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Univ. Prof. dr. Maarten Janssen University of Vienna Summer semester 2012 Week 16 (April 19,20)

Industrial Organization or Imperfect Competition Capacity constraint pricing and Consumer Search (start). Univ. Prof. dr. Maarten Janssen University of Vienna Summer semester 2012 Week 16 (April 19,20). Bertrand competition with capacity constraints.

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Univ. Prof. dr. Maarten Janssen University of Vienna Summer semester 2012 Week 16 (April 19,20)

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  1. Industrial Organization or Imperfect CompetitionCapacity constraint pricing and Consumer Search (start) Univ. Prof. dr. Maarten Janssen University of Vienna Summer semester 2012 Week 16 (April 19,20)

  2. Bertrand competition with capacity constraints • Same set-up as under homogeneous Bertrand competition, only difference is that firm i faces cannot sell more than Ki • Why would this make an important difference? • Undercutting argument presumes that you can enlarge your sales (then it can be profitable) • With capacity constraints, highest price firm may also sell positive amount • Notion of residual demand

  3. Residual demand • How to construct it. Suppose p2 > p1. All consumers like to buy from firm 1. • If D(p1) > K1, firm 2 gets residual demand: • Proportional rationing: Every consumer is served a little: K1/D(p1), D2 = D(p2) [1 - K1/D(p1)]. • Efficient rationing: (inverse) Residual demand for firm 2 is given by D2 = D(p2) - K1

  4. p D2(p2;p1) D(p) D Efficient Rationing • Also called parallel rationing • Efficient, because it would be same outcome if consumers would p 1 K2 K1

  5. p D2(p2;p1) D(p) D Proportional rationing • Also called randomized rationing • Not efficient as some high demand cosumers are not or only partially served p 1 K2 K1

  6. Analysis under efficient rationing linear demand, Small capacities • Linear demand: p = a- bQ • Small capacity: Ki < a/3b • No production cost • Claim: both firms set price p*i = a – b(K1 + K2), which is the maximal price they can get when selling their capacity • Cournot behavior!

  7. Proof of Claim • Suppose other firm charges p* • It is clear that it does not make sense to undercut • You sell your full capacity and can’t serve more consumers • Profit of firm i when pi > p*: • Πi(pi,p*) = pi(D(pi) – Kj) = pi(a - pi – bKj)/b • Maximize this wrt pi yields (a - 2pi – bKj)/b < - (a - 2bKi – bKj)/b < 0 (because of small capacities • Thus, for pi > p* firms want to set price as small as possible

  8. Analysis under efficient rationing linear demand, Large symmetric capacities I • Numeric example, Linear demand: p = 10- Q • No production cost, 5 < K1 = K2 < 10 • Claim 1: there is no sym. equilibrium in pure strategies • At any price p>0, individual firms have incentive to undercut as they do not sell up to capacity • But if p=0, then individual firms have incentive to increase price and make profit • Claim 2: there is no asym. equilibrium in pure strategies

  9. Analysis under efficient rationing linear demand, Large symmetric capacities II • Claim 3: there is an equilibrium in mixed strategies, F(p) • How to construct F(p)? • Trick: in a mixed strategy firm should be indifferent between any of the pure strategies given that competitor chooses mixed strategy • Πi(pi,F(p)) = pi { (1-F(pi))K + F(pi) (10-K- pi) } • Compact support: [p , p”] • At p” profit equals (10-K-p”)p” • Deviating to higher price should not be optimal: p” is monopoly price for residual demand: (10-K)/2 • Solve for mixed strategy distribution

  10. Types of Consumer Search • Common: consumers have to invest time and resources to get information about price and/or product • Sequential • After each search and information, consumer decides whether or not to continue searching • Simultaneous (fixed sample) • Have to decide once how many searches you make before getting results of any individual search • Sequential optimal of you get feedback quickly; otherwise simultaneous search optimal

  11. Search makes a difference • Consider Bertrand model • Each consumer has downward sloping demand • Add (very) small search cost ε> 0 • What difference does ε make? • All firms charging the (same) monopoly price is an equilibrium • How many times do consumers want to search? (Do they want to deviate?) • Is firms’ pricing optimal given strategies others (including search strategy consumers)? • Diamond result! (Diamond 1971) • No other type of equilibrium.

  12. Going back in Time • Stigler (1961) suggested that even for homogeneous products, markets seem to be characterised by price dispersion • Suggested this may be due to search costs • Some firms aim to get many consumers at low price, others go for the “tourists” • Consumers are also different: some search a lot, others not at all.

