Brain drain and income distribution

Abstract

In a context in which increased income inequality has raised much concern, and skilled workers move easily across countries, an important question arises: how does the brain drain affect income distribution in the source economy? We address this question and introduce two contributions to the literature on brain drain. First, we present and solve a simple stylized model to study whether and, if so, how the brain drain affects the distribution of income, in a context in which higher education is publicly financed with general taxes. Second, we explore empirically the effect of an increase in skilled emigration on income distribution. A key prediction of our theoretical model is the existence of a non-monotonic relationship between income inequality and emigration of skilled workers. Our empirical data confirm this result, showing a statistically significant inverse U-shaped form.

Appendix

Proof of Lemma 1

We first compute the derivatives of expressions (6) and (2) with respect to m. To save notation, we use the variable $\widehat{p}$ that measures the fraction of unskilled individuals, $\widehat{p}(m)=\frac{\widehat{a}(m)}{A}$.

$$\begin{aligned} \frac{\partial \widehat{a}}{\partial m}= & {} -\left( \frac{1+\beta }{\beta }\frac{\left( 1-t\right) w}{w^{e}}+\frac{\left( 1-e\right) c}{\beta w^{e}}\right) \frac{\left( w^{F}-\left( 1-t\right) w\right) }{w^{e}}, \nonumber \\ \frac{\partial \widehat{a}}{\partial m}= & {} -\hat{p}(m)A\frac{w^{F}-(1-t)w}{w^{e}}. \end{aligned}$$

(12)

$$\begin{aligned} \frac{\partial b_{2}}{\partial m}= & {} -tw\left( \frac{\frac{\partial \widehat{a}}{\partial m} }{A}\left( \left( 1-m\right) \widehat{a}\left( m\right) -1\right) +E\left[ a|a> \widehat{a}\left( m\right) \right] \right) . \end{aligned}$$

(13)

Part (a). Note that if $m=1,$ $\frac{\partial b_{2}}{\partial m}$ is negative. For $m<1$, $\frac{\partial b_{2}}{\partial m}$ is positive if and only if the following condition holds:

$$\begin{aligned} \frac{A^{2}-\widehat{a}\left( m\right) ^{2}}{2A}\leqslant & {} -\frac{ \frac{\partial \widehat{a}}{\partial m}}{A}\left( \left( 1-m\right) \widehat{ a}\left( m\right) -1\right) ,\text { or} \nonumber \\ \frac{A}{2}(1-\widehat{p}(m)^{2})\leqslant & {} \widehat{p}(m)\frac{w^{F}-(1-t)w}{w^{e}}((1-m)\widehat{a}(m)-1). \end{aligned}$$

(14)

By making $m=0$ and rearranging terms, we get the following expression:

$$\begin{aligned} \frac{A}{2}\left( \frac{1-\widehat{p}(0)^{2}}{\widehat{p}(0)}\right) \frac{1}{\widehat{a}(0)-1}< & {} \frac{w^{F}-(1-t)w}{(1-t)w}=\tilde{w}^{P}. \end{aligned}$$

(15)

The value of $\widehat{a}(0)$ is greater than 1, and the left hand side of condition (15) is $\Omega (X)$. If condition (15) holds, by applying the median value theorem we have that there is an $\overline{m}$ such that (14) is satisfied with equality. Then, for all $m\$smaller (higher) than $\overline{m},$ $\frac{ \partial b_{2}}{\partial m}$ is positive (negative).

Note that $\tilde{w}^{P}$ decreases with w, while $\Omega (X)$ increases with w since $\widehat{a}(0)$ decreases with w. Hence, the poorer the economy (i.e., the higher $\tilde{w}^{P}$ and the lower $\Omega (X)$), the more likely that the condition (15) is satisfied.

