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 � + ∞ 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 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 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) � 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 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 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) � 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: Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 10 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 10 / 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) Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 10 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 10 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 11 / 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? Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 12 / 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? Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 12 / 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 ? Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 12 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 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 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 13 / 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 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 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 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 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 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 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 ? Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 17 / 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 ( 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
No Clearing House Sequential Search - Reinganum Model Optimal Search Strategy � ¯ p Case 1: B (¯ p ) < c and E [ v ( 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
No Clearing House Sequential Search - Reinganum Model Optimal Search Strategy � ¯ p Case 1: B (¯ p ) < c and E [ v ( 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
No Clearing House Sequential Search - Reinganum Model Optimal Search Strategy 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
No Clearing House Sequential Search - Reinganum Model Optimal Search Strategy 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 r satisfies B ( r ) = c ⇔ p ( v ( p ) − v ( r )) f ( p ) dp = c Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 19 / 56
No Clearing House Sequential Search - Reinganum Model Optimal Search Strategy 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 r satisfies B ( r ) = c ⇔ p ( v ( p ) − v ( r )) f ( p ) dp = c Effect of search costs on r : 1 dr dc = q ( r ) F ( r ) > 0 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 19 / 56
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
No Clearing House Sequential Search - Reinganum Model Firm’s Problem 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
No Clearing House Sequential Search - Reinganum Model Optimal Price Setting Firm solves: max E [ π ( p )] p Solving the FOC: ǫ p ∗ = 1 + ǫ m Recall ǫ < − 1 so firm’s optimal price is just a constant % markup over cost F ( p ) = G ( p 1 + ǫ ǫ ǫ ⇒ ˆ ) for p ∈ [ m 1 + ǫ, ¯ 1 + ǫ ] m ǫ Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 21 / 56
No Clearing House Sequential Search - Reinganum Model Equilibrium Additional assumption: v − p ∗ ( ¯ m ) > c In response to ˆ F ( 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
No Clearing House Sequential Search - Reinganum Model Equilibrium Additional assumption: v − p ∗ ( ¯ m ) > c In response to ˆ F ( 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 ) = ˆ F ( p ) if p ∈ [ m 1+ ǫ , r ) and F ( r ) = 1 ǫ Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 22 / 56
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
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 ˆ F ( 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
No Clearing House Sequential Search - Reinganum Model Comparative Statics Variance in prices: σ 2 E [ p 2 ] − E [ p ] 2 = Effect of reservation price on variance in prices: d σ 2 2[1 − ˆ = F ( r )]( r − E [ p ]) ≥ 0 dr 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
No Clearing House MacMinn Model 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
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
No Clearing House MacMinn Model Firm’s problem 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
No Clearing House MacMinn Model Firm’s problem When a consumer obtains n > 1 price quotes, the n firms are effectively competing against each other in an auction. Revenue Equivalence Theorem requires: firms ex-ante symmetric 1 independent private values 2 efficient allocation - consumer buys from firm with lowest m 3 free exit 4 risk neutral 5 Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 26 / 56
No Clearing House MacMinn Model Firm’s problem 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 m j (1 − G ( m j )) n − 1 + (1 − G ( t )) n − 1 dt = m j Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 27 / 56
No Clearing House MacMinn Model Firm’s problem 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 ] � 1 − G ( t ) � ¯ � n − 1 m = 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
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
No Clearing House MacMinn Model Comparative Statics Special case when G ( m ) is uniform: p ( m ) = n − 1 m + 1 n ¯ m n � 2 � n − 1 σ 2 σ 2 p = m n Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 30 / 56
No Clearing House MacMinn Model Comparative Statics Special case when G ( m ) is uniform: p ( m ) = n − 1 m + 1 n ¯ m n � 2 � n − 1 σ 2 σ 2 p = m n higher variance in m ⇒ higher variance in p Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 30 / 56
No Clearing House MacMinn Model Comparative Statics Special case when G ( m ) is uniform: p ( m ) = n − 1 m + 1 n ¯ m n � 2 � n − 1 σ 2 σ 2 p = m 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
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
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
No Clearing House MacMinn Model Empirical Evidence - Search Cost 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
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
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
Burdett and Judd Model Burdett & Judd Aim: show equilibirium price dispersion with identical consumers and firms. Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 33 / 56
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
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
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