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Solutions 18.1 a. P = 53 Q PQ = 53Q Q2 MR = 53 2Q = MC = 5 Q = 24, P = 29, π = (P AC) Q = 576 b. MC = P = 5 P = 5, Q = 48 c. Competitive Consumers' Surplus = 2(48)2 = 1152. Under monopoly: Notice that the sum of consumer surplus, profits, and Deadweight loss under monopoly equals competitive consumer surplus. 18.2 Market demand Q = 70 P, MR = 70 2Q. a. AC = MC = 6. To maximize profits set MC = MR. 6 = 70 2Q 2Q = 64 Q = 32 P = 38 π = (P AC) Q = (38 6) 32 = 1024 b. TC = .25Q2 5Q + 300, MC = .5Q 5. Set MC = MR .5Q 5 = 70 2Q 2.5Q = 75 Q = 30 P = 40 π = TR TC = (30)(40) [.25(30)2 5(30) + 300] = 1200 375 = 825. c. TC = .0133Q3 5Q + 250. MC = .04Q2 5 MC = MR .04Q2 5 = 70 2Q or .04Q2 + 2Q 75 = 0. Quadratic formula gives Q = 25. If Q = 25, P = 45 TR = 1125 TC = 332.8 (MC = 20) π = 792.2 d. 18.3 a. AC = MC = 10, Q = 60 P, MR = 60 2Q. For profit max., MC = MR 10 = 60 2Q 2Q = 50 Q = 25 P = 35 π = TR TC = (25)(35) (25)(10) = 625. b. AC = MC = 10, Q = 45 .5P, MR = 90 4Q. For profit max., MC = MR 10 = 90 4Q 80 = 4Q Q = 20 P = 50 π = (20)(50) (20)(10) = 800. c. AC = MC = 10, Q = 100 2P, MR = 50 Q. For profit max., MC = MR 10 = 50 Q Q = 40 P = 30. π = (40)(30) (40)(10) = 800. Note: Here the inverse elasticity rule is clearly illustrated: Install Equation Editor and double - Part click here to view equation. a 1(35/25) = 1.4 .71 = (35 10)/35 b .5(50/20) = 1.25 .80 = (50 10)/50 c 2(30/40) = 1.5 .67 = (30 10)/30 d. The supply curve for a monopoly is a single point, namely, that quantity-price combination for which MC = MR. Any attempt to connect equilibrium points (price-quantity points) on the market demand curves has little meaning and brings about a strange shape. One reason for this is that as the demand curve shifts, its elasticity (and its MR curve) usually changes bringing about widely varying price and quantity changes. 18.4 a. b. No supply curve for monopoly; have to examine MR = MC intersection since a shift in demand is accompanied by a shift in MR curve. Case (1) and case (2) above show that P may rise or fall in response to an increase in demand. c. Can use inverse elasticity rule to examine this Install Equation Editor and double - click here to view equation. As e toward 1, P MR increases. Case 1 MC constant, MR constant If e , P MR , P . If e constant, P MR constant, P constant. If e , P MR , P . Case 2 MC falling, MR falls: If e , P MR , MR , P may rise or fall. If e constant, P MR constant. MR , P . If e , P MR , MR , P . Case 3 MC rising, MR rising If e , P MR , MR , P . If e constant, P MR constant, MR , P . If e , P MR , MR , P may rise or fall. 18.5 Q = (20 P)(1 + .1A .01A2) Install Equation Editor and double - 2 click here to view equation. Let K = 1 + 1A + .01A π = PQ TC = (20P P2)K (200 10P)K 15 A Install Equation Editor and double - click here to view equation. = (20 2P)K + 10K = 0. a. 20 2P = 10 P = 15 regardless of K or A If A = 0, Q = 5, TC = 65 π= 15 5 65 = 10 b. If P = 15, = 75K 50K 15 A = 25K 15 A = 10 + 1.5A 0.25A2 Install Equation Editor and double - click here to view equation. Q = 5(1 + .3 .09) = 5 1.21 = 6.05 PQ = 90.75 TC = 60.5 + 15 + 3 = 78.5 π = 12.25Can increase over the case A = 0. 18.6 A multiplant monopolist would still produce where MR = MC and would equalize MC among plants. This answer assumes the number of plants is fixed. If the number of plants is subject to choice by monopolist, this number should be chosen so that given the quantity to be produced, overall total costs are minimized. Install Equation Editor and double - 18.7 a. Q1 = 55 P1 R1 = (55 Q1)Q1 = 55Q1 click here to view equation. MR1 = 55 2Q1 = 5 Q1 = 25, P1 = 30 Install Equation Editor and double - Q2 = 70 2P2 click here to view equation. MR = 35 Q2 = 5 Q2 = 30, P2 = 20 π = (30 5) 25 + (20 5) 30 = 625 + 450 = 1075 b. Producer wants to maximize price differential in order to maximize profits but maximum price differential = $5. P1 = P2 + 5 π = (P1 5)(55 P1) + (P2 5)(70 2P2) = π + λ (5 P1 + P2) Install Equation Editor and double - click here to view equation. 