technology transfer, income distribution and the process of economic development

P1: LMW/PCY P2: KCU/JHR P3: VTL/JHR QC: MVG

Open economies review KL445-03-Zang May 5, 1997 9:46

Open economies review 8:3 245–270 (1997)c© 1997 Kluwer Academic Publishers. Printed in The Netherlands.

Technology Transfer, Income Distributionand the Process of Economic Development

HYOUNGSOO ZANG [email protected] World Bank, 1818 H Street, NW, Washington, DC 20433

Key words: income distribution, human capital, growth, complementarity

JEL Classification Number: O15, O40

Abstract

This paper studies the role of income distribution and technology transfer in the process of eco-nomic development. A novel aspect of the model is that the composition of human capital as well asthe level affect economic growth. Utilizing an overlapping-generations model in which income dis-tribution changes endogenously, we present an economic explanation for why some countries couldnot start modern economic growth; why some countries took off but have apparently stopped grow-ing after some time; and why some countries have successfully developed and continue to grow.

This paper studies the role of income distribution and technology transfer inthe process of economic development. The central features of the model arethat, due to capital market imperfections, the distribution of income determinesthe distribution of human capital, and that technological change is related toboth the level and composition of human capital. A novel aspect of the modelis that the distribution of human capital as well as the level affect economicgrowth. In the course of analysis, we present an economic explanation for whysome countries could not start modern economic growth; why some countriestook off but have apparently stopped growing after some time; and why somecountries have successfully developed and continue to grow.

Recently, enormous attention has been paid to the implications of incomedistribution on economic growth sparked by Galor and Zeira (1993) who havedemonstrated that due to borrowing constraints, income distribution affectsinvestment in human capital and consequently per-capita output in the long-run as well as in the short-run1. Alesina and Rodrik (1994) and Persson andTabellini (1994) have introduced political-economy approaches to the issue.

Several papers have presented mechanisms that may generate the Kuznetscurve, i.e., income inequality increases as per-capita income increases up toa certain level of development and then it decreases as an economy grows(Kuznets, 1955)2. Aghion and Bolton (1991, 1992) generate the Kuznets curvethrough the change in the rate of interest in a stochastic environment with cap-ital market imperfections. Banerjee and Newman (1993) show that the interplay



246 ZANG

between agent’s occupational choice and wealth distribution may be consis-tent with the Kuznets hypothesis. Galor and Tsiddon (1997a) develop a generalequilibrium model with intergenerational externalities in which the evolution ofincome inequality and output growth conforms with the Kuznets hypothesis.Perotti (1993) showed that, depending on the initial distribution of income thatdetermines the median voter’s preference over the degree of redistribution overtime, the trickle-down process may not work.

The Kuznets inverted-U relation seems to be empirically robust in cross-section studies and in time-series studies on most developed countries,whereas the relation is rarely found in time-series studies on most develop-ing countries3. In general, Latin American countries started to develop first,but they apparently stop growing with relatively high inequality. Most Africancountries still remain poor with less unequal income inequality than the LatinAmerican countries. In the course of analysis, this paper will address an eco-nomic explanation for the discrepancy between cross-section and time-seriesempirical evidence on the Kuznets hypothesis.

In the process of economic development, technology transfer from advancedcountries has been a major source of technological change for less-advancedcountries4. Successful imitation and implementation of the advanced technolo-gies, however, critically depend on the availability of domestic human capital.Accordingly, the first presumption of this model is that advanced technologiesmay be transferred to less-advanced countries only if the receiving countryhas a sufficient number of well-trained, elite group of people (high-tech hu-man capital) who absorb and imitate the advanced technologies5. This is inline with the vast theoretical and empirical literature on the conditional conver-gence hypothesis: there is no convergence unless some other characteristicsof development, notably initial levels of human capital, are not controlled6.

The second presumption of the model is that without accompanying broadhuman capital, e.g., mass education, further growth may be hindered by short-age of complementary skilled human capital. As Easterlin (1981) pointed out, asmall number of highly educated population without accompanying mass edu-cation is unlikely to sustain economic growth. Complementarity in the two kindsof human capital implies that as an economy grows the relative importance ofhigh-tech human capital diminishes and that of complementary human capitalin production comes in front.

We develop an overlapping-generations model where individuals live for twoperiods. In the first they acquire education and in the second they are em-ployed as either high-tech, skilled or unskilled workers according to their ed-ucation level. Due to borrowing constraints the acquisition of human capitalis limited to those individuals whose parents can provide a sufficient level ofwealth to finance their investment costs in human capital. The current distribu-tion of income determines the composition of high-tech, skilled and unskilledindividuals in the labor force in the next period. The economy is characterizedby multiple equilibria as a result of capital market imperfection. The economy’s



TECHNOLOGY TRANSFER, INCOME DISTRIBUTION 247

steady-state equilibrium is path dependent. The initial distribution of incomeof a country determines which steady-state the country ends up with.

