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Intermediate Macroeconomic Theory II, Fall 2024.
Problem Set 2. Due by December 2. 128 points.
1. (20 points) An employee has to choose between two contracts. Assume that the net real interest rate on saving and borrowing equals r > 0. Under contract A, she has gross incomes y and y' in the current and future periods, respectively, and has to pay taxes t and t' in the current and future periods, respectively. Under contract B, an employer offers the employee an option to increase income next year by x·(1 +r) units and reduce income this year by x units. Taxes are the same under both contracts.
(a) (10 points) Write down current and future budget constraints and the lifetime budget constraint under the two contracts. Which contract would the employee choose and why? (Hint: you should compare lifetime wealth under the two con-tracts.)
(b) (10 points) Assume that preferences over current and future consumption are U(c, c') = −2 1 (c − c¯) 2 − 1 2 β(c 0 − c¯) 2 , where ¯c is the bliss consumption level and β = 1+r/1. Find consumption in the current and future periods and saving under the two contracts. Compare consumption levels and saving under the two contracts.
2. (18 points) Assume a consumer has current-period income y = 120, future period income y' = 150, current and future taxes t = 60 and t' = 50, respectively, and faces a market real interest rate of r = 0. Consumer’s preferences over current and future consumption are U(c, c') = min (c, c'). The consumer faces a credit-market imperfection in that she cannot borrow at all, that is, s ≥ 0.
(a) (6 points) Calculate her optimal c, c', s.
(b) (6 points) Suppose that everything remains unchanged, except that now t = 40 and t' = 70. Calculate the effects on current and future consumption and optimal saving.
(c) (6 points) Calculate the marginal propensity to consume for this consumer fol-lowing the tax change, that is, the change in the current consumption following the change in taxes and disposable income that it entails. Define the Ricardian equivalence and comment if it holds in this case.
3. (50 points) Consumer has quadratic preferences and cares about consumption over two periods:
U(c0, c1) = −2/1(c0 − c¯) 2 − β2/1(c1 − c¯) 2 .
Assume that the real interest rate, r, is 1 9 , and the time discount factor, β, equals 0.9.
(a) (7 points) Consumer’s disposable income in period 0 equals 10, and in period 1 equals 20. There’s no uncertainty. Write down the Euler equation and find the optimal consumption levels in periods 0 and 1, and the optimal savings.
(b) Assume now that period 0 income stays at 10, while period 1 income is uncer- tain. There are two possible states of nature that might realize in period 1—with probability π = 3/1, income will equal 0 in period 1 if state 0 occurs whereas with probability 1 − π = 3/2 income will equal 30 in period 1 if state 1 occurs. Con-sumer has to make decision about her consumption and saving for period 0 before uncertainty is resolved. Consumer now maximizes expected utility
EU(c0, c˜1) = −2/1(c0 − c¯) 2 − πβ2/1(c1(0) − c¯) 2 − (1 − π)β2/1(c1(1) − c¯) 2 ,
where c1(k) is consumption in period 1, state k = 0, 1.
(i) (3 points) Write down the Euler equation and find the expected value and variance of income in period 1.
(ii) (6 points) Find the optimal consumption and saving in period 0, and con-sumption in period 1 in both states of nature.
(iii) (1 point) Does your answer for the optimal consumption in period 0 and savings differ from the answer to (3a), and why it does or why it doesn’t?
(c) Assume now that income in period 1 state 0 equals 0 with probability π = 0.99 and income in period 1 state 1 equals 2000 with probability 1 − π = 0.01.
(i) (3 points) Write down the Euler equation and find the expected value and variance of income in period 1.
(ii) (6 points) Find the optimal consumption and saving in period 0, and con-sumption in period 1 in both states of nature.
(iii) (1 point) Does your answer for the optimal consumption in period 0 and savings differ from the answer to (3b), and why it does or why it doesn’t?
Assume now that each period’s utility function is u(c) = ln(c). Continue assuming that the real interest rate, r, is 1/9, and the time discount factor, β, equals 0.90.
(d) (7 points) Write down the Euler equation and find the optimal consumption in periods 0 and 1 and optimal saving in period 0 given the data in (3a).
(e) (8 points) Write down the Euler equation and find the optimal consumption in periods 0 and 1 and optimal saving in period 0 given the data in (3b). Compare the optimal saving to the value you found in (3b) and argue why they are different (if different at all).
(f) (8 points) Write down the Euler equation and find the optimal consumption in periods 0 and 1 and optimal saving in period 0 given the data in (3c). Compare the optimal saving to the value you found in (3c) and argue why they are different (if different at all).
4. (18 points) Suppose there is a credit market with the fraction of a good borrowers, and the fraction of 1 − a bad borrowers, with the total number of borrowers equal Nb. Banks cannot differentiate between good and bad borrowers when making loans (asymmetric information) and loan out l units of goods to each borrower, good or bad. There are Nd depositors/savers in the economy. Banks attract deposits in the amount of L from each of them and promise to pay a net real interest rate of r1 to depositors. Banks charge net interest r2 on loans. Good borrowers are identical and always repay their loans, while a debt collection agency makes bad borrowers pay a fraction 0 ≤ f ≤ (1 + r2) of their loans (who, in the absence of the agency, would pay nothing). The banking sector is competitive, and the profit equals zero in equilibrium.
(a) (5 points) Using the bank balance sheet, find the relationship between Nd, Nb, l, and L.
(b) (10 points) Using the assumption of the competitive banking sector, find an expression for the interest rate on loans, r2, made by banks, as a function of a, f, and r1.
(c) (1 points) How will the interest rate change if the debt collection agency makes each borrower pay a higher fraction f of their loans taken?
(d) (2 points) What must f be for the interest rate on loans to equal r1?
5. (22 points) Consider the short-run model of aggregate economy we studied in class. Aggregate demand (AD) curve Y˜ t = ¯a−¯bm¯ (πt −π¯) was derived from the following two equations:
IS curve: Y˜ t = ¯a − ¯b(Rt − r¯)
Textbook MP curve: Rt − r¯ = ¯m(πt − π¯)
(a) Assume an alternative MP rule:
Alternative MP curve: Rt − r¯ = ¯m(πt − π¯) + ¯nY˜ t
(i) (6 points) Explain in words what this rule tries to achieve and how it com- pares to the standard textbook case. Derive the aggregate demand equation (AD') under the alternative rule.
(ii) (6 points) Plot (AS), the textbook (AD) and (AD') curves on the same graph. State which aggregate demand curve is steeper and why. (Hint: we are plotting πt against Y˜ t .)
(b) Assume now there is a temporary positive inflationary shock to the economy (¯o in the AS curve goes from 0 to a positive number temporarily).
(i) (4 points) Show how the economy responds over time using the AS/AD framework. (You should clearly label the axes and explain everything you want to show on your graph. You may use either AD or AD' to avoid clut-tering.)
(ii) (6 points) Show the path of real interest rates set by the central bank under the two alternative monetary policies. Which policy would result in a more prolonged adjustment of the real interest rate to its long-run value ¯r? (Your answer should be reflected in your graph.)