# Solution to Homework Set 7

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1 Solution to Homework Set 7 Managerial Economics Fall An industry consists of five firms with sales of \$00 000, \$ , \$ , \$ , and \$ a) points) Calculate the Herfindahl-Hirschman index HHI). The HHI is HHI = [ /15) + 5/15) + 4/15) + 3/15) + 1/15) ] = 444. b) 1 point) Calculate the four-firm concentration ratio C 4 ). The four-firm concentration ratio is = c) points) Based on the FTC and DOJ Horizontal Merger Guidelines described in the text, do you think the Department of Justice would attempt to block a horizontal merger between two firms with sales of \$ and \$ ? Explain. If the firms with sales of \$ and \$ were allowed to merge, the resulting HHI would increase by 71 to Since the pre-merger HHI exceeds that under the Guidelines 1 800) and the HHI increases by more than that permitted under the Guidelines 100), the merger is likely to be challenged.. 1 point) Suppose the own price elasticity of market demand for retail gasoline is -0.8, the Rothschild index is 0.6, and a typical gasoline retailer enjoys sales of \$1.45 million annually. What is the price elasticity of demand for a representative gasoline retailer s product? The elasticity of demand for a representative firm in the industry is 1.33, since 0.6 = 0.8/E F E F = 0.8/0.6 = Based on the information given, indicate whether the following industry is best characterized by the model of perfect competition, monopoly, monopolistic competition, or oligopoly. a) 1 point) Industry A has a four-firm concentration ratio of percent and a Herfindahl-Hirschman index of 75. A representative firm has a Lerner index of 0.45 and a Rothschild index of Industry A is a monopolistically competitive industry. 1

2 b) 1 point) Industry B has a four-firm concentration ratio of percent and Herfindahl-Hirschman index of 55. A representative firm has a Lerner index of and Rothschild index of Industry B is a perfectly competitive industry. c) 1 point) Industry C has a four-firm concentration ratio of 100 percent and Herfindahl-Hirschman index of A representative firm has a Lerner index of 0.4 and Rothschild index of 1.0. Industry C is a monopoly industry. d) 1 point) Industry D has a four-firm concentration index of 100 percent and Herfindahl-Hirschman index of A representative firm has a Lerner index equal to 0.43 and Rothschild index of Industry D is an oligopoly industry points) Suppose firm A recently entered into an agreement and plan of merger with firm B for \$7.5 billion. Prior to the merger, the market for the good produced by firm A and B consisted of five firms. The market was highly concentrated, with a Herfindahl-Hirschman index of Firm B s share of that market was 8 percent, while firm A comprised just 15 percent of the market. If approved, by how much would the postmerger Herfindahl-Hirschman index increase? Based only on this information, do you think the U.S. Justice Department would challenge the merger? Explain. If approved, the merger would raise the HHI by [ ) )] = 840 points. Since the pre-merger HHI is 3 61, which is greater than the Guidelines 1 800), and the HHI increases by 840 which is greater than the 100 points permitted in the Guidelines), it is unlikely that the merger will receive unconditional approval. 5. Multiple Choice; 3 points) You own two products, each of which is a substitute for the other. You raise price on the first product. What happens to marginal revenue? Assume that demand for each of the two goods can be written as a linear function of prices. To get all three points, justify your choice. a) MR for the first product falls but increases for the second. b) MR rises for both products. c) MR falls for both products. d) MR for the second product falls but increases for the first. Clearly, raising price of the first good reduces demand for it but increases demand for the second good. So we can write demand for the

3 first good as Q 1 = A 1 B 1 P 1 + C 1 P and it follows that P 1 = A 1 + C 1 P Q 1 B 1 MR 1 = A 1 + C 1 P Q 1 B 1. Analogously, MR = A +C P 1 Q B. This gives d is right). MR 1 = Q 1 > 0, and P 1 B 1 P 1 MR = C Q P 1 B B P 1 = C B C B < 0 6. points) Suppose elasticity of demand for you parking lot spaces is, and price is \$8 per day. If your marginal cost is zero, and your capacity is 80% full at 9am over the last month, are you optimizing? Why not)? If not, should you increase or decrease the price? No, you are not optimizing. If your marginal cost is zero, revenues are profits. To maximize revenue, you should choose a level of supply where the price elasticity of demand is 1. Because the elasticity currently is, increasing demand by decreasing the price would increase revenue. 7. points) Suppose elasticity of demand for you parking lot spaces is 0.5, and price is \$0 per day. If your marginal cost is zero, and your capacity is 96% full at 9am over the last month, are you optimizing? Why not)? If not, should you increase or decrease the price? No, you are not optimizing. If your marginal cost is zero, the revenues are profits. To maximize revenues, you should choose a level of supply where the price elasticity of demand is 1. Because the elasticity currently is 0.5, decreasing demand by increasing the price would increase revenue. Vertical Integration The following exercise analyzes the possibility of a vertical merger between an online CD-shop and a CD producer and its impact industry profits and social surplus. VI1: CD-Shop 4 points) Assume that the CD-shop can buy CDs a unit price of P buy. The cost of distributing a number Q shop of CDs be 3

