Let us make an in-depth study of the Proportional Tax Function and the Balanced Budget Multiplier.

If we consider the SKM without foreign trade, i.e., if we take a closed economy the BBM can be less than one if we change the usual assumption about the tax function.

Normally, in the SKM with government we take taxes as fixed.

In case of fixed tax (such as 7) disposable income (Yd) is the difference between national income and tax paid Yd = Y – T.

Now let us change the assumption about tax. Let us take taxes as proportional to income. Here we assume, more specifically, that taxes have both an autonomous and an induced (income-related) component. This means that there is an additional leakage from the circular flow of income.

If a certain portion of every extra rupee which is earned due to an increase in G is taxed away, consumption will not increase as much as it would otherwise (due to fall in disposable income). So the process of income generation will slow down and the increase in Y will be less than dG = dT, when the BBM is less than 1.

In fact, when taxes become a function of income, an increase in government purchases raises the level of disposable income only by (1 -1) times the increase in income. It is because a portion of t flows back to the government in the form of tax collections.

Since disposable income rises by only (1 – t) times the increase in government purchases (G) and since b times the change in disposable income is respent on consumption, the proportion b( 1 -1), instead of b will be respent on consumption. Thus the marginal propensity to consume national product is reduced and multiplier takes on a lower value given by the formula

This important proposition may now be proved.