## Abstract

This study analyzes the impact of international migration on economic growth of a source country in a stochastic setting. The model accounts for endogenous fertility decisions and distinguishes between public and private schooling systems. We find that economic growth crucially depends on the international migration since the migration possibility will affect fertility decisions and school expenditures. Relaxation of restrictions on the emigration of high-skilled workers will damage the economic growth of a source country in the long run, although a ‘brain gain’ may happen in the short run. Furthermore, the growth rate of a source country under a private education regime will be more sensitive to the probability of migration than a country under a public education regime.

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## Notes

See Bhagwati and Rodriguez (1975) for a literature survey of earlier works on this issue.

Parents would care about domestic and foreign children differently due to several reasons. For example, parents may care less about migrating children since they see them less often. On the other hand, parents may care more about migrating children because of a direct monetary reason such as remittances.

However, several modifications have been made because the intention of Kalemli-Ozcan (2003) was to study the implications of mortality.

Kalemli-Ozcan referred to this as the “insurance effect” since with the uncertainty of mortality of children, a self-insurance strategy for parents is to overshoot fertility.

The probability of migration for both low- and high-skilled workers will be determined by the policies adopted by the governments from both the source and home countries. However, in this paper, we assume that the probability of migration is exogenous, and we do not study the migration issue from a host country’s perspective.

The model can be easily extended to allow for the endogenous choice of migration by including the cost of migration and the innate ability into the human capital accumulation function. The results would be that young agents with high parental human capital and high innate ability (agents with high human capital accumulation) will emigrate. Hence, similar results can be obtained by assuming that high-skilled workers have higher probability to emigrate than low-skilled workers.

Notice that there is no intergroup mobility in our model. Similar model setting can be found in De la Croix and Doepke (2004). One possible way to allow for the intergoup mobility is to incorporate the innate ability into the human capital accumulation function and assume that the probability of migration depends on each agent’s human capital accumulation. However, this will complicate the model without changing our main results about the impacts of migration probability on the economic growth.

However, our computational results in the next section show that through the five periods (which is approximately 150 years),

*e*_{ rt }^{L}(*h*_{ rt }^{L}) is always lower than*e*_{ rt }^{H}(*h*_{ rt }^{H}) in every period.Given the exogenous real wage per unit of human capital (

*w*_{ j }), the elasticity of human capital for children with respect to school expenditure equals the income elasticity with respect to school expenditure.We choose the Philippines as our source country because, based on 1990 data, Carrington and Detragiache (1999) demonstrated that highly-educated migrants from the Asia-Pacific region were the second largest group of immigrants to the USA, and of this particular group, the Philippines was shown to be the major source country.

Source: per capita GDP, PPP (constant 1987 international dollar), World Development Index, World Bank.

Given that per capita GDP in the US is 7.975 times the level of per capita GDP in the Philippines,

*w*_{B}is calibrated as 7.975*w*_{A}.The data set composed by Deininger and Squire (1996) shows that the share of income of the bottom quintile was 5.2%, whilst the share of income of the top quintile was 52.1%.

Using these calibrated numbers along with other calibrated parameter values for fertility gives us a per capita GDP level equal to *2,522.6 at the end of the first period.

Source for income share and Gini coefficient: Deininger and Squire (1996), World Bank.

Public school enrollment was 59% of total enrollment in secondary schools for the Philippines in 1985.

Data source for public school enrollment rate and public spending on education: UNESCO, United Nations.

In the baseline model, everyone attends a public school. Hence, the proportion of public school expenditure to GNP becomes 0.014/0.59=2.373%.

Data source for economic growth rate and population growth rate: World Development Index, World Bank.

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## Additional information

*Responsible editor*: Klaus F. Zimmermann

## Appendices

### Appendix 1

In this appendix, we omit the time index *t* explicitly to make our derivations easier to read. We should first of all note that the first- and second-order partial derivatives of the utility function with respect to *N* are:

where *d*=*w*
_{B}−*aw*
_{A}.

The second-degree Taylor series expansions of the utility function around the mean *N*=*pn* is:

Substituting Eqs. (14) and (15) into Eq. (16) and using the statistical results that *E*(*N*−*pn*)=0 and *E*(*N*−*pn*)^{2}=*np*(1−*p*), we can derive the expectation of the utility function as:

where *w*=*pw*
_{B}+*a*(1−*p*)*w*
_{A}.

### Appendix 2

### Proof of proposition 3

We first analyze an economy under a private education regime and then go on to study an economy under a public education regime.

