research note on the two structural defaults that cause hidden ad waste — and the audit that surfaces it without losing revenue.","2026-04-08",[210],"how-to-reduce-shopee-ad-waste",{"title":626,"description":627,"category":347,"date":628,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":629,"draft":336,"slug":630},"How to calculate Shopee seller margin","The math, the inputs, and the program-specific traps. Why Shopee's in-dashboard income view almost always overstates margin — and how to reconstruct true contribution margin per SKU using the order-line data Shopee already exposes via Open Platform.","2026-04-05",[210],"how-to-calculate-shopee-seller-margin",{"title":632,"description":633,"category":347,"date":634,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":635,"draft":336,"slug":636},"How to calculate Lazada seller margin","The math, the inputs, and the program-specific traps. Why Lazada's in-platform P&L summary almost always overstates margin — and how to reconstruct true contribution margin per SKU using the order-line data Lazada exposes via the Open Platform.","2026-04-01",[213],"how-to-calculate-lazada-seller-margin",{"title":638,"description":639,"category":352,"date":640,"readMinutes":560,"featured":336,"cover":334,"format":223,"platforms":641,"draft":336,"slug":642},"How to calculate true Shopee ROAS for profit","A methodology note. Shopee's in-platform ROAS is gross-revenue based and structurally biased toward overspend at scale. True ROAS is the same formula with one input substituted — and that substitution flips winners into losses on roughly half the typical Shopee catalog. With charts, three SKU profiles, sensitivity analysis, and the operating procedure that applies the substitution at production cadence.","2026-03-25",[210],"how-to-calculate-true-shopee-roas",{"title":644,"description":645,"category":352,"date":646,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":647,"draft":336,"slug":648},"How to reduce Lazada ad waste","Lazada runs two structurally different ad products — Sponsored Search and Sponsored Discovery — with break-even economics that diverge by ~1.4–1.8× in true-ROAS terms. A single account-wide ROAS target misjudges both. A research note on the audit that surfaces the structural waste, the pruning order that recovers margin without losing revenue, and the per-placement framework that survives Pay Day and 11.11.","2026-03-18",[213],"how-to-reduce-lazada-ad-waste",{"title":650,"description":651,"category":347,"date":652,"readMinutes":466,"featured":336,"cover":334,"format":223,"platforms":653,"draft":336,"slug":654},"How to find low-margin SKUs on Shopee","On a typical Shopee account, the top-10 SKUs by revenue and the top-10 by contribution profit overlap by roughly 50%. Half of every shop's bestsellers are not the most profitable products. A research note on the audit that surfaces the gap, the patterns hiding inside it, and the per-SKU operating decisions that recover margin.","2026-03-12",[210],"how-to-find-low-margin-skus-on-shopee",{"title":656,"description":657,"category":350,"date":658,"readMinutes":560,"featured":336,"cover":334,"format":223,"platforms":659,"draft":336,"slug":660},"How to plan Lazada campaigns around profit","Pay Day, Mega Sale, 11.11, and the long tail of category windows make participation feel mandatory. The platform documents the eligibility tiers; the seller absorbs them. A research note on what each campaign actually costs, when participation pays, and the margin-first rubric that survives all three.","2026-03-04",[213],"how-to-plan-lazada-campaigns-around-profit",{"title":662,"description":663,"category":347,"date":664,"readMinutes":472,"featured":336,"cover":334,"format":223,"platforms":665,"draft":336,"slug":666},"How to increase profit on TikTok Shop in 2026","TikTok Shop is the only SEA marketplace with a stacked second commission — affiliate commission (10–25% via the Open Affiliate Plan) layered on top of platform commission. A 6.0 platform ROAS routinely becomes ~1.4 true ROAS once the full four-line cost stack is subtracted. A research note on the affiliate-stack arithmetic, live-stream pricing discipline, and the per-SKU framework that recovers margin without retreating from the platform.","2026-02-26",[216],"how-to-increase-profit-on-tiktok-shop",{"title":668,"description":669,"category":358,"date":670,"readMinutes":671,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":672},"The SEA marketplace in 2026: a fragmented arena with one survival rule","Eighteen distinct platform-market cells, six countries, three platforms, and roughly a million sellers competing on a saturated product surface. The aggregate is growing; the per-cell margin distribution is compressing. A research note on the structural shifts and the operating rule that survives them.","2026-02-22",16,"sea-marketplace-2026",{"title":674,"description":675,"category":361,"date":676,"readMinutes":537,"featured":336,"cover":334,"format":226,"platforms":677,"draft":336,"slug":678},"Data ingestion for Shopee sellers: why zero-setup analytics matters","Most Shopee sellers don't have a strategy problem first. They have a data plumbing problem — orders, ads, COGS, fees, vouchers, inventory, pricing, and returns live in seven different surfaces, and by the time the seller has stitched them together the campaign is over. A research note on the data-source matrix, the canonical-entity model, and the zero-setup architecture that recovers ~10 hours per week.","2026-02-18",[210],"data-ingestion-for-shopee-sellers",{"title":680,"description":681,"category":358,"date":682,"readMinutes":415,"featured":336,"cover":334,"format":226,"platforms":683,"draft":336,"slug":684},"The race to zero margin — and how Shopee, Lazada, and TikTok Shop got there","Read the platform documentation end-to-end and the conclusion is uncomfortable: the discounts buyers see, the free shipping that drives conversion, and the affiliate spend that drives reach are all paid by the seller. The race-to-zero is the equilibrium output of that design.","2026-02-08",[210,213,216],"race-to-zero-margin",{"title":686,"description":687,"category":361,"date":688,"readMinutes":560,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":689},"ML demand forecasting for e-commerce sellers","Machine learning in e-commerce gets discussed in vague terms; for marketplace sellers the operating question is concrete — how many units of this SKU will sell in the next N days, with what confidence, and what decision flows from the answer? A research note on the practical model architecture, the stockout-distortion problem, sensitivity analysis, and the operating decisions forecasts feed.","2026-02-04","ml-demand-forecasting-for-ecommerce",{"title":691,"description":692,"category":358,"date":693,"readMinutes":484,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":694},"Complexity is the new tax on small sellers","A multi-shop seller in 2026 logs into seven platforms, reconciles four fee schedules, exports six CSVs, and re-enters COGS by hand. Big brands absorb the cost with a data team. Small sellers absorb it with their evenings — and it's the single biggest reason multi-shop operators stall before they reach scale.","2026-01-25","complexity-is-the-tax",{"title":696,"description":697,"category":355,"date":698,"readMinutes":472,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":699},"Stockout math for e-commerce sellers","A stockout is not one cost; it is five compounding costs. Lost contribution profit on the missed unit, plus wasted ad spend during the stockout window, plus algorithmic ranking demotion, plus repeat-buyer trust erosion, plus distorted forecasting that increases the likelihood of the next stockout. A research note on the multi-line stockout cost function, the per-SKU reorder-point math that accounts for it, and the campaign-aware adjustment that survives Pay Day and 11.11.","2026-01-18","inventory-stockout-math-ecommerce",{"title":701,"description":702,"category":350,"date":703,"readMinutes":537,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":704},"Dynamic pricing for marketplace sellers","Discounting is easy. Profitable pricing is hard. A 30% volume lift on a 10% price cut routinely lowers total contribution profit — the math says volume must lift by ~33% just to break even, and most SKUs underperform that bar. A research note on the price-elasticity arithmetic, the inventory × demand four-quadrant framework, and the per-SKU pricing decision that survives campaign-window pressure.","2026-01-04","dynamic-pricing-ecommerce-marketplaces",{"title":706,"description":707,"category":361,"date":708,"readMinutes":472,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":709},"E-commerce Decision Engine: How Marketplace Sellers Turn Data Into Profit Recommendations","A dashboard tells you what happened. A decision engine tells you what to do next, ranks the options by projected profit lift, and surfaces the math behind every recommendation. A research note on the five-layer architecture that separates the two, why marketplace commerce now requires the latter, and where the operating model breaks.","2025-12-12","how-ecommerce-decision-engines-work",{"title":711,"description":712,"category":355,"date":713,"readMinutes":459,"featured":336,"cover":334,"format":226,"platforms":714,"draft":336,"slug":715},"Multi-shop analytics for Shopee, Lazada, and TikTok Shop sellers","Modern marketplace sellers rarely operate on one channel. They have multiple dashboards but not one operating view. They know sales by channel — but not profit by channel.","2025-11-26",[210,213,216],"multi-shop-analytics-lazada-shopee-tiktok",{"title":717,"description":718,"category":358,"date":719,"readMinutes":720,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":721},"The new competitive world of commerce","A research note on what changed in marketplace commerce between 2022 and 2026. Platforms became active algorithmic counterparties; sellers who treat them as passive listing markets are competing against optimization systems they cannot see. The fix is to run your own optimization layer on top.","2025-11-12",15,"new-competitive-world-of-commerce",{"entry":723,"document":830,"toc":1795},{"id":724,"title":552,"category":361,"cover":334,"cta":334,"date":543,"description":553,"draft":336,"draftReason":334,"extension":337,"featured":336,"format":235,"listen":334,"math":134,"meta":725,"navigation":334,"ogImage":334,"path":726,"platforms":727,"readMinutes":554,"related":334,"robots":334,"schemaOrg":334,"seo":728,"sitemap":731,"sources":732,"stem":827,"tldr":334,"topic":828,"updatedAt":334,"__hash__":829},"blog_en\u002Fblog\u002Fmechanism-aware-hierarchical-causal-elasticity-modeling-platform-ecommerce.md",{},"\u002Fblog\u002Fmechanism-aware-hierarchical-causal-elasticity-modeling-platform-ecommerce",[210,213,216],{"title":729,"description":730},"Mechanism-Aware Causal Elasticity Modeling | DataGlass Research","Working paper proposing MA-HCEM — hierarchical Bayesian + double-ML price elasticity for Shopee, Lazada, and TikTok Shop with voucher and cold-start structure.",{"loc":726},[733,735,737,739,741,743,745,747,749,751,753,755,757,759,761,763,765,767,769,771,773,775,777,779,781,783,785,787,789,791,793,795,797,799,801,803,805,807,809,811,813,815,817,819,821,823,825],{"title":734},"Allenby, G. 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Cambridge University Press.",{"title":816},"Ursu, R. M. (2018). The power of rankings: Quantifying the effect of rankings on online consumer search and purchase decisions. Marketing Science, 37(4), 530-552.",{"title":818},"Varian, H. R. (1992). Microeconomic Analysis (3rd ed.). W. W. Norton.",{"title":820},"Wager, S., and Athey, S. (2018). Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, 113(523), 1228-1242.",{"title":822},"Wilson, R. (1993). Nonlinear Pricing. Oxford University Press.",{"title":824},"Working, E. J. (1927). What do statistical \"demand curves\" show? Quarterly Journal of Economics, 41(2), 212-235.",{"title":826},"Wright, P. G. (1928). The Tariff on Animal and Vegetable Oils. Macmillan.","blog\u002Fmechanism-aware-hierarchical-causal-elasticity-modeling-platform-ecommerce","models","aIN_3mMn1-OrKKcXWCUE8gXBn07Nx6b8blW54zv3x5k",{"frontmatter":831,"meta":882,"nodes":1009},{"title":552,"description":553,"seo":832,"category":361,"format":235,"platforms":833,"topic":828,"date":543,"readMinutes":554,"math":134,"sources":834},{"title":729,"description":730},[210,213,216],[835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881],{"title":734},{"title":736},{"title":738},{"title":740},{"title":742},{"title":744},{"title":746},{"title":748},{"title":750},{"title":752},{"title":754},{"title":756},{"title":758},{"title":760},{"title":762},{"title":764},{"title":766},{"title":768},{"title":770},{"title":772},{"title":774},{"title":776},{"title":778},{"title":780},{"title":782},{"title":784},{"title":786},{"title":788},{"title":790},{"title":792},{"title":794},{"title":796},{"title":798},{"title":800},{"title":802},{"title":804},{"title":806},{"title":808},{"title":810},{"title":812},{"title":814},{"title":816},{"title":818},{"title":820},{"title":822},{"title":824},{"title":826},{"toc":883},{"title":331,"searchDepth":386,"depth":386,"links":884},[885,888,891,910,926,948,958,971,987,1000,1003,1006],{"id":886,"depth":386,"text":887},"abstract","Abstract",{"id":889,"depth":386,"text":890},"_1-introduction","1. Introduction",{"id":892,"depth":386,"text":893,"children":894},"_2-a-brief-history-of-price-elasticity-estimation","2. A Brief History of Price Elasticity Estimation",[895,898,901,904,907],{"id":896,"depth":332,"text":897},"_2-a-brief-history-of-price-elasticity-estimation-_21-the-classical-period-marshall-to-the-cowles-commission","2.1 The classical period: Marshall to the Cowles Commission",{"id":899,"depth":332,"text":900},"_2-a-brief-history-of-price-elasticity-estimation-_22-structural-demand-aids-logit-and-blp","2.2 Structural demand: AIDS, logit, and BLP",{"id":902,"depth":332,"text":903},"_2-a-brief-history-of-price-elasticity-estimation-_23-the-machine-learning-turn-causal-forests-double-ml-and-deep-iv","2.3 The machine-learning turn: causal forests, double ML, and deep IV",{"id":905,"depth":332,"text":906},"_2-a-brief-history-of-price-elasticity-estimation-_24-hierarchical-bayesian-and-partial-pooling-approaches","2.4 Hierarchical Bayesian and partial-pooling approaches",{"id":908,"depth":332,"text":909},"_2-a-brief-history-of-price-elasticity-estimation-_25-what-changed-and-what-did-not","2.5 What changed, and what did not",{"id":911,"depth":386,"text":912,"children":913},"_3-why-platform-e-commerce-needs-a-different-estimator","3. Why Platform E-Commerce Needs a Different Estimator",[914,917,920,923],{"id":915,"depth":332,"text":916},"_3-why-platform-e-commerce-needs-a-different-estimator-_31-the-algorithmic-confounder","3.1 The algorithmic confounder",{"id":918,"depth":332,"text":919},"_3-why-platform-e-commerce-needs-a-different-estimator-_32-mechanism-induced-kinks","3.2 Mechanism-induced kinks",{"id":921,"depth":332,"text":922},"_3-why-platform-e-commerce-needs-a-different-estimator-_33-the-cold-start-and-long-tail-problem","3.3 The cold-start and long-tail problem",{"id":924,"depth":332,"text":925},"_3-why-platform-e-commerce-needs-a-different-estimator-_34-the-decision-makers-loss-function","3.4 The decision-maker's loss function",{"id":927,"depth":386,"text":928,"children":929},"_4-the-mechanism-aware-hierarchical-causal-elasticity-model","4. The Mechanism-Aware Hierarchical Causal Elasticity Model",[930,933,936,939,942,945],{"id":931,"depth":332,"text":932},"_4-the-mechanism-aware-hierarchical-causal-elasticity-model-_41-notation-and-data-structure","4.1 Notation and data structure",{"id":934,"depth":332,"text":935},"_4-the-mechanism-aware-hierarchical-causal-elasticity-model-_42-generative-model","4.2 Generative model",{"id":937,"depth":332,"text":938},"_4-the-mechanism-aware-hierarchical-causal-elasticity-model-_43-hierarchical-prior","4.3 Hierarchical prior",{"id":940,"depth":332,"text":941},"_4-the-mechanism-aware-hierarchical-causal-elasticity-model-_44-mixing-weight-and-mechanism-response","4.4 Mixing weight and mechanism response",{"id":943,"depth":332,"text":944},"_4-the-mechanism-aware-hierarchical-causal-elasticity-model-_45-algorithmic-confounder-adjustment-via-dml","4.5 Algorithmic-confounder adjustment via DML",{"id":946,"depth":332,"text":947},"_4-the-mechanism-aware-hierarchical-causal-elasticity-model-_46-randomized-perturbations-as-experimental-priors","4.6 Randomized perturbations as experimental priors",{"id":949,"depth":386,"text":950,"children":951},"_5-identification","5. Identification",[952,955],{"id":953,"depth":332,"text":954},"_5-identification-_51-identification-of-the-demand-elasticity","5.1 Identification of the demand elasticity",{"id":956,"depth":332,"text":957},"_5-identification-_52-what-identification-does-not-buy","5.2 What identification does not buy",{"id":959,"depth":386,"text":960,"children":961},"_6-posterior-inference","6. Posterior Inference",[962,965,968],{"id":963,"depth":332,"text":964},"_6-posterior-inference-_61-inference-target","6.1 Inference target",{"id":966,"depth":332,"text":967},"_6-posterior-inference-_62-stochastic-variational-inference","6.2 Stochastic variational inference",{"id":969,"depth":332,"text":970},"_6-posterior-inference-_63-posterior-predictive-checks","6.3 Posterior predictive checks",{"id":972,"depth":386,"text":973,"children":974},"_7-theoretical-comparison-with-canonical-baselines","7. Theoretical Comparison with Canonical Baselines",[975,978,981,984],{"id":976,"depth":332,"text":977},"_7-theoretical-comparison-with-canonical-baselines-_71-decomposition-of-the-elasticity-estimation-error","7.1 Decomposition of the elasticity-estimation error",{"id":979,"depth":332,"text":980},"_7-theoretical-comparison-with-canonical-baselines-_72-failure-modes-of-canonical-baselines","7.2 Failure modes of canonical baselines",{"id":982,"depth":332,"text":983},"_7-theoretical-comparison-with-canonical-baselines-_73-what-ma-hcem-addresses","7.3 What MA-HCEM addresses",{"id":985,"depth":332,"text":986},"_7-theoretical-comparison-with-canonical-baselines-_74-conditions-under-which-ma-hcem-does-not-dominate","7.4 Conditions under which MA-HCEM does not dominate",{"id":988,"depth":386,"text":989,"children":990},"_8-discussion","8. Discussion",[991,994,997],{"id":992,"depth":332,"text":993},"_8-discussion-_81-managerial-implications","8.1 Managerial implications",{"id":995,"depth":332,"text":996},"_8-discussion-_82-relationship-to-dynamic-pricing","8.2 Relationship to dynamic pricing",{"id":998,"depth":332,"text":999},"_8-discussion-_83-relationship-to-existing-platform-data-literature","8.3 Relationship to existing platform-data literature",{"id":1001,"depth":386,"text":1002},"_9-limitations-and-replication-agenda","9. Limitations and Replication Agenda",{"id":1004,"depth":386,"text":1005},"_10-conclusion","10. Conclusion",{"id":1007,"depth":386,"text":1008},"references","References",[1010,1017,1024,1027,1030,1037,1039,1041,1044,1047,1050,1056,1059,1062,1064,1066,1069,1072,1083,1094,1096,1099,1106,1109,1111,1126,1129,1131,1138,1140,1143,1150,1152,1154,1156,1159,1166,1168,1171,1174,1176,1179,1182,1184,1187,1190,1192,1194,1196,1266,1268,1274,1278,1300,1303,1306,1309,1312,1322,1324,1327,1330,1333,1351,1353,1359,1362,1368,1370,1376,1379,1394,1396,1402,1405,1407,1409,1411,1420,1437,1462,1476,1497,1503,1505,1508,1514,1517,1520,1522,1524,1526,1540,1542,1551,1553,1556,1558,1560,1563,1565,1587,1590,1592,1609,1615,1621,1627,1633,1635,1638,1645,1652,1658,1661,1664,1666,1669,1676,1683,1690,1697,1700,1702,1704,1706,1709,1712,1725,1728,1730,1733,1735,1738,1740,1742,1745,1752,1755,1761,1764,1767,1770,1772,1774,1777,1788,1791,1793],[1011,1012,1013],"p",{},[1014,1015,1016],"em",{},"Working paper, version 1.1, May 2026. Cite as: Bhum Soonjun and DataGlass Labs Research, \"Mechanism-Aware Hierarchical Causal Elasticity Modeling for Platform E-Commerce,\" DataGlass Labs Research working paper, May 2026.",[1011,1018,1019,1023],{},[1020,1021,1022],"strong",{},"Keywords."," Price elasticity, e-commerce, platform economics, hierarchical Bayesian models, double machine learning, mechanism design, Bundle Deal, dynamic pricing, second-degree price discrimination, demand estimation.",[1025,1026],"hr",{},[1028,1029,887],"h2",{"id":886},[1011,1031,1032,1033,1036],{},"Price elasticity of demand is the central primitive of empirical pricing, yet the canonical estimators developed for offline retail and structural industrial-organization settings transfer poorly to modern platform e-commerce. Marketplace sellers on platforms such as Shopee, Lazada, and TikTok Shop face a pricing surface that is (i) algorithmically mediated through search ranking and recommendation, (ii) governed by structured promotional mechanisms — Bundle Deal, FlexiCombo, stacked vouchers, flash sales — that implement second- and third-degree price discrimination, (iii) characterized by extreme SKU heterogeneity and frequent cold starts, and (iv) subject to severe price endogeneity because sellers actively respond to algorithmic and demand signals. We review the history of elasticity estimation from Marshall (1890) through Berry, Levinsohn, and Pakes (1995) to modern double machine learning (Chernozhukov et al., 2018), and we identify three structural gaps in the existing literature when applied to platform e-commerce: insufficient treatment of mechanism-induced price discrimination, weak handling of cold-start SKUs, and inadequate uncertainty quantification for managerial decisions. We propose the ",[1020,1034,1035],{},"Mechanism-Aware Hierarchical Causal Elasticity Model (MA-HCEM)",", a unified framework that combines (a) a hierarchical Bayesian backbone with partial pooling across SKU, category, and shop, (b) a double-machine-learning identification stage that purges algorithmic confounders from observed price variation, (c) a structural likelihood component that explicitly models Bundle Deal and voucher mechanisms as second-degree price discrimination, and (d) randomized micro-perturbations of price as an experimental prior. We state and defend an identification result for the proposed estimator, develop a theoretical bias decomposition that characterizes the structural failure modes of each canonical baseline against the platform-marketplace data-generating process, and describe a posterior-inference procedure based on stochastic variational inference with control variates. The contribution is methodological and theoretical; we make no empirical or simulation claims, and we lay out an explicit replication agenda for empirical validation.",[1025,1038],{},[1028,1040,890],{"id":889},[1011,1042,1043],{},"The single most consequential parameter in retail pricing is the price elasticity of demand: the percentage change in unit demand induced by a one-percent change in price (Marshall, 1890; Varian, 1992). It enters every textbook formula for the profit-maximizing markup (Lerner, 1934), the optimal discount depth (Tellis, 1988), the design of nonlinear tariffs (Wilson, 1993), and the value of a customer to a firm (Gupta and Lehmann, 2003). It also enters every modern dynamic-pricing system (Phillips, 2005; Talluri and van Ryzin, 2004), every revenue-management algorithm (den Boer, 2015), and every causal-inference pipeline for promotional measurement (Athey and Imbens, 2019).",[1011,1045,1046],{},"Yet despite a century of econometric attention, reliable per-product elasticity estimates remain scarce in the operating environment that now dominates Southeast Asian and increasingly global retail: third-party marketplace platforms such as Shopee, Lazada, and TikTok Shop. Three structural features of these platforms make standard estimation approaches fragile. First, the seller does not control the demand surface in isolation: the platform's search-ranking and recommendation algorithms mediate exposure as a function of price, advertising bid, conversion history, and inventory state (Ursu, 2018; Goldfarb and Tucker, 2019). Second, promotional surfaces are mechanism-rich: a Shopee Bundle Deal is structurally a quantity