How does inequality aversion affect inequality and redistribution?

Abstract

We investigate the effects of inequality aversion on equilibrium labor supply, tax revenue, income inequality, and median voter outcomes in a society where agents have heterogeneous skill levels. These outcomes are compared to those which result from the behavior of selfish agents. A variant of Fehr-Schmidt preferences is employed that allows the externality from agents who are “ahead” to differ in magnitude from the externality from those who are “behind” in the income distribution. We find first, that inequality-averse preferences yield distributional outcomes that are analogous to tax-transfer schemes with selfish agents, and may either increase or decrease average consumption. Second, in a society of inequality-averse agents, a linear income tax can be welfare-enhancing. Third, inequality-averse preferences can lead to less redistribution at any given tax, with low-wage agents receiving smaller net subsidies and/or high-wage individuals paying less in net taxes. Finally, an inequality-averse median voter may prefer higher redistribution even if it means less utility from own consumption and leisure.

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Correspondence to Langchuan Peng.

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Appendix

Appendix

Claim: Wage income, wL(w,τ), is increasing in w if and only if

σ(1 − αH(w) + β[1 − H(w)]) > w(α + β)h(w) for all \(w\in [\underline {w},\overline {w} ]\).

Proof

Using (2) we have

$${wL(w,\tau )=w^{\frac{\sigma} {\sigma -1}}[\frac{(1-\tau )\{1-\alpha H(w)+\beta [1-H(w)]\}}{\xi} ]}^{\frac{1}{\sigma -1}}. $$

Now differentiate the above with respect to w to get a necessary and sufficient condition for labor income to be increasing in the wage:

$$\begin{array}{@{}rcl@{}} \frac{\partial (wL(w,\tau ))}{\partial w}&=&{w^{\frac{\sigma} {\sigma -1}}[\frac{(1-\tau )\{1-\alpha H\left( w \right)+\beta [1-H(w)]\}}{\xi} ]}^{\frac{1}{\sigma -1}-1}(\frac{1}{\sigma -1})\\&&\times(\frac{-(1-\tau )(\alpha +\beta )h(w)}{\xi} ) \end{array} $$
$$+{w^{\frac{1}{\sigma -1}}[\frac{(1-\tau )\{1-\alpha H(w)+\beta [1-H(w)]\}}{\xi} ]}^{\frac{1}{\sigma -1}}(\frac{\sigma} {\sigma -1})>0. $$

The parametric condition for \(\frac {\partial (wL(w,\tau ))}{\partial w}>0\) is thus

$${w^{\frac{1}{\sigma -1}}[\frac{(1-\tau )\{1-\alpha H(w)+\beta [1-H(w)]\}}{\xi} ]}^{\frac{1}{\sigma -1}}(\frac{\sigma} {\sigma -1})>$$
$${w^{\frac{\sigma} {\sigma -1}}[\frac{(1-\tau )\{1-\alpha H(w)+\beta [1-H(w)]\}}{\xi} ]}^{\frac{1}{\sigma -1}-1}(\frac{1}{\sigma -1})(\frac{(1-\tau )(\alpha +\beta )h(w)}{\xi} ). $$

Finally, rewriting the above yields the necessary and sufficient condition (given in (3)) that must be satisfied at all w in order for labor income to be monotonically increasing in the wage: σ(1 − αH(w) + β[1 − H(w)]) > w(α + β)h(w). □

Proof of Proposition 1

An inequality-averse individual i with an hourly wage wi provides weakly greater labor than a self-interested individual with the same hourly wage if and only if:

$$ {[\frac{(1-\tau )w_{i}\{1-\alpha H(w_{i})+\beta [1-H(w_{i})]\}}{\xi}]}^{\frac{1}{\sigma -1}}\ge {\mathrm{[[}\frac{(1-\tau )w_{i}}{\xi}]}^{\frac{1}{\sigma -1}}. $$
(8)

Rearranging the above yields the condition

$$ H(w_{i})\le \frac{\beta} {\alpha +\beta}. $$
(9)

Recall that \(\tilde {w}\) is defined as the wi such that (9) holds with equality. Since H is a strictly increasing function, (9) is satisfied if and only if \(w_{i}\le \tilde {w}\). □

