Homework 3
Due 2/6/2024
1) Suppose that labor and capital are both paid their marginal products, so the real wage is
w(t)=∂/∂L (F(K(t),A(t)L(t)))
(as opposed to F2(K(t), A(t)L(t))) and the (net) rate of return on capital is
a) Show that if F exhibits constant returns to scale then
b) Show that if F exhibits constant returns to scale then
so the sum of factor payments equals net output.
2) If wages and the return to capital are defined as in (1), show the fraction of output that goes to workers, i.e. (W(t)L(t))/(Y(t)), is called the share of labor. Likewise, the fraction of output that goes to the owners of capital (gross of depreciation), i.e. ((r(t)+δ)K(t))/(Y(t)), is called the share of capital. Show that the share of capital is the elasticity of f with respect to k. Recall from HW 1, that for a Cobb-Douglas production function KL1, this was .
3) Consider a Solow economy with a constant returns to scale production function F(K, AL). Suppose that labor and capital are both paid their marginal products, so the real wage is
and the rate of return on capital is
a) The return to capital is roughly constant over time, as are the shares of output going to capital and labor. Does a Solow economy on a balanced growth path exhibit these properties? What is the growth rate of w on a balanced growth path? What is dr/dt? (We really should not talk about the growth rate of r since r need not be positive.)
b) Suppose the economy begins with k(0) < k*. As k moves toward k*, does w grow at a rate greater than, less than, or equal to its growth rate on the balanced growth path. What about dr/dt?
