# How Should Public Spending Respond to Unemployment Fluctuations?

### This week's lecture material in ECON2800–Economic Slack

There is always some slack in the economy. Sometimes there is far too much of it—for instance during the Great Recession— and sometimes there is too little of it—in the aftermath of the coronavirus pandemic for instance. I am currently teaching a PhD course on the topic at Brown University. Today’s post provides a brief overview of the material covered this week. The topic for the week is “Optimal Fiscal Policy Over the Business Cycle”.

### Overview of this week’s material

Two weeks ago, we found that the US labor market is generally inefficiently slack, and especially exceedingly slack in slumps. That is, the unemployment gap is generally positive and especially large in slumps.

Last week, we showed that in a broad class of models, the optimal monetary policy is to adjust interest rates to eliminate the unemployment gap entirely. So the Fed should lower rates in bad times, when unemployment is inefficiently high, and raise rates in good times, when unemployment is inefficiently low. In fact, the optimal interest rate can be computed from two sufficient statistics: the current unemployment gap and monetary multiplier—the response of the unemployment rate to interest rates.

However, in several situations, monetary policy is inoperative—for instance when the zero lower bound on nominal interest rates is binding. In such a situation, as this week shows, fiscal policy should be adjusted to reduce the unemployment gap. In fact, optimal stimulus spending can be expressed as a function of a few sufficient statistics: unemployment gap (once again), elasticity of substitution between public and private goods (measuring the value of public goods), and the unemployment multiplier (response of unemployment rate to public expenditure).

### Breakdown of this week’s lecture videos

This week’s first lecture video discusses when public spending should be used for stabilization. Public spending distorts households’ consumption basket, increasing the share of public goods and decreasing the share of private goods. Monetary policy, on the other hand, does not create any distortions. It is therefore preferable to resort to monetary policy first for stabilization.

Sometimes, however, monetary policy cannot be used for stabilization. At the zero lower bound on nominal interest rates, interest rates cannot be reduced further, so monetary policy cannot be used to stabilize the economy. Countries that are part of a monetary union, such as countries of the Eurozone, cannot use monetary policy to respond to shocks that specifically hit their economy. Indeed, monetary policy is decided by the monetary union and cannot be tailored to individual members’ situation. In such cases, fiscal policy can help to stabilize the economy and reduce the unemployment gap.

The second video introduces the class of models that we will use to study optimal fiscal policy. The optimal fiscal-policy formula will apply to any model in this class. These models share three features:

A Beveridge curve generating unemployment

The Samuelsonian property that households value public goods but public goods are produced using the same resources as private goods

Inflation is exogenous or unaffected by fiscal policy

The third video describe the composition of the labor force in the model. Workers may be in four possible situations:

Unemployed

Employed and allocated to recruiting (either for the private or public sector)

Employed and producing public goods

Employed and producing private goods

The next video then introduces the marginal rate of substitution between public and private goods (MRS). The MRS measures the value of public goods relative to that of private goods. The famous Samuelson rule states that optimal public spending is such that MRS = 1. That is, the government should provide public goods until the marginal utilities from public and private goods are equalized.

The following video introduces the elasticity of substitution between public and private goods. This elasticity is one of the sufficient statistics that determines optimal public spending. It measures how well public goods can replace private goods. An elasticity of substitution of 0 means that public goods cannot replace private goods at all. An infinite elasticity of substitution means that public goods are perfect substitute for private goods .

The sixth video introduces the unemployment multiplier. The multiplier is another sufficient statistic that determines optimal public spending. The multiplier says by how many percentage points the unemployment rate falls when public employment increases by one percent of the labor force.

The seventh video begins the optimal policy analysis by describing the various channels through which public spending influences welfare.

More public spending means more public goods, which people enjoy.

More public spending means more public workers and—taking the employment rate as fixed—fewer private workers, which reduces private goods and welfare.

Public spending affects the unemployment rate and therefore the number of productive workers available in the economy, which affects production and welfare.

Channels 1 and 2 are present in the classic Samuelson analysis. Channel 3 is new to this analysis. It captures stabilization of the economy by public spending.

In the eight video we derive a formula for optimal public spending by solving the social planner’s problem. We find that this formula is composed of two elements: the classic Samuelson rule (ninth video), corrected by a new stabilization term (tenth video).

In the eleventh video, we rework the formula to express optimal stimulus spending—the amount of public spending above the spending given by the Samuelson rule—as a function of three sufficient statistics: elasticity of substitution between public and private goods, unemployment multiplier, and unemployment gap. From this new formula, we learn the following:

The Samuelson rule—which was derived in a neoclassical economy—remains valid in an economy with unemployment as long as unemployment is efficient or public spending has no effect on unemployment.