4 c Q shop. The total cost of selling Q shop CDs is the cost of CDs plus distribution cost. The shop can sell an amount Q shop of CDs according to the inverse demand function P shop = A BQ shop. What is the profit function π shop Q shop ) of the CD-shop? What is the profit maximizing amount Q shop that the CD-shop should try to sell? What is the price P shop it should set? The profit function is The FOC implies π shop Q shop ) = A BQ shop )Q shop c + P buy )Q shop. π shop Q shop ) Q shop = A BQ shop c P buy = 0 Q shop = A c P buy B P shop = A BQ shop = A A c P buy = A + c + P buy. VI: CD producer 4 points) Assume that the CD-producer has considerable fix cost F for producing an album artist wages, promotional spending, etc.) but the printing of CDs is virtually free. The cost function CQ prod ) simply is CQ prod ) = F. If the CD-producer can sell CDs according to the inverse demand function P prod = α βq prod, what is the profit function π prod Q prod )? What is the profit maximizing amount Q prod that the CD producer should try to sell? What is the price P prod it should set? The profit function is The FOC implies π prod Q prod ) = α βq prod )Q prod F. π prod Q prod ) Q prod = α βq prod = 0 Q prod = α β P prod = α βq prod = α α = α. 4

5 VI3: Market Equilibrium 7 points) Assume that the producer is the only source for CDs of certain artists for the CD-shop. Therefore, Q shop = Q prod and P buy = P prod. In VI1 you determined Q shop if P buy is given. Use this relationship to derive α and β of the inverse demand function that the CD producer is faced with! α and β should depend on A, B, and c. What are the maximized profits π prod? What are the maximized profits π shop, if the shop buys CDs at the price P buy = P prod that the producer has set? How many CDs will the CD shop sell at what price? Your results should not contain α and β any more. Solving for P buy gives Q shop = A c P buy B P buy = A c BQ shop P prod = A c BQ prod. Comparing with P prod = α βq prod gives α = A c β = B. Inserting the profit maximizing choices P prod = A c Q prod = A c 4B P shop = A+c+P prod Q shop = Q prod = 3A+c 4 into the two firms profit functions yields π prod = P prod Q prod F = A c) F 8B 3A+c π shop = P shop c P prod )Q shop = c A c ) A c 4 4B 3A c) = A c ) ) A c A c A c 4 4B = 4 4B = A c) 16B. 5

6 VI4: Merged firm 5 points) The two firms now merge. The merged firm will sell a new quantity Q m of CDs at a new unit price of P m. If CD production still requires fix cost F, and distribution of CDs to customers is the only variable cost and amounts to cq m, what is the merged firm s profit function π m Q m ) if consumer demand remains the same)? What quantity will it sell at what price? How high are the firm s maximized profits? The merged firm s profit function is The FOC π m Q m ) = A BQ m )Q m F cq m. π m Q m ) Q m = A c BQ m = 0 implies that the merged firm should choose Q m = A c B. The price at which it can sell this quantity is The maximized profits are P m = A+c. π m = A c A c A c) F = F. B 4B VI5: Comparison 6 points) Compare the profits of the firms in VI3 with the profits in VI4. Is there an incentive for the firms to merge and share the after-merger-profits? Compare the quantities of CDs that are sold to the consumers in the two cases. Compare prices charged to consumers as well. Which situation comes closer to the social optimum? Does this model motivate government intervention to reduce or increase the number of vertical mergers assume that firms always merge if this increases joint profits, but never merge if it does not increase profits)? Industry profits in the two firm equilibrium are π tot, = π prod +π shop = A c) 16B + A c) F 8B = 3A c) 16B F 6

