Problem Set 2

Due: Friday, February 21, 5:00 p.m. Eastern Time

Applied Econ 440.602: Macroeconomic Theory

Spring, 2025

1. Balanced-Budget Multipliers. Consider a closed economy where aggregate planned expenditure is given by:

and consumption and output are given by:

where I have used the same notation that we used in class. Also, maintain the assumptions that we did in class that c and d are positive and that mpc is between zero and one. An equilibrium requires that Y = Y pe. Assume that, initially, the government is running a balanced budget, meaning that T = G.

(a) Consider a proposal for the government to abandon its balanced budget and engage in deficit spending. That is, G goes up, but T remains fixed at its initial level. Assume that the central bank acts to keep the real interest rate constant: r = , an exogenous constant, both before and after any change in fiscal policy.

i. Derive an expression for the government-spending multiplier .

ii. Is this government-spending multiplier positive, negative, or zero? Explain how you know mathematically.

iii. Is this government-spending multiplier greater than, less than, or equal to one in absolute value? Explain how you know mathematically.

(b) Consider a proposal to expand government spending, while maintaining a balanced budget G = T. That is, G goes up, and T is simultaneously adjusted to increase dollar-for-dollar with G. Again, assume that the central bank acts to keep the real interest rate constant: r = , an exogenous constant, both before and after any change in fiscal policy.

i. Derive an expression for the government-spending multiplier .

ii. Is this government-spending multiplier positive, negative, or zero? Explain how you know mathematically.

iii. Is this government-spending multiplier greater than, less than, or equal to one in absolute value? Explain how you know mathematically.

(c) Now, let’s compare the above two proposals. Which one will result in a larger change in output? Explain how you know mathematically, and provide a brief economic explanation in words.

(d) When analyzing each proposal, we assumed that the central bank took action to keep interest rates constant. Assume that prices are totally sticky, so the real interest rate is equal to the nominal interest rate. When the fiscal authority increases G, what happens to Md , the demand curve in the market for money? What does the central bank have to do in the market for money to keep interest rates constant? Provide a supply-and-demand graph along with a brief explanation in words.

2. An Alternative IS Curve and a Deflationary Shock. This question will have you derive a version of the IS curve, which characterizes an equilibrium in the market for goods, under an alternative set of assumptions about investment behavior. Then, you’ll use this IS curve to develop an AD-AS framework to analyze the response to a deflationary shock.

Assume that the economy is closed, so equation (1) continues to describe aggregate planned expenditures. Also, like before, assume that consumption is given by equation (2). In class, we had assumed that investment only depended on interest rates. However, one might also expect investment to depend on income. For instance, one component of investment is expenditure on new housing, and someone with more income is likely to buy a larger house. Suppose that investment is given by:

where  is autonomous investment, r is the real interest rate on bonds, d is a parameter that governs the sensitivity of investment to interest rates, and mpi is the marginal propensity to invest. Assume that mpi > 0, and mpi + mpc < 1.

Monetary policy and aggregate supply take the same forms that we assumed in class. Specifically, the central bank follows the interest-rate rule:

The aggregate supply (AS) curve takes the usual form.

In each of the above two expressions, the notation is the same as we introduced in class.

(a) Derive the IS curve under these alternative assumptions. In other words, provide an expression for Y in terms of r. (This expression will also contain the exogenous quantities , , , and , as well as the parameters mpc, c, d, and mpi.)

(b) Your answer to part (a) should show that Y is decreasing in r. Let’s investigate this negative relationship more closely.

i. If r increases by one, then how much does Y decrease?

ii. Does a high value of mpi make Y more or less sensitive to changes in r? In a sentence or two, provide an economic interpretation.

(c) Derive the aggregate demand (AD) curve. That is, combine the IS curve with the monetary policy function (5) to write Y as a function of π.

(d) Assume that the economy is initially in a long-run equilibrium, with Y = Y p . Then, suppose that there’s a court decision that gives unions less bargaining power, making it harder for them to obtain higher wages. We will model this change as an decrease in ρ. In a sentence or two, explain why the decrease in union bargaining power would act as an deflationary shock.

(e) Depict the short-run response of the economy to the court decision using an AD-AS graph, holding all else equal. Label the axes appropriately, and label potential output appropriately. Label the initial aggregate demand curve as AD; label the initial aggregate supply curve as AS; label the new aggregate supply curve as AS′ . What happens to inflation and output?

