1. [45 points) Consider an economy where the production function is given by Y; = K3313? where is a constant value representing total factor productivity, Kt is the capital stoclt in time period t and L; is the total population in time period t. The law of motion of capital is given by Iii-{+1 = [1 Ejf + s, where 5 is the rate of depreciation and s is the savings rate. The population growth rate is n = [Lil-4, rate of depreciation is d = {1.1 and savings rate is s = {1.2 {a} [5 points} Divide the production function by the level of population to get per capita production function. Does the per capita. production function exhibit diminishing returns with respect to per capita capital? {b} [5 points] Rewrite the law of motion of capital in per capita terms. {c} [If] points} Does this eoonomy have a steady state level of per capita capital and per capita income? If yes, compute the steady state values of per capita. capital and per capita inoonie. {d} [5 points} Plot the phase diagram for the law of motion and plot the steady state value of per capita capital. {e} [5 points] Does this economy exhibit long run growth? If yes, why? If no, why not? {f} [ll] points} Now suppose that the population growth rate is an endogenous function of per capita capital, where: _ am as a: 2 \"(hi _ {run 1st }= 2 How many steady states will you have? What are they? {g} [5 points] If you were in the steady state obtained in part (c), would a donation of [1.3 per capita capital nnits push the economy to a better long on] per capita output? If yes, why? If no, why not? Would a donation of [1.5 per capita capital units do the job