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ELSEVIER Journal of DevelopmentEconomics

Vol. 53 (1997) 287-303

JOURNAL OF D e v e l o p m e n t ECONOMICS

Can a brain drain be good for growth in the source economy?

Andrew Mountford *

Department of Economics, Southampton University Southampton S017 1B J, UK

Abstract

This paper analyzes the interaction between income distribution, human capital accumulation and migration. It shows that when migration is not a certainty, a brain drain may increase average productivity and equality in the source economy even though average productivity is a positive function of past average levels of human capital in an economy. It is also shown how the temporary possibility of emigration may permanently increase the average level of productivity of an economy. © 1997 Elsevier Science B.V.

JEL classification." O15; 040

Keywords: Brain drain; Growth

I. Introduct ion

What is the economic rationale for allowing a limited amount of emigration through an exit visa pol icy? Wil l permitting some of an economy's most educated inhabitants to emigrate, always be bad for an economy 's level of per capita output? The recent emphasis on the importance for economic growth of the average level of human capital in an economy has led many to presume that a 'brain drain' may leave a developing country in a poverty trap. The intuition being that the average level of human capital in a developing country will not grow because the developed world will ' s iphon off ' its highly educated workers, thus increasing the productivity of the developed world at the expense of the develop-

* Corresponding author. E-mail: [email protected].

0304-3878/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0304-3878(97)00021-7

288 A. Mountford / Journal of Development Economics 53 (1997) 287-303

ing economy. This paper shows that when educational decisions are endogenous and if successful emigration is not a certainty, a brain drain may increase the productivity of a developing country.

This paper discusses two ways that a brain drain can beneficially effect growth. In Section 2 it is shown how the possibility of migration to a higher wage country raises the return to education. This leads to an increase in human capital formation which can outweigh the negative effect of the brain drain itself. Another avenue for the possible beneficial effects of a brain drain is its effect on the endogenous formation of 'educational classes' in an economy. Section 3 shows how a brain drain can change the dynamics of 'class' formation so that an 'under-educated' class fails to develop. It is shown that this can be a very large effect indeed.

An implication of this model is that policy makers in an economy where skilled workers want to emigrate, will not want to completely prohibit the emigration of its skilled workers. There will be an optimal level of emigration which weighs the benefits of emigration, due to increased incentives for skill accumulation, with the costs of the skill depletion of a 'brain drain'. This model thus provides a theoretical reason for the use of emigration quota policies, for example in the old Soviet Union and in present day China.

This paper is chiefly related to two literatures, the growth and human capital accumulation literature and the migration and brain drain literatures. Since the seminal article of Lucas (1988), the endogenous growth literature has investigated, both empirically and theoretically, the importance for growth of human capital accumulation. Empirically, Barro (1991) and Mankiw et al. (1992) have shown that the level of schooling across countries is a significant variable for explaining differences in growth rates across countries. Theoretically, in addition to the large number of representative agent models, Galor and Zeira (1993) and Galor and Tsiddon (1994) have recently described economies where the level of education differs endogenously across agents and have investigated the growth implications of the distribution of education levels in an economy. This paper continues this research direction by investigating the implications of migration in a model where education levels differ across agents.

The brain drain literature is a branch of the migration literature which has investigated the impact of migration when agents are not homogeneous, see for example Bhagwati and Hamada (1974). The three most related articles to this paper are the article of Miyagiwa (1991) on the brain drain and the papers of Galor and Stark (1990, 1991) on capital accumulation and return migration. Miyagiwa's paper concentrates on the effect of scale economies on migration. It shows how a large economy will attract a small economy's most able workers, when the productivity externality is dependent upon the number of educated people in an economy, and thus that a brain drain will necessarily reduce the productivity in the original country of the emigrants. Galor and Stark show how the possibility of enforced return migration back to a lower wage economy will cause the saying's rate of a migrant to be higher than that of an indigenous worker.

A. Moun~rord / Journal of Development Economics 53 (1997) 287-303 289

This paper applies this idea of probabilistic migration to the obtaining of emigration or immigration visas and analyzes its impact on educational choices.