Part (b). If condition (15) never holds, then $\frac{\partial b_{2}}{\partial m}$ is a non-increasing function of m. $\square$

Proof of Proposition 1

Recall expression (13). Condition (15) implies $\frac{\partial b_{2}}{\partial m}\ge 0$ whenever m is smaller or equal to $\overline{m}$. By differentiating $r\left( m\right)$ with respect to m we get:

$$\begin{aligned} \frac{\partial r}{\partial m}=\frac{\frac{\partial b_{2}}{\partial m}\left( y_{2u}-y_{2s}\right) }{\left( y_{2u}\right) ^{2}}. \end{aligned}$$

(16)

Then, for all $m\leqslant \overline{m}$, $\frac{\partial b_{2}}{\partial m} \geqslant 0$, hence, $\frac{\partial r}{\partial m}\leqslant 0$. And $\frac{ \partial r}{\partial m}>0$ for all $\overline{m}<m\leqslant 1$. Finally, when condition (15) does not hold, $b_{2}$ is a decreasing function of m. Hence, $\frac{\partial r}{\partial m}$ is positive for all $m\in \left[ 0,1\right]$.$\square$

Proof of Proposition 2

Part (a). Before showing that $\frac{\partial G(m)}{\partial m}|_{m=0}>0$, we use the government budget constraint in the second period to rewrite $\textit{GNI}(m)$

$$\begin{aligned} \textit{GNI}(m)= & {} w + \widehat{p}(m)w + (mw^{F}+(1-m)w)\frac{A}{2}(1-\widehat{p}(m)^{2}). \end{aligned}$$

We now compute some useful derivatives. Recall equation (13), i.e., $\frac{\partial b_{2}}{\partial m}$, and that $\widehat{p}(m)=\frac{\widehat{a}(m)}{A}$.

$$\begin{aligned} \frac{\partial \textit{GNI}(m)}{\partial m}= & {} \frac{\partial \widehat{p}\left( m\right) }{\partial m}w+\left( w^{F}-w\right) \frac{A}{2}\left( 1-\widehat{p}\left( m\right) ^{2}\right) - \left( mw^{F}+\left( 1-m\right) w\right) A\widehat{p}\left( m\right) \frac{\partial \widehat{p}\left( m\right) }{\partial m}, \nonumber \\= & {} \frac{\partial \widehat{p}(m)}{\partial m}(w -(mw^{F}+(1-m)w)\widehat{a}(m)) + \left( w^{F}-w\right) \frac{A}{2}\left( 1-\widehat{p}\left( m\right) ^{2}\right) . \end{aligned}$$

(17)

To save notation, let $\Delta$ be equal to

$$\begin{aligned} \Delta (m)= & {} \frac{1}{2}\left( \widehat{p}\left( m\right) ^{2}\left( 1-t\right) w+\left( 1-t\right) w+b_{2}\left( m\right) \right) +w^{e}\frac{A}{3}\left( 1-\widehat{p}\left( m\right) ^{3}\right) , \text{ and } \\ \frac{\partial \Delta (m)}{\partial m}= & {} \widehat{p}(m)\frac{\partial \widehat{p}(m)}{\partial m}((1-t)w-w^{e}\widehat{a}(m))+\frac{1}{2}\frac{\partial b_{2}(m)}{\partial m} +(w^{F}-(1-t)w)\frac{A}{3}(1-\widehat{p}(m)^{3}). \end{aligned}$$

We can write $\Phi (m) = 1-\frac{\Delta (m)}{\textit{GNI}(m)}$ and the derivative of $\Phi (m)$ with respect to m is

$$\begin{aligned} \frac{\partial \Phi (m)}{\partial m}= & {} -\frac{\frac{\partial \Delta (m)}{\partial m}{} \textit{GNI}(m)-\Delta (m)\frac{\partial \textit{GNI}(m)}{\partial m}}{\textit{GNI}(m)^{2}}. \end{aligned}$$

The derivative of G(m) with respect to m is equal to

$$\begin{aligned} \frac{\partial G(m)}{\partial m}= & {} -2\frac{\partial \Phi (m)}{\partial m},\\ \frac{\partial G(m)}{\partial m}\ge & {} 0 \Leftrightarrow \\ \frac{\partial \Delta (m)}{\partial m}{} \textit{GNI}(m)\ge & {} \Delta (m)\frac{\partial \textit{GNI}(m)}{\partial m}. \end{aligned}$$

We now evaluate $\frac{\partial \textit{GNI}(m)}{\partial m}$, $\frac{\partial \Delta (m)}{\partial m}$, $\Delta (m)$, and $\textit{GNI}(m)$ at $m=0$.

