Skill acquisition, credit constraints, and trade

International Review of Economics and Finance 18 (2009) 227–238

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International Review of Economics and Finance

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Skill acquisition, credit constraints, and trade☆

Tatyana Chesnokova a,⁎, Kala Krishna b,c,1

a Department of Economics, University of Auckland, Private Bag 92019, Auckland, New Zealandb Pennsylvania State University, United Statesc NBER, United States

a r t i c l e i n f o

☆ We are grateful to participants of the NBER winterand participants at seminars at the World Bank and D⁎ Corresponding author. Tel.: +64 9 3737599x88309

E-mail addresses: [email protected] (T.1 523 Kern Graduate Building, University Park PA 162 See for example, Goldin and Katz (2001) and Abra

1059-0560/$ – see front matter © 2008 Elsevier Inc. Adoi:10.1016/j.iref.2008.06.005

a b s t r a c t

Available online 24 June 2008
In this paper we develop a general equilibrium model where credit constraints limit the abilityof agents with heterogeneous abilities and wealth to acquire skills. We identify a new effect, theinduced Rybczynski effect, that works in the opposite direction from the normal supplyresponse and may result in relative supply being downward-sloping. We analyze the effects oftrade and show that under some conditions trade may reduce welfare. Finally, we study theeffects of trade on income distribution and inequality.
© 2008 Elsevier Inc. All rights reserved.

JEL classification:F16

Keywords:Credit constraintSkill formation

1. Introduction

What is the role of credit constraints in education andwhat is the role of international trade in this?Will welfare rise or fall withtrade and why? These are the questions addressed in this paper. They are clearly of immense relevance for policy since humancapital is not good collateral for loans and the ability to acquire skills can be severely limited by wealth.

There is clear evidence2 that the contribution of physical capital to growth fell considerably while that of human capital almostdoubled. This also ties into the old debate on the trade-off between inequality and growth. After all, if the rich save more than thepoor, assuming that the rental market for capital is not too distorted, then inequality may well promote physical capitalaccumulation and growth. However, as human capital becomes more important due to technological changes, and since there arediminishing returns to education at an individual level, inequality in the presence of credit constraints may well hinder growth.And indeed using cross country and panel data, Flug et al. (1998) show that the lack of financial markets seems to have a negativeeffect on the accumulation of human capital. All of this points to credit constraints in acquiring education being particularlyimportant for developing countries.

Even in the U.S., there is evidence that credit constraints play an important, though more subtle, role today. Intergenerationalincome correlation is reasonably high: Solon (1992) finds a ballpark figure of .4. Charles and Hurst (2003) find the pre-bequestcorrelation in log wealth to be .37. Empirical work on college attendance has consistently shown that parental income does predictcollege attendance and that the effect on college attendance of tuition is greater for lower income families. Should these facts betaken as evidence of credit constraints? Recent work by Cameron and Heckman (1998) questions such a simple interpretation ofthese facts. They show that college attendance does not depend on income once the child's skill endowment (as proxied for by theArmed Forces Qualifying Test (AFQT)) is controlled for. In other words, if fewer low income children go to college, they do sobecause by the time they are 18, they cannot get in, not because they cannot afford it. However, Keane and Wolpin (2001) arguethat credit constraints do matter. They estimate a structural model and argue that though borrowing constraints are tight, they are

meetings at Palo Alto, at the ERWIT workshop of the CEPR, at the Otago Workshop in International Trade,rexel University for comments on earlier drafts.; fax: +64 9 3737427.Chesnokova), [email protected] (K. Krishna).802, USA.movitz and David (2000).

ll rights reserved.

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http://dx.doi.org/10.1016/j.iref.2008.06.005

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228 T. Chesnokova, K. Krishna / International Review of Economics and Finance 18 (2009) 227–238

mitigated by agents adjusting labor supply. This is why changes in tuition affect college attendance by relatively little, and have agreater effect on poorer families.3

In this paper we develop a simple general equilibrium model where credit constraints limit the ability of agents withheterogeneous abilities and wealth to acquire skills. There are two tradable final goods and two factors, unskilled labor and skilledlabor, which is produced using skilled labor and unskilled labor. If expectations about future prices are taken as given, the responseof supply to price depends on the number of skilled agents in the economy. If there are relatively few skilled agents, the normalsupply response obtains. However, with many skilled agents, supply can be decreasing in price so that multiple equilibria mayexist. The intuition is that in addition to the normal supply response, there is an induced Rybczynski effect which could work ineither direction. When there are relatively few skilled agents, an increase in the price of the skill-intensive good, keeping as givenexpected future prices, raises the cost of education. This reduces the number of agents whowant to acquire training and thus raisesthe availability of unskilled labor. The availability of skilled labor for production also rises as less skilled labor is needed in training.Whether the relative availability of skilled labor rises or falls depends on the endowment of skilled workers to begin with: if thisendowment is large, the relative availability of skilled labor falls, while if it is small, it rises. In the former case, relative supply of theskill-intensive good can fall with price, while in the latter case it must rise.

