Department of Economics The Ohio State University Econ 805 Homework #3 Answers

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1 Prof James Peck Winter 004 Department of Economics The Ohio State University Econ 805 Homework #3 Answers 1. Varian, Chapter 13, prolem Answer: (a) The individual farmer s supply curve is found y nding the quantity at which price equals marginal cost. For farmer j, we have p = y j. Expressing this as quantity supplied, as a function of price, we have y j = p : () Since there are 100 identical farmers, the market supply curve is given y Y (p) = 50p. (c) Equating supply and demand, we have 00 50p = 50p; so the equilirium price is, p =. Plugging a price of into either the demand or the supply equation yields a quantity of 100. (d) Each farmer produces y j = 1 unit of output, and receives a pro t of p y j c(y j ), which equals 1. Assuming that this pro t is attriutale to the scarce land, and not a limited numer of competent farmers, then the long run equilirium rental price for a parcel of land is 1.. Varian, Chapter 14, prolem Answer: (a) The monopolist s pro ts can e written as = p(y; t)y cy: Di erentiating with respect to y, for xed t, we have the rst order t) p(y; t) + c = 0: (1) Since equation (1) must hold for each t, the derivative of the left side with respect to t must equal 0. 0 t) t) + + t) p(y; t) dt + p(y; t) 1

2 Solving, we have dy : () () If we have p(y; t) = a(y) + (t), then the expression simpli es. t) p(y; = 0 (t); = = a 0 p(y; = a 00 (y); Expression () ecomes dy dt = 0 (t) a 0 (y) + ya 00 (y) : 3. A monopolist manufacturer produces a product at constant marginal cost, c, and chooses a wholesale price (a price at which retailers can order the product from the manufacturer), p w. Given the wholesale price, p w, each of n retailers can order as much of the product they want, where the quantity ordered y retailer i is given y q i and the total quantity is given y Q. All output ordered y retailers is sold to consumers, with the retail price, p r, determined y the supply, Q, and the demand curve, D(p r ) = a p r. In other words, the manufacturer selects the wholesale price and the retail market is determined y quantity competition. Retailers have no costs, except for the wholesale price paid to the manufacturer for each unit. (a) Solve for the equilirium wholesale price, retail price, quantity ordered y each rm, and pro ts for each retailer and the manufacturer. () How does the equilirium retail price vary with n, and is the price greater than or less than the price a vertically integrated monopolist would charge? (A vertically integrated rm is a rm that acts as oth producer and retailer.) Notice that n asically measures the degree of competition in the retail market. Provide some intuition for this result. (c) Suppose that there is only one retailer, n = 1, and suppose that the manufacturer can o er the retailer a general nonlinear schedule specifying the payment made to the manufacturer as a function of the quantity ordered (as a take-it-or-leave-it o er). What pro t would the manufacturer receive from its optimal contract? Answer: (a) The optimal wholesale price depends on the solution to the retailer sugame, so we will solve that rst, for a xed wholesale price, and then go ack to the monopolist s prolem. Inverting the demand equation, we have p r = a Q ;

3 from which we derive the pro t function for retailer j, j = a Q q j p w q j (3) Di erentiating (3) with respect to q j, we have the rst order condition a Q + q j ( 1 ) pw = 0: Now we can impose symmetry to determine the supply of each retailer, q. or a nq q = p w ; q = a pw n + 1 : (4) To nish the prolem, we must determine the equilirium wholesale price. The monopolist s pro ts, taking into account how retail quantities depend on the wholesale price, are given y m = (p w c)q = n(pw c)(a p w ) : (5) n + 1 Di erentiating (5) with respect to p w and simplifying, we have the rst order condition, (p w c)( ) + (a p w ) = 0; from which we derive p w = a + c : (6) Sustituting (6) into (4), we nd the quantities, q = a a+c n + 1 n(a c) Q = (n + 1) = a c (n + 1) (7) (8) The retail price is p r = a n(a c) (n+1) = a n(a c) (n + 1) : (9) From (3), we compute the pro ts of retailer j as j = (a c) 4(n + 1) : (10) 3

