# The Effect of Interest Income Tax: The Substitution Effect and the Income Effect

Imagine that interest income tax t_{s} is imposed. This tax reduces the after-tax rate of return on saving. Namely, the after-tax interest rate reduces from r to r(1 — t_{s}). Would this decline depress savings?

Fig. 8.5 **The effect of interest income tax**

In Fig. 8.5, the budget line moves from AB to AB' to the left and downward. The optimal point moves from E to E'. E' is below E, but it could be to the right or left of E. This is because the substitution effect and the income effect offset each other, as with labor supply in Sect. 1. Namely, the substitution effect reduces the relative attractiveness of saving, while the income effect raises saving by depressing current consumption because of a decline in effective income. Note that consumption is a normal good. Thus, the income effect is always positive. Figure 8.5 illustrates the effect of interest income tax.

# The Cobb-Douglas Utility Function

For example, if we consider the Cobb-Douglas utility function, the substitution effect completely offsets the income effect; thus, saving becomes independent of interest income tax. This property is qualitatively the same as the impact of income tax on labor supply in Sect. 1.

Thus, the Cobb-Douglas utility function is specified as

The budget constraint with interest income tax is given as
Then, it is optimal to allocate income between c_{1} and c_{2} as follows:

Considering Eq. (8.7) or (8.8), we finally obtain

which is independent of the income tax rate, t_{s}. In this instance, an increase in the interest income tax does not depress saving. We do not have the disincentive effect of interest income tax on saving. This property is qualitatively the same as with labor supply, which is explained in Sect. 1.