Search and Price Dispersion Sibo Lu and Yuqian Wang Haas Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 1 / 56
Outline Introduction 1 No Clearing House 2 Basic Setup The Stigler Model Rothschild Critique and Diamond’s Paradox Sequential Search - Reinganum Model MacMinn Model Burdett and Judd Model Clearing House 3 Basic Setup Rosenthal Model Varian Model Baye and Morgan Asymmetric Consumers 4 Bounded Rationality / Unobserved Frictions 5 Conclusion 6 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 2 / 56
Introduction Questions Simple textbook models price of homogeneous product in competitive markets should be same However, empirical studies reveal that price dispersion is the rule (Varian, 1980, p. 651) Why? Cost of acquiring information about firms/transmitting information to consumers Search cost and other problem Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 3 / 56
Introduction Models and Approaches Search Theoretical Model/Marginal Search Cost Information Clearinghouse Others, e.g. limited rationality, asymmetric consumers Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 4 / 56
Introduction Motivating Examples Online Shopping Sequential search: Nike, then Reebox, then Addidas... Clearinghouse: Zappos, Amazon Labor Market Sequential search: worker looking for jobs over time Fixed sample search: PhD interview day Clearinghouse: LinkedIn, Monster, SimplyHired Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 5 / 56
No Clearing House Basic Setup Assumptions and variables A continuum of price-setting firms compete in homogeneous product market Unlimited capacity to supply, marginal cost m Mass of consumers be µ , indirect utility V ( p , M ) = v ( p ) + M Roy’s identity, we have q ( p ) = − v ′ ( p ) Consumer’s (indirect) utility V = v ( p ) + M − cn where c is serach cost per price quote if obtaining n price quotes Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 6 / 56
No Clearing House The Stigler Model Assumptions and Setups The Stigler Model For each consumer, K = q ( p ) = − v ′ ( p ) Fixed sample search, size n which is pre-determined Observed exogenous distribution of price, cdf F ( p ) on [ p , ¯ p ] Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 7 / 56
No Clearing House The Stigler Model Calculation Consumer Minimize the expected total cost E [ C ] = KE [ p ( n ) min ] + cn , since F ( n ) min = 1 − [1 − F ( p )] n We have � ¯ p pdF ( n ) E [ C ] = K min ( p ) + cn p � ¯ � � p [1 − F ( p )] n dp = K p + + cn p Consumer choose optimal n ∗ to minimize E [ C ] So the distribution of transaction price should be F ( n ∗ ) min ( p ) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56
No Clearing House The Stigler Model Calculation Marginal benefit of increasing sample size from n − 1 to n is [ E [ B ( n ) ] = ( E [ p ( n − 1) ] − E [ p ( n ) min ]) × K min The above is increasing in K and decreasing in n n ∗ is increasing in K A firm’s expected demand at price p is Q ( p ) = µ n ∗ (1 − F ( p )) n ∗ − 1 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56
No Clearing House The Stigler Model Propositions and Results Proposition 1 Suppose that a price distribution is a mean preserving spread of a price distribution F . 1 Then the expected transactions price of a consumer who obtains n > 1 price quotes is strictly lower under price distribution G than under F Proposition 2 Suppose that an optimizing consumer obtains more than one price quote when prices are distributed according to F , and that price distribution G is a mean preserving spread of F . Then the consumer’s expected total costs under G are strictly less than those under F intuition Consumers pay lower average prices and have lower expected total cost if prices are more dispersed � + ∞ 1 G is a mean preserving spread of F if (a) −∞ [ G ( p ) − F ( p )] dp = 0 and (b) � z −∞ [ G ( p ) − F ( p )] ≥ 0 for all z and strict for some z Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 9 / 56
No Clearing House The Stigler Model Empirical Works George Stigler, in his seminal article on the economics of information,advanced the following hypotheses: 1. The larger the fraction of the buyer’s expenditures on the commodity, the greater the savings from search and hence the greater the amount of search 2. The larger the fraction of repetitive (experienced) buyers in the market, the greater the effective amount of search (with positive correlation of successive prices) 3. The larger the fraction of repetitive sellers, the higher the correlation between successive prices, and hence, the larger the amount of accumulated search 4. The cost of search will be larger, the larger the geographic size of the market Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 10 / 56
