Consider N firms each with the constant-returns-to-scale production function Y = F (K, AL), or (using the intensive form) Y = ALf (k). Assume f'(•) > 0, f"(•) < 0. Assume that all firms can hire labor at wage wA and rent capital at cost r, and that all firms have the same value of A. (a) Consider the problem of a firm trying to produce Y units of output at minimum cost. Show that the cost-minimizing level of k is uniquely defined and is independent of Y, and that all firms therefore choose the same value of k. (b) Show that the total output of the N cost-minimizing firms equals the output that a single firm with the same production function has if it uses all the labor and capital used by the N firms.
2.1 Consider N firms each with the constant-returns-to-scale production function Y = F (K, AL), or (using the intensive form) Y = ALf (k). Assume f'(•) > 0, f"(•) < 0. Assume that all firms can hire labor at wage wA and rent capital at cost r, and that all firms have the same value of A.
(a) Consider the problem of a firm trying to produce Y units of output at minimum cost. Show that the cost-minimizing level of k is uniquely defined and is independent of Y, and that all firms therefore choose the same value of k.
(b) Show that the total output of the N cost-minimizing firms equals the output that a single firm with the same production function has if it uses all the labor and capital used by the N firms.
2.2 The elasticity of substitution with constant-relative-risk-aversion utility. Consider an individual who lives for two periods and whose utility is given by equation (2.43). Let P1 and P2 denote the prices of consumption in the two periods, and let W denote the value of the individual’s lifetime income; thus the budget constraint is P1C1 + P2C2 = W.
(a) What are the individual’s utility-maximizing choices of C1 and C2, given P1, P2, and W?
(b) The elasticity of substitution between consumption in the two periods is −[( P1/P2)/(C1/C2)][∂(C1/C2)/∂( P1/P2)], or −∂ ln (C1/C2)/∂ ln ( P1/P2). Show that with the utility function (2.43), the elasticity of substitution between C1 and C2 is 1/θ.
Growth, saving, and r − g. Piketty (2014) argues that a fall in the
(a) How, if at all, does this affect the k̇ = 0 curve?
(b) How, if at all, does this affect the ċ = 0 curve?
(c) At the time of the change, does c rise, fall, or stay the same, or is it not possible to tell?
(d) At the time of the change, does r − g rise, fall, or stay the same, or is it not possible to tell?
(e) In the long run, does r − g rise, fall, or stay the same, or is it not possible to tell?
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