In the steady state, however, non-monotonicity of supply and multiplicity of equilibria occur only in the presence of creditconstraints. An increase in price in the steady state raises the return to skilled labor today, and hence, the cost of education, but italso raises the return tomorrow. While the increase in the cost of education reduces the demand for training, the increase in thereturn tomorrow increases it. Since each agent must be able to train more than one worker for there to be skilled workers presentin the steady state, the latter effect dominates. Thus, an increase in price raises the number of trainees, reducing the availability ofunskilled workers for production. In the steady state, and in the absence of credit constraints, as the number of trainees rises, sodoes the number of skilled workers available for production. As a result, an increase in price raises the relative supply of the skill-intensive good. In the presence of credit constraints, relative supply need not be monotonic in price. For relative supply to bebackward bending, the increase in education cost resulting from an increase in price must constrain enough of the potential pool oftrainees.

There may or may not be multiple equilibria in the steady state with credit constraints: a key determinant is the distribution ofwealth. With such multiplicity, trade may even reduce welfare in steady state if (a) there is limited substitutability in consumptionand (b) the value of final good output evaluated at traded prices falls.

Finally, we study the evolution of the distribution of income. We show that in our model, an increase in the price of the skill-intensive good, if credit constraints do not bind, raises inequality: not only do the rich get richer and the poor get poorer, but thereare more of both of them. However, the presence of credit constraints can reduce the response of inequality to increases in price tothe extent that an increase in price results in fewer rich in the steady state. Most important, ourmodel suggests that if the rich drawfrom even a slightly better distribution of abilities at age 18 than do the poor, as suggested by the work of Cameron and Heckman(1998), there is a multiplier effect on inequality in the steady state. This highlights the role of access to decent public education inproviding both equity (as the income distribution is less unequal) and efficiency (as the more intrinsically able are cheaper to trainthan the less able).

1.1. Related literature

Our paper is related to the literature on endogenous skill formation in international trade. In an influential contribution, Findlayand Kierzkowski (1983) extend the standard Heckscher–Ohlin model by endogenizing the formation of human capital. They showthat trade amplifies initial differences in factor endowments through the Stolper–Samuelson effect: trade raises the reward of theabundant factor in each country.

Cartiglia (1997) incorporates credit constraints into a Findlay–Kierzkowski type model. He shows that trade should lead toconvergence in human capital endowments across countries rather than amplification of initial differences. Trade liberalization ina skill-scarce country, while reducing the return to education, also reduces the cost of educationwhich weakens credit constraints,resulting in a higher investment in human capital. This effect, in fact, outweighs the Stolper–Samuelson effect of Findlay–Kierzkowski, reversing their results.4

Ranjan (2001) points out a third effect that operates through changes in the distribution of income which influences theaccumulation of human capital and shows that trade may encourage skill acquisition in both countries. Ranjan (2003) looks at theeffect of trade liberalization on skill acquisition, the skilled–unskilled wage differential, and the distribution of wealth. Multiplesteady state equilibria also exist in his model and he shows that trade may induce convergence to the good equilibrium. However,his focus is on trade in intermediate goods as the single final good is not traded.

In this paper we derive a new effect, the induced Rybczynski effect, that helps explain how price changes affect the relativesupply of the skill-intensive good. We show how the relative supply curve can be downward-sloping and as a result, multipleequilibria are likely. We also provide simple conditions under which trade reduces welfare. Our work suggests that unlesscountries ensure access to education when liberalizing trade, they may lose their comparative advantage in skill-intensive goodsand even suffer welfare losses.

3 See Keane (2002) for a simple and clear summary of the issues here.4 A similar result obtains in Eicher (1999) via a domestic credit market.

229T. Chesnokova, K. Krishna / International Review of Economics and Finance 18 (2009) 227–238

The rest of the paper proceeds as follows. Section 2 lays out the model. Section 3 analyzes the autarky equilibrium. Section 4studies how trade affects welfare. Section 5 endogenizes the distribution of income. Section 6 provides concluding remarks. Detailsof the proofs are in the Appendix.

2. The model

There are two goods, X and Z, and one basic factor, unskilled labor, in the economy. Unskilled labor can be transformed into itsskilled counterpart by a skilled worker. However, if K unskilled workers are taken on by a skilled worker then only (1−AK) units ofthe skilled worker's time remain available to him.We assume that there are constant returns to scale in production and that good Zis relatively more skill-intensive than good X at all factor prices. The relative price of good Z is denoted by p.