7 while the profits of the merged firm are π m = A c) 4B = 4A c) 16B F F. As the profits of the merged firm are bigger than the profits of the two separate firms, the owners of the two firms have an incentive to mergeandshareprofits. Intermsofapproximatingthesocialoptimum P = MC = c), comparing with P shop = 3A+c 4 P m = A+c shows that as long as c < A, P shop > P m > c. Therefore the case of the merged firm comes closer to the social optimum. c A can be ruled out, as in this case, neither the shop, the producer, nor the merged firm would optimally produce positive output. Because P shop > P m, marked demand implies Q shop < Q m. The consumers get to buy more CDs at a lower price if the two firms merge. For all plausible choices of A and c, the owners of the two firms should agree to a merger. This increases social surplus, so government intervention should be unnecessary. How to grade VI5 you don t need to have found the correct results in VI3 and VI4, to get points): If you compare π prod +π shop VI3) with π m VI4): 1 pt If you compare P shop VI3) with P m VI4): 1 pt If you compare Q shop = Q prod VI3) with Q m VI4): 1 pt If you compare to the social optimum P = MC = c): 1 pt If your results only contain A,B,c, and F: 1 pt If you conclude that in VI4, profit and quantity are bigger and consumer price is smaller than in VI3: 1 pt 7

8 CES production function The constant elasticity of substitution ces) production function is defined as YK,L) = γαk ρ +1 α)l ρ ) 1/ρ. The elasticity of substitution is σ = 1 1 ρ. A firm faced with prices r and w for the inputs L and K has an incentive to maximize output from given expenditure M. The constrained nonlinear program that describes this problem is max K,L γαk ρ +1 α)l ρ ) 1/ρ s.t. wl+rk M. CES1 0 points; the solution was presented in lecture 5a ) Solve for the output maximizing levels of inputs K and L as functions of w, r, and M. Express K and L in terms of σ rather than ρ. Hint: K and L take the form K = α σ r σ X and L = 1 α) σ w σ X. Show this using the focs of the Lagrangian method and find X.) The Lagrangian is L = γαk ρ +1 α)l ρ ) 1/ρ +λ M wl rk). The first order conditions are λr = 1 ρ [YK,L)/γ]ρ1/ρ 1) αρk ρ 1 = λ[yk,l)/γ] 1 ρ αk ρ 1 λw = 1 ρ [YK,L)/γ]ρ1/ρ 1) 1 α)ρl ρ 1 = λ[yk,l)/γ] 1 ρ 1 α)l ρ 1 M = rk +wl. ) The first two imply r w = αkρ 1 and thus K 1 α)l ρ 1 L = 1 α)r σ. αw Inserting K = into the last foc ) σl gives This yields 1 α)r αw L = r = ) 1 α)r σ M = r +w) L. αw 1 α)r αw M ) σ = +w r r α w 1 α) σm α σ r 1 σ +1 α) σ w 1 σ w σm 1 α) ) σ +w w 1 α ) σ 8

9 and ) 1 α)r σ K = L = αw r ) σm α α σ r 1 σ +1 α) σ w 1 σ. CES 7 points) The above exercise yields L = K = w 1 α) σm α σ r 1 σ +1 α) σ w 1 σ r ) σm α α σ r 1 σ +1 α) σ w 1 σ. What is the maximized production level given prices r and w and expenditures M? Solve this expression for M to get the expenditures as a function of prices and the production level. Make sure to replace ρ with expressions of σ. Y = γm r ) ) ) σρ w σρ 1/ρ α σ r 1 σ +1 α) σ w 1 σ α +1 α). α 1 α Because ρ = 1 1 σ, ρσ equals σ 1, and Y = γm α σ r 1 σ +1 α) σ w 1 σ α σ r 1 σ +1 α) σ w 1 σ) 1/ρ = γm α σ r 1 σ +1 α) σ w 1 σ) 1/ρ 1 = γm α σ r 1 σ +1 α) σ w 1 σ) 1/σ 1). Solving for M, we get M = Y γ α σ r 1 σ +1 α) σ w 1 σ) 1/1 σ). If Y is the maximum production given wealth M, M is the minimized expenditure to achieve production Y. The unit cost function the minimum cost to produce one unit of output therefore is cr,w) = 1 γ α σ r 1 σ +1 α) σ w 1 σ) 1/1 σ). CES3 3 points) Make a convincing argument that if Y is the maximum production we can get with the money M, then M on the other hand is the minimized cost for producing at least Y units of output. Assume there was an alternative M < M that is enough to produce 9

10 Y. According to this assumption, there are amounts K and L that can produce Y. If we were endowed with M > M however, we could afford some combination K,L ) that satisfies K > K and L > L. With this, we could produce Y = YK,L ) > YK,L ) = Y. But if we can produce Y > Y from money M, Y was not the maximum we can produce from M in the first place. If Y really is the maximum production, our assumption that M < M is enough to produce Y must have been wrong. 10

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