problem. max Both k (0) and h (0) are given at time 0. (a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
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1 1. Endogenous Growth with Human Capital Consider the following endogenous growth model with both physical capital (k (t)) and human capital (h (t)) in continuous time. The representative household solves the problem max {c(t), k(t),ḣ(t)} 0 exp ( ρt) c (t)1 σ 1 σ dt s.t. k (t) + ḣ (t) = zk (t)α h (t) 1 α δk (t) δh (t) c (t), c (t) 0. Both k (0) and h (0) are given at time 0. (a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming problem. (b) Derive the Euler Equation for this problem. (c) Assume that at t = 0, the marginal product of k (0) and h (0) are equal to each other. Derive the restriction on k (0) and h (0). (d) Show that under the restriction in Part (c), the model features endogenous growth with a constant growth rate over time. (e) Under the same conditions that apply in Part (c) and Part (d), solve the optimal consumption at t (c (t)) as a function of k (t).
2 2. Capital and Labor Taxation with Capital Utilization Consider a one-sector real business cycle model. The household is endowed with 1 unit of time in every period. The preference of the representative household is given by: E 0 β t [η log (c t ) + (1 η) log (1 n t )], where c t and n t are private consumption and labor hours, respectively, and η (0, 1). The production technology is characterized by a Cobb-Douglas production function with endogenous capital utilization: y t = F (k t, n t, z t, h t ) z t (h t k t ) α n 1 α t. Here the continuous variable h t [0, 1] measures the utilization level of capital stock in t, and it is a choice variable in every period. The aggregate resource constraint reads: c t + k t+1 + G = y t + (1 δ (h t )) k t, where G is an exogenously-given constant government expenditure in every t. Unlike the standard RBC model, the depreciation rate of the capital is endogenously determined from a function: h ω t δ (h t ) = δ 0 + δ 1 ω, where δ 0 0, δ 1 0 and ω > 1. Thus, the depreciation rate of capital in period t is an increasing and convex function of the utilization level h t. The TFP z t is stochastic and evolves according to: log (z t+1 ) = ρ log (z t ) + ɛ t+1, where {ɛ t+1 } is an independent and identically distributed sequence of shocks drawn from a normal distribution N (0, σ 2 ɛ) and 0 ρ < 1. At the beginning of t = 0, k 0 > 0 is given. (a) Write the Bellman equation of the social planning problem of maximizing the welfare of the household. (b) Derive the FOC of choosing h t in Part (a). From the FOC in h t, derive the optimal h t as a function of (z t, k t, n t ) (assuming an interior solution of h t ).
3 (c) Suppose that the average behavior of the real-world data can be captured by the social planner solution. Assume that you have observed (h t, k t, n t, y t ) for a long period of time. Design a procedure to calibrate the production parameter (α) and TFP shock process (ρ). (d) Consider a sequential-trading environment with a fiscal authority. The household owns the capital, and supply capital and labor to the firm in each period. The representative firm maximizes one period profit by solving the problem max {k d t,nd t,ht} { z t ( ht k d t ) α ( n d t ) 1 α δ (ht ) k d t r t k d t w t n d t where δ (h t ) = δ 0 + δ 1 h ω t ω. Notice that we have assumed that the firm pays the depreciation ( δ (h t ) k d t ) after production. The household s budget constraint in a competitive equilibrium is c t + k t+1 = ( 1 τ k t ) rt k t + (1 τ n t ) w t n t + k t, where τ k t [0, 1] and τ n t [0, 1] are capital and labor income tax rates, respectively. The tax and spending policies are subject to a balanced-budget requirement, i.e., the tax revenue needs to equal to G in every period. Suppose that τ k t is determined through an exogenous fiscal rule: τ k t = Ψ k (K t, z t ), where K t is the aggregate capital stock in period t. τ n t is endogenously determined to balance the budget in t. Carefully define a recursive competitive equilibrium for this economy. Be sure to write down the household s dynamic programming problem, the firm s FOC, the government problem, and the equations which determine the pricing functions. You do NOT need to solve the pricing functions explicitly. (e) Suppose that at time 0 the fiscal authority is given a one-time opportunity to deviate from the fiscal rule Ψ and choose { τ k 0, τ n 0} to maximize the welfare of the representative household. The economy is otherwise the same as that in Part (d). In particular, the government still balances its budget for every t 0, and all future tax rates (i.e., for t 1) will again follow the fiscal rule in part (e). Without doing any actual calculation, discuss the optimal choice of { τ k 0, τ n 0} and give economic intuition. },
