2.1. Wages, the cost of migration, and the types of individuals

First, suppose that in economy $i$ $(i = MD,{\rm{ }}LD)$ , the wage per efficiency unit of labor, ${w^i},$ is fixed over time as follows from, for example, the assumption of a constant returns to scale technology with a single factor of production. Secondly, to introduce migration, we suppose that a young adult born in the less developed economy, who decides to migrate in period $t$ to the more developed economy, incurs a cost of migration equal to ${M_t}$ . We also assume that migration involves no losses of human capital, so that any unit of human capital acquired in the home country prior to migration is as productive as the skills acquired in the destination country.

Since by construction in this model, only one way of migration can be worthwhile, in the world economy under consideration there are potentially three types of individuals: (1) $MD$ , individuals born in the more developed economy, (2) $LD$ , individuals born in the less developed economy, who remain in their country of birth, and (3) $M$ , individuals born in the less developed economy, who migrate to the more developed economy. The conditions that lead to the decision to migrate are analyzed below in Section 2.5.

2.2. The formation of human capital

In any period $t$ , an adult of type $j$ $(j = MD,{\rm{ }}LD,{\rm{ }}M)$ born in economy $i$ $(i = MD,{\rm{ }}LD)$ , is characterized by a skill level ${h_t}$ that is distributed according to the cumulative density function ${F_t}( \cdot )$ over the strictly positive support $[h_t^{i,\min },{\rm{ }}h_t^{i,\max }].$ It is assumed throughout that in period $t$ , the average level of human capital in the more advanced economy is higher than that in the less advanced economy; $\bar h_t^{MD} \gt \bar h_t^{LD}$ .

In each period of life, individuals are endowed with one unit of time. In the first period, children devote their entire time to the acquisition of human capital. The acquired human capital increases if their time investment is supplemented with real resources invested in their education.

The human capital level of a child, who becomes an adult in period $t+1,$ depends on the parental real expenditure on that child’s education, ${e_t}$ , and on the average level of human capital of all adult individuals in economy in period $t$ , which is defined as $\bar h_t^i = \int {{h_t}dF_t^i({h^i})}, $ according to the human capital production function, or learning technology described by

(1) $$h_{t+1}^i = \Theta ({e_t},\bar h_t^i), \quad {\rm{ if \;}}i = MD,{\rm{ }}LD.$$

This learning technology captures an external spillover effect that arises from the average society’s level of human capital, ${\bar h_t}.$ Such formulation is consistent with the so-called global or atmospheric externality, which implies that an increase in the average level of human capital in the society as a whole increases the rate of return on investment in human capital for the children’s generation. First introduced by Tamura (Reference Tamura1991), the assumption that the average level of human capital in society is an input in the production of human capital for each individual became common in the literature. This externality has been utilized, for instance, by Tamura (Reference Tamura1996), Galor and Tsiddon (Reference Galor and Tsiddon1997), de la Croix and Doepke (Reference de la Croix and Doepke2003; Reference de la Croix and Doepke2004), Azarnert (Reference Azarnert2008; Reference Azarnert2009; Reference Azarnert2010a), and Aksan and Chakraborty (Reference Aksan and Chakraborty2014), among many others.

A particular form of human capital production function is specified below in equation (6).

2.3. The optimization of parents

Agents of any type derive utility from their own consumption in adulthood and from the total future income of their children in the economy where those children were born.Footnote ⁸ The utility function of an individual of any type $j$ ( $j = MD,{\rm{ }}LD,{\rm{ }}M$ ) born at time $t - 1$ is therefore

(2) $$U_t^j = (1 - \beta )\log C_t^j + \beta \log (I_{t+1}^{N,j}),$$

where $C_t^j$ is an individual’s own consumption, $I_{t+1}^{N,j}$ is the future income of that individual’s offspring, as defined in equation (4), and $\beta \in (0,{\rm{ }}1)$ captures the relative weight given to children.Footnote ⁹

In every period t, adult individuals are endowed with one unit of time which is allocated between childbearing and labor force participation. In either economy, the cost of feeding and raising children is measured in terms of work time foregone at $\delta $ per child. The cost of acquiring human capital in any economy $i$ is measured in units of the wage per efficiency unit of labor in that economy, ${w^i}$ . Migrants from the $LD$ to the $MD$ economy also pay the cost of migration, ${M_t}.$

Individuals are assumed to behave as atomistic agents, so that the migrants disregard the impact of their migration decision on the average levels of human capital, $\bar h_t^i$ , and consequently on the productivity of investment in children’s human capital in both their origin and destination economies. Similarly, all agents neglect the effect of their decision with respect to the number and quality of their offspring on the average level of human capital in the next generation.

To maximize utility, an adult of any type $j$ simultaneously chooses a current consumption, $C_t^j$ , the number of children, $N_t^j$ , and invests $e_t^j$ units of ${w^i}$ in each child’s education subject to the following budget constraint:

$$C_t^j + {\rm{ }}{w^i}(\delta {h_t} + e_t^j)N_t^j \le {w^i}{h_t},\quad\;\;{\rm{if}}\,j = i = MD,{\rm{ }}LD,$$

(3) $$C_t^j + {w^i}(\delta {h_t} + e_t^j)N_t^j \le {w^i}{h_t} - {M_t}, \;\;{\rm{if}}\,j = M{\rm{\; and \;}}i = MD.$$

The right-hand side of equation (3) represents an adult’s income (after subtracting migration costs for the $M$ –type agents), which is allocated between consumption and the total cost of rearing children.

