Kreps & Scheinkman with product differentiation: an expository note

Transcription

1 Kreps & Scheinkman with product differentiation: an expository note Stephen Martin Department of Economics Purdue University West Lafayette, IN April 2000; revised Decemer 200; August 2009, January 208 Astract Kreps and Scheinkman s (983) celerated result is that in a two-stage model of a market with homogeneous products in which firms noncooperatively pick capacities in the first stage and set prices in the second stage, the equilirium outcome is that of a one-shot Cournot game This note derives capacity est response functions for the first stage and extends the Kreps and Scheinkman result to the case of differentiated products This document saved as kspd208029wptex I am grateful to Norert Schulz and Xavier Wauthy for comments on an earlier version of these notes Responsiility for errors is my own

2 Contents Introduction 4 2 Demand 6 3 Cost 8 4 Benchmark cases 8 4 Bertrand duopoly 8 42 Cournot duopoly 9 5 Segments of the price est response function 9 6 Price est response functions 7 Results 6 7 Lemma 6 72 Lemma Lemma Lemma Theorem 23 8 Proof of Lemma 25 8 (,3,4) (2,4) (,2,4) and (,2,3,4) Summary 30 9 Proof of Lemma Cell (4,4) 3 92 Cell (3,4) Cell (2,4) Cell (,4) Cell (3,3) Cell (2,3) Cell (,3) Cell (2,2) Cell (,2) Cell (,) 49 0 Proof of Lemma (, ) (2, 2) 5 2

3 03 (2, ) 5 04 (, 2) 52 Proof of Lemma 4 53 Lower region: k 2 kb(c) 54 2 Upper region: a c k 2 θ Middle region: kb(c) k 2 a c 2 θ 2 [ 60 3 Left segment: 0 k [ 32 Middle segment: kb(c) r kb(c)] r (k2 ) 60 ] (k2 ) k kb(c) Right segment: kb(c) k Middle region: overall 64 2 Numerical Example 7 3 References 77 3

4 Introduction Kreps and Scheinkman (983) develop an extension of the Cournot and Bertrand duopoly models in which (983, p 327) Capacities are set in the first stage y the two producers Demand is then determined y Bertrand-like price competition, and production takes place at zero cost, suject to capacity constraints generated y the first-stage decisions Equilirium in this two-stage game has firms selecting capacities in the first stage that are just suffi cient to produce the Cournot equilirium outputs, and producing those outputs in the second period The Kreps and Scheinkman model is frequently cited as providing a justification for use of Cournot model, which is characterized as eing differentially unsatisfactory compared with the Bertrand model For example (Maggi, 996, p 240): The Cournot model of quantity competition has een the suject of considerale criticism in the theory of industrial organization If taken literally, the Cournot model assumes that firms dump their production on the market and that an auctioneer determines the price that clears the market In most industries there is nothing that resemles an auctioneer, and firms use prices as a strategic variale Such criticisms may e put forward in part to motivate interest in analyses of the Kreps and Scheinkman type This does not make them compelling All models are astractions from reality The mechanism y which price is determined is also unspecified in the standard model of a perfectly competitive market, a model that is not set aside on that account Nor are such criticisms needed to justify interest in the Kreps and Scheinkman model: it is a seminal example of analysis of rivalry in an imperfectly competitive market in which outcomes depend critically on the sequence in which decisions may e taken, and the way in which earlier decisions condition the payoffs associated with later decisions Friedman (982, p 505) compares the Cournot model, the Bertrand model, and a model of price-setting oligopoly with product differentiation, and concludes that the latter is to e preferred on the grounds that it relies on more satisfactory assumptions without eing computationally more complex 4

5 Kreps and Scheinkman assume that the product is homogeneous One implication of this assumption is that they must include in the model an assumed rationing rule that determines the quantity demanded of a higher-price firm if the capacity of a lower-price firm does not allow it to supply the entire quantity demanded at the lower price Their results depend on the particular form of rationing rule that is used, as they suggest (Kreps and Scheinkman, 983, p 328) and as Davidson and Deneckere (986) show formally If one extends the Kreps and Scheinkman model to differentiated products, the quantity demanded of each firm in the second stage of the game is well-defined for all price pairs This avoids the need to have a rationing rule as part of the model In this working paper, I show that the Kreps and Scheinkman result holds when the model is extended to the case of differentiated products There is a sense in which this result is intuitive One would not expect firms to hold excess capacity in equilirium If there is no excess capacity in the second stage, then when firms maximize profit in the first stage, they are maximizing a payoff function that has the same demand and cost structures as in the corresponding one-shot Cournot game, the difference eing that in the first stage of the two-stage game, firms select capacities rather than outputs This leads to the result that with product differentiation, firms capacity est response functions in the neighorhood of equilirium are isomorphic identical to the Cournot quantity est response functions It is thus to e expected that the equilirium of the two-stage game should reproduce the Cournot outcome This result is otained y Yin and Ng (997), who do not present the capacity est response functions The analysis reported here is tedious It has in common with many spatial models of imperfectly competitive markets that a complete treatment requires working through many cases that never occur in equilirium, and which one knows, or at least strongly suspects, in advance will not occur in equilirium It turns out nonetheless to e necessary to work these cases out in order to demonstrate that they do not occur in equilirium, and to verify that the most plausile suspect is in fact an equilirium Section 2 presents the model of demand for differentiated products that is used in these notes Section 3 gives the cost function For comparison and ackground, Sections 4 and 42 give areviated treatments of the Bertrand and Cournot duopoly models with product differentiation 5

6 As is standard in two-stage models, the analysis egins in the second stage and works ackward Section 5 introduces the possile segments of a firm s secondstage price est response function, and Section 6 works out the three possile shapes of the price est response functions Section 7 states the results, in the form of four Lemmas and a Theorem Lemma gives the relationship etween the capacity level chosen in the first stage and the shape of the firm s price est response function in the second period Lemma 2 gives the relationship etween the capacity levels chosen in the first period and the nature of the price est response functions in the neighorhood of equilirium Lemma 3, the proof of which is entirely mechanical, gives equilirium prices, quantities, and payoffs for the alternative second-stage equiliria Lemma 4 gives the properties of the (first-stage) capacity est response functions The Theorem is that the result of the Kreps and Scheinkman model holds when products are differentiated 2 Demand For duopoly, the quadratic representative consumer utility function U(q, q 2 ) = a(q + q 2 ) 2 (q2 + 2θq q 2 + q 2 2) + m, (2) where 0 θ, yields linear inverse demand equations 2 A Lagrangian function to descrie the constrained optimization prolem is L = a(q + q 2 ) 2 (q2 + 2θq q 2 + q 2 2) + m (22) +λ (Y m p q p 2 q 2 ), where Y is income, m all other goods, and p m = the price of all other goods The Kuhn-Tucker conditions are a (q + θq 2 ) λp 0 q [a (q + θq 2 ) λp ] = 0 q 0 (23) a (θq + q 2 ) λp 2 0 q 2 [a (θq + q 2 ) λp 2 ] = 0 q 2 0 (24) λ 0 m( λ ) = 0 m 0 (25) Y m p q p 2 q 2 0 λ (Y m p q p 2 q 2 ) = 0 λ 0 (26) 2 See Spence (976), Dixit (979), Vives (985) 6