There are L agents born in each period. Each agent lives for two periods and is endowedwith one unit of time in each period. Anagent is characterized by two parameters: γ, the probability of becoming a skilled worker upon undertaking the education, and y,his initial wealth. We assume that γ is distributed uniformly in the unit interval and y is distributed according to the distributionfunction F (·) in [ymin, ymax].5 Agents consume only at the end of their lifetimes and have identical homothetic preferences over Xand Z.

In the first period of life an agent makes career choices. He could remain an unskilled worker and work in both periods at theunskilled wage, wt. Alternatively, he could spend part of his time in the first period acquiring the skills that give him a chance atbecoming a skilled worker and allowing him, if he so chooses and is successful in his training, to earn the skilled wage in the secondperiod. Agents who try to become skilled but fail can work only as unskilled workers in the second period. Skilled workers couldalso choose to work as unskilled workers were it in their interests to do so.

The education system is such that unskilled workers have pay the skilled worker a tuition fee wtC up-front. The training takes

(1−β) units of trainees' time and they work for the remaining time as unskilled workers and earn βwt. There are no credit markets,hence, each agent has to finance the tuition fee from his wealth.

In each period of time t there are Mt skilled workers who are the successful trainees from the last period. Since there areconstant returns to scale in training and production, each skilled worker must maximize the value of profits he can obtain bysetting himself up in business. But as long as training occurs, the opportunity cost of a skilled worker's time as a teacher is theskilled worker's wage, wt

S. Hence, wSt ¼ wC

tA .

3. Autarky equilibrium

First, we analyze the autarky equilibrium in each period twhen expectations about future prices are given. Later we look at thesteady state autarky equilibrium.

An equilibrium in period t is characterized by a vector of prices (pt,wtS,wt). Since both goods are essential in consumption, both

goods must be produced in autarky. Therefore, the cost of making good X equals its price of unity.

The pr

5 We

1 ¼ cX wt ;wSt

� � ð1Þ

ice of Z equals its cost

pt ¼ cZ wt ;wSt

� �: ð2Þ

is pins down wt and wtS, and hence wt

C, for a given pt. From Eqs. (1) and (2) we know that since good Z is skilled labor
Thintensive, as pt rises, wt
S rises and wt falls according to the Stolper–Samuelson Theorem.A young agent in period t has two options. The first is to work both periods of his life as an unskilled worker. The second option

is to invest in skills hoping to become a skilled worker. Let γt denote the agent who is indifferent between these two optionsso

wt þwtþ1 Etptþ1ð Þ ¼ βwt−wCt þ γtw

Stþ1 Etptþ1ð Þ þ 1− γtð Þwtþ1 Etptþ1ð Þ ð3Þ

and

γt ¼ minwt 1−βð Þ þ AwS

t

wStþ1 Etptþ1ð Þ−wt Etptþ1ð Þ ;1

( )ð4Þ

, agents with γ ∈ [0, γt] work both periods of their life as unskilled workers, while agents with γ ∈ [γt , 1] want to invest in
Henceeducation. What are the effects of an increase in pt on γt? A higher pt leads to a lowerwt (which tends to reduce γt) and a higherwt
S

(which tends to increase γt).

Assumption 1. dcX=dwdcX=dwS N

1−βA for all wS≥w.

consider endogenous wealth distribution in Section 5.


This assumption guarantees that the second effect dominates and as a result AγAwS

tN0. In the analysis below we assume that

Assumption 1 holds.Given homothetic preferences, relative demand for good Z depends only on pt. Obtaining the relative supply of Z is slightly

more complicated. For a given Etpt+1 and a given number of skilled workers, the supply of X and Z can be derived as follows. Foreach pt we find γt which gives the set of agents who want to become skilled. The intersection of this set with the set of agents whohave wealth above wt

C determines how many agents are both willing and able to invest in education. Removing the skilled laborneeded for training them from the stock of skilled workers gives the supply of skilled labor available for production, LtS:

LSt ¼ Mt− A 1− γtð Þ 1−F wCt

� �� Þg those who choose to become unskilled workers today to the inherited stock of unskilled workers (which includes both
Addin
those who chose to become unskilled yesterday and those who failed at becoming skilled) and unskilled labor supplied by today'strainees, gives the supply of unskilled labor available for production, LtU:

LUt ¼ γt þ F wCt

� �1− γtð Þ þ β 1−F wC

t

� �� 1− γtð Þ þ 1−Mt

, the availability of skilled and unskilled labor for production gives the size of the standard Rybczynski box and relative
Finallysupply. Hence, every price pt corresponds to a point on the relative supply curve which is illustrated in Fig. 1(a) and (b). Price p1corresponds to wt
S=wt. When price is below p1, the return to skilled labor is less than the wage of an unskilled worker.As a result the option of working as an unskilled worker is more profitable than the option of being a skilled worker even for

those who are skilled. For all prices below p1 the supply of each good is zero. At price p1 skilled workers are indifferent betweentheir two options so that relative supply is horizontal.

