Answer to Question #168580 in Macroeconomics for Muhammad Talha Bilal

Question

Exercise 1 ([Indifference Curves). Consider the utility function U(C1, C2) = ln(C1) + ln(C2).

1. Using a program of your choice (say excel, or matlab) plot indifference curves in the space (C1, C2) for

U ̄ = -0.5, -1, and -1.5. Consider values of C1 in the interval (0, 1]. Set the range of the vertical axis to

[0, 2].

2. Find an analytical expression for the slope of the indifference curve and show that it is equal to the

(negative) of the marginal rate of substitution.

3. Show analytically that the indifference curves are convex.

4. For the 3 indifference curves plotted above, find the slope of the indifference curve at the point C1 = 1

and the corresponding value of C2. Explain why the indifference curves at C1 = 1 become flatter as

the level of welfare declines.

Exercise 2 (The Saving Schedule). Consider a two-period economy populated by identical households with

preferences defined over consumption in period 1, C1, and consumption in period 2, C2, and described by

the utility function

p

C1 +

p

C2.

Assume that households are endowed with Y1 kilos of apples in period 1 and with Y2 kilos of apples in

period 2. Let P1 and P2 denote the price of apples in periods 1 and 2. Households can save (or borrow)

at the nominal interest rate i. Let r denote the real interest rate, so that the gross real interest rate is

1 + r =

P1

P2

(1 + i). Let St denote saving in kilos of apples.

1. State the household’s budget constraints in periods 1 and 2.

2. Derive the household’s intertemporal budget constraint in terms of C1, C2, Y1, Y2, and r.

3. State the household’s utility maximization problem.

4. Find the optimal level of consumption in period 1, C1, in period 2, C2, and the associated level of

saving, S1. Express your answer in terms of Y1, Y2, and r.

5. Now assume that output is 10 kilos of apples in both periods (Y1 = Y2 = 10) and that the real interest

rate is 0 percent (r = 0). Find C1, C2, and S1. (Your answer should be 3 numbers.) Finally, compute

the same 3 numbers but under the assumption that the real interest rate is 10 percent (r = 0.1). Is

saving increasing in r? Provide intuition.

Exercise 3 (An Economy Driven by Natural-Rate Shocks). Consider a two-period sticky-price economy

populated by identical households with preferences defined over consumption in period 1, C1, and consumption

in period 2, C2, and described by the utility function

lnC1 + β lnC2,

1

where β = 1/1.1 is the subjective discount factor. In both periods, potential output (Y ) is equal to 10 kilos

of apples. Let P1 and P2 denote the price levels in periods 1 and 2, respectively. Assume prices are fixed

at P1 = P2 = 1 and that the economy is always in full employment in period 2 (the long run). The central

bank uses the nominal interest rate, denoted i, as its monetary instrument. The nominal interest rate is

subject to the zero lower bound (ZLB) constraint.

1. Assume further that the central bank sets the nominal interest rate so as to maximize employment

and minimize excess aggregate demand for goods. Denote this interest rate by i

∗

. Find i

∗

.

2. Now suppose that a financial panic causes households to become more patient. Specifically, suppose

that the subjective discount factor increases to 1/0.9. Suppose that the central bank is slow to react

and keeps the interest rate at i

∗

(the level of i obtained in question 1. Find the output gap, defined as

(Y /Y ̄

1 − 1)100, where Y1 denotes output in period 1.

3. Now suppose that contrary to the assumption in question 2, the central bank acts quickly and changes

the interest rate to minimize unemployment. Denote this interest rate by i

∗∗. Find i

∗∗ and the output

gap.

4. Consider the scenario of question 3, that is, i = i

∗∗. Suppose that the fiscal authority decides to

also intervene. Let G∗ denote the lowest level of government spending that eliminates involuntary

unemployment. Find G∗

. Calculate private consumption in period 1.

5. Suppose that the government miscalculates G∗ and instead sets government spending equal to G ̃, where

G ̃ is 10 percent higher than G∗

. Assume further that realizing this situation, the central bank changes

the interest rate to avoid excess aggregate demand, while still maintaining full employment. Find the

new interest rate and private consumption. Comment.

Answer 1

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