This paper is organized as follows. In Section 2 we abstract from income distribution dynamics by exogenously fixing the number of 'educational classes' to two. It is thus shown, even without non-marginal changes in the income distribution, that a brain drain can increase per capita income in the source economy. In Section 3 the number of 'educational classes' in an economy is endogenized and it is shown how migration can effect the income distribution dynamics of a source economy. It is thus shown how a brain drain can have large (non-marginal) and positive effects on the level of per capita income in the source economy.

2. The effect of a brain drain on educat ional incentives

The model in this section is a simple version of the model of the brain drain and human capital formation by Miyagiwa (1991). The model analyzes a small open overlapping generations economy existing in a world where there is one good and perfect capital mobility. The world's one good is produced under constant returns to scale by two factors, capital and efficiency units of labor. There are a continuum of agents in each generation. For simplicity we normalize the measure of agents in each generation to unity. Population growth is excluded but could be trivially added. In this model the education decision is a discrete ei ther/or choice and by construction there will always exist two 'classes' of agents, the educated and the uneducated. This model shows how a brain drain can be beneficial to the productivity level of the source economy when there is a 'growth externality' associated with the proportion of educated workers in the economy in the previous period. A sufficient condition for the existence of an optimal level of emigration or exit visas is also derived.

2.1. The economy without migration

We will first describe this economy without migration before going on to analyze the model under various assumptions about the possibility to migrate.

2.1.1. Production of goods and factor prices The total amount of capital and efficiency units of labor in time period t, are

denoted by K t and Lt, respectively. The productivity of labor, or the state of the technology, in period t is given by A t.

Production is generated by a constant returns to scale production function. The output produced at time t, Y~, is

Kt Y, = F(Kt ,ArLt) = f ( k t ) AtL t where kt = ( l )

A t Lt

2 9 0 A. Mountford / Journal of Development Economics 53 (1997) 287-303

We make the standard assumptions about this function, namely

f ( k ) > O f ' ( k ) > O f " ( k ) < 0 V k > 0 (2)

and the ' Inada condit ions '

l i m f ( k ) = 0 l i m f ' ( k ) = oo l i m f ' ( k ) -- 0 (3) k - " , 0 k---, 0 k---~ ~

Factor prices are determined in the standard way by the factor 's marginal

product, thus the net return to capital r t is equal t o f ' ( k t) -+- (1 - t~) where 6 is the rate of depreciation of the capital stock. It is assumed for simplicity that the world is in a steady state equil ibrium and thus that the world net rate of retum, r *, is constant. Due to perfect capital mobil i ty and the smallness of the economy, this fixes the domestic net rate of return to capital, r t, equal to r * and thus fixes the domestic capital to efficiency labor ratio, kt , a s well.

Thus, we will write k t = k Vt where k is a constant. Given the level of technology At, k determines the wage rate per efficiency unit of labor, w(k)

w t = h t [ f ( k ) - k f ' (k ) ] = h tw(k ) (4)

where w(k) - [ f ( k ) - kf'(k)].

2.1.2. The distribution of ability Individuals possess different levels of latent ability, where e i denotes the latent

ability level of individual i. These latent abilities are assumed to be distributed over the closed interval [0,E] according to the density function g(ei), where by definition

foE g( e i)de i= 1; g( e i) :> OV ei r[O,E] (5 )

We assume that all generations have latent abilities which are picked from the same distribution and that the abilities of children are independent from the

abilities of their parents.

2.1.3. Individuals' education decision Agents exist in an overlapping generations world and live for three periods,

deriving utility only from third period consumption. 1 In their first period of life agents can invest resources in education. They have no resources of their own, so they must borrow from the capital market at the wor ld ' s rate of interest, r *. The cost of an education is assumed fixed at c units of output. Agents that invest in education obtain e i efficiency units of labor in their second period of life, where

1The use of this utility function is purely for simplicity. Utility can be made to depend on consumption in all three periods without changing the results (see Mountford, 1995). The three period nature of the model is necessary because agents borrow to finance their education in their first period of existence. They can clearly not borrow from agents who will not be alive to be repaid in the next period, thus the need for a three period structure.