The total potential future income of that individual’s offspring is represented by the expression:

(4) $$I_{t + 1}^{N,j} = N_t^j{h_{t + 1}}{w^i},\quad{\rm{if}}\,\left\{ \matrix{ j = MD,{\rm{ }}M,{\rm{ }}i = MD, \hfill \cr j = i = LD. \hfill \cr} \right.$$

The next subsection discusses the solution for the parents’ optimization problem for a particular form of the human capital production function.

2.4. Choice of fertility and investment in education

From optimization, an adult’s consumption is:

(5) $$C_t^j = (1 - \beta )\left\{ \matrix{ {w^i}{h_t},\quad\quad\quad\;{\rm{ if \;}}\left\{ \matrix{ j = i = MD, \hfill \cr j = i = LD, \hfill \cr} \right. \hfill \cr {w^i}{h_t} - {M_t},\quad{\rm{ if \;}}j = M{\rm{\; and \;}}i = MD. \hfill \cr} \right.$$

That is, a fraction $1 - \beta $ of an adult’s net full income is allocated to consumption and hence a fraction $\beta $ is dedicated to childrearing.

To characterize optimal choices concerning fertility and investment in education, we postulate the following technology of human capital production:Footnote ¹⁰

(6) $$h_{t + 1}^j = A\left\{ {\matrix{ {{{(\mu + e_t^j)}^\gamma }{{(\bar h_t^i)}^{1 - \gamma }}, \quad\quad{\rm{ if \;}}j = i = MD,{\rm{ }}LD{\rm{,}}} \cr \!\!\!{{{(\mu + \alpha e_t^j)}^\gamma }{{(\bar h_t^i)}^{1 - \gamma }}, \quad \;\;{\rm{ if \;}}j = M,{\rm{ }}i = MD{\rm{ }}} \cr } } \right.$$

where $A\gt1$ , $\mu \gt0$ , $0 \lt \alpha \le 1,$ and $0 \lt \gamma \lt1$ . (6)

This learning technology implies that, while the efficiency of the educational investment in the offspring of the locally-born parents in either economy equals to unity, the efficiency of the investment in the education of the offspring of the migrants from the $LD$ economy born in the $MD$ economy equals to $0 \lt \alpha \le 1.$ In all other respects, children of the migrants from the $LD$ economy born in the $MD$ economy become similar to the offspring of the indigenous population in the $MD$ economy.

Based on equation (6), the optimal choice of investment in children’s education for an individual of any type in either economy is determined by the following expression:

(7) $$e_t^j = \left\{ {\matrix{ {{{\gamma \delta {h_t} - \mu } \over {1 - \gamma }}, \quad \quad \;{\rm{ if \;}}j = MD,{\rm{ }}LD,} \cr \!\!\!\!\!\!\!\!\!\!\!\!\!{{{\gamma \delta {h_t} - ({\mu \mathord{\left/ {\vphantom {\mu \alpha }} \right.} \alpha })} \over {1 - \gamma }},\quad {\rm{ if \;}}j = M.{\rm{ }}} \cr } } \right.{\rm{ }}$$

so that, according to (6),

(8) $$h_{t + 1}^j = A{\left( {{\gamma \over {1 - \gamma }}} \right)^\gamma }\left\{ {\matrix{ {{{\left( {\delta {h_t} - \mu } \right)}^\gamma }{{(\bar h_t^i)}^{1 - \gamma }}, \quad \;\; {\rm{ if \;}}j = MD,{\rm{ }}LD,{\rm{ }}i = MD,} \cr \!\!\!\!\!\!\!\!\!\!\!\!\!{{{\left( {\alpha \delta {h_t} - \mu } \right)}^\gamma }{{(\bar h_t^i)}^{1 - \gamma }}, \quad {\rm{ if \;}}j = M,{\rm{ }}i = MD{\rm{.}}} \cr } } \right.$$

Given the amount of resources allocated to children’s education, the desired fertility for agent of any type is:

(9) $$N_t^j = \beta (1 - \gamma )\left\{ {\matrix{ \!{{{\left( {\delta - ({\mu \mathord{\left/ {\vphantom {\mu {{h_t}}}} \right.} {{h_t}}})} \right)}^{{\rm{ - 1}}},}\quad\quad\quad\quad\quad\quad\quad\quad{\rm{ if }} \; j = i = MD, LD,} \cr \!\!{{{\left( {\delta - ({\mu \mathord{\left/ {\vphantom {\mu {\alpha {h_t}}}} \right.} {\alpha {h_t}}})} \right)}^{{\rm{ - 1}}}}\left( {1 - ({{{M_t}} \mathord{\left/ {\vphantom {{{M_t}} {{w^i}{h_t}}}} \right.} {{w^i}{h_t}}})} \right),\quad\!\!{\rm{ if \;}}j = M,{\rm{ }}i = MD.} \cr } } \right.$$

Equation (7) shows that the optimal choice of investment in the offspring’s education and consequently their human capital levels (Eq. 8) is positively related to the parent’s human capital, although parental human capital does not enter the learning technology directly. Equation (9) demonstrates the conventional negative relationship between the parental level of human capital and the choice of fertility.