7 Assume Y is suffi ciently large so that m > 0 Then (25) implies that and (26) implies Sustitute (27) in (23) and (24) to otain λ = (27) m = Y (p q + p 2 q 2 ) (28) a (q + θq 2 ) p 0 q [a (q + θq 2 ) p ] = 0 q 0 (29) a (θq + q 2 ) p 2 0 q 2 [a (θq + q 2 ) p 2 ] = 0 q 2 0 (20) Case : q > 0, q 2 > 0 Then (29) and (20) imply that the inverse demand curves are p = a (q + θq 2 ) (2) p 2 = a (θq + q 2 ) (22) The equations of the inverse demand curves can e inverted to otain the equations of the demand curves when consumption of oth varieties is positive:, q = a p θ(a p 2 ) ( θ 2 ) (23) q 2 = a p 2 θ(a p ) ( θ 2 (24) ) Case 2: q > 0, q 2 = 0 Then (29) implies that the inverse demand curve for variety is with corresponding demand curve p = a q, (25) q = a p (26) (20) implies Case 3: q = 0, q 2 > 0 p 2 a θq (27) 7

8 (20) implies with corresponding demand curve p 2 = a θq 2 (28) (29) implies that q 2 = a p 2 (29) p a θq 2 (220) 3 Cost Let k i = firm i s capacity ρ = long-run cost per unit of capacity The cost of capacity is fixed; once the firm gets to the second period, its cost function is C(q i ; k i ) = cq i + ρk i, q i k i (3) The units in which capacity is measured are normalized so that one unit of capacity allows production of one unit of output 4 Benchmark cases 4 Bertrand duopoly If firms compete in prices, marginal cost is x, and the quantity demanded of oth firms is positive, firm s profit function is π = (p x) ( θ)(a x) (p x) + θ(p 2 x) ( θ 2 (4) ) The first-order condition to maximize π with respect to p is 2(p x) θ(p 2 x) = ( θ)(a x) (42) and symmetric Bertrand equilirium prices are p B (x) = x + θ (a x) (43) 2 θ 8

9 Sustitute in the equation of the demand function to otain Bertrand equilirium quantities demanded: q B (x) = ( + θ)(2 θ) The Bertrand equilirium payoff with marginal cost x is a x (44) π B (x) = [p B(x) x] 2 ( θ 2 ) = + θ θ (a x) 2 (45) (2 θ) 2 42 Cournot duopoly If firms compete in quantities and marginal cost is x, firm s profit function is π = [a x (q + θq 2 )]q (46) The first-order to maximize π with respect to q condition is leading to symmetric equilirium outputs 2q + θq 2 = a x, (47) q C (x) = 2 + θ The corresponding Cournot equilirium prices are Cournot equilirium profit per firm is a x (48) p C (x) = x + a x 2 + θ (49) π C (x) = (a x) 2 (40) (2 + θ) 2 5 Segments of the price est response function Firms first choose capacities, then set prices, then produce the quantities demanded at those prices 9

10 Here we consider the nature of firm s price est response function, taking capacity as given There are at most four segments of the price est response function; the actual numer of segments may e two, three, or four, depending on the capacity level chosen in the first stage and on the parameters of the model Once k has een chosen, firm s profit function for output levels not greater than k, q = ( θ)a p + θp 2 ( θ 2 ) = ( θ)a (p c) + θ(p 2 c) ( θ 2 ) k, (5) is π = (p c) ( θ)a p + θp 2 ( θ 2 ) ρk (52) Call the first-order condition to maximize (52) when the capacity constraint (5) is not inding ranch one of firm s price est response function This is the Bertrand est response function with marginal cost equal to c, with equation that can e written 2(p c) θ(p 2 c) = ( θ)(a c) (53) If the capacity constraint is inding, firm s output equals capacity; the equation of the inding capacity constraint may e variously written or or q = ( θ)a p + θp 2 ( θ 2 ) p 2 = p ( θ)a + ( θ 2 )k θ = k (54) (55) p = θp 2 + ( θ)a ( θ 2 )k (56) Call this ranch two of firm s price est response function On ranch two of its est response function, firm s profit function is π = (p c ρ)k (57) If p 2 rises suffi ciently, the quantity demanded of firm 2 goes to zero At that point, the quantity demanded of firm is less than min(k, q m(c) ), where q m(c) is 0

11 the output of a single-variety monopolist with marginal cost c per unit The nonnegativity constraint q 2 0 ecomes inding, and it is the q 2 = 0 equation, θp + p 2 = ( θ)a (58) (from (24)) that is the equation of firm s price est response function Call this segment of the price est response function ranch three of firm s est response function Finally, if p 2 rises suffi ciently, the quantity demanded of firm equals min(k, q m(c) ), at which point firm s price est response function ecomes vertical: firm 2 s price is so high that firm can sell as close to monopoly output as its capacity level permits, without creating a positive demand for variety 2 Call this segment of the price est response function ranch four of firm s est response function 6 Price est response functions Four configurations are possile for firm s price est response function, depending on its capacity level and on the parameters of the model (2,4): For very low capacity levels (as specified in Lemma ), firm is capacity constrained until p 2 reaches such a high level that the quantity demanded of firm 2 is zero even when firm sells all it can produce, given its capacity; see Figure 6 3 (,2,4): For larger capacity levels, ut not exceeding a limit specified in Lemma, firm s price est response function egins with the unconstrained, ranch one segment, then moves on to ranch two and ranch four See Figure 62 (,2,3,4): For still higher values of k, ut not exceeding a level specified in Lemma, firm s price est response function has all four segments See Figure 63 (,3,4): For very high capacity levels, firm one is not capacity constrained, and sets price along its ranch one est response function, until p 2 reaches a level at which q 2 = 0 Firm produces the least output consistent with q 2 = 0 For suffi ciently high p 2, firm is ale to set the unconstrained monopoly price and sell the unconstrained monopoly output See Figure 64 Firm s price est response function is of form (, 3, 4) 3 Figures are drawn for a = 2, = c = ρ =, θ = /2

12 a p 2 c c q = k = 35 E (325) F (325) F (35) F (275) q = k = 325 q = k = 275 E (35) E (275) p a Figure 6: Firm s (2, 4) price est response function, k k A 2

13 a p 2 q = k, q 2 = 0 p 2 a θk F (k ) q = k q 2 > 0 q < k G (k ) c A c p a Figure 62: Firm s (, 2, 4) price est response function, k A k q m 3

14 a p 2 q = q m, q 2 = 0 p 2 a θk q F (k ) = k H q 2 > 0 G (k ) q < q m p 2 = 0 q < k c A c p a Figure 63: Firm s (, 2, 3, 4) price est response function, q m k k D 4

15 a p 2 c q 2 = 0, q q m q 2 = 0, q = q m H D q < k A c p a Figure 64: Firm s (, 3, 4) price est response function, k k D 5

16 7 Results 7 Lemma Lemma : Let k A a c + θ 2, (7) k D a c 2 θ 2 (72) Then (a) the relation etween first-stage capacity k i and the configuration of the secondstage price est response function is k i k A firm i s est response function is of the form (2,4) k A k i q m(c) firm i s est response function is of the form (,2,4) q m(c) k i k D firm i s est response function is of the form (,2,3,4) k D k i a c ρ firm i s est response function is of the form (,3,4) () in the second stage, there are 6 possile cominations of price est response functions of the two firms, one comination for each of the 6 regions shown Figure 7 6

