Preferences for redistribution and social structure

Abstract

We model inter-individual differences in preferences for redistribution as a function of (a) self-interest; (b) ideas about the deservingness of income differences due to luck, effort and talent; (c) subjective perceptions of the relative importance of these determinants for explaining the actual income distribution. Individuals base the latter on information obtained from their reference group. We analyse the consequences for redistributive preferences of homophilous reference group formation based on talent. Our model makes it possible to understand and integrate some of the main insights from the empirical literature. We illustrate with GSS data from 1987 how our model may help in structuring empirical work.

Appendices

Appendix

Data appendix: description of the variables

See Tables 6, 7, 8, 9, 10, 11, 12 and 13.

Table 6 Description of redistribution variables

Table 7 Dependent variables and factor loadings

Table 8 Description of variables: beliefs about determinants of success in life

Table 9 Description of variables: perceptions of incentives

Table 10 Summary statistics of perceptions and beliefs

Table 11 Beliefs: rotated factor loadings and uniqueness

Table 12 Description of socioeconomic variables

Table 13 Descriptive statistics independent variables

Mathematical appendix: proofs

1.1 A. Derivation of equation (10)

Implementing Eq. (5) for the naive idealist gives

$$\begin{aligned} \Omega _{(i)}^{\zeta 0} =&\frac{1}{|\mu (i)|}\int _{j\in \mu (i)}\left\{ (1-\tau )(1-\tau ^{a})(\beta a_{j}+\varepsilon _{j})+\tau (1-\tau ^{a})\beta \bar{a}_{(i)}\right. \\&\left. -\zeta (1-\tau ^{a})\beta a_{j}-(1-\zeta )(1-\tau ^{a})\beta \bar{a}_{(i)}\right\} ^{2}dj \end{aligned}$$

or

$$\begin{aligned} \Omega _{(i)}^{\zeta 0}=\frac{1}{|\mu (i)|}\int _{j\in \mu (i)}\left\{ (1-\tau )\varepsilon _{j}+\beta (1-\tau ^{a})(1-\tau -\zeta )(a_{j}-\bar{a}_{(i)})\right\} ^{2}dj \end{aligned}$$

Using condition 1, this immediately yields Eq. (10).

1.2 B. Comparative statics for sophisticated consumers

Because sgn$\left[ \frac{\partial \tau _{i}^{*}}{\partial z}\right] =$sgn$\left[ \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial z}\right] ,$ and

$$\begin{aligned} \Upsilon _{i}\equiv (1-\gamma )\beta \left[ \left( 1-2\tau \right) \bar{a}_{(i)}-\left( 1-\tau \right) a_{i}\right] -\gamma \frac{\partial \Omega _{(i)}^{\zeta }}{\partial \tau }=0 \end{aligned}$$

with

$$\begin{aligned} \frac{\partial \Omega _{(i)}^{\zeta }}{\partial \tau }&=-2\left( 1-\tau \right) Var\left( \varepsilon \right) _{(i)}-2\left( 1-\zeta -\tau \right) \left( 2-\zeta -2\tau \right) \left( 1-\tau \right) \beta ^{2}Var\left( a\right) _{(i)}, \end{aligned}$$

we have for self interest

$$\begin{aligned} \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial a_{i}}&=-\left( 1-\gamma \right) \left( 1-\tau _{i}^{*}\right) \beta \le 0 \end{aligned}$$

For the cultural traits

$$\begin{aligned} \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial \zeta } =&\,(2\gamma (1-\tau _{i}^{*})\beta ^{2}Var\left( a\right) _{(i)}\left[ 2\zeta -3\left( 1-\tau _{i}^{*}\right) \right]>0\Leftrightarrow \frac{2}{3}\zeta >\left( 1-\tau _{i}^{*}\right) \\ \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial \gamma } =&-\beta \left[ \left( 1-2\tau \right) \bar{a}_{(i)}-\left( 1-\tau \right) a_{i}\right] -\frac{\partial \Omega _{(i)}^{\zeta }}{\partial \tau }\\ =&\frac{\Upsilon _{i}-\beta \left[ \left( 1-2\tau \right) \bar{a}_{(i)}-\left( 1-\tau \right) a_{i}\right] }{\gamma }\\ =&\frac{\beta [\left( 1-\tau _{i}^{*}\right) a_{i}-\left( 1-2\tau _{i}^{*}\right) \bar{a}_{(i)}]}{\gamma }. \end{aligned}$$

For the perceptions and beliefs

$$\begin{aligned} \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial Var\left( \varepsilon \right) _{(i)}}&=\left( 2\gamma \left( 1-\tau _{i}^{*}\right) \right) \ge 0\\ \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial \bar{a}_{(i)}}&=\left( 1-\gamma \right) \left( 1-2\tau _{i}^{*}\right) \beta .\\ \frac{\partial \left( \Upsilon _{i}=0\right) }{\partial Var\left( a\right) _{(i)}}&=2\gamma \beta ^{2}\left( 1-\tau _{i}^{*}\right) (1-\zeta -\tau _{i}^{*})(2-2\tau _{i}^{*}-\zeta ). \end{aligned}$$

