Industrial Organization

Transcription

1 1 Industrial Organization Prof Arthur Fishman Textbook: Jean Tirole, theory of industrial organization, MIT press Concepts you should be familiar with: Games, Nash equilibrium of static games, pure and mixed strategies, dynamic games, subgame perfect equilibrium, repeated games (see last chapter of Tirole).

2 Lecture 1: Monopoly Monopoly solves: max q fp(q) c(q)g! pm C 0 (q(p m )) p m = 1 " where " is demand elasticity (Lerner Index). This is equivalent to the MR=MC rule. Lerner Index (Monopoly markup over marginal cost) inversely proportional to demand elasticity. Monopoly always produces in region where elasticity greater than 1.

3 monopoly price non decreasing in marginal cost. Thus increased MC leads to higher price, generally. price discriminating monopoly selling in distinct market charges higher price in market with lower elasticity monopoly price greater than competitive price! dead-weight loss of welfare (produces "too little") Limits on monopoly power - Durable goods 2 period example: consumers: unit demand per period, two types, V h and V l < V h

4 proportion of low demand type =: Assume: (1 )V h > V l > 0 zero production cost discount = 1 (for both consumers and monopoly) Subgame perfect equilibrium (consumers are forward looking): period 2 price =V l period 1 price: high demand consumers indi erent between buying at period 1 at p 1 and at period 2 at V l : 2V h p 1 = V h V l

5 ! p 1 = V h + V l (intertemporal price discrimination - but monopoly would prefer to be unable to price discriminate) NC = (1 )V h + V l Under Commitment: p 1 = 2V h ; p 2 V h Commitment pro t = C = 2(1 )V h > NC Thus inability to commit reduces monopoly pricing power. Getting around the "problem"

6 1. Leasing 2. Planned obsolescence (product breaks down after 1 period) 3. Compensate rst period buyers if price comes down (like commitment) More general formulation: good is in nitely durable, in nitely lived consumers, in nitely lived rm, unit production cost = c consumer valuation of a unit = discounted value of the services from a unit over the in nite horizon consumer valuations smoothly (e.g. uniformly) distributed over [c; 1)

7 consumer discount is Coase Conjecture: As! 1; monopoly pro t! 0 (price! c from rst period) Intuition: If prices are initially high, consumers with valuations near price prefer to wait for lower future price (the price will eventually go down to c; so if consumers are patient they will wait). If monopoly can commit not to reduce price, the equilibrium price is the static monopoly price each period (all purchases take place at period 1) Variation: New (in nitely lived) consumers enter market each period - high and low valuation consumers. Prices decline until the mass of low valuation consumers is high enough to warrant a sale. After that the price cycle starts again.

8 Lecture 2: Models of Imperfect Competition 1. Short Run Price competition - homogenous good N rms, constant unit costs, no xed costs, no capacity constraints consumers perfectly informed, buy only at lowest price Game: rms set prices simultaneously (one period game) Suppose c 1 = c 2 = :::c N = c: Then the unique equilibrium is: p 1 = :::: = c Di erent unit costs: c 1 > c 2 : Equilibrium: only rm 2 sells at price: max{ rm 1 s monopoly price, c 2 g

9 Later, we add capacity constraints to this game. 2. Short Run Quantity Competition (Cournot) Firms simultaneously choose quantities, q i ; to max their own pro t given other rms quantities and price is set as P ( P q i ) (price set to clear the market). Each rm chooses q i to max: i (q i ; q j ; ) = q i P (q i + q j ) C i (q i ) The FOC : i i (q i; q j ; ) = P (q i + q j ) C 0 i (q i) + q i P 0 (q i + q j ) = 0 gives rm i s reaction curve: R i (q j )

10 Assume that R i are downward sloping - this means that rm i 0 s marginal pro t is decreasing with the other rm s quantity. The equilibrium quantities are given by the intersection of the reaction functions. The FOC can also be written as: p p C 0 i = i " ; where i = q i Q That is, the Lerner index is proportional to the rm s market share. Example: Linear Demand: P (Q) = a bq; 2 rms, xed unit costs, c i for rm i; a > c 1 ; c 2