4 From (5), we compute the monopolist s pro ts as m = n(a c) 4(n + 1) : (11) () First, notice that for demand to e positive when the retail price equals the marginal production cost, we have a c > 0. This condition must hold in order for the market to e viale. From (9), it follows = (a c) (n + 1) < 0: In other words, more retailer competition leads to a lower retail price. This makes sense when you realize that the wholesale price does not depend on n. In the limit, as n approaches in nity, the retail price equals the wholesale price, and retailers receive zero pro ts. A vertically integrated monopolist would solve max (a Q )Q cq; (1) Q yielding the solution, Q = (a c) ; p r = a + c : (13) Thus, in the limit, as n approaches in nity, the retail price, total quantity, and monopolist s pro ts are identical to the vertically integrated case. With a small numer of retailers, however, the retail price is greater than the price a vertically integrated monopolist would charge. This phenomenon is called doule marginalization. The monopolist s wholesale price re ects a monopoly markup over marginal cost, and imperfect retail competition creates another markup of the retail price over the wholesale price. (c) With n = 1, and when the manufacturer s revenues must e p w Q, the monopolistic manufacturer and monopolistic retailer do not cancel each other s power, ut instead create a situation that is worse than dealing with just one monopolist. However, the doule marginalization prolem can e avoided if the manufacturer extracts a lump sum franchise fee, and then o ers the product at a marginal wholesale price equal to c. Let the payment from the retailer to the manufacturer e F + cq. Then the retailer s prolem is max (a Q )Q F cq: (14) Q Since the solution to (14) is the same as the solution to (1), the retailer chooses the vertically integrated monopoly quantity and retail price, given y (13). The highest value of F that the retailer will accept will yield the retailer zero pro ts: F = ( a Q = (a c) : 4 )Q cq = ( a (a c) (a c) ) 4 c (a c) (15)

5 Thus, the monopolist receives the same pro ts that the vertically integrated monopolist would receive. Since it is impossile to imagine a higher pro t, this contract must e optimal. Intuitively, the manufacturer sells the monopoly rights to the retailer and allows the retailer to order output at cost. 4. Two Cournot duopolists privately oserve their constant marginal production cost efore choosing the quantity they produce. The costs of the two rms are independent random variales, where for rm i = 1;, the marginal cost is given y c i = 0 with proaility 1 ; c i = 0 with proaility 1 : The market demand curve is given y X(p) = 80 p. Solve for the Bayesian Nash equilirium of this duopoly game. Answer: We will nd the reaction functions for rm 1, one function corresponding to a low-cost type and another function corresponding to a high-cost type. The reaction function gives a est response to the two outputs of rm. Then we will impose symmetry (after taking rst order conditions). When rm 1 s cost is zero, pro ts are L 1 = 1 (80 yl 1 y L )y L (80 yl 1 y H )y L 1 : Taking the partial derivative with respect to y1 L and setting it equal to zero, we can solve for y1 L : y1 L y L + y H = 40 : (16) 4 When rm 1 s cost is 0, pro ts are H 1 = 1 (80 yh 1 y L )y H (80 yh 1 y H )y H 1 0y H 1 : Taking the partial derivative with respect to y1 H and setting it equal to zero, we can solve for y1 H : y1 H y L + y H = 30 : (17) 4 Because of symmetry, we can impose y1 L = y L = y L and y1 H = y H = y H. Equation (16) can e simpli ed to y L = 3 y H 5 ; (18) and equation (17) can e simpli ed to y H = 4 y L 5 : (19) 5

6 Sustituting (18) into (19), we can solve for y H, and plugging the value of y H into (18), we compute y L : y H = 55 3 ; yl = 85 3 : 6