Demand and supply of differentiated products
Paul Schrimpf
Demand and supply of differentiated products
Paul Schrimpf
UBC Economics 567
January 28, 2021 Demand and supply of differentiated products 1 Introduction Paul Schrimpf
Introduction 2 Demand in product space Demand in product space Demand in 3 Demand in characteristic space characteristic space Early work Early work Model Model Model Estimation and Model identification Aggregate product Estimation and identification data Estimation steps Aggregate product data Pricing equation Estimation steps Micro data Pricing equation References Micro data References
4 References Demand and supply of differentiated products References
Paul Schrimpf
Introduction
Demand in product space
Demand in • Reviews: characteristic space • Ackerberg et al. (2007) section 1 (these slides use their Early work notation) Model Model • Aguirregabiria (2019) chapter 2 Estimation and identification • Reiss and Wolak (2007) sections 1-7, especially 7 Aggregate product data • Classic papers: Estimation steps Pricing equation • Berry (1994) Micro data • References Berry, Levinsohn, and Pakes (1995)
References Demand and supply of differentiated products
Paul Schrimpf
Introduction
Demand in product space
Demand in Section 1 characteristic space Early work Model Model Introduction Estimation and identification Aggregate product data Estimation steps Pricing equation Micro data
References
References Demand and supply of differentiated products Introduction
Paul Schrimpf
Introduction
Demand in product space • Typical market for consumer goods has many Demand in characteristic differentiated, but similar products, e.g. space Early work • Cars Model • Model Cereal Estimation and identification • Differentiated products are a source of market power Aggregate product data Estimation steps • Having many products can result in many parameters Pricing equation Micro data creating estimation difficulties and requiring References departures from textbook demand and supply models References Demand and supply of differentiated products Motivation
Paul Schrimpf
Introduction
Demand in product space • Counterfactuals that do not change production Demand in characteristic technology space Early work • Mergers Model • Model Tax changes Estimation and identification • Effects of new goods Aggregate product data Estimation steps • Cost-of-living indices Pricing equation Micro data • Product differentiation and market power References • Cross-price elasticities References Demand and supply of differentiated products
Paul Schrimpf
Introduction
Demand in product space
Demand in Section 2 characteristic space Early work Model Demand in product space Model Estimation and identification Aggregate product data Estimation steps Pricing equation Micro data
References
References Demand and supply of differentiated products Demand in product space 1
Paul Schrimpf • J products, each treated as separate good • Introduction Classical demand,
Demand in product space q1 =D1(p1, ..., pJ, z1, η1; β1) Demand in . . characteristic . = . space Early work Model qJ =DJ(p1, ..., pJ, zJ, ηJ; βJ), Model Estimation and identification Aggregate product data Estimation steps and supply (firms’ first-order conditions for prices): Pricing equation Micro data p =g (q , ..., q , w , ν ; θ ) References 1 1 1 J 1 1 1 . . References . = . d pJ =gJ(q1, ..., qJ, wJ, νJ; θJ),
where Demand and supply of differentiated products Demand in product space 2
Paul Schrimpf
Introduction • Demand in pj = price product space • qj = quantity Demand in • z = observed demand shifter characteristic j space • ηj = unobserved demand shock Early work Model • βj = demand parameters Model • Estimation and wj = observed supply shifter identification • Aggregate product νj = unobserved supply shock data • Estimation steps θj = supply parameters Pricing equation Micro data • Dj typically parametrically specified, e.g. References
References ln qj = βj0 + βj1p1 + ··· + βjJpJ + βjy ln y + Z1γ + νj Demand and supply of differentiated products Demand in product space
Paul Schrimpf • Use reduced form to find instruments
Introduction q q1 =Π1 (Z, W, ν, η; β, θ) Demand in . . product space . =. Demand in q characteristic q =Π (Z, W, ν, η; β, θ) space J J Early work p Model p1 =Π1 (Z, W, ν, η; β, θ) Model Estimation and . . identification . =. Aggregate product data p Estimation steps pJ =ΠJ (Z, W, ν, η; β, θ) Pricing equation Micro data
