Redistributive taxation, wealth distribution, and economic growth

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.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

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

  2. 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. 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. 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. 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. 6.

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

  7. 7.

    In the literature on economic growth and inequality, some studies have adopted this assumption (e.g., Alesina and Rodrik 1994; Rehme 1999, 2000; Tamai 2009).

  8. 8.

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

  9. 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. 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. 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. 12.

    See Appendix A.2. for derivation of Eq. (11) and Lambert (2001, Ch.2, 10) for details of the mathematical method.

  13. 13.

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

  14. 14.

    As referred in the Introduction, some issues are pointed out regarding the social welfare criterion. See the detail of these problems for Atkinson and Stiglitz (1980), Ch.11.2, Rehme (1999), the footnote 19, and Rehme (2003b, p.495).

References

  1. Aghion P, Caroli E, Garcia-Penalosa C (1999) Inequality and economic growth: the perspective of the new growth theories. J Econ Lit 37(4):1615–1660

    Article  Google Scholar 

  2. Alesina A, Rodrik D (1994) Distributive politics and economic growth. Q J Econ 109(2):465–490

    Article  Google Scholar 

  3. Atkinson AB, Bourguignon F (2000) Income distribution and economics. In: Atkinson AB, Bourguignon F (eds) Handbook of income distribution. Elsevier, Amsterdam

    Google Scholar 

  4. Atkinson AB, Stiglitz JE (1980) Lectures on public economics, International edn. McGraw-Hill, Singapore

    Google Scholar 

  5. Baliscan AM, Fuwa N (2003) Growth, inequality, and politics revisited: a developing-country case. Econ Lett 79(1):53–58

    Article  Google Scholar 

  6. Barro RJ (2000) Inequality and growth in a panel of countries. J Econ Growth 5(1):5–32

    Article  Google Scholar 

  7. Barro RJ, Sala-i-Martin X (1995) Economic growth. McGraw-Hill, New York

    Google Scholar 

  8. Bertola G (1993) Market structure and income distribution in endogenous growth models? Am Econ Rev 83(5):1184–1198

    Google Scholar 

  9. Caselli F, Ventura J (2000) A representative consumer theory of distribution. Am Econ Rev 90(4):909–926

    Article  Google Scholar 

  10. Champernowne D (1953) A model of income distribution. Econ J 63(250):318–351

    Article  Google Scholar 

  11. Corneo G, Jeanne O (1999) Pecuniary emulation, inequality and growth. Eur Econ Rev 43(9):1665–1678

    Article  Google Scholar 

  12. Cowell FA (2011) Measuring inequality, 3rd edn. Oxford University Press, Oxford

    Book  Google Scholar 

  13. Creedy J (1977) Pareto and the distribution of income. Rev Income Wealth 23(4):405–411

    Article  Google Scholar 

  14. Dahan M, Tsiddon D (1998) Demographic transition, income distribution, and economic growth. J Econ Growth 3(1):29–52

    Article  Google Scholar 

  15. Deininger K, Squire L (1998) New ways of looking at old issues: inequality and growth. J Dev Econ 57(2):259–287

    Article  Google Scholar 

  16. Forbes KJ (2000) A reassessment of the relationship between inequality and growth. Am Econ Rev 90(4):869–887

    Article  Google Scholar 

  17. Galor O, Zeira J (1993) Income distribution and macroeconomics. Rev Econ Stud 60(1):35–52

    Article  Google Scholar 

  18. García-Peñalosa C, Turnovsky SJ (2011) Taxation and income distribution dynamics in a neoclassical growth model. J Money Credit Bank 43(8):1543–1577

    Article  Google Scholar 

  19. Grüner HP (1995) Redistributive policy, inequality and growth. J Econ 62(1):1–23

    Article  Google Scholar 

  20. Hendel I, Shapiro J, Willen P (2005) Educational opportunity and income inequality. J Public Econ 89(5–6):841–870

    Article  Google Scholar 

  21. Knowles S (2005) Inequality and economic growth: the empirical relationship reconsidered in the light of comparable data. J Dev Stud 41(1):135–159

    Article  Google Scholar 

  22. Kuznets S (1955) Economic growth and income inequality. Am Econ Rev 45(1):1–28

  23. Kuznets S (1963) Quantitative aspects of the economic growth of nations: VIII. Distribution of income by size. Econ Dev Cult Change 11(2):1–79