If unemployment is inefficiently high and public spending reduces unemployment, then public spending should be above the Samuelson rule. That is, stimulus spending should be positive if the unemployment gap and unemployment multiplier are positive.

Conversely, if unemployment is inefficiently low and public spending reduces unemployment, then public spending should be below the Samuelson rule. That is, stimulus spending should be negative if the unemployment gap is negative and the unemployment multiplier is positive.

In the twelfth video, we continue working on the optimal spending formula to express optimal stimulus as a function not of current unemployment gap but of initial unemployment gap. This new formula is much more helpful to policymakers. It tells them how much stimulus spending should be, starting from a situation in which public spending follows the Samuelson rule but unemployment is suddenly inefficient.

In the next video, we derive additional properties of optimal stimulus spending based on the new formula:

Optimal stimulus spending is larger when the initial unemployment gap is larger.

Optimal stimulus spending is larger when the elasticity of substitution between public and private goods is larger. This is because public goods replace private goods better in that case, so public spending is less distortionary. In fact, when the elasticity of substitution is zero, stimulus spending should always be zero.

Maybe unexpectedly, optimal stimulus spending is a hump-shaped function of the unemployment multiplier. Optimal stimulus spending is zero for a zero multiplier, increasing in the multiplier for small multipliers, largest for a moderate multiplier, and decreasing in the multiplier beyond that. So larger multipliers do not necessarily mean larger stimulus spending. The reason is that when public spending becomes more powerful, less of it is required to reduce the unemployment gap.

In the last video, we discuss the amount of stabilization that can be achieved by public spending. We do that by computing the unemployment gap resulting from optimal stimulus spending, starting from some initial unemployment gap. Several results emerge:

With public spending it is optimal to reduce but not eliminate the unemployment gap. In that public spending differs from monetary policy.

It is optimal to reduce the unemployment gap more when the unemployment multiplier is larger. This is because public spending is more potent in that case. In fact, as the unemployment multiplier goes to infinity, the unemployment gap should disappear.

It is optimal to reduce the unemployment gap more when the elasticity of substitution between public and private goods is larger. This is because public goods replace private goods better in that case, so public spending is less distortionary. In fact, as the elasticity of substitution goes to infinity, distortions go to zero, so the unemployment gap should disappear.

### This week’s readings

This week’s analysis of fiscal policy follows “Optimal Public Expenditure With Inefficient Unemployment“, which Emmanuel Saez and I wrote. The paper builds a framework to study fiscal policy and unemployment and derives a sufficient-statistic formula for optimal stimulus spending. It explains when and how public spending should deviate from the standard Samuelson rule. The paper also reviews estimates of the sufficient statistics (unemployment gap, unemployment multiplier, elasticity of substitution between public and private goods) for the United States. Finally, the paper applies the formula to retroactively design the optimal stimulus package for the US Great Recession.

The Samuelson rule was derived by Paul Samuelson in ”The Pure Theory of Public Expenditure”. It is the standard formula for optimal public spending in public economics. It says that the government should provide public goods until the marginal utility from public goods equal that from private goods—or MRS = 1.

The Obama administration implemented a large stimulus package at the onset of the Great Recession. During the design phase, policymakers and academics argued about the size of the multiplier. Proponents of the stimulus package postulated that multipliers were larger in bad times, which would justify a larger package. We have since learned that multipliers are indeed larger in bad times—as showed for instance by Alan Auerbach and Yuriy Gorodnichenko in “Measuring the Output Responses to Fiscal Policy”. Furthermore, the models of slack studied in this course naturally generate multipliers that are larger when unemployment is higher—as I showed in “A Theory of Countercyclical Government Multiplier”.

However, this week’s lecture shows that fluctuations in the multiplier only have second-order effects on the optimal stimulus package. The unemployment gap has first-order effects on the optimal stimulus package. Furthermore, the optimal stimulus package is declining with the multiplier when the multiplier is above some intermediate value. So the effect of the multiplier on the optimal stimulus package is more subtle than people thought.

### The end!

This course is coming to an end, as this is the final week of the semester at Brown. All the lecture videos for the course are now organized in a publicly available playlist on YouTube. In addition, all the lecture notes and readings for the course are now collected in a publicly available repository on GitHub. The playlist and repository hopefully make it easy to go back to specific results and references, as well as catch up on specific material.