Income Redistribution

Abstract

Chapter 9 examined optimal taxation from the viewpoint of efficiency. Although this efficiency issue is an important topic of optimal taxation, the optimal tax literature also investigates the equity issue. This chapter examines income redistribution policy among heterogeneous individuals within the same generation.

Appendix: Optimal Linear Income Tax

1.1 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.

1.2 A2 The Model

The model is essentially the same as in Sects. 2 and 3 of the main text of this chapter and is in accordance with Mirrlees (1971) and Sheshinski (1972). For simplicity, suppose two individuals have the same preferences but different skills. Let u(c, L) be the common utility function, where c > 0 is consumption and $0\le L\le 1$ is labor. It is assumed that u₁ > 0 and u₂ < 0, where u is strictly concave. We also assume normality of consumption and leisure. The skill of an individual is denoted by w. Namely, for the rich individual w = w_H, and for the poor individual, w = w_L. From now on, subscript H refers to the rich individual and L the poor individual.

The wage rate earned by a w-person is assumed to be w. Hence, her or his gross income Y is wL. Each consumer chooses c, z, and L so as to solve the following:

$$ \begin{array}{l} \max\ \mathrm{u}\ \left(\mathrm{c},\;\mathrm{L}\right)\\ {}\mathrm{s}.\mathrm{t}.\ \mathrm{c}=\mathrm{Z}-\mathrm{T}\left(\mathrm{Z}\right),\mathrm{Z}=\mathrm{w}\mathrm{L},\end{array} $$

(10.A1)

where T is the tax function. We consider a linear tax function as follows:

$$ T\left(\mathrm{Y}\right)=-A+\left(1-\beta \right)Y $$

(10.A2)

where A is the minimum guaranteed income and 1 − β = t is the marginal tax rate.

We denote by c(βw; A) and L(βw; A) respectively w-person’s demand for consumption and her or his supply of labor. βw is the after-tax real wage rate and A is non-wage income. We also define an indirect utility function: v(βw; A) = u[c(βw; A)L((βw; A))].

Let R be a predetermined level of per capita government spending, so that the government’s budget constraint is T_H + T_L = R. Employing Eq. (10.A2), this constraint reduces to:

$$ R+2A=\left(1-\beta \right)\left[{w}_HL\left(\beta {w}_H;A\right)+{w}_LL\left(\beta {w}_L;A\right)\right] $$

(10.A3)

Let us draw a diagram of the tax possibility frontier. In Fig. 10.A1, curve BB shows the government budget constraint for given R. When β = 0, L is zero. From Eq. (10.A3), A = −R/2 (OA = R/2). When β = 1, A is also given by − R/2. For small values of β, L increases with β and A also increases. However, because a rise of β means a reduction of tax rate, the feasible guarantee eventually declines. M is the highest point of curve BB, and $\overline{\beta}$ is the associated β. For $\overline{\beta}<\beta$, the negative effect on revenue of a decrease in the marginal tax rate (1 − β) dominates the positive effect on revenue of an increase in work effort. We call curve BB the tax possibility frontier (TPF).

Fig. 10.A1

The tax possibility frontier

Mathematically, we have

$$ \frac{dA}{d{\beta}_{AB}}=\frac{\left(1-\beta \right)\left({w}_L{L}_{L\beta }+{w}_H{L}_{H\beta}\right)-\left({w}_L{L}_L+{w}_H{L}_H\right)}{2-\left(1-\beta \right)\left({w}_L{L}_{LA}+{w}_H{L}_{HA}\right)} $$

(10.A4)

where ${L}_{ij}=\partial {L}_i/\partial j\kern0.24em \left(\mathrm{i}=\mathrm{L},\ \mathrm{H}\ \mathrm{and}\ \mathrm{j}=\mathrm{A},\;\upbeta \right).$

Let us draw an indifference curve of an individual, curve I. Curve I is downward sloping. An increase in A raises utility and must be offset by a decrease in β so as to maintain the same utility. Considering the first order condition of utility maximization, we have

$$ dA/d{\beta}_I=-Y<0 $$

(10.A5)

It is interesting to note that curve I is not necessarily strictly convex.