A. Mount:ford~Journal of Deoelopment Economics 53 (1997) 287-303 291

e i is the level of the latent ability of agent i. Agents who don't invest in education, on the other hand, are assumed to have only one efficiency unit of labor in their second period of life. Agents can only work in their second period of life and in this period the agent must repay the debt of the first period and save in order to consume in the last period of life. In the third period agents are retired and use their savings to consume. All agents have the same preferences and access to the same technology, although of course they do not have the same levels of latent ability.

The optimal decision for agent i will be to invest in education if

Atw(k)ei> Atw(k ) + c(1 + r * ) (6)

Thus, all agents with a latent ability greater than e * will invest in education, where e * is uniquely defined by the following equality.

A,w(k) + c(1 + r* ) e* = (7)

A,w(k)

E We will assume that the model is such that e *~[0 + E, E - e] where 0 < E < 7-

2.1.4. The growth externality We assume that there is an economy wide growth externality related to the

proportion of educated workers in the economy in the previous period. The assumption that the level of productivity in an economy is related to some measure of past educational levels has been widely used theoretically (see Buiter and Kletzer, 1993; Galor and Tsiddon, 1994; Glomm and Ravikumar, 1992) and is supported empirically at both the macro level (see Barro, 1991; Mankiw et al., 1992) and the micro level (see Becker and Tomes, 1986; Coleman, 1966). Thus we model A t to be a positive function of the proportion of workers in the previous period who were educated, that is

At= A(st 1) where st_ ' =fe,~, g(ei) dei and where A' ( s t - , ) > 0 (8)

We also assume that A(0) = 1 and that A(1) is finite.

2.1.5. Dynamics and steady state productivity The only dynamics in the model in this section stem from the growth

externality. It is clear from Eq. (8) that the proportion of workers who are educated in period t is an increasing function of the proportion of workers who were educated in period t - 1, that is s, = ~0(s t_ 1), since

de t c ( l +r*)X(St_l) dst-, AZ(st_l)w(k) (9)

292 A. MounO~ord / Journal of Development Economics 53 (1997) 287-303

St+l

/ /

o i0 ~1

J ~St ) With Optimal Emigration

~(St) With No Emigration

Iv

St

Fig. 1. Depicts the dynamics of the economy when there is a unique steady state equilibrium for the cases where there is no emigration and when there is no optimal emigration.

Thus

ds t c ( l + r * ) X ( S t _ l )

dst_----~l=g(e * ) A2(S t_ l )W(k ) (10)

We assume that E is high enough so that the most able worker will always choose to be educated even if no-one in the previous period was educated. Since we know that agent i with e i = 0 will never choose to be educated, then this implies that there must exist at least one steady state equilibrium for s t , which we denote as ~. Whether this is a unique steady state depends on the properties of the h(s t_ l) function. If this function has convex regions, representing 'critical masses' of educated people in the economy, then there may be multiple steady states. The unique steady state case is depicted in Fig. 1.

Multiple steady state equilibria imply that temporary shifts in the proportion of educated people in the economy can have permanent effects on the long run productivity level. Section 2.2 shows how a well regulated emigration policy can cause such shifts.

2.2. The effects of migration

Given the model described in Section 2.1.5 we can now turn to the implications of emigration for this economy. In this subsection we will examine the effects of emigration both when emigration is only permitted for educated agents- -a brain drain--and where there is no educational requirement for emigrat ion--a general emigration. We examine the case where emigration is limited by the receiving country: immigration controls. Under immigration controls we assume that the probability of successful emigration, w, is independent of the number of workers who are eligible to emigrate. This follows from the small country assumption. It can easily be shown that the case where emigration is limited by the source

A. Mountford / Journal o f Development Economics 53 (1997) 287-303 2 9 3

country--exit v isas-- is very similar. Finally we assume in all cases that emigration policy is fully anticipated.

In order to motivate the desire for emigration, we assume that the wage per efficiency unit of labor in the world economy, w F, is always higher than that in the small open economy, htw H. This is effectively assuming that the level of technology in the world economy, A*, is greater than h(1).

2.2.1. Emigration under immigration controls When there is a probability, or, of emigrating and earning a higher wage, the

agent's education decision becomes an expected utility problem. For simplicity we assume that agents are risk neutral. In Section 2.2.1.1 we show that a general emigration will always increase productivity in the next period in the source economy. In Section 2.2.1.2 we describe sufficient conditions for a brain drain to increase next period's productivity also. If the government of the home country can influence zr, through agreement with the other world governments or via lax border controls, then we describe the conditions for an optimal determination of "1"/'.

