The theoretical literature on tax reform discusses a desirable tax system that assures the required revenue when multiple taxes are available. First, let us compare labor income tax and interest income tax with the equal revenue requirement.

Consider a simple two-period model. Let us denote the labor income tax rate by t_{w} and the interest income tax rate by t_{r}. The budget constraints of a representative consumer for each tax are written as

and

where c_{1} and c_{2} denote consumption in each period, Y is labor income, s is saving, and r is the rate of interest. Equations (9.1) and (9.2) include labor income tax, while Eqs. (9.3) and (9.4) include interest income tax. In this section, we assume that labor income Y is exogenously given. This is an important and crucial assumption. We also assume that the agent earns labor income only in period 1.

T. Ihori, Principles of Public Finance, Springer Texts in Business and Economics, DOI 10.1007/978-981-10-2389-7_9

where Eq. (9.5) represents the labor income tax case and Eq. (9.6) the interest income tax case. In both equations, the left-hand side is the present value of consumption and the right-hand side is the present value of labor income. Labor income tax reduces disposable labor income, while interest income tax raises the relative price of future consumption.

In order to collect the same amount of tax revenue, which tax is relatively desirable for a household? Figure 9.1 explains this problem. The vertical axis is future consumption and the horizontal axis is present consumption. Point E is the initial equilibrium point before tax where the before-tax budget line is tangent to an indifference curve. E_{w} is the equilibrium point under labor income tax and E_{r} is the equilibrium point under interest income tax. As shown in this figure, under the constraint of the same tax revenue, utility at E_{w} is always higher than utility at E_{r}.

Let us explain this result in Fig. 9.1. The vertical (or horizontal) gap between line AB and line DF corresponds to tax revenue in terms of second (or first) period consumption. If the equilibrium point is on line DF, the government may collect the same amount of tax revenue in either case. Thus, we have to compare two points associated with labor income tax and interest income tax on line DF. With regard to t_{w}, the relative price between c_{1} and c_{2} is not affected by tax; hence, the DF line is tangent to an indifference curve at E_{w}. In contrast, with regard to t_{r}, the relative price between c_{1} and c_{2} is affected by tax; hence, line AB^{0}, not AB, is tangent to an indifference curve at E_{r} on line DF.

Thus, utility at E_{r} is lower than utility at E_{w}. The optimal choice between c_{1} and c2 is distorted by interest income tax and creates an extra burden. This analysis suggests that labor income tax is better than interest income tax.