Before stating the proofs of Proposition 2 and 3, define average pre-tax income as

$$\text{I}(\alpha, \beta )={[\frac{(1-\tau )}{\xi} ]}^{\frac{1}{\sigma -1}}{\int}_{\underline{w}}^{\overline{w}} {z^{\frac{\sigma} {\sigma -1}}\{\left[ 1-\alpha H\left( z \right)+\beta \left[ 1-H\left( z \right) \right] \right]^{\frac{1}{\sigma -1}}\}h(z)dz} , $$

with I(0,0) corresponding to selfish agents. It is clear that I(α,β) is increasing in β and decreasing in α. For a given τ, both average tax revenue, S = τ I(α,β), and average consumption, (1 − τ)I(α,β), and are increasing in I(α,β), so the following proofs focus on I(α,β).

Proof of Proposition 2

(i).:

We know from above that I(0,0) < I(0,β). Suppose β ≤ 1. If I(β,β) ≥I(0,0), the result holds for α = β. If I(β,β) < I(0,0) < I(0,β), the result follows from the observation that I(α,β) is a continuous function. Next, suppose β > 1. If I(α,β) ≥I(0,0), the result holds for α = 1. If I(1,β) < I(0,0) < I(0,β), the result follows from the observation that I(α,β) is a continuous function.

(ii).:

Follows from 2(i) and Proposition 1.

Proof of Proposition 3

(i).:

We first establish that I(β,β) < I(0,0) for a symmetric distribution with σ > 2. The result then follows from the fact that I(α,β) is a continuous decreasing function of α. Take the derivative of ρ(β) ≡I(β) to yield

$$ \frac{d\rho (\beta )}{d\beta} =\frac{1}{\sigma -1}{[\frac{(1-\tau)}{\xi} ]}^{\frac{1}{\sigma -1}}{\int}_{\underline{w}}^{\overline{w}} {z^{\frac{\sigma} {\sigma -1}}\{{[1+\beta [1-2H(z)]]}^{\frac{1}{\sigma -1}-1}\}[1-2H(z)]h(z)dz}. $$
(10)

Let the median wage be wM. The above is negative if and only if

$$\begin{array}{@{}rcl@{}} &&{\int}_{\underline{w}}^{w_{M}} {z^{\frac{\sigma} {\sigma -1}}\{\left[ 1+\beta [1-2H(z)] \right]^{\frac{1}{\sigma -1}-1}\}[1-2H(z)]h(z)dz}\\ &&{\kern58pt} <{\int}_{w_{M}}^{\overline{w}} {z^{\frac{\sigma} {\sigma -1}}\{{[1+\beta [1-2H(z)]]}^{\frac{1}{\sigma -1}-1}\}[2H(z)-1]h(z)dz} . \end{array} $$
(11)

Using symmetry, which implies g(x) ≡ h(wMx) = h(wM + x), and reversing the limits of integration we arrive at (12) which is equivalent to (11):

$$\begin{array}{@{}rcl@{}} {\int}_{0}^{\frac{\overline{w} -\underline{w}}{2}} {{(w_{M}-x)}^{\frac{\sigma} {\sigma -1}}\{{[1+\beta \left[ 1-2H(w_{M}-x) \right]]}^{\frac{1}{\sigma -1}-1}\}[1-2H(w_{M}-x)]g(x)dx}\\ <{\int}_{0}^{\frac{\overline{w} -\underline{w}}{2}} {{(w_{M}+x)}^{\frac{\sigma} {\sigma -1}}\{{[1+\beta [2H(w_{M}-x)-1]]}^{\frac{1}{\sigma -1}-1}\}[1-2H(w_{M}-x)]g(x)dx}.\\ \end{array} $$
(12)

To show that (12) holds it is sufficient to show that (13) holds weakly for x = 0 and strictly for all \(x\in \left (0 \right .\frac {\overline {w} -\underline {w}}{2}]\).

