Answer Key: Problem Set 3

Transcription

1 Answer Key: Problem Set Econ Fall Question 1 a This is a standard monopoly problem; using MR = a 4Q, let MR = MC and solve: Q M = a c 4, P M = a + c, πm = (a c) 8 The Lerner index is then L M P M c P M = a c a + c b Given the rival s quantity q i, firm i s profit is P (q i + q i )q i cq i, or [a (q i + q i ) c]q i In a Cournot duopoly NE, firm i chooses q i to imize the profit, taking q i as given The FOC wrt q i is a 4q i q i c = 0 Solving for q i gives the best response quantity q i = a c q i 4 Since we have two unknowns (q 1 and q ) and two equations (FOC of each firm s profit imization), we can solve for q 1 and q This results in the equilibrium quantity q C i = q i C = (a c)/6 If you want to simplify this procedure, there is a shortcut: to use symmetry Imposing q i = q i in the FOC and letting q denote their value, we have a 4q q c = 0, or q = a c 6, 1

2 and thus q C i = q i C = (a c)/6 again Note that the symmetry is imposed after deriving the FOC, not before it Now the market quantity, the price, the market profit, and the Lerner index are Q C = q1 C + q C = a c, P C = a + c, π C = (a c), L C = a c 9 a + c, respectively Comparing parts a and b, we see Q C > Q M, P C < P M, π C < π M, L C < L M c We already know the results for the Bertrand duopoly; the industry quantity, the price, the industry price, and the Lerner index are Q B = a c, P B = c, π B = 0, L B = 0, respectively d Firms prefer Cournot competition because they can get positive profits π C / > 0, whereas the profits under price competition are zero Consumers prefer Bertrand competition since the price is lower and hence they can get more surplus Bertrand is more efficient because it yields larger quantity more specifically, greater total surplus e In the n-firm Cournot game, the profit function of firm i is P (q i + Q i )q i cq i where Q i = j i q j is the sum of the quantities produced by the others Hence the best response is characterized by the following problem: q i (a q i Q i c)q i The FOC wrt q i is a 4q i Q i c = 0 Because the game is symmetric, we have q 1 = q = = q n q and accordingly Q i = (n 1)q Hence the

3 last display becomes a 4q (n 1)q c = 0, from which we obtain the quantity of each firm, the industry quantity, and the price qi C = a c (), QC = n a c, P C = a + nc The firm profits and the industry profit are π C i = 1 ( ) a c, π C = n ( ) a c Check that the results are consistent with part b when we put n = Also the Cournot game approaches the perfect competition as n : Q C a c = Q e, p C c = P e, π C 0 as n, where Q e and P e denote the efficient quantity and price respectively Question a As in Question 1, the best response of firm i is characterized by q i (a bq i bq i c i )q i, whose FOC wrt q i is a c i q i bq i = 0 The best response, therefore, is q i = a c i q i

4 Since there are two firms, this equation must hold for i = 1, So we have the following system: q 1 = a c 1 q = a c q q 1 q a c 1 b q 1 = q1(q ) a c q q = q (q 1 ) O q 1 a c 1 a c b q 1 obtain Figure 1: Best responses and the Nash Equilibrium We no longer can impose q 1 = q since the game is not symmetric Plug one equation into the other to q 1 = a c 1 1 = a c 1 + c 4b or q 1 = a c 1 + c ( a c + q 1 4, q ) 1 Similarly q = a c + c 1 These two quantities characterize the pure strategy NE To ensure duopoly, ie, q 1, q > 0, assume a c 1 + 4

5 c > 0 and a c + c 1 > 0 (Actually the latter is implied by the former, and hence redundant) Since c 1 > c, it follows q 1 < q The industry quantity and price are Q = q 1 + q = a c 1 + c ( ) a c1 c P = a b + a c + c 1 = a + c 1 + c = a c 1 c b The firm profits are π 1 = q 1 (P c 1 ) = a c 1 + c = (a c 1 + c ) 9b π = (a c + c 1 ) 9b ( ) a + c1 + c c 1 > 0, > 0 under the assumption as above Since firm sells more quantity (q > q 1 ) at higher margin (P c > P c 1 ), firm makes larger profit Question a The firm solves The FOC wrt q is 11 q 1 = 0, which gives [(11 q)q ( + q) ] q }{{}}{{} revenue cost q m = 5, p m = 6 π m = (11 5) 5 ( + 5) = CS m = = 15 b Firm i solves q i [(11 q i Q i )q i ( + q i )] The FOC wrt q i is 11 q i Q i 1 = 0 Note that the fixed cost (= in this case) has no implication in 5

