1、 Suppose that the production function is Cobb-Douglas ，n denotes the population growthrate ， denotes the depreciation rate ，g denotes the rate of technological progress. Andcapital and labor are paid their marginal products. In addition, suppose that all capital income is saved and all labor income is consumed. Thus[](,)/KF K AL K K K δ=∂∂- a) Show that this economy converges to a balanced growth path.b) Find expressions for and as function of the parameters of the model.c) Show that k on the balanced growth path equal to the golden rule level of k.d) Find the elasticity of output per unit of effective labor on the balanced growth path withrespect to the rate of population growth, n.2、 Consider a RCK economy that is on its balanced growth path. Suppose the government switches to a policy of taxing investment income at rate . Thus the real interest rate that households face is now given by r(t)=(1-). Assume that the government returns the revenue it collects from this tax through lump-sum transfers.a) Assume that at time 0, the government announces it will tax investment income after time 0. How does the tax affect thelocus? What arethe dynamics after time 0?b) Assume that at time 0, the government also announces that after time 1, investmentincome will again be untaxed. What are the dynamics after time 0?c) Assume that at time 0, the government announces it will tax investment income from time1 to time 2. Before time 1 and after time 2, investment income will not be taxed. What are the dynamics after time 0?3、 Consider the following model:Assumea)Show that the economy converge to a balanced growth pathb)What are the growth rates of H, K and Y on the balanced growth path? And show that thegrowth rates of K and Y are greater than that of H.4、Consider an economy consisting of a constant population of infinitely lived individuals. Therepresentative individual maximizes the expected value ofthe instantaneous utility function is, Assume that C is always in the range where is positive.Output is linear in capital, plus an additive disturbance: , there is no depreciation; thus , and the interest rate is A. Assume A=. Finally, the disturbance follows a first-order autoregressive process:, where and `s are mean-zero, i.i.d. shocks.a)Find the F.O.C. relating and expectations of .b)Guess that consumption takes the form . Given this guess, what isas a function of ?c)What values must the parameters have for the first-order condition in part(a) tobe satisfied for all values of K and e.d)What are the effects of a one-time shock to on the paths of Y, K and C?5、Using IS-MP model describe how the rise of foreign interest rates affects the economy underthe following conditions:a)Under floating exchange rates, perfect capital mobility, and static real-exchange-rateexpectationsb)Under floating exchange rates, imperfect capital mobility, static real-exchange-rateexpectations and the assumption that net exports are the only component of plannedexpenditures affected by the exchange rate.c)Based on part(b),but under fixed exchange rates.6、Consider an economy consisting of some firms with flexible prices and some with rigid prices.Let denote the price set by a representative flexible-price firm and the price set by arepresentative rigid-price firm. Flexible-price firms set their price after m is known; rigid-price firms set their prices before m is known. Thus flexible-price firms set, and rigid-price firms set , where E denotesthe expectation of a variable as of when the rigid-price firms set their prices.Assume that fraction q of firms have rigid prices, so that p=q+(1-q).a)Find , in terms of m, Em and the parameters of the model.b)Find y in terms of m, Em and the parameters of the model. Can it be a support for “LucasCritique”? Why or why not?c)Can Taylor Model be a support for “Lucas Critique”? Compare your answer in part(b)with the Taylor Model. What are the differences on the conclusion?d)Compare your answer in part(b) with the Phillips Curve which as followed:. What are the differences on theconclusion? What`s the reason?e)Compare your answer in part(b) with the Fisher Model. What are the differences on theconclusion? What’s the reason?。