In the following section, I derive conditions that lead to the decision to migrate.

2.5. The decision to migrate

To characterize individual choice with respect to migration, recall that by definition in this model, in period t, the average level of human capital in the more developed economy exceeds that in the less developed economy, $\bar h_t^{MD} \gt \bar h_t^{LD}$ , which implies a higher return on investment in children’s education in the more developed economy. At the same time, the efficiency of educational investment in the offspring of immigrants is lower $(0 \lt \alpha \le 1).$ Also, in period t, the wage per unit of efficiency labor in the more developed economy is higher than in the less developed economy, ${w^{MD}} \gt {w^{LD}}$ , while migrants must cover the costs of migration, ${M_t}$ . Therefore, young adult individuals from the less developed country decide to migrate if their utility in the case of migration is higher than their utility in their country of birth; $U_t^M \gt U_t^{LD}$ .

Substituting an individual’s own consumption along with the income of that individual’s offspring in the case of migration, as shown in equations (5) and (4), respectively ( $j = M$ ), into the utility function (2) and comparing the result to the level of utility in the absence of migration ( $j = LD$ ) allows us to calculate the following threshold level of human capital:Footnote ¹¹

(10) $${\hat h_t} = {{{M_t}} \over {{w{\hskip0.5pt}^{MD}} - {w^{{\hskip0.5pt}LD}}{{\left( {\left( {{{{{\left( {{{\alpha \delta {h_t} \,- \,\mu } \over {\delta {h_t} \,- \,\mu }}} \right)}^{1 - \gamma }}} \mathord{\left/ {\vphantom {{{{\left( {{{\alpha \delta {h_t} - \mu } \over {\delta {h_t} - \mu }}} \right)}^{1 - \gamma}}} \alpha }} \right.} \alpha }} \right){{\left( {{{\bar h_t^{LD}} \over {\bar h_t^{MD}}}} \right)}^{1 - \gamma }}} \right)}^\beta }}}.$$

The threshold level defined in equation (10) divides the population in the less developed economy into two groups:

1. 1. Individuals with human capital levels lower than the threshold ${\hat h_t}$ who choose to remain in the $LD$ economy.

2. 2. Individuals with human capital levels greater than the threshold ${\hat h_t}$ who choose to migrate to the $MD$ economy.

The following proposition summarizes the main result of this section concerning the effect of the efficiency of the investment in immigrant children’s education on migration pattern.

Proof. Eq. (10).

3.1. The average level of human capital

In economy $i$ ( $i = MD,{\rm{ }}LD$ ), the average human capital level in period $t+1$ is defined as shown in equation (11):

(11) $$\bar h_{t + 1}^i \equiv \int {{h_{t + 1}}dF_{t + 1}^i(h)} = {{\int {N_t^j{h_{t + 1}}dF_t^i(h)} } \mathord{\left/ {\vphantom {{\int {N_t^j{h_{t + 1}}dF_t^i(h)} } {\int {N_t^jdF_t^i(h)} }}} \right.} {\int {N_t^jdF_t^i(h)} }},\,\,\,i = j = MD,{\rm{ }}LD.$$

Distinguishing parents of each type and given the number of children and the levels of human capital investment among all types of agents as determined in Section 2.4 and with human capital production function as given in equation (6), in the absence of migration (denoted by superscript $NM$ ), the average level of human capital in economy $i$ is as shown in equation (12):

(12) $$\bar h_{t + 1}^{i,NM} = A{\left( {{\gamma \over {1 - \gamma }}} \right)^\gamma }{{\int {{{(\delta {h_t} - \mu )}^{\gamma - 1}}{h_t}dF_t^i} } \over {\int {{{(\delta - {\mu \over {{h_t}}})}^{ - 1}}dF_t^i} }}{(\bar h_t^{i,NM})^{1 - \gamma }}{\rm{, \,\,\,\,if }}\,\,i = MD,{\rm{ }}LD.$$

Correspondingly, with free migration (denoted by superscript $FM$ ), the average levels of human capital in each of the economies are as shown in equations (13) and (14) for the $MD$ and the $LD$ economies, respectively:

$$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\rm{ }}\bar h_{t + 1}^{MD,FM} = A{\left( {{\gamma \over {1 - \gamma }}} \right)^\gamma }{(\bar h_t^{MD,FM})^{1 - \gamma }}\left[ {\int\limits_{} {{{(\delta {h_t} - \mu )}^{\gamma - 1}}{h_t}dF_t^{MD}} } \right.{\rm{ }}$$

(13) $$ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+ {{\left. {\int\limits_{{h_t} \gt {{\hat h}_t}} {(1 - {{{M_t}} \over {{w^{MD}}{h_t}}}){{(\alpha \delta {h_t} - \mu )}^{\gamma - 1}}{h_t}dF_t^{MD}} } \right]} \mathord{\left/ {\vphantom {{\left. {\int\limits_{{h_t} \gt {{\hat h}_t}} {(1 - {{{M_t}} \over {{w^{MD}}{h_t}}}){{(\alpha \delta {h_t} - \mu )}^{\gamma - 1}}{h_t}dF_t^{MD}} } \right]} {}}} \right.} {}}$$