11 i = [a b(q i + q j )]q i c i q i FOC gives rm i 0 s reaction function: R i (q j ) = a 2b c i 0:5q j The equilibrium quantities are given by the intersection of the reaction functions in q 1 ; q 2 space (see gure 1 in the gures le): q i = a 2c i + c j 3b Thus the equilibrium price is given as

12 p C = a + c 1 + c 2 3 (provided that p C > c 1 ; c 2 ) and i = (a 2c i + c j ) 2 9b Example 2: Linear demand, N rms, c 1 = c 2 = :::c N = c: In the same way as above we get: q i = a c (N + 1)b ; Q = Nq i = a c b N N + 1 ; p = a + Nc N + 1! c

13 The Cournot quantity is greater than the monopoly quantity but smaller than the competitive quantity. Thus welfare is between compeitive outcome and monopoly outcome. "Combining" price and quantity competition: Price competition with capacity constraints: Proposition: Suppose rm i cannot produce more than q i : Then if rms compete in prices and q 1 and q 2 are in the pure strategy region, then the equilibrium price of each rm is P (q 1 + q 2 ): Two stage game (Kreps, Scheinkman): Stage 1: linear demand, 2 rms, rm i can install q i units of capacity at cost of c per unit, where one unit of capacity can produce one unit of output

14 Stage 2: Firms engage in price competition, with zero marginal production costs subject to capacity constraints. Proposition ): Unique equilibrium: q i = cournot quantities for the cournot game without capacity constraints and zero production costs. Price Competition with Di erentiated Products (Location Models) 2 rms on the Hotelling Line of length 1. Firm 1 located at a; Firm 2 located at 1 b; (b is the distance from the right end). Measure 1 of consumers uniformly distributed on the line, consumer located at x; 0 x 1 : utility of consumer located at x is: U x = s t(x a) p 1 (s t(x (1 b) p 2 )); where s > 0 is "su ciently" large.

15 Location Game: Prices are exogenous, p 1 = p 2 = p and rms simultaneously choose where to locate. Unique equilibrium is both rms locate at middle of line (minimal di erentitation). Pricing Game: Suppose rms are located at ends of line and simultaneously choose prices. Solve by deriving pro t functions. Find consumer bx who is indi erent between the rms: s tbx p 1 = s t(1 bx) p 2 : Then the pro t function of rms 1 and 2 are: 1 = (p 1 c)bx; 2 = (p 2 c)(1 bx) Di erentiation with respect to p i ; d i = 0; gives the dp i reaction functions, solving gives: p 1 = p 2 = t: Other locations: Under this linear utility function, if the rms are located at the same place, the unqiue prices are p = c: If they are not at the same place, a pure strategy

16 equilibrium only exists if the rms are not too near each other. Therefore consider quadratic utility functions: U x = s t(x a) 2 p 1 (s t(x (1 b)) 2 p 2 ) Again nd consumer bx who is indi erent between the rms, which gives the pro t functions and the eqm prices as a function of a and b : p 1 = c+t(1 a b)(1+ a b 3 ); p 2 = c+t(1 a b)(1+ b a 3 ) We see that prices are higher the farther the rms are apart and p 1 = p 2 = c if the rms are located at the same location (homogenous good) Two stage game: At the rst stage the rms choose a and b; and at the second stage they choose prices. Then

17 it turns out that in eqm a = b = 0; i.e., the rms locate at the ends of the line (maximal di erentiation). The idea is that rms dont want to locate too near each other, in order to keep prices high.

18 Lecture 3: Long Term Competition (Tirole, Chap 6) in nitely repeated repeated price competition with homogenous products Firms want to maximize discounted future pro t. Firms have same unit cost, c; and same discount factor, ; < 1: Each rm chooses a price at each period to maximize sum of discounted future pro ts. Is it possible to sustain monopoly prices in the equilibrium of the repeated game?

