ECON 385

Intermediate Macroeconomic Theory II, Fall 2021.

Final exam. 138 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 0 in the current and future periods, respectively, and has to pay taxes t and t 0 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 − 2/1β(c 0 − c¯) 2 , where ¯c is the bliss consumption level and β = 1+ 1 r . Find consumption in the current and future periods and saving under the two contracts. Compare consumption levels and saving under the two contracts.

2. (10 points) A consumer receives income y in the current period and income y 0 in the future period, and pays taxes t and t 0 in the current and future periods, respectively. The consumer can lend/save at the real interest rate r1. The consumer can borrow at the real interest rate r2 > r1 only an amount x or less, where x < 1+r2/y'−t'. Use a diagram to clearly show the consumer’s budget set. Distinguish the cases when a consumer is: i) a saver, ii) a borrower whose borrowed amount does not exceed x, and iii) a borrower who chooses to borrow the full allowable amount x.

3. (36 points) Assume a consumer has current-period income y = 220, 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 0 = 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.

(d) (18 points) Now suppose alternatively that y = 120. Repeat the above parts, and explain any differences.

4. (10 points) Consider a onetime change in government policy that immediately and permanently increases the level of the labor force in an economy from L0 to L1 > L0 at some point in time t0. Assuming the economy with technological progress at a rate g starts in its initial steady state, use the Solow model to explain what happens to the economy over time and in the long run. In particular, draw two diagrams: 1) for real wages with time on the horizontal axis using a ratio scale; and 2) the Solow diagram that outlines the changes. Assume that the growth rate of population stays constant over time at a rate n.

5. (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 zero, and the time discount factor, β, equals 1. (Note (!) that consumption can be negative if preferences are quadratic.)

(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’re 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 = 2 3 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, c1) = − 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 (5a), 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 (5b), 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 zero, and the time discount factor, β, equals 1.

(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 (5a).

(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 (5b). Compare the optimal saving to the value you found in (5b) 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 (5c). Compare the optimal saving to the value you found in (5c) and argue why they are different (if different at all).

6. (12 points) Consider an economy that begins with output at potential and an inflation rate of ¯π, so the economy begins in steady state. A new chair of the central bank decides to lower the long-run inflation target to ¯π 0 < π¯. 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 should also list the equations for AD and AS curves.) Comment on your results.