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.

1 Motivation

In recent years, changes in the distribution of income—in particular, increased inequality—have raised much concern, due to the economic crisis. Berg and Ostry (2011) point out that in developing economies, extreme inequality can foster political instability, greater social vulnerability, and it can erode social cohesion. In this regard, the emigration of skilled workers is likely to play a key role in the resulting income distribution in the source countries.¹ Skilled migration undermines the capability of those left behind to fully capture the benefits of higher education. This is known as a drain effect. For instance, Desai et al. (2009) estimate an annual net of fiscal loss of roughly 2.5% of total fiscal revenue for the Indian economy. In addition, in several countries, higher education is publicly financed with general taxes; the optional nature of higher education then implies a (negative) redistribution from the taxpayers to those who benefit from this service, typically the rich (see Blanden and Machin 2004; Hansen 1970; Radner and Miller 1970). Hence, this type of migration can foster the reverse distribution of income.

On top of this effect, which works through the drain of skilled workers, it is important to consider the brain effect. Migration prospects increase the expected return on education, which in turn encourages more individuals to acquire a tertiary education. Since not all of them migrate, the stock of highly educated workers left behind can increase.² These highly educated workers will earn more income and pay more taxes, allowing for higher level of public spending, which, in turn, can improve the equality in the distribution of income.

For several reasons (e.g., descriptive economics, the design of higher educational policy, etc.), it is important to understand the channels through which the brain drain affects the distribution of income and to what extent this migration pattern can increase income inequality. In this paper, we study—theoretically and empirically—whether and, if so, how the brain drain affects the distribution of income.

To this end, we present a simple stylized model of the relationship between skilled emigration and the distribution of income. We consider a two-period economy with households composed of a parent and a child who decides whether to acquire tertiary education. If so, she studies during the first period and works in the second period. If she remains unskilled, she works in both periods. In this paper, we will refer to unskilled (skilled) households as those households whose children do not (do) get tertiary education. In the second period, migration takes place. We assume that emigration is not certain because the host country applies immigration control by issuing a limited number of working visas. In our model, tertiary education is partially publicly provided. Specifically, the tax-subsidy policy is exogenously given. Income tax revenue is used to finance lump-sum transfers and to subsidize only partially the cost of an education. Under this tax-expenditure scheme, lump-sum transfer redistributes income toward lower income households, while skilled emigration entails a fiscal loss. This way, we capture the positive impacts this migration pattern may have on lump-sum transfer (the brain effect), and the negative impacts skilled emigration may have on the distribution of income (the drain effect).

We now briefly summarize the main results of this work. Our model shows that the brain drain can simultaneously increase the stock of human capital left behind and the inequality in the distribution of income. The first result is in line with existing brain drain literature, i.e., the brain effect dominates the drain effect. Let us explain the consequences on the distribution of income. When the skilled emigration rate is sufficiently small, lump-sum households’ transfer rises with the skilled emigration. Because this sort of transfers represents a higher fraction of the income of an unskilled household than it does in that of a skilled household, the income dispersion between unskilled and skilled falls. But there is another effect at work. Since the expected return to education also increases, more individuals enroll on tertiary education. Skilled population becomes more heterogeneous (in terms of their productivities) because newly skilled workers are from the lower end of the ability distribution, thus increasing income dispersion within skilled households to such an extent that we see the Gini coefficient increase. Finally, our model highlights the possibility that further increments in the skilled emigration reduce the inequality in the distribution of income. As skilled emigration continuous to increase, this process keeps on adding new skilled workers from the lower end of the ability distribution, i.e., they are comparable to those already added to the stock of human capital; the skilled population turns relatively more homogeneous, leading to a drop in the Gini coefficient. To sum up, we find a non-monotonic relationship between the brain drain and the distribution of income.³

Our empirical analysis focuses on the relationship between inequality in income distribution (measured by the Gini coefficient) across disposable household income and emigration of skilled workers. We use the Standardized World Income Inequality Database (SWIID), which maximizes the comparability of income inequality statistics for the largest possible sample of countries and years (170 countries).⁴ Regarding skilled emigration, we use a dataset provided by Defoort (2008), which covers the years 1975 through 2000, with a periodicity of 5 years.⁵ Finally, to control for country-specific characteristics (e.g., income level), we use data from the World Bank’s database. The combination of these datasets makes it possible to consider an unbalanced panel data sample of 103 countries during the period 1980–2000.