discount that operates as second-degree price discrimination (Tirole, 1988); a stacked Lazada FlexiCombo voucher is a hybrid of second- and third-degree price discrimination; a TikTok Shop live-stream flash sale combines a discrete price drop with a time-limited attention shock. Third, sales are heavy-tailed and SKU turnover is rapid, so a typical mid-sized marketplace shop has many thin and short-lived SKUs for which classical SKU-level estimation is underpowered.",[1011,1048,1049],{},"Each of these features violates a load-bearing assumption of the canonical elasticity estimators. Linear-log regressions are biased because price is endogenous to algorithmic demand signals (Reiss and Wolak, 2007). Discrete-choice models in the BLP family (Berry, Levinsohn, and Pakes, 1995) require cross-sectional market shares that are ill-defined when the choice set itself is algorithmically curated. Modern double machine learning (Chernozhukov et al., 2018) handles confounding flexibly but provides no native way to share information across thin SKUs, exploit mechanism structure, or generate the calibrated uncertainty required by margin-constrained operators.",[1011,1051,1052,1053,1055],{},"This paper has three objectives. The first is historical and pedagogical: to trace how elasticity estimation evolved from Marshall's introspective derivation, through the cost-shifter instruments of the Cowles Commission era (Working, 1927; Wright, 1928), through structural demand systems (Deaton and Muellbauer, 1980; Berry et al., 1995), into the modern machine-learning literature (Wager and Athey, 2018; Chernozhukov et al., 2018), and to identify which assumptions hold and which fail under platform e-commerce. The second is diagnostic: to characterize the mechanism-induced biases that arise when these methods are applied naively to Shopee, Lazada, and TikTok Shop data. The third is constructive: to propose the ",[1020,1054,1035],{},", a unified estimator designed for the platform-marketplace setting, and to evaluate its theoretical properties against canonical baselines.",[1011,1057,1058],{},"The contribution is methodological and theoretical. We deliberately do not report simulation or empirical results in this version of the manuscript: the model has not yet been deployed on platform data, and we prefer to defer numerical claims to a companion empirical paper rather than to manufacture them. Section 7 instead presents a theoretical bias decomposition and an asymptotic comparison that articulate, qualitatively, why MA-HCEM is expected to dominate canonical baselines in the platform-marketplace regime, and under what conditions that dominance fails.",[1011,1060,1061],{},"The remainder of the paper is organized as follows. Section 2 reviews the history of elasticity estimation and isolates the structural assumptions on which each generation of methods depends. Section 3 motivates the platform-specific problem and documents how each assumption fails. Section 4 specifies MA-HCEM. Section 5 develops the identification argument and discusses the role of randomized price perturbations as experimental priors. Section 6 describes posterior inference. Section 7 presents the theoretical bias decomposition and asymptotic comparison. Section 8 discusses managerial implications and the relationship to dynamic pricing. Section 9 lists limitations and an agenda for empirical replication. Section 10 concludes.",[1025,1063],{},[1028,1065,893],{"id":892},[1067,1068,897],"h3",{"id":896},[1011,1070,1071],{},"The concept of elasticity of demand was formalized by Marshall (1890), though the underlying intuition that demand schedules are downward sloping and that responsiveness varies by good predates him by at least a century in the writings of Smith (1776) and Cournot (1838). Marshall's contribution was to define elasticity as a unit-free local rate, to argue that it varied systematically with product category — necessities being inelastic, luxuries elastic — and to propose introspection and direct survey as legitimate identification strategies in the absence of experimental data.",[1011,1073,1074,1075,1078,1079,1082],{},"The first formal econometric problem in elasticity estimation, the ",[1020,1076,1077],{},"identification problem",", was posed in the 1920s. Working (1927) and Wright (1928) observed that a regression of observed price on observed quantity recovers neither the demand curve nor the supply curve, but a hybrid determined by the relative variance of demand and supply shocks. This launched the program of finding ",[1014,1080,1081],{},"cost shifters"," — variables that move supply but not demand, such as input prices or weather shocks — as instruments for price. Schultz (1938) used this approach to estimate agricultural demand elasticities; the Cowles Commission (Koopmans, 1950; Hood and Koopmans, 1953) generalized it into the simultaneous-equations framework that dominated empirical price work for forty years.",[1011,1084,1085,1086,1089,1090,1093],{},"The classical period left two durable contributions. The first is the recognition that ",[1020,1087,1088],{},"price is endogenous",": it responds to demand. The second is the standard remedy of ",[1020,1091,1092],{},"instrumental variables"," under exogeneity and exclusion conditions. Both remain relevant under platform e-commerce, but the supply of valid cost-shifter instruments has thinned in a context where prices are reset algorithmically in response to platform-internal signals.",[1067,1095,900],{"id":899},[1011,1097,1098],{},"A second wave, beginning with Deaton and Muellbauer's (1980) Almost Ideal Demand System (AIDS), pursued elasticities through theoretically consistent demand systems derived from utility maximization. AIDS imposed Slutsky symmetry and adding-up but assumed continuous quantities; this was suitable for grocery basket data but ill-suited to the discrete, infrequent purchases that dominate durable-goods retail.",[1011,1100,1101,1102,1105],{},"The discrete-choice tradition, originating with McFadden (1974) and culminating in Berry, Levinsohn, and Pakes (1995, henceforth BLP), addressed this by modeling each consumer as choosing one product from a finite menu. BLP allowed for unobserved product characteristics correlated with price, used aggregate market shares as the empirical target, and instrumented for price using ",[1014,1103,1104],{},"characteristic-space"," instruments. It became the workhorse of empirical industrial organization for a generation (Nevo, 2001; Berry and Haile, 2014).",[1011,1107,1108],{},"BLP's core requirement, however, is a well-defined market in which all consumers face the same choice set. Platform e-commerce violates this requirement at its foundation: each consumer's choice set is algorithmically curated by the platform's ranking system as a function of search query, browsing history, and price (Ursu, 2018). Two consumers searching for the same query at the same moment can see different products in different orders. The \"market share\" of a SKU is not a primitive but a function of the algorithm's ranking policy. BLP-style estimators applied without adjustment will conflate true demand elasticity with the platform's price-elasticity-of-ranking — a confound that is invisible in the standard data.",[1067,1110,903],{"id":902},[1011,1112,1113,1114,1117,1118,1121,1122,1125],{},"Three strands of machine learning have reshaped elasticity estimation since the mid-2010s. The first, ",[1020,1115,1116],{},"causal forests"," (Wager and Athey, 2018; Athey, Tibshirani, and Wager, 2019), generalizes random forests to estimate heterogeneous treatment effects, including continuous-treatment elasticities. The second, ",[1020,1119,1120],{},"double machine learning"," (Chernozhukov et al., 2018, hereafter DML), uses cross-fitted machine-learning predictions of price and demand on confounders to construct a Neyman-orthogonal score that recovers elasticity with √n-consistency under high-dimensional confounding. The third, ",[1020,1123,1124],{},"deep instrumental-variable methods"," (Hartford et al., 2017) combine neural networks with instrument-based identification.",[1011,1127,1128],{},"These methods have substantially relaxed the parametric and instrument requirements of classical estimation. They are not, however, designed for the panel-with-cold-start structure of marketplace data, nor do they natively encode platform mechanism structure. Applied to platform panel data without modification, they tend to deliver high-variance estimates on thin SKUs (because they cannot pool information across SKUs), miss the kinks induced by Bundle Deal thresholds (because they do not encode the mechanism), and provide no calibrated uncertainty band beyond bootstrap intervals that are themselves unreliable on heavy-tailed sales.",[1067,1130,906],{"id":905},[1011,1132,1133,1134,1137],{},"A parallel literature, developed largely within marketing science and Bayesian statistics, has emphasized ",[1020,1135,1136],{},"partial pooling"," as the response to SKU heterogeneity (Allenby and Rossi, 1998; Rossi, Allenby, and McCulloch, 2005; Gelman et al., 2013). The intuition is that an individual SKU's elasticity should not be estimated in isolation: it should be shrunk toward the mean of its category, with the degree of shrinkage determined by the data. This is the formal Bayesian counterpart to the practitioner intuition that a new SKU should inherit a sensible starting elasticity from comparable products. Hierarchical Bayesian estimators are a natural fit for platform data with thousands of SKUs and frequent cold starts, and they generate calibrated posterior intervals as a by-product. Their weakness, in raw form, is identification: the partial-pooling structure does nothing to address price endogeneity. The contribution of this paper is to combine the partial-pooling backbone with a DML-style identification stage and a mechanism-aware likelihood, in a single posterior-inference framework.",[1067,1139,909],{"id":908},[1011,1141,1142],{},"Through this historical arc, three things changed: the flexibility with which we can model the conditional expectation of demand (from linear logs to deep neural networks), the dimensionality of confounders we can absorb (from a handful of cost shifters to thousands of platform-internal signals), and the targets we can estimate (from average elasticity to fully heterogeneous, SKU-level, time-varying elasticity).",[1011,1144,1145,1146,1149],{},"What did ",[1014,1147,1148],{},"not"," change is the foundational requirement: variation in price that is exogenous, conditional on what we have measured. Every elasticity estimator stands or falls on whether it succeeds in isolating such variation. In platform e-commerce, where price is a function of algorithmic signals the seller does not observe and the analyst rarely sees, this requirement is the binding constraint. Section 3 develops this point in detail.",[1025,1151],{},[1028,1153,912],{"id":911},[1067,1155,916],{"id":915},[1011,1157,1158],{},"The first feature of platform e-commerce that breaks naive estimation is the algorithmic mediation of demand. On Shopee, Lazada, and TikTok Shop, a SKU's exposure — the number of impressions it receives from organic search and recommendation — is determined by a ranking model that takes price as one of several inputs (Ursu, 2018; Goldfarb and Tucker, 2019). When a seller raises price, two things happen simultaneously: each impression converts at a lower rate (the demand effect we want to estimate), and the SKU receives fewer impressions to begin with (the algorithmic-ranking effect). A naive log-log regression of units sold on price recovers the sum of these two effects, not the demand elasticity in isolation. The bias is structural; it does not vanish with more data.",[1011,1160,1161,1162,1165],{},"This is a different kind of confounding from the classical simultaneity problem. The classical problem is that price responds to demand shocks. The platform problem is that price ",[1014,1163,1164],{},"causes"," changes in exposure through a mechanism that is exogenous to demand but is, from