The Dynamics of Wealth Inequality under Endogenous Fertility: A Remark on the Barro-Becker Model with Heterogenous Endowments

Abstract

Implicit in the seminal contribution of Barro-Becker [1], the lack of persistence of inequality in the presence of endogenous fertility is one of the most striking features of the models à la Barro-Becker. In this pedagogical note, we show how to uncover and interpret the latter property using standard optimization in contrast to the dynamic programming under homogeneity usually invoked in this literature.

Fertility, social mobility and long run inequality

“…As discussed in Cordoba and Ripoll (2014), the EGS is conceptually and quantitatively di¤erent from the elasticity of intertemporal substitution (EIS). 6 A positive constant U ensures a positive utility ‡ow in the low curvature case, > 1.…”

“…This is a natural assumption, as a number of legal and/or moral restrictions prevent parents from imposing debt obligations on their children. This problem is not a standard discounted dynamic programming problem due to the endogeneity of the discount factor (n), and the non-convexity introduced by 6 To see this, one can interpret adult consumption c as a composite good made of consumption ‡ows at various ages:…”

“…However, controlling for age for both parents and children, as well as for time e¤ects, allows our estimation of intergenerational persistence to include the whole lifetime pro…le of income of each child who has grown to form his / her own household. Fourth, the normalization of the child's age in estimation equation (6) implies that is the intergenerational income elasticity at age 40. As Lee and Solon indicate, this is the age at which measurement error in child's current income as a proxy for lifetime income has been found to be inconsequential.…”

“…We adopt the same normalization here. 16 We use equation (6) to estimate the intergenerational persistence of both income y and labor earnings e. Income is measured as PSID variable "total family money", which includes labor earnings of all family members, as well as any other money received by all members of the household.…”

“…In particular, Becker and Barro (1988) and Barro and Becker (1989) introduce optimal fertility choices within the optimal growth model and …nd that some of the most appealing conclusions obtained under the exogenous fertility assumption are seriously altered. 2 On the speci…c issue of inequality, the optimal fertility choice tends to eliminate any inequality and any persistence of inequality, a result highlighted by Bosi, Boucekkine, and Seegmuller (2011) in the context of a deterministic Barro-Becker model. In contrast, the version of the model with exogenous fertility predicts that any initial inequality is highly persistent, as shown by Chatterjee (1994).…”