You work for HiTek Bikes that designs and manufactures a high performance road bike frame, the Dura Carbon Ace. The demand curve for this frame is given by P = $6,600 - 10Q, where Q is the number of frames sold and P is the price. The total cost of production is TC = Q². Given this total cost curve, we know that marginal cost is MC = 2Q.
1. Verify numerically that when TC = Q² marginal cost is MC = 2Q.
2. What are the optimal output, price, and profits for the firm?
3. Now assume that the firm is divided into two profit centers. One division manufactures the product at a total cost of TC = Q² and then transfers it to a selling division that faces the firm's demand curve. The selling division has no other costs other than the transfer price for the product. Assume that the manufacturing division has the power to set the transfer price and that the selling division can only buy internally. The selling division, however, can select the quantity to purchase. What transfer price will the manufacturing unit select? What are the resulting profits of the two units?
4. From the firm's standpoint, what is the optimal transfer price?