Question 1. (70 points) Consider the matching model of the labor market that we studied in class. W ...
Question 1. (70 points) Consider the matching model of the labor market that we studied in class. We are going to use the following ingredients. E is employment, U is unemployment, L is the (exogenous or inelastic) labor supply, and L = E + U. Matches occur according to the function M (U, V ), where V are vacancies. Let ? ? V/U be the tightness ratio, f = M (U, V ) /U = M (1, ?) be the job finding rate and q = M (U, V ) /V = M (1/?, 1) be the vacancy filling rate. The law of motion of employment is ?E = M (U, V ) ? sE, where s is the exogenous separation rate. Let r denote the interest rate, A?w?c the profit to the employer from a filled job (w is the wage and c the non-wage cost of maintaining the job) and ?c the profit from an unfilled job. Finally, let E and U denote, respectively, the value for the worker of being employed and of being unemployed, and let J and V denote, respectively, the value for the employer of a filled job and a vacant job. Answer the following questions. 1. Agents’ behavior is characterized by the following equations: r = w E ? s E ? U E ; r = A ? w ? c J ? s J ? V J ; r = f E ? U U ; r = ? c V + q J ? V V . Interpret carefully these relations. 2. Assume that whenever a match occurs, employer and worker set the wage so that wi = argmax (Ei ? U)? (Ji ? V )1?? . Interpret this characterization. 3. Assume that the cost of creating a vacancy is zero and that there is free entry in vacancy creation so that V = 0. Interpret this condition. (Careful here: recall the abuse of notation where V at the numerator of ? is vacancies, not the value of a vacancy, which we are setting at zero.) Show that the five conditions just discussed yield a symmetric equilibrium where the wage curve is w = ? (A ? c + c?) . Interpret this equation. 4. Show that there exists an instantaneous equilibrium in (?,w) space given by the intersection of a job creation curve an the wage curve derived in 3 above. Interpret this equilibrium. 5. Now show that the model’s dynamics reduces to a di§erential equation in the unemployment rate u ? U/L. 6. Use the di§erential equation to discuss the e§ects of a permanent increase in each one of A, c, s, r.
