Fixed Sequential UBC Economics 565 November 25, 2015
Fixed Sequential Theory and overview of reduced form : Baye, Morgan, and Scholten (2006) Structural papers:, Moraga-González and Wildenbeest (2008), De los Santos, Hortaçsu, and Wildenbeest (2012), Wildenbeest (2011)
1 Fixed Sequential 2 Fixed Sequential 3 4 5
Fixed Sequential Section 1
Fixed Sequential Homogenous product and competitive market: Theory one Reality Explanations: Unobserved product heterogeneity Likely part of explanation, but it is largely tautological Imperfect information and costs Stigler (1961)
Fixed Sequential
Fixed Sequential
Fixed Sequential Section 2
Fixed Sequential Key point: relationship between primitives ( cost, market size, number of firms, demand elasticity) depends on modeling assumptions Types: 1 Search 1 Fixed : gather n s, choose lowest 2 Sequential : sequentially gather s, stop when low enough 3 : some consumers loyal to one firm, others buy from lowest 2 Bounded rationality: small departure from Nash equilibrium in firms pricing game can give large Quantal response equilibrium, ε-equilibrium, mistaken beliefs about distribution
Fixed Fixed Sequential Stigler (1961) Assumptions: 1 Distribution of s on [p, p], non-degenerate CDF F(p), known by consumers 2 Each consumer wants to buy K units 3 Search process: optimally choose fixed number of quotes, n; buy from firm with lowest
Fixed Sequential Number of quotes: ( ) E[p (1:n 1) ] E[p (1:n ) ] K c n increasing in K Firm expected demand: Fixed - model implications Q(p) = µn K(1 F(p)) n 1 ( ) E[p (1:n ) ] E[p (1:n +1) ] K Transaction costs decrease with If G is a mean preserving spread of F, then E G [p (1:n) ] < E F [p (1:n) ] for n > 1 Expected total costs are lower with greater If G is a mean preserving spread of F, then E G [p (1:n G ) ]K cn G < E F[p (1:n F ) ]K cn F for n > 1
Fixed - critique I Fixed Sequential Rothschild (1973) critique: 1 distribution of s is not endogenous 2 fixed may not be optimal for consumers For (2) need to be more specific about environment Fixed optimal if e.g. waiting time to obtain each quote Diamond (1971) in sequential or fixed model with homogenous firms and consumers, there is an equilibrium where all firms charge the monopoly Can obtain non-degenerate equilibrium distribution of s by introducing firm heterogeneity or consumer cost heterogeneity
Fixed Sequential Fixed - endogenous I Burdett and Judd (1983) : equilibrium with ex-ante identical consumers and firms Assumptions: 1 Consumers: unit demand with reservation v 2 Fixed sample 3 Firms: constant marginal cost m, optimal monopoly p 4 Consumer utility given p and n = 1 is positive Equilibrium: distribution, F(p), and distribution, P(n = i) for i = 1, 2,...
Fixed Sequential Fixed - endogenous I Implications: If F(p) non-degenerate, then P(n = 1), P(n = 2) > 0 and P(n > 2) = 0, let θ = P(n = 1), 1 θ = P(n = 2) Firm profits: { (v m)θ if p = v π(p) = (p m)p(consumer purchases) if p < v { (v m)θ if p = v = (p m) [θ + (1 θ)(1 F(p))] if p < v Firms indifferent among s implies: F(p) = 1 v p θ p m 1 θ
Fixed - endogenous II Fixed Sequential Consumers indifferent between n = 1 and n = 2 pins down θ (generally two equilibria with θ (0, 1) (there s also an equilibrium where firms charge monopoly and n = 1 for all consumers))
Fixed Sequential Sequential I Sequential : consumer pays cost c to obtain p F; can either buy at p (or any previous ) or again Optimal strategy = reservation p = min{p, z } where c = z p (z p)f(p)dp = z p F(p)dp With homogeneous firms and consumers unique equilibrium is for firms to charge the monopoly Equilibrium with: Heterogeneous firm marginal cost and elastic demand (i.e. not unit demand); or Heterogeneous costs (and assumptions about distribution of costs)
Fixed Sequential Finite number n > 1 of homogeneous firms Constant marginal cost c Clearinghouse charges ϕ 0 to firms to list their s Consumers with unit demand and reservation v S > 0 shoppers consult, buy at lowest if < v, else visits one other firm buys if < v, else does not buy L > 0 loyal consumers visit firm i, buy if p i < v Equilibrium with if L > 0 or ϕ > 0 Non- s all = v Distribution of s v
Fixed Sequential Section 3
: Using distributions to estimate costs Fixed Sequential Goal: estimate consumer costs Environment: online booksellers Homogeneous product Homogeneous firm costs Data: distribution of s Method: use distribution of s + assumption about form of to estimate distribution of consumer costs
Fixed Sequential
