Question

4. Golden Rule. Suppose that we have a standard Solow Model. There is no population or technology growth. (a) The rm problem is to maximize prots, t. The rm's problem can be written as: max KtC0;NtC0 t = AK t N1− t − wtNt − rtKt: The rm takes the factor prices as given. Find the rst order conditions characterizing the optimal rm behavior. (b) Use the FONCs from 4a to show that wtNt~Yt = 1 − , where Yt = AK t N1− t . (c) The household is endowed with labor and owns the capital stock and leases it to rms on a period-by- period basis. Firms remit any prots back to households. The household budget constraint is given by. Ct + It = wtNt + rtKt + t: Use your previous answers to show that the right hand side of the equation reduces to Yt = AK t N1− t . (d) Capital accumulation is standard: Kt+1 = (1 − )Kt + It where Kt is given in period t, as it is inherited from past decisions. Assume that the household consumes a constant fraction of its income each period, 1 − s, and supplies labor inelastically. Re-write the capital accumulation equation as a dierence equation relating kt+1 to kt and exogenous variables and parameters only, where k represents the per-worker level of capital. (e) Create a graph plotting kt+1 against kt. Argue that exists a steady state, k, at which kt+1 = kt. (f) Algebraically solve for the steady state stock, k, as well as steady state output, y, and consumption, c. (g) What value of s would maximize y? Do you think the household would like this saving rate? Why or why not? (h) What value of s would maximize current ct? Do you think it would be a good thing to have this saving rate? (i) What value of s would maximize steady state consumption, c? Please derive an analytical expression for the s that makes c as big as possible. (j) A reasonable value for is 0.33. With a saving rate between 10{15 percent, would the US be near the \Golden Rule" saving rate you found in (i)? If not, would you necessarily recommend that we increase our saving rate?

Answer 1