$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{ }}\left[ {\int\limits_{} {{{(\delta - ({\mu \mathord{\left/ {\vphantom {\mu {{h_t}}}} \right.} {{h_t}}}))}^{ - 1}}dF_t^{MD}} } \right.{\rm{ + }}\,\,\left. {\int\limits_{{h_t} \gt {{\hat h}_t}} {(1 - {{{M_t}} \over {{w^{MD}}{h_t}}}){{(\delta - ({\mu \mathord{\left/ {\vphantom {\mu {\alpha {h_t}}}} \right.} {\alpha {h_t}}}))}^{ - 1}}dF_t^{LD}} } \right]$$

and

(14) $${\rm{ }}\bar h_{t + 1}^{LD,FM} = A{\left( {{\gamma \over {1 - \gamma }}} \right)^\gamma }{(\bar h_t^{LD,FM})^{1 - \gamma }}\left[ {{{\int\limits_{{h_t} \lt {{\hat h}_t}} \!\!\!\!{{{(\delta {h_t} - \mu )}^{\gamma - 1}}{h_t}dF_t^{MD}} } \mathord{\left/ {\vphantom {{\int\limits_{{h_t} \lt {{\hat h}_t}} {{{(\delta {h_t} - \mu )}^{\gamma - 1}}{h_t}dF_t^{MD}} } {\int\limits_{{h_t} \lt {{\hat h}_t}} {{{(\delta - ({\mu \mathord{\left/ {\vphantom {\mu {{h_t}}}} \right.} {{h_t}}}))}^{ - 1}}dF_t^{LD}} }}} \right.} \!\!\!\!{\int\limits_{{h_t} \lt {{\hat h}_t}} \!\!\!\!{{{(\delta - ({\mu \mathord{\left/ {\vphantom {\mu {{h_t}}}} \right.} {{h_t}}}))}^{ - 1}}dF_t^{LD}} }}} \right]{\rm{.}}$$

where the threshold ${\hat h_t}$ is as determined in equation (10).

Comparing the levels of human capital under free migration ( $\bar h_{t+1}^{i,FM}$ ) to those in the absence of migration ( $\bar h_{t+1}^{i,NM}$ ), as shown in equations (13), (14) and (12), allows us to determine precisely whether migration in period $t$ increases or decreases the average level of human capital in period $t+1$ in either economy. Thus, if $\bar h_{t+1}^{i,FM} \gt \bar h_{t+1}^{i,NM}$ , free migration increases human capital in economy $i$ . In contrast, if $\bar h_{t+1}^{i,NM} \gt \bar h_{t+1}^{i,FM}$ , the average level of human capital in economy $i$ is higher in the absence of migration. An illustrative numerical example for a brain-diluting migration is provided below in Section 3.3.

3.2. Public policy interventions

If free uncontrolled migration leads to a reduction in human capital accumulation ( $\bar h_{t+1}^{i,NM} \gt \bar h_{t+1}^{i,FM}$ ), an appropriate public policy intervention in the $MD$ economy can help increase per-capita human capital. Thus, for example, we can consider the following three possibilities:

1. 1. Providing additional free educational services, financed by taxes levied “outside the model,” for immigrant children to increase the efficiency of the investment in their education.

2. 2. A lump-sum tax imposed on adult immigrants by the absentee government to finance a subsidy for the acquisition of human capital for immigrant children.

3. 3. A lump-sum tax imposed on adult immigrants by the absentee government to finance a subsidy for the acquisition of human capital for all children in the society.

Proceeding now to the public policy interventions, the average levels of human capital in each of the economies can be easily computed following the same steps as in equations (13) and (14). Thus, improving the efficiency of the investment in the education of immigrant children, captured here by an increase in the parameter $\alpha $ , as under Scenario 1, not only directly increases the human capital of immigrant children (Eq. 8, $j = M$ ), but also alters the parental quantity-quality trade-off in favor of quality enticing immigrant parents to increase their investment in the education of their offspring (Eq. 7, $j = M$ ) and reduce the investment in the number of children (Eq. 9, $j = M$ ), thereby generating an unambiguous positive effect on human capital accumulation and growth. On the other hand, as summarized in Proposition 1, a higher efficiency of the investment in immigrant children’s education reduces the threshold level of human capital ${\hat h_t}$ (Eq. 10), making migration more appealing to lower-skilled individuals from the $LD$ economy. As a decline in the threshold level ${\hat h_t}$ is associated with a reduction in the lower bound of human capital among migrants, an increase in $\alpha $ reduces the average level of human capital among immigrants, generating an equally unambiguous negative effect on human capital accumulation and growth the $MD$ economy. The net effect is therefore uncertain and depends on the relative strength of the two opposing effects. At the same time, for the $LD$ economy, it augments the magnitude of the outflow of relatively higher-skilled individuals, thereby reducing the average level of human capital in the parental generation and thus slowing down the process of human capital accumulation in that economy. An illustrative example highlighting the growth-depleting effect of an increase in the efficiency of investment in immigrant children’s education is provided in Table 1 in Section 3.3.

Table 1.