Fig. 1.


Next, we turn to the shape of the relative supply curve and the nature of equilibrium. Suppose that pt increases. Then γtincreases and, hence, the supply of unskilled labor in period t rises. As fewer agents want to be trained, skilled workers spend lesstime training them and the supply of skilled labor available for production increases. Therefore, both skilled and unskilled laboravailable for production rise so that their relative availability may rise or fall. The effects on relative supply of a price change can bedecomposed into two parts. First, there is the standard positive supply response. An increase in the relative price of the skill-intensive good raises the relative wage of skilled labor a la the Stolper–Samuelson Theorem. This results in substitution away fromskilled labor. For given factor supplies, this raises the relative supply of the skill-intensive good. Second, there is an inducedRybczynski effect: an increase in the availability of skilled relative to unskilled labor for production purposes raises the relativesupply of the skill-intensive good.

These two effects are illustrated in Fig. 2(a) and (b) where the supply of skilled labor is on the horizontal axis and the supply ofunskilled labor is on the vertical axis. The allocation of skilled and unskilled labor between sectors is at P. Now suppose that priceincreases. Then the ratio of unskilled to skilled labor in each sector increases. For fixed endowments of skilled and unskilled laborthe outcome is at P1. It is easy to see that at P1 the supply of good Z is higher and the supply of good X is lower compared to P. This isthe usual supply response for fixed factor endowments. However, the increase in pt also increases both endowments of skilled andunskilled labor. If the supplies of skilled and unskilled labor increase proportionately, then given the assumption that productionfunctions are homothetic, outputs of Z and X also increase proportionately. Hence, from P1 to P2 relative supply does not change.

Fig. 2(a) illustrates the case when the change in the endowment of skilled labor is proportionately more than the change in theendowment of unskilled labor. In this case, the ‘Rybczynski’ effect works in the same direction as the usual supply effect, movingthe equilibrium to P′. Fig. 1(a) depicts relative supply in this case. Since the relative demand curve can intersect the relative supplycurve at most once, there is a unique equilibrium.

Fig. 2.


Fig. 2(b) illustrates the case when the change in the endowment of skilled labor is proportionately less than the change in theendowment of unskilled labor. Now the ‘Rybczynski’ effect works in the opposite direction from the normal supply response. Thisis shown by P′ being closer to the origin OZ than P2. Since the induced Rybczynski effect now tends to decrease the relative supplyof good Z, relative supply may fall in response to higher prices which is depicted in Fig. 1(b). As a result, multiple equilibria mayarise in this case.

Next, differentiating LUtLStwith respect to pt we get

Note t

where

dLUtLSt

� �=dpt ¼

dγtdpt

1− F wCt

� �� þ dwCt

dptf wC

t

� �1− γtð Þ

� �LSt� �2 Mt 1þ A−βð Þ−2Að Þ

� �
Using Aγ
AptN0 we have d LUt

LSt=dptb0 if and only if MtbM ¼ 2A

1þA−β. Hence, whether the relative availability of unskilled labor increasesor decreases with an increase in price depends on the number of skilled workers in the current period. IfMt is sufficiently large, i.e.MtNM , then the relative availability of unskilled labor increases in response to higher prices and, as a result, the inducedRybczynski effect tends to decrease the relative supply of the skill-intensive good. Intuitively, withmany skilledworkers the supplyof skilled labor is relatively large. Then, any change in γt translates into a small percentage increase in the supply of skilled laborand, as a result, skilled labor becomes relatively less abundant and the relative supply of Z may fall!

Proposition 1 summarizes these results.

Proposition 1. If the number of skilled workers in period t is small enough, i.e., MtVM ¼ 2A1þA−βð Þ, then relative supply is increasing in

price. If MtNMthen relative supply need not be increasing in price and there may be multiple equilibria.

Next, we consider the autarky steady state equilibrium. In the steady state all prices are constant so γ is determined from

2w ¼ −wC þ βwþ γwS þ 1− γð Þw:

Thus

γ ¼ minw 1−βð Þ þ AwS

wS−w;1

�

hat the relationship between γ and p now has the opposite sign:

dγdp

¼ 1−β þ Að ÞwS−wð Þ2

wdwdp

−wS dwS

dp

� �b0

Lemma 1. In the steady state, γ NA.

