dynamic evolution of income distribution and credit-constrained human capital investment in open...

Journal of International Economics 55 (2001) 329–358www.elsevier.nl / locate /econbase

Dynamic evolution of income distribution and credit-constrained human capital investment in open economies

*Priya RanjanDepartment of Economics, University of California, 3151 Social Science Plaza, Irvine, CA 92697,

USA

Received 15 December 1999; received in revised form 9 October 2000; accepted 24 November 2000

Abstract

This paper shows how the degree of credit-market imperfections affects the steady-statedistributions of income and wealth, human capital investment, and the pattern ofcomparative advantage. The impact of trade liberalization on the accumulation of humancapital depends on how it affects (1) the incentives to accumulate human capital, (2) theborrowing constraints facing human capital accumulation, and (3) the distribution ofincome and wealth. If the degree of credit market imperfections is low in the skill-abundantcountries and high in the skill-scarce countries, then trade liberalization can increaseinvestments in human capital in both types of countries. 2001 Elsevier Science B.V. Allrights reserved.

Keywords: International trade; Credit-constraints; Human capital; Income distribution

JEL classification: F11; O15

1. Introduction

It is widely recognized that differences in human capital or skill across countriesplay an important role in determining differences in growth rates and per capitaincome levels across countries. As well, skill differentials form the basis of muchof the trade between the skill-abundant developed countries in the North and the

*Tel.: 11-949-824-1926; fax: 11-949-824-2182.E-mail address: [email protected] (P. Ranjan).

0022-1996/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PI I : S0022-1996( 01 )00103-9

330 P. Ranjan / Journal of International Economics 55 (2001) 329 –358

1skill-scarce developing countries in the South. While trade based on skilldifferential always provides static gains from specialization, the dynamic gainsdepend on how the investment in human capital is affected. This makes thequestion of how trade affects skill accumulation an important one.

In a pioneering paper, Findlay and Kierzkowski (1983) extended the staticHeckscher–Ohlin model of trade by endogenizing human capital accumulation toshow that trade amplifies initial differences in factor endowments. The channel ofinfluence is the Stolper–Samuelson effect of trade on factor prices which raises thereward of the abundant factor in each country. This provides further incentives toaccumulate human capital in skill-abundant countries and does the opposite inskill-scarce countries. Similar results were also obtained by Grossman andHelpman (1991). Two recent papers — Cartiglia (1997) and Eicher (1999) —show that trade leads to convergence in human capital endowments. A key elementof both these papers is that skill is used in the formation of skill (education sectoruses skilled labor), and therefore, any rise in the price of skill has an adverse effecton skill accumulation. In Cartiglia (1997) the credit market is missing, therefore,investment in human capital has to be self-financed from the initial endowment. Atrade-induced rise in the skilled wage in a skill-abundant country increases the costof education and hence exacerbates the borrowing constraint facing investment inhuman capital. In Eicher (1999) there is a domestic credit market where thesavings of the unskilled workers are used to finance the investment in humancapital. Now, a trade-induced decrease in the unskilled wage in a skill-abundantcountry reduces the resources available for financing investment in human capital,while a rise in the cost of education increases the need for resources to finance

2investment in human capital. Therefore, in both these papers trade liberalizationreduces the investment in human capital in a skill-abundant country. The oppositehappens in a skill-scarce country giving rise to the convergence in human capitalendowments after opening up to trade. This literature, however, has ignored theimpact of trade on investments in human capital coming through changes in thedistribution of income and wealth. This is a serious omission because thedistribution of wealth becomes an important determinant of investments in humancapital in the presence of borrowing constraints. The main contribution of ourpaper lies in bringing the distributions of income and wealth to the centre of thediscussion of the impact of trade on factor endowments. It endogenizes the wealth

1Looking at the exports of the South to the North, the share of manufactures as a proportion ofnon-fuel exports has gone up from 6% in 1955 to 71% in 1989. The North’s export to the Southcontinues to be dominated by manufactures; its share having risen from 73 to 79% over the sameperiod. The difference lies in the fact that the Northern manufactures are more skill intensive than theSouthern manufactures (source: Wood (1994)).

2Eicher (1999) has an endogenous growth model where the cost of human capital accumulation andthe cost of technical change interact to give a convergence in growth rates as well. We have just notedhis model’s implications for convergence in human capital endowments.

P. Ranjan / Journal of International Economics 55 (2001) 329 –358 331

distribution in the presence of credit market imperfections and shows that changesin the distribution of wealth brought about by trade have additional and, in somecases, offsetting effects on the accumulation of human capital.

It is first shown that there exists a unique invariant steady-state distribution ofwealth for a given degree of credit-market imperfection. A decrease in the degreeof credit-market imperfection implies an improvement in the steady state dis-tribution of wealth in the first order stochastic dominance sense. The degree ofcredit-market imperfections affects investments in human capital both directly andindirectly by improving the steady-state distribution of wealth, and hence, itbecomes a determinant of the pattern of comparative advantage. This new resultshows how an institutional variable like the degree of credit-market imperfectionscan become a determinant of the pattern of comparative advantage.

Next it is discussed how trade liberalization affects investments in humancapital by, among other things, altering the distribution of wealth. Tradeliberalization increases the incentive to invest in human capital in a skill-abundantcountry. However, an increase in the skilled wage raises the cost of education,which worsens the borrowing constraint in a world with credit-market imperfec-tions. In addition, the support of the steady-state distribution of wealth is widened.The widening of support at the bottom end has a negative impact on theinvestment in human capital. Intuitively, trade liberalization reduces the unskilledwage, which reduces the bequest of the unskilled, making their descendants moreprone to borrowing constraint. The net effect on investments in human capitaldepends on the relative strengths of these three effects. The opposite happens in askill-scarce country. There the negative incentive effect has to be balanced againstthe positive effect arising from the distributional changes and the relaxation ofborrowing constraint due to reduced cost of education. The support of the steadystate distribution of wealth shrinks to a smaller interval. The increase in the wealthof individuals at the bottom end caused by a rise in the unskilled wage has apositive impact on the investment in human capital of their descendants. Theunskilled are able to leave a larger bequest, which relaxes the borrowing constraintfor their descendants.

Finally, it is shown that if the degree of credit market imperfections is very lowin the skill-abundant countries and very high in the skill-scarce countries, thentrade liberalization is likely to increase investments in human capital in both typesof countries. In the presence of a very low degree of credit market imperfectionsthe positive rate of return effect will dominate in the skill-abundant countries,while in the presence of a very high-degree of credit market imperfections thepositive effect arising from distributional change and the relaxation of borrowingconstraint will dominate in the skill-scarce countries. Using data on the degree ofcontract enforcement as an indicator for the degree of credit market imperfections,we find that countries with low human capital endowment have a very high degreeof credit market imperfections, while countries with high human capital endow-ment have a very low degree of credit market imperfections. A companion paper


(Ranjan, 1997) using cross-country data finds a positive relationship betweenopenness and investments in human capital in both rich and poor countries.

The present paper differs from the existing literature as follows. Unlike Eicher(1999) which is a representative agent type model, we allow for heterogeneityamong agents in the inherited wealth and the ability to accumulate human capital.Cartiglia (1997) allows agents to have differing initial wealth but identical ability.Due to the assumption of identical ability in Cartiglia (1997), only the creditconstraints are binding, while the incentive effect of trade on investments inhuman capital that gives rise to divergence in Findlay and Kierzkowski (1983) andGrossman and Helpman (1991), is non-binding both before and after tradeliberalization. By allowing for heterogeneity in ability our paper allows both theincentive constraints and the borrowing constraint to be binding and is sufficient togenerate the ambiguous effect of trade on factor endowments. Also, unlikeCartiglia (1997) where credit markets are completely missing, we model credit-market imperfections explicitly as arising from information problems. The mostimportant difference is that the distribution of initial wealth, which is exogenous inCartiglia (1997) and does not matter in other models due to perfect credit marketassumption, evolves endogenously in our model through intergenerationalbequests, and provides additional channels of influence of trade on human capitalendowments. Like Cartiglia (1997) and Eicher (1999) we have also assumed thatthe cost of education depends on the skilled wage, however, the effect of tradeworking through changes in the distribution of wealth would arise even if the costof education is independent of the skilled wage.