1.3 C. Unidimensional reference group formation based on talent

Let the joint and marginal density functions of $\Phi \left( a,q\right)$ be denoted respectively $\phi \left( a,q\right) ,$ $\phi ^{a}\left( a\right)$ and $\phi ^{q}\left( q\right) .$ If $\delta =1$, individual i only takes into account professional talent in the formation of his reference group, so that Eq. (15) reduces to $\mu \left( i\right) =\left\{ j|\left| a_{i}-a_{j}\right| \le \pi \right\} .$ We derive for the perceived mean and variance of a : 

$$\begin{aligned} \bar{a}_{(i)}= & {} \frac{\int _{a_{i}-\pi }^{a_{i}+\pi }s\phi ^{a}\left( s\right) ds}{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }\\ Var\left( a\right) _{(i)}= & {} \frac{\int _{a_{i}-\pi }^{a_{i}+\pi }\left( z-\bar{a}_{(i)}\right) ^{2}\phi ^{a}\left( z\right) dz}{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }. \end{aligned}$$

These expressions immediately show that the beliefs of individual i will be influenced by his position in the distribution of talents. It is obvious that $\partial \bar{a}_{(i)}/\partial a_{i}>0$. The sign of $\partial Var(a)_{(i)}/\partial a_{i}$ is less straightforward, as it depends on the shape of $\phi ^{a}(a)$. If $\phi ^{a}(a)$ is the uniform density, natural talent $a_{i}$ has no effect on the perceived variance.

An increase in $\pi$ (i.e., an increase in the marginal utility of social relations $\xi$ or a decrease in the cost c) will lead to an extension of the reference group of the individual. This results in

$$\begin{aligned} \frac{\partial \bar{a}_{(i)}}{\partial \pi } =&\frac{\left( \left( a_{i}+\pi \right) \phi ^{a}\left( a_{i}+\pi \right) +\left( a_{i}-\pi \right) \phi ^{a}\left( a_{i}-\pi \right) \right) }{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }\\&-\frac{\left( \phi ^{a}\left( a_{i}+\pi \right) +\phi ^{a}\left( a_{i}-\pi \right) \right) \bar{a}_{(i)}}{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }, \end{aligned}$$

which is positive iff

$$\begin{aligned} \frac{\bar{a}_{(i)}-\left( a_{i}-\pi \right) }{\left( a_{i}+\pi \right) -\bar{a}_{(i)}}<\frac{\phi ^{a}\left( a_{i}+\pi \right) }{\phi ^{a}\left( a_{i}-\pi \right) }. \end{aligned}$$

Again, the effect of changes in $\pi$ will depend on the shape of $\phi ^{a}(a)$. If $\phi ^{a}(a)$ is uniform and if $\mu \left( i\right)$ is strictly within the support of $\phi ^{a}(a)$, the perceived mean $\bar{a}_{(i)}$ obviously does not change with changes in $\pi .$

The effect on the perceived variance is

$$\begin{aligned} \frac{\partial Var\left( a\right) _{(i)}}{\partial \pi }&=\frac{\left( \begin{array}{c} -2\int _{a_{i}-\pi }^{a_{i}+\pi }\left( z-\bar{a}_{(i)}\right) \frac{\partial \bar{a}_{(i)}}{\partial \pi }\phi ^{a}\left( z\right) dz\\ +\left( \begin{array}{c} \left( \left( a_{i}+\pi \right) -\bar{a}_{(i)}\right) ^{2}\phi ^{a}\left( a_{i}+\pi \right) \\ +\left( \left( a_{i}-\pi \right) -\bar{a}_{(i)}\right) ^{2}\phi ^{a}\left( a_{i}-\pi \right) \end{array}\right) \\ -\left( \phi ^{a}\left( a_{i}+\pi \right) +\phi ^{a}\left( a_{i}-\pi \right) \right) Var\left( a\right) _{(i)} \end{array}\right) }{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }\\&=\frac{\begin{array}{c} \left( \left( a_{i}+\pi -\bar{a}_{(i)}\right) ^{2}-Var\left( a\right) _{(i)}\right) \phi ^{a}\left( a_{i}+\pi \right) \\ +\left( a_{i}-\pi -\bar{a}_{(i)}^ {})^{2}-Var\left( a\right) _{(i)}\right) \phi ^{a}\left( a_{i}-\pi \right) \end{array}}{\Phi ^{a}\left( a_{i}+\pi \right) -\Phi ^{a}\left( a_{i}-\pi \right) }. \end{aligned}$$

This expression is positive iff

$$\begin{aligned}&((a_{i}+\pi -\overline{a}_{(i)})^{2}-Var(a)_{(i)})\phi ^{a}(a_{i}+\pi )\\&\qquad +((a_{i}-\pi -\bar{a}_{(i)}^ {})^{2}-Var\left( a\right) _{(i)})\phi ^{a}\left( a_{i}-\pi \right) >0 \end{aligned}$$

A sufficient condition for this is that $\Phi ^{a}$ is not too skewed around $a_{i}$, so that both

$$\begin{aligned} Var\left( a\right) _{(i)}-\left( \left( a_{i}+\pi \right) -\bar{a}_{(i)}\right) ^{2}<0 \end{aligned}$$

and

$$\begin{aligned} Var\left( a\right) _{(i)}-\left( \left( a_{i}-\pi \right) -\bar{a}_{(i)}\right) ^{2}<0. \end{aligned}$$

This condition is definitely satisfied if $\Phi ^{a}$ is uniform.