Proof. Using wS ¼ wC

A we can rewrite the return to the option of investing in skills as βw+(1−γ) w+wS (γ−A). Then, for γbA thisreturn is clearly less than the lifetime income of an unskilled worker. As profit goes to infinity (wS→∞) the proportion (1−A) wantto invest in education, i.e., γ→A. Therefore, in the steady state the proportion of agents who want to invest in education neverexceeds 1−A. □

Next, we turn to the shape of the relative supply curve and the nature of the steady state equilibrium. First, we show that if noagents are credit-constrained, i.e. for all agents wCby, then the relative supply curve is upward-sloping. Suppose that price rises.This results in a lower γ—more agents choose to invest in skills at a higher price and hence the supply of unskilled labor falls. Whatcan we say about the supply of skilled labor? Subtracting the skilled labor needed for training from the stock of skilled workersgives the supply of skilled labor available for production:

LS ¼ M−AK ¼ 12

1− γð Þ2� �

−A 1− γð Þ� �
whereM ¼ 1
2 1− γð Þ2 is the number of skilled workers and K=1− γ is the number of trainees. As price rises, the number of traineesincreases, as does the number of skilled workers. The total effect on the supply of skilled labor is positive— LS increases with pricesince:

dLS

dp¼ dγ

dpA− γð ÞN0

the inequality follows from Lemma 1. As a result, via the standard argument, the relative supply of the skill-intensive goodith p.
rises w
When there are some agents who are credit-constrained, the effects of an increase in price on the supplies of unskilled andskilled labor are ambiguous and depend on the distribution of wealth. If a higher price results in unskilled labor becomingrelatively more abundant, then the relative supply of the skill-intensive good may fall. If, for example, there are many agents whobecome credit-constrained at this higher price, then a large proportion of agents who would have invested in skills at lower price

Fig. 3.


can no longer afford to do so. Hence, the supply of unskilled labor rises, and the supply of skilled labor available for production ofgood Z falls. As a result, the relative supply of good Z may be lower at this higher price. Thus, when credit constraints operate, theshape of the relative supply curve depends on the distribution of wealth and can be either upward-sloping or downward-sloping.

Proposition 2 summarizes these results.

Proposition 2. Steady state relative supply need not be increasing in price and multiple steady state equilibria may exist.

4. Effects of trade

Having described the closed economy, we turn to the analysis of the effects of opening the economy up to trade. We analyze thewelfare effects of trade when the country is small and cannot affect the world price of good Z, denoted by pT. We consider thesteady state and as we know from the previous section, relative supply need not be increasing in price and multiple equilibria mayexist. As we show below, non-monotonicity of relative supply may result in trade equilibria where the country ends up importinggood Z at a world price higher than its autarky price and, as a result, loses from opening up to trade.

Consider the situation depicted in Fig. 3. Relative supply is non-monotonic and there are two stable steady state autarkyequilibria: E1A and E2

A. At equilibrium E1A , price, p1A , is low and relative supply of good Z is high, while at E2A price, p2A, is high and

supply is low. Suppose that initially the economy is in autarky equilibrium E1A. What happens when the country opens up to trade

and faces the world price of good Zwhich is higher than its autarky price? As the price rises, the skilled wage rises as well, so thatthe option of investing in skills becomes more attractive. As more agents decide to invest in skills, the up-front education fee goesup. Credit constraints become tighter and more agents are unable to afford education. Consequently, the supply of skilled labordecreases, resulting in lower output of good Z as well as lower relative supply, denoted by RST. As the price goes up, the relativedemand for good Z, denoted by RDT, falls. If a significant proportion of agents becomes credit-constrained at this higher price, then

Fig. 4.

Fig. 5.


the decrease in relative supply is considerable and exceeds the decrease in relative demand: opening up to trade allows only a fewrich agents to invest in skills and, as a result, the supply of the skill-intensive good falls dramatically and cannot satisfy domesticdemand at this price. Thus, the country imports good Z even though the world price is higher than the autarky price! Since thecountry loses its comparative advantage in good Z and has to import this good at a higher price, its aggregate welfare can be lowerwith trade.

Proposition 3 provides sufficient conditions for welfare to fall with tradewhen all agents have identical homothetic preferencesso that aggregation is possible.

Proposition 3. Opening up an economy to trade need not be welfare-improving. If a country imports the skill-intensive good at worldprices higher than its autarky price, substitution in consumption is small enough, and the value of the autarkic output at trade pricesexceeds that of the trade output, welfare must fall.