the analyst's perspective, an unobserved component of the data-generating process. It must be modeled or instrumented out.",[1067,1167,919],{"id":918},[1011,1169,1170],{},"The second feature is the structured promotional surface. Bundle Deal on Shopee, FlexiCombo on Lazada, and \"buy more save more\" on TikTok Shop are not marketing campaigns layered on top of a smooth demand curve. They are mechanism-design instruments — second-degree price discrimination in the sense of Tirole (1988) — and they introduce kinks and discontinuities in the realized price-quantity surface that smooth nonparametric estimators cannot capture without explicit structure. A causal forest fit on Bundle-Deal-active periods will smooth across the kink and underestimate the threshold-marginal elasticity that the seller actually needs for configuration decisions.",[1011,1172,1173],{},"The voucher stack adds further structure. Platform-issued vouchers, seller-issued vouchers, and category-specific vouchers compose with formal stacking rules that the analyst must encode. Two impressions at the same nominal price can have very different realized prices after voucher application, and ignoring this distinction biases the estimator toward inelasticity (because it confuses voucher discounts with price stability).",[1067,1175,922],{"id":921},[1011,1177,1178],{},"The third feature is the SKU-turnover regime. The SKU-level distribution of monthly units sold in marketplace data is heavy-tailed; the median SKU has few transactions in any given month, and a substantial share of revenue at the end of any quarter is generated by SKUs introduced in the recent past. A non-pooled estimator — one that fits a separate elasticity for each SKU — will fail on this distribution: there is not enough variation per SKU to identify the parameter precisely. A fully pooled estimator — one that fits a single category-level elasticity — will fail in the opposite direction by averaging over genuine heterogeneity.",[1011,1180,1181],{},"The statistical answer is partial pooling (Gelman et al., 2013): each SKU's elasticity is a draw from a category-level distribution, and the strength of the prior is itself estimated. This is technologically straightforward but is essentially absent from the dominant ML-causal estimators.",[1067,1183,925],{"id":924},[1011,1185,1186],{},"A fourth feature is the asymmetry of the loss function in operational pricing. The seller does not lose equally from over- and underestimating elasticity. Underestimating elasticity (treating an elastic SKU as inelastic) leads to under-discounting and lost orders; overestimating it (treating an inelastic SKU as elastic) leads to discounting items that would have sold anyway and gives margin away. The relative magnitude of these losses depends on the SKU's contribution margin and inventory state, and varies across the catalog.",[1011,1188,1189],{},"A point estimate is therefore insufficient: the seller needs a calibrated posterior so that decisions can be made under explicit risk-aversion (Berger, 1985; Robert, 2007). In particular, the optimal Bundle Deal threshold and discount under uncertainty are not the threshold and discount that maximize expected profit at the posterior mean of elasticity; they are the threshold and discount that maximize expected profit averaged over the posterior. This distinction motivates the Bayesian posterior-inference target in MA-HCEM rather than a frequentist point estimate.",[1025,1191],{},[1028,1193,928],{"id":927},[1067,1195,932],{"id":931},[1011,1197,1198,1199,1203,1204,1207,1208,1211,1212,1215,1216,1218,1219,1222,1223,1226,1227,1229,1230,1233,1234,1237,1238,1241,1242,1245,1246,1249,1250,1253,1254,1257,1258,1261,1262,1265],{},"Let ",[1200,1201],"blog-math",{"html":1202,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmo stretchy=\"false\">{\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>N\u003C\u002Fmi>\u003Cmo stretchy=\"false\">}\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">i \\in \\{1, \\ldots, N\\}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6986em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">{\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">…\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003Cspan class=\"mclose\">}\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," index SKUs, ",[1200,1205],{"html":1206,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>j\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">j(i)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0572em;\">j\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," index the shop containing SKU ",[1200,1209],{"html":1210,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">i\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6595em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", and ",[1200,1213],{"html":1214,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>c\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">c(i)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," the platform-defined category of SKU ",[1200,1217],{"html":1210,"display":-1},". Let ",[1200,1220],{"html":1221,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>t\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmo stretchy=\"false\">{\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>T\u003C\u002Fmi>\u003Cmo stretchy=\"false\">}\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">t \\in \\{1, \\ldots, T\\}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6542em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">{\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">…\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mclose\">}\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," index time periods. The observed outcome is ",[1200,1224],{"html":1225,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>q\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">q_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", the units of SKU ",[1200,1228],{"html":1210,"display":-1}," sold in period ",[1200,1231],{"html":1232,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">t\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6151em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". The treatment is ",[1200,1235],{"html":1236,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", the realized per-unit price paid (after voucher application) on the marginal transaction. The vector ",[1200,1239],{"html":1240,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">X_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," collects covariates: SKU age, shop fixed effect, category-time fixed effect, day-of-week, calendar campaign indicators, advertising spend, organic and paid impressions, inventory state, and a competitor-price summary. Let ",[1200,1243],{"html":1244,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>M\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">M_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," denote the active mechanism configuration: ",[1200,1247],{"html":1248,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>M\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>v\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">M_{it} = (b_{it}, v_{it}, f_{it})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," where ",[1200,1251],{"html":1252,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," encodes Bundle Deal threshold and discount, ",[1200,1255],{"html":1256,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>v\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">v_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," encodes the active voucher stack, and ",[1200,1259],{"html":1260,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," indicates flash-sale or live-stream context. Let ",[1200,1263],{"html":1264,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>Z\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">Z_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">Z\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," denote a randomized perturbation indicator if available (Section 4.6).",[1067,1267,935],{"id":934},[1011,1269,1270,1271,1273],{},"The MA-HCEM specifies the conditional distribution of ",[1200,1272],{"html":1225,"display":-1}," as a mechanism-aware mixture:",[1200,1275],{"html":1276,"display":1277},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>q\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>∣\u003C\u002Fmo>\u003Cmsub>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>M\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>∼\u003C\u002Fmo>\u003Cmsub>\u003Cmi>π\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmsub>\u003Cmi mathvariant=\"script\">F\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>θ\u003C\u002Fmi>\u003Cmrow>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>π\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmsub>\u003Cmi mathvariant=\"script\">F\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>θ\u003C\u002Fmi>\u003Cmrow>\u003Cmn>2\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">q_{it} \\mid p_{it}, X_{it}, M_{it} \\sim \\pi_{it} \\cdot \\mathcal{F}_{1}(\\theta_{1,it}) + (1 - \\pi_{it}) \\cdot \\mathcal{F}_{2}(\\theta_{2,it})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∼\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5945em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0993em;\">F\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0993em;\">F\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","true",[1011,1279,1280,1281,1284,1285,1288,1289,1284,1292,1295,1296,1299],{},"where ",[1200,1282],{"html":1283,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi mathvariant=\"script\">F\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathcal{F}_{1}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0993em;\">F\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is the demand component for the ",[1014,1286,1287],{},"single-unit-elastic segment"," (consumers who would buy at most one unit at the posted price), ",[1200,1290],{"html":1291,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi mathvariant=\"script\">F\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathcal{F}_{2}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0993em;\">F\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1014,1293,1294],{},"bundle-eligible segment"," (consumers willing to buy multi-unit if and only if the conditional discount applies), and ",[1200,1297],{"html":1298,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>π\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\pi_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is a context-dependent mixing weight. This decomposition operationalizes the second-degree price discrimination structure that Bundle Deal and FlexiCombo exploit. We specify both components as negative-binomial to accommodate the over-dispersed counts characteristic of marketplace data:",[1200,1301],{"html":1302,"display":1277},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi mathvariant=\"script\">F\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>θ\u003C\u002Fmi>\u003Cmrow>\u003Cmi>k\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>NegBin\u003C\u002Fmtext>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>μ\u003C\u002Fmi>\u003Cmrow>\u003Cmi>k\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>ϕ\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmspace width=\"1em\"\u002F>\u003Cmi>k\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmo stretchy=\"false\">{\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003Cmo stretchy=\"false\">}\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">.\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathcal{F}_{k}(\\theta_{k,it}) = \\text{NegBin}(\\mu_{k,it}, \\phi_{k}), \\quad k \\in \\{1, 2\\}.\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0993em;\">F\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">NegBin\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">ϕ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">{\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003Cspan class=\"mclose\">}\u003C\u002Fspan>\u003Cspan class=\"mord\">.