# A tax-modified multiplier

To explore how taxation works as a fiscal policy tool, a modification can be made to the aggregate demand model and multiplier. To keep things simple, we will assume a tax rate for the whole economy, capturing the proportion of income raised from tax. The coefficient *t* can be defined as the rate of tax – i.e. the proportion of national income levied by tax. Hence the total revenue received by the government from tax (*T*) is

*T* = *tY*

With a national income (*Y *) of £1200 billion, for example, and a tax rate of 1/3, the total tax revenue will be £400 billion.

The amount of income available for consumption (often referred to as disposable income) is now equal to income (*Y *) minus taxation (*tY *):

*Y* − *tY* = (1 − *t *)*Y*

It therefore follows that the tax-modified consumption function is

*C* = *a* + *b*(1 − *t *)*Y*

This tax-modified consumption function, shown in Figure 20, has a shallower slope than the consumption function shown in Figure 14. This new slope is represented by a new (tax-modified) marginal propensity to consume *b*(1 − *t *).

We can now consider the effect of modelling tax on the structure of the multiplier, which was shown earlier in this section to take the form 1/(1 − *b*). It intuitively follows that instead of having a denominator 1 − *b*, the tax-modified multiplier has a denominator 1 − *b*(1 − *t *). The marginal propensity to consume (*b*) is no longer the only determinant of the value of the multiplier. We now also need to take account of the leakage of tax, as shown in Figure 20, and the new multiplier is

## Activity 13

Suppose that the marginal propensity to consume (*b*) is equal to 0.75 and the tax rate (*t*) is 0.2. Calculate the size of the tax-modified multiplier.

### Answer

This calculation involves carrying out a number of steps.

*Step 1*: Calculate 1 − *t* = 1 − 0.2 = 0.8

*Step 2*: Calculate *b*(1 − *t*) = 0.75 × 0.8 = 0.6

*Step 3*: Calculate 1 − *b*(1 − *t*) = 1 − 0.6 = 0.4

*Step 4*: Calculate 1/(1 − *b*(1 − *t*)) = 1/0.4 = 2.5

The tax-modified multiplier has a value of 2.5. Without the leakage of taxation, using the calculation 1/(1 − *b*) = 1/(1 − 0.75), the multiplier would be equal to 4. So instead of a fourfold increase in income, in response to a fiscal stimulus, only a 2.5-fold increase is generated once the leakage of tax is taken into account.