American Export-Import Shipping Company operates a general cargo carrier service

Question

American Export-Import Shipping Company operates a general cargo carrier service between New York and several Western European ports. It hauls two major categories of freight: manufactured items and semi-manufactured raw materials. The demand functions for these two classes of goods are P1 = 100 - 2Q1 P2 = 80 - Q2 where Qi = tons of freight moved. The total cost function for American is TC =20 + 4(Q1 + Q2) a. Determine the firm's total profit function. b. What are the profit-maximizing levels of price and output for the two freight categories? c. At these levels of output, calculate the marginal revenue in each market. d. What are American's total profits if it is effectively able to charge different prices in the two markets? e. If American is required by law to charge the same per-ton rate to all users, calculate the new profit-maximizing level of price and output. What are the profits in this situation?

Explanation / Answer

A. TR = P1Q1 + P2Q2 TC = 10 +8(Q1+Q2) profit = TR - TC = p1Q1 +P2Q2 - (10 + 8(Q1+Q2)) = (60-2Q1)Q1 + (40-Q2)Q2 - 10 - 8(Q1+Q2) = 60Q1 - 2(Q1)^2 + 40Q2 - (Q2)^2 - 10 - 8Q1 - 8Q2 = 52Q1 - 2(Q1)^2 + 32Q2 - (Q2)^2 - 10 B. profit maximising P for retail outlets @ d(profit)/dQ1 = 0; 52 - 4Q1 =0, 52 = 4Q1, Q1=13; p1 = 60 - 2Q1; p1 = 60 - 2(13) = 60 - 26 = 34 profit maxmising Q for superior @ d(profit)/dQ2 = 0; 32 - 2Q2 = 0, 32 = 2Q2, Q2 = 16 p2 = 40 - 16 = 24 C. MR = dTR/dQ; dTR/dQ1 = p1 = 34; dTR/dQ2 = p2 = 24 D. you can use the original formula for this, but I am going to use the profit function I derived in A: profit = 52Q1 - 2(Q1)^2 + 32Q2 - (Q2)^2 - 10 = 52*13 - 2*(13)^2 + 32*16 - 16^2 - 10 = 676 - 338 + 512 - 256 -10 = 1188 - 604 = 584 E. TR = PQ1 + PQ2 = P(Q1+Q2) TC = 10+8(Q1+Q2) profit = TR - TC = P(Q1 + Q2) - 10 - 8(Q1 + Q2) = (P-8)(Q1+Q2) -10 profit max retail = d(profit)/d(Q1) = 0; P-8 = 0; P =8 profit max superior = d(profit)/d(Q2) = 0; P = 8 (confirmed) using the demand functions above for P1 and P2... P1 = 60 - 2Q1 = 8; 52 = 2Q1; Q1 = 16 P2 = 40 - Q2 = 8; 32 = Q2 However, as you know, if P = 8, (P-8)(Q1+Q2) = 0, so profit = -10; the firm is better off shutting down. This to me does not quite make sense, but there you go.

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