19 Let p m be the monopoly price and m = (p m c)q(p m ) be the one period monopoly pro t. Then if each rm s price is p m at each period, then a rms discounted pro t is m = m 2(1 ) This is an eqm only if a rm cannot gain by deviating to a price < p m : De ne the trigger strategy: At period 1, p i;1 = p m : At period t 2, p i;t = p m if and only if p 1; t 1 = p 2; t 1 = p m ; otherwise p i = c: If each rm follows this strategy, each rm s price is c following any deviation. Thus following a deviation future pro t of each rm is zero. Thus, given that rm j follows

20 the trigger strategy, then rm i 0 s pro t (evaluated at period 1) is m : If at period 1 its price is slightly less than p m ; then its pro t evaluated at period 1 is m : Thus p m is the optimal price at each period if and only if m m! 0:5 Similarly if there are n identical rms, p m is the eqm price if and only if n 1 n : Even then, p m is not the unique eqm price, any other price (including c) is also an equilibrium price.

21 Fluctuating Demand There are low and high demand periods: Q l and Q h. Demand at successive periods are distributed i:i:d: and the probability of each demand state is 0.5 at each period. Each rm learns the actual demand for that period at the beginning of the period. What is the most pro table eqm which can be sustained? Let p m l and p m h be the monopoly prices corresponding to Q l and Q h respecctively, let p l and p h be the equilibrium prices corresponding to Q l and Q h respectively, and let V be the discounted future pro t beginning at any period, before the current realized demand is learned. Then: V = f0:5 (p l c)q l (p l ) 2 +0:5 (p h c)q h (p h ) g=(1 ) 2 Can p l = p m l and p = p m h then be sustained in eqm? If so,

22 V m = (m l + m h )=4 1 where m l = (p m l c)q l (p m l ) and m h = (pm h c)q h (p m h ) For this to be an eqm, it must be optimal not to deviate from the monopoly price in any demand state. Again, if rms use the trigger strategy, then pro ts are zero after any deviation. Thus we must have: ; m l 2 V m; m h 2 V m LHS is the additional pro t from deviation, RHS is the future loss from deviation.

23 Obviously the binding constraint is m h 2 V m: Solving for gives: 0 = 2 m h 3 m h + m l Thus 2 3 > 0 > 0:5 Thus it is more di cult to sustain monopoly pro ts than if demand is constant. Partial Collusion is possible when 0:5 < 0 : Consider the maximization problem: Max pl ;p h f 1 1 (1 4 l(p l ) h(p h ))

24 s.t. and h (p h ) (1 4 l(p l ) h(p h )) l (p l ) (1 4 l(p l ) h(p h )) Again, only the rst constraint is binding. rewritten as: It can be h (p h ) K l (p l ); where K = for 0:5 We want h (p h ) and l (p l ) to be as large as possible. Note that the constraint is relaxed if l (p l ) increases, and l (p l ) is maximized if p l = p m l :

25 So choose p l = p m l and let p h satisfy h (p h ) = K l (p m l ) : Thus for 0:5 < 0 ; the most pro table equilibrium for the rms is : p l = p m l and p h < p m h : This means that it is possible that p l < p h : That is, prices are lower in the high demand state to sustain collusion.

26 Secret Price Cuts Suppose rms cannot observe rivals past prices. Is it still possible to support super-competitive pro ts in repeated games? Firms can still infer some information about rivals actions from their own pro ts. Consider the following example. There are two identical rms. At each period demand may be either "high" or "low". In the high demand state, monopoly pro t is m > 0: In the low state demand is zero.

27 The high demand state occurs with probability 1-; the low state with probability (i.i.d) The rms do not observe each others prices, do they observe the realized demand. nor Thus when a rm has zero pro t, this may be due either to low rivals prices or to low demand. Speci cally, if my price was the monopoly price and my pro t was zero, I dont observe if my rival undercut the price or demand was zero. Is it then possible to sustain monopoly pro t? Only if a rm always attributes zero pro t to zero demand. This is not sustainable. Why?