The plan of the rest of the paper is as follows. Section 2 sets up the basic model.Section 3 discusses the impact of trade liberalization on the steady statedistribution of wealth and investments in human capital. Section 4 presents someconcluding remarks.

2. The model

The model spelled out below captures the elements of trade based on differencesin skill endowments, making it suitable to analyze the impact of trade on skillaccumulation.

2.1. Technology and preferences

The economy can produce two final goods, H and L, with two factors ofproduction: skilled labor, S, and unskilled labor, U. Both final goods employ bothfactors of production with a constant returns to scale technology. H is thehigh-tech good which is more skill intensive than the low-tech good L at all factorprices. L is also the numeraire good. The production functions are given by


h hH 5 A F (S , U ) (1)h h

l lL 5 A F (S , U ). (2)l l

h lIn (1) and (2), A and A are the productivity parameters, changes in which cancapture biased technical changes.

The factor supplies are endogenously determined through the occupationalchoices of individuals. We assume a small open economy which takes the relativeprice of goods as given. The relative price of the high-tech good is denoted by p.Denote the skilled wage per efficiency unit of skill by w and unskilled wage bys

w . Given the above production structure, relative goods price fixes the relativel3factor price, v 5 w /w .s l

The education sector is modeled in a simple fashion to capture the fact that skillis used in the formation of skill. Denote the number of students by S and theamount of skill employed in the education sector to educate students by S , andE

assume the following functional form

S 5 QS . (3)E

In (3) Q is a parameter capturing the productivity of the education sector. Theabove functional form for the education sector leads to a cost of education per

4student equal to qw , where q 5 1/Q. The students in period t become skilleds

workers in period t 1 1. The total amount of skill in efficiency units is denoted by]S, which is allocated between production (S 1 S ) and education (S ) sectors:h l E]S 5 S 1 S 1 S .h l E

The amount of human capital or skill that an individual acquires upon going toschool depends on the talent that an individual is born with. We assume that theamount of skill acquired is independent of the educational inputs. We do so for tworeasons: (1) to keep the analysis simple, and (2) the empirical evidence on therelationship between educational input and student performance is far from

5conclusive.Each individual lives for two periods. The population born in each period is

normalized to have measure 1. There is no population growth. Each individual is]born with an endowment, a [ [a, a], of the numeraire good and a talent or abilityi ] ]to acquire human capital, s [ [s, s]. s can be thought of as the efficiency unitsi i]

of human capital that an individual acquires upon going to school. The talent ofeach individual is public knowledge. The labor income of a skilled individual withtalent s is s w . Denote the distribution function of talent by F(s), and thei i s

3This is true only in a diversified equilibrium where both goods are produced. Throughout the paperwe assume a diversified equilibrium.

4Eicher (1999) and Cartiglia (1997) model education sector in a similar fashion.5See Hanushek (1995) for a recent survey of empirical work on this issue.


corresponding density function by f(s). The distribution of endowments evolvesendogenously and is derived below.

In the first period an individual can either go to school or work as an unskilledworker. An individual who goes to school must pay an education cost of qw , ands

becomes a skilled worker in the second period. The unskilled work in bothperiods. For simplicity it is assumed that all consumption takes place in the second

6period only. The unskilled save their first period wage and endowment. Eachindividual has a child in the second period of his life. All parents work in thesecond period of their lives, while some children work as unskilled and others goto school in the first period of their lives. Parents care about their children andleave them a bequest. We assume that parents have warm glow preferences overbequests. That is parents derive utility by giving to their children, independently ofthe extent to which their children actually benefit from the bequest. Apart frombeing analytically tractable, the warm glow preference for bequests formulationseems to have better microfoundations than the Ricardian formulation (seeAndreoni (1989)).

In period t, young inherit a from their old parent. A simple form of utilityt

function is assumed with C as an index of consumption and a being the bequestb 12bV5 C a (4)

u 12uC 5 C C . (5)h l

In (5) C is the consumption of the high-tech good, C is the consumption of theh l

low-tech good. This specification of the utility function makes the indirect utilitylinear in income, and hence the expected utility is also linear in income, whichmakes the analysis of credit-market imperfections simple.

Denoting the market rate of interest by r, the incomes of unskilled and skilledindividuals in the second period of their lives can be written as

y 5 (2 1 r)w 1 (1 1 r)a (6)u l

y 5 sw 1 (1 1 r)(a 2 qw ). (7)s s s

If credit markets are perfect, then an individual decides to acquire skill or remainsunskilled depending solely on his ability. Let us look at this decision for a given r,w and w . The lifetime utility from becoming skilled for an individual withs l

endowment a and ability s is given byi i

iV 5 C8 3 (s w 1 (1 1 r)(a 2 qw )) (8)S i s i s

6Allowing for consumption in the first period will make the borrowing constraint more severebecause individuals may want to borrow for consumption smoothing as well. The results of the paperremain qualitatively similar after allowing for first period consumption. See Chiu (1998) for a modelthat allows for first period consumption.


where C8 is a constant which depends on the parameters of the utility function andthe product price ratio p. Similarly, the lifetime utility of an individual deciding toremain unskilled is given by

iV 5 C8 3 (w 1 (1 1 r)(a 1 w )). (9)U l i l

In equilibrium the marginal individual is indifferent between acquiring skill andi iremaining unskilled: V 5V . This implies a threshold level of ability, s* givenS U

by

(2 1 r)w 1 (1 1 r)qwl s]]]]]]]s* 5 (10)ws

such that all individuals with s . s* invest in human capital by going to school,i

while the others remain unskilled.

2.2. Imperfection in the credit-market

We make the following assumption about the rate of interest.

Assumption 1. Due to free international capital mobility, the individuals in thissmall open economy can lend any amount at the world rate of interest r.

They cannot borrow any amount at this rate of interest, however, due to theimperfections in the credit market described below.

A very simple form of imperfection in the credit market is assumed. The creditmarket is characterized by the possibility that a borrower may renege on a debt. Toabstract from the bankruptcy issues assume that the parameters are such that theborrower can always afford repayment. The borrower can renege at the time ofrepayment. The borrower succeeds in fleeing with probability p, in which case sheconsumes her entire second period skilled wage sw . The borrower is caught withs

probability 1 2 p, in which case her entire second period income is taken away by0the lender. Reneging, therefore, yields an expected payoff of pC sw , whiles

0repaying yields a payoff of C (sw 2 (1 1 r)(qw 2 a)). Therefore, lenders wills s

make loans that satisfy

0 0C (sw 2 (1 1 r)(qw 2 a)) $ pC sw . (11)s s s

Eq. (11) implies the following threshold level of wealth for each ability

1 2 p 1 2 p]] ]]S Da*(s) 5 qw 2 sw 5 w q 2 s . (12)s s s1 1 r 1 1 r

Eq. (12) can be written in an alternative form to show the threshold level of abilityrequired for each level of wealth as follows.


(1 1 r)(qw 2 a) (1 1 r)q (1 1 r)as˜ ]]]]] ]]] ]]]s(a) 5 5 2 . (13)1 2 p(1 2 p)w (1 2 p)ws s

(12) implies that for each level of s, individuals having a , a*(s) cannot borrowadequately to invest in human capital. Individuals having s . s* and a , a*(s)are rationed in the credit market in the sense that they would like to invest inhuman capital, but cannot borrow enough to do so. The parameter p captures thedegree of credit-market imperfections in the model. p 5 0 corresponds to thefirst-best case, where the borrowing constraint does not bind because from (10) fors . s* we have s . (1 1 r)q, therefore, if p 5 0, then a*(s) , 0 in (12). Thehigher the p the lower the amount that individuals can borrow against their futureearnings, and hence the more severe the borrowing constraint. In the extreme caseif p 5 1, individuals cannot borrow at all against their future earnings, andtherefore, education has to be completely self-financed.

