Problem set #2

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University of Minnesota-Twin Cities *

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Economics

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Feb 20, 2024

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Problem 1: Because malls are centrally located while outlets stores are not, demand tends to be different at the two types of retail venues. Suppose that demand for some product at malls is given by P = 100 – Q, so MR = 100-2Q, while demand at outlet stores is given by P = 50 – 0.5Q, while MR = 50 – Q. Suppose that the marginal cost of the product is 5 at both venues. (There are no fixed costs, so TC = 5Q). 1. Find the profit maximizing quantities and prices at the mall and outlet store. Calculate the associated profits. P at mall = 52.5. Q at Mall = 47.5 Profit at mall = 2,256.25 P outlet store = 27.5. Q at outlet store = 45 Profit at outlet store = 1,012.5 2. Suppose that one cannot segment the consumers using malls and outlet malls, so that one instead faces a single overall demand curve. Find the overall demand curve (that is consistent with the two segment curves given in the problem). Q = 200 – 1.5P P = 133.333 – 2/3Q 3. Find the profit maximizing quantity and price when one can set only a single price. How large are profits? How do these compare with the profits with segmentation? Q = 96.25 P = 69.1666 Profit = 6,176.04 Total profit is much higher than segmentation Problem 2: The market demand for a product is P = 100 – Q. Two firms – and only two firms – serve this industry. MC = 20 for each firm, and FC = 300 per firm. a) Find the Cournot/Nash equilibrium for this industry. How much output does each firm produce? What is the market price? How large are each firm’s profits? Quantity output for each firm = 26.666 Market price = 46.666 Profit per firm = 411.111

b) If the firms collude, what price do they charge to maximize profit? What is each firm’s output? Calculate each firm’s profits (assume that both firms must incur fixed costs even if they collude). Price = 60 Output per firm = 20 Profit per firm = 500 c) The last part of this question is about what we expect to happen in this industry. Suppose the two firms are deciding between producing the Cournot output (from part a) and the collusive output (from part b). Calculate the payoff (profit) for firm 1 if it plays “Cournot” while the other firm plays “collude.” And vice versa. If firm 1 plays “Cournot” while firm 2 plays “collude”, firm 1 will produce 30 while firm 2 will produce 20. With a new market price of 50, the profit for firm 1 will be 600 while profit for firm 2 is 300. d) Fill in the following payoff matrix. In each cell, put (profits for firm 1, profits for firm 2). Firm #2 plays Cournot Firm #2 plays collude Firm #1 plays Cournot 411.11 , 411.11 600 , 300 Firm #1 plays collude 300 , 600 500 , 500 e) With reference to this matrix, what do you expect to see happen? (Think this through by asking yourself, “suppose firm 2 plays Cournot. What would I (firm 1) prefer to do?” And so on…) It seems that it would be advantageous for firm 1 to always match whatever firm 1 chooses. So if a firm chooses to play cournot, the other firm will also play cournot

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