At early stages of development, sufficient high-tech human capital is nec-essary to initiate economic growth that requires imitation and adoption of ad-vanced technologies. Since the acquisition of high-tech human capital requiresa high investment cost, it is impossible for a very poor country to have high-tech human capital unless the country draws resources to some portion ofits population to invest in high-tech human capital, which implies that someunequal distribution of income may be necessary for a very poor country toinitiate economic growth. A very poor country with too equal an initial distribu-tion of income may stagnate forever since the country may not supply enoughhigh-tech human capital. As the economy develops and the technology gapshrinks, however, the existence of an ample complementary human capital inproduction gets more weight. In order to develop successfully, a less-advancedcountry should have sufficient high-tech human capital to imitate and absorbthe advanced technologies, and in addition it needs sufficient complementaryskilled human capital that facilitates the implementation of the technology. Ifthe income distribution of a poor country is so unequal that there is sufficienthigh-tech human capital, but that most of people cannot get any education,the country may not achieve further economic growth that should be accompa-nied by a large pool of human capital. Only economies with “moderate’’ incomeinequality follow the entire economic development path to prosperity.

Most of politico-economic approaches to the relationship between incomedistribution and growth seem to advocate for perfect equality as a way to pre-vent redistributive policies acting as a tax on accumulation or rent seekingactivities detrimental to growth (see Rodrik, 1994, 1996)7. The purely economicapproach taken in this paper should be considered as complementary and sup-plementary to the politico-economic models which underscore a democraticpolicy decision-making.

In our model, income distribution changes endogenously. The evolution ofincome distribution depends on technological change which is determined bythe initial distribution of human capital and consequently by the initial distri-bution of income. Very poor developing countries with too equal an incomedistribution remain very poor and relatively equal. They may not get the benefitfrom the other world’s technological innovation. These countries occupy thespots on the left tail of the (cross-sectional) Kuznets curve. Poor developingcountries with too unequal an initial income distribution cruise fast in the earlyphases of economic development with worsening distribution of income. Butthey may not be developed ultimately due to ironically their unequal income dis-tribution which in turn prevents the sufficient supply of human capital. Thesecountries are located around the top humped portion of the (cross-sectional)Kuznets curve. Countries with moderate income inequality can follow the en-tire economic development process to prosperity. Only these countries haveexperienced and shall experience the (temporal) Kuznets curve. Their incomeinequality first increases, and then it decreases.



248 ZANG

1. The basic model

1.1. Production

Consider a small open economy in a one-good world. The good, which can beused either for consumption or investment, is produced by a constant returns toscale technology which is stationary across time and identical across countries.Every individual has one unit of physical labor and can acquire human capitalif he invests in education. For simplicity, we assume that physical capital andphysical labor are perfect substitutes, whereas physical capital and humancapital exhibits complementarity in the production of goods. Our main resultsare not sensitive to the perfect substitutability of physical capital and physicallabor. What we need is that physical capital is more complementary to humancapital than to physical labor so that the wage rate for physical labor should notrise significantly as capital is accumulated. This assertion has been supportedby numerous theoretical and empirical studies8.

The output produced at time t,Yt , is

Yt = (Kt + Lt )α(λt Ht )

1−α ; α ∈ (0, 1), (1)

where Kt and Lt are the amount of physical capital and physical labor, respec-tively, employed at time t . Since everybody is endowed with one unit of physicallabor, Lt is the number of workers at time t . Ht is the aggregate units of hu-man capital. λt is the coefficient of the endogenous, human capital augmentingtechnological change at time t . There are two kinds of human capital: high-tech human capital, Hh, and skilled human capital, Hs. Both are employed inthe production of human capital such that the two kinds of human capital arecomplements rather than substitutes9.

Ht =(Hs

t

)β(Hh

t

)1−β. (2)

1.2. Factor prices

The labor and the goods markets are perfectly competitive. There is no interna-tional labor mobility. Capital is perfectly mobile across countries, so that firmsin a small country have free access to international capital markets at a sta-tionary world rate of interest, r > 0. The amount of physical capital is adjustedeach period so that:

r ≡ ∂Yt

∂Kt= α(Kt + Lt )

α−1(λt Ht )1−α = αz1−α; z≡ λt Ht

Kt + Lt. (3)

Hence, there is a constant ratio of total efficiency units of human capital tophysical units (i.e., the sum of physical capital and physical labor), z, which




implies a constant rate of return to a unit of physical capital. Given the structureof the production technology, the return to a unit of physical labor, wu, is equalto the return to a unit of physical capital, r .

The wage rate for a unit of skilled human capital, wst , is:

wst ≡

∂Yt

∂Hst= (1− α)z−αβλt

(Hh

t

/Hs

t

)1−β= (1− α)z−αβλtθ

1−βt ; θt ≡ Hh

t

/Hs

t . (4)

The wage rate for a unit of skilled human capital is increasing in λt and θt (theratio of the amount of high-tech human capital to that of skilled human capital).

