Search and Price Dispersion

Transcription

1 Search and Price Dispersion Sibo Lu and Yuqian Wang Haas Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 1 / 56

2 Outline 1 Introduction 2 No Clearing House Basic Setup The Stigler Model Rothschild Critique and Diamond s Paradox Sequential Search - Reinganum Model MacMinn Model 3 Burdett and Judd Model 4 Clearing House Basic Setup Rosenthal Model Varian Model Baye and Morgan 5 Asymmetric Consumers 6 Bounded Rationality / Unobserved Frictions 7 Conclusion Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 2 / 56

3 Questions Introduction Simple textbook models price of homogeneous product in competitive markets should be same Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 3 / 56

4 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) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 3 / 56

5 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 3 / 56

6 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

7 Models and Approaches Introduction Search Theoretical Model/Marginal Search Cost Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 4 / 56

8 Introduction Models and Approaches Search Theoretical Model/Marginal Search Cost Information Clearinghouse Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 4 / 56

9 Introduction Models and Approaches Search Theoretical Model/Marginal Search Cost Information Clearinghouse Others Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 4 / 56

10 Motivating Examples Introduction Online Shopping Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 5 / 56

11 Motivating Examples Introduction Online Shopping Sequential search: Nike, then Reebox, then Addidas... Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 5 / 56

12 Motivating Examples Introduction Online Shopping Sequential search: Nike, then Reebox, then Addidas... Clearinghouse: Zappos, Amazon Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 5 / 56

13 Motivating Examples Introduction Online Shopping Sequential search: Nike, then Reebox, then Addidas... Clearinghouse: Zappos, Amazon Labor Market Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 5 / 56

14 Motivating Examples Introduction Online Shopping Sequential search: Nike, then Reebox, then Addidas... Clearinghouse: Zappos, Amazon Labor Market Sequential search: worker looking for jobs over time Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 5 / 56

15 Motivating Examples Introduction 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 5 / 56

16 Motivating Examples Introduction 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

17 No Clearing House Assumptions and variables Basic Setup A continuum of price-setting firms compete in homogeneous product market Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 6 / 56

18 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 6 / 56

19 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 6 / 56

20 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) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 6 / 56

21 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

22 No Clearing House The Stigler Model Assumptions and Setups The Stigler Model For each cunsumer, K = q(p) = v (p) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 7 / 56

23 No Clearing House The Stigler Model Assumptions and Setups The Stigler Model For each cunsumer, K = q(p) = v (p) Fixed sample search, size n which is pre-determined Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 7 / 56

24 No Clearing House The Stigler Model Assumptions and Setups The Stigler Model For each cunsumer, 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

25 Calculation No Clearing House The Stigler Model Consumer Minimize the expected total cost Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56

26 No Clearing House The Stigler Model Calculation Consumer Minimize the expected total cost E[C] = KE[p (n) (n) min ] + cn, since F min = 1 [1 F (p)]n Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56

27 Calculation No Clearing House The Stigler Model Consumer Minimize the expected total cost E[C] = KE[p (n) (n) min ] + cn, since F min = 1 [1 F (p)]n We have E[C] = K = K p [ p p + pdf (n) min (p) + cn p p [1 F (p)] n dp ] + cn Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56

28 Calculation No Clearing House The Stigler Model Consumer Minimize the expected total cost E[C] = KE[p (n) (n) min ] + cn, since F min = 1 [1 F (p)]n We have E[C] = K = K p [ p p + pdf (n) min (p) + cn p p [1 F (p)] n dp Consumer choose optimal n to minimize E[C] ] + cn Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56

29 Calculation No Clearing House The Stigler Model Consumer Minimize the expected total cost E[C] = KE[p (n) (n) min ] + cn, since F min = 1 [1 F (p)]n We have E[C] = K = K p [ p p + pdf (n) min (p) + cn p p [1 F (p)] n dp Consumer choose optimal n to minimize E[C] ] + cn So the distribution of transaction price should be F (n ) min (p) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56

30 Calculation No Clearing House The Stigler Model Marginal benefit of increasing sample size from n 1 to n is [E[B (n) ] = (E[p (n 1) min ] E[p (n) min ]) K Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56

31 Calculation No Clearing House The Stigler Model Marginal benefit of increasing sample size from n 1 to n is [E[B (n) ] = (E[p (n 1) min ] E[p (n) min ]) K The above is increasing in K and decreasing in n Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56

