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ANSWERS TO SELECTED EXERCISES • 727 b With marginal cost of $30, the optimal prices and profits are PRICE PRICE BUNDLED PROFIT Sell separately $80.00 $80.00 — $200.00 Pure bundling — — $120.00 240.00 120.00 249.90 Mixed bundling 94.95 94.95 several brands with different prices and characteristics is one method of splitting the market into sets of customers with different price elasticities a To maximize profit p = 53Q - Q2 - 5Q, we find ⌬p/⌬Q = - 2Q + 48 = Q = 24, so P = 29 Profit is equal to 576 b P = 53 Q1 - Q2, p = PQ1 - C(Q) = 53Q1 - Q 21 Q1Q2 - 5Q1 a n d p2 = PQ2 - C(Q2) = 53Q2 Q1Q2 - Q22 - 5Q2 c The problem facing Firm is to maximize profit, given that the output of Firm will not change in reaction to the output decision of Firm Therefore, Firm chooses Q1 to maximize p1, as above The change in p1 with respect to a change in Q1 is 53 - 2Q1 - Q2 - = 0, implying Q1 = 24 - Q2/2 Since the problem is symmetric, the reaction function for Firm is Q2 = 24 - Q1/2 Now mixed bundling dominates all other strategies CHAPTER 11—APPENDIX We examine each case, then compare profits a Optimal quantities and prices with no external market for engines are QE = QA = 2000, PE = $8000, and PA = $18,000 For the engine-building division, TR = 2000 # $8000 = $16M, TC = 2(2000)2 = $8M, and pE = $8M For the automobile-assembly division, TR = 2000 # $18,000 = $36M, TC = $8000 # 2000 + 16M = $32M, and pA = $4M Total profits are $12M b Optimal quantities and prices with an external market for engines are QE = 1500, QA = 3000, PE = $6000, and PA = $17,000 For the engine-building division, TR = 1500 # $6000 = $9M, TC = 2(1500)2 = $4.5M, and p = $4.5M For the automobileassembly division, TR = 3000 # $17,000 = $51M, TC = (8000 + 6000)3000 = $42M, and p = $9M Total profits are $13.5M c Optimal quantities and prices with a monopoly market for engines are QE = 2200, QA = 1600, PE = $8800, and PA = $18,400, with 600 engines sold in the monopolized market for $9400 For the engine-building division, TR = 1600 # $8800 + 600 # 9400 = $19.72M, TC = 2(2200)2 = $9.68M, and p = $10.04M For the automobile-assembly division, TR = 1600 # $18,400 = TR = 1600 # $18,400 = $29.44M, TC = (8000 + 8800)1600 = $26.88M, and p = $2.56M Total profits are $12.6M The upstream division, building engines, earns the most profit when it has a monopoly on engines The downstream division, building automobiles, earns the most when there is a competitive market for engines Given the high cost of engines, the firm does best when engines are produced at the lowest cost with an external, competitive market for engines CHAPTER 12 Each firm earns economic profit by distinguishing its brand from all other brands If these competitors merge into a single firm, the resulting monopolist would not produce as many brands as would have been produced before the merger But, producing d Solve for the values of Q1 and Q2 that satisfy both reaction functions: Q1 = 24 - (1/2)(24 - Q1/2) So, Q1 = 16 and Q2 = 16 The price is P = 53 - Q1 - Q2 = 21 Profit is p1 = p2 = P · Qi - C(Qi) = 256 Total profit in the industry is p1 + p2 = 512 True The reaction curve of Firm will be q2 = 7.5 1/2q1 and the reaction curve of Firm will be q1 = 15 - 1/2q2 Substituting yields q2 = and q1 = 15 The price will be 15, which is the monopoly price a (i) In a Cournot equilibrium, when firm A has an increase in marginal cost, its reaction function shifts inward The quantity produced by firm A will decrease and the quantity produced by firm B will increase Total quantity produced will decrease and price will increase (ii) In a collusive equilibrium, the two firms will collectively act like a monopolist When the marginal cost of Firm A increases, Firm A will reduce its production to zero, because Firm B can produce at a lower marginal cost Because Firm B can produce the entire industry output at a marginal cost of $50, there will be no change in output or price However, the firms will have to come to some agreement on how to share the profit earned by B (iii) Because the good is homogeneous, both produce where price equals marginal cost Firm A increases price to $80 and firm B raises its price to $79.99 Assuming firm B can produce enough output, it will supply the entire market b (i) The increase in the marginal cost of both firms shifts both reaction functions inward Both firms decrease output, and price will increase (ii) When marginal cost increases, both firms will produce less and price will increase, as in the monopoly case (iii) Price will increase and quantity produced will decrease c (i) Both reaction functions shift outward and both firms produce more Price will increase (ii) Both firms will increase output, and price will also increase (iii) Both firms will produce more Because marginal cost is constant, price will not change
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