17 2:(,2,3,4){ k 2 k D q m k A : (,2,4) :(2,4) :(,3,4) 2:(,3,4) 2: 2:(,3,4) (,3,4) :(2,4) : :(,3,4) (,2,4) : :(2,4) (,2,4) :(,3,4) 2: 2:(,2,4) 2:(,2,4) (,2,4) : :(2,4) (,2,4) :(,3,4) 2:(2,4) 2:(2,4) 2:(2,4) k A q m k D }{{} :(,2,3,4) k Figure 7: Price est response function configurations, capacity space 7

18 72 Lemma 2 Lemma 2: Let k B(c) = a c ( + θ)(2 θ) (73) denote the capacity that is just suffi cient to allow a firm to produce the Bertrand equilirium output with marginal cost c and let kb(c)(k r j ) = ( ) a c 2 θ 2 θk j (74) e the capacity level that just allows firm i to produce its Bertrand est-response output when oth firms have marginal cost c and firm j produces output level k j There are four second-stage equilirium types: Region of capacity space Firm Firm 2 (, ) k kb(c), k 2) kb(c) ranch one ranch one (, 2) k kb(c) r 2), k 2 kb(c) ranch one ranch two (2, ) k kb(c), k 2 kb(c) r ) ranch two ranch one (2, 2) k kb(c) r 2), k 2 kb(c) r ) ranch two ranch two These regions are shown in Figure 72 8

19 k 2 k D kb(c) k 2 = k r B(c) (k ) (2, 2) k = k r B(c) (k 2) (2, ) (, ) kb(c) k D (, 2) k Figure 72: Segments of price est response functions that intersect in equilirium, capacity space; (i, j) indicates that in second-stage equilirium, firm s ranch i intersects with firm 2 s ranch j 9

20 73 Lemma 3 Lemma 3: Second-stage equilirium prices, outputs, and payoffs for the four equilirium types descried in Lemma 2 are (a) (, ) Firm Firm 2 p = c + θ (a c) p 2 θ 2 = c + θ (a c) 2 θ a c a c q = q (+θ)(2 θ) 2 = (+θ)(2 θ) θ (a c) π = 2 θ (a c) ρk (+θ)(2 θ) 2 π 2 = 2 ρk (+θ)(2 θ) 2 2 () (, 2) Firm Firm 2 p 2 = c + θ2 (a c θk 2 θ 2 2 ) p 22 = c + ( θ)(2+θ)(a c) 2( θ2 )k 2 θ 2 q 2 = a c θk 2 2 θ 2 π 2 = q 22 = k 2 θ2 (a c θk 2 ) 2 ρk (2 θ 2 ) 2 π 22 = ( ( θ)(2+θ)(a c) 2( θ 2 )k 2 2 θ 2 ) ρ k 2 (c) (2, ) Firm Firm 2 p 2 = c + ( θ)(2+θ)(a c) 2( θ2 )k p 2 θ 2 22 = c + θ2 (a c θk 2 θ 2 ) q 2 = k q 22 = π 2 = ( ( θ)(2+θ)(a c) 2( θ 2 )k 2 θ 2 ) ρ a c θk 2 θ 2 k π 22 = θ2 (2 θ 2 ) 2 (a c θk ) 2 ρk 2 (d) (2, 2) Firm Firm 2 p 22 = a (k + θk 2 ) p 222 = a (θk + k 2 ) q 22 = k q 222 = k 2 π 22 = [a c ρ (k + θk 2 )] k π 222 = [(a c ρ (θk + k 2 )] k 2 20

21 74 Lemma 4 Lemma 4: Let k S = a c ρ 2 2 [ θ2 ( θ)(2 + θ)(a c) (2 θ 2 ] )ρ θ 2 θ 2 4( θ 2 ) (75) Equilirium capacity est response functions are Firm j s capacity 0 k j k S k i (k j ) = kc(c+ρ) r (k 2) = 2 Firm i s est response capacity k S k j a c ρ k i (k j ) = ( θ)(2+θ)(a c) (2 θ2 )ρ 4( θ 2 ) ( a c ρ θk j ), where k r C(c+ρ) (k 2) is the capacity level that just allows firm i to produce its Cournot est-response output if oth firms have unit cost c + ρ and firm j is producing output k j The est response functions are shown in Figure 73 2

22 k 2 (2, ) Firm s capacity est response function k B(c) k 2 = kc(c+ρ) r (k ) (2, 2) (, 2) k = k r C(c+ρ) (k 2) k B(c) (, ) Firm 2 s capacity est response function k Figure 73: Capacity est response functions, Kreps & Scheinkman model with product differentiation, a = 2, = c = ρ =, θ = /2 22

23 75 Theorem Let k C(c+ρ) = a c ρ (76) 2 + θ denote the minimum capacity that permits a firm to produce the Cournot equilirium output of the one-shot game when oth firms have marginal cost c + ρ Theorem: In the unique noncooperative equilirium of the Kreps and Scheinkman model with product differentiation, firms select capacities k i = k C(c+ρ) in the first stage and set the Cournot equilirium prices in the second stage Proof: this follows from the facts that the segment of the capacity est response function that is isomorphic to the Cournot quantity est response function rises aove kb(c), the capacity level that permits a firm to produce the Bertrand equilirium output when marginal cost is c, k S > k B(c), and that Bertrand equilirium output with marginal cost c is greater than Cournot equilirium output with marginal cost c + ρ: k B(c) > k C(c+ρ) The capacity est response functions are shown in Figure 73 Figure 74 shows the price est response functions for the continuation game when the noncooperative equilirium capacity levels are chosen in the first stage 23

24 a p 2 q = k q 2 > 0 q 2 < k 2 A 2 G2 (k 2 ) c A q = k, q 2 = 0 p 2 a θk F (k ) F 2 (k 2 ) q 2 = k 2 q > 0 G (k ) q < k q 2 = k 2, q = 0 p a θk 2 c p a Figure 74: Second-stage (2, 2) equilirium for equilirium capacities; oth firms (, 2, 4) price est response functions 24

25 8 Proof of Lemma 8 (,3,4) Rewrite firm s ranch one profit function, (52), in terms of deviations from marginal cost: π = (p c) ( θ)(a c) (p c) + θ(p 2 c) ( θ 2 ) The first-order condition to maximize (8) with respect to p is: π = ( θ)(a c) 2(p c) + θ(p 2 c) p ( θ 2 ) ρk (8) = 0 (82) Note that (82) implies that when the first-order condition holds, firm s output is q = ( θ)(a c) (p c) + θ(p 2 c) ( θ 2 ) = p c ( θ 2 ) (83) so that along this segment of its est response function, firm s payoff is π = (p c) 2 ( θ 2 ) ρk (84) Solving (82) for p gives the equation of the ranch one (q < k ) segment of firm s est response function: p = c + 2 [( θ)(a c) + θ(p 2 c)] (85) Firm 2 would never charge a price elow p 2 = c For p 2 = c, p r (c) = c + ( θ) (a c) 2 is In the (,3,4) case, the initial point of firm s price est response function A : (p A, p 2A ) = (c + 2 ) ( θ) (a c), c (86) (Figure 64) 25

