# Intermediate Macro Exercise: How to Construct a Maximally Contractionary Budget Deficit Reduction

Consider an economy called Käseland, with gross output equal to approximately \$475 billion, and unemployment rate of 7.5%, so considerable underemployment of factors of production exists; consistent with this interpretation, the general nonfarm wage rate has been relatively constant, growing at only 1.2% on a 12 month basis through 2010, and the price level has risen by about 1.5% from the second half of 2009 to second half of 2010.

Suppose there is a budget deficit, that you wish to close. How do you maximize the negative impact on output?

First, consider the definition of a government budget balance. Assume interest costs away, since they are determined exogenously (or at least predetermined, for the current period):

BuS ≡ T-TR-G = (TA0 + t1Y) – (Tr0) – (GO0)

Where BuS is the budget balance, TA0 is lump sum taxes, t1 is the marginal tax rate, Y is output, or income, Tr0 is lump sum transfers, and GO0 is exogenous government spending on goods and services.

Effecting a positive change in the budget balance can be accomplished in three ways. To see this, consider the total differential of the budget balance (assuming the marginal tax rate is held constant).

Δ BuS = Δ TA + t1 Δ Y – Δ TR – Δ GO

One can (i) raise taxes; (ii) decrease transfers; or (iii) decrease government spending. For simplicity, consider cutting government spending (on civil servants, for instance), say by \$1 billion. In a demand determined model of output (i.e, Y = Z, where Z is aggregate demand), the resulting output reduction is:

Δ Y = γ Δ GO where γ ≡ [1-c1(1-t1) + m1]-1

where m1 is the marginal propensity to import from other economies, and should be relatively large in a small open economy like Kaseland. “Gamma” (γ) is the Keynesian multiplier. (Solving this simple model is laid out in detail in this handout). Recalling that the change in government spending on goods and service is less than zero, viz., Δ GO < 0, output falls by greater than a billion dollars.

One sees that an alternative means of closing the budget deficit is by raising taxes; for simplicity consider an increase in lump sum taxes.

Δ Y = -γ c1 Δ TA

Notice that because the parameter, c1 (the marginal propensity to consume out of disposable income, 0 ≤ c1 ≤ 1) premultiplies the change in lump sum taxes, then the corresponding reduction in output resulting from the increase in taxes is smaller than that for a cut in government spending on goods and services.

This has an additional ramification for the budget balance. Note that because tax receipts are endogenous, then the larger negative impact on output and hence income arising from government spending reduction manifests itself in a correspondingly larger decrease in tax receipts. To see this, repeat the expression for the impact on the budget balance:

Δ BuS = Δ TA + t1 Δ Y – Δ TR – Δ GO

For a decrease in government spending of \$1 billion, holding all other fiscal measures constant:

Δ BuS = t1 γ Δ GO – Δ GO

For an increase in lump sum taxes of \$1 billion, holding all other fiscal measures constant:

Δ BuS = Δ TA – t1 γ c1 Δ TA

Notice that the improvement of the budget balance is greater for an increase in lump sum taxes of \$1 billion than for a \$1 billion decrease in government spending.

Postscript, 4:53 Pacific: One can increase the contractionary effect of the budget balancing if one cuts taxes and increases the size of cuts to government spending to compensate, unless the elasticity of supply is sufficiently high. See a policy aimed at accomplishing this here Forbes (2/15/11).

Addendum 1: Typically, in examining the impact of tax policy changes, one analyzes changes in the marginal tax rate, or the parameter t1 (and not lump sum taxes) in this model. In order to accomplish this, one would need to examine the following expression for the budget balance, which uses the product rule for differentiation.

Δ BuS = (Δ t1) Y + t1 Δ Y

Holding all other fiscal policies constant. In addition, the product rule for differentiation has to be invoked for the impact on output:

Δ Y = – c1 γ 20]

Where Λ 0 is the sum of autonomous spending (e.g., investment unrelated to the level of income).

Without solving out for the analytical answer, what is true is that for a given improvement in the budget balance, a tax rate increase will have a smaller negative impact on output than a given cut in government spending on goods and services.

Addendum 2: The foregoing model has assumed perfectly elastic supply. Suppose the economy is at capacity, i.e., there are no underutilized factors of production. Then one would need to drop Y=Z, and substitute in AS=Z, where AS = fn(K, L). Then, under the assumption of sufficiently high elasticities of labor with respect to wages and capital with respect to the rental cost of capital (and assuming away accelerator effects), one can overturn the demand-determined model-based results obtained above. [1]

Addendum 3: The last IMF WEO assessed contractionary expansions. In that cross country assessment, spending (both goods and services and transfers) reductions effected smaller reductions in output than tax increases. The authors concluded that this outcome was due to the monetary authorities tending to offset spending cuts more than tax increases. In the model above, I have assumed Kaseland has no independent monetary authority (i.e., is a small open economy, with fixed exchange rate).

Addendum 4: If the marginal propensity to consume for government workers is higher than the marginal propensity to consume of the households that would be taxed at higher rates due to a tax increase, then the bottom line result would be strengthened.

Notes:

Statistics on employment, wages, in Wisconsin here; on CPI growth in the Midwest here (search Class A cities). Statistics on Wisconsin GSP (gross state product) here.

Implied impact on government payrolls of current proposals: Institute for Wisconsin’s Future, page 3.

Originally published at Econbrowser and reproduced here with permission.
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