60 2P1 = 4P2 80 and P1 = P2 + 5 Install Equation Editor and double - click here to view equation. 130 = 6P2 P2 = 21.66 P1 = 26.66 Install Equation Editor and double - click here to view equation. Install Equation Editor and double - c. P1 = P2 π = 140P 3P 625 2 click here to view equation. = 140 6P = 0 Install Equation Editor and double - P= click here to view equation. = 23a Q1 = 31b Q2 = 23a π = 1008a Install Equation Editor and double - d. If the firm adopts a linear tariff of the form click here to view equation. , it can maximize profit by setting m = 5, α1 = .5(55 5)(50) = 1250 α2 = .5(35 5)(60) = 900 and π = 2150. Notice that in this problem neither market can be uniquely identified as the "least willing" buyer so a solution similar to Example 18.5 is not possible. If the entry fee were constrained to be equal in the two markets, the firm could set m = 0, and charge a fee of 1225 (the most buyers in market 2 would pay). this would yield profits of 2450 125(5) = 1825 which is superior to profits yielded with T(Qi). 18.8 a. For P.C. MC = $10. For monopoly MC = $12. QD = 1000 50P P.C.: P = MC = $10. Thus Q = 1000 50(10) = 500. Install Equation Editor and double - Monopoly: P = 20 click here to view equation. Q, PQ = 20Q Install Equation Editor and double - click here to view equation. Q2 Install Equation Editor and double - Produce where MR = MC. MR = 20 click here to view equation. Q = 12. Q = 200, P = $16 b. See graph below. Loss of consumer surplus = Consumer surplus P.C. Consumer surplus monopoly = 2500 400 = 2100. c. Of this 2100 loss, 800 is a transfer into monopoly profit, 400 is a loss from increased costs under monopoly, and 900 is a "pure" deadweight loss. 18.9 a. The government wishes the monopoly to expand output toward P = MC. A lump- sum subsidy will have no effect on the monopolist's profit maximizing choice, so this will not achieve the goal. b. A subsidy per unit of output will effectively shift the MC curve downward. The figure illustrates this for the constant MC case. c. A subsidy (t) must be chosen so that the monopoly chooses the socially optimal quantity, given t. Since social optimality requires P = MC and profit Install Equation Editor and double - maximization requires that MR = MC t = click here to view equation. , Install Equation Editor and double - substitution yields click here to view equation. as was to be shown. Intuitively, the monopoly creates a gap between price and marginal cost and the optimal subsidy is chosen to equal that gap expressed as a ratio to price. 18.10 Since consumers only value X Q, firms can be treated as selling that commodity (i.e., batteries of a specific useful life). Firms seek to minimize the cost of producing X Q for any level of that output. Setting up the Lagrangian, = C(X)Q + λ (K X Q) yields the following first order conditions for a minimum: Install Equation Editor and double - click here to view equation. Install Equation Editor and double - click here to view equation. Install Equation Editor and double - click here to view equation. Combining the first two shows that C(X) C'(X)X = 0 or Install Equation Editor and double - click here to view equation. . Hence, the level of X chosen is independent of Q (and of market structure). The nature of the demand and cost functions here allow the durability decision to be separated from the output-pricing decision. (This may be the most general case for which such a result holds.)

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