32 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) min ] E[p (n) min ]) K The above is increasing in K and decreasing in n n is increasing in K Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56

33 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) min ] E[p (n) min ]) K 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) = µn (1 F (p)) n 1 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 8 / 56

34 No Clearing House Propositions and Results The Stigler Model 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 1 G is a mean preserving spread of F if (a) + [G(p) F (p)]dp = 0 and (b) [G(p) F (p)] 0 for all z and strict for some z z Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 9 / 56

35 No Clearing House Propositions and Results The Stigler Model 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 consunmer 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 1 G is a mean preserving spread of F if (a) + [G(p) F (p)]dp = 0 and (b) [G(p) F (p)] 0 for all z and strict for some z z Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 9 / 56

36 No Clearing House Propositions and Results The Stigler Model 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 consunmer 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) [G(p) F (p)] 0 for all z and strict for some z z Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 9 / 56

37 Empirical Works No Clearing House The Stigler Model George Stigler, in his seminal article on the economics of information,advanced the following hypotheses: Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 10 / 56

38 Empirical Works No Clearing House The Stigler Model 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 10 / 56

39 Empirical Works No Clearing House The Stigler Model 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) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 10 / 56

40 Empirical Works No Clearing House The Stigler Model 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 10 / 56

41 Empirical Works No Clearing House The Stigler Model 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

42 Empirical Works No Clearing House The Stigler Model Dispersion for Cheap versus Expensive Items Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 56

43 Empirical Works No Clearing House The Stigler Model Dispersion for Cheap versus Expensive Items Expensive (Large K in his model or high price) high marginal benefit of search more search low dispersion Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 56

44 Empirical Works No Clearing House The Stigler Model 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 56

45 Empirical Works No Clearing House The Stigler Model 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 56

46 Empirical Works No Clearing House The Stigler Model 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 56

47 Empirical Works No Clearing House The Stigler Model 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 56

48 Empirical Works No Clearing House The Stigler Model 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

49 Problems No Clearing House Rothschild Critique and Diamond s Paradox (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? Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 12 / 56

50 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? Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 12 / 56

51 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? Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 12 / 56

52 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

53 Diamond s Paradox No Clearing House Rothschild Critique and Diamond s Paradox Demand Curve: v (p) = q(p) and v (p) = q (p) < 0 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56

54 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56

55 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56

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

57 Diamond s Paradox No Clearing House Rothschild Critique and Diamond s Paradox It is a unique equilibrium for all firms to set price p, and consumer search only once Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56

58 Diamond s Paradox No Clearing House Rothschild Critique and 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56

59 Diamond s Paradox No Clearing House Rothschild Critique and 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56

60 Diamond s Paradox No Clearing House Rothschild Critique and 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 minp, p + c Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56

61 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 minp, p + c Perfect competition, but monopoly price in equilibrium, the reason is search cost Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56

62 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 minp, p + c Perfect competition, but monopoly price in equilibrium, the reason is search cost Different from previous model, no price dispersion Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 56

63 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 minp, 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

64 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

65 No Clearing House Sequential Search - Reinganum Model Consumer s Problem Identical demand: v (p) = q(p) = Kp ɛ with ɛ < 1, K > 0. q(p) > 0, q (p) = ɛkp ɛ 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

66 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

67 Consumer s Problem No Clearing House Sequential Search - Reinganum Model 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 p z p (v(p) v(z))f (p)dp v (p)f (p)dp (int. by parts) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 16 / 56

68 Consumer s Problem No Clearing House Sequential Search - Reinganum Model How does B(z) vary with z? Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 17 / 56

69 Consumer s Problem No Clearing House Sequential Search - Reinganum Model How does B(z) vary with z? By Fundamental Theorem of Calculus: B (z) = v (z)f (z) = q(p)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

70 No Clearing House Optimal Search Strategy Sequential Search - Reinganum Model Case 1: B( p) < c and E[v(p)] = p p v(p)f (p)dp < c recall consumers start with no price optimal strategy is to not search no transactions. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 18 / 56

71 No Clearing House Optimal Search Strategy Sequential Search - Reinganum Model Case 1: B( p) < c and E[v(p)] = p p v(p)f (p)dp < c recall consumers start with no price optimal strategy is to not search no transactions. Case 2: B( p) < c and E[v(p)] c optimal strategy is to search once. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 18 / 56