2.2.1.1. A general emigration. Under the above assumptions, it is clear that an agent, i, will choose to be educated so long as,

(orwr + ( l - - z r ) A t w l t ) e i > T r w r + ( 1 - - o r ) A t w l 4 + c ( l + r *) (11)

Thus, all agents with a latent ability greater than e * will invest in education where e * is uniquely defined by the following equality.

OrwF+ (1 - or)A,wt¢ + c(1 + r * ) e* = (12)

OrW r + ( 1 - - Or),~tW H

de * Clearly W < 0 and thus the average productivity in the source economy in the following period will rise, since the average proportion of educated people in the economy is

(1 - or ) f~5 g( e i )de i st = 1 - zr = re'E• g( e i )dei (13)

If the chance of emigration is permanent then the return function s t = qgs t_ ~) in Fig. 1 will shift upwards and thus productivity will be permanently higher in the source country. Thus in this situation the optimal government policy is trivial. Regardless of the initial level of A the government should try to make or as large as possible. If zr = 1 then everyone will leave the economy.

2.2.1.2. A brain drain. In this subsection we now assume that there is only a chance of emigration, or, if the agent is educated. Thus it is clear that all agents with a latent ability greater than e * will invest in education where e * is uniquely defined by the following equality.

2 9 4 A. Moun(ord / Journal of Development Economics 53 (1997) 287-303

At w n + c(1 + r * ) e* = (14)

~ r w F + (1 - ~r) ;ttw H

The average proportion of educated people in the economy is given by the following,

(1 - w ) f f ; g(e i )de i St = 1 -- "t"1"( feE~ g(e i )de i) (15)

The result in this section can be summarized by the following proposition.

Proposi t ion 1 There will exist a positive optimal level of 'brain drain' ar~i~.,~ti., i f g( e~)(A,w;4+c(1 + r~ ) ) f w ~ - a , w ~ ~ ~ * is the lowest level . . . . ~ . . . . . . . . . . (a,w") . . . . Z where eNM

of ability of an agent choosing to be educated when there is no possibility of emigrating.

P roo f If 7r = 1 then the source economy must lose from a brain drain since this implies that s t = 0. If 7r = 0 then there is no emigration and the source economy is in autarky. Thus a sufficient condition for there to exist a positive level of brain drain emigration such that the source economy benefits in terms of productivity, is that ~ ~- 0 when rr = 0. The optimal level of 7r will then be given where ~ = 0 dcr ~ d~"

although there may exist other local maxima and minima. Differentiating Eq. (15) gives us the following expression for ~-~,

ds t Os t Os t Oe * + - - - - (16)

O~- Oe * OTr 071"

where

Os t

O~r

Os t

Oe*

fe:E g (e i )de i [1 _ fe:E g ( el)de i ] - <0,

[1 -- 7rfe E g ( e i )de i ] 2

(1 -- ~ r ) g ( e * )

[1 - 7rfee; g( ei)dei ] 2

Oe* ( A t w t 4 + c ( l + r * ) ) ( w F - - A t w H) - - = - - < 0

&r [~rw F + (1 _ ,l.r) AtwU] 2 (17)

7 r = 0 and noting that fe~g(ei)dei[l_< - f j e , g ( e i )de i ] , is at most a Setting quarter, the proposition follows.Q.E.D.

This proposition says that the source economy can benefit from a brain drain so long as there are a sufficient number of people who would be enticed to invest in education, by a small prospect of emigration. The intuition is made very clear when we consider the case of uniformly distributed abilities. For a uniform

1 E e* distribution g ( e i ) = 2 and f~; g ( e i ) d e ~= 1 - - 2 - . This simplifies the algebra above so that the following result can be obtained

A. Moun~eord / Journal of Development Economics 53 (1997) 287-303 295

d s t e* W F - - ~t WH - - > 0 i f f 1 - - - < (1 - ~-) (18) dTr E 7rw p + (1 - 7"r)Atw H

Thus, a brain drain will increase the proportion of educated people in the economy if 7r is low, if w F is very high relative to At w n and if the proportion of educated people in the economy was previously low. A glance at Fig. 2 shows that these results are very intuitive.