26 Even though firm s ranch one est response price rises as p 2 rises from p 2 = 0, p rises relatively less than p 2, with the result that q rises, and q 2 falls, moving up along firm s ranch one price est response function This continues until p 2 reaches such a high level that q 2 falls to zero As p 2 rises from this point, D in Figure 64, firm s price rises, and q falls, moving along the q 2 = 0 line (firm s ranch three) It follows that for firm s price est response function to have the (,3,4) configuration, its capacity k must permit it to produce the quantity demanded of it at point D, the intersection of firm s ranch one and firm s ranch three The system of equations formed y the equation of ranch one, (85), and the equation of ranch three, (58), oth rewritten in terms of deviations from c, is with solution ( 2 θ θ ( p c p 2 c ) ( p c p 2 c ) ) = ( θ)(a c) = θ ( + θ 2 θ θ ( ) ; (87) ) (a c) (88) Firm s ranch one and ranch three intersect at point D : (p D, p 2D ) = (c + ) θ2 ( θ)(2 + θ) 2 (a c), c + 2 θ 2 θ 2 (a c) (89) From (83), the quantity demanded of firm at point D is q D = p c ( θ 2 ) = ( θ 2 ) θ 2 a c 2 (a c) = 2 θ 2 θ 2 (80) Note that q D > q m(c), the unconstrained monopoly output with marginal cost c per unit a c 2 θ 2 a c θ 2 a c = 2 2(2 θ 2 0 (8) ) The condition for firm s price est response function to have the (,3,4) configuration is then k k D (82) Firm s est response price runs along the q 2 = 0 line, (58), until p reaches the unconstrained monopoly price, which occurs for p = c + (p 2 c) ( θ)(a c) θ 26 = c + (a c) 2

27 p 2 = c + 2 (2 θ)(a c) p 2H The second segment of firm s Bertrand est response function is the straight line connecting point D : (p D, p 2D ) and point H : (p H, p 2H ) = (c + 2 (a c), c + 2 ) (2 θ)(a c) (83) For higher values of p 2, firm charges the unconstrained monopoly price and sells the unconstrained monopoly quantity If (82) is met, firm s est response function has three segments, ranch one from point A to point D, ranch three from point D to point H, and vertical at p = p H thereafter 82 (2,4) At point A, p 2 = c; the quantity demanded of firm at point A along firm s ranch one (q < k ) is q A = p A c ( θ 2 ) = a c + θ 2 (84) q A is less than the unconstrained monopoly output of a single variety: a c 2 a c + θ 2 = θ a c + θ 2 > 0 Let k A = a c (85) + θ 2 e the capacity that is just suffi cient to allow the firm to produce q A If k k A, (86) firm is on the capacity constrained (ranch two) segment of its price est response function until p 2 rises so much that q 2 falls to zero If p 2 = c on firm s ranch two, then from (56) firm s price is p = θc + ( θ)a ( θ 2 )k = c + ( θ)(a c) ( θ 2 )k p E (87) 27

28 If k k A, firm s est response function egins at point E (k ) : (p E, p 2E ) = ( c + ( θ)a ( θ 2 )k, c ) (88) (Figure 6) At what point do firm s ranch two and ranch three intersect? Solve the system of equations formed y (58) (q 2 = 0) and (56) (q = k ) ( θ θ ) ( ) ( ) ( ) p = ( θ)a ( θ 2 )k 0 p 2 ( p p 2 ) ( = a ) ( k θ The q 2 = 0 line and the q = k line intersect at point (89) ) (820) F (k ) : (p F, p 2F ) = (a k, a θk ) (82) (Figure 6) Firm s ranch two and ranch three segments intersect at point F (k ), with coordinates given y (82) If q 2 = 0, we otain the same results if firm s constrained optimization prolem is formulated with price or with quantity as firm s decision variale We proceed in terms of quantity For p 2 p 2F and k k A, firm maximizes suject to the capacity constraints (a c q )q k q and suject to the Kuhn-Tucker inequality for q 2 = 0 for the representative consumer constrained optimization prolem, p 2 a θq A Lagrangian for firm s constrained optimization prolem is L = (a c q )q + λ (k q ) + λ 2 (p 2 a + θq ) 28

29 The Kuhn-Tucker conditions are a c 2q λ + θλ 2 0 q [a c 2q λ + θλ 2 ] = 0 q 0 k q 0 λ (k q ) = 0 λ 0 p 2 a + θq 0 λ 2 (p 2 a + θq ) = 0 λ 2 0 Suppose q = k > 0 Then a c 2q λ + θλ 2 = 0 For this to e a solution, we must also have p 2 a θk = p 2F, which condition is met For p 2 > a θk, λ 2 = 0; then ( ) a c λ = 2 k > 0 2 When k k A and p 2 p 2F, firm s est response is to set price p F = a k and sell at capacity When capacity satisfies (86), the price est response function has a ranch two segment connecting point E (k ) to point F (k ) and is vertical thereafter Figure 6 shows firm s (2, 4) price est response functions for three alternative levels of k As k falls, firm s ranch two segment shifts right, and the point at which firm shifts from its ranch two to its ranch three moves up the q 2 = 0 line 83 (,2,4) and (,2,3,4) If k A k k D, firm s est response function egins on ranch one, ut its capacity is not suffi cient to allow it to produce along ranch one until it reaches ranch three If k A k k D, (822) then when p 2 rises suffi ciently from p 2 = c, firm one moves from ranch one to ranch two What is the point of intersection of ranch one and ranch two? The equations of ranch one and ranch two are 2(p c) θ(p 2 c) = ( θ)(a c) 29

30 and p θp 2 = ( θ)a ( θ 2 )k respectively Rewritten in terms of deviations from c, the system of equations formed y the equations of firm s ranch one and ranch two is ( 2 θ θ with solution ( p c p 2 c ) ( p c p 2 c ) ) = ( θ)(a c) = ( θ 2 )k ( 2/θ ) ( ) ( 0 ( θ 2 ) ( 0 ( θ)(a c) /θ The point of intersection of ranch one and ranch two is k ), (823) ) (824) G (k ): (p G, p 2G ) = ( c + ( θ 2 )k, c + 2( ) θ2 )k ( θ)(a c) θ (Figures 62 and 63) By (8), k D > q m(c) On the other hand, k A is less than q m(c) : (825) q m(c) k A = a c 2 ( ) a c + θ 2 a c + θ 2 = θ a c + θ 2 = > 0 For k A k q m(c), firm s est response function has three segments, ranch one from point A to point G (k ), ranch two from point G (k ) to point F (k ), and vertical (ranch four) thereafter; see Figure 83 For q m(c) k k D, firm s est response function has four segments, ranch one from point A to point G (k ), ranch two from point G (k ) to point F (k ), ranch three from point F (k ) to point H, and vertical (ranch four) thereafter; see Figure Summary The results otained aove are summarized in Tale 8 30

31 0 k k A (2,4) E (k ) F (k ) vertical k A k q m(c) (,2,4) A G (k ) F (k ) vertical q m(c) k k D (,2,3,4) A G (k ) F (k ) H vertical k D k (,3,4) A D H vertical Tale 8: Capacity and configuration of price est response function 9 Proof of Lemma 2 9 Cell (4,4) Cell (4,4) row 4, column 4 in Figure 7 is defined y the inequalities k D k, k D k 2 In cell (4,4) oth firms have est response functions with configuration (,3,4) Figure 9 shows an equilirium with oth firms on ranch one of their price est response functions When oth firms have (,3,4) est response functions, equilirium occurs at the intersection of the ranch one segments if point D lies aove firm 2 s ranch one and point D 2 lies to the right of firm s ranch one From (89), the coordinates of point D are (p D, p 2D ) = (c + ) θ2 ( θ)(2 + θ) 2 (a c), c + 2 θ 2 θ 2 (a c) is The equation of the ranch one segment of firm 2 s price est response function θ(p c) + 2(p 2 c) = ( θ)(a c) (9) The condition for point D to lie aove firm 2 s ranch one is θ(p D c) + 2(p 2D c) ( θ)(a c) (92) Sustituting the coordinates of point D, the condition is met if [ θ 2 ] [ ] ( θ)(2 + θ) θ 2 (a c) θ 2 θ 2 (a c) ( θ)(a c) θ + θ 2 θ θ 2 θ 2 3

