# Appendix: Optimal Linear Income Tax

# A1 Introduction

There are four main ingredients for a model of standard optimum linear income taxation: a social welfare function, a preference relation or labor supply function for individuals, an ability structure and distribution, and a revenue requirement for the government. As discussed by Atkinson and Stiglitz (1980), the standard conjectures may be summarized as follows.

- (i) The optimal marginal tax rate increases with the government’s inequality aversion.
- (ii) The optimal marginal tax rate decreases with the elasticity of labor supply.
- (iii) The optimal marginal tax rate increases with the spread in abilities.
- (iv) The optimal marginal tax rate increases with the government’s needs.

In this appendix, we intend to reexamine conjectures (i)-(iv) using a diagram of

the tax possibility frontier and the social indifference curve as explained in Sect. 3 of this chapter, based on Ihori (1987). All the conjectures, (i)-(iv), are not always analytically valid.

Section 3 of the main text has shown that the marginal tax rate is higher under Rawls’s criterion than under Bentham’s (see also, among others, Ihori (1981, 1987); Hellwig (1986)). The optimal marginal tax rate is bounded above by the Rawlsian rate, which in turn is bounded by the revenue-maximizing rate. Helpman and Sadka (1978) have reported that the effect of a mean-preserving spread in abilities cannot be determined in general.

The purpose of this appendix is to contribute to the understanding of the structure of the optimal linear income tax model. The appendix achieves this through diagrammatic examination of some comparative statics, using a diagram of the tax possibility frontier and the social indifference curve, as in the main text of the chapter.

The appendix is organized as follows. Section A2 recapitulates Sheshinski’s (1972) formulation of the linear income tax problem and presents a diagram of the tax possibility frontier and the social indifference curve. Section A3 analyzes the response of the parameters of optimal linear income tax to changes in the social objective function from Bentham’s sum-of-utilities to Rawls’s max-min. It is shown that conjecture (i) is analytically established. Section A4 analyzes the response of the parameters to changes in government budgetary needs, using Sheshinski’s (1971) educational — investment model. It is shown that conjecture

(iv) cannot be valid in general. Finally, Sect. A5 concludes this appendix.

Section A3 corresponds to the comparative statics of the social welfare function (movement of the social indifference curve), and Sect. A4 corresponds to the comparative statics of the tax requirement (movement of the tax possibility frontier). It is shown that once the tax possibility frontier moves, the analytical results become ambiguous in general. Note that conjectures (ii) and (iii) correspond to the situation where both the tax possibility frontier and the social welfare function move.