28 What is the best possible attainable equilibrium? Consider the following: As long as each rm earns monopoly pro t, it continues to charge the monopoly price - the "cooperative phase". If pro t is zero, enter a "punishment phase" in which price = marginal cost for T periods, and then go back to monopoly pricing. Note: Here punishment occurs in equilibrium, in contrast to the case where past prices are perfectly observed. Let V C be expected future pro ts during the cooperative phase and V P expected future pro ts during the punishment phase: (1) V C = (1 )( 1 2 m + V C ) + (0 + V P ) (2) V P = T V C

29 There is no incentive to deviate during the cooperative phase if: (3) V C (1 )( m + V P ) + V P which may be written: (4) (V C V P ) 1 2 m (interpretation?) Solving (1) and (2) gives: V C = (1 )( 1 2 m ) 1 (1 ) T +1 V P = T (1 )( 1 2 m ) 1 (1 ) T +1

30 Thus the "no deviation" condition (4) becomes: (5) 2(1 ) + (2 1) T +1 1 Thus the highest sustainable pro t is the solution to the problem: maxv C ; s:t:(5) Since V C is decreasing in T; we look for the smallest T that satis es (5) ((5) implies that T > 0) (5) can be written: (6) 2(1 )(1 T ) + T 1

31 which is possible only if (1 ) 1 2 which is possible only if < 1 2 That is, super-competitive pro t is possible only if the low state does not occur "too frequently". For example, if = 1 3 ; then 3 4

32 Collusion and Multi Market Interaction Example I: Market 1- rms A and B Market 2 - rms A B and C If 0:5 < 2=3; collusion cannot be sustained in market 2 idea of multi market collusion - give rm C a share s > 1=3 in market 2. Firm C doesnt deviate from the monopoly price if s m 1 m! s 1 (firms A and B get share =2

33 The punishment strategy of rms A and B: If there is a deviation in either market, the price is c in both markets forever after. The no deviation condition for A and B: 1 1 [m 2 + m 2 ] 2m! 3=5 Thus for 3=5 2=3; multi market interaction enables collusion which is otherwise unattainable. Example 2: Market Sharing Two markets, two rms A and B.

34 Firm A installed in market 1, rm B in market 2. Each rm has marginal cost c in home market, but the cost c + t in other market (e.g., transportation cost). Demand in each market is Q = 1 p What is the most pro table outcome for the two rms? Each rm is a monopoly in its own market (where its cost is only c) The home-market monopoly price is p m = 1 + c 2 Pro t from non deviation is : V C = (1 c)2 4(1 )

35 If following a deviation, there is Bertrand competition and the pro t is zero in each market, then the pro t from deviation is: V D = (1 c)2 4 + (1 c)(1 c 2t) 4 V C V D if and only if: 1 c 2t 2(1 c t) < 1 2 Thus the threat of losing both markets helps sustain collusion even if < 1 2 (which is not true if the rms compete in only one market). In other words, deviation in the less pro table market is punished by also losing the more pro table market.

36 Market Preemption I: Preemption by pre-investing in capacity Incumbent rm, potential entrant linear demand: P = 1 Q three stages: stage 1 - incumbent constructs capacity, bq 1 at cost of k per unit. stage 2 - incumbent can add capacity, entrant decides whether to enter; entry costs e: stage 3- if there is entry, rms compete in quantities. Entrant s marginal cost is k + c; incumbent s marginal cost is c for q 1 bq 1, and c + k for for q > bq 1 : If no entry, only incumbent produces

37 The idea is that by paying k in advance, which becomes a sunk cost, the incumbent has an advantage at the production - competition stage. In equilibrium, the incumbent never adds capacity at stage 3 - given the eqm q 1 ; it should construct all its capacity at the rst stage. Thus the pro t function of the incumbent at stage 3 is 1 = (1 q 1 q 2 c)q 1 The incumbent s reaction function at stage 3 is given by d 1 dq 1 = 1 2q 1 q 2 c = 0! q 1 (q 2 ) = 1 2 (1 q 2 c)

