Inequality and Government Size: A Political Economy Theory and OECD Evidence

Abstract

The median voter theory of government size predicts that greater inequality leads to greater demand for redistribution and larger government Meltzer and Richard in J Polit Econ 89(5), 914–927 (1981). However, this prediction is often rejected empirically. This paper distinguishes between income inequality induced by differences in labor productivity and income inequality induced by differences in capital income. Whilst the standard argument applies to productivity-induced income inequality, greater capital income inequality leads to smaller government if, as often observed, capital income is difficult to tax. Using OECD data, government size and capital income inequality (proxied by the top 1% income share) are found to be negatively related in both fixed effects and instrumental variable regressions. Moreover, controlling for capital income inequality yields a positive and significant relationship between government size and labor income inequality, as originally conjectured.

This chapter is coauthored with Andrew Pickering and Paulo Santos Monteiro at Department of Economics and Related Studies, University of York.

Appendix

1.1 Derivation of Equations (2.12) and (2.13)

The problem of the median voter m is to choose the tax rate so as to maximize

$$\begin{aligned} u^{m}(c^{m},l^{m})=u^{m}\Big [\left( 1-t\right) x^{m}n^{m}+R^{m}+t\bar{y},1-n^{m}\Big ], \end{aligned}$$

(2.17)

and the first-order condition for the median voter with respect to the tax rate is

$$\begin{aligned} \left( \bar{y}-y^{m}+t\frac{d \bar{y}}{d t}\right) u_{c}+\Big [\left( 1-t\right) x^m u_{c}-u_{l}\Big ]\left( \frac{d n^{m}}{d t}\right) =0. \end{aligned}$$

(2.18)

Thus, making use of equation (2.3), the tax rate chosen by the median voter must satisfy

$$\begin{aligned} \bar{y}-y^{m}+t\left( \frac{d \bar{y}}{d t}\right) =0. \end{aligned}$$

(2.19)

Changes in the tax rate t affect average income via two channels: its effect on the opportunity cost of leisure, and its effect on transfers (from the government’s budget constraint $r=t\bar{y}$). In particular, we have that

$$\begin{aligned} \begin{aligned} \frac{d\bar{y}}{dt}&= \frac{\partial \bar{y}}{\partial r} \frac{dr}{dt}-\frac{\partial \bar{y}}{\partial {\tau }}, \\&= \frac{\partial \bar{y}}{\partial r} \left( \bar{y}+t\frac{d\bar{y}}{dt}\right) -\frac{\partial \bar{y}}{\partial {\tau }}. \end{aligned} \end{aligned}$$

(2.20)

with $\tau =1-t$. Thus, the total derivative of average income with respect to changes in the tax rate is given by

$$\begin{aligned} \frac{d\bar{y}}{dt}=\frac{\bar{y}_r\bar{y}-\bar{y}_\tau }{1-t\bar{y}_r}<0, \end{aligned}$$

(2.21)

with $\bar{y}_r=\frac{\partial \bar{y}}{\partial r}$ and $\bar{y}_\tau =\frac{\partial \bar{y}}{\partial {\tau }}$.

Finally, making use of (2.21) to substitute in (2.19), we obtain

$$\begin{aligned} \begin{aligned} 0&= \bar{y}-y^{m}+t\left( \frac{\bar{y}_r\bar{y}-\bar{y}_\tau }{1-t\bar{y}_r}\right) ,\\&= \left( \bar{y}-y^{m}\right) \left( 1-t\right) +\left[ \frac{\eta _r\bar{y}\left( 1-t\right) -\eta _\tau \bar{y}t}{1-\eta _r}\right] , \end{aligned} \end{aligned}$$

(2.22)

where $\eta _r=\bar{y}_r\left( r/\bar{y}\right)$ and $\eta _\tau =\bar{y}_\tau \left( \tau /\bar{y}\right)$ are the partial elasticities of average income. Solving the above equation for t, yields

$$\begin{aligned} t=\frac{m-1+\eta _r}{m-1+\eta _r+m\eta _\tau }, \end{aligned}$$

(2.23)

with $m=\bar{y}/y^m$.