References • Cost shifters of product j excluded from demand and References supply of product k, but in reduced form • Cost data often not available • If available, unlikely to be product specific • Attributes of other products • Hausman (1996) uses prices of other products • Hard to justify, especially with prices Demand and supply of differentiated products Demand in product space 1
Paul Schrimpf • Advantages of product space: Introduction • Demand in Flexible substitution patterns product space • Does not require detailed product attribute data Demand in characteristic • Problems with product space: space Early work 1 Representative agent and aggregation issues Model • With heterogeneous preferences, aggregate market Model Estimation and demand need not meet restrictions on individual identification Aggregate product demand derived from economic theory data Estimation steps • Cannot use restrictions easily to improve estimates Pricing equation • Can use simulation to aggregate (Pakes, 1986) Micro data 2 References 2 Too many parameters, O(J )
References • Can limit by restricting cross-price elasticities, e.g. Pinkse, Slade, and Brett (2002) 3 Too many instruments needed, J 4 Cannot analyze new goods Demand and supply of differentiated products
Paul Schrimpf
Introduction
Demand in product space
Demand in Section 3 characteristic space Early work Model Model Demand in characteristic space Estimation and identification Aggregate product data Estimation steps Pricing equation Micro data
References
References Demand and supply of differentiated products Demand in characteristic space
Paul Schrimpf
Introduction
Demand in • Motivation: product space • Why do firms differentiate products? Demand in • characteristic Because consumers have heterogeneous tastes for space product characteristics Early work Model • E.g. cars: tastes for size, safety, fuel efficiency, etc Model Estimation and • identification Main idea: model consumer preferences for Aggregate product data characteristics and treat products as bundles of Estimation steps Pricing equation characteristics Micro data • Early work: Lancaster (1971), McFadden (1973) References
References • Key extension to early work: Berry, Levinsohn, and Pakes (1995) Demand and supply of differentiated products Early work in characteristic Paul Schrimpf space • Introduction Consumer chooses one or none of J products
Demand in • Utility of consumer i from product j product space
Demand in uij = xjβ + εij characteristic space with εij iid across i and j (usually Type I extreme value) Early work Model • Implies aggregate demand (for logit) Model Estimation and identification exp(xjβ) Aggregate product qj = PJ data 1 + exp(x β) Estimation steps k=1 k Pricing equation Micro data • Problem: restrictive substitution “independence of References irrelevant alternatives” References • Two goods with the same shares have the same cross price elasticities with any third good (think about a luxury and bargain good with equal shares) • Goods with same shares should have same markups • Solution: add heterogeneity in β and/or allow correlation across j in εij Demand and supply of differentiated products Model 1
Paul Schrimpf
Introduction
Demand in product space • Consumers i, goods j, markets t Demand in • characteristic Utility: (include good 0 = buy nothing) space Early work unobserved Model z}|{ unobserved Model z}|{ Estimation and uijt = U( ˜xjt , ξjt , zit , νit , pjt ; θ) identification |{z} Aggregate product |{z} |{z} data observed observed observed Estimation steps Pricing equation Micro data RK RR RL References • xjt = (˜xjt, pjt)) ∈ , zit ∈ , νit ∈
References • Choose j if uijt > uikt ∀k ̸= j Demand and supply of differentiated products Model 1 Paul Schrimpf • Usually U(·) linear:
Introduction 1×K K×1 1×1 Demand in z}|{ z }| { z}|{ product space u = x θ + ξ +ε ijt jt |{z}it jt ijt Demand in o u characteristic =θ¯+θ zit+θ νit space Early work Model Model Estimation and identification for j = 1...J and normalize ui0t = 0 Aggregate product data • ε Estimation steps Assume ijt i.i.d. double exponential Pricing equation Micro data • Assume νit ∼ fν(·; θ), e.g. independent normal References • Write as product specific + observed interactions + References unobserved interactions
u = δ +x θo z + x θu ν + ε ijt |{z}j jt |{z} it jt |{z} it ijt K×R K×L =xjtθ¯+ξjt Demand and supply of differentiated products Endogeneity
Paul Schrimpf
Introduction
Demand in product space • Demand in Usually assume E[νit|xjt, zit] = 0 and E[εijt|xjt, zit] = 0 characteristic • space Not interested in counterfactuals with respect to Early work changes in zit, so can treat as residual, i.e. Model Model Estimation and identification νit = θit − E[θit|zit] Aggregate product data Estimation steps • Market average νit or εijt plausibly correlated with pjt or Pricing equation Micro data other product characteristics, but this correlation