It can be easily seen from (12) that the threshold level of collateral is decreasingin the ability of individuals. What this implies is that the higher the ability of anindividual the less likely the individual is to be credit constrained. The intuition forthis result is simple. The threshold level of wealth in (12) is the gap between thecost of education and the amount that individuals can borrow against their futureearnings. The cost of education is unrelated to the ability of an individual,however, the amount they can borrow against their future earnings is positivelyrelated to their ability because their future earnings is positively related to theirability. Further, (13) makes it clear that wealthy individuals (those with high a) are

˜less likely to be credit rationed because s is lower for them.Eqs. (13) and (10) together imply that in the presence of credit market

˜imperfections only individuals with s . max s(a), s* invest in human capital;h jothers remain unskilled.

2.3. Distributional dynamics

Now we look at the evolution of wealth distribution for this economy. We beginby considering the long run evolution of lineage wealth for a single lineage in thiseconomy. We show that the probability distribution of lineage wealth converges toa unique stationary distribution. This stationary distribution can be interpreted asthe steady state wealth distribution for the economy since all lineage wealthprocesses are identically and independently distributed, and since there is acontinuum of lineages. To establish the convergence of the probability distributionof lineage wealth to a unique stationary distribution we appeal to results ofconvergence for monotonic Markov processes in Hopenhayn and Prescott (1992).Denoting the bequest function by b(a, s), the evolution of lineage wealth can bewritten as

a 5 b(a , s ). (14)t11 t t


Further, using the superscript u to denote the bequest of an unskilled parent and sto denote the bequest of a skilled parent, b(a, s) can be written as

u ˜b (a) 5 (1 2 b )((1 1 r)a 1 (2 1 r)w ) if s , max s(a), s* (15a)h jl

s ˜b (a, s) 5 (1 2 b )(sw 1 (1 1 r)(a 2 qw )) if s $ max s(a), s* . (15b)h js s

It should be noted from (15a) that the bequest of an unskilled parent does not]depend on the level of ability of the parent. Let a and a be the highest and lowest

]sustainable wealth levels given as follows

](1 2 b )(sw 2 (1 1 r)qw )s s] ]]]]]]]]a 5 (16)1 2 (1 2 b )(1 1 r)

(1 2 b )(2 1 r)wl]]]]]]a 5 (17)

] 1 2 (1 2 b )(1 1 r)

]In order for a and a to be non-negative it is assumed that (1 2 b )(1 1 r) , 1. Since]

an individual with ability s* is indifferent between becoming skilled andremaining unskilled, the labor income of a skilled individual (net of the cost ofeducation) with ability s* is equal to the labor income of an unskilled individual:s*w 2 (1 1 r)qw 5 (2 1 r)w . Therefore, (17) can be written ass s l

(1 2 b )(s*w 2 (1 1 r)qw )s s]]]]]]]]a 5 . (18)

] 1 2 (1 2 b )(1 1 r)

˜ˆ ˆ ˆDefine s as the level of ability such that a*(s ) 5a or alternatively, s(a) 5 s.] ]

Assume the following.

]ˆAssumption 2. s , s* , s ,s]

]s ,s ensures that individuals born with the highest possible ability can alwaysfind ways to invest in human capital no matter how poor they are born. Thiscondition provides intergenerational mobility in the model. If this condition is notsatisfied, then there will be a poverty trap in the model: once an individual in anygeneration is unskilled, all his descendants are going to remain unskilled. Since,due to random ability shocks everyone has a positive probability of becomingunskilled, in the long run all individuals become unskilled, and the steady state

ˆdistribution of wealth is concentrated at a single point a. s* , s ensures that]

borrowing constraint binds for at least some individuals.Now we are ready to show the existence of a unique invariant distribution when

]Assumption 2 is satisfied. Let A 5 a, a and let V denote the set of Borel subsetsf g]of A. Given that s is i.i.d, the stochastic process of lineage wealth described by(14) is a stationary Markov process. The corresponding transition function: P:A 3 V → 0, 1 is simply defined byf g


P(a, B) 5 prob b(a, s) [ B , for all Borel subsets B [ V. (19)h j

The long run dynamic behavior implied by P(., .) is described by determiningthe existence of a unique invariant distribution G. For any wealth distribution G(.),let TG(.) be the Markov transformation of G defined by:

TG(B) 5EP(a, B) dG(a) for all Borel subsets B , A.

A wealth distribution G on A is invariant for P if for all Borel subsets B , A,one gets the following

TG(B) 5 G(B).

Fig. 1 gives an idea of why an invariant distribution G(.) exists for our Markovuprocess of lineage wealth. We have plotted two bequest lines: b (a) for the

s ]unskilled; and b (a, s) for the highest ability skilled. There is going to be acontinuum of bequest lines in between for the skilled corresponding to each abilitylevel ( . s*) of the skilled. It can be seen from Fig. 1 that if the lineage wealth at

]some date t is above a or below a, then in finite time it will come back to the]]interval a, a . Once lineage wealth falls in this interval, it will remain in thisf g]

interval forever. Thus, after a sufficient amount of time has elapsed, all lineages]will find their wealth in the interval a, a . Fig. 1 suggests that wealth lineages willf g]] ]move from any subset of a, a to any other measurable subset of a, a . This isf g f g] ]

Fig. 1. Steady-state wealth distribution.


what gives a unique invariant distribution for lineage wealth. More formally thefollowing can be proved.

Proposition 1. Under Assumption 2 there exists a unique invariant distribution Gfor the Markov process corresponding to P(a, B). As well, for any initial wealth

n ndistribution G , the sequence (T ) (G ) ((T ) is the nth iterate of T ) converges to0 0

G.

The proof is a straightforward application of Hopenhayn and Prescott’s (1992)analysis of existence, uniqueness and convergence properties of monotonicstochastic processes. The proof is contained in Appendix A.

2.4. Pattern of comparative advantage

The model for the small open economy can be solved as follows. Given anexogenous product price ratio, p, factor prices are determined through theStolper–Samuelson relationship between product prices and factor prices. Giventhe factor prices, the cost of education qw is determined. From Proposition 1 wes

know that starting from any initial distribution of wealth the economy willconverge to a unique steady state distribution of wealth. Suppose that the steady

]state distribution of wealth is G(a) defined over the support [a, a]. Recalling that]

˜s 5 s(a) is the level of ability above which an individual is unconstrained, the]

fraction of population investing in human capital in steady state is given below

] ˆa s

˜S 5 1 2EF(max s*, s(a) ) dG(a) 5 1 2 F(s*) 2EG(a*(s))f(s) ds. (20)h ja s *]

The term under the integral on the right hand side in (20) captures the fraction ofpopulation that would like to invest in human capital, but is borrowing con-strained. The amount of skill available in efficiency units is

]s s

]S 5E [1 2 G(a*(s))]sf(s) ds 1Esf(s) ds. (21)

s * s

Once the endowment ratio is known, production of each good and the volume oftrade can be easily calculated given the production functions and the utilityfunction. A complete closed form solution with specific functional forms is givenin Appendix A.

Next we perform comparative statics with respect to p, the degree of creditmarket imperfection. It is easy to see what happens if p 5 0. As discussed earlier,this corresponds to the first-best case, where borrowing constraint is not bindingfor any individual with ability greater than s*. Therefore, if p 5 0, then S 5 1 2


]] s ˆF(s*) and S 5 e sf(s) ds. The condition s* , s mentioned in Assumption 2s *

earlier ensures that the borrowing constraint binds for at least some individualswhen p . 0.

