1. An economy consists of a representative household that makes choices over consumption 0,; and labor Ni to maximize [: 5t l10g(0t) HLJ (NtlleH subject to KH1 = (Rf; + (1 5)) K. + mmt c.t + H. where Hi 2 (Rc + (1 5)) is the gross rate of return on capital, Wt is the real wage and Ht represent prots (dividends) paid out by rms. We thus assume that households own the capital stock which they rent to rms in this economy. Firms in this economy choose capital and labor inputs to maximize mznmmmm subject to Yi I (BtNt)l_aKE-X. The goods market satises: It + at = Y}.- where capital accumulation is: Km = (1 6m; + It (a) Derive the equilibrium conditions for this economy. b Assume that at rows at a constant rate . Alon the balanced- E} 9 E? growth path, show that MPK2r+6 where M PK is the marginal product of capital and r is an appropri ately dened long-run real rate of interest. Interpret this condition. How does 7' vary with 9? Provide intuition for this result. (c) Show that along the balanced-growth path at = {1 Surly} where SLR is the longrun rate of savings. Derive an expression for SLR as a function of the underlying parameters in the model (9, ,6,o,o:). 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