(f) Suppose that the central bank responds to the court decision by trying to stabilize output. That is, the central bank adjusts r in order to keep Y at Y p . Use an AD-AS graph to depict the court decision and the central bank’s response in the short run. Label the axes appropriately, and label potential output appropriately. Label the initial aggregate demand and supply curves as AD and AS; label the new aggregate demand and supply curves as AD′ and AS′ . When the central bank acts to stabilize output, what happens to inflation, relative to the case where the central bank doesn’t adjust r at all?

(g) Suppose that, instead of trying to stabilize output, the central bank responds to the court decision by trying to stabilize inflation. That is, the central bank adjusts r in order to keep π at its initial level. Use an AD-AS graph to depict the court decision and the central bank’s response in the short run. Label the axes appropriately, and label potential output appropriately. Label the initial aggregate demand and supply curves as AD and AS; label the new aggregate demand and supply curves as AD′ and AS′ . When the central bank acts to stabilize inflation, what happens to output, relative to the case where the central bank doesn’t adjust r at all?

(h) Now, suppose that the central bank ultimately decides to take a hands-off approach, meaning that it doesn’t adjust r at all after the court decision. For the economy to transition to a new long-run equilibrium, what curve(s) shift(s), and why? Draw an AD-AS graph depicting the shift, with the curves and axes appropriately labeled.

3. Exploring Okun’s Law. The relationship between real output and unemployment is known as Okun’s law. There are several versions of Okun’s law, and this exercise will have you explore them empirically. Start off by going to FRED to get the following data: unemployment (UNRATE) and real GDP (GDPC1). The raw unemployment data is monthly, but we’ll have to make it quarterly to match the GDP data. When you pull up the unemployment data in FRED, click on “Edit Graph,” go to “Modify Frequency,” and select “Quarterly,” with aggregation method “End of Period.” Your sample for the variables should run from 1948 to present, when both variables are available. (Some, but not all, of the questions below will specialize to the pre-2020 sample to avoid large outliers.) Throughout, Let ut denote the unemployment rate at date t, let Yt denote real GDP at date t, and let %∆Yt ≡ (Yt − Yt−1) /Yt−1 denote the GDP growth rate at date t.

(a) The version of Okun’s law presented in Mishkin says that the growth rate of output %∆Yt is systematically related to the change in unemployment ∆ut ≡ ut − ut−1.

i. Construct two scatter plots with %∆Yt on the horizontal axis and ∆ut on the vertical axis. For the first plot, use only data from 1948 to 2019. For the second plot, use the full sample from 1948 to present.

ii. Using the 1948–2019 sample, compute the correlation coefficient between %∆Yt and ∆ut.

iii. Using the 1948–2019 sample, estimate the regression:

where ϵt is a residual. What is the estimated value of β1?

(b) Some economists prefer to look at how deviations from trend in GDP are related to deviations from trend in unemployment. To do so, it’s necessary to specify what, exactly, we mean by trend. We’ll use the Hodrick-Prescott (HP) filter. For any variable xt, the HP trend τt is the solution to the following minimization problem:

where λ > 0 is a parameter that controls how smooth the trend is. Notice that the first sum is minimized by setting τt = xt, and the second sum is minimized by a linear trend (by setting ∆τt equal to a constant). Hence, a low value of λ produces a more flexible trend that closely tracks all the fluctuations in the data, and a high value of λ produces a more rigid trend that’s closer to a straight line. When working with quarterly data, it’s standard practice to set λ = 1, 600, and that’s what you should do when applying the HP filter in this exercise. In Stata, you can compute the HP trend using the command tsfilter hp. In MATLAB, you can compute the HP trend using the command hpfilter. (You can also download an add-in for Microsoft Excel that will compute the HP filter, but I encourage you to do the exercise using a statistical or mathematical coding language.)

i. Let τt (u) denote the HP trend of ut. Let u ∗ t ≡ ut −τt (u) denote the deviation in unemployment from the HP trend. Plot u*t over the sample from 1948–2019.

ii. Let τt (y) denote the HP trend of log (Yt), where log (·) denotes the natural logarithm. Let y*t ≡ 100 × h log (Yt) − τt (y) i denote the deviation in log GDP from the HP trend, multiplied by 100. (By looking at the log deviation times 100, we can interpret y*t as the approximate percent deviation from trend in real GDP.) Plot y*t over the sample from 1948–2019.

iii. Using the 1948–2019 sample, construct a scatter plot with y*t on the horizontal axis and u*t on the vertical axis.

iv. Using the 1948–2019 sample, what is the correlation coefficient between y*t and u*t?

v. Using the 1948–2019 sample, estimate the regression:

where ϵt is a residual. What is the estimated value of β1?