In Fig. 1 we draw a downward sloping curve to show the extent of borrowingconstraint. The height of this curve is equal to (1 2 b )(1 1 r)(a 2 qw ) 1 (1 2s

˜ ˜b )s(a)w , where s(a) (derived in Eq. (13)) is the threshold level of ability aboves

which the borrowing constraint does not bind. The vertical gap between this curveu sand the bequest line corresponding to b (a) ( 5 b (a, s*)), given by (1 2

˜b )w (s(a) 2 s*), can be understood as the extent of borrowing constraint for eachs

˜level of wealth. If s(a) . s* for a particular a, it implies that the probability of˜ ˜this individual being credit constrained is F(s(a)) 2 F(s*). If s(a) , s*, in-

dividuals with that level of wealth are unconstrained. The higher the wealth thelower the vertical gap between the two curves, and hence the lower the probabilityof being borrowing constrained. In Fig. 1 individuals with a . a9 are uncon-

˜strained. The higher the p the higher the ability threshold, s(a), for each level ofwealth, and thus higher the negatively sloped curve. Thus a higher p would implya larger gap between the negatively sloped curve and the bequest line for theunskilled in Fig. 1, implying a larger probability of being borrowing constrained.Therefore, each lineage has lower wealth for a longer time. The following Lemmais proved in Appendix A.

Lemma 1. If p9 . p, then the steady state distribution of wealth under p

dominates the steady state distribution of wealth under p9 in the first orderstochastic dominance sense: G (a) $ G (a) for all a.p 9 p

We can see the impact of a greater degree of credit market imperfections on theˆinvestments in human capital from Eq. (20). The level of talent, s, above which

ˆcredit constraint is non binding is increasing in p (≠s /≠p . 0). Further, ≠a*(s) /≠p . 0 from (12). These two combined with the result in Lemma 1 imply that

]dS /dp , 0 and dS /dp , 0. This gives us the result summarized in Lemma 2below.

Lemma 2. If p9 . p, then the economy with lower degree of credit marketimperfections has greater skill endowment in the steady state.

Lemma 1 and Lemma 2 imply the following proposition which is proved inAppendix A.

Proposition 2. Ceteris paribus, if the degree of credit-market imperfections in aneconomy is less than that in the average economy, then under the condition ofbalanced trade the former exports the skill intensive good.

This result shows the link between an institutional variable like the degree of


credit market imperfections and the pattern of trade based on skill endowmentdifferences.

It is straightforward to show that if the distribution of talent in an economydominates that in another economy in a first order stochastic sense, then the formerhas greater fraction of population investing in human capital and greater skillendowment in efficiency units. Therefore, differences in the distribution of abilitycan become a source of comparative advantage as well.

3. Impact of trade liberalization

The impact of trade liberalization for a small open economy will work throughchanges in the relative price of the high-tech good, p. For an economy exportingskill intensive good, trade liberalization means an increase in p, which raises theskilled wage w and reduces the unskilled wage w through the standard Stolper–s l

Samuelson effect. We recall from (20) that the fraction of population investing inhuman capital is

s

S 5 1 2 F(s*) 2EG(a*(s))f(s) ds

s *

ˆwhere s is the level of talent above which the borrowing constraint does not bind.ˆTherefore, the impact of trade liberalization depends on how s*, s, a*(s) and

G(a) are affected. Let us use the subscript A to denote the value of a variablebefore liberalization and T to denote its value after liberalization. From (16) and(17) we see that the support of the steady state distribution of wealth widens from

] ][a , a ] to [a , a ]. The impact of trade liberalization on the investment in humanA A T T] ]capital depends on, among other things, how the steady-state distribution of wealthchanges from G (a) to G (a). The change in the investment in human capital uponA T

trade-liberalization is given by

ˆ ˆs sT A

* * * *S 2 S 5F(s ) 2 F(s ) 2 EG (a (s))f(s) ds 2EG (a (s))f(s) dsT A A T T T A A#%%%"!%%%$ 3 4

s sI * *T A#%%%%%%%%%%"!%%%%%%%%%%$

II

(22)

The first term on the r.h.s. in (22) is the change in the fraction of population thatwould like to invest, while the second term is the change in the fraction ofpopulation that can afford to invest. The magnitude of the second term depends onhow the borrowing constraint and the steady-state distribution of wealth change,which in turn depend crucially on the degree of credit market imperfections.

It can be easily seen from (10) that s* is decreasing in w and increasing in w .s l


If the reward from acquiring skill (w 2 (1 1 r)qw ) rises, and the opportunity costs s

((2 1 r)w ) falls, people with lower ability will find it worthwhile to invest inl

* * * *education. Therefore, s 2 s . 0, and hence F(s ) 2 F(s ) . 0. This capturesA T A T

the positive rate of return effect on investment in human capital. If p 5 0, then thisis the only effect of trade liberalization on investment in human capital in a skillabundant country. When p . 0, the second term in (22) is non-zero and isdiscussed below.

* *From Eq. (12) we note that a (s) . a (s). The intuition is as follows. AnT A

increase in the skilled wage increases the cost of education for all individualsidentically. However, the increase in the future earnings potential of the skilleddepends on the amount of talent they possess. From (12) it is clear that at lowlevels of talent, the increase in the future earnings potential is less than theincrease in the cost of education, giving rise to an increase in the threshold level ofwealth. Therefore, the borrowing constraint becomes tighter for each level oftalent. The impact of this on the investment in human capital is qualitativelysimilar to the impact of a higher degree of credit market imperfections discussed inLemmas 1 and 2. This effect by itself will reduce the investment in human capitaland worsen the distribution of income in the first-order stochastic sense. Thiseffect is qualitatively similar to the effects in Cartiglia (1997) and Eicher (1999)arising because the cost of education depends on the skilled wage. It should benoted that the magnitude of this effect depends, among other things, on the degree

* *of credit market imperfections: ≠(a (s) 2 a (s)) /≠p . 0. The greater the degreeT A

of credit market imperfections the greater the increase in borrowing constraints.Finally, as was mentioned earlier, the support of the steady state distribution of

] ]wealth widens from [a , a ] to [a , a ]. This change can be viewed as a shift inA A T T] ] ] ] ]the probability mass from the center ([a , a ]) to the tails ([a , a ] and [a , a ]) ofA A T A A T] ] ]the wealth distribution. It is easy to see that all individuals in the interval [a , a ]T A] ]]are more credit constrained than anyone in the interval [a , a ] due to lower wealthA A]as well as increased cost of education. Therefore, investment in human capital islikely to fall because of this distributional change. What happens is that a declinein the unskilled wage reduces the bequest of the unskilled, which makes their

]descendants more prone to borrowing constraint. The individuals in the interval [a,]a ] have greater wealth, but face a higher cost of education. In general, the impactT

of distributional change depends on how the entire distribution of wealth changes.The net impact on the investment in human capital depends on the relative

strengths of these effects. If p is very small, then the rate of return effect is goingto dominate the distributional effect (in the limit when p 5 0 the distributionaleffect vanishes), and hence the investment in human capital is going to increase.As seen earlier, the worsening of borrowing constraint is positively related with p,therefore, the smaller the p the smaller the negative effect coming from theworsening of borrowing constraint and distributional change. Thus, for a small p

the net impact of trade liberalization on the investment in human capital for askill-abundant country is likely to be positive.


The opposite happens in a country having a comparative advantage in theunskilled intensive good. In this case the return on investment in human capitaldeclines and, therefore, the first term in (22) is negative. However, there is adecrease in the fraction of population that is borrowing constrained because of tworeasons: the borrowing constraint declines for each level of talent due a decrease inthe skilled wage, and there is a positive distributional effect coming from anincrease in the unskilled wage that enables the unskilled to leave a higher bequest,which allows their descendants to overcome the borrowing constraint. The neteffect again depends on the relative strengths of these effects. The larger thedegree of credit market imperfections, the greater the decrease in the fraction ofpopulation that is borrowing constrained, and hence the net effect is more likely tobe positive.

