The tax multiplier reflects the fact that a given autonomous change in taxes does NOT result in an equal change in aggregate expenditures. Taxes change disposable income, which causes changes in both consumption expenditures and saving. And only consumption expenditures affect aggregate expenditures. The expenditures multiplier, however, reflects the fact that a given autonomous change in an expenditure results in an equal change in aggregate expenditures.

The tax multiplier is actually a family of multipliers that differ based on which components of the Keynesian model are assumed to be induced by aggregate production and income. The simple tax multiplier, as the name suggestions, is the simplest variation and includes only induced consumption. Every other component -- investment expenditures, government purchases, taxes, exports, and imports -- are assumed to be autonomous.

More complex tax multipliers include different combinations of induced components, ranging all of the way up to the "complete" tax multiplier that realistically includes all induced components. Induced consumption, investment, and government purchases all increase the value of the expenditures multiplier. Induced taxes and imports both decrease the value of the expenditures multiplier.

The Simple Tax Multiplier

The formula for this simple tax multiplier. (m[tax]), is:

m[tax]

=

- MPC

x

1 MPS

=

- MPC MPS

This formula is almost identical to that for the simple expenditures multiplier. The only difference is the inclusion of the negative marginal propensity to consume (- MPC).

If, for example, the MPC is 0.75 (and the MPS is 0.25), then an autonomous $1 trillion change in taxes results in an opposite change in aggregate production of $3 trillion.

Two Differences

• First, a change in taxes causes an opposite change in the disposable income of the household sector. An increase in taxes decreases disposable income and an decrease in taxes increases disposable income. This is why the simple tax multiplier has a negative value.


• Second, the household sector reacts to the change in disposable income caused by the change in taxes by changing both consumption and saving. How much consumption changes is based on the MPC. The MPC means that for each one dollar change in taxes, consumption and thus aggregate expenditures change by a only fraction. The fraction is equal to the MPC. The reason, of course, is that the taxes affect income and income is divided between saving and taxes.

If the marginal propensity to consume is 0.75, then consumption increases by $750 billion. This $750 billion change in consumption then triggers the multiplier process much like that for an autonomous change in investment expenditures. The difference, however, is the full $1 trillion change in investment triggers the multiplier process, but only 75 percent of the change in taxes works its way into the multiplier.

More Complex Tax Multipliers

Here is the formula for just such a multiplier, which can be labeled m[tax-all].

m[tax-all]

=

- MPC {1 - [MPC + MPI + MPG - (MPC x MPT) - MPM]}

Let's run through the cast of characters in this formula.

• MPC is the marginal propensity to consume.


• MPI is the marginal propensity to invest.


• MPG is the marginal propensity for government purchases.


• MPT is the marginal propensity to tax.


• MPM is the marginal propensity to import.

This particular tax multiplier formulation includes all induced components. However, several other tax multipliers that include different combinations of these induced components can be identified. One multiplier can include only induced consumption and induced investment. Another can include induced consumption, induced government purchases, and induced taxes. The possibilities are almost endless.

Other Multipliers

• Expenditures Multiplier: The expenditures multiplier measures changes in aggregate production caused by changes in an autonomous expenditure. Like the tax multiplier this comes in several varieties, simple and complex, depending on which expenditures and other components are induced by aggregate production and income. It differs from the tax multiplier in that aggregate expenditures change by full amount of the autonomous change.


• Balanced-Budget Multiplier: The balanced-budget multiplier measures the combined impact on aggregate production of equal changes in government purchases and taxes. The simple balanced-budget multiplier has a value equal to one.