The linear income tax problem may be written as

$$ \begin{array}{l} \max\ \mathrm{W}\left(\mathrm{A},\;\upbeta \right)\\ {}\mathrm{s}.\mathrm{t}.+\left(10.\mathrm{A}3\right),\end{array} $$

(10.A6)

where W(A, β) is the social welfare function . The social welfare function is given by

$$ W\left(A,\beta \right)=\frac{1}{1-\nu}\left[\nu {\left(A,\beta {w}_L\right)}^{1-\nu }+\nu {\left(A,\beta {w}_H\right)}^{1-\nu }-2\right] $$

(10.A7)

where $\nu \ge 0$. With $\nu =0$, we have the Bentham (utilitarian) objective. With $\nu =\infty$, we have the Rawls (maximin) case. For higher values of ν the function is more concave. We now illustrate a social indifference curve in the (A, β) plane. The slope of the social indifference curve is given by

$$ \frac{dA}{d{\beta}_w}=-\frac{\nu_L^{-\nu }{\nu}_{L\beta }+{\nu}_H^{-\nu }{\nu}_{H\beta }}{\nu_L^{-\nu }{\nu}_{LA}+{\nu}_H^{-\nu }{\nu}_{HA}} $$

(10.A8)

where ${\nu}_{ij}=\partial {\nu}_i/\partial j$ (i = L, H and j = A, β). The social indifference curve is not necessarily convex. Figure 10.A1 illustrates the social optimal point E where curve W is tangent to curve BB. Once we know the tax possibility frontier and the social indifference curve, we can attain the optimal point.

1.3 A3 Shift of the Social Welfare Function

In this section, we investigate the comparative statics of the weight of the social welfare function. When ν changes, the social indifference curve moves, but the tax possibility frontier does not. The optimal point moves on the initial tax possibility frontier. As shown by the movement from W to W’ in Fig. 10.A2, if the absolute slope of the social indifference curve increases with the same values of A and β, the optimal point moves to the right: the optimal level of A decreases and the optimal level of β increases. Thus, it is useful to differentiate dA/dβ with respect to v.

Fig. 10.A2

Shift of the social welfare function

$$ \begin{array}{l}\frac{d^2A}{d\beta d\nu}=\frac{1}{{\left({\nu}_L^{-\nu }{\nu}_{LA}+{\nu}_H^{-\nu }{\nu}_{HA}\right)}^2}\Big[\left({\nu}_L^{-\nu }{\nu}_{LA}+{\nu}_H^{-\nu }{\nu}_{HA}\right)\left({\nu}_L^{-\nu }{\nu}_{L\beta } \log {\nu}_L-{\nu}_H^{-\nu }{\nu}_{H\beta } \log {\nu}_H\right)\\ {}\kern1.32em +\left({\nu}_L^{-\nu }{\nu}_{L\beta }+{\nu}_H^{-\nu }{\nu}_{H\beta}\right)\left({\nu}_L^{-\nu }{\nu}_{LA} \log {\nu}_L-{\nu}_H^{-\nu }{\nu}_{HA} \log {\nu}_H\right)\Big]\\ {}\kern1.20em =\frac{1}{{\left({\nu}_L^{-\nu }{\nu}_{LA}+{\nu}_H^{-\nu }{\nu}_{HA}\right)}^2}{\nu}_L^{-\nu }{\nu}_H^{-\nu}\left({\nu}_{HA}{\nu}_{L\beta }-{\nu}_{H\beta }{\nu}_{LA}\right)\left( \log {\nu}_H- \log {\nu}_L\right)\end{array} $$

(10.A9)

We know that w_L < w_H and ${\nu}_{\mathrm{H}}>{\nu}_{\mathrm{L}}$. We show that

$$ {v}_{L\beta }{\nu}_{HA}-{v}_{H\beta }{\nu}_{LA}<0 $$

or

$$ \frac{\nu_{L\beta }}{\nu_{LA}}<\frac{\nu_{H\beta }}{\nu_{HA}} $$

Using the envelope theorem, it is straightforward to see that

$$ \frac{\nu_{\beta }}{\nu_A}=c $$

Since c is an increasing function of w, it follows that the above inequality holds.