If the incentive effect dominates in the skill-abundant countries, while thepositive effect arising from distributional changes dominates in the skill-scarcecountries, then a trade liberalization will increase skill endowments in all tradingpartners. This outcome is more likely if the degree of credit market imperfectionsis very small in the skill-abundant country and very large in the skill-scarcecountry. In the general case, it is difficult to give precise parametric restrictionsunder which there is an increase in investment in human capital in both skill-abundant and skill-scarce countries because the investment in human capitaldepends on the entire shape of the distribution of wealth. However, below we givea numerical example to illustrate the possibility of trade liberalization increasinginvestment in human capital in both the skill-scarce and the skill-abundantcountries, and then we provide some empirical evidence to support our contentionthat the degree of credit market imperfections is very high in the South and low inthe North.

3.1. A numerical example

Below we construct a tractable example by taking a simple distribution of talent,which under some restrictions on parameters, yields multiple non-overlappingintervals as the support of the steady-state distribution of wealth. Most importantlyfor our purposes here, the transition probabilities between these intervals and theprobability of investing in human capital do not depend on the exact wealth of theindividual, but only on the interval in which the wealth of the individual belongs.The key parameters in the numerical example take the following values. Theability or talent shock takes three values: s 5 2, s 5 1.5, and s 5 1, each withh m l

probability 1 /3. (1 2 b ) 5 0.2, (1 1 r) 5 1.1, q 5 0.5. The other details of the7numerical example are gathered in an Appendix.

Table 1 summarizes the results of the numerical exercise. In each case the

7This appendix is not being published to conserve space. It is available from the author’s website athttp: / /orion.oac.uci.edu/ |pranjan / research.html


Table 1aResults of numerical simulation

Fraction of population Fraction of population Fraction of populationwanting to invest borrowing constrained actually investing

South before 1 4/9 5/9liberalizationSouth after 2 /3 1 /15 3/5liberalizationNorth before 2 /3 0 2/3liberalizationNorth after 1 1/5 4/5liberalization

a Fraction of population actually investing5fraction of population wanting to invest2fraction ofpopulation borrowing constrained.

fraction of population actually investing is the difference between the fraction ofpopulation that wants to invest and the fraction that is borrowing constrained.

3.1.1. Case of a Southern countryThe parameters are such that before liberalization everyone wants to invest in

human capital. However, due to a high degree of credit market imperfections(p 5 0.78) not every one with talent s or s can invest in human capital. There isl m

a level of wealth a(s ) such that only those with wealth a . a(s ) can invest inm m

human capital upon receiving a shock of s . a(s ) is similarly defined for abilitym l

s . The steady-state distribution of wealth is defined over three intervals I 5 [a,l l ]]a ], I 5 [a , a ], and I 5 [a(s ), a]. The dynamics of wealth distribution arel1 m m1 h1 h l

depicted in Fig. 2 which captures the fact that those with wealth in I can investl

only if they get a shock of s , those in I can invest only after receiving a shock ofh m

s or s , and those in I always invest. Therefore, some individuals with wealth inm h h

the intervals I and I are borrowing constrained. As shown in Table 1 the fractionl m

of population investing in human capital is 5 /9 in this case.Now when this economy opens up to trade, due to its comparative advantage in

the unskilled intensive good, trade liberalization raises unskilled wage and reducesskilled wage. The dynamics of wealth distribution after liberalization are depictedin Fig. 3. The steady-state-distribution of wealth is given over the following three

]intervals now: I 5 [a , a(s ) ), I 5 [a(s ) , a ], I 5 [a , a ]. Now individualsl T m T m m T l2 h m2 T]with ability s do not want to invest because the return on education decreases.l

This will reduce the fraction of population investing in human capital. However, adecrease in the skilled wage, combined with an increase in the unskilled wagemakes it possible for some individuals, who were unable to do so earlier, to investin human capital after receiving a shock of s . This reduces the fraction ofm

population that is borrowing constrained, and hence increases investment in humancapital. In Fig. 3 the only people who are borrowing constrained are those in theinterval I receiving an ability realization of s . The net effect of tradel m


Fig. 2. South: (pre-liberalization).

liberalization, which is a sum of these two effects, is to increase the fraction ofpopulation investing in human capital from 5/9 to 3/5. Numerical exercise alsoconfirms that the larger the degree of credit market imperfections the larger the

Fig. 3. South: (post-liberalization).


fraction of population that is borrowing constrained, and hence the larger thedecrease in the fraction of population that is borrowing constrained upon tradeliberalization. Thus, the net effect of trade liberalization on human capitalinvestment is more likely to be positive in a Southern economy the larger itsdegree of credit market imperfections.

3.1.2. Case of a Northern countrySuppose in the North parameters are such that before liberalization only those

receiving talent realizations of s or s want to invest. Also, the borrowingm h

constraint is non-binding for all individuals before liberalization (p 5 0.67 hasbeen assumed for North in the numerical example). In this case the fraction ofpopulation investing in human capital before liberalization is 2 /3.

The dynamics of wealth distribution after liberalization in the North are depictedin Fig. 4. The steady-state distribution of wealth in this case is defined over the

]intervals: I 5 [a , a(s ) ), I 5 [a(s ) , a(s ) ), and I 5 [a(s ) , a ]. Fig. 4l T m T m m T l T h l T T]captures the fact that after trade liberalization the rate of return on human capitalincreases in such a way that everyone would like to invest in human capital.However, those with wealth in the interval I cannot afford to invest if their abilityl

realization is s or s , while those in the interval I cannot afford to invest if theirm l m

ability realization is s . This happens because due to a decline in the unskilledl

wage some children of unskilled parents find their bequest inadequate to investwhen their ability realization is low. This should be contrasted with the fact thatbefore liberalization no one with ability s was borrowing constrained (whilem

Fig. 4. North: (post-liberalization).


those with ability s did not want to invest before liberalization). Therefore, tradel

liberalization again produces two opposing effects. The positive rate of returneffect has to be balanced against the negative effect coming from the increase inthe fraction of population that is borrowing constrained. As shown in Table 1 thenet effect for the chosen parametric configuration is positive: trade liberalizationincreases the fraction of population investing in human capital from 2/3 to 4/5.The numerical exercise also confirms that the negative effect arising from anincrease in the fraction of population that is borrowing constrained is larger thelarger the degree of credit market imperfections.

The proposition below summarizes the result on the impact of trade liberaliza-tion on credit-constrained investments in human capital when the distribution ofwealth evolves endogenously.

Proposition 3. The net effect of trade liberalization on investments in humancapital depends on the relative strengths of the opposing effects coming fromchanges in the incentives to invest, and changes in the borrowing constraint andthe distribution of wealth. If the degree of credit market imperfections is very lowin the skill-abundant countries and very high in the skill-scarce countries, thentrade liberalization is likely to increase the investment in human capital in both.

The result above contrasts with the earlier results predicting divergence as inFindlay and Kierzkowski (1983) and Grossman and Helpman (1991) or aconvergence as in Cartiglia (1997) and Eicher (1999). Also, in the last two papersit was shown that trade liberalization led to a decrease in the investment in humancapital in the North. In contrast, our results imply that it is possible for theinvestment in human capital to increase in both the North and the South upon tradeliberalization.

3.2. Some empirical evidence in support of the results

The degree of credit market imperfections in our theoretical model was capturedby the probability of successful default. Empirically, this is going to depend onhow well the contracts are enforced in a country. There are some data on thedegree of contract enforcement collected by ICRG (see Knack and Keefer (1995)for details). We use one of their measures: Rule of Law (RLW). This indicator runsfrom 0 to 6. This variable reflects the degree to which the citizens of a country arewilling to accept the established institutions to make and implement laws andadjudicate disputes. Therefore, it should be highly correlated with the degree ofcredit market imperfections. We find that this variable has a high positivecorrelation of 0.64 with the human capital endowments measured by the averageyears of secondary schooling of adult population. Also, for countries with humancapital endowment in the bottom quartile (less than 0.44 years of averagesecondary schooling) the average value of RLW was 2.28, while for countries in


8the top quartile (greater than 1.72 years) the value was 4.7. The differencebetween the two groups is almost one and half times the standard deviation for theentire sample. These numbers suggest that the degree of credit market imperfec-tions is very high in countries with low skill endowment and very low in countrieswith high skill endowment, which is consistent with the conclusion in Proposition3.