38 The entrants pro t (not including e) is 2 = (1 q 1 q 2 c k)q 2 ; which gives its reaction curve if it enters as: q 2 (q 1 ) = 1 2 (1 q 1 c k); It thus enters if and only if 2 = (1 q 1 q 2 c k)q 2 e 0! q 1 eq 1 = 1 c k 2e 0:5 There are three cases. Let q m be the incumbents monopoly quantity: q1 m = 2 1 (1 c k) Case A: if q m 1 > eq 1 ; then entry is blockaded Case B: Entry is inevitable- the maximum that incumbent would ever produce is less than eq 1

39 Speci cally, suppose that for any q; the incumbent s marginal cost is c ( k was paid at stage 1) and entry occurs. Then the eqm quantities are the intersection of the reaction functions q 1 = 1 2 (1 q 2 c) and q 2 = 1=2(1 q 1 c k), which gives q 1 = q V 1 = 1 3 (1 c + k) The incumbent never produces more than q1 V, even if its capacity is greater than q1 V - thus there de nitely is entry if q1 V < eq 1 Knowing that entry is inevitable, the incumbent chooses capacity as in a one stage Stackelberg game, in which rst rm 1 produces, then rm 2 produces in reaction to q 1 : That is, the incumbent s optimal capacity is the solution of the maximization problem: max q1 f(1 q 1 q 2 (q 1 ) c k)q 1 ]

40 diagram 1:jpg ; where q 2 (q 1 ) = 1=2(1 q 1 c k) is the entrant s reaction function. The solution to this is: q s 1 = qm 1 (provided that qs 1 < qv 1 ; which is the case if k 1 k Case C: q1 s < eq 1 q1 V ; in this case the incumbent can prevent entry by choosing capacity eq 1 : It does so if and only if eq 1 is more pro table than q1 s ; and otherwise chooses capacity q1 s:

41 Brand Proliferation 2 markets, A and B, 2 rms, 1 and 2. A and B are imperfect substitutes Firm 1 is a protected monopoly in market A Either rm can enter Market B at a cost of e m 1 (1) is rm 10 s pro t if it is a monopoly in only product A and there are no rms which sell product B m 1 (2) is rm 10 s pro t from being a monopoly in both markets. To focus on the interesting case, suppose m 1 (1) > m 1 (2)

42 Thus, without the threat of entry into market B, rm 1 would only be in A d 1 (1) and d 1 (2) are rm 1 s pro t as a duopoly when it is present only in market A and present in both markets respectively. If both rms are in market B; price competition with zero pro t. Thus assume d 1 (2) < d 1 (1): Why? Because if both rms are in market B, competition is intense, p B is very low, which lowers demand and prices in market A if the two goods are imperfect substitutes. Firm 2 s pro t from entry is e if rm 1 is also present, and d 2 (1) e > 0 if rm 1 is present only in market A

43 Three stage game Stage 1: rm 1 decides whether to enter market B. stage 2: rm 2 decides whether to enter market B stage 3: rm 1 decides whether to exit market B if it has previously entered. Exist is costly and costs X: Analysis If X is su ciently large, the subgame perfect eqm is that rm 1 enters both markets and rm 2 doesnt enter. Given that rm 1 cannot exit, rm 2 knows that it will earn e from entry and hence stays out. Here rm 1 enters both markets only to preempt rm 2. This is known as brand proliferation.

44 However if X is small enough, then rm 1 will not enter and rm 2 will. Why? If rm 2 enters and rm 1 remains, rm 1 s pro t (not counting the entry cost, which is sunk after entry) is d 1 (2) while if it exits its pro t is d 1 (1) X > d 1 (2) if X is small enough. Thus, in subgame perfect equilibrium, brand proliferation is a "winning" strategy only if exit is costly. Examples of costly exit: long term contracts with suppliers, long term labor contracts, etc. Example of Brand Proliferation: The ready to eat breakfast cereal market. Up to the 1970 s, this market was highly concentrated (four major players). There was no new entry, but the existing rms frequently added new brands. There was no evidence for usual barriers to entry (e.g., economies to scale) in the entry. This brand proliferation was interpreted by the federal trade commission as erecting barriers to entry.