#### Under a private education regime

Given all parameter values, Eq. 11 implies that the number of children is constant. Hence, from Eq. 12, *e*
_{
rt
} is a linear function of *h*
_{
rt
} and can be expressed as *e*
_{
rt
}=*ɛh*
_{
rt
}, where *ɛ* is a positive number. Equation 2 tells us that human capital in the next period will be \(h_{{rt + 1}} = \lambda {\left( {\varepsilon h_{{rt}} } \right)}^{\gamma } h^{{1 - \gamma }}_{{rt}} = \lambda \varepsilon ^{\gamma } h_{{rt}}\). With homogeneous agents, *H*
_{
rt
}=*h*
_{
rt
} for all *t*. Therefore, the growth rate of average human capital under a private education regime is:

Equation 18 implies that *g*
_{
r
}
^{H} is constant.

#### Under a public education regime

Under a public education regime, the human capital accumulation function becomes:

When agents are homogeneous, *H*
_{
ut
}=*h*
_{
ut
} for all *t*. This implies that the growth rate under a public education is:

Because Eqs. 9 and 13 show that given the probability of migration, the tax rate and fertility are constant, Eq. 20 implies that *g*
_{
u
}
^{H} is constant.

QED.

### Appendix 3

### Proof of proposition 4

We first consider an economy under a private education regime. Then we study an economy under a public education regime.

#### Under a private education regime

The left-hand side of Eq. 11 is a function of *n*
_{
rt
} and can be expressed by *ξ*(*n*
_{
rt
}). The right-hand side depends on *n*
_{
rt
} and *p* and can be represented by *μ*(*n*
_{
rt
},*p*). Hence, we can rewrite Eq. 11 as

Taking the derivative of both sides of Eq. 21 with respect to *p*, we get

Note firstly that

secondly that

and thirdly that

Substituting the definitions of *w* and *d* into the numerator of Eq. 25, we can get

Define *p** such that \(\frac{{a{\left( {1 - p*} \right)}}}{{p*}} = \frac{{w_{{\text{B}}} }}{{w_{{\text{A}}} }}\). Hence, if \(\frac{{\partial \mu }} {{\partial p}} < 0\;if\;p > p*\). Then one must have *n*
_{
rt
}′(*p*)<0 if *p*>*p** for Eq. 22 to hold. The situation will be reversed (\(\frac{{\partial \mu }}{{\partial p}} > 0\) and *n*
_{
rt
}′(*p*)>0) if *p*<*p**.

From Eq. 12, the derivative of *e*
_{
rt
} with respect to *n*
_{
rt
} is:

Using the implicit differentiation of *e*
_{
rt
}, we can get that \(e^{\prime }_{{rt}} {\left( p \right)} = \frac{{de_{{rt}} }}{{dn_{{rt}} }}n^{\prime }_{{rt}} {\left( p \right)} > 0\) if *p*>*p** and vice versa.

#### Under a public education regime

The left-hand side of Eq. 13 is a function of *n*
_{
ut
} and can be expressed by *ξ*(*n*
_{
ut
}). The right-hand side depends on *n*
_{
ut
} and *p* and can be represented by *μ*(*n*
_{
ut
},*p*). Hence, we can rewrite Eq. 13 as:

Taking the derivative of both sides of Eq. 27 with respect to *p*, we get

Note firstly that

secondly that

and thirdly that

Hence, if \(p > p*,\;\frac{{\partial \mu }} {{\partial p}} < 0\). Then from Eq. 28, we know that *n*
_{
ut
}′(*p*)<0. The situation will be reversed [\(\frac{{\partial \mu }}{{\partial p}} > 0\) and *n*
_{
ut
}′(*p*)>0] if *p*<*p**.

From Eq. 6, the derivative of *e*
_{
ut
} with respect to *n*
_{
ut
} is

Using the implicit differentiation of *e*
_{
ut
}, we can get that \(e^{\prime }_{{ut}} {\left( p \right)} = \frac{{de_{{ut}} }}{{dn_{{ut}} }}n^{\prime }_{{ut}} {\left( p \right)} > 0\) if *p*>*p** and vice versa.

QED.