72 No Clearing House Optimal Search Strategy Sequential Search - Reinganum Model Case 3: B( p) c recall B (z) < 0 consumers search until they obtain a price quote at or below a reservation price r Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 19 / 56

73 No Clearing House Optimal Search Strategy Sequential Search - Reinganum Model Case 3: B( p) c recall B (z) < 0 consumers search until they obtain a price quote at or below a reservation price r r satisfies B(r) = c r (v(p) v(r))f (p)dp = c p Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 19 / 56

74 No Clearing House Optimal Search Strategy Sequential Search - Reinganum Model Case 3: B( p) c recall B (z) < 0 consumers search until they obtain a price quote at or below a reservation price r r satisfies B(r) = c r (v(p) v(r))f (p)dp = c p Effect of search costs on r: dr dc = 1 q(r)f (r) > 0 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 19 / 56

75 No Clearing House Sequential Search - Reinganum Model Firm s Problem Each firm has a marginal cost of production m m is drawn from an atomless distribution G(m) with support [m, m] Each firm anticipates consumers search strategy and optimal prices set by other firms. Suppose a fraction 0 λ < 1 of firms price above r and there are µ consumers on average per firm. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 20 / 56

76 Firm s Problem No Clearing House Sequential Search - Reinganum Model Let E[π(p)] be a firm s profit as a function of the price it sets. All firms with p r have an equal chance of being picked by a consumer, so: µ E[π(p)] = (p m)q(p) 1 λ For p > r? Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 20 / 56

77 No Clearing House Sequential Search - Reinganum Model Optimal Price Setting Firm solves: Solving the FOC: max E[π(p)] p p = ɛ 1 + ɛ m Recall ɛ < 1 so firm s optimal price is just a constant % markup over cost ˆF (p) = G(p 1 + ɛ ) for p [m ɛ ɛ 1 + ɛ, m ɛ 1 + ɛ ] Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 21 / 56

78 No Clearing House Sequential Search - Reinganum Model Equilibrium Additional assumption: v p ( m) > c In response to ˆF (p), consumers choose an optimal reservation price r. However if r < p ( m) then some firms would have no sales not NE Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 22 / 56

79 No Clearing House Sequential Search - Reinganum Model Equilibrium Additional assumption: v p ( m) > c In response to ˆF (p), consumers choose an optimal reservation price r. However if r < p ( m) then some firms would have no sales not NE Instead firms with marginal costs s.t. p (m) > r will choose to price at r. So F (p) = ˆF (p) if p [m ɛ 1+ɛ, r) and F (r) = 1 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 22 / 56

80 No Clearing House Sequential Search - Reinganum Model Equilibrium Need to check that given F (p), r is still consumers optimal reservation price: Recall B(r) = c and B(r) = E[v(p) v(r) p < r] Prob[p < r] Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 23 / 56

81 No Clearing House Sequential Search - Reinganum Model Equilibrium Need to check that given F (p), r is still consumers optimal reservation price: Recall B(r) = c and B(r) = E[v(p) v(r) p < r] Prob[p < r] So B(z) is unchanged from ˆF (p) to F (p) and thus r is still optimal. Recall also that p (m) is independent of λ, µ. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 23 / 56

82 Comparative Statics No Clearing House Sequential Search - Reinganum Model Variance in prices: σ 2 = E[p 2 ] E[p] 2 Effect of reservation price on variance in prices: dσ 2 dr = 2[1 ˆF (r)](r E[p]) 0 and inequality holds strictly if r < p ( m). And dr dc > 0 so an increase in search costs increases the variance of equilibirium prices. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 24 / 56

83 MacMinn Model No Clearing House MacMinn Model Aim: show price dispersion when consumers conduct fixed sample search and firms optimally set prices. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 25 / 56

84 No Clearing House MacMinn Model MacMinn Model Aim: show price dispersion when consumers conduct fixed sample search and firms optimally set prices. identical consumers demand 1 unit of a good with valuation v marginal cost of search c > 0 per price quote firms have private marginal costs m G(m), atomless with support [m, m] m < v Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 25 / 56

85 Firm s problem No Clearing House MacMinn Model When a consumer obtains n > 1 price quotes, the n firms are effectively competing against each other in an auction. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 26 / 56

86 Firm s problem No Clearing House MacMinn Model When a consumer obtains n > 1 price quotes, the n firms are effectively competing against each other in an auction. Revenue Equivalence Theorem requires: 1 firms ex-ante symmetric 2 independent private values 3 efficient allocation - consumer buys from firm with lowest m 4 free exit 5 risk neutral Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 26 / 56