We relegate the formal proof to the Appendix and focus on the intuition here. The outcome in Fig. 3 is depicted in Fig. 4 in awaythat highlights the conditions given above. The production possibility frontier (PPF) envelope in the steady state in the presence ofcredit constraints need not have the usual shape: it can be bowed in.6 At pT, and similarly at pA, there is a given availability of skilledand unskilled labor for production in the steady state. This defines the usual PPF at the given price. These PPFs are tangent to theirrespective price lines at QT and QA respectively as depicted in Fig. 4. The relative demand for good Z exceeds the relative supply atpT but equals it at pA as shown. If there is no substitution in consumption, demand always lies along the ray OA and if, as depicted,the line connecting QT and QA is flatter than pT, then welfare must fall with trade. Note that limited substitution in consumption isessential: if substitution in consumption is large enough, say if the indifference curves are close to being linear so that they are nottoo far from the autarky price line, then the result does not go through.

The intuition is simple: when the price of the skill-intensive good rises through trade, credit constraints are tightened and itsrelative supply falls. Since the supply of the skill-intensive good is too low to begin with, the distortion is worsened by trade andwelfare falls. This suggests that countries, that have a comparative advantage in the skill-intensive good need to ensure access toeducation to fully reap the gains from trade.

5. Endogenizing the distribution of income

In this sectionwe endogenize the distribution of income and consider a small open economy.7 Suppose now that all agents haveidentical Cobb–Douglas preferences given by

6 It m1983). T

7 Hen

U ¼ cδXc1−δz

� �1−θbθ

where cX is the consumption of good X, cZ is the consumption of good Z, and b is bequests which are modelled as a ‘warm glow’.Now the initial wealth of an agent is a bequest from his parents: yt+1=bt.

As a benchmark case we first consider the situation where credit constraints do not bind in the steady state, i.e. for all agentsinitial wealth exceeds education tuition. In this case, the income distribution does not affect the equilibrium, but the equilibrium

ust also lie inside the steady state PPF envelope in the absence of credit constraints (which has the usual bowed out shape as in Findlay & Kierzkowski,his makes sense as credit constraints result in inefficiency: the wrong people are trained, which limits production possibilities.ce, the price and wages are fixed.


affects the income distribution so that there is only a one way causation. An agent's total income, which is allocated across thegoods and bequests at the end of his second period of life is the sum of his bequest, bt, and his earned income. Let Ytj (bt) denote thistotal income for j=u, f, s since earned income can take only three values, that of unskilled (u), the failed skilled (f), and the skilled(s). Thus, bequests of those born in period t who inherit bt are given by b(bt, j)=θ (Ytj (bt)) or

8 This

b bt ; jð Þ ¼θ bt þ βw −wC þw� �

; for j ¼ fθ bt þ 2wð Þ; for j ¼ u

θ bt þ βw−wC þwS� �

; for j ¼ s

8<:

te that for a given p, earned income is lowest for the failed skilled and highest for the skilled. Moreover, an increase in p
NoraiseswS and reducesw. Thus, as p increases the nominal income of the unskilled falls as does that of the failures, and so does theirreal income. Income of the skilled may increase or decrease depending on p.
Bequest lines are depicted in Fig. 5. Note that there is no way for bequests to exceed bH or fall below bL: if an agent's ancestors

were all skilled, the bequest he would get would be bH and if all of them were failures, it would be bL. The bequests that can be

made by agents of type j= f are contained in the interval [bL, bL], those made by agents of type j=u are contained in [b

M, bM], while

thosemade by agents of type s are contained in [bH, bH]. Since the bequest line is flatter than the 45° line, the Feller condition is met

and as the distribution of abilities is common across all agents, the Markov process converges to a unique invariant bequest andassociated total income distribution.

To analyze this further, we assume that the intervals defined above are disjoint as in Fig. 5.8 Let pL, pM, pH (for low, middle andhigh) denote the fraction of agents in the steady state who are in each of these intervals. Then we can calculate the steady statelevels of these variables for given values of γ. Thus, we can obtain the steady state distribution of bequests and wealth at a givenp since for each p we get a unique γ. We can show the following.

Proposition 4. An increase in p raises inequality in the distribution of income in the absence of credit constraints both because the poorget poorer and the rich richer and because there are more of them.

Proof. In the Appendix. □Being rich or poor is just a matter of luck with no long term consequences so that one thinks of inequality as being less

pernicious in the above setting.Now consider the situation where credit constraints do bind for some agents: only the subset of agents who have bequests

over wC, the up-front costs of education, can be trained. Suppose that wC lies between b L and bM (Case A). The unskilled are thus

all those with ability below γ (which have a mass of γ) as well as those whose parents failed at becoming skilled but whose γexceeds γ (which have a mass of (1− γ)pL). Note that with credit constraints, there is a two way causation: not only does theequilibrium affect the income distribution, but the income distribution affects the equilibrium through the extent of creditconstraints. Let 1 γð Þ ¼ 1− γ2ð Þ