  24. Lambert PJ (2001) The distribution and redistribution of income, 3rd edn. Manchester University Press, Manchester

    Google Scholar 

  25. Li H, Zou H (1998) Income inequality is not harmful for growth: theory and evidence. Rev Dev Econ 2(3):318–334

    Article  Google Scholar 

  26. OECD (2008) Growing Unequal? Income Distribution and Poverty in OECD Countries. OECD, Paris

  27. Okuno N, Yakita A (1981) Public investment and income distribution: a note. Q J Econ 96(1):171–176

    Article  Google Scholar 

  28. Owen AL, Weil DN (1998) Intergenerational earnings mobility, inequality and growth. J Monet Econ 41(1):71–104

    Article  Google Scholar 

  29. Perotti R (1993) Political equilibrium, income distribution, and growth? Rev Econ Stud 60(4):755–776

    Article  Google Scholar 

  30. Perotti R (1996) Growth, income distribution, and democracy: what the data say. J Econ Growth 1(2):149–187

    Article  Google Scholar 

  31. Persson T, Tabellini G (1994) Is inequality harmful for growth? Am Econ Rev 84(3):600–621

    Google Scholar 

  32. Rehme G (1999) Public policies and education, economic growth and income distribution. Economics Working Papers eco99/14, European University Institute

  33. Rehme G (2000) Economic growth and (re-)distributive policies: a comparative dynamic analysis. Economics Working Papers eco2000/13, European University Institute

  34. Rehme G (2003a) (Re-)Distribution of personal incomes, education and economic performance across countries. In: Eicher T, Turnovsky SJ (eds) Growth and inequality: theory and policy implications. MIT Press, Cambridge

    Google Scholar 

  35. Rehme G (2003b) Education policies, economic growth, and wage inequality. FinanzArchiv 59(4):479–503

    Article  Google Scholar 

  36. Rehme G (2007) Education, economic growth and measured income inequality. Economica 74(295):493–514

    Article  Google Scholar 

  37. Rehme G (2011) Endogenous policy and cross-country growth empirics. Scott J Polit Econ 58(2):262–296

    Article  Google Scholar 

  38. Romer PM (1986) Increasing returns and long-run growth. J Polit Econ 94(5):1002–1037

    Article  Google Scholar 

  39. Tamai T (2009) Inequality, unemployment, and endogenous growth in a political economy with a minimum wage. J Econ 97(3):217–232

    Article  Google Scholar 

  40. Tamai T (2010) Public goods provision, redistributive taxation, and wealth accumulation. J Public Econ 94(11–12):1067–1072

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Toshiki Tamai.

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

$$\begin{aligned} \frac{{\dot{c}}_i}{c_i}&= \frac{{\dot{C}}_i}{C_i} - \frac{{\dot{W}}_i}{W_i} = - \frac{(1-\tau )(\sigma -1)A + \rho }{\sigma } + c_i - \frac{\tau A}{w_i}, \end{aligned}$$
(29)
$$\begin{aligned} \frac{{\dot{c}}}{c}&= \frac{{\dot{C}}}{C} - \frac{{\dot{W}}}{W} = \frac{(1 - \sigma - \tau )A - \rho }{\sigma } + c, \end{aligned}$$
(30)
$$\begin{aligned} \frac{\dot{w}_i}{w_i}&= \frac{{\dot{W}}_i}{W_i} - \frac{{\dot{W}}}{W} = - \tau A - c_i + \frac{\tau A}{w_i} + c. \end{aligned}$$
(31)

In the balanced growth equilibrium, the equations above are

$$\begin{aligned} 0&= - \frac{(1 - \tau )(\sigma - 1)A + \rho }{\sigma } + c_i - \frac{\tau A}{w_i}, \end{aligned}$$
(32)
$$\begin{aligned} 0&= \frac{(1 - \sigma - \tau )A - \rho }{\sigma } + c, \end{aligned}$$
(33)
$$\begin{aligned} 0&= - \tau A - c_i + \frac{\tau A}{w_i} + c. \end{aligned}$$
(34)

By (33), we obtain

$$\begin{aligned} c^* = \frac{(\sigma - 1 + \tau )A + \rho }{\sigma }. \end{aligned}$$
(35)

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

$$\begin{aligned} c_i=\frac{(1-\tau )(\sigma -1)A+\rho }{\sigma }+\frac{\tau A}{w_i}. \end{aligned}$$
(36)

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

$$\begin{aligned} \frac{{\dot{w}}_i}{w_i}&= - \tau A - c_i + \frac{\tau A}{w_i} + \frac{(\sigma - 1 + \tau )A + \rho }{\sigma } \nonumber \\&= \frac{(1 - \tau )(\sigma - 1)A + \rho }{\sigma } - c_i + \frac{\tau A}{w_i}. \end{aligned}$$
(37)

By (37)

$$\begin{aligned} \frac{{\dot{c}}_i}{c_i} = - \frac{\dot{w}_i}{w_i} = - \frac{(1-\tau )(\sigma -1)A+\rho }{\sigma }+c_i - \frac{\tau A}{w_i}. \end{aligned}$$
(38)

