please state your assumptions clearly

4. (20) Consider the standard growth model in discrete time. There is a large number of identical households normalized to 1. Each household wants to maximize life-time discounted utility U((chio) = Estu(a), BE (0, 1). Each household has an initial capital ko > 0 at time 0, and one unit of productive time in each period that can be devoted to work. Final output is produced using capital and labor, according to a production function, F, which has the standard properties discussed in class, most notably, it is increasing in both arguments and exhibits CRS. This technology is owned by firms (whose measure does not really matter because of the CRS assumption). Output can be consumed (c) or invested (i,). Households own the capital (so they make the investment decision), and they rent it out to firms. Let & e (0, 1) denote the depreciation rate of capital. Households own the firms, i.e., they are claimants to the firms' profits, but these profits will be zero in equilibrium. The function a also has the usual nice properties, which I will not spell out here since you will not need them explicitly. In this economy there is a government that collects taxes and (for simplicity) throws the tax revenues into the ocean. The government can implement one of the following two alternative taxation systems, let us call them System A and System B. System A is a proportional tax, 7 E (0, 1], on agents' capital income. In other words, if the government implements System A, it collects a fraction 7 of all the income that agents earn by renting out their capital to firms. System B is a proportional tax, r E (0, 1], on agents' investment. In other words, if the government implements System B, it collects a fraction r of all the resources that agents choose to allocate into investment. (a) Write down the problem of the household recursively, under both taxation systems.' Pay special attention to the budget constraints. These constraints will not be the same under the two specifica- tions. Also, notice that I am not asking you to define a RCE in detail; just state the representative agent's problem within a RCE environment. (b) Describe the steady state equilibrium capital stock under taxation System A, for any given 7 e [0, 1]. Denote this object by KA(T). (c) Describe the steady state equilibrium capital stock under taxation System B, for any given 7 E [0, 1]. Denote this object by KE(T). (d) Assume that F(K, N) = K" NI-, a E (0, 1). Provide closed form solutions for the terms KA(T), K(T), described in parts (b),(c). Hint: Here, it is more convenient to work directly with F, i.e., do not work with the auxiliary function f that we introduced in the lectures. (e) Plot the terms KA, KB, calculated in part (d), against 7 6 [0, 1] and in the same graph. Discuss briefly. (f) Describe the government's total tax revenue in steady state under System B, Ts. Plot To as a function of the tax rate 7 (this is the so-called Laffer curve). Discuss the shape (i.e., the monotonicity) of the Laffer curve for the various values of a and T.3. (20) Consider a standard optimal growth model in continuous time in which the aggregate production function is given by: Y (t) = FIK (t) , N ()] where F () has standard properties and N (t) is growing at the rate n > 0.The depreciation rate of capital is given by 6 > 0.The single household inelastically supplies labor each period and then chooses consumption and savings in order to maximize: -MU (c () N (1 ) at where c (t) = NO 190 is per-capita consumption and U (.) has the functional form: U (c(t)) = 1-6:041 Inc (t) ; 0 = 1 It is assumed that all parameter values are such that a well-behaved equilibrium exists. In addition to output produced via the production function, output arrives exogenously every period at the rate of d units per person. Given this environment, do the following: (a) Express the social planner problem in intensive (i.e. per-capita) form - show your derivation. (b) Write down the Hamiltonian for this problem and derive the necessary conditions; include the transversality condition. (c) Derive the phase diagram for this economy - be sure to explain your derivation. (d) Define the steady-state. What fraction of the exogenous output, o, is consumed in steady-state? Why?1. (10) Each period, an infinitely-lived agent divides his endowment of 1 unit of time between human capital production (a non-market activity) and work in order to maximize the present discounted value of lifetime earnings. Let h, denote the human capital stock at time t and let 1 -4 be time spent working. Income each period is given by My (1 - 4) w where w is the rental rate of human capital. New human capital (i.e. investment in human capital) is produced via the production function (h,4)"; are (0, 1). Note that human capital depreciates at the rate o and the real interest rate is given by the constant, r. (a) Express the agent's maximization problem as a dynamic programming problem and identify the states and controls. (b) Derive and interpret the Euler equations associated with this problem. (c) Assume that a steady-state exists so that he = h and 4 = 1. Solve for these steady-state values. (d) What is the impact of the two prices (w, r) on the steady-state values? Explain. 2. (20) Consider a representative agent economy in which preferences are given by: * ( [ - arm ] ) . BE (0 . 1) . 720 where E denotes the expectations operator and , is an i.i.d. random variable which affects the disutility of labor supply. Output in the economy is produced by firms that use labor and an inelastically supplied unit of non- depreciating capital (owned by firms - you can think of this as land). The choice of labor is made to maximize profits each period: max N = It - wthe he Me = h hi-", aE (0, 1) where y denotes output and w is the wage. The profits are returned to the households. In addition to labor supply, agents also trade one period bonds (risk-free) that cost p at time t and return 1 unit of consumption in period t + 1. Given this environment, do the following: (a) Express the household's problem as a dynamic programming problem and derive the associated necessary conditions. Note that households take as given firm profits, the wage and the price of bonds (i.e. it is a standard competitive economy). (b) Find the competitive equilibrium allocation by solving the social planner's problem for this economy. (c) Solve for the policy functions which describe equilibrium consumption and labor. Provide an expla nation for the implied behavior of these variables. (d) Determine the solution for the equilibrium price of bonds. Explain how the preference shock, , affects the price of bonds