Macro Qualifying Exam
June 22-23, 2015
Instructions.
The exam consists of two parts. Please answer 3 out of 4 questions in Part I, and 8 out of 9 questions in Part II. Start your answer to each question on a fresh sheet of paper. Clearly label the problem number and your assigned ID at the top of each page.
Try to answer as many parts of each question as possible. It is OK to skip part of a question and still try to answer later parts to the extent this is possible. Nevertheless, verbal answers that do not engage the math (when math is expected) will receive little or no credit.
You are strongly encouraged to work out your initial algebra attempts on scrap paper so your final answer is clean and easy to grade; your final answer should nevertheless include all relevant steps. Messy or confusing answers will be marked down.
Keep in mind, you will not receive any credit for answering a different question than the one being asked.
For this reason, it is very important that you read each question carefully . Be as precise in your answers as possible.
You are encouraged to read over all questions for each part before choosing which ones to answer.
You have 5 hours to complete the exam. Part I will count for two-thirds of the total points and part II will count for one third, so please allocate your time wisely. Try not to spend too much time bogged down on any one question; you are better off moving on and trying to return to it later.
Good Luck!
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Exercise 1 Consider the problem of maximizing
∞
X
β t ln ( c t
) , 0 < δ < 1 , t =0 subject to k t +1
= Dk
α t
+ (1 − δ ) k t
− c t
, for all t ≥ 0 , 0 < α < 1 , D > 0 k
0
> 0 given.
1. (10 points) Write the Bellman equation so the choice variable “today” is the value of the state variable
“tomorrow”. Use “primes” to indicate the value of variables in the following period.
2. (10 points) Define the Bellman operator (i.e., with math). In what way is it an “operator”? In particular, What are its inputs and what are its outputs?
For parts 3 through 6 below, assume that δ = 0 in the problem above.
3. (20 points) Starting from an initial guess V (0) ( k ) = 0, iterate twice on the Bellman operator. Do this analytically with algebra. The result of this process should give you V ( e ) ( k ).
4. (10 points) Next, guess that the value function takes the form V ( k ) = Ak α , where A is a constant in
R
. Use guess and verify to determine if this guess is correct.
5. (10 points) Next, guess that the value function takes the form V ( k ) = a + b ln( k ), where a and b are both constants in
R
. Use guess and verify to determine if this guess is correct.
6. (25 points) Using the correct guess, derive an expression for the policy function, written in terms of fundamental parameters of the model.
7. (15 points) Next, assume, in contrast to above, that 0 < δ < 1. Now repeat the guess and verify exercise above using the correct guess used in part 6. Does the guess work? Go as far as you can.
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Exercise 2 Consider the continuous-time Solow model where population, N , grows at constant rate n and
TFP, A , grows at constant rate a . Let K denote the aggregate stock of capital, and let
F ( K, N, A ) = ( AN )
1 − α
K
α be the aggregate production function.
Moreover, let v be the savings rate, and let δ be the depreciation rate for the capital stock. The fundamental (aggregate) differential equation in Neoclassical growth theory is then
˙
= v ( AN )
1 − α
K
α
− δK .
1. (15 points) Define m =
K
. Showing all relevant steps, derive the corresponding differential equation
AN governing the evolution of m.
2. (15 points) What happens to m in the long run? Justify your answer fully.
3. (15 points) Solve explicitly for the long run level of m.
4. (15 points) What happens to capital intensity (
K
L
) over time? Be specific and justify your answer.
5. (10 points) What does the model predict about the ability of poor countries to catch up to rich countries? Be specific and justify your answer.
6. (10 points) Define a balanced growth path. Does the model generate balanced growth?
Be specific and justify your answer.
7. (10 points) What happens to the model economy in the long run if a = n = 0? Be specific and justify your answer.
8. (10 points) What does the model have to say about the fundamental determinants of economic growth?
Be specific and justify your answer.
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Exercise 3 Consider the following decentralized Neoclassical economy. The representative household has instantaneous preferences over consumption streams represented by the following utility function:
 u [ c ( t )] =

