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Unformatted text preview: Intermediate Microeconomics, Econ 401 Winter 2008 Problem Set No 22: Solutions Q1. Adverse Selection and Buying Stocks. Reconsider the problem of buying stocks. Suppose the value of the stock can be either high, medium, or low. Each of the three values is equally likely (i.e., 1 3 ). Values and costs are as follows V alue v c High 100 90 Medium 90 80 Low 80 70 So this is the same setup as in the lecture, but with three possible values instead of two. Again, the buyer makes a price offer p and the seller accepts if p ≥ c j . a) Calculate the expected utility of the buyer if he offers p = 90 , p = 80 , p = 70 . Note that we assume, as in lecture, that the buyer is risk neutral, so u ( v ) = v . Here, the seller knows the cost of his stock with certainty, but the buyer only knows the distribution of possible costs. If the buyer offers p = 90, all three types of seller would like to sell him the stock, because 90 is equal to or greater than all three costs. The buyer’s expected utility is a weighted average of the three possible stock values minus the price paid, weighted by their probabilities: EU ( p = 90) = 1 3 ( u (100)- u (90)) + 1 3 ( u (90)- u (90)) + 1 3 ( u (80)- u (90)) EU ( p = 90) = 1 3 (100- 90) + 1 3 (90- 90) + 1 3 (80- 90) EU ( p = 90) = 1 3 (10 + 0- 10) EU ( p = 90) = If the buyer offers p = 80, then only the two lowest-cost types of seller will want to sell him their stock. The buyer will pay 80 but get a stock that could be worth either 90 or 80. The type of seller that has a high-type stock will not sell his stock, therefore the buyer expects to purchase stock only 2/3 of the time. Therefore the buyer’s expected utility is EU ( p = 80) = 1 3 ( u (90)- u (80)) + 1 3 ( u (80)- u (80)) EU ( p = 80) = 1 3 (90- 80) + 1 3 (80- 80) EU ( p = 80) = 1 3 (10 + 0) EU ( p = 80) = 10 3 1 If the buyer offers p = 70, then only the lowest-cost type of seller will want to sell him their stock. Therefore the buyer only purchases stock 1/3 of the time, and his expected utility is EU ( p = 80) = 1 3 ( u (80)- u (70)) EU ( p = 70) = 1 3 (80- 70) EU ( p = 70) = 10 3 b) The three prices from before are the only candidates for optimal offers. Why? If the buyer were to offer a price below 70, the seller wouldn’t sell stock, because p < c . If the buyer offered a price above 90, all three types of seller would want to sell stock. The buyer’s expected utility would decrease because the weighted average of values would be the same as with p = 90, but we’d subtract a larger price. The same argument works for prices between 70 and 80, and between 80 and 90. c) Which of the three prices is actually the optimal offer? Of the three prices, p = 70 and p = 80 lead to the highest expected utility for the buyer, at 10/3....
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- Winter '08