45 Bundling and Leverage of Market Power 2 markets, A and B. 2 rms, 1 and 2. Firm 1 is a protected monopoly in market A. Firm 2 is present in market B. demand for the 2 goods (markets) are independent. Speci cally, each consumer is willing to pay a for A and b for B, where a and b are independently and uniformly distributed over [0,1] Any rm in market B has to pay a xed cost f > 0 to be operative. If rm 1 enters, there is Bertrand price competition and each rm s pro t is zero (gross of the xed cost f > 0):

46 If rm 1 stays out of market B, then the two goods are sold independently at monopoly prices p m a and p m b. Firm 1s pro t is 1/4 and rm 2 s pro t is 1=4 e: Firm 1 will not enter market B if its sells the same product as rm 2. Suppose rm 1 produces both products and only sells the two products as a bundle at a price p ab : and let p b be the price of B alone set by rm 2 in this case.

47 Consumers which buy product B (and not the bundle) are those for whom: (i) b p b a + b p ab! a p ab p b and (ii) b p b 0 Thus rm 2 s demand is (1 p b )(p ab p b ): The FOC for rm 2 is thus: (1 2p b )(p ab p b ) p b (1 p b ) = 0

48 and rm 2 s best response function is: p br b = 1 + p ab (1 p ab + p 2 ab )1=2 Consumers who buy the bundle are those for whom: a + b p ab b p b AND a + b p ab 0 which gives rm s demand as (1 p ab + p b ) p 2 b 2

49 this gives p br a = 1 + p b 2 p 2 b 2 Thus the eqm prices are p ab = 0:61; p b = 0:24 and pro ts are 1 = 0:369 f; 2 = 0:067 f Whereas if rm 1 stays out, its pro t is Thus, even if rm 2 doesnt exit, entry is pro table for rm 1 if f < 0:119: If 0:067 < f < 0:119 rm 1 will enter with a bundle and rm 2 will exit. Notice that in this case, rm 1 s price for the pundle is the monopoly price p m ab :

50 Example: In 2004 the European Commission ned Microsoft for leveraging its market power from its PC operating system (in which it controlled 90%) to the market for work group server OS. In the latter market, its market share rose from 20 to 60 %: The commission rule that this spectacular increase was due, in part, to Microsoft s deliberate restriction ofthe interoperability between windows PCs and non-microsoft work group servers. This restriction may be seen as virtual bundling between microsoft PC OS and its server OS. Similarly, Microsofts actual bundling of windows media player with windows OS was found to be anticompetitive by the Commission.

51 Predatory pricing - limit pricing 1) Deep pockets - Incumbent can a ord to lose money until the entrant goes bankrupt because it has better access to nancing. 2) Milgrom Roberts Model 2 periods Incumbent rm ( rm 1) monoply in period 1 entry by rm 2 can occur in period 2 incumbent can have either low cost (with probability x) or high cost (probability 1 x)

52 the interesting case: entry is pro table for the entrant only if incumbent is high cost the incumbent might try to use its rst period price to signal that it is low cost Separating eqm: entry occurs only if incumbent is low cost (in eqm the entrant is not "fooled") Let ML and M H be the monopoly pro t of the low and high type respectively (when they charge their monopoly prices, where the low cost monopoly price is lower than the high cost monopoly price)

53 Let the eqm price of the low cost type be ep (which may or may not be di erent from its monopoly price). In a separating eqm the high cost type charges its monopoly price at period 1 (why?) ep must satisfy: (S:1) M L ( ep) + M L M L + D1 L where DL 1 is the low cost type s duopoly pro t if entry occurs. That is, it is optimal for the low cost type to charge the "separating price. Similarly, it must be optimal for the high cost incumbent not to charge the separating price (and thus reveal its true type) (S:2) M H ( ep) + M H M H + D1 H If both these conditions are met, then ep deters entry, and so there is entry only if the incumbent is low cost.