Fixed I Fixed Sequential Firm marginal cost r, continuum of firms with equilibrium distribution F p Consumer cost c i F c Number of es ( ) ( ) E[p (1:n(ci) 1) ] E[p (1:n(ci)) ] K c i E[p (1:n(ci)) ] E[p (1:n(ci)+1) ] Define n = E[p (1:n 1) ] E[p (1:n) ]; F P observed, so n identified Let q n = F c ( n 1 ) F c ( n ) = portion of consumers who obtain n s q n not observed Assume F c such that q n = 0 for all n > K (could be relaxed, but complicates econometrics)
Fixed Sequential Fixed II Firms indifferent among s p [p, p], so [ K (p r) q 1 = (p r) q k k ( 1 F p (p) ) ] k 1 k=1 Observed s p j, j = 1,..., n f (p r) q 1 = (p r) identifies q 1,..., q K and r [ K k=1 q k k ( 1 ˆF p (p j ) ) ] k 1 Knowing q 1,..., q K can solve for F c ( 1 ),..., F c ( K ) Estimate using likelihood ( efficiently weighted GMM)
Fixed Sequential Estimates for Billingsley using 20 s, K = 3, and 5 moments q 1 = 0.633, q 2 = 0.309, q 3 = 0.058
Fixed Sequential Sequential I Consumer cost c i F c Reservation, p i = p(c i ) = min{z(c i ), p} where c i = z(ci ) Let G(p) = CDF of p i Firm indifference: p (z(c i ) p)f(p)dp = z(ci ) p F(p)dp (p r) (1 G(p)) = (p r) (1 G(p)) Data: n f s, but n f 1 indifference conditions, so need some restriction Parametric assumption about F c (in fixed model, assumption about K played a similar role) Or fix r and estimate F c nonparametrically Estimate by MLE
Fixed Sequential
Fixed Sequential
Fixed Sequential
Results Fixed Sequential For text books: Stokey-Lucas, Lazear, Billingsley, Duffie Fixed model: Median cost $2.50 (quantiles above median not identified) 25%tile $0.68 - $2.50 Selling cost r 65% of median Sequential model: Median cost $9.22-$29.40 Search cost such that z(c i ) = p, $4.56 $19.19 Selling cost r 40% of median Check whether parametric assumption driving sequential results: fix r and estimate nonparametrically
Fixed Sequential
Fixed Sequential Section 4
Fixed Sequential I Moraga-González and Wildenbeest (2008): Oligopoly version of fixed model MLE instead of nonparametric EL Chen, Hong, and Shum (2007): Model selection test to choose between fixed and sequential Test is inconclusive Moraga-González, Sándor, and Wildenbeest (2012) /Moraga-González and Wildenbeest (2008) fixed model with multiple markets Data: multiple markets with common cost distribution, but different reservation s, firm costs, and/or number of firms Semi-nonparametric estimator Application: memory chips
Fixed Sequential II De los Santos, Hortaçsu, and Wildenbeest (2012) Data on web browsing and purchases to test sequential vs fixed Key difference: behavior in sequential model depends on s observed so far; in fixed model it does not Context: online book stores Results: favor fixed model; also evidence of unobserved product heterogeneity (store loyalty) Hortaçsu and Syverson (2004) Context: mutual funds Model with frictions and product heterogeneity Results: Investors value observable nonportfolio product attributes Small costs can rationalize Wildenbeest (2011) Vertical product differentiation and frictions
Fixed Sequential III Fixed model ML estimation Context: grocery items Results: supermarket heterogeneity more important than frictions Honka (2014): & switching costs in auto insurance Fixed model Consumer knows of current insurer, and s of k others Pays switching cost if change insurer Finds costs more important than switching costs for customer retention & consumer welfare Search with learning: De los Santos, Hortacsu, and Wildenbeest (2012),
Fixed Sequential Section 5
Fixed Sequential Search model with unknown distribution Model based on Rothschild (1974) Applied to S&P 500 mutual funds Highlights differences with model with known distribution
Model I Fixed Sequential N products with utilities S N = {u 1,..., u N }, where u 1 > u 2 > > u N Consumer believes possible utilities S G = {ũ 1,..., ũ G } with S N S G Search technology: each independent and ũ g drawn with probability p g Consumer does not know p g, has Dirichlet prior with parameters α 1,..., α G, f( p 1,..., p g ) = Γ( α g ) α p g 1 g γ(αg ) which implies E[ p j ] = α j αg
Fixed Sequential Model II Bayesian updating: after seeing ũ g, n g times, so f(p n) f(n p)f(p) ( n g )! ng! n p g Γ( α g ) g γ(αg ) Γ( α g + n g ) α p g +n g 1 g γ(αg + n g ) E[ p j n 1,..., n g ] = α j + n j αg + n g p α g 1 g Sequential and at end buy best good found Search cost c, best good found so far u r
Model III Fixed Sequential Continue ing if E[max{ũ, u r } n] u r > c (ũ g u r )E[ p g n] > c ũ g >u r