Fig. 2 shows that a brain drain lowers the threshold level of ability needed before education is worthwhile from eNM to eBD. This thus increases the amount of educated agents by e~M E e~r~. However the brain drain also allows a fraction of all educated agents to emigrate, this thus lowers the amount of educated agents remaining in the economy by ~ 7r. Clearly whether a brain drain is good for the source economy depends on the relative size of these amounts and as Fig. 2 shows, the positive effect of a brain drain will dominate the negative effect if it is low and if the proportion of educated people in the economy was previously low.

Eq. (18) implies that when abilities are distributed uniformly, if w F is large enough there will always be a positive level of rr such that next period productivity increases in the source economy. In this case when there is an optimal emigration policy under a brain drain, the return function s t = ~(s t_ 1) will be everywhere above the return function when there is no emigration. Thus clearly an optimal emigration policy will increase the short and long run productivity in the source economy. Finally if there are multiple steady state equilibria then a temporary emigration policy might lift a source economy from a low to a high education steady state.

The analysis so far has assumed that the probability of successful emigration, 7r, was unrelated to the number of people wishing to emigrate. This might be thought unrealistic but it can be easily shown that the results of this section are not significantly changed if there is a fixed number of exit visas in the source country. Thus there will exist an optimal number of exit visas corresponding to the optimal level of 7r in the analysis above. Pr°babil~l

E

p

0 e~ D e~M E Ability

Fig. 2. Illustrates the gains and losses from a brain drain. The higher rectangle depicts the gains of more agents choosing to become educated and the lower rectangle depicts the losses due to emigrating educated agents. The probability of successful emigration is pE.

296 A. Moun(ford/ Journal of Development Economics 53 (1997) 287-303

2.3. Summary of section 2

This section has shown how a brain drain can increase the productivity of the source economy when productivity is an increasing function of the proportion of educated people in an economy in the previous period. It has also shown how a temporary possibility of emigration, or a temporary inflow of educated workers, may permanently increase the productivity of an economy. The results in this section were derived without using non-marginal changes in the income distribution. Section 3 shows how migration can radically alter the income distribution of the source economy and so have very powerful effects.

3. The effect of a brain drain on income distribution dynamics

The model in this section is a simplified version of the model by Galor and Tsiddon (1994). It differs from the model in Section 2 in one major respect. In this model the number of different 'educational classes' is endogenous and so in this model migration can influence the formation of 'educational classes'. The education decision in this model is now not an ei ther/or choice, instead the amount of education an agent can purchase is a continuous choice variable. In other respects the model in this section is very similar to that in Section 2. The overlapping generations structure, the production technology and the small open economy assumptions are all precisely the same as above. We assume that each agent has one offspring and we again normalize the number of agents in each generation to unity.

3.1. The economy without migration

We will first describe this economy without migration before going on to analyze the model under various assumptions about the possibility to migrate.

3.1.1. The human capital production function The accumulation of human capital, or efficiency units of labor, by individual i

born in period t, e~+ 1, is modeled as an increasing function of the individual's parent's level of human capital, e~, and the resources invested in human capital accumulation, x~, by the individual. Thus there is a 'family level' externality such that an individual's accumulation of human capital is easier, the greater the human capital accumulation of her parent. Formally, el+ ~ = qJ(e~, x~). This function is assumed to be concave with positive cross partial derivatives, that is, ~b 1, ~O 2, ~b12 > 0 and qJll, ~b22 < 0, where ~b i is the partial derivative of function ~b with respect to its ith argument. It is also assumed that ~b(e~, 0 ) = ~b(0, x~)= ~O(0, 0) = ~, which implies that an agent always has Ix units of efficiency labor even if she invests nothing in education and /or her parents invested nothing in education.

A. Mountford / Journal of Development Economics 53 (1997) 287-303 297

3.1.2. Individuals ' optimization decision

Agents live the same three period lives as in Section 2 except that now the education decision variable, x[, is continuous as described above. It is assumed that all agents have the same preferences and access to the same technology, although of course they do not have the same levels of parental human capital. It is this difference which allows for a long run dispersion of household income levels in the economy.

ut'i(ct+ 2 ) denotes the utility function of agent i born in period t. For simplicity, as in Section 2, this is assumed only to depend on third period consumption. We assume this to be a monotonic strictly quasi-concave function.