References absorbed into ξjt and/or market fixed effects
References Demand and supply of differentiated products Endogeneity
Paul Schrimpf
Introduction
Demand in product space Demand in • Problem is ξjt characteristic space • Prices and other flexible product characteristics must Early work Model be correlated with ξjt Model • If ξ serially correlated, then likely also correlated with Estimation and jt identification inflexible product characteristics Aggregate product data • Need instrument, wjt such that E[ξjt|wjt] = 0 Estimation steps Pricing equation • Cost shifters Micro data • Characteristics of other products References
References Demand and supply of differentiated products Estimation and identification
Paul Schrimpf
Introduction
Demand in product space
Demand in characteristic • Depends on data: space Early work • Aggregate product market shares and characteristics Model • Model Individual characteristics and choices Estimation and identification • Additional assumptions: Aggregate product data • Use supply and equilibrium assumptions to get a Estimation steps Pricing equation pricing equation Micro data
References
References Demand and supply of differentiated products Aggregate data 1 Paul Schrimpf • Often only have data on product characteristics and Introduction market shares Demand in product space • Maybe also distribution of some individual
Demand in characteristics for each market (e.g. income and characteristic space education from CPS or census) Early work Model • Instrument w such that E[ξj|w] = 0 Model Estimation and • ν ∼ (·; θ ) identification Distribution of fν ν Aggregate product data • Combination of estimated market level distribution of Estimation steps observed individual characteristics and parametric Pricing equation Micro data distributions of unobserved individual characteristics References • e.g. νit = (educit, incomeit, eit) References µ ˆ e − θν Fν,t(s, y, e; θν ) = Ft(s, y) Φ σ | {z } θν empirical distribution
ˆFt(s, y) estimated from CPS or other similar data set Demand and supply of differentiated products Aggregate data 2
Paul Schrimpf
Introduction
Demand in product space
Demand in characteristic space • Early work Assume εijt ∼ double exponential (aka Gumbel or type I Model extreme value) as in logit Model Estimation and • identification Computationally convenient, but other distributions Aggregate product data feasible too Estimation steps Pricing equation Micro data
References
References Demand and supply of differentiated products Estimation outline
Paul Schrimpf
Introduction • Demand in Estimate θ from moment condition product space Demand in E[ξ(·; θ)|w] = 0 characteristic space Early work • Where ξ(·; θ) is such that model predicted market Model 1 Model shares = observed market shares Estimation and identification 1 Compute shares given θ, σ(·; θ, δ) Aggregate product data 2 Find δ(·; θ) = xjtθ¯ + ξ(·; θ) such that observed shares, sjt Estimation steps Pricing equation = model shares, σ(·; θ, δ), then Micro data
References ξ(·; θ) = δ(·; θ) − xjtθ¯ References
1In this slide · means the data. I will leave the · out of the notation in subsequent slides. I will also leave out t subscripts. Demand and supply of differentiated products Computing model shares Paul Schrimpf • Integrate over ν
Introduction Z u exp(δj + xjθ ν) Demand in σj(θ, δ) = P dFν(ν) product space j u 1 + exp(δk + xkθ ν) Demand in k=1 characteristic space • Integral typically has no closed form, so compute Early work Model numerically, usually by Monte Carlo integration Model Estimation and identification Ns Aggregate product X u data exp(δj + xjθ νr) Estimation steps σj(θ, δ) = Pj Pricing equation u 1 + exp(δ + x θ νr) Micro data r=1 k=1 k k
References where ν are N random draws from f References r s ν • Issues about how best to compute integral — simulation vs quadrature, type of simulation (Skrainka and Judd, 2011) • Simulation (more generally approximation) of integral affects distribution of estimator Demand and supply of differentiated products Solving for δ and ξ
Paul Schrimpf • Introduction Want δ s.t. σj(θ, δ) = ˆsj Demand in • Berry, Levinsohn, and Pakes (1995) show product space
Demand in characteristic T(δ) = δ + log(ˆsj) − log(σj(θ, δ)) space Early work Model is a contraction Model Estimation and • Unique fixed point δ such that identification Aggregate product δ = δ + log(ˆsj) − log(σj(θ, δ)), i.e. ˆsj = σj(θ, δ) data Estimation steps • Can compute δ(θ) by repeatedly applying contraction Pricing equation Micro data (in theory and practice often faster to use other