The wage rate for a unit of high-tech human capital, wht , is:

wht ≡

∂Yt

∂Hht

= (1− α)z−α(1− β)λt(Hh

t

/Hs

t

)−β = (1− α)z−α(1− β)λtθ−βt . (5)

The wage rate for a unit of high-tech human capital is increasing in λt anddecreasing in θt .

The relative wage ratio of high-tech human capital to skilled human capital attime t, vt , is:

vt ≡ wht

wst= 1− β

βθ−1

t , (6)

which is decreasing in θt and independent of λt . Since everybody is endowedwith a unit of physical labor, a skilled worker gets wu+ws

t and a high-tech workercollects wu + wh

t .

1.3. Technology

There is no direct cost associated with technology transfer across countries.But, the level of technology of a less-advanced country λt , is determined by theprevious-period level of high-tech human capital of the less-advanced countryand by the technology level of the leader country. To emphasize the importanceof the minimum level of high-tech human capital in adopting advanced technolo-gies and to sharpen results, we assume that there is a minimum requirementof high-tech human capital, Hh

min, to start to adopt and imitate the advancedtechnologies10. The level of high-tech human capital of a country determineshow far the country’s technology could improve relative to the technology levelof the leader country. The technology transfer (or adoption) function is:

λt+1 ={λt +Max

{δ[(

Hht /Hh,leader

t

)φλleader

t − λt], 0}

if Hht ≥ Hh

min,

λmin if Hht < Hh

min,(7)

where δ ∈ (0, 1), φ ∈ (0, 1), Hht < Hh,leader

t and λmin<(Hhmin/Hh,leader)φλleader

t for all t .



250 ZANG

The parameter δ governs the speed of convergence to a steady-state level oftechnology that is affected by φ. Since φ and δ are less than 1, the technologytransfer function shows decreasing returns to scale to the level of high-techhuman capital of the technology-absorbing country. Note that the technologytransfer function captures the spillover effect of the accumulation of high-techhuman capital in the previous period on the current level of technology whichfacilitates the adoption and imitation of foreign advanced technologies in thecurrent period. In this sense, there is an intergenerational externality becausethe proceeds of the technological progress cannot be claimed by the previousgeneration. Unlike other models in which the world technology is available andappropriable for all countries at the same degree, our model assumes that thedegree of appropriation by a country depends positively on the level of high-tech human capital of the country. A conditional catch-up effect is present inour model rather than the global catch-up effect. The relative level of high-techhuman capital of an imitating country to that of the leader country sets an upperlimit to the country’s ultimate technology level.

For the moment, we assume that the level of high-tech human capital of theleader country and so the level of world technology is on a steady-state. That is,Hh,leader

t = Hh,leaderand λleadert = λleaderfor all t . We will discuss the implications of

a steady growth of the world technology in Section 2.3. The technology transferfunction for the case of stationary world technology is11:

λt+1 ={λt +Max

{δ[(

Hht

/Hh,leader

)φλleader− λt

], 0}

if Hht ≥ Hh

min,

λmin if Hht < Hh

min.(8)

Note that the steady-state level of technology of an imitating country fora given steady-state level of high-tech human capital of the country, λ(H h),is:

λ(H h) = (Hh/Hh,leader)φλleader. (9)

Since both φ and H h/Hh,leaderare less than 1, the relative level of the steady-statetechnology of a backward country compared to the leader country, λ/λleader,is proportionally greater than the relative level of the steady-state high-techhuman capital, H h/Hh,leader. This reflects the well-cited advantage of “startinglate’’. The smaller φ is, the greater the benefit from imitation is.

1.4. Individuals’ decision problem

Individuals live two periods in overlapping generations. Each parent at timet has one child and each child has one parent12. There is a continuum ofindividuals of size L in each generation. For simplicity, individuals work andconsume in the second period of their lives only. They work as unskilled workersin the second period if they did not invest in human capital in the first period of




life, whereas if they did when young, they work as skilled workers or high-techworkers depending on how much they spend on investment in human capital.Investment in human capital in order to become skilled workers costs s, andthat to become high-tech workers costs h; 0 < s < h. Both investment costsare assumed to be stationary over time and across countries.

The benefit of being a skilled worker is always greater than the investmentcost associated with the training of a skilled worker. Namely,

wst > s(1+ r ) ∀ t. (10)

However, the wage rate for high-tech human capital is not guaranteed to bealways sufficiently higher than that for skilled human capital to cover the dif-ference in investment costs. If the wage differential, wh

t − wst , is the same as

the cost differential in human capital investment, (h− s)(1+ r ), then individualsare indifferent to whether they become skilled or high-tech workers. This couldhappen if the level of technology rises sufficiently and/or if there are relativelymany high-tech workers.

