# The Theory of Taxation

**8**

## Taxation and Labor Supply

### A Model of Labor Supply

The second half of this book considers the microeconomic aspects of public finance, which are standard topics of modern public finance. With regard to a useful textbook on microeconomics, see Varian (2014) among others. It is also useful to read standard textbooks on public finance such as Rosen (2014) and Stiglitz (2015).

In this chapter, we first investigate the microeconomic effects of tax on labor supply and other economic activities. In order to investigate this issue, it is necessary to formulate a way in which to determine labor supply in a simple microeconomic model. A household optimally allocates its available time Z between labor supply L and leisure x. Thus,

Z is exogenously given, say, as 24 h per day or 356 days per year. An increase in L means a decrease in x by the same amount.

Labor supply has the benefit of earning income, although it has the cost of sacrificing leisure. Optimal labor supply is determined at the point where the marginal benefit and marginal cost of labor supply are equal. Let us explain this point briefly.

The standard utility function is given as

where c is consumption and L is labor. Utility U increases with c and decreases with L (increases with leisure x).

When a tax is imposed on labor income, the budget constraint is given as © Springer Science+Business Media Singapore 2017

T. Ihori, *Principles of Public Finance,* Springer Texts in Business and Economics, DOI 10.1007/978-981-10-2389-7_8

where t is income tax rate and w is wage rate. The right-hand side of Eq. (8.2) means the disposable income. wL is before-tax labor income and tw is the payment of taxes on labor income. (1 — t)w means the after-tax effective wage rate. For simplicity, we assume a proportional linear tax structure in this section.

The household optimally determines its labor supply to maximize its utility (1) subject to its budget constraint (2). In Fig. 8.1, the vertical axis is consumption and the horizontal axis is labor supply. The budget line before tax is line OA, the slope of which is the wage rate, w. As shown in Fig. 8.1, the point E_{0}, where the budget line OA is tangent to an indifference curve, is initially optimum. At this point, the marginal benefit of raising labor supply, the slope of the budget line, is equal to the marginal cost, the slope of the indifference curve.

Alternatively, with t > 0 mathematically, we have as the first order condition

where U_{L} denotes the marginal disutility of labor and U_{c} denotes the marginal utility of consumption. The left-hand side of Eq. (8.3) is the marginal benefit of labor, which corresponds to the slope of the budget constraint. The right-hand side of Eq. (8.3) is the marginal cost of labor, which corresponds to the slope of the indifference curve.

How does income tax affect labor supply? Imagine that initially any labor income tax is not imposed. Then, a labor income tax is imposed. In Fig. 8.1, E_{0 }corresponds to the initial instance oft = 0. When t > 0 is imposed, budget line OA moves to OA' with a flatter slope. This means that the effective wage rate, (1 — t)w, declines by an increase in t. Thus, the equilibrium point moves from E_{0} to E_{1}. It is easy to see that E_{1} is below E_{0}, but could be to the left or right of E_{1}. This is because the substitution and income effects offset each other to some extent.

**Fig. 8.1 Tax and labor supply**