2. An economy consists of a representative household that makes choices over consumption C, and labor N, to maximize subject to Kill = (RF + (1 - 5)) K, + WIN, - C: +II, where Rt = (Ry + (1 -6)) is the gross rate of return on capital, W is the real wage and II, represent profits (dividends) paid out by firms. We thus assume that households own the capital stock which they rent to firms in this economy at the rental rate Ry. Firms in this economy choose capital and labor inputs to maximize If = Y, - RK, - WIN. subject to Y = ( 0, N. ) - KP. The goods market satisfies: where capital accumulation is: Kit = (1 - 5)K, + 1 (a) Derive the equilibrium conditions for this economy. Note that in gen- eral d is a random variable so the equilibrium conditions should be properly expressed taking into account expectations over future out- comes (i.e. as in the Euler equation for the household consumption- savings choice). (b) Consider a balanced-growth path where of grows at a constant rate g. Along this balanced-growth path, show that MPK =r+6 where MPK is the marginal product of capital and r is an appropri- ately defined long-run real rate of interest. Interpret this condition. How does r vary with g? Provide intuition for this result. (c) Show that along the balanced-growth path Ct = (1 - SLR)Y where sLR is the long-run rate of savings. Derive an expression for aLA as a function of the underlying parameters in the model (g. 8. 6,5, a). What is the effect of an increase in 8 on the savings rate? Explain. (d) Is this savings rate the same as the golden rule rate of savings derived in the Solow model? Explain. (e) Show that the equilibrium labor is constant along the balanced- growth path. Provide intuition for why labor is constant even when wages are growing at the rate of technology