For the moment, we suppose that the wage differential is high enough tocover the investment cost differential between high-tech human capital andskilled human capital, i.e.,

wht > ws

t + (h− s)(1+ r ). (11)

Thus, all individuals prefer to invest in high-tech human capital. In Section 2.2,we will examine in detail the case where wh

t = wst + (h− s)(1+ r ).

Firms and individuals can lend any amount at the international interest rater . However, unlike firms, individuals cannot borrow neither in the internationalmarkets nor in the domestic market. Thus, although investment in human capi-tal is beneficial, individuals cannot invest in skilled (or high-tech) human capitalunless they receive sufficient funds to cover the cost of education s (or h)13. Thisimperfection in the consumer credit market may be due to monitoring costs as-sociated with default risks, virtual inexistence of institutionalized consumer loanmarket, as is the case in many less developed countries, etc. The economy maybe Pareto-inefficient since not all individuals may not invest in human capital,even if such an investment raises net income. This Pareto-inefficiency is a resultof the capital market imperfection14.

A parent decides about the family’s consumption level and implicitly via be-quests which are rendered to children when they are young, whether the chil-dren acquire education. Individuals born at time t derive their lifetime utilityfrom consumption in the second period of their lives, ct+1, and from bequeststo their offspring, bt+1:15

Ut = U (ct+1, bt+1), (12)

The utility function is identical across individuals and satisfies the neo-classicalproperties in addition to homotheticity. Since it is homothetic, the ratio of



252 ZANG

consumption to bequest is independent of income. Let γ be the proportionof income devoted for family consumption, 0 < γ < 1. Then, individuals leave(1 − γ ) percent of their income as a bequest to their offspring, and conse-quently individuals with higher income leave higher bequest to their child. Notethat, from the construction of the homothetic utility function, income distribu-tion (yt ) is perfectly correlated with wealth distribution (bt ) in this model. Inparticular,

bt = (1− γ )yt . (13)

Unskilled workers. Consider individuals who are born and inherit an amountbt at time t . If this amount is less than a cost of education, s, the individu-als cannot invest in human capital and consequently they become unskilledworkers in the next period. The income of a family of unskilled workers, yu

t+1,consists of the return to their savings in addition to their wage earnings, namely,yu

t+1 = bt (1+ r ) + wu. Thus, noting the homotheticity of the utility function, theunskilled worker born at time t leaves at time t + 1 a bequest of size:

but+1 = (1− γ )[bt (1+ r )+ wu]. (14)

Skilled workers. Individuals who inherit more than s but less than h acquireskilled human capital and invest in physical capital the remaining portion of theirbequests. The income of a skilled worker born at time t is ys

t+1 = (bt − s)(1+ r )+ wu + ws

t , and his/her bequests at time t + 1 are,

bst+1 = (1− γ )

[(bt − s)(1+ r )+ wu + ws

t

]. (15)

High-tech workers. Individual who inherit more than h acquire high-tech hu-man capital and invest in physical capital the remaining portion of their be-quests. A high-tech worker born at time t earns yh

t+1 = (bt − h)(1+ r )+ wu+wht ,

and leaves bequests at time t + 1,

bht+1 = (1− γ )

[(bt − h)(1+ r )+ wu + wh

t

]. (16)

The following restrictions on parameters guarantee that the evolution of be-quest from generation to generation is stable, and that the cost, s, is high enoughor the wage rate for unskilled workers is low enough to prevent unskilled work-ers’ offspring from investing in human capital, which is more realistic and inter-esting case than otherwise.

(1− γ )(1+ r ) < 1 and(1− γ )wu

1− (1− γ )(1+ r )< s. (17)




2. The dynamic evolution of the economic system

2.1. Case I

From the construction of the model, an income distribution determines the num-bers of unskilled workers (Lu), skilled workers (Ls) and high-tech workers (Lh)

in the next period completely, and consequently, the next-period compositionof human capital stocks, θ , and the next-period relative wage ratio of high-techhuman capital to skilled human capital, v. Under conditions (8), (10) and (11),they are constant as long as wage rates for skilled and high-tech workers arein an appropriate range, which will be discussed below.

The distribution of income in the previous period determines the next-perioddistribution of inheritances and so income as well:

but+1 = (1− γ )[bt (1+ r )+ wu] if bt < s,

bst+1 = (1− γ )

[(bt − s)(1+ r )+ wu + ws

t

]if s ≤ bt < h,

bht+1 = (1− γ )

[(bt − h)(1+ r )+ wu + wh

t

]if bt ≥ h.

(18)

Figure 1(a) illustrates a typical dynamic evolution of bequest (and so that ofincome distribution) through time as presented in (18). The lower, the middleand the upper curves describe the dynamic evolution of bequest for unskilled,skilled, and high-tech workers, respectively. Wage rates for skilled and high-tech human capital should be in the following ranges to produce figure 1(a):

s

1− γ − wu ≤ ws

t <h

1− γ + (h− s)(1+ r )− wu. (19)

wht ≥

h

1− γ − wu. (20)

Figure 1(a). Dynamic evolution of bequests: Case I.