32 a p 2 c A 2 q < k q 2 = 0, q q m A H D q 2 = 0, q = q m H 2 D 2 q = 0, q 2 = q m q = 0, q q m q 2 < k 2 c p a Figure 9: (, ) second-stage equilirium, oth firms (, 3, 4) price est response functions 32

33 4 + θ θ 2 2 θ θ θ 2 2 θ θ 0, which is always the case In the same way, point D 2 is always to the right of firm one s ranch two in cell (4,4) When oth firms have (,3,4) price est response functions, equilirium always occurs at the intersection of the ranch one segments 92 Cell (3,4) Cell (3,4) row 4, column 3 of Figure 7 is defined y the inequalities q m k k D, k D k 2 In this region of capacity space, firm s price est response function is of form (,2,3,4) Firm 2 s price est response function is of form (,3,4) This configuration is shown in Figure 92 One condition for (,) equilirium in cell (4,3) is that point D 2 e to the right of firm s ranch one; y the argument of Section 9, this condition is always met The second condition is that point G e aove firm two s ranch one The coordinates of point G (k ) are or (p G, p 2G ) = ( c + ( θ 2 )k, c + 2( ) θ2 )k ( θ)(a c) θ From (9), the condition for point G to lie aove firm 2 s ranch one is θ(p G c) + 2(p 2G c) ( θ)(a c); (93) or θ( θ 2 )k + 2 2( θ2 )k ( θ)(a c) θ k a c ( + θ)(2 θ) ( θ)(a c) k B(c), (94) 33

34 a p 2 A 2 q = q m, q 2 = 0 p 2 a θk q = k q 2 > 0 q < k c A G (k ) F (k ) H q < q m p 2 = 0 H 2 D 2 q = 0, q 2 = q m q = 0, q q m q 2 < k 2 c p a Figure 92: (, ) equilirium, cell (4,3): q m k k D, k D k 2 34

35 where, from (44), the capacity level kb(c) is the capacity level that is just suffi cient to allow the firm to produce Bertrand equilirium output when oth firms have marginal cost c The range of k in cell (3,4) is q m k k D Since q m kb(c) = a c 2 ( + θ)(2 θ) ( ) 2 a c = ( + θ)(2 θ) = θ( θ) a c 2 ( + θ) (2 θ) > 0, a c condition (94) is always met in cell (3,4) By similar arguments, in cell (4,3), equilirium is of type (,) for k 2 k B(c) and the condition is always met 93 Cell (2,4) Cell (2,4) row 4, column 2 of Figure 7 is defined y the inequalities k A k q m, k D k 2 In this region of capacity space, firm s price est response function is of form (,2,4) Firm 2 s price est response function is of form (,3,4) From the discussion of cell (3,4), q m > kb(c) We also have Hence ( + θ)(2 θ) + θ k B(c) k A = a c ( 2 θ 2 θ a c ( + θ)(2 θ) 2 a c 2 + θ ) a c = k A k B(c) q m > 0 and the cases, k kb(c), k kb(c) can oth occur in cell (2,4) 35 =

36 a p 2 A 2 q = k, q 2 = 0 p 2 a θk q = k q 2 > 0 G (k ) F (k ) H 2 D 2 q = 0, q 2 = q m q = 0, q q m q 2 < k 2 A q < k p a Figure 93: (,) equilirium, firm (,2,4) price est response function, firm 2 (,3,4) price est response function 36

37 By the argument of Section 92, for k kb(c) second-stage equilirium is of type (,), as shown in Figure 93 For k kb(c), second-stage equilirium is of type (2,) as shown in Figure 94 In the same way, second-stage equilirium in cell (4,2) is of type (,) for k 2 kb(c), and second-stage equilirium is of type (,2) in cell (4,2) for k 2 kb(c) 94 Cell (,4) Cell (,4) row 4, column of Figure 7 is defined y the inequalities 0 k k A, k D k 2 In this region of capacity space, firm s price est response function is of form (2,4) Firm 2 s price est response function is of form (,3,4) The conditions for (2,) second-stage equilirium in cell (,4), as shown in Figure 95, are that point F e aove firm 2 s ranch one and that point D 2 e to the right of firm s ranch 2 The coordinates of point F are (p F, p 2F ) = (a k, a θk ), and the condition for point F to e aove firm 2 s ranch one is θ(p F c) + 2(p 2F c) ( θ)(a c) θ(a c k ) + 2(a c θk ) ( θ)(a c) θ(a c) + θk + 2(a c) 2θk ( θ)(a c) ( θ)(a c) θ(a c) + 2(a c) θk k a c θ (95) (a c)/ is the long-run equilirium output of a perfectly competitive industry; (95) will e met for all k of interest The coordinates of point D 2 are (p D, p 2D ) = ( ( θ)(2 + θ) c + 2 θ 2 (a c), c + θ2 2 θ 37 ) 2 (a c)

38 a p 2 A 2 q = k, q 2 = 0 p 2 a θk q = k q 2 > 0 F (k ) H 2 D 2 q = 0, q 2 = q m q = 0, q q m G (k ) q 2 < k 2 A q < k p a Figure 94: (2,) equilirium, firm (,2,4) price est response function, firm 2 (,3,4) price est response function 38

39 a p 2 A 2 F (k ) q = 0, q 2 = q m H 2 q 2 < k 2 D 2 q = 0, q q m q = k E (k ) p a Figure 95: Firm s price est response function, k k A 39

40 The condition for point D 2 to e to the right of firm s ranch two is θ(p 2D c) + p D c ( θ)(a c) ( θ 2 )k θ θ2 ( θ)(2 + θ) 2 (a c) + 2 θ 2 θ 2 (a c) ( θ)(a c) ( θ 2 )k [ ( θ 2 )k ( θ) + θ + θ 2 θ θ ] 2 θ 2 (a c) ( + θ)k 0 and this condition is always met In cell (,4), second-stage equilirium is of type (2,) In cell (4,), secondstage equilirium is of type (,2) 95 Cell (3,3) Cell (3,3) row 3, column 3 of Figure 7 is defined y the inequalities q m(c) k k D, q m(c) k 2 k D Figure 96 shows a second-stage equilirium with oth firms on ranch one of their (,2,3,4) est response functions, producing less than capacity This comination of price est response functions occurs in the (3,3) cell of Figure 7 The conditions for second-stage equilirium to have this configuration are that point G (k ) e aove firm 2 s ranch one and point G 2 (k 2 ) e to the right of firm s ranch one From Section 92, the conditions for this are k kb(c), k 2 kb(c) Since q m(c) k B(c) = a c 2 θ( θ) a c 2 ( + θ) (2 θ) a c ( + θ)(2 θ) 0, this condition is always satisfied, and in cell (3,3) the second-stage equilirium is always of type is (2,2) 96 Cell (2,3) Cell (2,3) row 3, column 2 of Figure 7 is defined y the inequalities k A k q m, q m(c) k 2 k D Firm s est response function is of form (,2,4) Firm 2 s est response function is of form (,2,3,4) = 40

