Homework #5 - Econ 57 (Due on /30) Answer Key. Consider a Cournot duopoly with linear inverse demand curve p(q) = a q, where q denotes aggregate output. Both firms have a common constant marginal cost c > 0, and where a > c. Assume that firms do an equity swap of γ, i.e., each firm i receives a share 0 < γ in firm j s profits, where j i. (a) Find the Cournot equilbrium output, (q C, q C ). Firm i s profit-maximization problem (PMP) is given by max q i ( γ)(a q i q j c)q }{{} i + γ(a q i q j c)q }{{} j Firm i s profit Firm j s profit Taking first-order conditions with respect to q i yields ( γ)(a q i q j c) ( γ)q i γq j = 0 In a symmetric equilibrium q C i = q C j = q C, which lets us simplify the above expression as follows ( γ)(a q C c) ( γ)q C γq C = 0 which, solving for q C yields q C = ( γ)(a c) 3 γ (b) Evaluate equilibrium output q C i at γ = 0 and γ =. Interpret. When firms do not benefit from each other s profits, γ = 0, equilibrium output q C becomes a c, thus coinciding with that under the standard Cournot model 3 with linear inverse demand curve p(q) = a q. In contrast, when firms fully share their profits, γ = /, equilibrium output q C becomes a c 4, thus coinciding with half of monopoly output (or cartel output). (c) Determine if q C i increases or decreases in γ. Differentiating q C with respect to share γ yields q C γ = a c 3 γ + ( γ)(a c) (3 γ)
which simplifies to a c (3 γ) which is clearly negative since a > c by definition. Intuitively, as firms share more of each other s profits, their individual PMP resembles the joint PMP in a cartel, leading each of them to reduce its production. For illustrative purpose, figure depicts q C as a function of the profit share γ. For simplicity, we consider a = and c = 0 which yields a output q C = γ 3 γ. (d) Find equilibrium profits, π C, and determine whether they increase or decrease in γ. Equilibrium profits are π C = ( γ)(a c) (3 γ) which increase in γ since π C γ = ( γ)(a c) (3 γ) 3 is positive given that γ by definition. Intuitively, as firms take more into account each other s profits, their individual profits approach those they would obtain under a cartel agreement, which are larger than under a standard Cournot model.. Consider a homogeneous good industry with two potential firms. Market demand is given by Q = ( p), where is the market size and Q is the industry output. Firms have zero constant marginal costs but incur a fixed cost k ( 0, 9 ) if they are active. The time structure of the game is as follows: First, they decide wether or not to enter. Then, they compete in the product market. For the following three different forms of competition, find equilibrium quantities, prices, profits, consumer surplus and welfare: (a) Firms independently and simultaneously choose quantities (Cournot competition).
To analyze the case of quantity competition, we first find the inverse demand function (solving for p), which is p = Q. A firm i chooses q i to maximize π i = The first-oder conditions are [ q ] i + q j q i k π i q i = q i q j = 0 At the symmetric equilibrium q i = q j = q the solution will be given by q C = 3. Equilibrium prices and individual profits will be p C = 3 and πc = 9 k > 0. It is easy to find consumer surplus and total welfare as C C = ( p)( ( ( p) = ) 3) 3 = and W C = C C + π C = 4 k. 9 9 (b) Firms non-cooperatively choose prices (Bertrand competition); Given that products are homogeneous, competition would bring about zero (short-run) profits, which would not allow both firms to cover fixed costs. Therefore, at the (long-run) equilibrium in pure strategies only one firm will be active in this market. This firm will choose Q so as to maximize monopoly profits π i = [ Q ] Q k Taking first-oder conditions with respect to Q, we find that equilibrium output, price and profits will be Q B =, pb =, and πb = [ ] k = k > 0. Consumer surplus and total welfare are 4 CB = ( p)( p) = ( ) ( ) = and W B = C B + π B = 3 k. (c) Firms set quantities (or prices, it is equivalent) so as to jointly maximize their profits (cartel). Firms will set outputs q and q so as to maximize the joint profits π + π = [ q ] + q (q + q ) k Clearly, this gives rise to the same solution as in the Bertrand case (where a single firm sets aggregate output Q as a monopolist) but with duplication in the fixed costs. Consumer surplus and total welfare are C M = ( p)( p) = ( ) ( ) = 3
