Econ 680 Sample Questions
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Econ 680
Sample Questions
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❼ Below are sample exam questions. These questions are taken from past exam questions.
❼ I will share an answer key a few days before the exam. I will NOT reply to e-mails requesting
for an earlier release of the answer key.
❼ The purpose of the answer key is help you check your answers. The answer key will have
little value if you do not attempt these questions beforehand.
❼ The actual exam may have fewer questions.
❼ Below are instructions you will find on the cover page of the actual exam. Make sure to read
them at least once before the actual exam.
– You may not use your books or notes. You may use a simple calculator to answer nu- merical questions.
– Round numbers (if needed) to 4 decimal places for greater accuracy.
– You are required to show your work on each problem on this exam.
– Mysterious or unsupported answers will not receive full credit. A correct an- swer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit.
Life Expectancy: As life expectancy increases, people tend to save more for retirement (holding income and the real interest rate constant). Suppose that this is the case in an imaginary country called Isoland.
(a) (2 points) Consider the capital market equilibrium described by capital supply and capital
demand functions. How does the increase in life expectancy affect the capital market equilibrium outcomes? Explain your answer using a well-labeled diagram of the capital market.
(b) (5 points) In this part, you are asked to analyze this problem in a two-period model with
consumption and saving. Assume that income is exogenous in both periods and denoted by y1 and y2 . Assume that the representative consumer’s preferences are given as follows:
u(c1 ,c2 ) = log(c1 ) + β log(c2 ), (1)
where β captures how much individuals value future consumption relative to today’s con- sumption, and log(.) is the natural logarithm. Find the optimal savings as a function of the real interest rate, r .
(c) (3 points) How would you justify your answer in part (a) using the model developed in part (b)?
Lucas’ Tree: Consider an economy with infinitely lived identical households where apple is the only good that is produced and consumed. Throughout the question, assume that every price is relative to the price of an apple and there is no inflation. To simplify the analysis, you may impose that the price of the apple is always 1, i.e. Pt = 1 for every period t. Because the households are identical, I describe the environment below for a representative household.
Suppose that there is only one type of assets: an apple tree. Let at denote the quantity of the apple trees the representative household owns at the end of period t and St denote the price of the apple tree (in terms of apples) in period t. The representative household can freely buy and sell apple trees in the financial markets. Each apple tree produces Z apples at the beginning of every period that is constant over time. Specifically, if the representative household has at−1 apple trees at the beginning of period t, he collects Zat−1 apples. Note that he does not necessarily consume Zat−1 apples in period t. He is free to buy and sell apples in the goods market but he cannot store his apples. The consumer also makes a lump-sum tax payment of Tt apples to the government. Finally, there is no other form of income, expenditure or asset except for the ones indicated above.
The representative household derives utility from consumption of apples, ct, and maximizes life time utility subject to a sequence of period budget constraints. The period utility function has the (natural) log form: u(ct) = log(ct). Utility from future consumption is discounted by β < 1.
(a) (6 points) Set up the Lagrangian for the utility maximization problem. Indicate the choice
variables of the representative household.
(b) (6 points) Obtain the first order conditions for period t variables of the Lagrangian you
described in the previous part.
For the rest of the question, focus on the steady state equilibrium and assume the following. β = 0.9, Z = 100, T = 10 . The population is normalized to one, i. e . there is only the representative household. There are 10 apple trees and all of them are owned by the representative household in equilibrium.
(c) (8 points) Calculate the steady state price of the apple tree, S . Explain your answer carefully.
(d) (6 points) What is the steady state level of consumption, c?
Habit Persistence: Habit persistence is meant to indicate that individuals become “habit- uated” to previous levels of consumption. To capture this behavior, suppose that the utility derived in period-t depends on the consumption level in period-t-1. For example, we can write the life time utility function as follows:
∞
U = 工 βiu(ct+i,ct− 1+i)
i=0
(2)
Note that when a consumer arrives in period t, ct−1 cannot be changed because it happened in the past.
(a) In a model in which a stock can be traded every period (such as the one we studied in class), how is the pricing equation for St, the nominal stock price, altered due to the assumption of habit persistence? In particular, discuss how consumption decision affects the pricing kernel with habit persistence.
(b) Suppose that the period utility function is u(ct,ct−1,ct−2). How is the pricing equation
for St, the nominal stock price, altered due to the assumption of habit persistence? Does habit persistence imply more forward-looking behavior or backward-looking behavior, or both? Explain your answer briefly.
Optimal Tax on Revenue: Consider a two-period consumption-saving model with govern- ment and production. The representative household’s preferences are given by the following utility function:
U(c1 ,c2 ) = log(c1 ) + log(c2 ) (3)
The household inelastically supplies one unit of labor in both periods (n1(s) = n2(s) = 1) and earns
w1 and w2 in period 1 and 2, respectively. There is a perfectly competitive representative firm
which produces according to 100 ^ni(d), where ni(d) is the demand for labor in period i = {1, 2}.
Note that the wage rate is determined in equilibrium in the labor market.
Assume that government spending is equal to 10 in the first period, g1 = 10, and 22 in the second period, g2 = 22. All of the variables are indicated in terms of the consumption good.
Suppose both the household and the government start with zero initial assets, and the real interest rate is always 10% regardless of the changes in the tax policy.
For the purposes of this question, also assume that the government can only collect tax from the representative firm. More specifically, the government imposes a proportional tax on the
firm’s revenue, i.e., for each unit produced and sold, the firm pays τi fraction of its total revenue
to the government so that its after-tax revenue is equal to (1 − τi)100^ni(d) .
(a) (10 points) If τ1 = 0, what value of τ2 satisfies the government’s lifetime budget con-straint?
(b) (10 points) Suppose now that the government is free to impose a proportional tax on the firm’s revenue in both periods, i.e. τ1 is not necessarily equal to zero. Solve for the equilibrium optimal consumption in both periods as a function of τ1 and τ2 .
(c) (10 points) Suppose that the government wants to maximize the representative house- hold’s utility by choosing τ1 and τ2 subject to its own lifetime budget constraint. Set up the Lagrangian for this problem and calculate the optimal tax rates. Is the consumer better off compared to part (a)? Briefly comment on the Ricardian equivalence.
Lump-sum vs. Proportional Tax: Consider the two-period model with consumption and leisure decisions. Assume that a representative individual has the following utility function:
U = u(c1 ,n1 ) + 0.9u(c2 ,n2 ). (4)
and
u(ct,nt) = ct − ,
(5)
where and ct and nt denote consumption and labor supply of the individual (potentially ex- ceeding 1) for t = {1, 2}. Assume the real wage rate (in terms of the consumption good) is equal to 1.1 in the first period and 1 in the second period. The individual pays a labor income tax in both periods, which is equal to τ in both periods. He starts his life with no assets and can freely save and borrow in the first period. Let a denote his savings at the end of the first period. He receives interest income r for his savings at the beginning of the second period. At the end of the second period, he leaves no assets behind, i.e. no savings in the second period.
(a) Write down the budget constraint for each period and obtain the life-time budget con-straint.
(b) Set up the Lagrangian for the utility maximization problem and obtain the first order conditions.
(c) What is equilibrium interest rate, r?
(d) What value of τ maximizes the present discounted value of government tax revenue?
(e) Suppose that the present discounted value of government spending is equal to 0.48. What
value of τ generates this tax revenue and maximizes consumer utility?
(f) Suppose that the present discounted value of government spending is equal to 0.48 and the
government replaces the labor income tax in both periods with a lump-sum tax T = 0.48 in the first period. Is the consumer better off? Clearly show your work.
2022-10-02