2. An economy consists of a representative household that makes choices over consumption Ct and labor Me to maximize 28 ' 108 ( 01 ) - 1 It. ( Ni ) 1 to subject to Kt+1 = (Ri + (1 -8)) Kt + WiNt - Ct + IIt where Rt = (R* + (1 -8)) is the gross rate of return on capital, Wt is the real wage and IIt represent profits (dividends) paid out by firms. We thus assume that households own the capital stock which they rent to firms in this economy. Firms in this economy choose capital and labor inputs to maximize IIt = Yt - RYKt - WiNt subject to Yt = ( 0+ Ni ) 1- " KQ. The goods market satisfies: It + Ct = Yt. where capital accumulation is: Kt+1 = (1 -o)Kt + It (a) Derive the equilibrium conditions for this economy. (b) Assume that Ot grows at a constant rate g. Along the balanced- growth path, show that MPK =r+6 where r is an appropriately 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) Yt where SLR is the long-run rate of savings. Derive an expression for SLR as a function of the underlying parameters in the model (gB , 6, 0, @). What is the effect of an increase inS 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