2 denote the expected yield of the successes and let / γð Þ ¼ 1− γð Þ22 denote the expected yield of failures

when the cutoff level is γ. Then in the steady state:

pL ¼ 1− γð Þ22

1−pL� � ¼ / γð Þ 1−pL

� �pM ¼ γ þ 1− γð ÞpL

pH ¼1− γ2� �

21−pL� � ¼ 1 γð Þ 1−pL

� �

te that in this instance, the children of the skilled and the unskilled are not constrained by bequests, only the children of the
Nounfortunate failures are. However, since their children can become skilled, the mixing condition is still met.
Since ϕ and ς are increasing in p, pL rises, and pM falls, though pH may rise or fall. As the price rises, the income of the skilledrises while the incomes of the unskilled and failures fall.

Proposition 5. An increase in p raises inequality in the distribution of income in the presence of credit constraints because the poor getpoorer and the rich richer but can reduce inequality to the extent that there are fewer rich in the steady state.

If wC lies between bM and bH (Case B) then the children of the unskilled cannot become skilled and the economy falls into a

poverty trap. This can be seen since

pL ¼ / γð ÞpHpM ¼ 1− pH þ γpH

pH ¼ 1 γð ÞpH

so the only solution is pH=pL=0.

requires that θ be small enough.


A non-degenerate equilibrium can be restored by perturbing the economy a bit. The perturbation can be interpreted as thegovernment giving aid by removing the credit constraint of a fraction σ of those who are constrained, independent of their ability.This makes sense since ability need not be easily verifiable so that targeting the most able, as would be efficient, may not be easy. Ifwe do so then we get

then e

pH ¼ 1 γð Þ pH þ σ 1−pH� ��

pL ¼ / γð Þ pH þ σ 1−pH� ��

pM ¼ γ þ 1−pH� �

1− γð Þ 1−σð Þ

ving for pH gives
Sol
pH ¼ σ1 γð Þ1−σ1 γð Þ

e untalented (γ ) as well as the credit-constrained talented ((1−pH) (1− γ)) who do not get their credit constraints relaxed
Thearn unskilled wages in both periods which explains pM. The successful able children of parents who earned skilled wages as wellas the successful able children of the rest of the population who had their credit constraints relaxed make up pH. Their failed ablechildren make up pL. Note that even a small σ moves the economy away from the poverty trap.
Finally, we askwhen inequalitymight bemore pernicious. Herewe look at the consequences of amore subtlemanifestation ofcredit constraints. According to Cameron and Heckman (1998) conditioning on ability (test scores), income seems to play no rolein college attendance. This suggests that income mainly operates through affecting the ability of children at the time ofgraduation from high school. The poor, oneway or the other, seem to not invest asmuch in their children as the rich do.Wemodelthis by having the skilled draw from a different distribution of talent, say one which is uniform over [λ, 1]. Then we show thefollowing.

Proposition 6. An improvement in the distributions that the skilled draw from has a multiplier effect on the steady state distribution ofincome and skills. The elasticity of pHwith respect to λ is increasing in λ, suggesting that when the school systems accessed by the rich andpoor differ greatly, we should see very significant inequality.

Proof. In the Appendix. □

6. Conclusions

In this paper we develop a model where credit constraints limit the ability of agents with heterogeneous abilities and wealth toacquire skills. We show that if expectations about future prices are taken as given, then the response of supply to price depends onthe number of skilled agents in the economy. If there are relatively few skilled agents, the normal supply response obtains.However, with many skilled agents, supply can be decreasing in price so that multiple equilibria may exist. In the steady state,however, such non-monotonicity of supply and a multiplicity of equilibria obtain only in the presence of credit constraints. Theremay or may not be multiple equilibria in the steady state: a key determinant is the distribution of wealth. With such multiplicity,we show that under some conditions trade may reduce welfare.

Appendix

Proof of Proposition 3. Let e(P, u) be the usual expenditure function. Let p, Q, C, u denote the price, output and consumptionvectors while u denotes utility. The superscripts A and T refer to the autarky and trade outcomes. We know that

e pA;uA� � ¼ pAQA ¼ pACA and QA ¼ CA

e pT;uT� � ¼ pTQT ¼ pTCT

e pA;uT� �

b pACT;

e pT;uA� �

b pTCA ¼ pTQA:

Thus,

e pT;uT� �−e pT;uA� �

> pTCT−pTCA ¼ pTQT−pTQA:

If there is no substitution in consumption, the first inequality is an equality. Hence, if

pTQANpTQT ð3Þ

(pT, uT)−e(pT, uA)b0 so uTbuA. By continuity, if substitution in consumption is small enough and Eq. (3) holds, uTbuA. □