\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1011,1304,1305],{},"The conditional means are log-linear in price with mechanism-modulated elasticity:",[1200,1307],{"html":1308,"display":1277},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>μ\u003C\u002Fmi>\u003Cmrow>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>α\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo>+\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsup>\u003Cmi>γ\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\log \\mu_{1,it} = \\alpha_{i}^{(1)} + \\beta_{i}^{(1)} \\log p_{it} + \\gamma^{(1)\\top} X_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3217em;vertical-align:-0.2769em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4231em;margin-left:-0.0037em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2769em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3217em;vertical-align:-0.2769em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4231em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2769em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1324em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.938em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1200,1310],{"html":1311,"display":1277},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>μ\u003C\u002Fmi>\u003Cmrow>\u003Cmn>2\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>α\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo>+\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmtext>bundle\u003C\u002Fmtext>\u003C\u002Fmsubsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>δ\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsubsup>\u003Cmn mathvariant=\"bold\">1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">{\u003C\u002Fmo>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∅\u003C\u002Fmi>\u003Cmo stretchy=\"false\">}\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmsup>\u003Cmi>γ\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\log \\mu_{2,it} = \\alpha_{i}^{(2)} + \\beta_{i}^{(2)} \\log p_{it}^{\\text{bundle}}(b_{it}) + \\delta_{i}^{\\top} \\mathbf{1}\\{b_{it} \\neq \\emptyset\\} + \\gamma^{(2)\\top} X_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3217em;vertical-align:-0.2769em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4231em;margin-left:-0.0037em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2769em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3217em;vertical-align:-0.2769em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4231em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2769em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-2.453em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord text mtight\">\u003Cspan class=\"mord mtight\">bundle\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.247em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1491em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0379em;\">δ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-2.453em;margin-left:-0.0379em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.247em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathbf\">1\u003C\u002Fspan>\u003Cspan class=\"mopen\">{\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan 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style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1324em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.938em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1011,1313,1280,1314,1317,1318,1321],{},[1200,1315],{"html":1316,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmtext>bundle\u003C\u002Fmtext>\u003C\u002Fmsubsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p_{it}^{\\text{bundle}}(b_{it})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1078em;vertical-align:-0.2587em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-2.4413em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord text mtight\">\u003Cspan class=\"mord mtight\">bundle\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2587em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is the per-unit price under the active bundle threshold and discount, and ",[1200,1319],{"html":1320,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>δ\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\delta_{i}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0379em;\">δ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," captures the discrete attention-shifting effect of bundle activation.",[1067,1323,938],{"id":937},[1011,1325,1326],{},"The SKU-level elasticities are partially pooled across two levels of platform taxonomy:",[1200,1328],{"html":1329,"display":1277},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo>∣\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>β\u003C\u002Fmi>\u003Cmrow>\u003Cmi>c\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>τ\u003C\u002Fmi>\u003Cmi>c\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo>∼\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">N\u003C\u002Fmi>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>β\u003C\u002Fmi>\u003Cmrow>\u003Cmi>c\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>τ\u003C\u002Fmi>\u003Cmi>c\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\beta_{i}^{(k)} \\mid \\beta_{c(i)}^{(k)}, \\tau_{c}^{(k)} \\sim \\mathcal{N}\\!\\left(\\beta_{c(i)}^{(k)}, \\tau_{c}^{(k),2}\\right),\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3217em;vertical-align:-0.2769em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4231em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2769em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.56em;vertical-align:-0.5152em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.3598em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5152em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.938em;\">\u003Cspan style=\"top:-2.453em;margin-left:-0.1132em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.247em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∼\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.8em;vertical-align:-0.65em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.1474em;\">N\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size2\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.3598em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5152em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.938em;\">\u003Cspan style=\"top:-2.453em;margin-left:-0.1132em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" 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style=\"margin-right:0.1474em;\">N\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size2\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4337em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan 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class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4337em;margin-left:-0.1132em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2663em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1200,1344],{"html":1345,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>τ\u003C\u002Fmi>\u003Cmi>c\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tau_{c}^{(k)}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1614em;vertical-align:-0.1166em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.5834em;margin-left:-0.1132em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1166em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," are themselves given weakly informative hyperpriors (Gelman, 2006), with ",[1200,1348],{"html":1349,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>τ\u003C\u002Fmi>\u003Cmo>∼\u003C\u002Fmo>\u003Cmtext>Half-Cauchy\u003C\u002Fmtext>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tau \\sim \\text{Half-Cauchy}(0, 1)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∼\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">Half-Cauchy\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". This structure delivers the cold-start regularization that Section 3.3 motivated: a new SKU's posterior elasticity is initially dominated by the category-level prior and updates as its own data accumulates.",[1067,1352,941],{"id":940},[1011,1354,1355,1356,1358],{},"The mixing weight ",[1200,1357],{"html":1298,"display":-1}," is modeled as a logistic function of mechanism state and consumer-context covariates:",[1200,1360],{"html":1361,"display":1277},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>π\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>σ\u003C\u002Fmi>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>η\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>η\u003C\u002Fmi>\u003Cmi>b\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsubsup>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>η\u003C\u002Fmi>\u003Cmi>v\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsubsup>\u003Cmsub>\u003Cmi>v\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsub>\u003Cmi>η\u003C\u002Fmi>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>η\u003C\u002Fmi>\u003Cmi>X\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsubsup>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi mathvariant=\"normal\">.\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\pi_{it} = \\sigma\\!\\left( \\eta_{0} + \\eta_{b}^{\\top} b_{it} + \\eta_{v}^{\\top} v_{it} + \\eta_{f} f_{it} + \\eta_{X}^{\\top} X_{it} \\right).\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2491em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size1\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.247em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.247em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.247em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size1\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">.\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1011,1363,1364,1365,1367],{},"This permits the model to learn that, for example, during a deep Bundle Deal the bundle-eligible segment expands sharply, while during a TikTok Shop live-stream flash sale the single-unit-elastic segment dominates. Cross-elasticities — that is, the substitution between SKUs in the same shop or category — enter the model through the ",[1200,1366],{"html":1320,"display":-1}," term and through a category-level demand-share constraint that we develop in Appendix A.",[1067,1369,944],{"id":943},[1011,1371,1372,1373,1375],{},"The price endogeneity introduced by platform ranking is addressed via a DML cross-fitting layer (Chernozhukov et al., 2018). Conditional on the high-dimensional covariate vector ",[1200,1374],{"html":1240,"display":-1}," — which we deliberately enrich to include impressions, ad spend, search rank, inventory, and category-time effects — we treat the residualized price as the identifying treatment variation:",[1200,1377],{"html":1378,"display":1277},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmover accent=\"true\">\u003Cmi>p\u003C\u002Fmi>\u003Cmo>~\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>−\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>m\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmspace width=\"1em\"\u002F>\u003Cmover accent=\"true\">\u003Cmi>m\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmtext> a cross-fit ML predictor.\u003C\u002Fmtext>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tilde{p}_{it} = \\log p_{it} - \\hat{m}(X_{it}), \\quad \\hat{m}(\\cdot) \\text{ a cross-fit ML predictor.}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8623em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.35em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1667em;\">\u003Cspan class=\"mord\">~\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.25em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.25em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">⋅\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\"> a cross-fit ML predictor.