Question

(29 points) GHH preferences Consider an economy where the representative consumer has a utility function u (C, L) over consumption 0 and leisure L. Assume preferences satisfy the standard prop- erties we saw in class. The consumer has an endowment of H units of time that they allocate to leisure or labor. The consumer also receives dividends, D, from the representative rm. The representative consumer provides labor, NS, at wage rate It}, and receives dividends D, from the representative rm. The representative rm has a production technology given by Y = AF (K, N) where K is the xed capital input and N is labor input. (a) (2 points) Set up the consumer's problem. (b) (2 points) Solve the consumer's problem.in order to nd a set of equations that implicitly allow you to nd optimal labor supply N *, optimal consumption, 0*, and optimal leisure, L", as a function of (m, D). Assume for the remaining parts of this exercise that the utility function is given by _ (H LY\") MC, L) = log (0 1+ 7 over consumption 0 and leisure L, where 95,7 > 0. (c) (4 points) Use this specication for the utility function to nd optimal labor supply N*, optimal consumption, 0*, and optimal leisure, L*, as a function of (w, D) from the optimality conditions you derived in 4a. (d) (2 points) Use the above optimality conditions to assess how the optimal supply of labor, N 3*, responds to a change in dividends. (e) (2 points) How does this differ from the example we saw in class when u (C, L) = logC+IylogL, 0 0 (h) (2 points) Derive the optimal amount of labor that the firm wants to hire as a function of capital, K, and the wage rate, w. What is the name of this function? (i) (2 points) Define (as we did in class) a competitive equilibrium for this economy. (j) (3 points) Solve for the competitive equilibrium level of labor. (k) (3 points) Use the above equilibrium conditions to assess how the equilibrium of labor, Ns*, responds to a change in total factor productivity, A. (1) (3 points) How does this differ from the example we saw in class? Why