  13. Varian’s model of sales (1981) – Solution to Diamond paradox I • Two types of consumers • Shoppers compare all prices (fraction λ) and buy at shop with lowest price • Loyal consumers go to only one shop; suppose every shop has equal number of loyals (fraction 1-λ) • All have same willingness to pay v • Firms simultaneously set prices to max profits • No production cost • Firms are only strategic decision-makers • What is an equilibrium? • Set of prices or price distributions such that no firm individually benefits by deviating (Nash)

  14. Varian’s model of sales (1981) – Solution to Diamond paradox II • No equilibrium in pure strategies • Due to the presence of shoppers • How to derive sym. equilibrium in mixed strategies F(p)? • No atoms in distribution • Write down profit equation of individual firm given that all other firms charge F(p) • Π(p) = { λ(1-F(p))N-1 + (1- λ)/N } p • No wholes in the distribution; otherwise there are prices p1 < p2 with F(p1) = F(p2) implying π(p1) ≠ π(p2) • If p’ is max price charged (F(p’) = 1), then p’ = v • F(p) solves π(p) = π(v) = (1- λ)v/N

  15. First, search is exogenous IJ&R 2001 • Consumer wants to have house painted • May ask one or several (N) firms to do job • Each firm is asked (active) with prob α • An active firm thinks that with probability (1-α)N-1 it is a monopolist • Firms have cost c • Consumer has willingness to pay v (suppose this is known). • Which price will a firm charge?

  16. First, search is exogenous IIJ&R 2001 • No price equilibrium in pure strategies • How to construct equilibrium in mixed strategies? • F(p) cum. symmetric equilibrium distribution function • Write down expected profits ind. firm given F(p) • Equate with certain profits of charging upper bound; what is upper bound? • Derive mixed strategy distribution • Mixed strategies interpreted as price dispersion: identical goods are sold at different prices

  17. Comparative statics wrt N • Individual profits declining in N • As ind. profits are α(1-α)N-1 v • Industry profits Nα(1-α)N-1 v are also declining in N • By choosing α can be made to mimic empirical relation of industry profits to number of firms

  18. Endogenous Fixed Sample Search B&J 1983, J&M 2004 • Consumers can decide how many firms to search 0, 1, 2, …; • each search has cost s; • willingness to pay v • N Firms choose prices as before • Symmetric Nash equilibrium where • Consumer search behaviour is optimal given strategy of firms • Firm pricing behaviour is optimal given strategy of other firms and consumers

  19. Endogenous Fixed sample SearchRuling out equilibria • Can there be a sym. equilibrium in which firms charge pure strategies? • No. Suppose they did. Consumers will only search once. But if they do, firms have incentive to set p=v and then consumers won’t search • Due to unit demand assumption; cf., Diamond result • Can there be a sym. equilibrium where all consumers search at least two times? • No. Suppose they did. No firms would like to charge highest price in F(p). All price equal marginal cost, but then consumers would like to search only once • Same argument if consumers choose either at least two searches or not to search at all • Mixed strategy eq where some consumers search only once

  20. Endogenous Fixed sample SearchRuling out equilibria • If consumers search one time, their exp pay-off is v - E(p) – c • If they don’t search exp pay-off is 0 • If they search k times v- E(min[p1,.., pk]) – kc • Consumers can’t randomize between one search and no search at all (firms would price at v) • Consumers can’t randomize between 1 and 3 or more firms: v - E(p) – c = v- E(min[p1,.., pk]) – kc. But E(min[p1,.., pk]) is decreasing in k and at a decreasing rate. Searching more than once and less than three times would be better • Consumers have to randomize between 1 and 2 times.

  21. Endogenous Fixed sample Searchmixed strategy equilibria • If consumers search once or twice (with prob. α, resp 1- α: • Π(p) = { 2(1- α)(1-F(p))/N + α /N } p • Π(v) = αv /N • F(p) = 1- α(v-p) / [2(1- α)p] • E(p) = ∫ pdF(p) = (αv / 2(1- α)) ln [(2- α)/α] • How to determine α? • v - E(p) – s = v- E(min[p1,p2]) – 2s • E(min[p1,p2]) = ∫ pdF(min[p1,p2]) = 2 ∫ p(1-F(p))f(p) dp • No explicit solution for α (only implicitly defined, or numerically) • Equilibrium solution independent of N

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