Numerical example: Five cases under discussion

Source: Author’s calculations. In the computations of the average levels of human capital it has been assumed that from period $+2$ on, the efficiency of the educational investment in the immigrants’ children is equal to that of the offspring of the local indigenous population $(\alpha =1).$

With the lump-sum tax, $T_t^{S2}$ , levied in the $MD$ economy on immigrants from the $LD$ economy with a redistribution of the proceeds as a subsidy at rate $s_t^{S2}$ for the education of immigrant children, as under Scenario 2 (denoted by superscript $S2$ ), the required budget constraint for the redistribution is as follows:

(15) $$T_t^{S2}\int\limits_{{h_t} \gt \hat h_t^{S2}} {N_{t - 1}^{LD}dF_{t - 1}^{LD}({h^{LD}})} = {w^{MD}}s_t^{S2}\int\limits_{} {N_t^{M,S2}e_t^{M,S2}dF_t^{M,S2}({h^M}).} {\rm{ }}$$

With the adjusted $\bar h_t^{i,S2},$ the difference in this case is that the threshold becomes:

(16) $$\hat h_t^{S2} = {{{M_t} + T_t^{S2}} \over {{w^{MD}} - {w^{LD}}{{\left( {\left( {{{{{\left( {{{\alpha (1 + S_t^{S2})\delta {h_t} - \mu } \over {\delta {h_t} - \mu }}} \right)}^{1 - \gamma}}} \mathord{\left/ {\vphantom {{{{\left( {{{\alpha (1 + s_t^{S2})\delta {h_t} - \mu } \over {\delta {h_t} - \mu }}} \right)}^{1 - \gamma}}} {\alpha (1 + s_t^{S2})}}} \right.} {\alpha (1 + S_t^{S2})}}} \right){{\left( {{{\bar h_t^{LD,S2}} \over {\bar h_t^{MD,S2}}}} \right)}^{1 - \gamma }}} \right)}^\beta }}},$$

while the optimal fertility choice, the investment in the offspring’s education and resulting children’s levels of human capital among immigrants are, respectively:

(17) $$N_t^{M,S2} = {{\beta (1 - \gamma )} \over {\delta - ({\mu \mathord{\left/ {\vphantom {\mu {\alpha (1 + s_t^{S2}){h_t})}}} \right.} {\alpha (1 + s_t^{S2}){h_t})}}}}\left( {1 - {{{M_t} + T_t^{S2}} \over {{w^{MD}}{h_t}}}} \right){\rm{, }}$$

(18) $$e_t^{M,S2} = {{\gamma \delta {h_t} - ({\mu \mathord{\left/ {\vphantom {\mu {\alpha (1 + s_t^{S2})}}} \right.} {\alpha (1 + s_t^{S2})}})} \over {1 - \gamma }},{\rm{ }}$$

and

(19) $$h_{t + 1}^{M,S2} = A{\left( {{\gamma \over {1 - \gamma }}(\alpha (1 + s_t^{S2})\delta {h_t} - \mu )} \right)^\gamma }{(\bar h_t^{MD,S2})^{1 - \gamma }}.$$

With the lump-sum tax, $T_t^{S3}$ , imposed on immigrants to finance a subsidy for the acquisition of human capital for all children in the more developed society at rate $s_t^{S3}$ as under Scenario 3 (denoted by superscript $S3$ ), the required budget constraint for the redistribution is:

(20) $$T_t^{S3}\!\!\int\limits_{{h_t} \gt \hat h_t^{S3}} \!\!{N_{t - 1}^{LD}dF_{t - 1}^{LD}({h^{LD}})}\! = \!{w^{MD}}s_t^{S3}\left( \!{\int\limits_{} {N_t^{MD,S3}e_t^{MD,S3}dF_t^{MD}({h^{MD}})}\! + \!\!\int\limits_{} \!{N_t^{M,S3}e_t^{M,S3}\!dF_t^M({h^M})} \!} \right){\rm{. }}$$

In this case, with the adjusted $\bar h_t^{i,S3}$ , the threshold is:

(21) $$\hat h_t^{S3} = {{{M_t} + T_t^{S3}} \over {{w^{MD}} - {w^{LD}}{{\left( {\left( {{{{{\left( {{{\alpha (1 + s_t^{S3})\delta {h_t} - \mu } \over {\delta {h_t} - \mu }}} \right)}^{1 - \gamma}}} \mathord{\left/ {\vphantom {{{{\left( {{{\alpha (1 + s_t^{S3})\delta {h_t} - \mu } \over {\delta {h_t} - \mu }}} \right)}^{1 - \gamma}}} {\alpha (1 + s_t^{S3})}}} \right.} {\alpha (1 + s_t^{S3})}}} \right){{\left( {{{\bar h_t^{LD,S3}} \over {\bar h_t^{MD,S3}}}} \right)}^{1 - \gamma }}} \right)}^\beta }}}$$

and the results of the optimization for all individuals in the $MD$ economy are affected, as shown in equations (22) – (24):