 c ( t )
1 − θ − 1
, θ = 1
1 − θ ln c ( t ) , θ = 1
Households consume and save in order to accumulate wealth a ( t ) over an infinite horizon, given an initial stock of wealth a (0) > 0. They supply labor inelastically to firms, and discount the future at a rate ρ > 0.
There is a large number of identical firms producing according to a Cobb-Douglas( α ) production function using capital and labor. Capital does not depreciate. There is no technological change, and we denote total factor productivity by A > 0. All firms are price-takers in factors market, and the market prices for labor and capital are denoted by w ( t ) and r ( t ) respectively. Given an initial size N (0), the household size grows at an exogenous rate n > 0. Government taxes proportionally the rate of return on assets, r ( t ), at a constant rate τ ∈ (0 , 1), and remits the proceedings lump-sum to the private sector. Denote such lump-sum transfers as z ( t ). The government runs a balanced budget.
1. (10 points) Write the representative household’s budget constraint, and use it to write down the representative household’s infinite horizon problem. What is the terminal condition that you must impose on wealth if households are allowed to borrow?
2. (25 points) Set up the current-value Hamiltonian for the problem. Write down the first-order necessary conditions for an optimal control solution to the infinite-horizon problem, and use them to write down the consumption Euler equation. What is the role of the so-called transversality condition in the solution?
3. (20 points) Define a competitive equilibrium for this economy, making sure to include in your definition the equilibrium allocation of labor, capital, consumption and wealth. What must be the relation between a ( t ) and capital per worker k ( t ) at a competitive equilibrium? Will the rate of return to wealth be different from the rate of return to capital? Why or why not?
4. (15 points) Write down the dynamical system describing the economy at a competitive equilibrium.
Solve for steady state values of consumption and capital stock per worker. How does the tax rate affect the steady state values k ss
, c ss
?
5. (20 points) Linearize the dynamical system you obtained above around the steady state. Either qualitatively, or by actually calculating the eigenvalues of the Jacobian matrix of the system, evaluated at the steady state, show that the steady state is saddle-path stable.
6. (10 points) Suppose that the economy starts off the steady state. Describe in words how, given the initial condition on capital stock per worker k (0), the economy approaches the steady state. Is there any jump variable in this model? What is its role in the process?
4

Exercise 4 Consider the following one-sector growth model in continuous time indexed by t ∈ [0 , ∞ ). At each t , the representative household has preferences over consumption streams given by
 u [ c ( t )] =

 c ( t )
1 − θ − 1
, θ = 1
1 − θ ln c ( t ) , θ = 1 and maximizes its lifetime utility, discounted at a rate ρ > 0.
Labor supply is inelastic to the wage.
Population is constant, and capital stock does not depreciate. Output per capita is produced according to: y ( t ) = Ak ( t )
α
γ ( t )
1 − α
, α ∈ (0 , 1) (1) where A > 0, k denotes capital per worker and γ is a flow of government spending (per worker) on public services. Public expenditure is financed by a proportional tax on income (output) τ , which is paid by consumers. The government cannot borrow, hence its budget must always be balanced. Assume from the start that consumers accumulate no debt, so that their assets are equal to capital stock at all times.
Consumers treat the amount of public good γ ( t ) as a given each period.
1. (20 points) State the infinite-horizon utility maximization problem for the representative consumer, and write down the first order necessary conditions for an optimal control using the current value
Hamiltonian.
2. (15 points) (i) Define a competitive equilibrium (CE) for this economy. (ii) Use the balanced budget requirement to solve for the growth rate of consumption, which will be equal to the growth rate of output per worker along a balanced growth path (BGP), as a function of A, α, τ, θ, ρ .
3. (15 points) Show that, at a CE, output per capita is given by an AK-type production function: (i) the marginal product of capital is constant, and (ii) total factor productivity depends on the tax rate.
4. (15 points) (i) Find the tax rate that maximizes the growth rate of consumption per capita, equal to the growth rate of real GDP per worker at a BGP. What would happen if τ = 0? And if τ = 1?
(ii) Suppose the public good did not enter the aggregate production function. What would be the growth-maximizing tax rate in this case?
5. (20 points ) State and solve (that is, find the BGP growth rate corresponding to the solution of) the problem of a social planner maximizing the present discounted value of the representative consumer’s utility taking into account that the public good amount is not given, but depends on income per worker.
6. (15 points) (i) Comparing the solution to the planner’s problem with the CE solution, show that the growth rate you found at a CE is below the efficient growth rate, and provide intuition for the result. (ii)
How does the growth-maximizing tax rate at the efficient path compare with the growth-maximizing tax rate at a CE?
5

Your answer to each question in part 2 should not exceed 15 lines, not including equations.
1. Explain in words what the Bellman operator is. What do we do with it? Why?
2. Consider the following empirical question: What portion of US economic growth over the past decade is attributable to population growth?
What empirical approach could be used to answer this question?
What data would be needed to apply the approach to the posed question? Be specific.
3. Describe the regression equation estimated by Mankiw, Romer and Weil (1992). What research question did the authors seek to answer? What data do they use?
4. What is a stochastic process? What mathematical assumptions do we need to impose on such a process to incorporate it into a dynamic programming problem?
5. Explain the reason why, by assuming constant returns to scale in production together with perfectly competitive product and factor markets, the Neoclassical growth model is not able to generate endogenous growth.
6. What is the main reason why the labor market does not clear when there is a matching function representing the trade process occurring between job seekers and firms with vacancies? Explain.
7. Consider the basic Schumpeterian growth model (Aghion and Howitt, 1990). Can there be excessive growth in equilibrium? Why or why not?
8. By referring to the Galor and Zeira (1993) model, explain why wealth inequality can generate constraints to human capital accumulation, and affect economic growth through this channel.
9. Consider the basic partial equilibrium model of investment by Tobin (1969). What is the main variable determining investment demand in this model? Does it have anything to do with the interest rate?
Explain why, in this sense, the Tobin model provides a microeconomic foundation to the basic idea of autonomous investment one finds in Keynes’ General Theory.
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