## Abstract

This paper examines the relationship between wealth distribution and economic growth in an endogenous growth model with heterogeneous households and redistributive taxation. In this paper, we incorporate an endogenous determination of redistributive policy into the model, focusing on the relation between pre- and post-tax inequality. Endogenous redistributive policy affects wealth distribution and economic growth. Therefore, the relation between post-tax inequality and economic growth is different from that between pre-tax inequality and economic growth. Results show that there exists a negative correlation between pre-tax inequality and economic growth, whereas there exists an inverted-U relationship between post-tax inequality and economic growth in a voting equilibrium.

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## Notes

- 1.
See Atkinson and Bourguignon (2000) for issues related to the Kuznets hypothesis.

- 2.
See Aghion et al. (1999) for a general review and survey of this issue. Numerous theoretical approaches exist in the literature: a focus on the imperfection of the credit market (e.g., Dahan and Tsiddon 1998; Galor and Zeira 1993; Hendel et al. 2005; Owen and Weil 1998), a political economy approach (e.g., Alesina and Rodrik 1994; Bertola 1993; Perotti 1993; Persson and Tabellini 1994), a representative consumer theory of distribution (e.g., Caselli and Ventura 2000; García-Peñalosa and Turnovsky 2011) and others.

- 3.
Corneo and Jeanne (1999) examine a two-class growth model in which agents concern themselves with both consumption and social status. Rehme (2007) examines the effects of education on growth and inequality in an endogenous growth model with low-skilled and high-skilled people. Tamai (2009) incorporates a heterogeneous labor-skill endowment and an endogenous determination of minimum wage in his model.

- 4.
Grüner (1995) also examines the relation among redistributive policy, inequality, and growth using a simple overlapping-generations model with human capital accumulation and heterogeneous learning skill.

- 5.
As described in this paper, a dot above the letter denotes the derivative with respect to time, e.g., \({\dot{x}}:=dx/dt\).

- 6.
When \(\sigma = 1\), we have \(U_i =\int ^{\infty }_{0}\log C_i \exp (- \rho t) dt\).

- 7.
- 8.
Recently, Tamai (2010) investigated the interaction between public goods provisions and wealth accumulation using the extended Romer model with this redistributive transfer.

- 9.
Cowell (2011, Ch.4) in particular states that the Pareto distribution is the most useful application as an approximate description of the distribution of incomes and wealth among the rich and the moderately rich, although the Pareto formulation has proved to be extremely versatile across the social sciences. See Cowell (2011, Ch.4) for details of the properties of a Pareto distribution.

- 10.
The expected value does not exist if \(\alpha \le 1\). In addition, the variance does not exist if \(\alpha \le 2\). Creedy (1977) finds the Pareto index between \(1.7\) and \(3\).

- 11.
The average value is derived from \(W = \int ^{\infty }_{W_0} W_i dF(W_i)\). Solving \(F(W_m) = 1/2\) with respect to \(W_m\), the median is obtained.

- 12.
- 13.
The derivation process of the indirect utility function is given in Appendix A.3.

- 14.

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## Additional information

I thank two anonymous referees and Giacomo Corneo, the Editor of this journal, for their helpful comments and suggestions. I am also grateful to Shinya Fujita, Akira Kamiguchi and the seminar participants at Nagoya University for their advice and comments. This work was supported by a Grant-in-Aid for Young Scientists (B) (No. 22730268) of the Japan Society for the Promotion of Science.

## Appendix A

### Appendix A

### A.1 Proof of Lemma 1

Using Eqs. (5)–(7), the logarithmic derivation of \(c_i\), \(c\), and \(w_i\) with respect to time gives

In the balanced growth equilibrium, the equations above are

By (33), we obtain

Equation (35) shows that the stationary value of \(c\) is determined uniquely. The stationary point \(c^*\) is unstable. Therefore, the value of \(c\) jumps from its initial value to \(c^*\) because Eq. (30) shows that the dynamic equation of \(c\) has a positive eigenvalue; \(\left. \partial {\dot{c}} /\partial c \right| _{c = c^*} > 0\). Using (35) and (32) or (34), we arrive at

Equation (36) shows that \(c_i\) depends on \(w_i\).

We now consider the dynamic motion of \(w_i\) to investigate the existence of a stationary value of \(w_i\). Inserting \(c^*\) into \(c\) in (31), we have

By (37)

For \(w_i\) at \(t = 0\) (\(w_i^* = w_i(0)\)), \(c_i\) should be chosen as (36):

Therefore, there exists a unique balanced growth equilibrium and the economy is always in the balanced growth equilibrium.

### A.2 Derivation of Eqs. (11) and (12)

The Lorenz curve shows the quantile share information for a given income distribution. Let \(p\) as a cumulative share of people from the lowest to the highest incomes. For given income level \(Y\), the cumulative distribution function gives a unique value \(p\) such as \(p = F(Y)\). The Lorenz curve can be defined as

Note that \(F^{\prime }(\cdot ) = f(\cdot )\) where \(f(\cdot )\) is the probability density function. Using the chain rule, we obtain the derivative of Eq. (39):

The definition of the Gini coefficient is \(G_W = 1 - 2 \int ^{1}_{0} L dp\). Using (39) and (40), we have

Note that we apply integration by part to the first line of above equation. In the same way as Eq. (39), we define \(L_X\) as

Then, we have

We can use the same way as derivation of \(G_W\). Therefore, we obtain

### A.3 Derivation of the indirect utility function

Because the economy is in a balanced growth equilibrium (Lemma 1), consumption is growing at a constant growth rate given as (5). Therefore, household \(i\)’s consumption at time \(t\) becomes

where \(\gamma := [(1 - \tau )A - \rho ]/\sigma \). Inserting (43) into (1), we obtain

### A.4 Proof of Proposition 3

Total differentiation of (26) gives

Here, \(G_W \rightarrow 1 \Leftrightarrow \alpha \rightarrow 1 \Leftrightarrow w_m^* = 0\) and \(G_W \rightarrow 0 \Leftrightarrow \alpha \rightarrow \infty \Leftrightarrow w_m^* = 1\). Calculating the limiting values of \(\partial G_M/\partial G_W\), one obtains

and

The results presented above show that \(G_M\) is monotonically increasing in \(G_W\), or \(G_M\) is locally decreasing in \(G_W\), although \(G_M\) is increasing in \(G_W\) around both ends. By Proposition 4, \(\partial G_W/\partial \gamma < 0\) holds. Therefore, the relation between the inequality and economic growth represents a downward or locally inverted-U shaped curve with the Gini coefficient on the vertical axis and economic growth rate on horizontal axis.

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Tamai, T. Redistributive taxation, wealth distribution, and economic growth.
*J Econ* **115, **133–152 (2015). https://doi.org/10.1007/s00712-014-0424-2

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### Keywords

- Redistributive tax
- Inequality
- Economic growth

### JEL Classification

- O11
- O15
- H23