$$\begin{array}{@{}rcl@{}} &&{(w_{M}-x)}^{\frac{\sigma} {\sigma -1}}{[1+\beta \left[ 1-2H(w_{M}-x) \right]]}^{\frac{1}{\sigma -1}-1}\\ &&{\kern95pt} \le {(w_{M}+x)}^{\frac{\sigma} {\sigma -1}}{[1+\beta [2H(w_{M}-x)-1]]}^{\frac{1}{\sigma -1}-1}. \end{array} $$
(13)

Clearly, (13) holds weakly for x = 0 since H(wM) = 1/2. We now show that (13) holds strictly for \(x\in \left (0 \right .\frac {\overline {w} -\underline {w}}{2}]\), which implies \(\frac {\overline {w} -\underline {w}}{2}\le w_{M}\). So we have 2H(wMx) < 1 and, given that \(\frac {1}{\sigma -1}-1<0\) for σ > 2, it follows that

$$ {[1+\beta [1-2H(w_{M}-x)]]}^{\frac{1}{\sigma -1}-1}<{[1+\beta [2H(w_{M}-x)-1]]}^{\frac{1}{\sigma -1}-1}. $$
(14)

Finally, (14) and (wMx) < (wM + x) implies that (12) holds strictly for

$$x\in \left( 0 \right.\frac{\overline{w} -\underline{w}}{2}]. $$

(ii). Follows from part (i) and Proposition 1. □

Proof of the Lemma

Observe that γ(wM,α,β) can be written as

$$\begin{array}{@{}rcl@{}} \gamma (w_{M},\alpha, \beta )&\equiv& \frac{1}{\sigma -1}(\frac{1}{2})[\beta \int\limits_{w_{M}}^{\overline{w}} {zL(z,\tau )} 2h(z)dz-\alpha \int\limits_{\underline{w}}^{w_{M}} {zL(z,\tau )} 2h(z)dz]\\ &+&(\frac{\beta} {2})[{\int}_{w_{M}}^{\overline{w}} {zL(z,\tau )} 2h(z)dz-w_{M}L(w_{M},\tau )]\\ &&+(\frac{\alpha} {2} )[w_{M}L(w_{M},\tau )-{\int}_{\underline{w}}^{w_{M}} {zL(z,\tau )} 2h(z)dz]. \end{array} $$
(15)

Average income of those above the median is given by \({\int }_{w_{M}}^{\overline {w}} {zL\left (z,\tau \right )} 2h(z)dz\) and average income of those below the median is given by \({\int }_{\underline {w}}^{w_{M}} {zL(z,\tau )} 2h(z)dz\). For all β > 0 and α ∈ [0,β] the sum in the first bracketed term is positive since average income of those above the median is greater than average income of those below the median. The sum in the second bracketed term is positive because average income of those above the median is greater than the median income. The sum in the third bracketed term is positive because median income is greater than the average income of those below the median. □

Proof of Proposition 5

Let τM denote the most preferred tax for an other-regarding median agent and let τS denote the most preferred tax for a selfish median agent. The condition α = β implies that labor income for the selfish median voter equals that of the inequality averse median voter; namely, wML(wM,τ). Thus, τS must satisfy

$$ -w_{M}L(w_{M},\tau^{S})+{\int}_{\underline{w}}^{\overline{w}} {[zL(z,\tau^{S})+\tau^{S}z\frac{\partial L(z,\tau^{S})}{\partial \tau} ]} h(z)dz= 0. $$
(16)

Now if we evaluate the expression in (7) at τ = τS we get

$$-w_{M}L(w_{M},\tau^{S})+{\int}_{\underline{w}}^{\overline{w}} {[zL(z,\tau^{S})+\tau^{S}z\frac{\partial L(z,\tau^{S})}{\partial \tau} ]} h(z)dz+ \gamma (w_{M},\alpha, \beta )>0. $$
(17)

Given that τM must satisfy (6) it follows that τM > τS. □

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Murray, M.N., Peng, L. & Santore, R. How does inequality aversion affect inequality and redistribution?. J Econ Inequal 16, 507–525 (2018). https://doi.org/10.1007/s10888-018-9389-7

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Keywords

  • Income distribution
  • Inequality aversion
  • Redistribution