GS/ECON 5010 Answers to Assignment 3 W2005

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Norma Anthony
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1 GS/ECON 500 Answers to Assignment 3 W005 Q. What are the market price, and aggregate quantity sold, in long run equilibrium in a perfectly competitive market f which the demand function has the equation Q = 000 5p (where Q is aggregate quantity demanded, and p the price), if there is free entry by identical firms to the industry, each of which has the long run total cost function T C = 600q 80q + 4q 3 where q is the quantity produced by the firm? A. Given the total cost function, the average cost function is AC = q + 4q ( ) and the marginal cost function is MC = q q ( ) From ( ), AC = 8q 80 ( 3) which shows that the average cost curve is U shaped : it decreases with q if and only if 8q < 80, q < 0. At q = 0, AC = (0) + 4(0) = 00 and MC = (0) + (0) = 00 so that the marginal cost curve cuts the average cost curve at the minimum average cost, as required. In long run equilibrium, with free entry, each firm must be operating at its minimum average cost, which here is AC = 00, attained at an output level of q = 0. Due to free entry (and the U shaped average cost curve), the long run supply curve is hizontal, at a height of 00. The equilibrium price must be 00, no matter what the demand curve. So p = 00, which means, from the equation of the demand curve, that aggregate quantity demanded Q is 000 5(00) = 000. In long run equilibrium, there will be 00 identical firms in the industry, each producing 0 units at a cost of 00 per unit.

2 Q. Suppose that a market contains million identical consumers, each of whom has preferences which can be represented by the utility function U(X, q, q ) = X + (q α + q α ) β where X is consumption of a numéraire good, and q and q are consumption of goods produced by firms # and # respectively, and where α < and β <. Firms and each have the identical total cost function, C(q, w) = cq, where c is a constant. i Which levels of output f the two firms would maximize their combined profits? ii What levels of output would firms # and # produce if they behaved as Cournot duopolists? (You may assume that q = q in the solutions to each of the problems.) A. Given the preferences of consumers, what is each consumer s inverse demand function? The first der conditions f consumer utility maximization imply that U X = = λp X = λ ( ) where U i is the marginal utility with respect to q i. U = αβ(q α + q α ) β q α = λp ( ) U = αβ(q α + q α ) β q α = λp ( 3) Since good X is numéraire, equation ( ) implies that λ =, so that equations ( ) and ( 3) define the inverse demand functions of the two firms. If firm produces q million units of output, and firm produces q million, then equations ( ) and ( 3) define the prices p and p that will clear the markets f the two goods. So if firm produces q million units, its profits will be (p c)q, where p = αβ(q α + q α ) β q α and similarly, firm will earn profits of (p c)q, where p = αβ(q α + q α ) β q α i Joint profits of the two firms are (p c)q + (p c)q. If q = q = q, then p = p = αβ[q α ] β q α = αβ β q αβ so that joint profits are (αβ β q αβ c)q = [(αβ α(β ) q αβ ) cq] ( 4)

3 Maximizing expression ( 4) with respect to q implies that α β β q αβ = c ( 5) where q = c γ [α β β ] γ ( 6) γ = αβ ii If firm behaves as a Cournot duopolist, it chooses its own output q to maximize its own profit (p c)q, taking the other firm s output q as given. That means it tries to maximize with respect to q. αβ(q α + q α ) β q α cq The first der condition f firm s profit maximization is α β[(q α + q α ) β q α ][(β )q α + (q α + q α )] c = 0 ( 7) Firm has an analogous optimality condition α β[(q α + q α ) β q α ][(β )q α + (q α + q α )] c = 0 ( 8) In Cournot Nash equilibrium, both firms are on their reaction functions, defined by equations ( 7) and ( 8). If the equilibrium is symmetric, so that q = q = q, equation ( 7) ( ( 8)) becomes α β[q α β q α [(β )q α + q α ] = c ( 9) so that α β(β + ) β q αβ = c ( 0) q c = c γ [α β(β + ) β ] γ ( ) Comparing ( 6) and ( 0), q c β = [ β + ]γ q Since β <, β = β + β < β +. But since γ < 0, therefe, q c > q. Q3. Solve f the quantity produced by each firm, the price, and each firm s profits, if there were J firms acting as Cournot oligopolists, each producing a homogeneous good, f which the market demand is linear p = a bq 3

