Problem 7.3: Minimum Wage in the DMP Model. (30 points) Consider the Diamond-Mortensen-Pissarides model of the labor market. There is a mass u of unemployed workers. Workers receive b if they do not match with a firm. Firms can post vacancies at a cost k. If a firm and a worker match, they produce output worth y. Let j be the equilibrium number of vacancies in the economy, and w the equilibrium wage. The matching function is given by a m(u,j) = uj'~ Suppose that the wage rate is determined by Nash Bargaining, so that w = Sy + (1 3)b where 3 is the bargaining weight of the worker. (a) Consider a firm that has decided to post a vacancy. What are its expected benefits of doing so (not including the vacancy cost)? [Hint: You will need to calculate the probability that the firm finds a worker.] (b) Find the number of vacancies j such that the expected benefits of posting a vacancy are equal to the cost k. Argue that this is the equilibrium number of vacancies. (c) A benevolent social planner would choose vacancies equal to P {(1ai(yb)]i . Suppose a > 3. Argue that the equilibrium is not efficient and provide some intuition for why it is not. (d) Suppose the social planner could introduce a minimum wage. Should she do that? What would be the optimal level of the minimum wage