References method)
References • ξj(θ) = δj(θ) − xjθ¯ • Important identifying assumption: only θ s.t. 0 ξj(θ) = ξj is true θ0 Demand and supply of differentiated products Estimating θ
Paul Schrimpf • Conditional moment restriction E[ξj(θ)|w] = 0 • Empirical unconditional moments: Introduction Demand in XJ XT product space 1 GJ,T,N,Ns = ξjt(θ)f(wt) Demand in JT characteristic j=1 t=1 space Early work where Model Model • f(w) = vector of function of w Estimation and identification • J = number of products Aggregate product data • T = number of markets Estimation steps Pricing equation • N = number of observations in each market underlying Micro data ˆsj References • Ns = number of simulations References • Asymptotic properties (consistency, distribution), depend on which of J, T, N, and Ns are →∞, see Berry, Linton, and Pakes (2004) • Reynaert and Verboven (2014): using optimal instruments greatly improves efficiency and stability Demand and supply of differentiated products Pricing equation 1 Paul Schrimpf • More moments give more precise estimates
Introduction • Assumption about form of equilibrium allows use of Demand in firm first order condition (pricing equation) as product space additional moment Demand in characteristic • space Nash equilibrium in prices Early work Model • Log linear marginal cost Model Estimation and identification k Aggregate product log mcj = rjθ + ωj data Estimation steps Pricing equation Micro data • r = observed product characteristics, input prices, References j maybe quantity, etc References • ωj = unobserved productivity, possibly endogenous
• Firm f producing set of product Jf, X max pj − Cj(·) Msj(·, p) pj:j∈Jf j∈Jf Demand and supply of differentiated products Pricing equation 2
Paul Schrimpf • Introduction First order condition: Demand in X product space ∂σl(·) σj(·) + (pl − mcl) = 0 Demand in ∂pj characteristic l∈Jf space Early work Model • Collect as Model Estimation and identification s + (p − mc)∆ = 0 Aggregate product data Estimation steps • Rearrange and use log linear marginal cost Pricing equation Micro data −1 c References log(p − ∆ σ) − rθ = ω(θ)
References • Conditional moment restriction E[ω(θ)|w] = 0 P • 1 Add empirical moments to G, JT jt ωjt(θ)f(wt) Demand and supply of differentiated products Micro data
Paul Schrimpf • Berry, Levinsohn, and Pakes (2004) • Data on individual choices and characteristics Introduction o u Demand in uijt = δj +xjt θ zit + xjt θ νit + εijt product space |{z} |{z} |{z} K×R K×L Demand in =xjtθ¯+ξjt characteristic space Early work • Random coefficients discrete choice model, socan o u Model identify and estimate δ, θ , and θ without Model Estimation and assumptions about ξ and x identification Aggregate product • Ichimura and Thompson (1998) give conditions for data Estimation steps nonparametric identification of random coefficients Pricing equation Micro data binary choice models
References • Estimate by MLE or (usually) GMM References • Still need θ¯ for price elasticities, etc
δj = xjtθ¯ + ξjt
• Use IV • Use IV with a pricing equation Demand and supply of differentiated products
Paul Schrimpf
Introduction
Demand in product space
Demand in Section 4 characteristic space Early work Model Model References Estimation and identification Aggregate product data Estimation steps Pricing equation Micro data
References
References Demand and supply of differentiated products Ackerberg, D., C. Lanier Benkard, S. Berry, and A. Pakes. Paul Schrimpf 2007. “Econometric tools for analyzing market
Introduction outcomes.” Handbook of econometrics 6:4171–4276. URL
Demand in http://www.sciencedirect.com/science/article/ product space pii/S1573441207060631. Ungated URL Demand in characteristic http://people.stern.nyu.edu/acollard/Tools.pdf. space Early work Aguirregabiria, Victor. 2019. “Empirical Industrial Model Model Organization: Models, Methods, and Applications.” URL Estimation and identification http: Aggregate product data Estimation steps //aguirregabiria.net/wpapers/book_dynamic_io.pdf. Pricing equation Micro data Berry, S., J. Levinsohn, and A. Pakes. 2004. “Differentiated References Products Demand Systems from a Combination of Micro References and Macro Data: The New Car Market.” Journal of Political Economy 112 (1):68–105. URL http://www.jstor.org/stable/10.1086/379939. Demand and supply of differentiated Berry, Steve, Oliver B. Linton, and Ariel Pakes. 2004. “Limit products Theorems for Estimating the Parameters of Differentiated Paul Schrimpf Product Demand Systems.” The Review of Economic Studies
Introduction 71 (3):613–654. URL http://restud.oxfordjournals.
