(a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming

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1 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 function takes the form Y (t) = zk (t) α G (t) 1 α. The social planner solves the problem max {c(t),g(t), k(t)} 0 exp ( ρt) c (t)1 σ 1 σ dt s.t. k (t) = zk (t) α G (t) 1 α δk (t) c (t) G (t), c (t) 0. (a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming problem. (b) Derive the optimal G (t) as a function of k (t). (c) Derive the Euler Equation. (d) Show that the model features a constant growth rate over time. (e) Solve the optimal consumption at t (c (t)) as a function of k (t).

2 2. Uncertain Business Cycles 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 log (c t ) η n1+γ t, 1 + γ t=0 where c t and n t are private consumption and labor hours, respectively, and η > 0. The production technology is characterized by a standard Cobb-Douglas production function: y t = z t kt α nt 1 α. The aggregate resource constraint reads: c t + k t+1 + G = y t + (1 δ) k t, where government purchases (G) is exogenous and time-invariant, and δ (0, 1] is the depreciation rate of capital. The TFP z t is stochastic and evolves according to: log (z t+1 ) = ρ z log (z t ) + σ t ɛ t+1, where {ɛ t+1 } t=0 is an independent and identically distributed sequence of shocks drawn from a standard normal distribution N (0, 1) and 0 ρ z < 1. Unlike the standard TFP shock process, the conditional standard deviation term, σ t, is stochastic and follows an AR(1) process log (σ t+1 ) = ρ σ log (σ t ) + u t+1, where {u t+1 } t=0 is an independent and identically distributed sequence of shocks drawn from a standard normal distribution N (0, 1), and 0 ρ σ < 1. At the beginning of t = 0, k 0 > 0 is given, and the household observes z 0 and σ 0 before making decisions. (a) (i) Write the Bellman equation of the social planner who maximizes the welfare of the household. (ii) Derive the FOCs and the Euler Equation of social planner s problem. (iii) Is it possible for the social planner to choose n = 0 or n = 1 at the optimal solution? (b) Suppose that the average behavior of the real-world data can be captured by the one-sector model in Part (a). Design a procedure to calibrate the government purchases parameter G.

3 (c) For this part only, assume that δ = 1 and G = 0. Solve the value function of the social planner s dynamic programming problem. (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 z t k d α ( ) t n d 1 α } t rt kt d w t n d t. {kt d,nd t} The household s budget constraint in a competitive equilibrium is c t + k t+1 = (1 τ t ) ((r t δ) k t + w t n t ) + k t, where τ t [0, 1] is the income tax rate. Notice that the capital depreciation is tax-deductible. The government tax and spending policies are subject to a balanced-budget requirement, i.e., the tax revenue needs to equal to G in every period. 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. (e) Suppose that the fiscal authority chooses the optimal time-invariant G to maximize the welfare of the representative household in the recursive competitive equilibrium in Part (d). Without doing any actual calculation, discuss the optimal choice of G and give economic intuition.

4 3. Three Overlapping Generations Consider a pure-exchange economy with three 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, ω m units when middle-aged, and ω o units when old. The initial old consumer has utility function log c 2 0 and is endowed with amount m 2 of fiat money. The initial middle-aged consumer has utility function log c log c 1 1 and is endowed with amount m 1 of fiat money. The consumer born in period t = 0, 1,... has utility function log c t t + log c t t+1 + log c t t+2. (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. (d) For each consumer, specify the equation(s) needed to solve for the consumer s savings function(s) in terms of interest rates, parameters, and the initial price level only. (e) Specify how to use the savings functions to calculate the sequential markets equilibrium taking the initial price level as given.

5 4. Job Search Consider the problem of an unemployed worker. Each period the worker receives a job offer from cdf F (w), w [0, B]. The worker can either accept the offer, in which case the worker gets that wage forever, or the worker can reject the offer and keep searching. There is an unemployment benefit b. The worker has linear utility and discount factor β. (a) Specify the worker s problem as a recursive dynamic program, characterize the solution in terms of a reservation wage, and sketch the value function. (b) Show that w b = β 1 β where w denotes the reservation wage. B w (w w)df (w ), (c) Prove that an increase in the unemployment benefit leads to an increase in the reservation wage. (d) Solve for the average number of periods that a worker is unemployed before accepting a job as a function of the reservation wage. (e) Now suppose that an employed worker is laid off with probability δ at the end of each period. Specify the worker s problem as a recursive dynamic program and characterize the solution in terms of a reservation wage.