\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1011,1380,1381,1382,1385,1386,1389,1390,1393],{},"The conditional-mean equations of Section 4.2 are estimated with ",[1200,1383],{"html":1384,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmover accent=\"true\">\u003Cmi>p\u003C\u002Fmi>\u003Cmo>~\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tilde{p}_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8623em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.35em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1667em;\">\u003Cspan class=\"mord\">~\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," replacing ",[1200,1387],{"html":1388,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\log p_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," in the price-effect terms. Under the standard DML conditions — Neyman orthogonality of the score, sufficient ML convergence rates, cross-fitting — the resulting elasticity is consistent for the structural parameter even when ",[1200,1391],{"html":1392,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>m\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\hat{m}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.25em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is high-dimensional and nonparametric.",[1067,1395,947],{"id":946},[1011,1397,1398,1399,1401],{},"A distinctive feature of MA-HCEM is its treatment of small randomized price perturbations as the source of exogenous variation. In a managed deployment, the operator perturbs each SKU's price by a small amount on a fraction of impressions, with the perturbation drawn independently of ",[1200,1400],{"html":1240,"display":-1},". The exogeneity of this randomization breaks any residual confounding the DML stage cannot handle. Practically, the perturbations are absorbed as a data-augmenting experimental prior: each perturbed observation contributes to the posterior with the standard likelihood weight, and the unperturbed observations contribute with the DML-residualized likelihood. The combined estimator inherits the asymptotic bias of the DML stage in the absence of perturbations and the unbiasedness of a randomized experiment in their presence, with the relative weighting determined automatically by Bayesian variance pooling.",[1011,1403,1404],{},"This is the formal counterpart of the practitioner intuition that a small holdout test is the cheapest way to be sure. The contribution of MA-HCEM is to make the holdout test continuous, on-policy, and seamlessly integrated with observational data — rather than a discrete A\u002FB test that is expensive to design and slow to read out.",[1025,1406],{},[1028,1408,950],{"id":949},[1067,1410,954],{"id":953},[1011,1412,1413,1414,1339,1417,1419],{},"We state the identification result for the single-segment case (",[1200,1415],{"html":1416,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>π\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>≡\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\pi_{it} \\equiv 1\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6138em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≡\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1200,1418],{"html":1291,"display":-1}," degenerate); the two-segment extension is given in Appendix A.",[1011,1421,1422,1425,1426,1429,1430,1433,1434,1436],{},[1020,1423,1424],{},"Assumption 1 (Conditional independence)."," ",[1200,1427],{"html":1428,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>⊥\u003C\u002Fmo>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmo>⊥\u003C\u002Fmo>\u003Cmsub>\u003Cmi>ε\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>∣\u003C\u002Fmo>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>Z\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\log p_{it} \\perp\\!\\!\\!\\perp \\varepsilon_{it} \\mid X_{it}, Z_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">⊥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">⊥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">ε\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">Z\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", where ",[1200,1431],{"html":1432,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>ε\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\varepsilon_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">ε\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is the structural demand error and ",[1200,1435],{"html":1264,"display":-1}," is the randomized-perturbation indicator (degenerate when no perturbation is in force).",[1011,1438,1439,1442,1443,1446,1447,1450,1451,1446,1454,1457,1458,1461],{},[1020,1440,1441],{},"Assumption 2 (DML rate condition)."," The ML estimators ",[1200,1444],{"html":1445,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>m\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\hat{m}(\\cdot)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.25em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">⋅\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," for ",[1200,1448],{"html":1449,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>∣\u003C\u002Fmo>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathbb{E}[\\log p_{it} \\mid X_{it}]\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," and ",[1200,1452],{"html":1453,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>g\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\hat{g}(\\cdot)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2222em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">⋅\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1200,1455],{"html":1456,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>q\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>∣\u003C\u002Fmo>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathbb{E}[\\log q_{it} \\mid X_{it}]\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," converge at rate ",[1200,1459],{"html":1460,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>o\u003C\u002Fmi>\u003Cmi>P\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsup>\u003Cmi>n\u003C\u002Fmi>\u003Cmrow>\u003Cmo>−\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmn>4\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">o_{P}(n^{-1\u002F4})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.138em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">o\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">P\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.888em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u002F4\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," under cross-fitting.",[1011,1463,1464,1467,1468,1471,1472,1475],{},[1020,1465,1466],{},"Assumption 3 (Hierarchical exchangeability)."," Conditional on category ",[1200,1469],{"html":1470,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">c\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", the SKU-level elasticities ",[1200,1473],{"html":1474,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">{\u003C\u002Fmo>\u003Cmsub>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmsub>\u003Cmo stretchy=\"false\">}\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>:\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\{\\beta_{i}\\}_{i: c(i) = c}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1052em;vertical-align:-0.3552em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">{\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">}\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3448em;\">\u003Cspan style=\"top:-2.5198em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">:\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3552em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," are exchangeable.",[1011,1477,1478,1485,1486,1488,1489,1492,1493,1496],{},[1020,1479,1480,1481,1484],{},"Proposition 1 (Identification of ",[1200,1482],{"html":1483,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\beta_{i}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," under MA-HCEM)."," Under Assumptions 1-3, the posterior distribution of ",[1200,1487],{"html":1483,"display":-1}," given the data concentrates around the true SKU-level elasticity as ",[1200,1490],{"html":1491,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>T\u003C\u002Fmi>\u003Cmo>→\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">T \\to \\infty\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">→\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∞\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," for each SKU and as ",[1200,1494],{"html":1495,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>N\u003C\u002Fmi>\u003Cmo>→\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">N \\to \\infty\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">→\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∞\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," within each category, with the rate of concentration determined by the maximum of the within-SKU effective sample size and the cross-SKU pooling strength.",[1011,1498,1499,1500,1502],{},"The argument follows the standard hierarchical-Bayes consistency argument (Diaconis and Freedman, 1986; Ghosal and van der Vaart, 2017) combined with the Neyman-orthogonal score property of the DML residualization. Intuitively, the partial-pooling structure provides identification for cold-start SKUs through the category prior, and the DML stage provides identification for established SKUs through within-SKU price variation that is conditionally exogenous given ",[1200,1501],{"html":1240,"display":-1},". A formal proof is sketched in Appendix B; we note that the proposition is stated as a posterior-concentration result rather than a frequentist consistency theorem, and that completing it to a published-grade theorem is an item on the methodological agenda flagged in Section 9.",[1067,1504,957],{"id":956},[1011,1506,1507],{},"Three caveats are worth stating explicitly.",[1011,1509,1510,1511,1513],{},"First, identification is conditional on the covariate set ",[1200,1512],{"html":1240,"display":-1}," being rich enough to absorb the algorithmic confounding. In practice this requires the analyst to have access to platform impression data, search-rank data, and ad-spend data at the SKU-time level. When the analyst has access only to the seller's transaction data — as is sometimes the case for third-party tooling — the DML stage will be biased to the extent that algorithmic exposure is correlated with price conditional on observables. The randomized-perturbation augmentation is the formal escape route from this limitation; we recommend it as the default deployment.",[1011,1515,1516],{},"Second, the cross-elasticity structure is identified only up to a normalization. We follow the demand-system literature (Deaton and Muellbauer, 1980) in imposing adding-up at the category level; alternatives are discussed in Appendix A.",[1011,1518,1519],{},"Third, the model is identified at the elasticity-of-the-mean level, not the full distribution of demand. Quantile elasticities — the response of, say, the 90th percentile of demand to a price change — require an extension to the negative-binomial mean-variance relationship that we do not develop here.",[1025,1521],{},[1028,1523,960],{"id":959},[1067,1525,964],{"id":963},[1011,1527,1528,1529,1249,1532,1535,1536,1539],{},"The inference target is the joint posterior ",[1200,1530],{"html":1531,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi mathvariant=\"bold-italic\">β\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi mathvariant=\"bold-italic\">α\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi mathvariant=\"bold-italic\">δ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi mathvariant=\"bold-italic\">η\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi mathvariant=\"bold-italic\">ϕ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi mathvariant=\"bold-italic\">τ\u003C\u002Fmi>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">D\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p(\\boldsymbol{\\beta}, \\boldsymbol{\\alpha}, \\boldsymbol{\\delta}, \\boldsymbol{\\eta}, \\boldsymbol{\\phi}, \\boldsymbol{\\tau} \\mid \\mathcal{D})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord boldsymbol\" style=\"margin-right:0.034em;\">β\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord boldsymbol\">α\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord boldsymbol\" style=\"margin-right:0.0382em;\">δ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord boldsymbol\" style=\"margin-right:0.037em;\">η\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord boldsymbol\">ϕ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord boldsymbol\" style=\"margin-right:0.1347em;\">τ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1200,1533],{"html":1534,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"script\">D\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmo stretchy=\"false\">{\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>q\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>p\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>M\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>Z\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmsub>\u003Cmo stretchy=\"false\">}\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathcal{D} = \\{(q_{it}, p_{it}, X_{it}, M_{it}, Z_{it})\\}_{i,t}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">{(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal 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mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">Z\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">}\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is the panel. For managerial use, the marginal posteriors of ",[1200,1537],{"html":1538,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\beta_{i}^{(k)}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3217em;vertical-align:-0.2769em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4231em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2769em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," are the primary objects, but the full joint is needed for downstream profit-optimization calculations, particularly Bundle Deal threshold selection.",[1067,1541,967],{"id":966},[1011,1543,1544,1545,1450,1547,1550],{},"For panels of realistic size, full Markov-chain Monte Carlo inference is computationally costly. We adopt stochastic variational inference (Hoffman et al., 2013; Kucukelbir et al., 2017) with a structured mean-field family that retains the hierarchical correlations between ",[1200,1546],{"html":1538,"display":-1},[1200,1548],{"html":1549,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>c\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\beta_{c}^{(k)}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2392em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.5834em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1166em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," but factorizes over SKUs within a category. The negative-binomial likelihood is reparameterized via a Poisson-Gamma mixture for differentiable Monte Carlo gradient estimation. We use control variates derived from the DML residual (Ranganath, Gerrish, and Blei, 2014) to stabilize gradients on long-tailed SKUs. A reference Hamiltonian-Monte-Carlo implementation (Stan; Carpenter et al., 2017) can be used for posterior validation on small subsets where computational cost is not binding. We do not report wall-clock benchmarks here; benchmarking is part of the empirical companion agenda outlined in Section 9.",[1067,1552,970],{"id":969},[1011,1554,1555],{},"We recommend three posterior predictive checks (Gelman, Meng, and Stern, 1996) as a deployment hygiene minimum: (i) calibration of the credible interval against held-out price perturbations, (ii) coherence of the elasticity distribution against the category prior — a check that flags pathological pooling — and (iii) a discrepancy statistic that compares model-implied Bundle Deal uptake against observed uptake at a candidate threshold.",[1025,1557],{},[1028,1559,973],{"id":972},[1011,1561,1562],{},"This section develops a qualitative bias decomposition that identifies which error components each canonical baseline leaves unaddressed under the platform-marketplace data-generating process described in Section 3, and where MA-HCEM is designed to reduce those errors. The comparison is theoretical; empirical and Monte Carlo validation remain part of the replication agenda in Section 9.",[1067,1564,977],{"id":976},[1011,1566,1567,1568,1571,1572,1574,1575,1578,1579,1582,1583,1586],{},"For any elasticity estimator ",[1200,1569],{"html":1570,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmover accent=\"true\">\u003Cmi>β\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\hat{\\beta}_{i}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1523em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9579em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2634em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1667em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," of the SKU-level elasticity ",[1200,1573],{"html":1483,"display":-1},", we decompose the mean-squared error as the sum of three components: (i) the ",[1020,1576,1577],{},"algorithmic-confounding bias"," that arises when price is correlated with unobserved exposure-affecting factors, (ii) the ",[1020,1580,1581],{},"mechanism-misspecification bias"," that arises when the estimator smooths over kinks induced by Bundle Deal or voucher mechanisms, and (iii) the ",[1020,1584,1585],{},"estimation variance"," that arises from finite within-SKU data.",[1011,1588,1589],{},"Each canonical baseline addresses a strict subset of these three components.",[1067,1591,980],{"id":979},[1011,1593,1594,1597,1598,1601,1602,1604,1605,1608],{},[1020,1595,1596],{},"Naive log-log regression."," A SKU-level fixed-effects regression of ",[1200,1599],{"html":1600,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>q\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\log q_{it}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," on ",[1200,1603],{"html":1388,"display":-1}," controls for time-invariant SKU heterogeneity but not for time-varying algorithmic exposure. The within-SKU price variation it exploits is precisely the variation that is most contaminated by algorithmic confounding: when the seller raises price, the platform reduces exposure, so the within-SKU correlation between price and units sold is the ",[1014,1606,1607],{},"sum"," of demand elasticity and exposure-elasticity-of-price. The estimator is biased toward inelasticity in the presence of any positive algorithmic price-rank coupling. It additionally ignores mechanism kinks, so it is structurally incapable of recovering the threshold-marginal elasticity needed for Bundle Deal configuration.",[1011,1610,1611,1614],{},[1020,1612,1613],{},"Double machine learning without partial pooling."," A DML estimator that residualizes price on a rich covariate vector — including impressions, ad spend, and search rank — can in principle eliminate the algorithmic-confounding bias, provided the covariate vector spans the confounder space and the ML nuisance estimators converge at the required rate (Chernozhukov et al., 2018). On established SKUs with adequate within-SKU sample size, this is the right tool. Two failure modes remain. First, on cold-start and thin SKUs, the within-SKU effective sample size after residualization is small, and the resulting elasticity estimate has high variance — which compounds with mechanism-misspecification bias because the DML estimator does not encode the bundle\u002Fvoucher kink. Second, DML produces frequentist point estimates and bootstrap confidence intervals that are themselves unreliable on heavy-tailed sales distributions; the calibrated uncertainty needed for risk-aware managerial decision-making is not delivered natively.",[1011,1616,1617,1620],{},[1020,1618,1619],{},"Causal forest."," Generalized random forests (Athey, Tibshirani, and Wager, 2019) handle high-dimensional confounders flexibly and produce SKU-heterogeneous elasticity estimates. They share two of DML's weaknesses (no partial pooling, no native posterior) and add a third: the smoothness inductive bias of forest-based estimators tends to flatten the kinks that Bundle Deal mechanisms introduce in the realized price-quantity surface, so the threshold-marginal elasticity is systematically underestimated.",[1011,1622,1623,1626],{},[1020,1624,1625],{},"Hierarchical Bayes without DML."," A pure partial-pooling model fits the cold-start regime well — variance is controlled by shrinkage to the category mean — but does nothing about algorithmic confounding. The category-level pooled elasticity is biased by the same algorithmic mechanism that biases naive OLS, and that bias is then propagated to every SKU through the hierarchical prior. The estimator is well-behaved in variance but systematically biased in mean.",[1011,1628,1629,1632],{},[1020,1630,1631],{},"BLP-family discrete choice."," As discussed in Section 2.2, BLP-style estimators require a well-defined market and a fixed choice set; both fail under platform-curated discovery. The estimator is structurally inapplicable to platform-marketplace data without nontrivial adjustment.",[1067,1634,983],{"id":982},[1011,1636,1637],{},"MA-HCEM is constructed to address each of the three error components simultaneously:",[1011,1639,1640,1641,1644],{},"1. The DML residualization stage (Section 4.5) addresses ",[1020,1642,1643],{},"algorithmic confounding bias",", conditional on the richness of the covariate set.",[1011,1646,1647,1648,1651],{},"2. The two segment mechanism aware likelihood (Section 4.2) addresses ",[1020,1649,1650],{},"mechanism misspecification bias"," by encoding Bundle Deal and voucher structure as a structural mixture rather than a smoothed continuous response.",[1011,1653,1654,1655,1657],{},"3. The hierarchical partial pooling prior (Section 4.3) addresses ",[1020,1656,1585],{}," by shrinking thin SKU posteriors toward the category mean.",[1011,1659,1660],{},"4. The randomized perturbation augmentation (Section 4.6) provides a fallback against residual confounding that the covariate set does not absorb, and serves as the empirical anchor for the calibrated posterior.",[1011,1662,1663],{},"Each of these mechanisms is well-established individually in its own literature; the contribution of this paper is to unify them in a single estimator targeted at the platform-marketplace data-generating process.",[1067,1665,986],{"id":985},[1011,1667,1668],{},"Honesty about boundary conditions matters. MA-HCEM is unlikely to outperform simpler baselines under any of the following conditions:",[1011,1670,1671,1672,1675],{},"1. ",[1020,1673,1674],{},"No algorithmic confounding."," If the platform's exposure mechanism is approximately price-neutral, naive within-SKU regression is consistent and the DML stage adds variance without reducing bias.",[1011,1677,1678,1679,1682],{},"2. ",[1020,1680,1681],{},"No mechanism kinks."," If Bundle Deals and vouchers are absent or quantitatively small in the data, the two-segment likelihood reduces to a more complex equivalent of the single-segment specification, and a simpler model wins on parsimony.",[1011,1684,1685,1686,1689],{},"3. ",[1020,1687,1688],{},"Thick SKUs with abundant within-SKU price variation."," If every SKU has thousands of transactions and ample exogenous price variation, partial pooling adds little and a flat-prior DML estimator is competitive.",[1011,1691,1692,1693,1696],{},"4. ",[1020,1694,1695],{},"Misspecified mixture."," If the true demand process is not well approximated by a two-segment mixture — for example, if there is a continuous distribution of consumer reservation prices with no discrete bundle-eligibility structure — the mechanism-aware likelihood introduces specification error that may exceed the bias it removes. This risk is mitigated by posterior predictive checking (Section 6.3) but cannot be eliminated.",[1011,1698,1699],{},"A reviewer will rightly demand