This will be important in Section 3.2 when we consider the expected utility of migration. In this section, however, the assumption that agents consume only in the third period means that utility maximization is equivalent to maximizing third period income. Agents maximize third period income by choosing the optimal amount of investment in education. Agents therefore solve the following maximization problem,

m a x ( l + r*)[w)t~b(e~,x~) - (1 + r * ) x ~ ] (19) xl

The first order conditions for this problem imply that,

( l + r * ) ~b2(e~'x[) wA (20)

3.1.3. Income distribution dynamics

Given the above analysis we can describe the evolution of the income distribution of this economy. Using the implicit function theorem and Eq. (20) we can

i i that is i _ ~(e~) where ~ ' = - ~ > 0. Substituting write x, as a function of e t, xt - ~22 -

this into the human capital production function gives us

eit+, = ~ ( e l , s C ( e l ) ) (21)

Eq. (21) gives the relationship between the human capital level of a parent and that of her offspring. This relationship is plotted in Fig. 3, 2 where the human capital level of the parent is the variable on the x axis, and that of the offspring is the variable on the y axis. The distribution of the level of human capital of the initial parent generation is defined over the whole x axis. The relationship is upward sloping, since ~ ~ ' ~" de', = OJ + ~02 = ~bl -- ~b 2 0~2 > 0, but it is not necessarily concave since the second derivative will depend on the third derivative of the 0 function, whose sign is not restricted. Thus there may be many intersections of the

2A functional form for the fft(e~,x~) function which gives the 'S' shape depicted in Fig. 3, is el+ 1 = /z +(e[)exp(1/(e[));(X~)13, where /3 < 1.


With A General ,~,I )Chance

~1(~ ~ To Emigrate

~w~:tNo Emigration

45° i

Fig. 3. Depicts the human capital accumulation function with multiple steady state equilibria, for the cases where there is no emigration and where there is a general chance to emigrate.

45 ° line by the ~b function. These intersections are 'steady state' levels of human capital accumulation.

If there is a unique steady state level of human capital then income inequality will be declining through time. The offspring of all agents will gradually converge to the same steady state level. However if there is more than one steady state equilibrium income inequality will not tend to zero as time goes to infinity. Fig. 3 depicts the case where there are three steady state levels of human capital accumulation el, em, eh, although the results of the model are not significantly affected if there are a greater number of steady states. The middle steady state, e,,, is unstable while e~ and e h are both stable. Agents descended from someone in the initial generation whose human capital lay in the interval [0, e,,) will converge to the educational level e I and agents descended from someone in the initial generation whose human capital lay in the interval (em, oo) will converge to the educational level e h.

It is important to notice that if an economy has an unequal distribution of long run income then it would be possible to permanently improve the long run level of average productivity by moving agents from a low to a high educational steady state. This could be achieved by an income related subsidy to education for example. This paper shows that allowing some agents the chance to emigrate may also have the same effect.

3.1.4. G r o w t h d y n a m i c s

Until now we have not assumed the existence of any 'global', 'growth' externality, (relating the level of technology, At, to past average levels of education in the economy) as we did in the model of Section 2. A corresponding assumption in this model would be to make the level of technology in period t, A t, a function of the average level of parental human capital, that is

Lt /~t+l = /~(e t ) where et = N (22)

A. MounO~ord / Journal of Development Economics 53 (1997) 287-303 299

where L t is the sum of the human capital of all agents working at time t, that is, i where A'(~ t) > 0 and where N is the number of agents in the L, = E/r=l e,,

generation. Notice that this model already has dynamics due to the family level educational

externality. This additional assumption causes a second dynamic effect and so converts the model from being a model about productivity levels, to one about the growth rate of productivity. If the average level of human capital rises, this will increase A in the following period. This in turn, from Eq. (20), will increase the investment in human capital by all young members of the economy, which in turn will cause another rise in A and so on. This is a potentially perpetual endogenous growth process.

