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ps3soln - Problem Set 3 Solutions ECON105 Industrial...

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Problem Set 3 Solutions ECON105 Industrial Organization and Firm Strategy Professor Michael Noel University of California San Diego 1. A monopolist produces a durable good. All production takes place at constant MC= c in the first period and the good lasts until the end of the second period. There is a continuum of consumers uniformly distributed on [0, 1], indexed by θ . A consumer receives utility u = 2 θ - p if she purchases the good in the first period (and enjoys it for two periods) and u = θ - p if she buys in the second period. The monopolist can charge different prices in each period. a. Assume the monopolist can credibly commit to a specific second period price while still at the beginning of the first period. i. Set up the maximization program. First, let’s consider the intuition behind the setup. Given any prices p 1 and p 2 , there will be a consumer with θ=θ 1 within [0, 1], such those with higher θ’s will buy in the 1 st period and those directly below will buy in the 2 nd period if at all (if we change prices, then θ 1 will be at a different spot on the interval, but it will always exist somewhere). This consumer with θ 1 is indifferent between when to buy: 2θ 1 -p 1 = θ 1 p 2 . This equation gives us the Incentive Compatibility constraint for θ 1 and, while it binds with equality for θ 1 , it also automatically holds with strict inequality for all people with even higher θ’s: 2θ -p 1 > θ – p 2 for all θ > θ 1 . I.e., those with θ>θ 1 strictly prefer to buy in the first period than buying in the second. Therefore, the mass of consumers with θ’s higher than θ 1 will be the monopolists quantity sold in the first period alone: q 1 =1- θ 1 . There is also some consumer with θ= θ 2 such that those directly above will buy in the 2 nd period and those directly below won’t buy at all. This particular consumer with θ 2 is indifferent between buying in the 2 nd period and not at all: θ 2 p 2
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