4 3. Inequality and Growth in a Neoclassical Model Consider an infinite-horizon neoclassical growth model with a continuum of households of two types, i = 1 and 2. Each household type represents 1 unit of the economy s population, and shares the same time-separable preference with discount factor β (0, 1): U ( c i) = β t log ( ct) i. Every type is endowed with one unit of labor (n = 1) in every t and supply it inelastically. There is a continuum of representative firm, with the production function Y t = zk α t N 1 α t, α (0, 1), where z is the exogenous technology, and K t and N t are the aggregate capital and labor used by the firm, respectively. The resource constraint at time t is C t + K t+1 = Y t + (1 δ) K t, where C t = c 1 t + c 2 t is the aggregate consumption, and δ (0, 1] is the capital depreciation rate. At the beginning of t = 0, K 0 > 0 is given. (a) Consider the social planner s problem. The social planner maximizes a social welfare function λ 1 U ( c 1) + λ 2 U ( c 2), or equivalently, β [ t λ 1 log ( ) c 1 t + λ 2 log ( )] c 2 t, ( where λ i is a ) positive weight with λ 1 + λ 2 = 1. The social planner chooses {c i t} 2 i=1, K t+1 subject to the resource constraint. Write the Bellman equation of the social planning problem and derive the Euler equation. Argue that the optimal solution of aggregate capital (K t+1 ) t=1 is identical to the representativeagent Ramsey growth model ) with the same K 0. Find the steady state of the economy ({c i } 2i=1, C, K. (b) Suppose that the average behavior of the real-world data can be captured by the social planner solution. Assume that you have observed ({c it} ) 2i=1, C t, K t, Y t for a long period of time. calibrate { λ i} 2. i=1 From your solution in Part (a), design a procedure to
5 (c) Let the capital depreciate fully in every t, i.e., δ = 1. Find the policy function of optimal capital accumulation in the planner s problem in closed form. (d) Consider a competitive equilibrium setup with a fiscal authority in discrete time. Like in the standard model, the households own the capital and labor and rent them to the firm in each period. The competitive firm makes decisions of renting capital and labor to maximize one-period profit. The fiscal authority chooses income tax rate (τ t ) and transfer (T R t ) for each t, subject to a balanced-budget requirement T R t = τ t Y t. τ t is determined through an exogenous constitution rule τ t = Ψ (K 1 t, K 2 t ), where K i t is the average capital stock for type i in period t. Like in a representativeagent model, notice that K i t is equal to k i t in equilibrium in every t, but a (small) household in type i may not take this into account in making its own decision on ( k i t+1 ). The budget constraint of the household i is c i t + k i t+1 = (1 δ) k i t + (1 τ t ) y i t T R t, where y i t is the total pre-tax income of the household i. The initial capital (k 1 0, k 2 0) is given, with 0 < k 1 0 < k 2 0 and k k 2 0 = K 0. Carefully define a recursive competitive equilibrium for this economy. Be sure to write down households dynamic programming problem and functional forms of pricing functions explicitly.
6 4. Overlapping Generations with Growth in Population and Money Consider a pure-exchange economy with two overlapping generations in each period. There is a single nonstorable consumption good in each period. Each consumer is endowed with ω y units of the good when young and ω o units when old. Each initial old consumer has standard utility function v(c 1 0 ). Each consumer born in period t = 0, 1,... has standard utility function u(c t t, c t t+1). Let N t denote the measure of consumers born in period t; it evolves according to N t = nn t 1. Let M t denote the aggregate supply of fiat money in period t; it evolves according to M t = gm t 1. Newly printed money is equally distributed among the old consumers in each period, so each old consumer in period t is endowed with amount (M t M t 1 )/N t 1 of fiat money. (a) Define an Arrow Debreu equilibrium. (b) Define a sequential markets equilibrium. (c) Characterize the solution to each consumer s problem in a sequential markets equilibrium. (Be sure to eliminate any Lagrange multipliers.) (d) Consider a stationary equilibrium. Specify the 5 equations that the objects {c y, c o, b, r} must satisfy. Briefly argue that there are two stationary equilibria. (e) Characterize the golden rule allocation (that is, set up the problem of maximizing the utility of some future generation subject to the stationary resource constraint and characterize the solution). Which stationary equilibrium satisfies the golden rule? If the supply of money is not zero in that stationary equilibrium, what is the value of g?
7 5. Cash-Credit Goods Economy and the Friedman Rule Consider a standard cash-credit goods economy. Good 1 must be purchased with cash, while good 2 must be purchased with credit. Bonds are not state-contingent and earn gross return R(s t ). At the beginning of each period, assets are traded in a centralized securities market. Then money is used to purchase good 1. Finally, labor is paid in cash. The representative consumer has expected utility function s βt π(s t ) ( log c 1 (s t ) + log c 2 (s t ) l(s t ) 2 /2 ). t The resource constraint is c 1 (s t ) + c 2 (s t ) + g(s t ) = z(s t )l(s t ). The government purchases goods g(s t ), taxes labor income at rate τ(s t ), and finances changes in money supply M(s t ) through open-market operations. (a) Define a sequential markets equilibrium. (b) Characterize the solution to the consumer s problem. (You need not eliminate the Lagrange multipliers.) (c) Derive the implementability constraint. (d) Set up the Ramsey problem and characterize the solution. (e) Prove that the Friedman rule (R(s t ) = 1) is optimal.
(a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Government Purchases and Endogenous Growth Consider the following endogenous growth model with government purchases (G) in continuous time. Government purchases enhance production, and the production
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