Our theoretical results depend on the link between wealth inequality andinvestment in human capital due to credit constraints, and the impact of trade onwealth inequality coming through the Stolper–Samuelson effect on wages.Williamson (1993) using data for 35 countries from 1960 to 1980 finds thatsecondary enrolment ratio is negatively correlated with inequality in the dis-tribution of income (ratio of share of top 20% to bottom 40%) after controlling for

9other variables. Our companion paper Ranjan (1997), using cross-country data onsecondary and tertiary enrolment ratio and income inequality data from Deiningerand Squire (1996) finds that greater inequality as measured by either Ginicoefficient or fractile ratio (ratio of share of top 20% to bottom 20%) issignificantly negatively related with investment in human capital after controllingfor other variables like per capita income, regional dummies etc. Finally, Ranjan(1997) using several measures of openness finds positive correlation betweenopenness and investments in human capital for both rich and poor countries aftercontrolling for other determinants of investment in human capital, which isbroadly consistent with the theoretical possibility shown in the numerical examplethat trade can increase investment in human capital in both trading partners.

4. Concluding remarks

This paper constructs a dynamic general equilibrium model where the pattern ofcomparative advantage depends on the degree of credit market imperfectionsaffecting human capital investments. Endogenizing wealth distribution by allowingfor intergenerational transfers, the paper identifies a novel channel of influence oftrade liberalization on investment in human capital. Changes in the distribution ofwealth brought about by trade liberalization have additional and, in some casesoffsetting effects on the accumulation of human capital. This makes it possible fortrade liberalization to increase investments in human capital in both skill-abundant

8The data on average years of secondary schooling are from Barro and Lee (1994). The data onRLW are averages for the period 1982–1985. Similar results were obtained using other indicatorscollected by ICRG like Repudiation of Contracts by Governments, Risk of Expropriation etc. Also,similar results were obtained using data on average years of tertiary education rather than secondaryeducation.

9Lloyd-Ellis (2000) finds a statistically significant negative impact of income inequality measured byGini coefficient on secondary enrolment ratio in a sample of 35 countries.


and skill-scarce countries. Therefore, trade liberalization could potentially yielddynamic gains to all trading partners.

Before ending the paper we discuss the reasons for introducing several specialelements in the model and their implications. The reason for introducingheterogeneous ability is to allow for two-way (both upward and downward)intergenerational mobility, which results in a unique invariant steady statedistribution of income. In the absence of heterogeneous ability, there will be nointergenerational mobility, and the steady state distribution of income will bedetermined by the initial distribution of income as in Galor and Zeira (1993).However, heterogeneous ability alone is not sufficient to generate a unique steadystate distribution. It is the partially open credit market along with heterogeneousability that provides two-way intergenerational mobility. Partially open creditmarket allows high ability individuals to invest in human capital even if they areborn poor. If the credit market is completely missing, then heterogeneous abilitywill result only in downward mobility and no upward mobility. In this case insteady state everyone will become unskilled. Therefore, both heterogeneous abilityand partially open credit market are essential features of the model. Also, theassumption that the ability of each individual is publicly known is a simplifyingone, given our story of imperfection in the credit market. Without this assumptionthe contract in the credit market is much more complicated. However, modellingcredit market imperfections in an alternative way a la Galor and Zeira (1993) willobviate the need for this assumption without changing any of the qualitative resultsin the paper. The reason for choosing our story of credit market imperfections isanalytical tractability of the distributional dynamics. In our story, the bequest leftby a skilled person does not depend directly on the degree of credit marketimperfections, p, but using Galor and Zeira’s (1993) story will mean that forborrowers (a , qw ) the bequest depends on the borrowing rate of interest i. Fors

lenders the relevant rate of interest will continue to be r. Therefore, even amongthe educated, bequest functions differ depending on whether they are borrowers orlenders making the distributional dynamics more complicated.

The assumption that skill is used in the formation of skill is also not crucial forgetting the new insights about trade affecting investment in human capital throughchanges in the distribution of wealth. Even if the direct cost of education is just afixed amount of numeraire good, the changes in the steady state distribution ofwealth will produce similar effects. Also, we can completely get rid of the directcost of education, and have just the opportunity cost which is foregone wages. Ifwe allow consumption in the first period, similar results will appear. Nowborrowing constraint will affect consumption possibilities of those born poor whowould like to invest in education. If they cannot borrow for consumption in thefirst period, high marginal utility of first period consumption may not justifyinvestments in human capital. This formulation of the model will make itapplicable to the cases where there are no direct costs of education, such asprimary education in most countries or public education more generally.


One shortcoming of the paper is that the model developed here is one of a smallopen economy rather than a two country case. The main reason for doing this isanalytical tractability. The small open economy assumption allows us to solve themodel for an exogenous product price ratio and rate of interest. This makes thedistributional dynamics a simple linear Markov process. If the rate of interest andproduct prices are endogenous, the wealth dynamics will become non-linearmaking it extremely difficult to study the steady-state wealth distribution. Futureresearch will try to address this issue.

Acknowledgements

I would like to thank an anonymous referee and a co-editor for helpfulcomments. I would also like to thank J. Bhagwati, R. Findlay, A. Newman, andseminar participants at Rice, Toronto, UC-Berkeley, UC-Irvine, UCLA, andUC-Riverside for helpful suggestions and discussions on earlier versions of thispaper. All remaining errors are mine.

Appendix A

Proof of Proposition 1. The proof proceeds in several steps. We first establish themonotonicity and monotone mixing property of the transition function and thenuse results derived in Hopenhayn and Prescott (1992) to prove the existence,uniqueness and convergence to an invariant distribution.

Lemma A.1. (Monotonicity of P): the transition function P(a, B) is increasing inits first argument a in the following ( first order stochastic dominance) sense: for

2all (a, a9) [ A , a # a9 ⇒ ;x [ A, P(a9, a, x ) # P(a, a, x ).f g f g] ]

uProof. Denote the bequest of an unskilled parent by b (a) and of a skilled parentsby b (a, s).

sDefinition. For each a [ A and x [ A define s0(a) as follows: b (a, s0(a)) 5 x.

s s uClearly, if s . s0(a), then b (a, s) . x. Also, b (a, s) . b (a) ;s $ s*, fromsthe incentive compatibility for investing in skill. Also, since b (a, s) is increasing

in a, if a9 $ a, then s0(a9) # s0(a). Further define I as an indicator function asfollows


uI 5 1 if b (a) , xub (a),x

5 0 otherwise˜I 5 1 if s0(a) . max s(a), s*h j˜s 0(a).max s(a), s *h j

5 0 otherwise2For all (x, a) [ A ,

s 0(a)

˜P(a, [a, x]) 5 F max s(a), s* I 1 I E f(s) dsuh h jj ˜b (a),x s 0(a).max s(a), s *h j]˜max s(a), s *h j

(A.1)

(A.1) can also be written as

˜P(a, [a, x]) 5 F max s(a), s* I 1uh h jj b (a),x] (A.2)˜F(s0(a)) 2 F max s(a), s* If h h jjg ˜s 0(a).max s(a), s *h j

We can write a similar expression for P(a9, [a, x]). Furthermore, if I 5 0,ub (a9),x]s uthen I 5 0 because b (a9, s) . b (a9) in the relevant range.˜s 0(a9).max s(a9), s *h jTherefore, if I 5 0, then P(a9, [a, x]) 5 0, hence P(a9, [a, x]) # P(a, [a, x])ub (a9),x ] ] ]is trivially true. Thus, we only need to consider cases corresponding to I 5ub (a9),x

1.