4 where Q q + q + + q J was industry output, if each firm had the (same) total cost function C(q) = cq f some positive constant c? A4 Each firm s profits pq i c(q i ) would be π i = (a b[q + q + + q J ])q i c(q i ) (3 ) maximizing π i with respect to q i yields the first der condition a bq i = (b + c)q i (3 ) where Q i j i q j is the sum of all other firms quantities produced. In a symmetric Nash equilibrium, q = q = = q J = q, so that Q i = (J )q and equation (3 ) becomes [(J + )b + c]q = a (3 3) q = a (J + )b + c Industry output Q equals Jq in a symmetric Nash equilibrium, so that Q = aj (J + )b + c (3 4) (3 5) Since p = a bq, and the profit of each firm is b + c p = a (J + )b + c π i = p i q i cq i = a (b + c) [(J + )b + c] a c [(J + )b + c] = [ a (b + c) [(J + )b + c] ] So (as in the case of constant marginal costs), total industry profit decreases with the number of firms J in the industry. Q4. What would be the equilibrium price, and the equilibrium profits of each firm, in a market with two Bertrand oligopolists, producing goods which are imperfect substitutes f each other, with quantity demanded of the products of the two firms being q = pr p r + pr 4

5 q = pr p r + pr where r < 0, if each firm s cost of producing c units were cq (where c is a positive constant).. (You may restrict attention to symmetric equilibria, in which p = p.) A4. The profit of firm, if it charges a price p (and if firm charges a price of p ), is (p c)q = pr p r + pr cpr p r + pr (4 ) Maximizing this profit with respect to p gives the first der condition p r [p r + pr ] [rpr c(r )p r c(r )p r p + crp r ] = 0 (4 ) In a symmetric equilibrium, in which p = p = p, equation (4 ) becomes rp = c(r ) p = r c (4 3) r Q5. Another model of duopoly is that of von Stackelberg, in which firms choose output levels sequentially. That is, firm chooses its output. Firm observes what output level fiirm has chosen, and then chooses its own output level. What output levels would the firms choose, if they behaved in this manner, if they both produced an identical product f which the market inverse demand function had the equation if each firm had a total cost function where q i is the output level of firm i? p = (q + q ) T C = 4 + q i if q i > 0 0 if q i = 0 A5. This problem must be solved backwards. First, what is firm s reaction to firm producing an output level of q? If q > 0, then π = pq T C(q ) = [ (q + q )]q (4 + q ) (5 ) Maximizing π with respect to q yields the first der condition q q = 0 (5 ) 5

6 q = 0 q (5 3) But firm will choose to produce a positive level of output only if it earns a positive profit. What is its profit if firm has chosen an output level of q, and if firm has responded by choosing q = 0 (q /)? In this case the price is q q, which means that Substituting back into (5 ), p = q (0 q ) = q (5 4) π = [0 q ][p ] 4 = [0 q ] 4 (5 5) So firm can earn a positive profit only if which is the same thing as [0 q ] > 4 0 q > q < 6 So if q 6, then firm s best response is to produce nothing at all, since the fixed costs (of 4) imply that it would lose money at any positive level of production. If q < 6, firm should produce the output level defined by equation (5 3). Now consider firm s decision. It knows that if it produces an output level of q < 6, then firm will follow by producing 0 q, resulting in a price of q. So firm s profit, if it chooses an output level of q initially, will be π = pq T C(q ) = [0 q ]q 4 (5 6) Maximizining (5 6) with respect to q yields the first der condition resulting in profits of q = 0 π = [0 0 ]0 4 = 46 On the other hand, if firm produces an output of 6 me, then firm will shut down completely. That would result in a price of q, and a profit to firm of ( q )q q 4 = 0q q 4 (5 7) The expression (5 7) is decreasing in q when q 6. That means that, if firm were to find it profitable to have q 6, that q = 6 would be the best level of output to choose. That level is the smallest level of output f firm which will induce firm to shut down completely. At q = 6 (and q = 0), equation (5 7) shows that π = 0(6) 56 4 = 60 Since 60 > 46, then the best strategy f firm is to produce an output just high enough that firm cannot make a profit. The Stackelberg equilibrium here has q = 6 and q = 0. 6

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