Problem Set
Instructions: You need to show enough details of your work to receive full credit. When asked to draw a graph, label things clearly. (I recommend you to draw graphs by hand. Do not copy and paste any graphs from the lecture notes, otherwise you will receive zero credit.)
1 Applying the Theory: Income Tax and Labor Supply [15]
1. Based on the 2004 paper by Edward Prescott, what is the main reason for the differ- ences in average hours worked between European countries and the U.S.? Explain.
2. Laffer Curve:
(a) What is Laffer Curve?
(b) What is the shape of Laffer Curve? Draw it in a graph (You do not need to draw it accurately, just the shape in general.)
(c) What are the tax policy implications of Laffer Curve? (This is a relatively open question.)
2 The Two-Period Model of Households [60+5]
Consider a household who lives for two periods: the current period and the future period. The income of the household is y in the current period and y' in the future period.
In both periods, the household has to pay lump-sum taxes to the government. The tax liability is t for the current period and tj for the future period. A credit market is available to the household, so the household can lend (save) or borrow in the current period with an interest rate r.
The household will receive or pay back the principal plus interest in the future period. The amount of savings in the current period is denoted by s. (A negative s implies the household is borrowing.) The household's preferences over the current con- sumption c and the future consumption c' are represented by a utility function U(c, c'), which is both monotonic and concave.
1. In the (c, c') space, draw two of the household's indifference curves with different utility levels. What are the properties of the indifference curves? Label clearly which one of the two indifference curves represents a higher utility level.
2. What is the household's budget constraint for the current period?
3. What is the household's budget constraint for the future period?
4. Derive the household's lifetime budget constraint by combining the budget con- straints of both periods and eliminating the savings s.
(Hint: For each period bud- get constraint, try to put savings s on one side and everything else on the other side, and then connect the two resulting inequalities through s. In the end, it is better to put all the terms about expenditures c and c' on one side and all the terms about income y - t and y' - t' on the other side.)
5. Are there other constraints that the household's choice must obey? If so, what are them?
6. In the (c, c') space, draw the set of feasible choices of the household. Label clearly the following:
(1) the slope of the budget line;
(2) the intercepts on the vertical and horizontal axes;
(3) the area of feasible choices;
(4) the endowment point E.
7. What is the household's optimization problem? Be clear about the following:
(1) the objective function;
(2) the choice variables;
(3) the constraints if any.
8. Assume there is an interior solution to the household's problem. In the (c, c') space, use indifference curves and the set of feasible choices to find the solution to the household's optimization problem, i.e., the optimal choice of the household.
9. Based on your graph in the last question, write down the optimality conditions that must be satisfied at the household's optimal choice. Briefly explain why they must hold. (Hint: you should have two conditions.)
10. Suppose both the current and future consumption goods are normal goods, discuss and explain without any graphs what will happen to the household's current con- sumption c, future consumption c' and savings s if there is
(a) an increase in current income y;
(b) an increase in future income y';
(c) an increase in interest rate r, assuming the household is a lender;
(d) an increase in interest rate r, assuming the household is a borrower.
11. Suppose both the current and future consumption goods are normal goods, and the substitution effect is stronger than the income effect. In the (c, c') space, show graphically the effects of the following changes to the household's optimal choice. If applicable, be clear about which part of the change is due to the income effect and which part is due to the substitution effect.
(a) an increase in future income y';
(b) an increase in interest rate r, assuming the household is a lender;
(c) an increase in interest rate r, assuming the household is a borrower. From now on, let the utility function of the household be
U(c, c') = log(c) + β log(c'),
where β is a parameter between 0 and 1, and assume that there is always an interior solution to the household's problem.
12. What is the marginal rate of substitution of current consumption for future con- sumption MRSc,c' given this utility function? How does it change with c and c'?
13. Solve the household's optimization problem with the lifetime budget constraint. That is, find the household's optimal current consumption c and future consump- tion cj in terms of (y, y', t, t', r, β). You can solve the household's problem with EI- THER of the two methods below. (You DO NOT need to do both.)
(a) Solve the system of two equations in two unknowns ( c and c') defined by the two optimality conditions in Question 9.
(b) Solve the household's problem with the method of Lagrange multipliers.
14. From your answer to the last question, what is the household's savings s in the current period? Your answer should be in terms of
(y, y', t, t', r, β).
15. The parameter β in the utility function governs how patient the household is. From your answer to the last two questions, how do the household's current consumption c, future consumption c', and savings s respond to an increase of β (i.e., when the household becomes more patient)?
(Hint: The value of 1 x increases with x when x is between 0 and 1.)