Case I. I 5 1 and I 5 1. In this case P(a9, [a, x]) 5u ˜b (a9),x s 0(a9).max s(a9), s *h j ]F(s0(a9)). Since I $ I , therefore, I 5 1. Further, s0(a) $u u ub (a),x b (a9),x b (a),x

˜s0(a9) . max s(a9), s* . If I 5 1, then P(a, [a, x]) 5 F(s0(a)) $h j ˜s 0(a).max s(a), s *h j ]˜F(s0(a9)) 5 P(a9, [a, x]). Otherwise, P(a, [a, x]) 5 F max s(a), s* $ F(s0(a)) $h h jj] ]

F(s0(a9)) 5 P(a9, [a, x]).]

Case II. I 5 1 and I 5 0. In this case P(a9, [a, x]) 5u ˜b (a9),x s 0(a9).max s(a9), s *h j ]˜F(max s(a9), s* ). Again, I 5 1. Now, if I 5 1, then P(a, [a,uh j ˜b (a),x s 0(a).max s(a), s *h j ]

˜ ˜ ˜ ˜x]) 5 F(s0(a)) . F max s(a), s* . Since s(a) . s(a9), we get F max s(a), s* $h h jj h h jj˜F max s(a9), s* , hence P(a, [a, x]) . P(a9, [a, x]). If I 5 0, thenh h jj ˜s 0(a).max s(a), s *h j] ]

˜ ˜P(a, [a, x]) 5 F max s(a), s* $ F max s(a9), s* 5 P(a9, [a, x]). hh h jj h h jj] ]

Lemma A.2. (Monotone mixing condition): the monotonic transition function Psatisfies the following property:

]˜for an a [ (a, a) there exists an integer m such that:]

m m] ]˜ ˜P (a, [a, a]) . 0 and P (a, [a, a ]) . 0;] ]

mP (a, B) above denotes the probability of reaching B from a after mgenerations. What the condition above implies is that even the poorest individual

˜will have his lineage wealth end up above a after m consecutive high ability


descendants; similarly, even the richest individual will have his lineage wealth end˜up below a after m consecutive low ability descendants or borrowing constrained

descendants. The fact that this condition is satisfied should be obvious from Fig. 1.The formal proof is given below.

] ]˜Proof. Take any a [ (a, a). Then since (b(a, s) 2 a) is always strictly positive and]

˜continuous on [a, a ], it remains uniformly bounded below by some positive] n ] ˜number w. Thus, there exists an integer n such that: n $ n ⇔b (a, s) . a, where1 1 ]n ] ]b (., s) denotes the nth iterate of b(., s). n is less than or equal to the smallest1

˜integer n such that n.w . a.]˜Similarly, since [a 2 b(a, s)] is strictly positive and continuous on [a, a], it

]remains bounded below by some positive number c. Thus, there exists an integer

n n] ˜n such that: n $ n ⇔b (a, s) , a, where b (., s) denotes the nth iterate of b(.,0 0 ] ] ] ˜s). n is less than or equal to the smallest integer n such that a 2 n.w , a.0]Let m 5 max(n , n ). Recalling that f(s) denotes the density function of s, we0 1

have

m m] ]˜P (a, [a, a]) $ ( f(s)) . 0; and] hm m] ˜P (a, [a, a ]) $ ( f(s)) . 0.

] ]

A.1. Existence of invariant distribution

The existence of an invariant distribution G for the Markov process defined byP(a, B) follows from the monotonicity of P established in Lemma A.1 and fromCorollary 4 in Hopenhayn and Prescott (1992).

A.2. Uniqueness and convergence

This follows from the monotonicity of P established in Lemma A.1, itsmonotone mixing property established in Lemma A.2 and Theorem 2 inHopenhayn and Prescott (1992).

A.3. Closed form solution for specific functional forms

A.3.1. The modelProduction functions for the two goods are as follows:

h g 12gH 5 A S U (A.3)h h

l f 12fL 5 A S U . (A.4)l l

Assume g . f to make H relatively more skill intensive at all factor prices.Perfect mobility of factors within sectors implies the following:


≠H ≠L] ]p 5 5 w (A.5)s≠S ≠S

≠H ≠L] ]p 5 5 w (A.6)l≠U ≠U

(A.5) and (A.6) can be solved for s 5 S /U and s 5 S /U to get the following:h h h l l l

l 1 /g 2f f /g 2f f 21 /g 2fA f 1 2 g 21 /g 2f] ] ]]s 5 p (A.7)S D S D S Dh h g 1 2 fAl 1 /g 2f g /g 2f g 21 /g 2fA f 1 2 g 21 /g 2f] ] ]]s 5 p . (A.8)S D S D S Dl h g 1 2 fA

Thus, given the relative price ratio, p, factor intensities in the two sectors aredetermined by (A.7) and (A.8). (A.7) and (A.8) in turn determine the absolutereturns to the two factors of production:

g(f 21) /g 2ffh 12f /g 2f l g 21 /g 2f ]w 5 (A ) (A ) fS Ds g

((g 21)(f 21)) /g 2f1 2 g 12f /g 2f]]3 p (A.9)S D1 2 f

gf /g 2f ((g 21)f ) /g 2ff 1 2 gh 2f /g 2f l g /g 2f 2f /g 2f] ]]w 5 (A ) (A ) (1 2 f) pS D S Dl g 1 2 f

(A.10)

Therefore, we see that w is positively related and w negatively related with thes lhrelative price of the high-tech good. As well, the relationship between A and the

two wages is exactly the same as between the relative price p and the two wages.Therefore, the impact of a skill-biased technical progress captured by an increase

hin A on wages is the same as the impact of an increase in the relative price of thehigh-tech good.

For a given p there is a unique G(a) and the corresponding fraction of]

population investing in skill, S, and skill endowment in efficiency unit, S are givenby (20) and (21), respectively. From the education sector production function theamount of skill used in the education sector is

1]S 5 S. (A.11)E Q

]Therefore, the amount of skill involved in direct production is S 5S 2 S . TheP E

factor endowment ratio relevant for goods production is given by S /2U, whereP

U 5 1 2 S is the fraction of population remaining unskilled in each generation.Therefore, given the relative product price, p, the relative factor supplies aredetermined.


Since the factor supplies are known, domestic production of H and L can bedetermined from the following relationship

SP]ls 1 (1 2 l)s 5 5 s. (A.12)h l 2U

In (A.12) l 5 U /2U is the fraction of unskilled labor used in the high-tech sector.h

The condition required for obtaining a diversified equilibrium is s [ (s , s ). If thel h

world price is such that this condition is violated then the economy will specializein one of the two goods.

(A.12) implies that l 5 (s 2 s ) /(s 2 s ). This determines U and U . Since U ,l h l h l h

U , s , and s are known, production of H and L are determined from thel h l

production functions in (A.3) and (A.4).What is left to determine is the consumption of H and L. The utility function of

consumers in the second period of their lives is

u 12uC 5 C C . (A.13)h l

ˆDenote the average level of steady state wealth by a. Denoting the aggregateincome of each generation in the second period of their lives by Y9, we get thefollowing:

]ˆY9 5 (1 1 r)a 1 (2 1 r)(1 2 S)w 1Sw 2 (1 1 r)Sqw . (A.14)l s s

If the aggregate income is Y9, then bY9 is spent on consumption and (12b )Y9 isspent on bequests. Let us define Y 5 bY9. (A.13) implies the following for thelevels of consumption of the two goods

uY]H 5OC 5 ; L 5OC 5 (1 2u )Y. (A.15)C h C lp

Therefore, we have determined the domestic availability and domestic consump-tion of both goods. The gap between domestic availability and domestic consump-tion is met through international trade. The material balance conditions for the twogoods are as follows

L 1 M 5 L (A.16)P C

H 2 X 5 H . (A.17)P C

In (A.16) if M . 0, then it implies that the numeraire good is imported, whileM , 0 implies that the numeraire good is exported. X in (A.17) has a similarinterpretation for the high-tech good.

It can be easily checked that (A.16) implies (A.17) so that if the market forlow-tech good clears, the market for high-tech good clears as well.