We use an empirical approach based on a methodology first introduced by Arellano and Bover (1995) and extended by Blundell and Bond (1998) and Blundell et al. (2000). The generalized method of moments instrumental variable for linear dynamic panel data models (GMM-IV), which properly accounts for the possibility of persistence in the inequality variable, allows us to take into account potential problems of weak exogeneity of the (skilled) emigration rate. In addition to skilled emigration rate, we use other independent variables (e.g., tertiary education enrollment rates, etc.) that may also affect income distribution. Our data provide support for the key implication of the theoretical model: the relationship between the Gini coefficient and the skilled emigration exhibits an inverse U-shaped form, thus suggesting the existence of an empirical threshold level of skilled emigration above which the brain drain improves income distribution in sending countries. Moreover, the data suggest that the poorer the source economy, the more likely it is that the brain drain improves equality in the distribution of income.

The current paper is related to the brain drain literature. The effects that international emigration of skilled workers have on source economies has been widely analyzed in the brain drain literature. Most of the theoretical studies focus on the effect that this type of emigration has on human capital formation (see Miyagiwa 1991; Mountford 1997; Romero 2013; Vidal 1998). In an environment where tertiary education is privately financed, Mountford (1997) also studies the effect of skilled emigration on income distribution, showing that, in the long run, the brain drain may improve income distribution in the source economy. On the other hand, several authors have studied the relationship between income inequality and incentives to migrate, concluding that higher income inequality corresponds to low levels of emigration of the most talented skilled workers (Borjas 1987; Borjas and Bratsberg 1996, among others).⁶ The present work expands on the brain drain literature by explicitly modeling public financing of higher education. This allows us to make two contribution to this literature, (a) we show that the brain drain can foster the accumulation of human capital in the source country, while it may increase income inequality and (b) we provide empirical evidence that supports a key prediction of our theoretical model, that is, the non-monotonic relationship between skilled emigration and the Gini coefficient.

The rest of the paper is organized as follows. We first outline our theoretical model in Sect. 2. We present comparative static results in Sect. 3. The data and our empirical approach are described in Sect. 4. In Sect. 5, we present our empirical results, and in Sect. 6, we conclude. Proofs are gathered in the Appendix.

2 The model

Consider a two-period economy with a continuum of households of mass 1. Each household is composed of one parent and one child. Both the parent and the child live two periods of length 1 each. Households derive utility from consumption and spend their disposable income on it. Their preferences are summarized by a utility function $u(y_{1}, y_{2})$, where $y_{j}$ is the disposable income in period j, with $j=1,2$. To simplify, we take $u(y_{1},y_{2})=y_{1}+\beta y_{2}$, where parameter $\beta$ is a discount factor.

Parents are homogeneous in their productivity, or in their human capital endowments. They work in both periods and earn a wage equal to w.⁷ For simplicity, we assume that agents take the market wage rate as given, and the production function is linear for effective labor.

Children differ in their ability to learn. Each child has an ability a, where a is uniformly distributed on $\left[ 0,A\right]$.⁸ In the first period, a child decides whether to obtain a tertiary education or to enter the workforce. Without a tertiary education, the child remains unskilled with a wage w in each period. If the child gets a tertiary education, in the first period, she studies and does not work. Acquiring tertiary education has a cost c. The government partially subsidizes tertiary education, and e denotes the subsidy. Thus, a household whose child gets tertiary education pays $\left( 1-e\right) c$. In the second period, the child becomes skilled and works a skilled job. Firms match skilled workers’ wages with their productivity; therefore, a skilled worker with productivity a earns a wage equal to wa.

Notice that we are assuming that an investment in education acts as a screening device. That is, acquiring tertiary education merely reflects worker (productivity) ability. Empirical evidence suggests that the extent of investment in education as a screening device is culture specific and depends on institutions. Brown and Sessions (1999) provide empirical evidence supporting the hypothesis that education may also enhance inherent abilities. Because we would need to relay on numerical solutions if this possibility is taken into account, we use this simpler specification of education.⁹ On the other hand, modelling “education as screening device” is a typical assumption in the brain drain literature (see, e.g., Chau and Stark 1999; Miyagiwa 1991; Mountford 1997).

Because we are interested in analyzing the effects of the brain drain on the distribution of income, we only allow skilled workers to emigrate.¹⁰ If a skilled worker with ability a emigrates, she earns a wage, net of tax and emigration costs, equal to $w^{F}\!a$.¹¹ We require that $w^{F}>w$, implying that all skilled workers want to emigrate. The host country, however, applies immigration controls, and hence, not all skilled workers can emigrate. Under such immigration controls, there is a probability m of emigrating, with $m\in \left[ 0,1 \right]$, which is independent of the number of workers who are eligible to work abroad.