41 a p 2 A 2 q = q m, q 2 = 0 p 2 a θk q F (k ) = k H q 2 > 0 q 2 = q m, q = 0 p a θk q < q m G (k ) q 2 = 0 q 2 = k 2 H 2 q > 0 q 2 < k 2 F 2 (k 2 ) G 2 (k 2 ) q 2 < q m q = 0 q < k A p a Figure 96: (,) equilirium, oth firms (,2,3,4) price est response functions 4

42 a p 2 A 2 q = k, q 2 = 0 p 2 a θk q = k q 2 > 0 q 2 < k 2 G 2 (k 2 ) F (k ) H 2 F 2 (k 2 ) q 2 < q m p = 0 q 2 = k 2 q > 0 q 2 = q m, q = 0 p a θk G (k ) A q < k p a Figure 97: (2,) second-stage equilirium, firm (,2,4) price est response function, firm 2 (,2,3,4) price est response function 42

43 Figure 97 shows a (2,) second-stage equilirium when firm has a (,2,4) price est response function and firm 2 has a (,2,3,4) price est response function The conditions for such an equilirium are first that point F (k ) e aove firm 2 s ranch one while point G (k ) is elow firm 2 s ranch one, and second that point G 2 (k 2 ) e to the right of firm s ranch two From Section 94, point F (k ) is aove firm 2 s ranch one for all k of interest From Section 92, the condition for point G (k ) to e elow firm 2 s ranch one is that k k B(c) The coordinates of point G 2 (k 2 ) are ( c + 2( ) θ2 )k 2 ( θ)(a c), c + ( θ 2 )k 2 θ The condition for point G 2 (k 2 ) to e to the right of firm s ranch two is p c θ(p 2 c) ( θ)(a c) ( θ 2 )k 2( θ 2 )k 2 ( θ)(a c) θ θ( θ 2 )k 2 ( θ)(a c) ( θ 2 )k 2( θ 2 )k 2 ( θ)(a c) θ 2 ( θ 2 )k 2 θ( θ)(a c) θ( θ 2 )k θ( θ 2 )k + ( θ 2 )(2 θ 2 )k 2 θ( θ)(a c) + ( θ)(a c) θ( θ 2 )k + ( θ 2 )(2 θ 2 )k 2 ( θ 2 )(a c) θk + (2 θ 2 )k 2 a c k 2 ( ) a c 2 θ 2 θk k r B(c)(k ), (96) where kb(c) r (k ) is the capacity level that is just suffi cient to allow firm 2 to produce its Bertrand est response output if firm 2 s marginal cost is c and firm s output is k If (96) is met, second-stage equilirium in cell (2,3) is of type (2,) If k 2 kb(c) r (k ), second-stage equilirium in cell (2,3) is of type (,) 43

44 a p 2 A 2 F (275) q 2 = k 2, q = 0 p a θk 2 q 2 = k 2 q > 0 F 2 (k 2 ) G2 (k 2 ) q 2 < k 2 q = k = 275 E (275) p a Figure 98: (2,2) second-stage equilirium, cell (,3) 44

45 97 Cell (,3) Cell (,3) row 3, column in Figure 7 is defined for q m(c) k 2 k D, 0 k k A Firm s est response function is of form (2,4) Firm 2 s est response function is of form (,2,3,4) The conditions for equilirium to occur at the intersection of the ranch two segments of the price est response functions in cell (,3) are that point F (k ) e aove firm 2 s ranch two, that point F 2 (k 2 ) e to the right of firm s ranch two, and that point G 2 (k 2 ) e to the left of firm s ranch two The coordinates of point F (k ) are (p F, p 2F ) = (a k, a θk ) The equation of firm 2 s ranch two is θ(p c) + p 2 c = ( θ)(a c) ( θ 2 )k 2 The condition for point F (k ) to e aove firm 2 s ranch two is θ(a c k ) + a c θk ( θ)(a c) ( θ 2 )k 2 θ(a c) + θk + a c θk ( θ)(a c) ( θ 2 )k 2 k 2 0, and this is satisfied for all k 2 In the same way, point F 2 (k 2 ) is always to the right of firm s ranch two From Section 96, the condition for point G 2 (k 2 ) to e to the left of firm s ranch two is k 2 k r B(c)(k ), and if this condition is met, the second-stage equilirium in cell (,3) is of type (2,2) See Figure 98 If instead k 2 kb(c) r (k ), second-stage equilirium in cell (,3) is of type (2,) In the same way, in cell (3,2), second-stage equilirium is of type (,2) if k kb(c) r (k 2) and of type (2,2) if k kb(c) r (k 2) 45

46 a p 2 q = k q 2 > 0 q 2 < k 2 A 2 G2 (k 2 ) c A q = k, q 2 = 0 p 2 a θk F (k ) F 2 (k 2 ) q 2 = k 2 q > 0 G (k ) q < k q 2 = k 2, q = 0 p a θk 2 c p a Figure 99: Second-stage (2, 2) equilirium for equilirium capacities; oth firms (, 2, 4) price est response functions 46

47 98 Cell (2,2) Cell (2,2) is defined y the inequalities k A k q m(c), k A k 2 q m(c) Figure 99, which is Figure 74 reproduced here for convenience, shows a second-stage (2,2) equilirium in cell (2,2) The condition for this to occur is that point F (k ) e aove firm 2 s ranch two, G (k ) elow firm 2 s ranch two, F 2 (k 2 ) to the right of firm s ranch two, and G 2 (k 2 ) to the left of firm s ranch two From Section 97, F (k ) and F 2 (k 2 ) always have the required positions, while G (k ) and G 2 (k 2 ) have the required positions if If (97) is not met, k k B(c)(k 2 ), k 2 k B(c)(k ) (97) k k B(c)(k 2 ), k 2 k B(c)(k ), (98) cell (2,2) second-stage equilirium is of type (,) If point F (k ) is aove firm 2 s ranch two, point G (k ) elow firm 2 s ranch two, and point G 2 (k 2 ) to the right of firm s ranch two, then second-stage equilirium is of type (2,) From Section 96, the condition for point G 2 (k 2 ) to e to the right of firm s ranch two is k 2 k r B(c)(k ) Thus the conditions for (2,) second-stage equilirium in cell (2,2) are k k B(c)(k 2 ), k 2 k r B(c)(k ) (99) In the same way, the conditions for (,2) second-stage equilirium in cell (2,2) are k kb(c)(k r 2 ), k 2 kb(c)(k ) (90) 99 Cell (,2) Cell (,2) row 2, column in Figure 7 is defined y the inequalities 0 k k A, k A k 2 q m(c) Firm s price est response function is of form (2,4) Firm 2 s est response function is of form (,2,4) Figure 90 shows a (2,2) second-stage equilirium in cell (,2) The conditions for the type of equilirium to occur are that point F (k ) e aove firm 2 s 47

48 a p 2 A 2 F (k ) q 2 = k 2, q = 0 p a θk 2 q 2 = k 2 q > 0 F 2 (k 2 ) G2 (k 2 ) q 2 < k 2 q = k E (k ) p a Figure 90: cell (,2): firm (2,4), firm 2 (,2,4) 48