Substituting for L in (A.16) using (A.15) we getC

L 5 Y 2uY 2 M. (A.18)P


Denote the net borrowing or lending of each generation by I where

ˆI 5 a 1 (1 2 S)w 2 Sqw . (A.19)l s

Using (A.19) above, (A.14) can be rewritten as]

Y9 5 Uw 1Sw 1 (1 1 r)I. (A.20)l s

Further, from the constant returns to scale production function we get thefollowing equality between the value of output and factor payments

]L 1 pH 5 2Uw 1 w (S 2 S ). (A.21)P P l s E

Now substitute for L from (A.18) in (A.21) to getP

]pH 5 2Uw 1 w (S 2 S ) 2 Y 1uY 1 M. (A.22)P l s E

ˆSubstitute for Y in (A.22) from (A.20) using the fact that Y 5 bY9 and a 5 (1 2

b )Y9 and that the total cost of education is Sqw 5 w S , to get the followings s E

pH 5 pH 1 M 2 rI. (A.23)P C

Using the balance of payment condition pX 1 rI 5 M (A.23) can be written as(A.17). In (A.23) I . 0 implies that each generation is a lender in the capitalmarket. However, in steady state in every period there will be a net inflow ofcapital because the repayment inclusive of interest payment is more than thelending. Thus, the economy will run a trade deficit (M 2 pX . 0). The oppositehappens for an economy with I , 0.

Proof of Lemma 1. Let T and T be the Markov operators associated with thep p 9

degrees of credit market imperfections p and p9, respectively. Similarly, denotethe unique invariant distributions associated with the degrees of credit marketimperfections p and p9 by G (a) and G (a), respectively. Using the transitionp p 9

function defined earlier we can write the following]a

T G (x) 2 T G (x) 5E P (a, [a, x]) 2 P (a, [a, x]) dG (a). (A.24)f gp 9 p 9 p p 9 p 9 p p 9] ]a]

Next we show that P (a, [a, x]) $ P (a, [a, x]) ;a. Again, as in the proof ofp 9 p] ] sLemma A.1 define s0(a) as the level of talent such that b (a, s0(a)) 5 x. Using thetransition function defined in (A.2) above and subscripting the variables whichdepend on the degree of credit-market imperfection, p, we can write the following.

˜P (a, [a, x]) 5 F max s (a), s* I 1uh h jjp p b (a),x] (A.25)˜F(s0(a)) 2 F max s (a), s* If h h jjg ˜p s 0(a).max s (a), s *h jp

P (a, [a, x]) can be defined similarly. It should be noted that I and s0(a) doup 9 b (a),x]not depend on p because the amount of bequest does not depend on p directly.


˜ ˜Further, it can be easily seen that s (a) . s (a). As well, as seen in the proof ofp 9 p

Lemma A.1 if I 5 0 then I 5 0 and I 5 0.u ˜ ˜b (a),x s 0(a).max s (a), s * s 0(a).max s (a), s *h j h jp 9 p

Therefore, we only need to consider cases corresponding to I 5 1.ub (a),x

Case I. I 5 0 and I 5 0. In this case P (a, [a,˜ ˜s 0(a).max s (a), s * s 0(a).max s (a), s * p 9h j h jp 9 p ]˜ ˜x]) 2 P (a, [a, x]) 5 F max s (a), s* 2 F max s (a), s* $ 0.h h jj h h jjp p 9 p]

Case II. I 5 1 and I 5 1. In this case P (a, [a,˜ ˜s 0(a).max s (a), s * s 0(a).max s (a), s * p 9h j h jp 9 p ]x]) 2 P (a, [a, x]) 5 0.p ]

Case III. I 5 1 and I 5 0. In this case P (a, [a,˜ ˜s 0(a).max s (a), s * s 0(a).max s (a), s * p 9h j h jp 9 p ]˜x]) 2 P (a, [a, x]) 5 F(s0(a)) 2 F max s (a), s* $ 0 because s0(a) .h h jjp p]

˜ ˜max s (a), s* $ F max s (a), s* .h j h h jjp 9 p

Case IV. I 5 0 and I 5 1. In this case P (a, [a,˜ ˜s 0(a).max s (a), s * s 0(a).max s (a), s * p 9h j h jp 9 p ]˜x]) 2 P (a, [a, x]) 5 F max s (a), s* 2 F(s0(a)) $ 0 because s0(a) ,h h jjp p 9]

˜max s (a), s* by definition.h jp 9

Therefore, P (a, [a, x]) 2 P (a, [a, x]) $ 0 ;a.p 9 p] ]Thus, we get T G (x) 2 T G (x) $ 0 or T G (x) 5 G (x) $ T G (x). Sincep 9 p 9 p p 9 p 9 p 9 p 9 p p 9

the Markov operator T is increasing (implied by the monotonicity of thep

transition function P(a, B), we also get T G (x) $ T (T G (x)). It follows thatp p 9 p p p 9n nG (x) $ (T ) G (x), where (T ) is the nth iterate of T . We know fromp 9 p p 9 p p

nProposition 1 that for n → ` (T ) G (x) converges to G (x). Therefore, we havep p 9 p

proved that G (x) $ G (x).p 9 p

Proof of Proposition 2. The proof uses closed form solution for the examplegiven above as well as Lemma 1 proved above and Lemma 2 in the text.

Suppose the world is populated by a continuum of small open economiesindexed by n. The only difference between these economies lies in the degree ofcredit market imperfections p, giving rise to a different steady state distribution ofwealth, denoted by G (a) for the nth economy. Now suppose the world relativen

price of the high-tech good settles at p, which is taken as given by all economies.Given this world price, all variables for a typical small open economy aredetermined as explained in the example above. Given the identical and homotheticpreferences, all economies consume the two goods in the ratio H /L 5u /(1 2C C

u )p. The production and availability ratios differ across economies depending onˆtheir endogenous skill endowment. Let us call an economy n the average economy

in the following sense. Its distribution of wealth is denoted by G (a) and then]corresponding skill endowment is S . Given the above skill endowment, then

production levels of the two goods H and L can be derived from Eq. (A.12) inp P

the example. The production levels for the average economy are such that the


availability ratio for the two goods coincides with the consumption ratio. Since inˆ ˆ ˆsteady state a 5 a 5 a, the production ratio is the same as the consumptiont t11

ratio, i.e.

nH H uP C] ] ]]]5 5 . (A.26)n L (1 2u )pL CP

Denote the export of high-tech good by X and the import of low-tech good by M.(A.26) implies that the average economy does not export or import anything

ˆ ˆn n(X 5 M 5 0). From the balance of payment condition this further implies that theaverage economy does not borrow or lend in the international capital market.

]Now, suppose there is another economy n with smaller degree of credit market]ˆimperfections than n. From Lemma 2 it is clear that n will have greater skill

] ]]ˆendowment than n : S .S . Using the familiar Rybczynski theorem one can easilyˆn n

] ]ˆ ˆn n n nsee that H . H and L , L . Therefore,P P P P

] ˆn nH H uP P] ] ]]]. 5 (A.27)] nn (1 2u )pLL PP

](A.27) implies that in the absence of trade the economy n will have an excesssupply of the high-tech good, therefore, the autarchy price of the high-tech good in

]this economy will be lower than the world price. Thus, the economy n has acomparative advantage in the high-tech good. After opening up to trade, thefollowing has to be true.

] ]n nH 2 X uP]]] ]]]5 (A.28)] ]n n (1 2u )pL 1 MP

] ]n n ](A.27) and (A.28) above imply that X . 0 and M . 0, if economy n has] ]n nbalanced trade, i.e. pX 2 M 5 0. If this economy is a net importer of capital,

] ]n nthen it has to run a trade surplus, in that case it is possible that X . 0 and M , 0.On the other hand, if this economy is a net exporter of capital, then it runs a trade

]ndeficit, and in this case it is possible that it imports both goods, i.e. X , 0 and]nM . 0. Thus in the case of balanced trade, the Heckscher–Ohlin pattern of trade

is verified. In the case of unbalanced trade, it is possible for the economy to exportor import both the goods, however, it will never be the case that the pattern oftrade is the opposite of that predicted by the Heckscher–Ohlin theorem. h

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