6 the FOC; fixed cost only affects entry decision, not the quantity conditional on entry Symmetry implies Q i = (n 1)q i, which reduces the FOC to 10 ()q i = 0 (Note again that if you make this substitution before deriving the FOC, you will not get the right solution; remember that firm i does not get to choose the quantities of its competitors!) Therefore q C i (n) = 10 Q C (n) = nq C i (n) = 10n P C (n) = 11 Q C (n) = 11 + n π C (n) = nπ i (n) = (P C (n) 1)Q C (n) }{{} CS C (n) = 1 ( ) 10n variable profit n }{{} fixed cost = 100n () n To see how these quantities change with the number of firms, take the derivatives wrt n: dqi C dn = 10 () < 0 dq C dn = 10 () > 0 dπ C dn d(cs C ) dn = 100(1 n) () < 0 = 100n () > 0 Therefore the firm quantities and the total profit are decreasing in n, whereas the total quantity and the consumer surplus are increasing in n c Suppose now the number of firms is determined endogenously; ie, we are going to determine n Under free-entry, the firms enter the market until the profit is zero; π i = 0, or 100 () = 0 using the result in part b This gives n = At this value of n, the industry profit is 0 and the consumer surplus is CS 686 (plug n = 607 into the formula in part b) The total surplus, therefore, is 686 6

7 d To find the efficient number of firms, we need to solve (CS + π), n ie, n [ ( ) ] 1 10n + 100n () n The FOC wrt n is 100n 100(1 n) + () () = 0, from which we obtain n = 50 1/ 1 68 Plugging this into the formula we derived in part b, we get CS + π 4095 e The inefficiency of free entry is due to business-stealing effect; part of an additional entrant s profit comes at the expense of existing firms When firms make entry decisions, they do not take into consideration the decrease in their opponent s quantities In other words, the private benefit for an additional firm exceeds the social benefit f When the fixed cost is eliminated, the socially optimal market quantity would be the one at which P = MC holds Therefore the socially optimal market quantity is Q e = 10 Under no fixed cost, we can see (with a slight modification of part b) that the profit of each firm is 100/() > 0, for each n Since the profit is always positive, firms enter indefinitely under free entry; the number of firms tends to infinity But would it result in the optimal quantity, rather than letting the firms merely steal the sales of each other? The answer is yes We have seen that, in this Cournot game, having more entrants drives the price down to the marginal cost; indeed, we can derive (as in part b) the market quantity (under zero fixed cost) as 10n/(), which tends to Q e = 10 as n 1 Since the quantity approaches Q e, the price is driven down to the marginal cost, as in the efficient case g Since the total surplus achieved its imum at n 68, we may compare the total surplus values at n = and and then pick the one that yields the greater total surplus In part d the total surplus function 1 You might have guessed or observed that the market quantity (conditional on the number of firms n) under zero fixed cost is the same as in part b where the fixed cost was positive This is correct; remember that, given the number of entrants, fixed costs do not affect the quantity Firms take fixed costs into account only when deciding whether to enter Technically, this method works because the function (without the integer constraint) is smooth and has a unique peak, 7

8 was T S(n) 1 ( ) 10n + 100n () n Plugging in n = and, we obtain T S() < = T S() Therefore the efficient number of firms, under the integer constraint, is T S(n) O n Figure : Total surplus as a function of the number of firms Question 4 a First consider the demand of product A In order to derive the residual demand function, let q A = a b 1 p A + b p B According to the information, for every dollar increase in the price of Product A, for any given price of product B, product A loses twenty units of sales to products outside the candidate market and ten units of sales to product B This means b 1 = = 0 Since the same is true for product B (ie, for every dollar increase in p B, for any given price of product A, product B loses ten units of sales to product A), which is the case here technical Although this condition can be checked analytically, we omit the proof since it is either tedious or 8

9 b = 10 Hence we have q A = a 0p A + 10p B In the current situation we have p A = p B = 100 and q A = 100 Plugging these into the last display we find that a = 00 Similarly q B = 00 0p B + 10p A b Since we are given that mc = 60, we have that the imization problem for firm i = A, B (when firms choose prices) is p i (00 0p i + 10p i )(p i 60), from which we get the FOC wrt p i : 00 60p i + 10p i = 0 Imposing symmetry p i = p i p, we get p = 100 Therefore p A = p B = 100, which is consistent with each sells for $100 c Suppose now that a monopolist sells both products, choosing the prices p A and p B The imization problem is thus {(00 0p A + 10p B )(p A 60) + (00 0p B + 10p A )(p B 60)}, p A,p B and the relevant FOCs are 00 60p A + 10p B p B 600 = 0, 00 60p B + 10p A p A 600 = 0 Imposing symmetry p A = p B p, we get p = 110 Therefore p A = p B = 110, which is consistent with [a] monopolist controlling products A and B would raise both of their prices by ten percent, to $110 As this example shows, price competition does not necessarily mean Bertrand game Remember that the Bertrand game assumes homogeneity of the products; any slight price differentiation causes complete substitution between the firms In this example, however, a decrease in p A does not steal all the sales from product B 9