Status, Wealth Distribution, and Endogenous Economic Growth

Abstract

Based on a stochastic dynamic general equilibrium model with heterogeneous households, this paper analyzes the effects of positional preferences on the interaction between the distribution of wealth and endogenous economic growth. Households exhibit positional preferences, i.e., they derive utility not only from their own consumption level, they are also concerned about their wealth-rank, i.e., their relative wealth position in the distribution of wealth in the society. Contrary to adopting ad hoc assumptions regarding the specification of the wealth rankings, we demonstrate that the distribution of wealth ranks necessarily follows a power law. Under this power law, individual wealth growth rates differ, although the aggregate distribution converges to a stationary distribution in the balanced growth path. Differences in individual growth rates are due to differences in the return to households’ assets (e.g., differences in the return to one’s human capital—considered as part of the total capital). Moreover, the endogenous growth rate is shown to strongly depend on the power-law exponent, which is itself an endogenous variable that is determined by the standard deviation of the individual growth rates and the minimum level of the wealth distribution.

Appendix

The Gini coefficient $\Gamma$ of a continuous distribution F(w) is given by

$$\displaystyle{\Gamma (w) = 2\int _{w_{min}}^{\infty }F(w)(1 - F(w))dw\frac{1} {2\mu }\,,}$$

where μ denotes the distribution’s mean. If the distribution F(w) follows a power-law with $F(w) = \left (\frac{w_{min}} {w} \right )^{\kappa }$ and w _min  ≤ w, $\mu = \frac{\kappa } {\kappa -1}w_{min}$ and the equation above is solved by

$$\displaystyle\begin{array}{rcl} \Gamma (w)& =& \int _{w_{min}}^{\infty }\left (\frac{w_{min}} {w} \right )^{\kappa } -\left ( \frac{b} {w}\right )^{2\kappa }dw\frac{1} {\mu } {}\\ & =& \left [- \frac{1} {1-\kappa }w_{min} + \frac{1} {1 - 2\kappa }w_{min}\right ]/\mu {}\\ & =& \frac{\kappa } {(1-\kappa )(1 - 2\kappa )}w_{min} \frac{\kappa -1} {\kappa w_{min}}\,, {}\\ \end{array}$$

yielding the Gini coefficient

$$\displaystyle{\Gamma (w) = \frac{1} {2\kappa - 1}\,.}$$