$$\begin{aligned} \frac{\partial \textit{GNI}(m)}{\partial m}|_{m=0}= & {} -\widehat{p}(m)|_{m=0}\tilde{w}^{P}w(1{-}\widehat{a}(m)|_{m=0}){+}(w^{F}{-}w)\frac{A}{2}(1{-}\widehat{p}(m)^{2}|_{m=0}).\\ \frac{\partial \Delta (m)}{\partial m}|_{m=0}= & {} -\widehat{p}(m)^{2}|_{m=0}\tilde{w}^{P}(1-t)w(1-\widehat{a}(m)|_{m=0}) \\&+\frac{1}{2}tw[\widehat{p}(m)|_{m=0}\tilde{w}^{P}(\widehat{a}(m)|_{m=0}-1)- \frac{A}{2}(1-\widehat{p}(m)^{2}|_{m=0})] \\&+(w^{F}-(1-t)w)\frac{A}{3}(1-\widehat{p}(m)^{3}|_{m=0}). \\ \Delta (m)|_{m=0}= & {} \frac{1}{2}[\widehat{p}(m)^{2}|_{m=0}(1-t)w+(1-t)w\\&+tw(1+\widehat{p}(m)|_{m=0}+\frac{A}{2}(1-\widehat{p}(m)^{2}|_{m=0})) ] \\&+(1-t)w\frac{A}{3}(1-\widehat{p}(m)^{3}|_{m=0}), \\ \textit{GNI}(m)|_{m=0}= & {} w + w\widehat{p}(m)|_{m=0}+w\frac{A}{2}(1-\widehat{p}(m)^{2}|_{m=0}). \end{aligned}$$

Note that $\frac{\partial \textit{GNI}(m)}{\partial m}|_{m=0}>0$, since $(1-\widehat{a}(m)|_{m=0})<0$. We have:

$$\begin{aligned} \frac{\partial \Delta (m)}{\partial m}|_{m=0}{} \textit{GNI}(m)|_{m=0}> & {} \Delta (m)|_{m=0}\frac{\partial \textit{GNI}(m)}{\partial m}|_{m=0} \Leftrightarrow \nonumber \\ \varphi (m)|_{m=0}= & {} \frac{\frac{\partial \Delta (m)}{\partial m}|_{m=0}}{\frac{\partial \textit{GNI}(m)}{\partial m}|_{m=0}}\left( \frac{\Delta (m)|_{m=0}}{\textit{GNI}(m)|_{m=0}}\right) ^{-1} > 1. \end{aligned}$$

(18)

Notice that these expressions, and hence $\varphi (m)|_{m=0}$, are written in terms of $\widehat{p}(m)|_{m=0}$ (the percentage of unskilled households when emigration is not allowed). Since $\widehat{a}(m)$ decreases with m, as the skilled migration rate tends to zero, $\widehat{p}(m)$ approaches to 1. We then take limit of $\varphi$ for $\widehat{p}(m)|_{m=0}$ that tends to 1:

$$\begin{aligned} \lim _{\widehat{p}(m)|_{m=0}\rightarrow 1}\varphi (\widehat{p}(m)|_{m=0})= & {} \frac{\tilde{w}^{P}w(A-1)(2-t)}{\tilde{w}^{P}w(A-1)2}\frac{2w}{w} = 2-t>1, \end{aligned}$$

(19)

indicating that, in the limits, condition (18) holds. Because $\varphi (\widehat{p}(m)|_{m=0})$ is continuous in $\widehat{p}(m)|_{m=0}$, in the vicinity of 1, values of $\widehat{p}(m)|_{m=0}<1$ satisfy condition (18).

Part (b). By applying an analogous reasoning, evaluating the resulting expressions at $m=1$ and taking limits for $\widehat{p}(m)|_{m=1}$ that tends to zero, we have that $\frac{\partial G(m)}{\partial m}<0$ if and only if

$$\begin{aligned} \frac{\frac{w^{F}}{w}\frac{A}{3}-(1-t)\frac{A}{3}-t\frac{A}{4}}{\frac{w^{F}}{w}\frac{A}{3}+\frac{1}{2}} < \frac{(\frac{w^{F}}{w}-1)\frac{A}{2}}{\frac{w^{F}}{w}\frac{A}{2}+1}. \end{aligned}$$

(20)

The poorer the economy (i.e., the higher $\frac{w^{F}}{w}$), the more likely that the condition above is satisfied and hence the Gini coefficient decreases with m. However, when w approaches to $w^{F}$, and hence $\frac{w^{F}}{w}$ tends to 1, this condition is not satisfied and the Gini coefficient increases with m. $\square$