In both periods, the government levies a tax on household income, where t denotes the tax-rate and $t\in (0,1]$. In the first period, the government uses the tax revenue to subsidize tertiary education and to finance transfers, where each household receives a lump-sum transfer equal to $b_{1}$ . In the second period, the government uses the tax revenue to finance lump-sum transfer $b_{2}$. Let S denote the proportion of agents enrolled in tertiary education and $\widehat{a}$ denote the cutoff value above which individuals decide to obtain this education. In the first period, the government budget constraint is:

$$\begin{aligned} b_{1}+ecS=t\left( w+\left( 1-S\right) w\right) , \end{aligned}$$

(1)

and, in the second period, it is:

$$\begin{aligned} b_{2}=t\left( w+\left( 1-S\right) w+w\left( 1-m\right) E\left[ a|a>\widehat{a }\right] \right) , \end{aligned}$$

(2)

where $E\left[ a|a>\widehat{a}\right]$ is the average productivity of the skilled population. The tax-expenditure policy is $q=(t,e,b_{1},b_{2})$, where the tax rate t and the subsidy e are exogenously given, while the transfers $b_{1}$ and $b_{2}$ are endogenously determined by the number of agents enrolled in tertiary education. We assume that $e\in (0,\overline{e}]$, with $\overline{e}=\min \{t\frac{w}{c},1\}$. The upper bound $\overline{e}$ implies that $b_{1}$ cannot be negative. If $e=t\frac{w}{c}$ (with $t\frac{w}{c}<1$) and $S=1$, then $b_{1}=0$. Hence, $b_{1}\ge 0$ and $b_{2}>0$ for all $S\in [0,1]$.¹² We also assume that the government cannot transfer public funds between periods. This approach is consistent with interpreting the two-period model as a steady-state of a multi-period model.

Consider a household whose child has ability a and does not acquire a tertiary education. Let $y_{1u}$ and $y_{2u}$ denote the household’s first and second-period disposable incomes, respectively. These incomes are equal to:

$$\begin{aligned} y_{ju}=2\left( 1-t\right) w+b_{j},\text { with }j=\left\{ 1,2\right\} . \end{aligned}$$

(3)

Both incomes are based on after-tax incomes of the parent and the child and by household transfers. Now, consider that the child acquires a tertiary education. Let $y_{1s}$ be the first-period income, which is equal to:

$$\begin{aligned} y_{1s}=\left( 1-t\right) w+b_{1}-\left( 1-e\right) c. \end{aligned}$$

(4)

In this case, the child does not work in the first period, and hence, $y_{1s}$ is determined by the parent’s income, the household’s transfer, and the educational cost net of the subsidy. In the second period, the child attempts to emigrate, and the expected income of a skilled household is:

$$\begin{aligned} y_{2s}=\left( 1-t\right) w+w^{e}a+b_{2}, \end{aligned}$$

(5)

where $w^{e}$ is the expected wage, that is, $\left( 1-m\right) \left( 1-t\right) w+mw^{F}\!$.

Summarizing, the timing of the model is as follows: in the first period, children decide whether to acquire a tertiary education. In the second period, emigration occurs, agents work, pay taxes, and households receive transfers.

2.1 Education decisions

For a given policy $q=(t,e,b_{1},b_{2})$, a child decides to acquire a tertiary education if and only if the following inequality holds: $u( y_{1s},y_{2s}) >u( y_{1u},y_{2u})$, which is equivalent to requiring that $a>\widehat{a}(\bullet )$, where:

$$\begin{aligned} \widehat{a}\left( \bullet \right) =\frac{1+\beta }{\beta }\frac{\left( 1-t\right) w}{w^{e}}+\frac{\left( 1-e\right) c}{\beta w^{e}}. \end{aligned}$$

(6)

Because we are interested in studying how the brain drain affects income distribution, we focus only on the parameter m, and the ability threshold denoted by $\widehat{a}(m)$. This cutoff value divides the population into two groups: unskilled and skilled households (see Fig. 1). Note that $\widehat{a}\left( m\right)$ is negatively related with m; as m increases, $\widehat{a}\left( m\right)$ decreases. That is, an increase in the probability of migrating m increases the expected return to tertiary education, which encourages more agents to become skilled, thus leading to a rise in the number of skilled agents. This is the brain effect.

Fig. 1

Children’s education decision, $\widehat{a}(m)$ cutoff value

We assume that $A>\widehat{a}\left( 0\right)$, implying that the most able agent always finds it profitable to get a tertiary education. Consider a policy q and a probability of migrating m,  then the proportion of agents enrolled in tertiary education is equal to:

$$\begin{aligned} S=1-\frac{\widehat{a}\left( m\right) }{A}. \end{aligned}$$

(7)

3 Comparative static results

The goal of this section is to study how the brain drain affects the distribution of income in the source economy. To this end, we proceed as follows. First, we study the effects of m on second-period transfers, and then we analyze the effects on the distribution of income.