254 ZANG

The LHS of (19) comes from that bst+1(s) ≥ s and the RHS of (19) is drawn from

that bst+1(h) < h. And the condition (20) is derived from that bh

t+1(h) ≥ h. Theseconditions ensure that offsprings from skilled or high-tech workers remain sogeneration after generation.

If a country’s wages rates satisfy (19) and (20) for all t and the technologylevel of the leader country is stationary at λleader, then the technology level ofthe country converges to a steady-state level, λ = (H h/Hh,leader)φλleader, as longas the economy’s high-tech human capital exceeds the minimum level, Hh

min, asthe number of high-tech workers is constant at H h > Hh

min. In the meantime, themiddle and the upper curves in figure 1(a) move upwards until the technologylevel converges to the steady-state level, λ, because bothws

t andwht are affected

positively by λt as in (4) and (5), and because the composition of human capital,θ , is constant in this case.

The bequests of three kinds of dynasties within a country converge to thesteady-state levels of bequests, bu, bs and bh, respectively, for unskilled, skilledand high-tech workers:

bu = (1−γ )wu

1−(1−γ )(1+r ) ,

bs = (1−γ )[wu+ws−s(1+r )]1−(1−γ )(1+r ) ,

bh = (1−γ )[wu+wh−h(1+r )]1−(1−γ )(1+r ) ,

(21)

where ws and wh are the steady-state wage rates for skilled and high-tech hu-man capital, respectively. The steady-state pe-capita bequest, b, is calculatedas:

b = l ubu + l sbs + l hbh, (22)

where l u, l s, and l h are the proportion of unskilled, skilled and high-tech workersin the labor force in the steady-state, respectively. These numbers are constantas long as (19) and (20) are satisfied.

Substitution of (21) into (22), noting (13), yields the steady-state per-workerincome as:

yI = 1

1− (1− γ )(1+ r )

×{l uwu + l s[wu + ws − s(1+ r )] + l h[wu + wh − h(1+ r )]}. (23)

2.2. Case II

Note that the LHS of (19) and (20) are guaranteed to be satisfied. If wage ratesdo not satisfy them, changes in the number of skilled and high-tech workers willadjust the wage ratio, θt , which will make in turn the wage rates to adjust ac-cording to (4) and (5) until they satisfy the LHS of (19) and (20) again16. However,




the RHS or (19) is not guaranteed to be satisfied. Once the wage rate for skilledworkers satisfies that

wst ≥

h

1− γ + (h− s)(1+ r )− wu, (24)

then some skilled workers become high-tech workers. As the number of high-tech workers is increasing, the wage rate for skilled workers increases againdue to increase in θt . In addition, an increase in the number of high-tech workersimproves the level of technology ultimately, which increases ws

t further. Con-sequently, more skilled workers will become high-tech workers as the level oftechnology increases. Thus, once the wage rate for skilled human capital sat-isfies (24) instead of the RHS of (19), it will no longer return to satisfy the RHSof (19) again.

As the relative proportion of high-tech workers increases, the relative wageratio for high-tech human capital, vt decreases as can be seen in (6). As technol-ogy improves and as the wage rate for skilled human capital increases, the ef-fect of the fixed investment cost differential diminishes and consequently moreworkers become to own high-tech human capital. The wage rate for skilledhuman capital will rise until it satisfies:

wht − ws

t = (h− s)(1+ r ), (25)

in which individuals are indifferent between being skilled workers and beinghigh-tech workers since the wage differential,wh

t −wst , is equal to the investment

cost differential, (h− s)(1+ r ).Figure 1(b) illustrates the case where the wage rate for skilled human capital

starts to satisfy (24).17 The curve bst+1 will move upwards until it meets the

Figure 1(b). Dynamic evolution of bequests: Case II.



256 ZANG

curve bht+1 to become the curve bs+h

t+1 . Both skilled and high-tech workers getthe same net benefit from their human capital investment, and their bequestswill converge to the same level of bs+h. In the long-run there will be two kindsof dynasties; the rich and the poor. For Case II, noting (21), (22), (25) and (13),the steady-state per-worker income is:

yII = 1

1− (1− γ )(1+ r ){l uwu + (1− l u)[wu + wh − h(1+ r )]}

= 1

1− (1− γ )(1+ r ){l uwu + (1− l u)[wu + ws − s(1+ r )]}. (26)

To examine the condition (24) in more detail, we substitute (4) into (24) to get:

λt ≥{

h

1− γ + (h− s)(1+ r )− wu

}(1− α)−1zαβ−1θ

β−1t ≡ ζ(θt ). (27)