87 Firm s problem No Clearing House MacMinn Model Use Revenue Equivalence Theorem and 2nd Price Auction to calculate firms expected revenues R(m). Firms bid their private values. So for firm j, if m 0 = min{m i } i j : R(m j ) = Prob[m j < m 0 ] E[m 0 m j < m 0 ] = m j (1 G(m j )) n 1 + m m j (1 G(t)) n 1 dt Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 27 / 56

88 Firm s problem No Clearing House MacMinn Model For a given price p j, expected revenue is: R(m j ) = p j Prob[m j < m 0 ] = p j (1 G(m j )) n 1 We can therefore solve for equilibrium price p j as a function of m j : p j (m j ) = E[m 0 m j < m 0 ] m ( ) 1 G(t) n 1 = m j + dt 1 G(m j ) m j Thus G(m) results in distribution of prices F (p(m)). Notice that p(m) is increasing in m so allocation is efficient. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 28 / 56

89 No Clearing House MacMinn Model Consumer s problem & Equilibrium Optimal sample size n is set by: E[B (n+1) ] < c E[B (n) ] where E[B (n) ] is the expected benefit from increasing sample size from n 1 to n, as in the Stigler Model. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 29 / 56

90 Comparative Statics No Clearing House MacMinn Model Special case when G(m) is uniform: p(m) = n 1 n m + 1 n m ( ) n 1 2 σp 2 = σm 2 n Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 30 / 56

91 Comparative Statics No Clearing House MacMinn Model Special case when G(m) is uniform: p(m) = n 1 n m + 1 n m ( ) n 1 2 σp 2 = σm 2 n higher variance in m higher variance in p Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 30 / 56

92 Comparative Statics No Clearing House MacMinn Model Special case when G(m) is uniform: p(m) = n 1 n m + 1 n m ( ) n 1 2 σp 2 = σm 2 n higher variance in m higher variance in p larger sample size n higher variance in p Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 30 / 56

93 No Clearing House MacMinn Model MacMinn vs. Reinganum Sequential search: lower search costs lower reservation price, e.g. from r to r. firms with p r do not change their prices firms with p > r lower their prices to r so dispersion decreases. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 31 / 56

94 No Clearing House MacMinn Model MacMinn vs. Reinganum Sequential search: lower search costs lower reservation price, e.g. from r to r. firms with p r do not change their prices firms with p > r lower their prices to r so dispersion decreases. Fixed sample search: increase in n increases competition faced by all firms E[m 0 m j < m 0 ] m j decreasing in n Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 31 / 56

95 No Clearing House Empirical Evidence - Search Cost MacMinn Model Online vs. Offline selection bias different search behaviors mixed results re price dispersion Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 32 / 56

96 No Clearing House MacMinn Model Empirical Evidence - Search Cost Online vs. Offline selection bias different search behaviors mixed results re price dispersion Geographic distance Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 32 / 56

97 No Clearing House MacMinn Model Empirical Evidence - Search Cost Online vs. Offline selection bias different search behaviors mixed results re price dispersion Geographic distance Estimates of search costs from structural estimation: $1.31 to $29.40 for online listings of economics and stats textbooks (Hong and Shum 2006) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 32 / 56

98 Burdett & Judd Burdett and Judd Model Aim: show equilibirium price dispersion with identical consumers and firms. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 33 / 56

99 Burdett and Judd Model Burdett & Judd Aim: show equilibirium price dispersion with identical consumers and firms. Consumers demand 1 unit of valuation v > 0 Fixed sample search Firms have identical marginal cost c < v v max{p} c Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 33 / 56

100 Burdett and Judd Model Burdett & Judd Equilibrium characterized by: optimal price distribution F (p) optimal search distribution < θ n > n=1 where θ i is fraction of consumers obtaining i price quotes Price dispersion originates from existence of a mixed search strategy equilibirium. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 34 / 56

101 Burdett and Judd Model Burdett & Judd No pure search strategy: θ 1 all firms price at identical monopoly price p = v θ 1 = 0 multiple identical firms compete so p = c Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 35 / 56

102 Burdett and Judd Model Burdett & Judd No pure search strategy: θ 1 all firms price at identical monopoly price p = v θ 1 = 0 multiple identical firms compete so p = c Therefore firms face no competition with probability θ 1 (0, 1) per customer. Firms randomize prices so that each is indifferent between charging p or v, for p in support of F (p). Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 35 / 56

