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Question 3 (15 points) Consider the following static version of the Mortensen-Pissarides model. Labor force is normalized to 1. There is a large number of firms who can enter the market and search for a worker. Firms who engage in search first have to pay a fixed cost k. If a measure v of firms enters the market, a CRS matching function m(1, v) gives us the total measure of matches in the economy. Within each match, the firm and worker bargain for the wage, w, with / denoting the bargaining power of the worker. If they agree, they can move on to production, which will deliver output equal to y. If they disagree both parties get nothing. Assume that &/y Bula). Each household has an initial capital stock zo at time 0, and one unit of produc- tive time in each period, that can be devoted to work. Final output is produced using capital and labor services, where F is a CRS production function. This technology is owned by firms whose number will be determined in equilibrium. Output can be consumed (c) or invested (,). We assume that households own the capital stock (so they make the investment decision) and rent out capital services to the firms. The depreciation rate of the capital stock (z,) is denoted by 6.' Finally, we assume that households own the firms, i.e. they are claimants to the firms' profits. The functions u and F have the usual nice properties. a) First consider an Arrow-Debreu world. Describe the households' and firms' problems and carefully define an AD equilibrium. How many firms oper- ate in this equilibrium? b) Now focus on an alternative environment with spot (sequential) markets. Describe the households' and firms' problems and carefully define a sequential markets equilibrium (SME). c) Write down the problem of the household recursively. Be sure to care- fully define the state variables and distinguish between aggregate and individual states. Define a recursive competitive equilibrium (RCE)