Demand in org/content/71/3/613.abstract. product space
Demand in Berry, Steven, James Levinsohn, and Ariel Pakes. 1995. characteristic space “Automobile Prices in Market Equilibrium.” Econometrica Early work 63 (4):pp. 841–890. URL Model Model http://www.jstor.org/stable/2171802. Estimation and identification Aggregate product Berry, Steven T. 1994. “Estimating Discrete-Choice Models of data Estimation steps Product Differentiation.” The RAND Journal of Economics Pricing equation Micro data 25 (2):pp. 242–262. URL References http://www.jstor.org/stable/2555829. References Hausman, J.A. 1996. “Valuation of new goods under perfect and imperfect competition.” In The economics of new goods. University of Chicago Press, 207–248. URL http://www.nber.org/chapters/c6068.pdf. Demand and supply of differentiated Ichimura, Hidehiko and T.Scott Thompson. 1998. “Maximum products likelihood estimation of a binary choice model with Paul Schrimpf random coefficients of unknown distribution.” Journal of Introduction Econometrics 86 (2):269 – 295. URL Demand in product space http://www.sciencedirect.com/science/article/
Demand in pii/S0304407697001176. characteristic space Lancaster, K. 1971. Consumer demand: A new approach. Early work Model Columbia University Press (New York). Model Estimation and identification McFadden, D. 1973. “Conditional logit analysis of qualitative Aggregate product data choice behavior.” Frontiers in Econometrics :105–142URL Estimation steps Pricing equation http://elsa.berkeley.edu/pub/reprints/mcfadden/ Micro data zarembka.pdf. References
References Pakes, Ariel. 1986. “Patents as Options: Some Estimates of the Value of Holding European Patent Stocks.” Econometrica 54 (4):pp. 755–784. URL http://www.jstor.org/stable/1912835. Demand and supply of differentiated products Pinkse, J., M.E. Slade, and C. Brett. 2002. “Spatial price Paul Schrimpf competition: a semiparametric approach.” Econometrica Introduction 70 (3):1111–1153. URL http://onlinelibrary.wiley.com/ Demand in doi/10.1111/1468-0262.00320/abstract. product space Demand in Reiss, P.C. and F.A. Wolak. 2007. “Structural econometric characteristic space modeling: Rationales and examples from industrial Early work Model organization.” Handbook of econometrics 6:4277–4415. URL Model Estimation and http://www.sciencedirect.com.ezproxy.library. identification Aggregate product ubc.ca/science/article/pii/S1573441207060643. data Estimation steps Pricing equation Reynaert, Mathias and Frank Verboven. 2014. “Improving Micro data the performance of random coefficients demand models: References The role of optimal instruments.” Journal of Econometrics References 179 (1):83 – 98. URL http://www.sciencedirect.com/ science/article/pii/S0304407613002649. Demand and supply of differentiated products
Paul Schrimpf
Introduction
Demand in product space Skrainka, B. and K. Judd. 2011. “High performance Demand in characteristic quadrature rules: How numerical integration affects a space Early work popular model of product differentiation.” Available at Model Model SSRN 1870703. URL http://papers.ssrn.com/sol3/ Estimation and identification papers.cfm?abstract_id=1870703. Aggregate product data Estimation steps Pricing equation Micro data
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