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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.