Fixed Sequential Market shares I Observe: market shares, product characteristics Consumers have different costs c i F(c) Challenge: many histories can lead to the same choice; need to integrate over all histories to compute market shares Define k r = longest a consumer with best draw u r will continue ing kr = max 1, 1 g u r )α g c ũ g >u r (ũ α g g Show that market shares can be written as a function of just the k 1,..., k N k r is integer valued and decreasing in c
Example Fixed Sequential
Example Fixed Sequential
Fixed Sequential Paper has simulations comparing elasticity in with learning versus without learning models Simulations also show that ignoring learning can lead to bias
Fixed Sequential Application: S&P 500 mutual funds u g = (fixed fee per $10,000 invested) log c i N(µ 0 + µ 1 t, δ 0 + δ 1 t) Search probabilities depend on fund age: ρ jt = Aγ jt A γ kt Rational prior: α jt = ρ jt N 0 Consumers prior not identified from market share data alone
Application: S&P 500 mutual funds Fixed Sequential
Application: S&P 500 mutual funds Fixed Sequential
Application: S&P 500 mutual funds Fixed Sequential
Application: S&P 500 mutual funds Fixed Sequential
Fixed Sequential Baye, Michael R., John Morgan, and Patrick Scholten. 2006., Search, and Dispersion. Working Papers 2006-11, Indiana University, Kelley School of Business, Department of Business Economics and Public Policy. URL http://econpapers.repec.org/repec:iuk: wpaper:2006-11. Burdett, Kenneth and Kenneth L. Judd. 1983. Equilibrium Dispersion. Econometrica 51 (4):pp. 955 969. URL http://www.jstor.org/stable/1912045. Chen, Xiaohong, Han Hong, and Matthew Shum. 2007. Nonparametric likelihood ratio model selection tests between parametric likelihood and moment condition models. Journal of Econometrics 141 (1):109 140. URL http://www.sciencedirect.com/science/article/ pii/s0304407607000103. <ce:title>semiparametric methods in econometrics</ce:title>.
Fixed Sequential De los Santos, Babur, Ali Hortacsu, and Matthijs R Wildenbeest. 2012. Search with Learning. Tech. rep. URL http: //www.bus.indiana.edu/riharbau/repec/iuk/wpaper/ bepp2012-03-delossantos-hortacsu-wildenbeest. pdf. De los Santos, Babur, Ali Hortaçsu, and Matthijs R. Wildenbeest. 2012. Testing models of consumer using data on web browsing and purchasing behavior. The American Economic Review 102 (6):pp. 2955 80. URL http://www.aeaweb.org/articles.php?f=s&doi=10. 1257/aer.102.6.2955. Diamond, Peter A. 1971. A model of adjustment. Journal of Economic Theory 3 (2):156 168. URL http://www.sciencedirect.com/science/article/ pii/0022053171900135.
Fixed Sequential Hong, Han and Matthew Shum. 2006. Using Distributions to Estimate Search Costs. The RAND Journal of Economics 37 (2):pp. 257 275. URL http://www.jstor.org/stable/25046241. Honka, Elisabeth. 2014. Quantifying and switching costs in the US auto insurance industry. The RAND Journal of Economics 45 (4):847 884. URL http://dx.doi.org/10.1111/1756-2171.12073. Hortaçsu, Ali and Chad Syverson. 2004. Product Differentiation, Search Costs, and Competition in the Mutual Fund Industry: A Case Study of S&P 500 Index Funds. The Quarterly Journal of Economics 119 (2):403 456. URL http://qje.oxfordjournals.org/content/119/2/ 403.abstract.
Fixed Sequential, Sergei. 2013. Search With Dirichlet Priors: Estimation and Implications for Consumer Demand. Journal of Business & Economic Statistics 31 (2):226 239. URL http://dx.doi.org/10.1080/07350015.2013.764696. Moraga-González, José Luis and Matthijs R. Wildenbeest. 2008. Maximum likelihood estimation of costs. European Economic Review 52 (5):820 848. URL http://www.sciencedirect.com/science/article/ pii/s001429210700102x. Moraga-González, José Luis, Zsolt Sándor, and Matthijs R. Wildenbeest. 2012. SEMI-NONPARAMETRIC ESTIMATION OF CONSUMER SEARCH COSTS. Journal of Applied Econometrics :n/a n/aurl http://dx.doi.org/10.1002/jae.2290.
Fixed Sequential Rothschild, Michael. 1973. Market Organization with Imperfect : A Survey. Journal of Political Economy 81 (6):pp. 1283 1308. URL http://www.jstor.org/stable/1830741.. 1974. Searching for the Lowest When the Distribution of s Is Unknown. Journal of Political Economy 82 (4):689 711. URL http://www.jstor.org/stable/1837141. Stigler, George J. 1961. The Economics of. Journal of Political Economy 69 (3):pp. 213 225. URL http://www.jstor.org/stable/1829263. Wildenbeest, Matthijs R. 2011. An model of with vertically differentiated products. The RAND Journal of Economics 42 (4):729 757. URL http: //dx.doi.org/10.1111/j.1756-2171.2011.00152.x.