(22) $$N_t^j = \beta (1 - \gamma )\left\{ {\matrix{ {{{\left( {\delta - ({\mu \mathord{\left/ {\vphantom {\mu {(1 + s_t^{S3}){h_t}}}} \right.} {(1 + s_t^{S3}){h_t}}})} \right)}^{{\rm{ - 1}}}}{\rm{, \quad \quad \quad \quad \quad \quad if }}j = MD,} \cr \!\!\!{{{\left( {\delta - ({\mu \mathord{\left/ {\vphantom {\mu {\alpha (1 + s_t^{S3}){h_t}}}} \right.} {\alpha (1 + s_t^{S3}){h_t}}})} \right)}^{{\rm{ - 1}}}}\left( {1 - {{{M_t} + T_t^{S3}} \over {{w^{MD}}{h_t}}}} \right){\rm{, \quad \!\!\!\! if }}j = M.{\rm{ }}} \cr } } \right.$$

(23) $$e_t^j = \left\{ {\matrix{ {{{\gamma \delta {h_t} - ({\mu \mathord{\left/ {\vphantom {\mu {(1 + s_t^{S3})}}} \right.} {(1 + s_t^{S3})}})} \over {1 - \gamma }},{\rm{\quad\;\; if \;}}j = MD,{\rm{ }}} \cr \!\!\! {{{\gamma \delta {h_t} - ({\mu \mathord{\left/ {\vphantom {\mu {\alpha (1 + s_t^{S3})}}} \right.} {\alpha (1 + s_t^{S3})}})} \over {1 - \gamma }},\quad{\rm{ if \;}}j = M.{\rm{ }}} \cr } } \right.$$

and

(24) $$h_{t + 1}^j = A{\left( {{\gamma \over {1 - \gamma }}} \right)^\gamma }\left\{ {\matrix{ {{{\left( {(1 + s_t^{S3})\delta {h_t} - \mu } \right)}^\gamma }{{(\bar h_t^{MD,S3})}^{1 - \gamma }},\quad\; \; {\rm{ if \;}}j = MD,} \cr \!\!\!{{{\left( {\alpha (1 + s_t^{S3})\delta {h_t} - \mu } \right)}^\gamma }{{(\bar h_t^{MD,S3})}^{1 - \gamma }},\quad{\rm{ if \;}}j = M{\rm{. }}} \cr } } \right.$$

Therefore, taxing migrants, as under Scenarios 2 and 3, increases the threshold levels, $\hat h_t^{S2}$ and $\hat h_t^{S3}$ , as shown in equations (16) and (21). As a consequence, as opposite of the effect of the increase in the efficiency of the investment in immigrant children’s schooling $(\alpha ),$ it reduces migration yielding a positive effect on the average skill levels in either economy.

As for the subsidy for the acquisition of human capital among immigrant children, $s_t^{S2},$ as under Scenario 2, it reduces the threshold, $\hat h_t^{S2},$ and increases the level of human capital among immigrant children (Eq. 19) with a positive overall effect on human capital accumulation in the $MD$ economy if the later effect dominates. Following the same intuition as under Scenario 1, educational subsidy, $s_t^{S2},$ also generates an unambiguously negative effect on human capital accumulation in the $LD$ economy. Providing a subsidy for acquiring education for all children in the $MD$ economy, whether born to immigrant and or local parents, as outlined in Scenario 3, limits the effect of the subsidy, $s_t^{S3},$ on the threshold level, $\hat h_t^{S3},$ thereby reducing the negative effect on human capital accumulation in either economy.

Furthermore, as a subsidy is provided not only to the offspring of immigrants but also to children of the local parents, the overall positive effect on human capital accumulation in the $MD$ economy can be strong enough to outweigh the negative effect of the lower-skilled immigration. An illustrative example for a brain dilution taxation that reduces the negative effect of a lower-skilled immigration on human capital accumulation in the $MD$ economy if the revenue is redistributed to support human capital investment in immigrant children, and improves human capital accumulation if the proceeds are used as educational subsidies for all children in the society follows in Table 1 in Section 3.3.

3.3. Numerical illustration: Migration, human capital, population growth, and public policy interventions

To characterize the effect of migration on the evolution of human capital in either economy, suppose that, in period $t$ , in the presence of migration, the adult population in the $MD$ economy includes both individuals born in this economy and migrants, while in the $LD$ economy the adult population consists solely of individuals who remain in their country of origin:

(25) $$L_t^i \equiv \left\{ {\matrix{ {\int {N_{t - 1}^idF_{t - 1}^i(h)} + \int\limits_{{h_t} \gt {{\hat h}_t}} {N_{t - 1}^{LD}dF_{t - 1}^{LD}({h^{LD}}),} {\rm{\;\;\;\; if \;}}i = MD,} \cr {\int\limits_{{h_t} \lt {{\hat h}_t}} {N_{t - 1}^idF_{t - 1}^i(h),} {\rm{ \quad \quad \quad \quad \quad \quad \quad \quad \quad if \;}}i = LD.{\rm{ }}} \cr } } \right.$$

Also assume for simplicity that, in period $t$ , in the absence of migration, the size of the adult population in either economy is normalized to unity; $L_t^{i,NM} \equiv \int {N_{t - 1}^i dF_{t - 1}^i(h)} =1$ , $i = LD,{\rm{ }}MD$ .

Consider the following five cases: (1) the absence of migration, (2) free migration, (3) public policy intervention: Scenario 1, (4) public policy intervention: Scenario 2, and (5) public policy intervention: Scenario 3, (denoted by $NM,{\rm{ }}FM, \,$ $S1,{\rm{ }}S2,{\rm{\; and \;}}S2,$ respectively).