### Appendix 4

### Proof of proposition 7

We should first of all note that:

Under a private education regime, the partial differentiation of *H*
_{
rt+1} with respect to *p*
^{H} can be written as:

Since *e*
_{
rt
}
^{L}′(*p*
^{H})=0, Eq. 34 can be expressed as:

Because \(\frac{{\partial \theta ^{{\text{L}}}_{{rt + 1}} }}{{\partial p^{{\text{L}}} }} < 0\) and \(\theta ^{{\text{L}}}_{{rt + 1}} \frac{{\partial h^{{\text{L}}}_{{rt + 1}} }}{{\partial e^{{\text{L}}}_{{rt}} }}e^{{\text{L}}}_{{rt}} \prime {\left( {p^{{\text{L}}} } \right)} > 0\) if *p*
^{H}>*p**, a sufficient condition for \(\frac{{\partial H_{{rt + 1}} }}{{\partial p^{{\text{H}}} }} > 0\) under a private education regime is that \(\frac{{\partial \theta ^{{\text{L}}}_{{rt + 1}} }}{{\partial p^{{\text{H}}} }}\) is not too large (that is, an increase in *p*
^{H} will not cause a large increase in *θ*
_{
rt+1}
^{L}).

Under a public education regime with heterogeneous agents, public school expenditure is:

From Eq. 36, we can derive that \(e^{\prime }_{{ut}} {\left( {p^{{\text{H}}} } \right)} = \frac{{\partial e_{{ut}} }}{{\partial n^{{\text{H}}}_{{ut}} }}n^{\prime }_{{ut}} {\left( {p^{{\text{H}}} } \right)} > 0\).

The partial differentiation of *H*
_{
ut+1} with respect to *p*
^{H} can be written as:

Because\(\frac{{\partial \theta ^{{\text{L}}}_{{ut + 1}} }}{{\partial p^{{\text{H}}} }} > 0,\,\frac{{\partial h^{{\text{L}}}_{{ut + 1}} }}{{\partial e_{{ut}} }} > 0,\,\frac{{\partial h^{{\text{H}}}_{{ut + 1}} }}{{\partial e_{{ut}} }} > 0\), and *e*
_{
ut
}
^{H}′(*p*
^{H}) if *p*
^{H}>*p**, a sufficient condition for \(\frac{{\partial H_{{ut + 1}} }} {{\partial p^{{\text{H}}} }} > 0\) under a public education regime is that \(\frac{{\partial \theta ^{{\text{L}}}_{{ut + 1}} }}{{\partial p^{{\text{H}}} }}\) is not too large (that is, an increase in *p*
^{H} will not cause a large increase in *θ*
_{
ut+1}
^{L}).

QED.

### Appendix 5

### Proof of proposition 8

We should first of all note that:

Under a private education regime, the partial differentiation of *H*
_{
rt+1} with respect to *p*
^{L} can be written as:

Since *e*
_{
rt
}
^{H}′(*p*
^{L})=0, Eq. 39 can be expressed as:

Because \(\frac{{\partial \theta ^{{\text{L}}}_{{rt + 1}} }} {{\partial p^{{\text{L}}} }} < 0\) and \(\theta ^{{\text{L}}}_{{rt + 1}} \frac{{\partial h^{{\text{L}}}_{{rt + 1}} }} {{\partial e^{{\text{L}}}_{{rt}} }}e^{{\text{L}}}_{{rt}} \prime {\left( {p^{{\text{L}}} } \right)} > 0\) if *p*
^{L}>*p**, from Eq. 40, we have \(\frac{{\partial H_{{rt + 1}} }} {{\partial p^{{\text{L}}} }} > 0\).

Under a public education regime, from Eq. 36, we can derive that

The partial differentiation of *H*
_{
ut+1} with respect to *p*
^{L} can be written as:

Because \(\frac{{\partial \theta ^{{\text{L}}}_{{ut + 1}} }}{{\partial p^{{\text{L}}} }} < 0,\,\frac{{\partial h^{{\text{L}}}_{{ut + 1}} }}{{\partial e_{{ut}} }} > 0,\,\frac{{\partial h^{{\text{H}}}_{{ut + 1}} }}{{\partial e_{{ut}} }} > 0\), and *e*
_{
ut
}
^{H}′(*p*
^{L})>0 if *p*
^{L}>*p**, we can get that \(\frac{{\partial H_{{ut + 1}} }} {{\partial p^{{\text{L}}} }} > 0\) from Eq. 41.

QED.

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### Cite this article

Chen, HJ. International migration and economic growth: a source country perspective.
*J Popul Econ* **19**, 725–748 (2006). https://doi.org/10.1007/s00148-005-0023-1

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DOI: https://doi.org/10.1007/s00148-005-0023-1

### Keywords

- Migration
- Brain drain
- Economic growth

### JEL Classification

- F22
- J24
- O15