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

$$\begin{aligned} c_i^* = \frac{(1 - \tau )(\sigma - 1)A + \rho }{\sigma } + \frac{\tau A}{w_i^*}. \end{aligned}$$

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

$$\begin{aligned} p = F(Y) \Rightarrow L(p) = \frac{1}{W} \int ^{Y}_{W_0} W_i dF(W_i) = \frac{1}{W} \int ^{Y}_{W_0} W_i f(W_i)dW_i. \end{aligned}$$
(39)

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):

$$\begin{aligned} \frac{dL}{dp} = \frac{dL}{dW_i} \frac{dW_i}{dp} = \frac{W_i}{W}. \end{aligned}$$
(40)

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

$$\begin{aligned} G_W&= 1 - 2 \int ^{1}_{0} L dp \\&= 2 \int ^{1}_{0} p \frac{dL}{dp} dp - 1 = \frac{2}{W} \int ^\infty _{W_0} W_i F(W_i)f(W_i)dW_i - 1 = \frac{1}{2\alpha -1}. \end{aligned}$$

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

$$\begin{aligned} p = F(Y) \Rightarrow L_X&= \frac{1}{W} \int ^{Y}_{W_0} [W_i - T(W_i)] dF(W_i) \nonumber \\&= \frac{1}{W} \int ^{Y}_{W_0} [W_i - T(W_i)] f(W_i)dW_i. \end{aligned}$$
(41)

Then, we have

$$\begin{aligned} \frac{dL_X}{dp} = \frac{dL_X}{dW_i} \frac{dW_i}{dp} = \frac{(1-\tau )W_i - \tau W}{W}. \end{aligned}$$
(42)

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

$$\begin{aligned} G_X&= 1 - 2 \int ^{1}_{0} L_X dp = 2 \int ^{1}_{0} p \frac{dL_X}{dp} dp - 1 \\&= \frac{2}{W} \int ^\infty _{W_0} [(1 - \tau )W_i + \tau W] F(W_i)f(W_i)dW_i - 1 = \frac{1-\tau }{2\alpha -1}. \end{aligned}$$

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

$$\begin{aligned} C^*_i(t) = C_i^*(0) \exp ( \gamma t ) = c_i^*(0) W_i^*(0)\exp (\gamma t), \end{aligned}$$
(43)

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

$$\begin{aligned} V_i^*&= \int ^{\infty }_0 \frac{C_i^*(t)^{1 - \sigma } - 1}{1 - \sigma } \exp (-\rho t)dt \\&= \frac{1}{1 - \sigma } \left[ \int ^{\infty }_0 \{c_i^*(0)W_i^*(0)\}^{1 - \sigma } \exp \{ -((\sigma - 1)\gamma \!+\! \rho )t \} dt - \int ^{\infty }_0 \exp (-\rho t)dt \right] \\&= \frac{1}{1 - \sigma } \left[ \frac{\{c_i^*(0)W_i^*(0)\}^{1 - \sigma }}{(\sigma - 1)\gamma + \rho } + \frac{1}{\rho } \right] \\&= \frac{\left[ \{ (\sigma - 1)(1 - \tau )A + \rho \} W^*_i + \sigma \tau AW^* \right] ^{1 - \sigma }\sigma ^\sigma }{(1 - \sigma )[(\sigma - 1)(1 - \tau )A + \rho ]} - \frac{1}{(1 - \sigma )\rho }. \end{aligned}$$

A.4 Proof of Proposition 3

Total differentiation of (26) gives

$$\begin{aligned} \frac{\partial G_M}{\partial G_W} = \left[ 1 - \frac{(\sigma - 1)A + \rho }{2G_W \{ A - (1 - w^*_m)\rho \}\{(1 - w^*_m)\sigma + w^*_m \} } \frac{dw_m^*}{d\alpha } \right] \frac{G_M}{G_W}. \end{aligned}$$

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

$$\begin{aligned} \lim _{G_W \rightarrow 0} \frac{\partial G_M}{\partial G_W}&= \lim _{\alpha \rightarrow \infty } \left[ \frac{A - (1 - w_m^*)\rho }{\{(1 - w_m^*)\sigma + w^*_m\}A} - \frac{\{(\sigma - 1)A + \rho \}(2\alpha - 1)}{2\{(1 - w_m^*)\sigma + w^*_m\}^2 A}\frac{dw^*_m}{d\alpha } \right] \\&= 1, \end{aligned}$$

and

$$\begin{aligned} \lim _{G_W \rightarrow 1} \frac{\partial G_M}{\partial G_W} = - \frac{(1 + \sigma )\rho - A}{\sigma ^2A} > 0. \end{aligned}$$

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Redistributive tax
  • Inequality
  • Economic growth

JEL Classification

  • O11
  • O15
  • H23