To provide a numerical example, suppose that the parameters in the model are as follows: $\delta =0.15$ , $\beta =0.4$ , $\mu =1$ , $\gamma =0.5$ , $\alpha =0.85$ , ${w^{MD}}=1.4$ , ${w^{LD}}=0.7$ , and $A=3.$ Also suppose that in period $t,$ the cost of migration from the $LD$ economy to the $MD$ economy equals to 38.26 $({M_t}=38.26).$ Further, for comparability, suppose that, under Scenario 1, the absentee government provides additional free educational services for immigrant children, financed by the taxes levied “outside the model,” to equalize the efficiency of the investment in their education to that of the offspring of the locally born parents $({\alpha ^{S1}}{\rm{ }}=1),$ whereas, under Scenario 2, the government provides a subsidy for immigrant children, financed by the taxes imposed on adult immigrants, $T_t^{S2},$ to similarly enhance the efficiency of the investment in their schooling to the same level as that of the local children; $\alpha (1 + s_t^{S2})=1.$ Finally, suppose that, under Scenario 3, the subsidy, financed by a lump-sum tax levied on adult immigrants, is provided for all children born in the economy at rate 0.004; $s_t^{S3}=0.004.$

To complete the description of the world, suppose that the distribution of human capital is such that in period $t$ , in the more developed economy, human capital is distributed equally around $\bar h_t^{MD}=60$ with the lower bound $\bar h_t^{MD,\min }=40$ and the upper bound $\bar h_t^{MD,\max }=80$ . In a similar manner, in the less advanced economy, the distribution of human capital is such that human capital is distributed equally around $\bar h_t^{LD}=40$ with the lower bound $\bar h_t^{LD,\min }=20$ and the upper bound $\bar h_t^{LD,\max }=60$ .

Table 1 presents the results for the five cases under discussion. As can be seen, free migration ( $FM$ ) in period $t$ decreases the average level of human capital in either economy in period $t+1$ , relative to the case, when migration is absent ( $NM$ ) ( $\bar h_{t+1}^{MD,NM} \gt \bar h_{t+1}^{MD,FM}$ and $\bar h_{t+1}^{LD,NM} \gt \bar h_{t+1}^{LD,FM}$ ). If free migration is human capital diluting as in Case 2 ( $FM$ ), we can consider the public policy interventions to help improve human capital accumulation, as specified above under Scenarios (1) to (3) in Section 3.2. However, following the intuition presented in Section 3.2, providing free educational services for immigrant children, as under Scenario 1 ( $S1$ ), can work against its stated goal, further reducing human capital accumulation in the host economy $(\bar h_{t+1}^{MD,FM} \gt \bar h_{t+1}^{MD,S1}$ ). In contrast, a lump-sum tax imposed on adult immigrants to finance a subsidy for the acquisition of human capital for immigrant children, as under Scenario 2 ( $S2$ ), can help mitigate a negative effect of human capital depleting migration $(\bar h_{t+1}^{MD,S2} \gt \bar h_{t+1}^{MD,FM}$ ). Furthermore, if the proceeds are redistributed to finance an educational subsidy for all children, as under Scenario 3 ( $S3$ ), public policy intervention can help improve human capital accumulation in the $MD$ economy not only relative to the case of free migration ( $FM$ ), but also relative to the situation when migration is absent ( $NM$ ) $(\bar h_{t+1}^{MD,S3} \gt \bar h_{t+1}^{MD,NM} \gt \bar h_{t+1}^{MD,S2} \gt \bar h_{t+1}^{MD,FM} \gt \bar h_{t+1}^{MD,S1}$ ). With regard to the sending economy, the effect of migration and public policy interventions in the $MD$ economy on human capital accumulation in this country follows the pattern observed in the $MD$ economy except for Case 5, when the inter-temporal increase in the average level of human capital in the $LD$ economy from period $t$ to period $t+1$ is positive, but smaller than in the absence of migration $(\bar h_{t+1}^{LD,NM} \gt \bar h_{t+1}^{LD,S3} \gt \bar h_t^{LD} \gt \bar h_{t+1}^{MD,FM} \gt \bar h_{t+1}^{LD,S2} \gt \bar h_{t+1}^{LD,S1}$ ).

As follows from the property of the learning technology (6) with respect to the average level of human capital in the society, the effect of migration and public policy interventions on human capital levels in each of the economies evolves further from one generation to the next. With time, these effects tend to intensify, leading to greater disparities across the countries regarding their average levels of human capital. Notably, under Scenario 3, the outcome closely resembles that which would occur in the absence of migration. A comparison of period $t+5$ to period $t$ is presented in Table 1.