16. What is the permanent income hypothesis? Is the prediction of the permanent in- come hypothesis consistent with the data in reality?
17. Bonus Question [5] (This is a question that you should be able to answer with the knowledge and tools learned in the class, but it will require some additional think- ing.):
In our standard two-period model of households, the credit market is as- sumed to be perfect in the sense that a household can lend and, more importantly, borrow any amount it likes as long as it can pay back the debt in the future.
In reality, however, there may be many frictions in the credit market, and one common type is the borrowing limit. Consider our standard two-period model, but now let's add a borrowing limit of zero to the household.
That is, the household now cannot borrow at all (but can still lend). In terms of constraints, this is equivalent to adding another constraint, s ≥ 0, for the household's set of feasible choices.
Question: Draw the household's set of feasible choices in the (c, c') space with this additional borrowing limit constraint. Will the household's optimal choice be af- fected by this zero borrowing limit?
(Hint: It depends on whether the household is a borrower or lender without the borrowing limit.)
3 The Two-Period Model of Firms [25]
Consider a firm that produces the consumption good in two periods: the current pe- riod and the future period. The production function in the current period is
Y = zF(K, N)
where Y is the current output, z is the current total factor productivity, K is the current capital stock, and N is the current labor hired. The production function in the future period is
Y' = z' F(K', N')
where Y is the future output, z is the future total factor productivity, K is the future capital stock, and N is the future labor hired.
The firm is a price-taker, and the real wage of labor is w in the current period and w' in the future period. The interest rate in the credit market is r.
The capital stock in the current period K is given (i.e., not chosen by the firm), but the firm can invest I units of output in the current period to increase its capital stock in the future period K'. Capital depreciates after being used in production, and the depreciation rate is d.
So the future capital stock is determined by the following law of motion for capital.
K' = (1 - d)K + I.
At the end of the future period, all the capital left after the production (1 - d)K' can be converted one-for-one back into the consumption good, which can be sold in the same way as the output produced.
1. What is the firm's profit in the current period π in units of the current consumption good? (Hint: Profit=Revenue-Cost.)
2. What is the firm's profit in the future period πj in units of the future consumption good? (Hint: Profit=Revenue-Cost.)
3. Since the firm operates in both periods, the firm needs to take into consideration the profits from both periods when making decisions. What should the firm try to maximize in this two-period world? (i.e., what is the objective function of the firm?)
4. Write down the firm's optimization problem. Be clear about the following:
(1) the objective function (You should replace the output Y and Y' with the correspond- ing production functions here.);
(2) the choice variables;
(3) the constraints if any. (Hint: There are two ways of writing the firm's optimization problem depending on whether you want to substitute K' with (1 - d)K + I. Both ways are correct.)
5. Now let us "derive" the optimality conditions of the firm using the marginal cost- benefit analysis.
(a) Current labor hired (N):
i. What is the benefit of hiring one more unit of labor in the current period? (Hint: What can the firm do with one extra unit of labor?)
ii. What is the cost of hiring one more unit of labor in the current period? (Hint: Labor is not free, right?)
iii. At the optimum, the benefit and cost above should be equal, so the firm cannot be better off by changing the amount of current labor hired. What is the optimality condition implied by this? (Hint: Benefit=Cost.)
(b) Future labor hired (N'):
i. What is the benefit of hiring one more unit of labor in the future period?
ii. What is the cost of hiring one more unit of labor in the future period?
iii. At the optimum, the benefit and cost above should be equal, so the firm cannot be better off by changing the amount of future labor hired. What is the optimality condition implied by this?
(c) Investment (I):
i. What are the benefits of investing one more unit of the consumption good in the current period? (Hint: There are two sources of benefits in the future period.)
ii. What is the cost of investing one more unit of the consumption good in the current period? (Hint: The cost is incurred in the current period.)
iii. At the optimum, the benefit and cost above should be equal, so the firm cannot be better off by changing the amount of investment.
What is the optimality condition implied by this? (Hint: Be careful! The benefits are in the future period, but the cost is in the current period, so you need to make them comparable first.)
6. Based on the optimality conditions you "derived" in the last question, suppose
(1) the current and future marginal product of labor MPN and MPN' are all decreasing in the amount of labor hired N and N';
(2) the future marginal product of capital MPK' is decreasing in the amount of future capital Kj but increasing in the amount of future labor hired N', answer the following questions and explain why:
(a) How would the current labor hired N change when there is an increase of cur- rent wage w?
(b) How would the future labor hired N' change when there is a decrease of future wage w'?
(c) How would the investment I change when there is an increase of interest rate r?
(d) How would the investment I change when there is a decrease of future wage w'? (Hint: w' → N' → I)