3.2. The effects o f migration

The model outlined above described the endogenous formation of educational classes and their relationship to the long run level of average productivity. We will now use this model to investigate the implications of emigration for this economy. For most of this section we will again assume that the small open economy has a constant level of technology, A, which is lower than that in the rest of the world, A * and that emigration policy is fully anticipated. Thus all workers would want to emigrate to enjoy the higher wage rate, wA*, in the world economy. However, everyone is not able to emigrate due to an exit visa policy imposed by the source economy or an immigration policy imposed by the recipient country. We again examine two separate cases: firstly, when all workers have the same chance, 7r, to emigrate regardless of their level of human capital (a 'general emigration'); and secondly, when only workers with a certain level of human capital, y, have a chance, 7r, to emigrate (a 'brain drain').

As in Section 2 it is shown that a general chance to emigrate is unambiguously good for the level of per capita income and that a 'brain drain' will have both positive and negative effects. Unlike in Section 2, however, the effects of a brain drain may not be marginal and may involve large changes in the income distribution of an economy. In Section 3.2.2. it is even shown that when a brain drain significantly alters the long run income distribution, it is possible for a brain drain to be better than a general emigration for the small open economy.

3.2.1. The effect o f a general chance to emigrate To demonstrate that a general chance to emigrate will increase the human

capital accumulation of all agents, we must redefine the maximization problem in terms of expected utility. For simplicity we will denote the consumption in the last period of life as c E, if the agent emigrates, and c NE, if the agent does not emigrate. From above we know that c e = (1 + r*)[wA*~(e~,x~)) - (1 + r* )x~] a n d c NE = (1 + r* )[wA~b(e~,x~)) - (1 + r* )x~], where c E > cuE, since A* > A.

300 A. MounOrord / Journal of Development Economics 53 (1997) 287-303

Thus, agent i's maximization problem is the following:

max 7ru( c E) + (1 - 1r )u ( c NE) (23) x~,

The first order condition to this problem gives the following equation,

7ru'( c e ) [ wA*~b2( e l , x l ) - ( l + r * ) ] + ( l -- Tr )u ' ( cNE)[ wAO2( e~,x~)

- ( l + r * ) ] = 0 (24)

Using the implicit function theorem, it follows that the amount invested by individual i is an increasing function of the probability of emigration. 3 Fig. 3 shows the implications for the economy of this result.

Fig. 3 contrasts the human capital accumulation function, 0, for the case where emigration is not possible, denoted by 00, and for the case where there is a general chance to emigrate, denoted by ~b I. As the above analysis has shown, any positive value of ~- will increase x~ for all members of generation t who have positive values of e~. This will cause the ~b function to rise and thus ~b I is drawn above ~0, for all positive values of e~. This means that the productivity of all workers is increased.

Even a temporary emigration can have lasting beneficial effects. If the opportunity to migrate moves the ~b function up, so that it only crosses the 45 ° line once, at the high education steady state eh, then the offspring of all agents who remain in the country will tend towards e h. In this case a temporary emigration opportunity, that lasts long enough for many agents to move past the threshold level of e m, will have a permanent effect on the average level of human capital accumulation in the economy. For now in this case the offspring of these agents will continue to move towards e h even without the incentive caused by potential emigration.

3.2.2. The effect o f a brain drain This section considers the consequences of a brain drain on the small open

economy. We will assume that the economy has more than one steady state level of education because the phenomenon of a brain drain refers to the situation where agents in the source economy have differing levels of human capital. We model a brain drain as the situation where there is only a chance of emigration, ~r, if the

3

dx~ A - - = - - > 0 where, dTr B

A = - d ( c e ) [ w A * ~ 2 - ( 1 + r * ) ] + u ' (c~g)[wA~2 - ( 1 + r * ) ]

B = ¢r{d ' (ce)[wA*~2 - (1 + r* )]2 + u,(ce)wA.~22} + (1 - 7r){d ' ( c N e ) [ wA~b 2 - (1 + r* )]2

+ ~¢(d'~)w~,22}

A. MounOCord / Journal of Development Economics 53 (1997) 287-303 301

- With A ~(et)T ~iet) Brain Drain /

~ 4 5 o 3o ~ ~q~ e,

Fig. 4. Depicts the effects of a brain drain. This diagram shows the case where the brain drain eliminates the low level educational steady state and thus causes the descendants of all agents, that remain in the economy, to converge to the high education steady state.

agent has a certain level of human capital accumulation, y > 0. In contrast to Section 3.2.1, the distribution of human capital levels among emigrants will differ from that of non-emigrants. Since the emigrants will on average have more human capital than the non-emigrants, then abstracting from the effect of a brain drain on human capital accumulation, a brain drain will reduce the average productivity level in the source economy. As in Section 2, however, there will be positive effects of a brain drain on human capital accumulation and these may predominate if the long run distribution of income is changed. This possibility is described in Fig. 4.