Hence, it is easy to see that the sign of Eq. (10.A9) is negative. The absolute value of the slope of the social indifference curve decreases with ν. Thus, the optimal value of β decreases with ν, and the optimal value of A increases with ν. When the social function approaches the Rawls criterion as $\nu \to \infty$, the optimal tax parameters converge to the Rawls optimal tax parameters. When the social welfare function approaches the Bentham criterion as $\nu \to 0$, the optimal tax parameters converge to the Bentham optimal tax parameters. We confirm analytically the conjecture that the optimal marginal tax rate increases with the government’s inequality aversion.

1.4 A4 Shift of the Tax Possibility Frontier

Let us examine how the optimal point changes when R is increased. In this situation, the tax possibility frontier moves but the social indifference curve does not. From now on, we concentrate on the case of the educational investment model in accordance with Sheshinski (1971). Remember that the educational investment model is a special instance of the labor incentive model. We have

$$ \mathrm{u}\left(\mathrm{c},\;\mathrm{L}\right)=\mathrm{u}\left[\mathrm{c}-\mathrm{g}\left(\mathrm{L}\right)\right] $$

and first order conditions that are

$$ {\nu}_A=u^{\prime}\kern0.24em \mathrm{and} $$

(10.A10a)

$$ {\nu}_{\beta }=u^{\prime }c $$

(10.A10b)

g(L) is the cost of education where g(ċ) is convex (i.e., there are increasing marginal costs). It is unnecessary to assume that u is strictly concave here.

In the educational investment model, the income effect on labor supply is assumed as y_A = 0. Substituting y_A = 0 into Eq. (10.A4), the slope of the tax possibility frontier is given by

$$ \frac{dA}{d{\beta}_B}=\frac{\left(1-\beta \right)\left({w}_L{L}_{L\beta }+{w}_H{L}_{H\beta}\right)-\left({w}_L{L}_L+{w}_H{L}_H\right)}{2} $$

(10.A11)

Because L depends only upon β, the slope of curve BB is determined solely by the level of β.

A celebrated property of the educational investment model is that the slope of curve BB is independent of A; thus, we can explore intuitive implications of an increase in R. An increase in R moves curve BB downwards. It is easy to see that the combination of dA/dR > 0 and dβ/dR > 0 is not feasible. The marginal tax rate should be moderate in the sense that a decrease in 1 − β = t does not induce people to work in such a way that it increases the tax revenue. An extra resource left to the government is unavailable as the result of increases in A and β. Thus, we have the following three possibilities:

• (a) $\frac{dA}{dR}<0\kern0.24em \mathrm{and}\kern0.24em \frac{d\beta }{dR}\ge 0$

• (b) $\frac{dA}{dR}<0\kern0.24em \mathrm{and}\kern0.24em \frac{d\beta }{dR}<0,\kern0.24em \mathrm{and}$

• (c) $\frac{dA}{dR}\ge 0\kern0.24em \mathrm{and}\kern0.24em \frac{d\beta }{dR}<0$

We consider the special instances of $\nu =\infty$ (the maximin case) and $\nu =0$ (the utilitarian case) with regard to the social welfare function . Note that the analysis conducted for the case of $\nu =0$ is also valid for all $\nu \ge 0$.

1.4.1 A4.1 The Maximin Case

When the social welfare function is given by the maximin criterion , the relevant social indifference curve is given by the least prosperous person’s indifference curve I_L, which is strictly concave. Considering y_A = 0, the curve’s slope depends only on β. Because the slope of curve BB also depends only on β, the new optimal point E’ in Fig. 10.A3 is just below the initial optimal point E. The optimal marginal tax rate in the maximin criterion, $1-{\beta}_M^{*}$, is independent of R. Thus, we have case (a).

Fig. 10.A3

The maximin case

According to the maximin criterion , an increase in the tax revenue requirement should be financed by a decrease in the income guarantee, while the marginal rate should be kept constant. Observe that this policy implication is valid as long as social welfare is given by the representative individual of some given target ability ŵ. As shown in the prior section, it is true that the optimal marginal tax rate is higher in the maximin case than in any other. The optimal level of β increases with target ability ŵ. However, a less progressive tax structure is desirable when R is increased.