3.1 Effects of the brain drain on second-period transfers

Let $\Omega (X)$ be a function that only depends on the parameters $X=(w,c,t,e,\beta ,A)$. Specifically, $\Omega (X)$ represents a threshold level which allows us to state the following comparative static results. We derive its analytical expression in the Appendix. Let $\tilde{w}^{P}$ denote the wage premium $\frac{w^{F}-(1-t)w}{(1-t)w}$.

Lemma 1

• (a) If $\tilde{w}^{P}\ge \Omega (X)$, there is a threshold value $\overline{m}$ such that for m lower (higher) than $\overline{m},$ the transfer in the second period increases (decreases) with m.

• (b) If $\tilde{w}^{P}<\Omega (X)$, then $b_{2}$ is decreasing in m for all $m\in \left[ 0,1\right]$.

Proof

See the “Appendix”. $\square$

Consider Part (a) of Lemma 1. When the wage premium is high enough, $\tilde{w}^{P}\ge \Omega (X)$, there exists a range of values of m,  that is, $\left[ 0,\overline{m}\right]$, for which the brain effect dominates the drain effect. The brain drain increases the stock of human capital left behind. Therefore, as m increases, the second-period revenue also increases, leading to an increase in $b_{2}\left( m\right)$. This result is consistent with previous results in the brain drain literature (Vidal 1998; Chau and Stark 1999; and Beine et al. 2001)

The wage premium is decreasing in w and $\Omega (X)$ is increasing in w, thus implying that the poorer the source economy, as measured by the wage differential, the more likely that $\tilde{w}^{P}\ge \Omega (X)$ holds, and hence, the more likely $b_{2}\left( m\right)$ is to increase in m,  for a small enough m. Additionally, the poorer the economy, the smaller the size of the tertiary educated population. This trend is in accordance with empirical literature that shows that an increase in the magnitude of the brain drain would increase the average productivity of countries that exhibit low current skilled emigration rates and low levels of human capital (see Beine et al. 2008). On the other hand, Part (b) of Lemma 1, that is, when $b_{2}$ always decreases with m, is more likely to hold for relatively wealthier economies.

3.2 Effects of skilled emigration on income distribution

For expositional purposes we start this subsection by analyzing how the brain drain affects the income gap between unskilled and skilled households. Then we study the effects on the whole income distribution, measured by the Gini coefficient.

Let r(m) denote the ratio between the income going to the richest skilled household and the income accruing in the unskilled households.

Consider two households. In one household, a child has ability $a_{1},$ with $a_{1}<\widehat{a}\left( 1\right)$. In another household, a child has the highest ability A in the population. The child with ability $a_{1}$ remains unskilled, while the other child becomes skilled. For simplicity, we assume that the child with ability A does not migrate. The numerator of the expression below is the second-period income of the skilled household and the denominator is the corresponding income of the unskilled household.¹³

$$\begin{aligned} r(m) =\frac{(1-t) w( 1+A) +b_{2}(m)}{2( 1-t) w+b_{2}(m)}. \end{aligned}$$

(8)

The higher the $r\left( m\right)$, the higher is the income gap between the richest skilled household and unskilled households.

Proposition 1

If $\tilde{w}^{P}\ge \Omega (X)$, then $r\left( m\right)$ has a U-shaped form. Otherwise, $r\left( m\right)$ is increasing in m for all $m\in \left[ 0,1\right]$.

Proof

See the “Appendix”. $\square$

Consider the case where $r\left( m\right)$ has a U-shaped form, as in Fig. 2. Let $\overline{m}$ denote the value of the skilled emigration rate associated with the minimum value of the inequality ratio. Note that this cutoff value is the same as that for which $b_{2}$ reaches its maximum value. When m is smaller than $\overline{m},$ a rise in m increases $b_{2}\left( m\right)$, leading to an increase in all household incomes. This increment is relatively higher for unskilled households than for skilled households. Thus, the rise in $b_{2}\left( m\right)$ reduces $r\left( m\right)$. In contrast, when m is higher than $\overline{m},$ we see that r(m) is increasing in m.