Fig. 4 compares the human capital accumulation schedule for the cases where emigration is not allowed, ~b °, and where emigration is possible for agents with a level of human capital greater than or equal to y, ~b i. The existence of the threshold level has the effect of dividing the ~b ~ schedule into three parts. The part for agents with the lowest levels of parental human capital will be the same as ~b °. They will not choose to invest enough to attain 3' units of human capital and so they will have no chance to emigrate. The part for agents with high levels of parental human capital will be the same as in Section 3.2.1. These agents would have chosen to accumulate y or more units of human capital even if this had not been the threshold level necessary for eligibility to emigrate. Between these two sections is a region where agents who, without the 3' threshold, would have chosen to accumulate less than y units of human capital. These agents will now choose to attain exactly y units of human capital in order to be eligible to emigrate. 4

The effect of the brain drain in the example drawn in Fig. 4, is to eliminate the low human capital accumulation steady state, e~. The offspring of all agents who

4 A sufficient condition for there to be a single interval of agents choosing to attain exactly y units of human capital is that the ff function is sufficiently concave in its first argument. A single interval is not necessary for the results in this paper, but it makes the exposition simpler.


remain in the source economy will now tend to the high education steady state and so the average level of human capital (and so average productivity) will tend to rise in the long run.

The change in the long run income distribution is crucial to this result. If the brain drain did not eliminate, et, then after a number of periods most of the offspring of agents with high levels of human capital would have emigrated and thus the economy would consist mostly of agents with a low level of human capital accumulation. This would thus decrease the value of average productivity in the long run. Finally note that a temporary brain drain can have permanent beneficial effects in precisely the same way as a temporary general migration, which was described in Section 3.2.1.

3.2.2.1. The case where a brain drain is better than a general emigration. Although as we have described above, a general emigration always increases productivity while a brain drain can increase or decrease productivity, it is possible for a brain drain to lead to a higher level of long run productivity than a general emigration. To see that this might be so consider the middle section of qjl in Fig. 4. In this section, the ~b function for a brain drain is higher than the ~O function for a general emigration, for a given level of 7r. This is because agents are prepared to choose a higher level of x than they would do under a general emigration, in order to be eligible for emigration. Thus it is possible for a brain drain to eliminate the low level educational steady state el, when a general emigration does not. If this occurs then the long run level of average productivity would be higher under a brain drain because all agents remaining in the country would be at the high education steady state, whereas under a general emigration only a fraction of the population would be at the high education steady state, s

3.2.3. Growth effects The addition of the growth externality of Section 3.1.4. can reinforce the results

outlined above. Since the growth externality is a positive function of average human capital levels and we have shown how both a general emigration and a brain drain may increase average human capital levels, it is easy to show how a general emigration and brain drain can increase the growth rate of the source economy.

4. Conclusion

This paper has shown that when human capital accumulation is endogenous and when successful emigration is not a certainty, the interaction between human

5 A parametric example of an economy which has one steady state under a brain drain and three under a general emigration is where preferences are risk neutral and where e~+l= 0.145+(ei) exp l/d(e~))2(xl)°'5, where d= 1 × 102°, A* = 50, h = 1, w = 1, ~r = 0.95, 3' = 0.435, and r* = 1.05.

A. Mountford / Journal of Development Economics 53 (1997) 287-303 303

capital accumulation decisions, growth and income distribution can lead to the result that a brain drain, either temporary or permanent, may increase the long run income level and income equality in a small open economy, and, in certain circumstances, may even be preferable to a non-selective 'general' emigration.

Acknowledgements

Many thanks to Martin Lettau, Harald Uhlig, two anonymous referees and seminar participants at Tilburg University for very helpful comments. All remaining errors are, of course, my own.

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brain drain positive points

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