A producer has the possibility of discriminating between the domestic and foreign markets for a product where the demands, respectively, are (4.5) Total cost = 2000 + 100 where Q=Q +Q. What price will the producer charge in order to maximize profits (a) with discrimination between markets and (b) without discrimination? (c) Compare the profit differential between discrimination and nondiscrimination. a) To maximize profits under price discrimination, the producer will set prices so that MC MR in each market. Thus, MC = MR, = MR With TC = 2000 + 100. dTC dQ Hence MC will be the same at all levels of output. In the domestic market, Q = 21 -0.1P, P = 210-100, TR, (210-10Q)Q, 2100,- 100 Hence, Q = 21-0.1P Q = 50-0.4P and When MR, MC, When Q = 10, In the foreign market, Hence, Thus MC = MR = TR dQ; Hence, and Q=Q+Q=21-0.1P+50-0.4P = 71-0.5P c) With discrimination, P=142-20 TR=(142-20) Q = 1420-20 MR = When MRMC. When Q = 33, P=142-2(33) = 76 When no discrimination takes place, the price falls somewhere between the relatively high price of the domestic market and the relatively low price of the foreign market. Notice, however, that the quantity sold remains the same: at P = 76, Q, 134, Q- 19.6, and Q = 33. dTR dQ Thus, Without discrimination, = 142-4Q 142-40 = 10 USES OF THE DERIVATIVE IN MATHEMATICS AND ECONOMICS TC = 2000 + 100, where Q = Q + Q Q=33 TR = TR + TR = PQ + PQ = 110(10) +67.5(23)=2652.50 TC= 2000+10(10+23) = 2330 ==TR-TC=2652.50-2330-322.50 [CHAP. 4 TR=PQ=76(33) = 2508 TC = 2330 since costs do not change with or without discrimination. Thus, -2508-2330 = 178. Profits are higher with discrimination (322.50) than without discrimination. i. Update the cost function so to include a general "shipping cost function" for each market (i.e. the current cost function will reflect productioncosts, and we're going to add a function to describe the cost of shipping to the domestic market and another to describe the cost of shipping to the foreign market). ii. Solve parts a) and b) using that generalized form.