Fig. 2

Relationship between inequality ratio and skilled emigration rate

We now turn to study the effects on the Gini coefficient. Let G denote this coefficient. Because we are concerned with the distribution of income across households, a suitable measure of income in the source economy is the gross national income (GNI). The variable $\widehat{p}$ denotes the fraction of unskilled households, i.e., $\widehat{p}(m)=\frac{\widehat{a}(m)}{A}$. The GNI and the Gini coefficient are given by:

$$\begin{aligned} \textit{GNI}(m) =\widehat{p}(m) y_{2u} {+} (1-\widehat{p}(m))((1-t)w+b_{2}(m))+w^{e}E\left[ a|a>\widehat{a}(m)\right] ,\quad \end{aligned}$$

(9)

and,

$$\begin{aligned} G\left( m\right)= & {} 1- 2\Phi (m), \end{aligned}$$

(10)

where,

$$\begin{aligned} \Phi \left( m\right)= & {} 1-\frac{\frac{1}{2}(\hat{p}(m)^{2}(1-t)w+(1-t)w+b_{2}(m))+w^{e}\frac{A}{3}(1-\hat{p}(m)^{3})}{\textit{GNI}(m)}. \end{aligned}$$

Figure 3 has a graphical representation of the Gini coefficient. Notice that $\Phi (m)$ measures the area below the Lorenz Curve, which is composed of two parts. For the poorest p fraction of the population, with $p\le \widehat{p}(m)$, the curve is a straight line. This fraction corresponds to unskilled households with a constant income equal to $y_{2u}$ . The income of skilled households, that is, $y_{2s},$ depends on children’s abilities. Therefore, $y_{2s}$ differs across skilled households. This is captured by the convex part of the curve. The break point of the Lorenz Curve measures the fraction of GNI that goes to unskilled households.

Fig. 3

Lorenz curve

The sign of the derivative of $G\left( m\right)$ with respect to m is ambiguous. The (net) effect on G depends on (a) how income dispersion changes between unskilled and skilled households, i.e., r(m) and (b) how it changes within skilled households.

Proposition 2

There may be a non-monotonic relationship between the Gini coefficient, G, and the skilled emigration rate, m.

• (a) If $\widehat{p}(m)$ approaches 1, when m tends to zero, $\frac{\partial G(m)}{\partial m}|_{m=0}>0$.

• (b) If $\widehat{p}(m)$ approaches zero, when m is sufficiently high, G(m) may decrease when m increases.

Proof

See the “Appendix”. $\square$

The first part of Proposition (2) shows that that for m sufficiently small, G(m) increases with m. Note that this result does not depend on the impact of m on second period transfer. Assume that the brain drain increases fiscal revenue. When the brain effect dominates, increases in m lead to a drop in r(m), which is consistent with a rise of the linear part of the Lorenz Curve and hence with an improvement in the equality of the distribution of income. However, as m rises, new individuals from the lower end of the ability distribution become skilled, increasing income dispersion within skilled households to such an extent that we see the Gini coefficient increase. Consider now the case where the drain effect dominates, fiscal revenue decreases with skilled emigration. In this case, the brain drain increases the Gini coefficient since a rise in m increases the income gap between skilled and unskilled households and increases income dispersion within skilled households.

The non-linearity of the relationship between G and m allows for the possibility that the Gini coefficient decreases with m. As m continues to increase, the migration process keeps on adding new skilled workers from the lower end of the ability distribution, comparable to those already added to the stock of human capital, thus reducing income dispersion within skilled households. This process hence leads to a drop in the Gini coefficient. In the Appendix, we show that it is more likely that skilled emigration improves the income distribution in relatively poor source economies.

4 An empirical analysis for the link between inequality and skilled emigration

We aim to empirically assess the relationship between the Gini coefficient and the skilled emigration.¹⁴ A key prediction of our theoretical model is the existence of a non-monotonic relationship between these two variables. In particular, our model shows that for m sufficiently small, the Gini coefficient increases with skilled emigration. However, for m sufficiently high, this coefficient might fall.

4.1 Data

We use the Standardized World Income Inequality Database (SWIID v. 3.1). This database maximizes the comparability of income inequality statistics for the largest possible sample of countries and years by using a transparent procedure (Solt 2009).¹⁵ It provides information for the Gini coefficient before and after taxes and transfers, that is, a gross and net Gini index for a sample of 170 countries and a time coverage of 50 years for the period 1960–2010. Of particular interest for us is the Gini coefficient corresponding to after-tax-transfer household income (i.e., the net Gini index).

Note that our testable predictions on the impact of m on G are based on a definition of income that includes the earnings once the individual has left the home country (see Eq. (9)). But the data on the Gini coefficient are based on survey data that may include remittances. If we assume that only a fraction $\rho$ of emigrants’ earnings is sent back to the household of origin of the emigrants and we modify Eq. (13), our predictions do not change. We carry out simulations considering that remittances represent 5, 25 and 75% of the income earned abroad and, in all cases, we find that the relationship between G and m has an inverse U-shaped form.

