This problem considers a variation on the Solow model. Suppose that instead of the population growing at a constant rate, that people have fewer children as they become wealthier. In particular, as always suppose that output is produced via a Cobb-Douglas production function Y = KL 1 , where technology is A = 1 and thus its growth rate is g = 0 for simplicity. Consumers save a constant fraction s of their income, and the capital stock depreciates at rate . The capital stock evolves according to the following equation

dK/ dt K = sY K Define the per-capita variables as k = K/L, y = Y/L, and c = C/L. However, instead of being a constant, now the population growth rate is proportional to the marginal product of capital (MPK): L/ L = n MPK where s > n > 0. Since the MPK falls as capital increases, this captures the declining population growth rates of wealthier nations.

(a) Derive the equation of motion for per-capita quantities of capital k dk/ dt = d(K/L)/ dt .

(b) Determine the steady state per-capita quantities of capital, output, consumption, and population growth rate.

(c) What are the growth rates of aggregate capital stock K, aggregate output Y , and aggregate consumption C in the steady state?

(d) What are the short run (transitional dynamics) and long run (steady state) effects of an increase in n on the per-capita quantities of capital, output, consumption, and the population growth rate?

( y = Y/ L = k^ L/ L = n MPK = nK^(1)L^(1) = n(K/ L)^(1) = nk^(1) Then k = (KL LK)/ L^2 = K /L K/ L* L/ L )