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=KL1, 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 dtdKK=sYK 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) : LL=nMPK 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 kdtdk=dtd(K/L). (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? (Hint: Notice that y=LY=kLL=nMPK=nK1L1=n(LK)1=nk1 Then k=L2KLLK=LKLKLL) 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=KL1, 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 dtdKK=sYK 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) : LL=nMPK 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 kdtdk=dtd(K/L). (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? (Hint: Notice that y=LY=kLL=nMPK=nK1L1=n(LK)1=nk1 Then k=L2KLLK=LKLKLL)