With respect to the (skilled) emigration rates, we use the dataset published by Defoort (2008), which covers the years 1975, 1980, 1985, 1990, 1995 and 2000 for 195 source countries and considers six possible destination countries: Australia, Canada, France, Germany, United Kingdom and the USA.¹⁶ We are aware that considering only six destination countries can limit the scope of our empirical analysis. The reason is that, in some source countries, because of existing colonial ties or for language reasons, an important number of their emigrants might have other destination countries than those considered by Defoort. However, these six destination countries represent 75% of the OECD total immigration stock (Docquier et al. 2016) and the 77% of the OECD skilled immigration stock in 2000 (Beine et al. 2011).¹⁷ It is important to point out that we do not use data corresponding to the year 1975, because for this particular year data are less reliable, since for many countries a number of interpolations were required (Defoort 2008).

Skilled workers are defined as those with a post-secondary certificate. This way the skilled emigration rate is the ratio between the number of skilled emigrants aged 25 and older to the six major receiving countries and the total number of natives (residents and emigrants) older than 25. This database does not include the unskilled emigration rate. We follow Ugarte and Verardi (2012) and compute this variable by combining the databases of Defoort (2008) and that of Barro and Lee (2001) which provides information about total (native) population aged 25 and older classified according to their educational attainment.

Finally, in our estimations we control for some country-specific characteristics. These variables are from the World Bank database. Based on a country’s Gross National Income per capita, the World Bank classifies the country as low-, middle- or high-income. This database provides information for 215 countries and for the period 1987–2012.¹⁸ We construct three dummy variables:

• 1. $\textit{Low-Income}_{i,t}$: This variable equals 1 if country i at period t is classified as a Low-Income country,

• 2. $\textit{Middle-Income}_{i,t}$: This variable takes the value 1 if country i at period t is classified as either a Lower- or Upper-Middle-Income country, and

• 3. $\textit{High-Income}_{i,t}$: This variable equals 1 if country i at period t is a High-Income country.

We also control for the level of tertiary school enrollment and the population size. Including these variables reduces our sample to an unbalanced panel data framework of 103 countries for the period 1980–2000. Table 1 lists the 103 countries included in our estimations; while Table 2 provides summary statistics of our variables of interests.

Table 1 Countries included in the estimation

Table 2 Summary statistics

4.2 The econometric approach

Notice that our two-period model can be interpreted as a simplification of the steady state in a multi-period model. Thus, our empirical investigation relies on the estimation of an econometric model that combines a time-series dimension and a cross-section variation of the data. Specifically, we use a methodology introduced by Arellano and Bover (1995) and extended by Blundell and Bond (1998) and Blundell et al. (2000): the generalized method of moments instrumental variable for linear dynamic panel data models (GMM-IV). Several reasons justify the use of this approach rather than a cross-sectional analysis. First, panel data estimation can include country- and time-fixed effects to control for country-specific unobservable effects. In addition, because of the existence of the Nickell’s bias in within-estimator for panel data with short time dimension (Nickell 1981), the GMM method renders the most appropriate estimation approach.

Second, past inequality is correlated with the current level of the Gini coeffient (Bourguignon and Verdier 1997; Chong 2001; Engerman and Sokoloff 1997). The Arellano–Bover/Blundell–Bond’s linear dynamic panel data method properly accounts for the possibility of persistence in the inequality variable. Moreover, controlling for past inequality allows us to take into account other factors that may affect the current level of income inequality, which is not included in our control variables, such as political institutions, historical, cultural, and natural factors.

Finally, a pure cross-section panel may generate biased estimations because of a lack of strict exogeneity of the emigration rates. Let us explain this issue in more detail. We argue that high income inequality may be associated with (relatively) high skilled-emigration rate. But which causes which? It depends on where we temporally place our variables of interest. On the one hand, our model shows that current skilled emigration can increase future levels of the inequality in the income distribution. On the other hand, Borjas (1987) shows that current income inequality can determine the skill composition of the immigration flow. Hence, shocks that affect the current value of the Gini index can have some effects on the subsequent values of the emigration rates. This is the reason we model emigration rates as predetermined variables.

Which instrument should we use? In a related empirical study, Docquier et al. (2016) argue that it is difficult to find an exogenous instrument for the emigration variable, and suggest using an internal instrument instead. We follow them and use the Arellano and Bover (1995) and Blundell and Bond (1998) system GMM for linear dynamic model which uses internal instruments. How does this procedure instrument the skilled (and unskilled) emigration rates at period $t-5$? It uses lagged differences (of the emigration rates) as an instrument for the level equation. That is, the instruments depend on emigration rates at period $t-10$ and older. As a result, we do not expect in such a long period of time the problem of lack of strict exogeneity continues to exist.

