1. 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 particular1 as alwan suppose that output is produced via a Cobb-Douglas production function V : K'kLF'I. where technology is A : 1 and thus its growth rate is g = II] for simplicity. Consumers save a constant fraction s of their income1 and the capital stock depreciates at rate 5. The capital stock evolves according to the following equation [EX ~ EI':.*Y6K (ft '1. '3 Dene the per-capita variables as k = KfL, y = 17L. and c = (TIL. However, instead of being a constant1 now the population growth rate is proportional to the marginal product of capital (It! PK ): L = n - 1U PK L where i P n :3 [1. Since the EU PK falls as capital increases, this captures the declining population growth rates of wealthier nations. {3) Derive the equation of motion for per-capita quantities of capital 5: E % = CHISEL}. (b) Determine the stead}.r state per-capita quantities of capital, output, consumption. and population growth rate. {as} Vv'hat are the growth rates of aggregate capital stock K. aggregate output Y. and aggregate consumption C in the steady state? (d) 1Ella'hat are the short run {transitional dynamics} and long run l{stead}.r 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 L I' \"-1 '9' = E = k\" E = n - MPK : firJzKa1L1\" : no- () : nailed1 '._ KLLK _ g _ g; Then R _ L3 _ L L L}