103 Assumptions Clearing House Basic Setup An information clearinghouse provides a subset of consumers with a list of prices charged by different firms in the market Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 36 / 56

104 Clearing House Basic Setup Assumptions An information clearinghouse provides a subset of consumers with a list of prices charged by different firms in the market n firms in the market selling homogeneous product at constant marginal cost m Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 36 / 56

105 Clearing House Basic Setup Assumptions An information clearinghouse provides a subset of consumers with a list of prices charged by different firms in the market n firms in the market selling homogeneous product at constant marginal cost m Firm i charge price p i for its product and decide whether list this price at the clearing house at the cost of φ Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 36 / 56

106 Clearing House Basic Setup Assumptions An information clearinghouse provides a subset of consumers with a list of prices charged by different firms in the market n firms in the market selling homogeneous product at constant marginal cost m Firm i charge price p i for its product and decide whether list this price at the clearing house at the cost of φ All consumers have unit demand with a maximal willingness to pay of v > m Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 36 / 56

107 Clearing House Basic Setup Assumptions An information clearinghouse provides a subset of consumers with a list of prices charged by different firms in the market n firms in the market selling homogeneous product at constant marginal cost m Firm i charge price p i for its product and decide whether list this price at the clearing house at the cost of φ All consumers have unit demand with a maximal willingness to pay of v > m S of consumers are price-sensitive shoppers Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 36 / 56

108 Clearing House Basic Setup Assumptions An information clearinghouse provides a subset of consumers with a list of prices charged by different firms in the market n firms in the market selling homogeneous product at constant marginal cost m Firm i charge price p i for its product and decide whether list this price at the clearing house at the cost of φ All consumers have unit demand with a maximal willingness to pay of v > m S of consumers are price-sensitive shoppers L of consumers per firm directly purchase if its price doesn t exceed v or do not buy at all Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 36 / 56

109 General Treatment Clearing House Basic Setup Proposition 3 Let 0 φ < n 1 n (v m)s. Then in a symmetric equilibrium of the general clearinghouse model, we have Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 37 / 56

110 General Treatment Clearing House Basic Setup Proposition 3 Let 0 φ < n 1 n (v m)s. Then in a symmetric equilibrium of the general clearinghouse model, we have 1.Each firm lists its price at the clearinghouse with probability ( n n 1 α = 1 φ ) 1 n 1 (v m)s Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 37 / 56

111 General Treatment Clearing House Basic Setup Proposition 3 Let 0 φ < n 1 n (v m)s. Then in a symmetric equilibrium of the general clearinghouse model, we have 1.Each firm lists its price at the clearinghouse with probability ( n n 1 α = 1 φ ) 1 n 1 (v m)s 2.If a firm lists its price at the clearinghouse, it charges a price drawn from the distribution F (p) = 1 ( n ) n 1φ + (v p)l 1 n 1 1 on [p 0, v] α (p m)s, where p 0 = m + (v m) L L+S + n n 1 L+S φ Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 37 / 56

112 General Treatment Clearing House Basic Setup Proposition 3 Let 0 φ < n 1 n (v m)s. Then in a symmetric equilibrium of the general clearinghouse model, we have 1.Each firm lists its price at the clearinghouse with probability ( n n 1 α = 1 φ ) 1 n 1 (v m)s 2.If a firm lists its price at the clearinghouse, it charges a price drawn from the distribution F (p) = 1 ( n ) n 1φ + (v p)l 1 n 1 1 on [p 0, v] α (p m)s, where p 0 = m + (v m) L L+S + n n 1 L+S φ 3.If a firm does not list its at the clearinghouse, it charges a price equal to v Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 37 / 56

113 General Treatment Clearing House Basic Setup Proposition 3 Let 0 φ < n 1 n (v m)s. Then in a symmetric equilibrium of the general clearinghouse model, we have 1.Each firm lists its price at the clearinghouse with probability ( n n 1 α = 1 φ ) 1 n 1 (v m)s 2.If a firm lists its price at the clearinghouse, it charges a price drawn from the distribution F (p) = 1 ( n ) n 1φ + (v p)l 1 n 1 1 on [p 0, v] α (p m)s, where p 0 = m + (v m) L L+S + n n 1 L+S φ 3.If a firm does not list its at the clearinghouse, it charges a price equal to v 4.Each firm earns equilibrium expected profits equal to Eπ = (v m)l + 1 n 1 φ Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 37 / 56