5 Results

We regress the Gini index inequality on its lagged value, on the emigration rate of the skilled workers, and on other additional regressors. The model we estimate is specified as follows:

$$\begin{aligned} G_{i,g,t}=\beta _{0} + \beta _{1}m_{i,t-5}+\beta _{2}m_{i,t-5}^{2} +\beta _{3} um_{i,t-5} + \beta _{4} G_{i,t-5} + \gamma X_{i,t-5}+ \alpha _{i} + \delta _{t} + u_{i,t}, \end{aligned}$$

(11)

where $\beta _{0}$ is the intercept, $\alpha _{i}$ and $\delta _{t}$ capture country i’s fixed effect, and the temporal fixed effects, respectively. For country i, $G_{i,t}$ and $m_{i,t-5}$ denote the level of inequality of the Gini index at period t and the skilled emigration rate at period $t-5$, respectively. The lag of the Gini coefficient is considered to control for the presence of persistence in the inequality variable. We introduce a lagged value of the unskilled emigration rate, $um_{i,t-5}$, as an explanatory variable to control for the distributional effects this type of emigration may have.

In order to test the non-monotonic relationship between the Gini index and the skilled emigration rate, we consider a second order polynomial function.¹⁹ Because our theoretical model predicts that the Gini index first increases with the skilled emigration, and then it eventually falls, we expect that the signs of the estimated coefficients $\beta _{1}$ and $\beta _{2}$ be positive and negative, respectively.

$X_{i,t-5}$ is a vector that includes tertiary education enrollment rates, which may affect income redistribution, since higher enrollment rates tend to be associated with lower inequality values (Bruno et al. 1996). The quantitative importance of this effect makes it relevant to consider the educational variables as control variables in our analysis. We do not consider other control variables such as ethnic diversity, degree of urbanization, cost of education, quality of public health, and education institutions. Some of these variables are relatively stable over time while others exhibit considerable inertia across years. Therefore, it is unclear whether the explicit inclusion of these other variables in the regression would improve the quality of fit significantly and reduce the degree of misspecification bias. To diminish the possibility that the emigration rate picks up a country-size effect, we control for the (log of) population size (Docquier et al. 2016).

On the other hand, controlling for per capita gross domestic product could be a source of heterogeneity because of its interaction with the skilled emigration rate. Moreover, such an inclusion can introduce a strong problem of endogeneity, causing the interpretation for the conditional effect of emigration to be more difficult (Beine et al. 2011). To avoid this problem, we introduce our country-income dummy variables defined in the previous subsection. This classification allows us to control for the heterogeneity introduced by different income levels. Note that our country-income dummy variables are time dependent; meaning that, over the years, country i can move upwards or downwards in the income classification. This is consistent with the estimation of the country-fixed effects.

Table 3 Two step system GMM estimation

In Table 3, column (A), we report our empirical results. Note that a lag of the Gini variable has a positive and strongly significant coefficient, which confirms the existence of persistence in the inequality in income distribution. Although, the estimated coefficient of the (lag of) education variable does not have the expected sign (positive), it is not statistically significant. The estimated coefficients of the other controls variables are not significant, with the exception of $\textit{Middle-Income}_{i,t}$, which is positive and significant at 5%.

The estimated coefficients $\widehat{\beta }_{1}$ and $\widehat{\beta }_{2}$ confirm the key prediction of our theoretical model, the existence of a non-monotonic relationship between the Gini coefficient and the skilled emigration rate. This result implies that there exists an empirical threshold value $m^{*}$, equal to 0.30, above which the relationship between the Gini coefficient and the skilled emigration rate turns negative.²⁰ In 2000, 10% of the source countries had an skilled emigration rate above this threshold.²¹ Specifically, some Central American and some African countries have seen their equality in the income distribution improve because of the brain drain. Note, however, that this estimation cannot assess whether the distributional effects vary across source economies depending on their level of income. This leads us to the next estimation.

Since our assumption that all skilled workers want to emigrate may seem reasonable for less developed or developing countries, we estimate the model (5) for the subsample of low- and middle-income countries. We observe that the estimates generally confirm the existence of a non-monotonic relationship between G and m (see Table 3, column (B)).²²

Our theoretical model shows that the wealthier the source economy, the more likely the brain drain increases the inequality in the distribution of income. We test for this possibility by interacting our country-income dummy variables with the skilled emigration rate and with its squared value. This way, for country i, $\textit{SER-J-Income}_{i,t-5}$ (Squared $\textit{SER-J-Income}_{i,t-5}$) is the product between $m_{i,t-5}$ ($m_{i,t-5}^{2}$) and the corresponding country-income dummy variable for group J (with $J =\{\textit{Low-Income}, \textit{Middle-Income}, \textit{High-Income}\}$). We expect that the threshold value for the skilled emigration rate, $m^{*}$, increases across the income categories.

