Question

Please explain and show me the process with answer. Thank you!

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. T he 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 constant return to scale. This is because if k is increased by constant proportion lambda: F(k) will rise by lambda as well. Proof: F(k) = A K F(lambda k) = A(lambda k) = lambda AK = lambda F(k) Marginal product of capital: MPK = partial differential Y/partial differential K production function: Y = AK therefore partial differential Y/partial differential K = A = constant The neoclassical assumption about MPK is that MPK must be decreasing in K. But here MPK is constant and hence does not follow neo classical assumption. b) The growth rate of capital is given as g_k = K^dot/K = sY - delta K/K = sA K - delta K/K = sA - delta therefore g_k = sA - delta Given A and delta: a rise in s will rise g_k. c) The growth rate of output: g_Y = g_A + g_K Now A is constant: therefore g_Y = g_K = SA - delta