If a developing country’s initial level of high-tech human capital is greater thanHh

min, then the technology level of the country starts to increase and convergesto a steady-state level. In the mean time, the wage rates for skilled and high-tech human capital satisfy (19) and (20), respectively. There is a constant ratioof high-tech human capital to skilled human capital, θ , and a constant numberof high-tech workers, Hh, in the economy. But, as the technology level of theeconomy increases over time, it may be the case that the wage rate for skilledhuman capital goes up to satisfy (24), i.e., Case II, instead of the RHS of (19),i.e., Case I. The possibility depends positively on the initial level of high-techhuman capital given the composition of human capital, θ . The condition (27)shows this point clearly. The higher the initial level of high-tech human capitalis (and consequently the higher the level of technology (λ) is) and the higher theratio of high-tech human capital to skilled human capital (θ) is, the more likelythe wage rate for skilled human capital satisfies (24) or (27). The downwardsloping curve drawn in figures 2(a) and (b) illustrates the boundary of (27), ζ(θ).

To find out the steady-state level of high-tech human capital, that of technol-ogy and that of the composition of human capital in Case II, we need to examine(25) more carefully. Substitution of (4) and (5) into (25) yields:

λt = (h− s)(1+ r )

(1− α)z−αθβt

(1− β)− βθt≡ ω(θt ). (28)

Given the level of technology at time t, λt , the ratio of the number of high-tech workers to that of skilled workers, θt , is positively related to λt . However,there is an upper bound on θt , i.e., (1 − β)/β, to which θt converges as thetechnology level approaches infinity, if it is feasible. Due to the complementarityin human capital, expanding high-tech human capital without accompanyingskilled human capital causes the deterioration of the relative wage benefitsfor high-tech human capital, which draws an upper bound on the proportion




Figure 2(a). Determination of the steady-state level of technology and human capital distribution:Case I.

Figure 2(b). Determination of the steady-state level of technology and human capital distribution:Case II.

of high-tech workers among the labor force with human capital. The upwardsloping curve, ω(θ), in figures 2(a) and (b) illustrates the relation (28). The curveapproaches to infinity when θ converges to (1− β)/β. We can find the lowerbound for the steady-state ratio of high-tech human capital to skilled humancapital, θ , by solving (27) and (28) (see Appendix 3).



258 ZANG

The following propositions characterize the relationship between the steady-state values and the initial size of human capital pool (i.e., the number of indi-viduals with some human capital).

Proposition 1. A larger initial number of individuals with some human capital,Ls+h, always results in a higher steady-state level of high-tech human capital,Hh, and consequently that of technology than a smaller one does as long as ws

tstarts to satisfies (24) and φ ≤ β.18

Proof: See Appendix 4. 2

Remark. The proposition implies that the steady-state level of high-tech hu-man capital and that of technology is not necessarily higher when the initiallevel of high-tech human capital is higher. Rather, the size of human capitalpool determines the ultimate level of technology. This is a very interesting re-sult because it suggests that education policy of developing countries shouldfocus on the total number of educated people rather than on that of “highlyeducated’’ people in order to attain a higher steady-state level of technology.

Proposition 2. Under the same conditions as in Proposition 1, the largerthe initial number of individuals with some human capital, Ls+h, the larger thesteady-state level of human capital stock, H .


Proposition 3. Under the same conditions as in Proposition 1, the largerthe initial number of individuals with some human capital, Ls+h, the higher thesteady-state ratio of high-tech human capital to skilled human capital, θ .

Proof: From (28), after substituting steady-state values, λ and θ , we can easilyshow that dθ/dλ > 0. Using Proposition 1, dθ/dLs+h = d(θ/dλ)(dλ/dLs+h) > 0.

2

Proposition 4. Under the same conditions as in Proposition 1, the largerthe initial number of individuals with some human capital, Ls+h, the higherthe steady-state wage rates for both high-tech human capital (wh) and skilledhuman capital (ws), and thus, the higher the steady-state level of per-worker( per-capita) income; yII .


Figures 2(a) and (b) illustrate the determination of the steady-state level ofhigh-tech human capital, Hh (and consequently that of technology, λ), and thatof the ratio of high-tech human capital to skilled human capital, θ , for Case Iand Case II, respectively. We will discuss the determination mechanism withexamples in Section 3. The curve λ(Hh) in the 3rd quadrants in figures 2(a)




and (b) illustrating the level of technology as a function of the level of high-techhuman capital, is increasing in Hh and strictly convex with the flat segment(λmin) associated with the levels of high-tech human capital that are less thanthe critical level, Hh

min. As the technology level converges to a steady-state level,the curves λ(Hh) converge to the curve λ(Hh).