Table 4 Two step system GMM estimation

Table 4 shows these results. We observe no change in the sign and significance level of our control variables (the lag of the Gini index, population size, etc.). In line with our theoretical predictions, we see that the quadratic relationship between our variables of interest is statistically significant for both Low- and Middle-Income groups. For the Low-Income group, the estimated threshold value $m^{*}$ is equal to 0.27, smaller than 0.3 (the previous estimated threshold level), and even smaller than the estimated threshold for Middle-Income countries: 0.32. In 2000, 21% of countries clustered in the Low-Income group had an skilled emigration rate above 0.27; while this percentage fell to 7% for those countries classified as Middle-Income. Concerning the High-Income countries, this estimate suggests that the emigration of skilled workers does not affect their (next period) distribution of income. This may have to do with the fact that in these countries, skilled emigration is not a significant phenomenon since they are net-receivers of skilled immigrants.

5.1 Robustness check

In this section we conduct a robustness check using a new database “The IAB brain drain data” (Brücker et al. 2013), which covers information for 20 OECD destination countries for the years 1980–2010 (with a periodicity of 5 years).

The combination of this data set with that of the World Bank and the SWIID database makes it possible to consider an unbalanced panel data sample of 109 countries during the period 1980–2005. We do not use data corresponding to the year 2010 because for this particular year data are less reliable. Brücker et al. (2013) point out that in most cases interpolation were required because the last year of the Census was not available at the time they started the data collection.

Table 5 Results based on Brücker et al. (2013) database

Table 5 presents the results corresponding to two alternative specifications. In column (1) we estimate Eq. (11), while in column (2) we interact our country-income dummy variables with the skilled emigration rate and with its squared value. Overall the estimated coefficients of our variables of interest are characterized by the same sign and significance patterns as those of the previous section. Hence, empirical evidence suggests that the Gini coefficient and the skilled emigration rate exhibit an inverse U-shaped form. Since The IAB brain drain data covers 20 destinations countries, these estimates provide an important robustness check of our main result.

6 Concluding remarks

This paper studied, theoretically and empirically, the effect of the brain drain on the distribution of income in a source economy. We presented a simple stylized model in which tertiary education is only partially subsidized, in a context where skilled workers can emigrate. A key prediction of the model is the non-monotonic relationship between the Gini coefficient and the skilled emigration rate. The brain drain can increase the stock of human capital left behind (this is the brain gain hypothesis), while it increases the inequality in the distribution of income. When the skilled emigration rate, m, is sufficiently small, the brain effect dominates the drain effect, leading to an increase in second-period transfers. As transfers represent a higher fraction in the income of an unskilled household compared to that of a skilled household, the income gap between unskilled and skilled households falls. But there is another, opposing effect at work. For m sufficiently small, as the skilled emigration rate increases newly skilled workers are from the lower end of the ability distribution, skilled population becomes more heterogeneous, which, in turn, increases income dispersion within skilled households to such an extent that the Gini coefficient increases. Finally, the model highlights the possibility that further increments in the skilled emigration rate reduce both the stock of human capital left behind and the inequality in the income distribution. This has to do with the fact that for m sufficiently high, as the skilled emigration continues to increase, the emigration process keeps on adding new skilled individuals from the lower end of the ability distribution, comparable to those already skilled, which makes the distribution of income within skilled population more homogeneous, leading to a fall in the Gini coefficient.

Our empirical approach tests the relationship between household income distribution (measured by the Gini coefficient) and the brain drain (measured by skilled emigration rate). We find evidence for a significant inverse U-shaped form between the Gini index and the skilled emigration. In particular, our estimates show that the brain drain seems to improve the equality in the distribution of income for (some of) the poorest countries in our sample; while for relatively richer countries skilled emigration increases the inequality in the distribution of income. We can summarize our results as follows: countries with a low skilled emigration rate benefit from the brain drain in terms of human capital accumulation while their distribution of income becomes more unequal. On the other hand, those source countries with a high level of skilled emigration may see their human capital stock decrease, but it is likely that they benefit from a more equal distribution of income.

These results open the possibility that the brain drain increases the social welfare of those left behind, while increases the inequality in the distribution of income. Our model shows that the brain drain can increase per capita income in the source country, while it increases the inequality in the distribution of income. This highlights the trade-off between efficiency and equity effects. If we use a Sen-type social welfare function, the efficiency and equity effects do not move in the same direction: while a ”brain-gain” increases the social welfare, the dispersion effects move in the opposite direction. As a result, it is not clear whether the migration of skilled workers entails a ”welfare-gain” or a ”welfare-loss” in the source country. Future research is needed to explore the effects of the brain drain on the social welfare of the source country.

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. $\square$