Question

1. (The AK Model) Consider an economy with an aggregate production function given by y = F(K) = AK Capital is the only relevant factor of production. A is fixed and represents the productivity of capital. The law of motion for capital is just as in the neoclassical model where s and δ are the savings rate and depreciation rate, respectively. a) Show whether F(K) exhibits constant, decreasing or increasing returns to scale. Com- pute the marginal product of capital. Does this function satisfy the neoclassical assumptions? b) Compute the growth rate (gK) of capital K in this economy. What happens to gk if the savings rate is higher? c) Compute the growth rate (gy) of output. Compute the capital share of output. d) Make a graph of the curve In(Kt) with time on the x-axis. Carefully show the shape and label the slope of this curve.

Answer 1

(A) f(k) represents constants return to scale this is because if 'k' is increased by constant proposition lambda: f(k) will rise by lambda as well f(k) = Ak F(lambda k) = A(lambda k) = lambda Ak = lambda F(k) marginal product of capital: Mpk = partial differential r/partial differential k Production function: r = Ak therefore partial differential r/partial differential k = A(constant) the neoclassical assumption about Mpk in that Mpk must be decreasing in k But here Mpk is constant and hence does not follow neoclassical assumption (b) the growth rate of capital is given as g_k = k/k delta y - delta k/k = SAk - delta k/k = SA - delta \r\n therefore g_k = SA - delta then A and 'delta' is a rise in 's' will rise g_k (c) the growth rate of output: g_r = g_A + g_k Now 'A' is constant therefore g_r = g_k = SA - delta the equilibrium mpk = pk therefore A = pk therefore r = pk k therefore the