It is very important to note here that the same value of θ is compatible withmany different numbers of high-tech workers, Hh. To see this point clearly, werewrite θ as:

θ ≡ Hh

Hs= Hh

Ls+h − Hh≡ θ(Hh; Ls+h), (29)

where Ls+h is the number of individuals with some human capital, skilled or high-tech, which is the same as the (initial) number of individuals who inherit morethan s. For each Ls+h, there is a value of Hh that produces the same θ . For agiven θ , a larger Ls+h allows higher Hh and so higher λ. As more individuals canacquire some human capital, there will be larger potential for higher steady-statelevels of high-tech human capital. The upper bound for θ , i.e., (1−β)/β, also setsupper bounds for Hh given Ls+h by (29). The upper bound for Hh, (1− β)Ls+h,is greater for a larger Ls+h. The curves θ(Hh; Ls+h

i ) drawn in the 4th quadrantsin the figures represent the ratio of high-tech human capital to skilled humancapital for a given number of workers with human capital, Ls+h

i ; Ls+h2 > Ls+h

1 , asgiven in (29). The curves θ(Hh; Ls+h

i ) are upward sloping and strictly concavewith limHh→Ls+h(∂θ/∂Hh) = ∞.

2.3. Steady growth of the world technology

When the world technology or the technology level of the leader country growssteadily, the technology level of an technology-absorbing country may alsogrow steadily as long as there was initially enough high-tech human capital(>Hh

min) in the economy. Consequently, after some time, λt is large enoughto make possible some descendants of skilled workers to invest in high-techhuman capital. Thus, eventually, the level of technology will make the wagerates for both skilled and high-tech workers to satisfy (25), when individuals areindifferent between investment in high-tech human capital and that in skilledhuman capital.

The analysis in this case will be exactly the same as that in Section 2.2. Theonly difference is that there is no limit in the technology levels of both the leaderand technology-absorbing countries. However, the production technology setsthe upper bound on the relative proportion of high-tech human capital, (1−β)/β.The perpetual increase in technology will make the level of high-tech humancapital to converge to its upper bound, (1− β)Ls+h. Since the upper bound forthe level of high-tech human capital is higher for a country with more individualswith human capital, Ls+h, the technology level of the country will be also higher.



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3. Path-dependent steady-states and initial distribution of income

3.1. Determination of steady-state: Case I

If a poor developing country has a very equal initial distribution of income sothat the level of high-tech human capital of the country falls short of the mini-mum level, λmin, then the country cannot utilize the technological progress fromabroad and remains poor. Their income distribution also remains unchangedand equal.

Figure 2(a) illustrates the determination of and the evolution towards thesteady-state of a country when the initial income distribution makes possiblefor the country to supply enough high-tech human capital (>λmin), but (24) or(27) is not satisfied. The initial location of the country is denoted by A. As thecountry’s technology level increases toward a steady-state, it moves to A′ andfinally to A, the steady-state. The levels and composition of human capital donot change at all along this process. The steady-state per-worker income thecountry ends up with, given by (23), depends not only on the level of technologybut also on the composition of labor force, i.e., initial distribution of income andinitial per-worker income. There is no general pattern for the relationship be-tween initial income inequality and the steady-state per-worker income in thiscase. Thus, countries with low levels of initial income may end up with any com-bination of steady-state per-worker income and income inequality dependingon their initial conditions.

3.2. Determination of steady-state: Case II

Consider two countries, B and C, with the same initial level of per-capita incomeand the same size of population. The income of country B is more unequallydistributed than that of country C so that there are initially more unskilled work-ers and more high-tech workers in country B than in country C. Thus, countryC has initially more individuals with some human capital (Ls+h

2 ) than countryB does (Ls+h

1 ). This implies that the initial ratio of high-tech human capital toskilled human capital is higher for country B because two countries’ aggregateresources are assumed to be same.

The locations of two countries are denoted as B and C in figure 2(b). Bothcountries’ initial levels of high-tech human capital are big enough to satisfy (24)in some time. For country B, higher initial Hh and θ let ws satisfy (24) far earlierthan for country C. Country C does not go into the inside of the boundary, ζ(θ),until her technology level approaches a steady-state level for a given initial Hh.But, once the country C starts to increase the number of high-tech workers,the technology level of country C will surpass that of country B easily since thecountry C has more potential resources for high-tech human capital.

The steady-states of the countries are denoted and B and C. Even thoughthe initial level of high-tech human capital and so that of technology is higherfor country B where income is more unequally distributed, the steady-state




level of high-tech human capital and that of technology is higher for country Cwhere income is less unequally distributed than country B. The country C hasrelatively more high-tech human capital in the steady-state than country B does,although the country C started with relatively less high-tech human capital.