54 2 cases: If ep = p m L satis es both these conditions, then it is costless for the low type to deter entry. If ep = p m L does not satisfy these conditions, but there is ep < p m L which does, then the low cost type invests in entry deterrence; that is, he takes a loss at period 1 (relative to monopoly pro t) in order to remain a monopoly in the future. The idea is that it is more costly for the high cost type to charge a low price than for the the low cost type (holds under some demand conditions; see Tirole). Consumer beliefs: The preceding must be consistent with consumer beliefs.

55 Pooling Equilibria In this case the low and high cost type behave the same (so the entrant cant learn anything from the rst period price) This is an eqm only if there is no entry in equilibrium (why?). A necessary for a pooling eqm is (P:1) xd 2 L + (1 x)d2 H < 0 where DL 2 and D2 H are the entrants duopoly pro ts against an L and H incumbent respectively. Let bp be the pooling price. Then the eqm conditions are (P.1) and the following:

56 (P:2) M L ( bp) + M L M L + D1 L (P:3) M H ( bp) + M H M H + D1 H These conditions state that it is optimal for both types to charge the pooling price and deter entry. Again, bp may or may not be the low cost type monopoly price.

57 Example quantity competition; demand: P = 1 Q incumbent ( rm 1): c 1 = 0 (probability0.5) or c 1 = 4 ; entrant: c 2 = 1: Let = 1. incumbents cost period 2 pro ts if rm 2 enters period 2 pro c 1 = 0 1 = 13, 2 = :9 1 = 25; c 1 = 4 1 = 1, 2 = 7 1 = 9; Separating eqm : Claim: The following is a separating eqm: If c 1 = 0; q 1 = 5:83 (whereas the monopoly quantity is 5), if c 1 = 0; q 1 = 3. In this eqm, the entrant enters if q 1 > 5:83 and does not enter if q 1 5:83: Here the low cost incumbent produces more than its monopoly quantity at period 1 to deter entry.

58 In order to show that this is indeed an eqm, it needs to be checked that the low cost incumbent s pro t is greater when q 1 = 5:83 (which deters entry) then by producing its monopoly quantity and the reverse for the high cost incumbent. If low cost incumbent produces 5 at period 1, there will be entry and thus its total pro t is 25+13=38. If q 1 = 5:83 its total pro t is ( ) =49.31>38. Thus q 1 = 5:83 is optimal for the low cost incumbent. If the high cost incumbent produces 5:83 at period 1, its total pro t is ( ) =9.99, if it produces its monopoly output ( q 1 =3) at period 1, its total pro t is 10. Thus q 1 =3 is optimal for the high cost incumbent. Search Models

59 Note: In all of the models below, rms compete in prices and rms products are assumed to be homogenous, except for the last one (Wolinsky), in which products are di erentiated. Salop and Stiglitz - Bargains and Ripo s Every consumer is willing to buy a unit of the good if and only if the price is V: There are two types of consumers. One type - informed buyers - always know all the prices and only buys at the lowest price. The other type - random buyers - choose a rm at random and buys if the price V: The number of informed consumers is I; the number of uninformed consumers is U: Each rm has a U shaped average cost curve, where q is the quantity with the lowest AC, such that AC(q) = p.in equilibrium there are two prices, V and p. There is free entry of rms, such that each rms pro t is zero, and N is the total number of rms in equilibrium, where the number of low priced rms is N l and the number of high priced rms is N h, N l + N h = N: The informed consumers only buy at

60 the low price, the uninformed consumers buy randomly. In equilibrium, q = I + U U and AC( ) = N l N l + N h N l + N h V: See gure 2 (Salop Stiglitz) in gures le. Varian - A Model of Sales This model is similar to Salop and Stiglitz, except that there is a xed number of rms, N; and each rm has a strictly decreasing AC curve. Again, as in the Salop and Stiglitz model there are I informed consumers and U random buyers. The rm with the lowest price gets all the informed buyers. The rms compete in prices, but there is no pure strategy equilibrium (why?). The equilibrium is in mixed strategies, such that each rm has an equal chance of being the lowest price. The equilibrium price distribution is continuous so that with probability 1 one rm has the lowest price. Therefore this model can be interpreted as a model of sales. Sequential Search - Diamond Paradox