No Clearing House The Stigler Model Empirical Works Dispersion for ”Cheap” versus ”Expensive” Items ”Expensive”(Large K in his model or high price) → high marginal benefit of search → more search → low dispersion (Stigler, 1961)Government purchase of coal VS Household purchase of automobile (Pratt, Wise and Zeckhauser, 1979) Regress the standard deviation of (log) price for a given item on the sample (log) mean price for the same item Others Dispersion and Purchase Frequency (Sorensen, 2000) Market for priscription drug Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 56
No Clearing House Rothschild Critique and Diamond’s Paradox Problems (Rothschild, 1973)Why consumers have fixed sample size search? exogenous? Optimal? Do we need to consider the update of information, such as exceptionally low price from an early search? Only consider the consumers’ effect on distribution of transaction price. But what about the firms’ side effect? Is the ex-ante price distribution F really exogenous? Why firms do not optimize their profits by setting price p ? ”partial-partial equilibrium” approach Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 12 / 56
No Clearing House Rothschild Critique and Diamond’s Paradox Diamond’s Paradox Demand Curve: − v ′ ( p ) = q ( p ) and − v ′′ ( p ) = q ′ ( p ) < 0 Sequential Search Monopoly Price p ∗ , here we assume that consumers buy quantity according to p , not a constant K v ( p ∗ ) > c Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56
No Clearing House Rothschild Critique and Diamond’s Paradox Diamond’s Paradox It is a unique equilibrium for all firms to set price p ∗ , and consumer search only once (Existence)It is an equilibrium (Uniqueness)For any firm, setting price above p ∗ is a dominated strategy (Uniqueness)If lowest price p ′ < p ∗ , it has incentive to deviate to min p ∗ , p ′ + c Perfect competition, but monopoly price in equilibrium, the reason is search cost Different from previous model, no price dispersion taking Rothschild’s criticism into account, and increase in search intensity can lead to increases or decreases in the level of equilibrium price dispersion, depending on the model. Since Stigler didn’t consider firm’s optimization behavior, it challenges his hypotheses. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56
No Clearing House Sequential Search - Reinganum Model Sequential Search - Reinganum Model Identical consumers search firm by firm and choose a stopping rule; search is costly Firms have heterogeneous marginal costs and set prices Aim: show existence of a dispersed price equilibrium. Stopping rule is optimal given firms’ optimal prices and vice versa Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 14 / 56
No Clearing House Sequential Search - Reinganum Model Consumer’s Problem Identical demand: − v ′ ( p ) = q ( p ) = Kp ǫ with ǫ < − 1 , K > 0. q ( p ) > 0 , q ′ ( p ) = ǫ Kp ǫ − 1 < 0 Search costs c > 0 per additional firm. Free-recall i.e. customers can always go back. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 15 / 56
No Clearing House Sequential Search - Reinganum Model Consumer’s Problem Assume for now a given distribution of prices F ( p ). F ( p ) is atomless with support [ p , ¯ p ]. Let z = min( p 1 , p 2 , ..., p n ) be the lowest price found after n searches. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 16 / 56
No Clearing House Sequential Search - Reinganum Model Consumer’s Problem Assume for now a given distribution of prices F ( p ). F ( p ) is atomless with support [ p , ¯ p ]. Let z = min( p 1 , p 2 , ..., p n ) be the lowest price found after n searches. Then the expected benefit of one more search is: B ( z ) = E [ v ( p ) − v ( z ) | p < z ] Prob [ p < z ] � z = ( v ( p ) − v ( z )) f ( p ) dp p � z − v ′ ( p ) F ( p ) dp = (int. by parts) p Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 16 / 56
No Clearing House Sequential Search - Reinganum Model Consumer’s Problem How does B ( z ) vary with z ? By Fundamental Theorem of Calculus: B ′ ( z ) = − v ′ ( z ) F ( z ) = q ( z ) F ( z ) > 0 ∀ z > p So lower z ⇒ lower benefit of additional search. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 17 / 56
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