1.4.2 A4.2 The Utilitarian Case

Let us now consider the utilitarian case. Substituting $\nu =0$ and Eq. (10.A10a,b) into Eq. (10.A8), the slope of the social indifference curve is reduced to

$$ \frac{dA}{d{\beta}_U}=-\frac{u{\prime}_L{Y}_L+u{\prime}_H{Y}_H}{u{\prime}_L+u{\prime}_H} $$

(10.A12)

Social indifference curve U is downward-sloping. However, curve U is not necessarily concave.

Let us now examine the comparative statics of an increase in R. If dA/dβ_U is independent of A as in the maximin case, the new optimum E’ is just below E in Fig. 10.A4, given by a point such as F. If dA/dβ_U is an increasing function of A at point F, the slope of curve U is steeper than that of curve BB; thus, the new optimal point is to the southeast of E. Consequently, if d²A/dβdA_U ≥ 0, we have case (a), and a less progressive tax structure is desirable.

Fig. 10.A4

The utilitarian case

We have

$$ \begin{array}{l}\frac{d^2A}{d\beta dA}=\frac{1}{{\left(u{\prime}_L+{u\prime}_H\right)}^2}\left[\left(u{{\prime\prime}}_L{Y}_L+u^{{\prime\prime} }{}_HY_H\right)\right(u{\prime}_L+{u\prime}_H\left)-\left(u{\prime}_L{Y}_L+u^{\prime }{}_HY_H\right)\right(u{{\prime\prime}}_L+{u{\prime\prime}}_H\left)\right]\\ {}=\frac{1}{{\left(u{\prime}_L+{u\prime}_H\right)}^2}\left({r}_L-{r}_H\right)\left({Y}_L-{Y}_H\right)u{\prime}_L{u\prime}_H\end{array} $$

(10.A13)

where $r\equiv -u^{{\prime\prime} }/u^{\prime }$ is absolute risk aversion.

If absolute risk aversion is constant, Eq. (10.A13) is zero; hence, we have dβ/dR = 0 and dA/dR < 0, as in the maximin case. If absolute risk aversion is non-decreasing, Eq. (10.A13) is positive and we have case (a).

If the utility function is such that the marginal utility of higher ability persons is relatively more weighted as the result of a decrease in A (an increase in R), the slope of the social indifference curve U becomes steeper; hence, case (a) is more likely, and a less progressive tax structure is optimal. It should be stressed that if the new optimal point is between M’ and F, it is still possible to have a less progressive tax structure: case (b). Even when r is increasing (but not so rapidly), a less progressive tax structure could be optimal.

1.5 A5 Conclusion

As an extension of Sect. 3 of the main text of the chapter, this appendix has considered the role of the tax possibility frontier and the social indifference curve in a comparative statics analysis. It is shown that when the social indifference curve moves, the comparative statics result is analytically well investigated. We confirm the conventional conjecture that the optimal marginal tax rate increases with a government’s inequality aversion.

However, if the tax possibility frontier moves, the comparative statics result is rather ambiguous. Even if we employ an extreme case of the educational investment model , we cannot always confirm analytically the conventional conjecture that the optimal marginal tax rate increases with a government’s budgetary needs .

Questions

• 10.1 Consider the three-person economy. Each person’s income is given by 10, 30, and 50 respectively. What is the income tax schedule to realize the perfect equality of after-tax income?

• 10.2 Consider the following Cobb Douglas utility function :

  $$ \mathrm{U}={\mathrm{c}}^{\upalpha}{\left(\mathrm{Z}-\mathrm{L}\right)}^{1-\upalpha}, $$

  where Z is available time endowment, L is labor supply , and c is consumption. Suppose the government imposes a linear income tax and a lump sum transfer:

  $$ \mathrm{c}=\left(1-\mathrm{t}\right)\mathrm{w}\mathrm{L}+\mathrm{A}, $$

  where t is the tax rate and A is transfer. What is the optimal labor supply function? Draw the tax possibility curve .