and W M = C M + π M = 3 k. (d) Compare the social welfare resulting from the three forms of competition analyzed in parts (a)-(c). Comparing W C and W B, we find that W C W B if and only if 4 k 9 3 5 k, or if k satisfies k, as depicted in the upper region of figure 3. 7 (Note that, by definition, k.) 9 In addition, note that the cartel (or monopoly) yields the lowest social welfare since W M < W C and W M < W B. Indeed W M < W C since 3 k < 4 9 k for all k > 5 7 (which holds by definition since k > 0), while W M < W B given that 3 k < 3 k for all values of k. 3. Consider an industry with n identical firms competing a la Cournot. uppose that the inverse demenad function is P (Q) = a bq, where Q is total industry output, and a, b > 0. Each firm has a marginal costs, c, where c < a, and no fixed costs. (a) (no merger) Find the equilibrium output that each firm produces at the symmetric Cournot equilibrium, What is the aggregate output and the equilibrium price? What are the profits that every firm obtains in the Cournot equilibrium? What is the equilibrium social welfare? at the symmetric equilibrium with n firms, we have that each firm i maximizes max (a bq i bq i ) q i cqi () where Q i q j denotes the aggregate production of all other j i firms. taking F.O.C. with respect to q i, we obtain a bq i bq i c = 0 () An at the symmetric equilibrium Q i = (n )q i. Hence, the above FOCs become a b(n )qi q i bqi q i c = 0 or a c = b(n + )qi. olving for q i, we find that the individual output level in equilibrium is q i = a c (n + )b (3) Hence, aggregate output level in equilibrium is Q = nqi q i = n n+ 4 a c; while the b
equilibrium price is p = a+cn and equilibrium profits are n+ π i = (a c). b(n+) Q The level of social welfare with n firms, Wn, is defined by W n = [a bq] dq cq. Calculating the integral, i.e., yields Q [a bq] dq cq = aq b Q, and substituting for Q 0 W n = n(n + )(a c) n(n + ) (4) 0 (b) (Merger) Now let m out of n firms merge. how that the merger is profitable if and only if it involves a sufficiently large number of firms. Assume that m out of n firms merge. While before the merger there are n firms in this industry,, after there are n m +. In order to examine whether the merger is profitable for the m mergerd firms, we need to show that the profit after the merger, π n m+, satisfies π n m+ mπ n.that is (a c) m(a c)(n + ) (5) (n m + ) solving for n, we obtain that n < m m, note that this condition is compatible with the fact that the merger must involve a subset of all firms, i.e., m n. alternative interpretation of the condition can be found by first solving for m, which yields m > n + 3 5+4n m; and, second, finding the market share that this minimal number of firms represents, i.e., α = m n = 3+n 5+4n n (c) Are the profits of the nonmerged firms larger when m of their competitors merge than when they do not? The number of firms producing the Cournot output decreases (since the merged firms produce a smaller output), implying that each of the nonmerged firms earns larger profits after the merger. This condition holds for mergers of any size, i.e., both when condition n < m m and otherwise. This surprising result is often referred as the merger paradox as it is the nonmerged firms the ones seeing their profits increase for all paremeters values. 4. Consider a tackelberg duopoly model with the following timing: () firm chooses a quantity q 0; () firm observes q and then chooses a quantity q 0; (3) the payoff to firm i is given by the profit function π i (q i, q j ) = q i [P (Q) c], An 5
where P (Q) = a Q is the market-clearing price when the aggregate quantity on the market is Q = q + q. (a) Use backward induction to find the production levels as a function of the marginal cost to the firms of c and of parameter a. Using backwards-induction, firm s profit maximization is max q [a q q c] (6) with FOC a q q c = 0 (7) solving for q, we obtain firm s best response function q (q ) = a c q () Moving to the first period, firm s maximization problem is max q [a c q q(q )] c] (9) with FOC a c q = 0 or q = a c and substituting into firm s best response function q = a c 4. (b) olve for each firm s profit levels π and π, in the backwards-induction outcome, when Firm moves first. Give the sign for and interpret π i c i = {, }. for every firm The equilibrium price is p = a+3c 4 and profits are π = (a c) and π = (a c) 6. Differentiating both firms profits with respect to c, π c = a c < 0 4 (0) π c = a c < 0 () whivh indicates that as costs rise, the profits for both firms fall, with the profit for firm falling at twice the rate as the profit for firm. (c) Consider now the following variation in the tackelberg game: Firm is given the following choice before either firm chooses production levels. For some set value of K > 0, it is told that if it pays K, it will get to be the first-mover in the tackelberg game. If it does not pay K, it will be the tackelberg follower. 6
. As a function of c and a, solve for the highest K, K, that Firm would pay for the first-move advantage. Firm would be willing to pay any K, such that its profits as the leader are weekly higher than its profits as the follower, i.e., (a c) K (a c) 6 () olving for K, firm would be willing to pay any K such that K (a c). 6. Comparative statics. olve for and give an interpretation of K K and. c a olving for the two derivatives K c = a c < 0 (3) K a = a c > 0 (4) The first derivative implies that as cots go up, firm would be willing to pay less in order to be tackelberg leader, which is consistent with our results from the last part as the profits for the leader fall faster than the profits for the follower. The second derivative implies that as the size of the market increases, firm is willing to pay more to be the tackelberg leader as its profit level will increase more as the stackelberg leader than as the follower. 7