As can be seen,

$${{{{\bar h_{t + 5}^{MD,S1}} \mathord{\left/ {\vphantom {{\bar h_{t + 5}^{MD,S1}} {\bar h_{t + 5}^{LD,S1}}}} \right.} {\bar h_{t + 5}^{LD,S1}}}} \over {{{\bar h_t^{MD,S1}} \mathord{\left/ {\vphantom {{\bar h_t^{MD,S1}} {\bar h_t^{LD,S1}}}} \right.} {\bar h_t^{LD,S1}}}}} \gt {{{{\bar h_{t + 5}^{MD,S2}} \mathord{\left/ {\vphantom {{\bar h_{t + 5}^{MD,S2}} {\bar h_{t + 5}^{LD,S2}}}} \right.} {\bar h_{t + 5}^{LD,S2}}}} \over {{{\bar h_t^{MD,S2}} \mathord{\left/ {\vphantom {{\bar h_t^{MD,S2}} {\bar h_t^{LD,S2}}}} \right.} {\bar h_t^{LD,S2}}}}} \gt {{{{\bar h_{t + 5}^{MD,FM}} \mathord{\left/ {\vphantom {{\bar h_{t + 5}^{MD,FM}} {\bar h_{t + 5}^{LD,FM}}}} \right.} {\bar h_{t + 5}^{LD,FM}}}} \over {{{\bar h_t^{MD,FM}} \mathord{\left/ {\vphantom {{\bar h_t^{MD,FM}} {\bar h_t^{LD,FM}}}} \right.} {\bar h_t^{LD,FM}}}}} \gt {{{{\bar h_{t + 5}^{MD,S3}} \mathord{\left/ {\vphantom {{\bar h_{t + 5}^{MD,S3}} {\bar h_{t + 5}^{LD,S3}}}} \right.} {\bar h_{t + 5}^{LD,S3}}}} \over {{{\bar h_t^{MD,S3}} \mathord{\left/ {\vphantom {{\bar h_t^{MD,S3}} {\bar h_t^{LD,S3}}}} \right.} {\bar h_t^{LD,S3}}}}} \gt {{{{\bar h_{t + 5}^{MD,NM}} \mathord{\left/ {\vphantom {{\bar h_{t + 5}^{MD,NM}} {\bar h_{t + 5}^{LD,NM}}}} \right.} {\bar h_{t + 5}^{LD,NM}}}} \over {{{\bar h_t^{MD,NM}} \mathord{\left/ {\vphantom {{\bar h_t^{MD,NM}} {\bar h_t^{LD,NM}}}} \right.} {\bar h_t^{LD,NM}}}}}$$

As for the relative size of the countries in terms of population, as follows from the normalization of the size of adult population in either economy to unity, in period $t$ , migration mechanically reduces the size of the source country relative to the destination country. However, with time, population size of the less developed source economy will increase relative to that of the more developed host economy due to a higher fertility in the former. As a consequence, at some point, the total population in the $LD$ economy will overtake that in the $MD$ economy.Footnote ¹² From this period on, the economies will diverge in terms of both average human capital and the size of population. A numerical example is provided in Table (A1) in Appendix.

By mitigating the loss of human capital caused by brain drain in the source country, the policy intervention discussed under Scenario 3 constrains the relative growth of its population vis-à-vis the host country, slowing down the divergence in population sizes between the two countries compared to what would occur under unrestricted migration and the other two scenarios:

$${{{{L_{t + 5}^{MD,NM}} \mathord{\left/ {\vphantom {{L_{t + 5}^{MD,NM}} {L_{t + 5}^{LD,NM}}}} \right.} {L_{t + 5}^{LD,NM}}}} \over {{{L_t^{MD,NM}} \mathord{\left/ {\vphantom {{L_t^{MD,NM}} {L_t^{LD,NM}}}} \right.} {L_t^{LD,NM}}}}} \lt {{{{L_{t + 5}^{MD,S3}} \mathord{\left/ {\vphantom {{L_{t + 5}^{MD,S3}} {L_{t + 5}^{LD,S3}}}} \right.} {L_{t + 5}^{LD,S3}}}} \over {{{L_t^{MD,S3}} \mathord{\left/ {\vphantom {{L_t^{MD,S3}} {L_t^{LD,S3}}}} \right.} {L_t^{LD,S3}}}}} \lt {{{{L_{t + 5}^{MD,FM}} \mathord{\left/ {\vphantom {{L_{t + 5}^{MD,FM}} {L_{t + 5}^{LD,FM}}}} \right.} {L_{t + 5}^{LD,FM}}}} \over {{{L_t^{MD,FM}} \mathord{\left/ {\vphantom {{L_t^{MD,FM}} {L_t^{LD,FM}}}} \right.} {L_t^{LD,FM}}}}} \lt {{{{L_{t + 5}^{MD,S2}} \mathord{\left/ {\vphantom {{L_{t + 5}^{MD,S2}} {L_{t + 5}^{LD,S2}}}} \right.} {L_{t + 5}^{LD,S2}}}} \over {{{L_t^{MD,S2}} \mathord{\left/ {\vphantom {{L_t^{MD,S2}} {L_t^{LD,S2}}}} \right.} {L_t^{LD,S2}}}}} \lt {{{{L_{t + 5}^{MD,S1}} \mathord{\left/ {\vphantom {{L_{t + 5}^{MD,S1}} {L_{t + 5}^{LD,S1}}}} \right.} {L_{t + 5}^{LD,S1}}}} \over {{{L_t^{MD,S1}} \mathord{\left/ {\vphantom {{L_t^{MD,S1}} {L_t^{LD,S1}}}} \right.} {L_t^{LD,S1}}}}}$$

Thus, the brain dilution tax proposed in this study may also help counteract a potential increase in global inequality that can be exacerbated by brain drain migration, as poorer economies grow larger in terms of population (Mountford and Rapoport, Reference Mountford and Rapoport2011).