114 Explanation Clearing House Basic Setup Several forces influence firm s strategy Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 38 / 56

115 Clearing House Basic Setup Explanation Several forces influence firm s strategy Firms wish to charge v to extract maximal profits from the loyal segment Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 38 / 56

116 Clearing House Basic Setup Explanation Several forces influence firm s strategy Firms wish to charge v to extract maximal profits from the loyal segment But this is not equilibrium because if all firms do so, a firm could just slightly undercut the price and gain all shoppers Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 38 / 56

117 Clearing House Basic Setup Explanation Several forces influence firm s strategy Firms wish to charge v to extract maximal profits from the loyal segment But this is not equilibrium because if all firms do so, a firm could just slightly undercut the price and gain all shoppers However, once prices get sufficiently low, a firm is better off by simply charging v and giving up their on shoppers Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 38 / 56

118 Clearing House Basic Setup Explanation Several forces influence firm s strategy Firms wish to charge v to extract maximal profits from the loyal segment But this is not equilibrium because if all firms do so, a firm could just slightly undercut the price and gain all shoppers However, once prices get sufficiently low, a firm is better off by simply charging v and giving up their on shoppers The only equilibrium is mixed strategy, firms randomize their prices, sometimes pricing relatively low to attract shoppers and other times pricing fairly high to maintain margins on loyals Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 38 / 56

119 Assumptions Clearing House Rosenthal Model Environment is similar to previous setup Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 39 / 56

120 Clearing House Rosenthal Model Assumptions Environment is similar to previous setup But each firm enjoys a mass L of loyal consumers Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 39 / 56

121 Clearing House Rosenthal Model Assumptions Environment is similar to previous setup But each firm enjoys a mass L of loyal consumers Costless to list prices on the clearing house: φ = 0 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 39 / 56

122 Results Clearing House Rosenthal Model Follows from Proposition 3 and set φ = 0 and get α = 1 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 40 / 56

123 Results Clearing House Rosenthal Model Follows from Proposition 3 and set φ = 0 and get α = 1 The equilibrium distribution of price is ( (v p) F (p) = 1 (p m) L S ) 1 n 1 where L p 0 = m + (v m) L + S Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 40 / 56

124 Results Clearing House Rosenthal Model Follows from Proposition 3 and set φ = 0 and get α = 1 The equilibrium distribution of price is where Mixed Strategy equilibrium ( (v p) F (p) = 1 (p m) L S L p 0 = m + (v m) L + S ) 1 n 1 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 40 / 56

125 Results Clearing House Rosenthal Model Loyal customers expect to pay the price E[p] = v p 0 pdf (p) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 40 / 56

126 Results Clearing House Rosenthal Model Loyal customers expect to pay the price Shoppers expect to pay [ E E[p] = p (n) min ] = v p 0 pdf (p) v p 0 pdf (n) min (p) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 40 / 56

127 Results Clearing House Rosenthal Model Loyal customers expect to pay the price Shoppers expect to pay [ E E[p] = p (n) min ] = v p 0 pdf (p) v p 0 pdf (n) min (p) As the number of competing firms increases, the expected transactions price paid by all consumers go up Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 40 / 56

128 Results Clearing House Rosenthal Model Loyal customers expect to pay the price Shoppers expect to pay [ E E[p] = p (n) min ] = v p 0 pdf (p) v p 0 pdf (n) min (p) As the number of competing firms increases, the expected transactions price paid by all consumers go up It is partly because we assume entry brings more loyals into the market. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 40 / 56

129 Results Clearing House Rosenthal Model Loyal customers expect to pay the price Shoppers expect to pay [ E E[p] = p (n) min ] = v p 0 pdf (p) v p 0 pdf (n) min (p) As the number of competing firms increases, the expected transactions price paid by all consumers go up It is partly because we assume entry brings more loyals into the market. For loyals, they are expected to pay more, for shoppers, the proof need a bit more work Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 40 / 56

130 Setup Clearing House Varian Model Environment is similar to previous setup Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 41 / 56

131 Clearing House Varian Model Setup Environment is similar to previous setup S informed consumers and L = U n uninformed consumers Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 41 / 56