49 ranch two, point F 2 (k 2 ) e to the right of firm s ranch (which conditions are always satisfied), and that point E (k ) e elow firm 2 s ranch two, while point G 2 (k 2 ) e to the right of firm s ranch two Point E (k ), which is on the horizontal axis, is always elow firm 2 s ranch two From Section 96, the condition for point G 2 (k 2 ) to e to the right of firm s ranch two is k 2 kb(c)(k r ) (9) If on the other hand k 2 kb(c)(k r ), (92) second stage equilirium in cell (,2) is of type (2,2) In the same way, if k kb(c)(k r 2 ), (93) then second-stage equilirium in cell (2,) is of type (2,2), while if k k r B(c)(k 2 ), (94) then second-stage equilirium in cell (2,) is of type (,2) 90 Cell (,) Cell (,) is defined y the inequalities 0 k k A, 0 k 2 k A Figure 9 shows a second-stage (2,2) equilirium in cell (,) The conditions for cell (,2) second-stage equilirium to have this form are that point F (k ) e aove firm 2 s ranch two while point F 2 (k 2 ) is to the right of firm s ranch two These conditions are always satisfied 0 Proof of Lemma 3 0 (, ) In (, ) equilirium, equilirium prices are the Bertrand equilirium prices with marginal cost equal to c: (see (43)); equilirium quantities are p = c + θ (a c); (0) 2 θ q = ( + θ)(2 θ) 49 a c ; (02)

50 a p 2 c c q 2 = k 2 = 35 F (35) F 2 (35) E 2 (35) q = k = 35 E (35) p a Figure 9: (2, 2) equilirium, oth firms (2, 4) price est response functions 50

51 equilirium payoffs are 02 (2, 2) π i = θ (a c) 2 ρk ( + θ)(2 θ) 2 i (03) If oth firms are on their quantity constraint lines, equilirium prices are ( ) ( ) ( ) ( ) θ p = ( θ) a ( θ 2 k ) θ p 2 k 2 (04) with solution p 22 = a (k + θk 2 ) (05) p 222 = a (θk + k 2 ), (06) which simply recovers the equations of the inverse demand curves (2), (22) Equilirium outputs are k and k 2 Second-stage payoff functions are 03 (2, ) If point C (k ) is elow the q 2 < k 2 line, π 22 = [a c ρ (k + θk 2 )] k (07) π 222 = [a c ρ (θk + k 2 )] k 2 (08) and point C 2 (k 2 ) is to the right of the q = k line, k k B(c), (09) k 2 k r B(c)(k ), (00) then equilirium occurs where firm s ranch two intersects firm 2 s ranch one This case is symmetric with the (, 2) equilirium Firm is capacity constrained; the equation of firm s price est response function is (56) p = θp 2 + ( θ)a ( θ 2 )k, or, rewritten in terms of deviations from c, p c θ(p 2 c) = ( θ)(a c) ( θ 2 )k 5

52 Firm 2 is not capacity constrained; the equation of firm 2 s price est response function is (9) θ(p c) + 2(p 2 c) = ( θ)(a c) Equilirium prices solve ( ) ( θ p c θ 2 p 2 c ) ( = ( θ)(a c) ) ( ( θ 2 )k 0 ), leading to p 2 = c + ( θ)(2 + θ)(a c) 2( θ2 )k 2 θ 2 (0) p 22 = c + θ2 2 θ 2 (a c θk ) (02) Equilirium quantities demanded are q 2 = k q 22 = a c θk 2 θ 2 (03) Second-stage payoffs are [ ( θ)(2 + θ)(a c) 2( θ 2 ] )k π 2 = 2 θ 2 ρ k (04) 04 (, 2) π 22 = θ2 (a c θk ) 2 (2 θ 2 ρk ) 2 2 (05) The equation of the ranch one segment of firm s price est response function is 2(p c) θ(p 2 c) = ( θ)(a c) (06) The equation of the ranch two segment of firm 2 s est response function is p 2 = θp + ( θ)a ( θ 2 )k 2 (07) Rewrite (07) in terms of deviations from c: θ(p c) + p 2 c = ( θ)(a c) ( θ 2 )k 2 52

53 The system of equations is ( ) ( ) ( 2 θ p c = ( θ)(a c) θ p 2 c ) ( 0 ( θ 2 )k 2 ) (08) Prices in (, 2) equilirium are p 2 = c + θ2 2 θ 2 (a c θk 2) (09) p 22 = c + ( θ)(2 + θ)(a c) 2( θ2 )k 2 2 θ 2 (020) Firm is on ranch one of its est response function; from (83), the quantity demanded of firm is q 2 = p c ( θ 2 ) = θ 2 ( θ 2 ) 2 θ 2 (a c θk 2) = a c θk 2 2 θ 2 (02) Firm s second-stage payoff is π 2 = Firm 2 s second-stage payoff is Proof of Lemma 4 θ2 (a c θk 2 ) 2 (2 θ 2 ρk ) 2 (022) π 22 = (p 22 c ρ)k 2 = [ ( θ)(2 + θ)(a c) 2( θ 2 ] )k 2 2 θ 2 ρ k 2 (023) By the way in which the first-stage payoff functions are derived, they must e continuous in capacities I have verified this, ut omit the details of these parts of the proof 53

54 Lower region: k 2 k B(c) Let k 2 kb(c) Firm s payoff function is π (k, k 2 ) = { [a c ρ (k + θk 2 )] k 0 k kb(c) r (k 2) (2, 2) θ 2 (a c θk 2 ) 2 ρk (2 θ 2 ) 2 kb(c) r (k 2) k (, 2) () If k kb(c) r (k 2), second-stage equilirium occurs where oth firms are on ranch two, producing at capacity By (05), firm s equilirium price is given y the equation of its inverse demand curve, writing capacities in place of quantities demanded: Its payoff is p = a (k + θk 2 ) π (k, k 2 ) = (p c ρ)k = (a c ρ (k + θk 2 ))k (2) If k k r B(c) (k 2), then in second-stage equilirium, firm is on its ranch one (q k ), while firm 2 is on its ranch two (q 2 = k 2 ) From (022), firm s equilirium payoff is π (k, k 2 ) = θ2 (a c θk 2 ) 2 (2 θ 2 ρk ) 2 (3) For the numerical example, firm s payoff function for k 2 = 4 is shown in Figure Comparing (2) and (46), for k k r B(c) (k 2), firm s payoff function has the same functional form as firm s payoff function in the standard Cournot duopoly model with product differentiation and marginal cost c+ρ, (46), and the capacity that maximizes firm s payoff in the left-hand region k kb(c) r (k 2), which with a certain ause of notation we denote as k r Cour(k 2 ) = 2 ( a c ρ θk 2 ), (4) has the same functional form as the Cournot est response output with marginal cost c + ρ, (47) 54

55 π k = 2 θ 2 ( a c θk 2 ) k Figure : Firm s profit function, k 2 = 4 55

56 In order for (4) to give the value that maximizes firm s equilirium profit in the (2, 2) region, the capacity level identified y (4) must e in the (2, 2) region The condition for this is k (k 2 ) = ( ) a c ρ θk 2 k r 2 B(c)(k 2 ) (5) k 2 θ2 (a c) + (2 θ 2 )ρ θ 3 k LLint (6) We know that in the region now under analysis Comparing k LLint and k B(c), k 2 k B(c) = 2 θ 2 [ θ ( + θ)(2 θ) k LLint k B(c) = ( + θ) (2 θ) Hence in the case we are now considering a c k 2 k B(c) k LLint, a c + ] ρ θ 2 > 0 and the gloal maximum of (2) occurs within the (2, 2) region To the right of the line k = k r B(c) (k 2) firm s payoff, (3), θ 2 (a c θk 2 ) 2 (2 θ 2 ρk ) 2, is its payoff for (,2) equilirium This is maximized y making k as small as possile within the relevant region, that is, y setting k = kb(c) Since the payoff function is continuous, and rising moving to the left from k = kb(c) r (k 2), the gloal maximum of the payoff function for the left-hand segment occurs within the left-hand segment, and firm maximizes its payoff for k 2 kb(c) y setting the capacity (4), kcour r = ( ) a c ρ θk 2, 2 which is the equation of the segment of firm s capacity est response function 0 k 2 k B(c) 56

