Human Capital Inequality with Subsistence Consumption

Abstract

This paper analyzed the evolution of human capital inequality in an overlapping generations model that incorporates Stone–Geary preferences with subsistence consumption. Many studies have revealed an inverted U-shaped relationship between human capital inequality and economic development, namely the human capital Kuznets curve. However, this curve is inconsistent with empirical evidence in a US context, where the data suggest a U-shaped evolution of earnings inequality. We show that the utility function has an increasing elasticity of substitution, and the time allocated to human capital investment decreases in the early stages of human capital accumulation and increases afterward. Thus, we obtain a U-shaped curve for human capital inequality if the elasticity of substitution is large. Eventually, decreasing returns to human capital accumulation reduce inequality.

Appendix: Derivation of \(\sigma (n_{i,t},c_{i,t+1})\)

From $MRS_{i,t}=\left( \frac{c_{i,t+1}-\bar{c}}{n_{i,t}} \right) ^\theta =\left[ \frac{c_{i,t+1}}{n_{i,t}}\left( 1-\frac{\bar{c}}{c_{i,t+1}} \right) \right] ^\theta$, we have

$$\begin{aligned} \frac{MRS_{i,t}}{c_{i,t+1}/n_{i,t}}=\left( \frac{c_{i,t+1}}{n_{i,t}} \right) ^{\theta -1}\left( 1-\frac{\bar{c}}{c_{i,t+1}} \right) ^\theta , \end{aligned}$$

(11)

such that

$$\begin{aligned} \frac{dMRS_{i,t}}{d(c_{i,t+1}/n_{i,t})}=\theta \left[ \frac{c_{i,t+1}}{n_{i,t}} \left( 1-\frac{\bar{c}}{c_{i,t+1}} \right) \right] ^{\theta -1}\left[ 1-\frac{\bar{c}}{c_{i,t+1}}+\frac{c_{i,t+1}}{n_{i,t}}\frac{d(1-\bar{c}/c_{i,t+1})}{d(c_{i,t+1}/n_{i,t})} \right] . \end{aligned}$$

(12)

From Eqs. 11 and 12, $\sigma (n_{i,t},c_{i,t+1})$ can be expressed as follows:

$$\begin{aligned} \sigma (n_{i,t},c_{i,t+1}) &=\frac{MRS_{i,t}}{c_{i,t+1}/n_{i,t}}\frac{d(c_{i,t+1}/n_{i,t})}{dMRS_{i,t}} \\ &=\frac{1}{\theta }\left[ 1+\frac{c_{i,t+1}/n_{i,t}}{1-\bar{c}/c_{i,t+1}}\frac{d(1-\bar{c}/c_{i,t+1})}{d(c_{i,t+1}/n_{i,t})} \right] ^{-1}. \end{aligned}$$

(13)

To calculate the remaining derivative $\frac{d(1-\bar{c}/c_{i,t+1})}{d(c_{i,t+1}/n_{i,t})}$, we transform the problem from the standard variables $(n_{i,t},c_{i,t+1})$ into variables $(\phi _{i,t},\upsilon _{i,t})$, as follows:

$$\begin{aligned} \phi _{i,t}&\equiv \frac{c_{i,t+1}}{n_{i,t}}, \end{aligned}$$

(14)

$$\begin{aligned} \upsilon _{i,t}&\equiv \frac{n_{i,t}^{1-\theta }+(c_{i,t+1}-\bar{c})^{1-\theta }}{1-\theta }, \end{aligned}$$

(15)

where $\phi _{i,t}$ is the ratio of $c_{i,t+1}$ to $n_{i,t}$ and $\upsilon _{i,t}$ is a monotonic transformation of utility. The elasticity of substitution measures substitutability between $n_{i,t}$ and $c_{i,t+1}$ along an indifference curve. Hence, we use $u(n_{i,t},c_{i,t+1})=constant$ or $d\upsilon _{i,t}=0$.

From Eqs. 14 and 15, we obtain $1-\frac{\bar{c}}{c_{i,t+1}}=\left[ \frac{(1-\theta )\upsilon _{i,t}}{c_{i,t+1}^{1-\theta }}-\phi _{i,t}^{\theta -1} \right] ^{\frac{1}{1-\theta }}$. Therefore,

$$\begin{aligned} \frac{d(1-\bar{c}/c_{i,t+1})}{d(c_{i,t+1}/n_{i,t})} &=\frac{d(1-\bar{c}/c_{i,t+1})}{d\phi _{i,t}} \\ &=\left[ \frac{(1-\theta )\upsilon _{i,t}}{c_{i,t+1}^{1-\theta }}-\phi _{i,t}^{\theta -1} \right] ^{\frac{1}{1-\theta }-1}\left[ \upsilon _{i,t}(\theta -1)c_{i,t+1}^{\theta -2}\frac{dc_{i,t+1}}{d\phi _{i,t}}+\phi _{i,t}^{\theta -2} \right] . \end{aligned}$$

(16)

From Eqs. 14 and 15, we have $\upsilon _{i,t}=\frac{(c_{i,t+1}/\phi _{i,t})^{1-\theta }+(c_{i,t+1}-\bar{c})^{1-\theta }}{1-\theta }$. Setting $d\upsilon _{i,t}=0$ yields

$$\begin{aligned} \frac{dc_{i,t+1}}{d\phi _{i,t}}=\frac{c_{i,t+1}^{1-\theta }\phi _{i,t}^{\theta -2}}{(c_{i,t+1}-\bar{c})^{-\theta }+c_{i,t+1}^{-\theta }\phi _{i,t}^{\theta -1}}. \end{aligned}$$

(17)

Combining Eqs. 13 through 17, we have

$$\begin{aligned} \sigma (n_{i,t},c_{i,t+1})=\frac{1}{\theta }\left[ 1-\frac{\bar{c}}{\left[ n_{i,t}^{\theta -1}(c_{i,t+1}-\bar{c})^{1-\theta }+1 \right] c_{i,t+1}} \right] . \end{aligned}$$

(18)

From $MRS_{i,t}=\left( \frac{c_{i,t+1}-\bar{c}}{n_{i,t}} \right) ^\theta$, $n_{i,t}=(c_{i,t+1}-\bar{c})MRS_{i,t}^{-\frac{1}{\theta }}$. Plugging this into Eq. 18 yields Eq. 2.