132 Setup Clearing House Varian Model Environment is similar to previous setup S informed consumers and L = U n uninformed consumers φ = 0 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 41 / 56

133 Setup Clearing House Varian Model Environment is similar to previous setup S informed consumers and L = U n φ = 0 uninformed consumers We have α = 1 and the equilibrium distribution of prices is F (p) = 1 ( ) 1 U n 1 (v p) n (p m) S on [p 0, v] where p 0 = m + (v m) U n U n + S Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 41 / 56

134 Questions and Results Clearing House Varian Model What if consumers could make optimal decisions? The equilibrium persist if consumers have different cost of accessing the clearinghouse. And the value of information is VOI (n) = E[p] E[p (n) min ] Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 42 / 56

135 Questions and Results Clearing House Varian Model What if consumers could make optimal decisions? The equilibrium persist if consumers have different cost of accessing the clearinghouse. And the value of information is VOI (n) = E[p] E[p (n) min ] If consumers information costs are zero, all consumers choose to become informed and all firms price at marginal cost, if consumers information costs are sufficiently high, no consumers choose to become informed and all firms charge the monopoly price v. So price dispersion is not a monotonic function of consumers information cost Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 42 / 56

136 Clearing House Clearinghouse with listing cost Baye and Morgan Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 43 / 56

137 Clearing House Baye and Morgan Clearinghouse with listing cost Clearinghouse enters to serve all markets: each firm can pay φ 0 to list on clearinghouse each consumer can pay κ 0 to shop at clearinghouse Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 43 / 56

138 Clearing House Baye and Morgan Clearinghouse with listing cost In equilibrium: clearinghouse optimally sets φ, κ to maximize expected profit φnα + Sκ consumers choose whether or not to access clearinghouse each firm sets its price and chooses whether or not to list on clearinghouse Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 43 / 56

139 Clearing House Baye and Morgan Equilibrium Results Baye & Morgan can be seen as a special case of the general clearinghouse model. In equilibirium: φ > 0 κ = 0 L = 0 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 44 / 56

140 Equilibrium Results Clearing House Baye and Morgan Apply Proposition 3 to obtain equilibrium listing probability α and clearinghouse price distribution F (p). F (p) atomless with support [p 0, v], p 0 < v clearinghouse higher competition lower prices Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 45 / 56

141 Equilibrium Results Clearing House Baye and Morgan Apply Proposition 3 to obtain equilibrium listing probability α and clearinghouse price distribution F (p). F (p) atomless with support [p 0, v], p 0 < v clearinghouse higher competition lower prices κ = 0, φ > 0 lower κ more customers on clearinghouse & fewer local customers higher incentives for firm to list lower φ pricing is more competitive lower local prices Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 45 / 56

142 Equilibrium Results Clearing House Baye and Morgan Apply Proposition 3 to obtain equilibrium listing probability α and clearinghouse price distribution F (p). F (p) atomless with support [p 0, v], p 0 < v clearinghouse higher competition lower prices κ = 0, φ > 0 lower κ more customers on clearinghouse & fewer local customers higher incentives for firm to list lower φ pricing is more competitive lower local prices e.g. Amazon, Zappos, Ebay Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 45 / 56

143 Equilibrium Results Clearing House Baye and Morgan Price dispersion persists even when search costs are 0. Rather price dispersion exists because it is costly for firms to transmit price information to consumers i.e. list on clearinghouse. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 46 / 56

144 Clearing House Empirical Evidence - Competition Baye and Morgan Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 47 / 56

145 Asymmetric Consumers Asymmetric Consumers in Duopoly Market Two firms competing in the market i = 1, 2 Customers demand 1 unit Mass L i customers are loyal to firm i, with L 1 L 2 Mass S customers buy at lowest price on clearinghouse Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 48 / 56

146 Asymmetric Consumers Firm s Problem Let A i = 1 if firm i lists on clearinghouse and 0 otherwise. Expected profits if firm i posts price p, given firm j s actions: E[π i (p A i = 0)] = (L i + (1 α j ) S 2 )(p m) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 49 / 56

147 Asymmetric Consumers Firm s Problem Let A i = 1 if firm i lists on clearinghouse and 0 otherwise. Expected profits if firm i posts price p, given firm j s actions: E[π i (p A i = 0)] = (L i + (1 α j ) S 2 )(p m) E[π i (p A i = 1)] = [L i + S(1 α j ) + Sα j (1 F j (p)](p m) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 49 / 56