On the optimal rate structure of an individual income tax.

• Author: Ching-Huei Chang

1. Introduction

   What is the optimal rate structure of a personal income tax? In the literature on optimal taxation this important question has been approached by introducing a linear income-tax formula into a revised Ramseyian model which takes into account both efficiency and equity objectives of taxation. The assumption of a linear tax enables us to examine the degree of tax progression through the behavior of the average rate, but it does not throw light on how the marginal rate should vary with income |1, Lecture 13~. A classic paper by Mirrlees |7~, which was later expounded on by Atkinson and Stiglitz |1, 412-22~, solves this problem by explicitly considering a general income tax schedule. However, both linear and general tax formulae provide little guide for policy making, since in practice the tax rates are graduated by a bracket system. That is, the income scale is divided into segments or brackets and rates are applied only to the income in each bracket. Rates increase (or decrease) by small percentage points from one bracket to the next, in order to avoid large and abrupt changes in tax burden as income rises.

   The present paper attempts to demonstrate how we can easily transform a segment-graduated tax system into a multiple linear system and then incorporate it into the standard optimum-taxation model. We then proceed to derive the conditions governing the optimal value of an upper (or lower) limit of a tax bracket and the marginal tax rate applied to the bracket. Combining these conditions together will enable us to infer what the optimal rate structure should be.

2. Identical Tastes and Earnings

   We begin with the simplest case. All individuals are identical in tastes and ability to earn wage income. Designate by u (Y, 1 - L) the "representative" individual's utility function, where Y is the after-tax income, L (0 |is less than~ L |is less than~ 1) is the labor supply, and u is strictly quasi-concave with respect to its argument. By definition, Y = Z - T(Z), where Z is the before-tax income and T denotes tax payment, which is a function of Z. With zero unearned income, Z = WL, where the wage rate per hour W is a constant.

   It should be noted that in this paper we follow the optimum taxation literature by assuming that the individual is the basic unit of taxation |1, Lecture 13; 2~. This assumption may not be so unrealistic, considering so many countries in the world (such as Canada, Japan, Australia, New Zealand, England, Singapore, Taiwan, and etc.) that treat the individual rather than the family as the primary unit of account for personal taxation. The readers who are interested in the optimal linears tax system for married couples should refer to the paper by Boskin and Sheshinski |3~. Equally noteworthy is the assumption that an individual's labor supply is primarily measured by hours of work in a given period of time. The results obtained below may be subject to changes or revisions once a non-neoclassical model of labor supply (like a life cycle model) is used or if other dimensions of job (like work effort intensity and job choice) are taken into account |4~.

   The personal income tax rates are graduated by a bracket system. Figure 1 depicts the rate structure of the tax system. In general, the tax payment of an individual whose before-tax income falls in the jth bracket is equal to

   T(Z) = |t.sub.1~(|Z.sub.1~ - |Z.sub.0~) + |t.sub.2~(|Z.sub.2~ - |Z.sub.2~) + . . . + |t.sub.j~(|Z.sub.j~ - |Z.sub.j-1~) + |T.sub.0~, (1)

   where |T.sub.0~ is a lump-sum tax, |Z.sub.i~(i = 1, . . . , j) is the upper-limit income in the ith bracket, |t.sub.i~ is the marginal tax rate applied to the income of ith bracket, and |Z.sub.0~ denotes the threshold level of income that is exempt from taxation.(1) With a constant wage rate, the tax formula above can be written as

   T(Z) = W||t.sub.1~(|H.sub.1~ - |H.sub.0~) + |t.sub.2~(|H.sub.2~ - |H.sub.1~) + . . . + |t.sub.j~(L - |H.sub.j-1~)~ + |T.sub.0~, (1|prime~)

   where |H.sub.i~ is the quantity of the labor supply corresponding to |Z.sub.i~. By substitution, the after-tax income can be specified as

   Y = W|H.sub.0~ + |summation of~ |w.sub.i~(|H.sub.i~ - |H.sub.i-1~) where i=1 to m-1 + |w.sub.j~(L - |H.sub.j-1~) - |T.sub.0~, (2)

   where |w.sub.i~ |is equivalent to~ (1 - |t.sub.i~)W is the after-tax wage along the ith bracket (i = 1, . . . , j). The above equation characterizes the stepwise budget constraint, OABCD . . . , facing the representative individual in Figure 1. Interposition of the indifference curves derived from the individual's utility function will result in an inner equilibrium along a linear segment or a corner solution at a kink point of this budget constraint.

   Obviously, with a budget constraint like the one given in (2), it is difficult to approach the individual's optimization, and eventually the optimal taxation problem, by means of a simple mathematical method. For ease of analysis, we rewrite (1|prime~) as

   T = |t.sub.i~WL - |G.sub.i~, for |H.sub.i - 1~ |is less than~ L |is less than~ |H.sub.i~, i = 1, . . . , I, (3)

   where |G.sub.i~ |is equivalent to~ W|(|t.sub.i~ - |t.sub.i-1~)|H.sub.i-1~ + . . . + (|t.sub.2~ - |t.sub.1~)|H.sub.1~ + |t.sub.1~|H.sub.0~~ - |T.sub.0~ and |t.sub.i~ is the highest marginal rate.(2) Equation (3) represents the familiar linear tax formula, with t denoting the marginal rate and G the guaranteed income level or characteristic income, as it is called by Hausman |5~.(3) After (3) is rewritten in this way, the individual's budget constraint is given by

   Y = |w.sub.i~L + |G.sub.i~, for |H.sub.i-1~ |is less than~ L |is less than or equal to~ H, i = 1, . . . , I, (4)

   which is also of a linear form. As will become clear later, the formulations in (3) and (4) are very helpful in the...