57 2 Upper region: 2 θ 2 a c k 2 π (k, k 2 ) = [ ] ( θ)(2+θ) (a c) ρ 2 θ2 k 2 θ 2 2 θ 2 k 0 k kb(c) (2, ) θ (a c) 2 ρk (+θ)(2 θ) 2 kb(c) k (, ) (7) Firm s payoff function in the left-hand region is its payoff in (2, ) equilirium, [ ( θ)(2 + θ) π 2 = 2 θ 2 (a c) ρ 2 ] θ2 2 θ 2 k k (8) The gloal maximum of (8) is at k r 2 = ( θ)(2 + θ)(a c) (2 θ2 )ρ 4( θ 2 (9) ) This is interior to the left-hand region, for which k kb(c) = a c : (+θ)(2 θ) k r 2 k B(c) k r 2 = [ θ 2 a c + 2 ] θ2 ρ > 0 (0) 4( + θ) 2 θ + θ may e negative, if ρ is suffi ciently large We will assume that kr 2 > 0 This assumption will e used several times elow to determine the signs of various expressions Firm s payoff at (9) is π T L = 2 θ2 2 θ 2 ( k r 2) 2 = 2 θ 2 [ ( θ)(2 + θ)(a c) (2 θ 2 ] 2 )ρ 8 θ 2 (2 θ 2 () ) Firm s payoff in the right-hand region, (03), is its payoff in (, ) equilirium, θ (a c) 2 ρk ( + θ)(2 θ) 2, 57

58 π k = k B(c) k Figure 2: Firm s profit function, k 2 2 θ 2 a c 58

59 and this is maximized y making k as small as possile within the relevant range, that is, y setting k = kb(c) By continuity of the payoff function and the fact that the payoff function rises moving left from the oundary k = kb(c), the gloal maximum in the top region occurs at (9) Figure 2 shows firm s payoff function for k 2 a c 2 θ 2 Figure 3 shows the lower (0 k 2 kb(c) a c ) and upper ( k 2 θ 2 2 a c ρ ) segments of the capacity est response functions k 2 (2, ) Firm s capacity Firm 2 s capacity est response function est response function k B(c) (2, 2) k B(c) (, ) (, 2) k Figure 3: Capacity est response functions, upper and lower segments, Kreps & Scheinkman model with product differentiation 59

60 3 Middle region: kb(c) k 2 a c 2 θ 2 The equation of the oundary etween the (2, 2) region and the (2, ) region is θk + (2 θ 2 )k 2 = a c (2) or k 2 = kb(c)(k r ) or k = [ k r B(c)] (k2 ) If k = 0 along this line, then k 2 = payoff function has three segments: 2 θ 2 a c For k B(c) k 2 a c 2 θ 2, firm s π (k, k 2 ) = (3) [ (a c ρ (k + θk 2 ))k 0 k kb(c)] r (k2 ) (2, 2) ( ) ] ( θ)(2+θ) (a c) ρ 2 θ2 k 2 θ 2 2 θ 2 k [kb(c) r (k2 ) k kb(c) (2, ) θ (a c) 2 ρk (+θ)(2 θ) 2 kb(c) k (, ) 3 Left segment: 0 k [ k r B(c)] (k2 ) In this region, firm s payoff is that of (2, 2) equilirium; this case has een analyzed in Section Firm s profit-maximizing capacity is (4), k r Cour = 2 ( a c ρ θk 2 ), if kcour r lies with the left-hand range of the middle region 0 k [ a c θ The condition for kcour r to e within the left-hand range of the middle region is or 2 ( ) a c ρ θk 2 [ a c θ k 2 (2 θ)(a c) + θρ (4 3θ 2 ) 60 (2 θ 2 )k 2 ], (2 θ 2 )k 2 ] k MLint (4)

61 In the middle region kb(c) k 2 a c 2 θ 2 Compare k MLint with the upper end of the range of k 2 that defines the middle region: [ a c θ ( θ)(2 + θ)(a c) (2 θ 2 ] )ρ 2 θ 2 k MLint = 4 3θ 2 2 θ 2 > 0, where the sign depends on the assumption that (4), firm one s est-response capacity in the upper region, is positive Now compare k MLint with the lower end of the range of k 2 that defines the middle region: k MLint kb(c) = [ θ 4 3θ 2 θ 2 ( + θ) (2 θ) a c + ρ ] > 0 Hence kb(c) < k MLint < a c 2 θ 2 For firm 2 capacity levels falling in the range k B(c) k 2 k MLint, (5) kcour r is interior to the left-hand segment of the middle region, firm s payoff function has an interior maximum at kcour r on the left-hand segment, and firm s payoff at this maximum is In contrast, for ( ) 2 a c ρ θk 2 4 k MLint k 2 a c 2 θ 2, the local maximum of firm s payofffunction on the [ range 0 k θ is at the right-hand order of this range, k = a c ] (2 θ 2 )k θ 2 [ a c (2 θ 2 )k 2 ] 6

62 32 Middle segment: [ ] kb(c) r (k2 ) k kb(c) Firm s payoff function in the middle segment of the middle region is its payoff in (2, ) equilirium, π 2 = [ ( θ)(2 + θ) 2 θ 2 (a c) ρ 2 ] θ2 2 θ 2 k k The gloal maximum occurs for (9) k r 2 = ( θ)(2 + θ)(a c) (2 θ2 )ρ 4( θ 2 ) We know from our discussion of the upper region (see (0) and the associated text) that k2 r < kb(c) The condition for [ ] a c (2 θ 2 )k 2 k r θ 2, so that k2 r lies within the middle range of the region, is k 2 k MMint can also e written 4 + 2θ θ2 a c θ ρ 4( + θ)(2 θ 2 + ) 4( θ 2 ) k MMint (6) k MMint = 2 θ 2 ( a c We know that kb(c) k 2 a c 2 θ 2 Compare k MMint and kb(c) : k MMint k B(c) = [ θ 4( + θ) ) θk2 r (7) θ 2 (2 θ) ( 2 θ 2) a c + θ ] ρ > 0 62

63 Compare k MMint and 2 θ 2 a c : a c 2 θ 2 k MMint = θ ( θ)(2 + θ)(a c) (2 θ 2 )ρ 4( + θ) ( + θ)(2 θ 2 ) > 0 where once again the sign depends on the assumption that (4), firm one s est-response capacity in the upper region, is positive Hence kb(c) k MMint a c 2 θ 2 For kb(c) k 2 k MMint, the maximum of firm s profit function on the range [ ] a c (2 θ 2 )k 2 k kb(c) θ occurs at the left-hand oundary, k = [ a c θ (2 θ 2 )k 2 ], while for k MMint k 2 a c 2 θ 2, firm s profit function on the range [ ] a c (2 θ 2 )k 2 k kb(c) θ has an interior maximum at k r 2 : 2 = ( θ)(2 + θ)(a c) (2 θ2 )ρ 4( θ 2, ) k r and firm s payoff at this capacity level is 2 θ2 2 θ 2 ( k r 2) 2 63