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Consider the market for used cars and assume that the cars can be of one of two possible quality levels low

Consider the market for used cars and assume that the cars can be of one of two possible quality levels: low (the ’lemons’) and high (the ’plums’). The seller knows the true quality of the car (the seller’s type is given by the quality level) but the buyers only know the probability distribution of the cars in the market. Assume that the buyer’s initial (prior) beliefs are as follows: probability p for a car being of high quality, and probability (1 − p) for a car being of low quality. If the seller sells his car, then his payoff is equal to the price paid by the buyer. If the seller does not sell the car, his payoff will be equal to his valuation of the used car: $10000 if it is of high quality and $6000 if it is of low quality. The buyer’s valuation of cars with high quality is $12000, and his valuation of low-quality cars is $7000. However, he only learns the true quality after the purchase. If the buyer purchases a car, then his payoff is given by the difference between his true valuation and the price he pays. If he does not buy a car, his payoff is zero. (a) First, consider a market with complete information for both sides of the market. Explain why, given the buyers’ and sellers’ valuations, all cars will be sold. Is this outcome Pareto-optimal? (b) Now consider the market with incomplete information as described above. What would be a seller’s pooling strategy (to offer all the cars at the same price)? If sellers follow this pooling strategy, what would be their best response (strategy and beliefs) given their prior belief p? Given this best response of the buyers, is it optimal for the seller to follow the pooling strategy given p? (c) Find values for prior belief p such that there is a perfect Bayesian Nash equilibrium with pooling. Is this outcome Pareto-optimal, i.e. are all those cars sold that would be sold with complete information?

Feb 06 2020 View more View Less

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