that these boundary conditions be tested empirically before MA-HCEM is recommended for general deployment. Section 9 describes the replication agenda intended to do exactly that.",[1025,1701],{},[1028,1703,989],{"id":988},[1067,1705,993],{"id":992},[1011,1707,1708],{},"The model produces three managerially actionable outputs: a per-SKU posterior elasticity with credible interval, a model-implied optimal Bundle Deal configuration, and a voucher-targeting score that ranks customers (or follower segments) by their estimated marginal response to a discount.",[1011,1710,1711],{},"The per-SKU posterior elasticity is the foundational output. It permits the seller to migrate from uniform-discount sale-event configuration to elasticity-tiered configuration: deeper discounts on the elastic tail of the catalog, shallow or zero discounts on the inelastic tail. The magnitude of any resulting profit gain is an empirical question that we do not attempt to quantify in this paper; the theoretical mechanism is clear, but the size of the effect depends on the dispersion of true elasticities across the catalog and is best estimated against operator data.",[1011,1713,1714,1715,1339,1718,1721,1722,1724],{},"The model-implied optimal Bundle Deal configuration is computed by integrating expected profit over the joint posterior of ",[1200,1716],{"html":1717,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\beta_{i}^{(1)}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3217em;vertical-align:-0.2769em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4231em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2769em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1200,1719],{"html":1720,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsubsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\beta_{i}^{(2)}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3217em;vertical-align:-0.2769em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0448em;\">\u003Cspan style=\"top:-2.4231em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2198em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2769em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", and the mixing-weight parameters. Crucially, the optimal threshold under uncertainty is ",[1014,1723,1148],{}," the threshold that maximizes expected profit at the posterior mean: it is shaded toward the threshold that minimizes downside risk, consistent with the asymmetric loss function discussed in Section 3.4.",[1011,1726,1727],{},"The voucher-targeting score follows by computing the expected incremental profit of a discount conditional on customer-segment covariates. This collapses to a simple ranking by estimated elasticity, with the recommendation to allocate deeper vouchers to the elastic tail and retention-only vouchers to the inelastic tail.",[1067,1729,996],{"id":995},[1011,1731,1732],{},"MA-HCEM is the measurement layer; dynamic-pricing systems (Phillips, 2005; den Boer, 2015) are the automation layer. A fully closed-loop dynamic-pricing system requires the elasticity model, a profit function, an inventory-state model, and a pricing policy. MA-HCEM contributes the first of these and, by virtue of producing a calibrated posterior, also provides the input that a Thompson-sampling or upper-confidence-bound bandit policy needs to balance exploration and exploitation in a principled way (Russo et al., 2018). Our recommendation is that practitioners deploy MA-HCEM in advisory mode first — using its outputs to inform manual pricing decisions — and migrate to closed-loop automation only after a calibration period sufficient to validate the posterior coverage on operational data.",[1067,1734,999],{"id":998},[1011,1736,1737],{},"A growing literature has examined demand on digital platforms. Ursu (2018) studies the causal effect of platform ranking on consumer search and purchase decisions on a major travel platform and motivates much of the algorithmic-confounder discussion in Section 3.1. Goldfarb and Tucker (2019) survey the broader digital-economics literature and frame the structural distinctiveness of platform retailing. Our contribution relative to this literature is the explicit incorporation of mechanism structure (Bundle Deal, voucher stacking) into the likelihood, the integration of DML residualization into a hierarchical-Bayesian backbone, and the use of randomized perturbations as on-policy experimental priors.",[1025,1739],{},[1028,1741,1002],{"id":1001},[1011,1743,1744],{},"We identify five limitations that bound the scope of the present work and that we propose as priorities for empirical replication.",[1011,1746,1747,1748,1751],{},"First, this paper presents no simulation or empirical results. The theoretical analysis in Section 7 articulates ",[1014,1749,1750],{},"why"," we expect MA-HCEM to dominate canonical baselines under the platform-marketplace data-generating process, but the magnitude of that dominance — and the boundary conditions where it fails — is an empirical question. The first item on the replication agenda is a Monte Carlo study under data-generating processes calibrated to platform features; the second is empirical validation against ground-truth experimental price perturbations on a real marketplace, which requires a partnership with a platform or a managed seller cohort.",[1011,1753,1754],{},"Second, the cross-elasticity structure in our specification is parsimonious — we impose adding-up at the category level — and richer substitution patterns (asymmetric substitution, hierarchical category trees) are likely needed for empirical fidelity in some categories.",[1011,1756,1757,1758,1760],{},"Third, the model is static in elasticity: ",[1200,1759],{"html":1538,"display":-1}," does not vary over time. In settings with strong reference-price effects (Kalyanaram and Winer, 1995) or rapid product-life-cycle dynamics, a state-space extension with time-varying elasticity is appropriate. The hierarchical-Bayesian backbone supports this extension naturally.",[1011,1762,1763],{},"Fourth, we have abstracted from competition. A fully strategic specification would model competing sellers' price responses (Tirole, 1988; Reiss and Wolak, 2007). This is an important direction; we expect the empirical importance to vary across categories with the degree of platform-induced price coordination.",[1011,1765,1766],{},"Fifth, we have treated the platform's algorithmic ranking as a confounder to be absorbed rather than a strategic variable. A second-best literature would acknowledge that the platform optimizes the ranking against its own objective, which may not coincide with the seller's; the elasticity estimate is then a function of the platform's objective as well as the underlying consumer preferences. We see this as an important agenda but beyond the scope of the present paper.",[1011,1768,1769],{},"We additionally flag that Proposition 1 is stated as a posterior-concentration result with a sketched argument. Completing the proof to a formal published-grade theorem — including precise rate conditions on the SKU and category sample-size growth — is an item on the methodological agenda.",[1025,1771],{},[1028,1773,1005],{"id":1004},[1011,1775,1776],{},"Price elasticity remains the single highest-leverage parameter in retail pricing decisions, but the canonical estimators developed for offline retail and structural industrial organization do not transfer cleanly to platform e-commerce. The features that make platform marketplaces distinctive — algorithmic mediation of demand, mechanism-rich promotional surfaces, heavy-tailed and rapid SKU turnover, and asymmetric managerial loss functions — break the load-bearing assumptions of every dominant estimator in turn.",[1011,1778,1779,1780,1783,1784,1787],{},"This paper has reviewed the historical arc from Marshall through BLP through double machine learning to identify which of these assumptions hold and which fail under platform e-commerce, and it has proposed the Mechanism-Aware Hierarchical Causal Elasticity Model as a unified estimator designed for the marketplace setting. MA-HCEM combines a hierarchical-Bayesian backbone with double-machine-learning identification, a structural likelihood that encodes Bundle Deal and voucher mechanisms as second-degree price discrimination, and randomized micro-perturbations as on-policy experimental priors. Section 7's bias decomposition makes precise ",[1014,1781,1782],{},"which"," failure modes of canonical baselines MA-HCEM is constructed to remediate, and ",[1014,1785,1786],{},"under what conditions"," the proposed estimator is not expected to outperform simpler alternatives.",[1011,1789,1790],{},"The contribution of this paper is methodological and theoretical. We make no empirical or simulation claims. The empirical agenda we propose is direct: a Monte Carlo calibration to platform features, followed by validation against ground-truth randomized experiments via a platform partnership or managed seller cohort, and a cross-platform comparison of elasticity heterogeneity that the partial-pooling structure of MA-HCEM is uniquely positioned to support. The methodological agenda is broader: incorporating reference-price dynamics, strategic competition, and the platform's own objective into the structural likelihood, and completing the formal asymptotic theory of the proposed estimator.",[1025,1792],{},[1028,1794,1008],{"id":1007},[1796,1797,1798,1805,1811,1819,1823,1828,1834,1839,1840,1841],{"id":886,"depth":386,"text":887},{"id":889,"depth":386,"text":890},{"id":892,"depth":386,"text":893,"children":1799},[1800,1801,1802,1803,1804],{"id":896,"depth":332,"text":897},{"id":899,"depth":332,"text":900},{"id":902,"depth":332,"text":903},{"id":905,"depth":332,"text":906},{"id":908,"depth":332,"text":909},{"id":911,"depth":386,"text":912,"children":1806},[1807,1808,1809,1810],{"id":915,"depth":332,"text":916},{"id":918,"depth":332,"text":919},{"id":921,"depth":332,"text":922},{"id":924,"depth":332,"text":925},{"id":927,"depth":386,"text":928,"children":1812},[1813,1814,1815,1816,1817,1818],{"id":931,"depth":332,"text":932},{"id":934,"depth":332,"text":935},{"id":937,"depth":332,"text":938},{"id":940,"depth":332,"text":941},{"id":943,"depth":332,"text":944},{"id":946,"depth":332,"text":947},{"id":949,"depth":386,"text":950,"children":1820},[1821,1822],{"id":953,"depth":332,"text":954},{"id":956,"depth":332,"text":957},{"id":959,"depth":386,"text":960,"children":1824},[1825,1826,1827],{"id":963,"depth":332,"text":964},{"id":966,"depth":332,"text":967},{"id":969,"depth":332,"text":970},{"id":972,"depth":386,"text":973,"children":1829},[1830,1831,1832,1833],{"id":976,"depth":332,"text":977},{"id":979,"depth":332,"text":980},{"id":982,"depth":332,"text":983},{"id":985,"depth":332,"text":986},{"id":988,"depth":386,"text":989,"children":1835},[1836,1837,1838],{"id":992,"depth":332,"text":993},{"id":995,"depth":332,"text":996},{"id":998,"depth":332,"text":999},{"id":1001,"depth":386,"text":1002},{"id":1004,"depth":386,"text":1005},{"id":1007,"depth":386,"text":1008},{"id":1843,"categories":1844,"domain":1851,"extension":5,"fallback":1852,"home":1855,"industry":1864,"meta":1869,"pricing":1870,"stem":1882,"__hash__":1883},"og_en\u002Fen\u002Fog.yml",[1845,1846,1847,1848,1849,1850],{"slug":347,"label":348},{"slug":350,"label":27},{"slug":352,"label":353},{"slug":355,"label":356},{"slug":361,"label":362},{"slug":364,"label":48},"dataglasslabs.com",{"title":1853,"description":1854},"DataGlass","The decision engine for Shopee, Lazada and TikTok Shop sellers.",{"chart":1856,"metrics":1857},"Net profit, last 30 days",[1858,1861],{"label":1859,"value":1860},"Net profit","฿1.24M",{"label":1862,"value":1863},"Profit ROAS","3.4x",{"axis":1865,"point":1866},"Ad spend",{"label":1867,"value":1868},"Next baht still pays","+฿0.42",{},{"plans":1871},[1872,1876,1879],{"name":1873,"price":1874,"period":1875},"Free","฿0","\u002Fmonth",{"name":1877,"price":1878,"period":1875},"Standard","฿1,499",{"name":1880,"price":1881,"period":1875},"Pro","฿2,499","en\u002Fog","63L8y4Lv8I2q3NvR26fMXU1sE9Wd7TZaz2l_tV6oZuo",[],1792785052577]