Proof of Proposition 4. pM equals the mass of all agents who choose to be unskilled, or γ. 1 Similarly, pL must equal the mass ofagents who are failures, or ∫1γ 1−γð Þdγ, and pH must equal the mass of agents who become skilled, or ∫1γγdγ. Thus,

pL ¼ 1− γð Þ22

pM ¼ γ

pH ¼ 1− γð Þ22

:

sy to verify that dpL

dγ b0; dpM

dγ N0; dpH

dγ b0.
It is eaSince γ falls as p rises, dpL
dp N0;dpM

dp b0; dpH

dp N0, so that an increase in p moves mass from the middle to the tails in thebequest distribution. Note that the location of the tails themselves will also change as p also affects incomes, and henceYtj (bt). □

Proof of Proposition 6. First, consider the case when for all agents credit constraints do not bind. The mass of agents who chooseto be unskilled are those with abilities below γ. The ones who have skilled parents, who are (pH) of the total in steady state, have aprobability γ−λ

1−λ

� �of being in the region [λ, γ] which is less than γ, that of agents who do not have skilled parents. Thus, pM is a

convex combination of these two probabilities. Those who choose to try and get trained are those with abilities above γ. The oneswho have skilled parents, who are (pH) of the total in steady state, have a probability 1− γð Þ2

2

h i1

1−λ of being in the region [γ, 1] whilethose who do not have skilled parents, who are (1−pH) of the total in steady state, have a probability 1− γð Þ2

2

h i. Making similar

calculations for pH yields:

pL ¼ 1−γð Þ22

" #1−pH� �þ pH

1−λ

�

pM ¼ γ 1−pH� �þ pH

γ−λ1−λ

� �

pH ¼ 1−γð Þ22

" #1−pH� �þ pH

1−λ

�:

Solving this gives

pH ¼

1− γð Þ22

" #

1−1−γð Þ22

!λ

1−λ

� �" #

pL ¼ 1− γð Þ22

" #ð1þ

λ1−λ

� � 1− γ2� �

2

24

35

1−1−γ2� �

2

0@

1A λ

1−λ

� �24

35

266666664

377777775

pM ¼ γ− 1−γð Þ

λ1−λ

� � 1− γ2� �

2

24

35

1−1−γ2ð Þ2

� �λ

1−λ

� � � :

ilar, if credit constraints operate we have
Sim
pL ¼ 1−γð Þ22

" #pM þ pH

1−λ

�

pM ¼ pL þ γpM þ pHγ−λ1−λ

� �

pH ¼1−γ2� �

2

24

35 pM þ pH

1−λ

�:


Solving this gives

pH ¼ 1

1þ 1− γð Þ22

1− γ2� �

2

−λ

1−λ

� �266664

377775

pL ¼

1− γð Þ22

" #

1þ 1− γð Þ22

" #−

1− γ2� �

2λ

1−λ

� �

Note that pL and pH are higher and, hence, pM is lower, thanwhen they draw from the same distribution. □

References

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Cameron, S. V., & Heckman, J. J. (1998). Life cycle schooling and dynamic selection bias: models and evidence for five cohorts of American males. Journal of PoliticalEconomy, 106, 262−333.

Cartiglia, F. (1997). Credit constraints and human capital accumulation in the open economy. Journal of International Economics, 43, 221−236.Charles, K. K., & Hurst, E. (2003). The correlation of wealth across generations. Journal of Political Economy, 111, 1155−1182.Eicher, T. S. (1999). Trade, development and converging growth rates: dynamic gains from trade reconsidered. Journal of International Economics, 48, 179−198.Findlay, R., & Kierzkowski, H. (1983). International trade and human capital: a simple general equilibrium model. Journal of Political Economy, 91, 957−978.Flug, K., Spilimbergo, A., & Wachtenheim, E. (1998). Investment in education: do economic volatility and credit constraints matter? Journal of Development

Economics, 55, 465−481.Goldin, C., & Katz, L. F. (2001). The legacy of U.S. educational leadership: notes on distribution and economic growth in the 20th century. American Economic Review,

91, 18−23.Keane, M. (2002). Financial aid, borrowing constraints and college attendance: evidence from structural estimates. American Economic Review, Papers and

Proceedings, 92, 293−297.Keane, M., & Wolpin, K. (2001). The effects of parental transfers and borrowing constraints on educational attainment. International Economic Review, 42,

1051−1103.Ranjan, P. (2001). Dynamic evolution of income distribution and credit-constrained human capital investment in an open economy. Journal of International

Economics, 55, 329−358.Ranjan, P. (2003). Trade induced convergence through human capital accumulation in credit constrained economies. Journal of Development Economics, 72,

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Documents

Skill acquisition, credit constraints, and trade