2. Consider a real business cycle model in which the representative agent chooses capital and and labor to maximize the utility of consumption (c) and leisure ((1)), where the time endowment is unity and labor is . u(c, 1-l) subject to stochastic productivity shocks (A). Output (y) is given by y= Aka (1-0)-a where the firm rents capital from the household at rental rate r. (a) Write the firm's profit maximization problem and solve for the values of the wage (w) and the rental rate (r). (b) Write the expression for the agent's budget constraint using recursive notation (primes for one-period-ahead values) Let the rate of depreciation on capital be 5. Why can't the representative agent in a closed economy use bonds to smooth consumption? (c) What are the state variables in the consumer's optimization problem? Write the value function for the consumer, using recursive notation and take first order conditions. Write the expression for the envelope condition and write an expression for the Euler equation and one for the labor supply decision. (d) Explain the permanent income theory of consumption. Use this theory to compare the effect of a transitory increase in A on consumption with a permanent increase. (e) Now, consider three different specifications of utility, each of which is used in macro models. u(c,1 ) Inc+ 3 = (1-0)1-7 1-7 u(c. 1 ) Inc - vl 3 > 0,7 > 1 (BL) (IDL) u(c, 1 )Inc+3= (1-0)1-7 1-7 (GHH) where (BL) represents the baseline specification, (IDL) is the specification with indi- visible labor, and the (GHH) is due to Greenwood, Hercowitz and Huffman. Write the equations for the equilibrium relationship between consumption and leisure for each specification. (f) Define a balanced growth equilibrium. Which, if any, of the specifications have a labor- leisure choice which is consistent with balanced growth? Explain. (g) Compare the response of labor supply to a transitory increase in A which raises the wage using the baseline model and the GHH model. 1. Diamond-Mortensen-Pissarides with on-the-job search Time: Discrete, infinite horizon Demography: A mass of 1 of workers with infinite lives. There is a large mass of firms who create individual and identical vacancies. The number of vacancies, u, is controlled by free-entry. Preferences: Workers and firms are risk neutral (i.e. u(x) = r). The common discount rate is r. The value of leisure for workers is b. The cost of holding a vacancy for firms is a utils per period. Productive Technology: A firm matched to a worker produces p units of the consump- tion good per period. With probability A each period, jobs (filled or vacant) experience a catastrophic productivity shock and the job is destroyed. Matching Technology: In this arrangement, workers are always in the market. Whether they have a job or not does not stop them getting another job. As they can only have one job at a time if an employed worker meets a firm with a vacancy, the worker quits the current job and switches employment to the new firm. Firms cannot commit to paying a higher wage than the current firm. (Wages are determined by Nash bargaining and symmetry will mean they all pay the same wage.) With probability m(u) each period workers encounter vacancies where again v is the mass of vacancies. The function m(.) is increasing concave and m(v) < 1 for all v. Also limom'(v) = 1, lime m'(v) = 0, and m(v) > vm' (v). The rate at which vacancies encounter workers is then m(u)/u which is decreasing in v. (Assume that job destruction and matching are mutually exclusive so m(v) + 0) equilibrium exist? Explain. (e) Obtain an expression for steady-state unemployment. (f) How does unemployment change with the separation rate, X? Briefly explain. 5) EITC and Married Women a) The most common model of married women's labor supply is the secondary earner model where the woman takes the husband's earnings as exogenous and part of her non-labor income. Ignoring the EITC, present the secondary earner model graphically and discuss the comparative static results associated with an increase in her wage and an increase in the earnings of her husband. For the rest of the problem, use the secondary earner model to analyze the impact of the EITC on the labor force participation of married women. b) Consider a woman married to a man who is not working (e.g. his earnings =0). Draw her budget constraint with and without the EITC. What is the theoretical prediction of the impact of the EITC on the labor force participation for this woman? c) Now consider a woman married to a man who works and has earnings in the phase-out range of the EITC. Draw her budget constraint with and without the EITC. What is the theoretical prediction of the impact of the EITC on the labor force participation for this woman? d) Lastly, what if woman is married to a man who works and has earnings above the phase-out range of the EITC. Draw her budget constraint with and without the EITC. What is the theoretical prediction of the impact of the EITC on the labor force participation for this woman? e) Use your information in (b)-(d) to summarize the predictions for impact of EITC on the labor force participation of married women. How does this compare to incentives for single women?