61 Suppose all consumers have unit demand, such that each consumer is willing to pay up to V for a unit. There are many small rms and consumers know only the actual price distribution but must pay a search cost s to nd out the price of any rm. Consumers must search sequentially; that is, a consumer who searches samples randomly one price at a time at the cost of s perd price. However, the rst price may be sampled without cost. The question is, what is the equilibrium price distribution? The answer is that in equilibrium the price of every rm is V (the monopoly price), regardless of whether s is large or small. Of course there is no actual search in this equilibrium. Several models try to resolve the Diamond Paradox: Reinganum s model In this model, consumers search sequentially as above, where the cost of each price sampled is s; but each consumer has a downward sloping demand curve (so that

62 the lower the price he pays, the greater the quantity he buys). Let the demand curve be D(p); where D(p) is the quantity a consumer demands when the price is p: Let CS(p) be the consumer surplus from buying at the price p and let the utility from buying at price p and searching n times be CS(p) ns: There are two types of rms. One type has a marginal cost of c l per unit, the other type has a marginal cost of c h per unit, where c h > c l : The proportion of low cost rms is : Let pl be the monopoly pricce of low cost rms and ph the monopoly price of high cost rms, p m h > pm l : Equilibrium: The price of low cost rms is p m l (same reason as in Diamond Paradox). But the price of the high cost rms is p h which depends on s: Speci cally, when s is below a threshold value, p h < p m h : p h is de ned by the following condition:

63 CS(ph) = s + CS(p m l ) + (1 )CS(p h ) In this equation the right hand side is the expected utility from searching once more for a consumer who has found a rm with the price p h and the left hand side is the utility from accepting p h : Note that here too there is no search in equilibrium ( p h is accepted without search). Stahl s Model (American Economic Review) a variant on Varian N rms, xed marginal cost, no xed costs two types of consumers, shoppers and searchers. searchers search sequentially, costs s per search

64 No pure strategy eqm (similar to Varian); eqm prices are mixed strategies (so theres a conintuous price distribution) - lowest price only sells to shoppers, the rest sell to searchers (who dont search in eqm) Average price increasing in s (higher s means searchers more willing to accept high prices rather than search) interesting point: Higher N leads to higher prices (in limit, monopoly price). The reason is that higher N means less chance of being lowest price, which reduces incentive to charge low prices Burdett and Judd (Econometrica) non sequential search consumers decide in advance how many prices to sample; each price costs s

65 N identical rms, constant marginal cost, no xed costs A Diamond eqm always exists But also mixed strategy equilibria with price distribution; some consumers sample 2 prices, some 1 price, such that E 2 = E 1 s (where E i is the average price from sampling i prices) In this sense there is search in eqm Wolinsky s Model (Di erentiated Products) Sequential search. Di erentiated products, N rms each consumer gets random utility u from a randomly selected rm, where u is distributed according to F (); with pdf f():

66 Suppose the eqm price is p e : The consumer search rule is given by a stopping rule, a; such that you search until you get utility a: a is given by (assuming that each rms price is p e ) : 1Z a (u a)f(u)du = s Thus a is decreasing in s (the higher the search cost, the less "choosy" the consumer is) deriving p e :Suppose price of all other rms is p e : Then if a rms price is p; it sells to consumers who "visit" it if: u p a p e! u > p + a p e So it sells with probability: 1 F (p + a p e ) and thus its expected pro t is:

67 E = p(1 F (p + a p e ) So p must be chosen to maximize E! p = 1 F (a) ; and in eqm, p f(a) = p e Thus p e is decreasing in a: But a is decreasing in s; and thus the equilibrium price is decreasing in s: Su cient conditon for this to be an eqm price is that is increasing. f() 1 F ()