1.2 Proof of Proposition 2.1

We begin with the following decomposition of average income

$$\begin{aligned} \bar{y}=p\left( \mathcal {K}\right) \bar{y}\left( \mathcal {K}\right) +\left( 1-p\left( \mathcal {K}\right) \right) \bar{y}\left( \sim \mathcal {K}\right) , \end{aligned}$$

(2.24)

where $\bar{y}\left( \mathcal {K}\right)$ is the average income of the individuals in set $\mathcal {K}$ and $\bar{y}\left( \sim \mathcal {K}\right)$ is the average income of the individuals not in set $\mathcal {K}$. From Assumption 2.2 we have that $\bar{y}^{\mathcal {K}}>y^m$.

Taking the total derivative of $\bar{y}$ with respect to $R\left( \mathcal {K}\right)$, the capital income of the individuals in set $\mathcal {K}$ in Eq. (2.24) we obtain

$$\begin{aligned} \begin{aligned} \frac{d\bar{y}}{dR\left( \mathcal {K}\right) }&= p\left( \mathcal {K}\right) \left( \frac{\partial \bar{y}\left( \mathcal {K}\right) }{\partial R\left( \mathcal {K}\right) }+\frac{\partial \bar{y}\left( \mathcal {K}\right) }{\partial r}\frac{d\bar{y}}{dR\left( \mathcal {K}\right) }t\right) +\left( 1-p\left( \mathcal {K}\right) \right) \left( \frac{\partial \bar{y}\left( \sim \mathcal {K}\right) }{\partial r}\frac{d\bar{y}}{dR\left( \mathcal {K}\right) }t\right) ,\\&= p\left( \mathcal {K}\right) \frac{\partial \bar{y}\left( \mathcal {K}\right) }{\partial R\left( \mathcal {K}\right) }+\frac{\partial \bar{y}}{\partial r}\frac{d\bar{y}}{dR\left( \mathcal {K}\right) }t,\\&= p\left( \mathcal {K}\right) \frac{\partial \bar{y}\left( \mathcal {K}\right) }{\partial R\left( \mathcal {K}\right) }+\eta _r\frac{d\bar{y}}{dR\left( \mathcal {K}\right) }, \end{aligned} \end{aligned}$$

(2.25)

where we used the fact that $\eta _r=\frac{\partial \bar{y}}{\partial r}\frac{r}{\bar{y}}=\frac{\partial \bar{y}}{\partial r}\frac{t\bar{y}}{\bar{y}}=\frac{\partial \bar{y}}{\partial r}t$. Using (2.25) to solve for $\frac{d\bar{y}}{dR\left( \mathcal {K}\right) }$, we obtain

$$\begin{aligned} \frac{d\bar{y}}{dR\left( \mathcal {K}\right) }=\frac{p\left( \mathcal {K}\right) }{1-\eta _r}\frac{\partial \bar{y}\left( \mathcal {K}\right) }{\partial R\left( \mathcal {K}\right) }<0, \end{aligned}$$

(2.26)

since leisure is a normal good. Thus, average income $\bar{y}$ must fall.

In turn, we have that

$$\begin{aligned} \frac{d y^m}{dR\left( \mathcal {K}\right) }=\frac{\partial y^m}{\partial r}\frac{\partial \bar{y}}{\partial R\left( \mathcal {K}\right) } t>0. \end{aligned}$$

(2.27)

Thus, we have established that $\bar{y}$ must fall and $y^m$ must increase following an increase in the capital-income going to the top capital-income recipients. Therefore, $m=\bar{y}/y^m$ falls and the increase in capital income inequality lowers labor income inequality. The upshot is that the increase in the capital income going to the top capital-income recipients results in a lower t, the labor income tax chosen by the median voter.