We note here that countries with a larger initial number of individuals withsome human capital always end up with a higher steady-state level of high-tech human capital and consequently a higher steady-state level of technologythan countries with a smaller one as long as ws

t starts to satisfies (24) andφ ≤ β (Proposition 1). Furthermore, the steady-state ratio of high-tech humancapital to skilled human capital, θ , is higher for countries with a larger Ls+h

(Proposition 3). What really matters in this case is the initial number of individu-als inherited more than s or those with some human capital (Ls+h). The countrywith a greater Ls+h not only attains a higher steady-state level of technology butalso has a larger number of rich dynasties. In addition, the steady-state level ofincome for rich dynasties is greater as Ls+h is greater (Proposition 4). Therefore,per-capita (per-worker) income will also be higher for the country with a largerpool of human capital or with a smaller initial number of very poor individualswho cannot invest in any kind of human capital (L − Ls+h) (Proposition 4).

3.3. The evolution of income inequality

Theil (1967) and Bourguignon (1979) proposed an inequality index defined as:

J =n∑

i=1

pi ln(pi /yi ); 0≤ J ≤ 1, (30)

where pi and yi are the population share and income share of income group i ,respectively.

Looking at only steady-states and using (22), (30) becomes in our Case I:

JI = ln

[l ubu + l sbs + l hbh

(bu)lu(bs)l

s(bh)l

h

]. (31)

That is, the measure of income inequality is the logarithm of the ratio of arith-metic mean income to geometric mean income. In our Case II, (30) becomes:

JII = ln

[l ubu + (1− l u)bs+h

(bu)lu(bs+h)1−l u

]. (32)

Utilizing the inequality index, we can show that the evolution of income in-equality of a country with “moderate’’ inequality in the model may follow the(temporal) Kuznets curve (see Appendix 7). Income inequality of the coun-try first increases as modern economic growth starts due to widening gaps inwages between unskilled workers and skilled and high-tech workers. As the



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technology level of the country continues to improve, more individuals can gethigh-level education, and consequently the wage gap between skilled and high-tech workers diminishes. After some turning-point in the process of economicdevelopment, income inequality starts to decline.

The theoretical model presented here, however, implies that not all countriescan follow the entire evolution process. Developing countries with too equalan initial distribution may stagnate forever with relatively equal distribution ofincome. These countries appear on the left tail of the (cross-sectional) Kuznetscurve. On the other hand, developing countries with too unequal distributionof income may not be ultimately developed. These countries are found aroundthe top portion of the (cross-sectional) Kuznets curve. Only economies withmoderate income inequality can experience the entire (temporal) Kuznets curve.In contrast, time-series data from other countries may not show the invertedU-relation between income and income inequality. The exact type of the relationdepends on initial conditions.

4. Concluding remarks

Our model suggests an important policy recommendation for intermediately de-veloped countries which are rich enough to supply sufficient high-tech humancapital to imitate and absorb more advanced technologies. The most impor-tant concern for these countries is to have their income distribution as equalas possible so that the size of human capital pool should be extended as faras possible, which should be the source of continuing supply of high-tech hu-man capital. This could be done in a way by a subsidy to private educationand a direct provision of public education. Countries with only a small numberof well-educated people without accompanying mass education are unlikely togrow. This suggestion may also apply to some highly developed countries sinceour model can be extended such that the technological change of the leadercountry also depends on the composition of as well as the level of human capi-tal of the country. For instance, the United States may lose her competitivenessshould recent increases in income inequality result in significant decreases in in-vestments in human capital of poor people. These people are potential sourcesof her continuing prosperity.

Appendices

Appendix 1. A CES specification of the production function for human capital

Now, suppose that human capital is produced by a CES specification:

Ht =[β(Hs

t

)−ρ + (1− β)(Hht

)−ρ]− 1ρ ; −1< ρ 6= 0, (A1)




with the elasticity of substitution σ = 1/(1+ ρ). Then, the wage rates are:

wst = (1− α)z−αβλt

(Hs

t /Ht)−(1+ρ)

= (1− α)z−αβλtη−(1+ρ)t ; ηt ≡ Hs

t /Ht . (A2)

wht = (1− α)z−α(1− β)λt

(Hh

t /Ht)−(1+ρ)

= (1− α)z−α(1− β)λt (1− ηt )−(1+ρ). (A3)

The relative wage ratio for human capitals at time t, vt , is

vt ≡ wht

wst= 1− β

βθ−(1+ρ)t . (A4)

Note that if ρ = 0, (A4) is the same as (6). The condition (28) becomes:

λt = (h− s)(1+ r )

(1− α)z−α[(1− β)(1− ηt )

−(1+ρ) − βη−(1+ρ)t

]−1. (A5)

Using ηt ≡ 1/(1+ θt ) and 1− ηt ≡ θt/(1+ θt ), (A5) can be restated as:

λt = (h− s)(1+ r )

(1− α)z−α[(1− β)

(θt

1+ θt

)−(1+ρ)− β

(1

1+ θt

)−(1+ρ)]−1

. (A6)

Given the level of technology, λt , (A6) determines the composition of humancapital at time t, θt . A higher value of λt is compatible with a lower value ofηt or a higher value of θt . That is, a higher level of technology makes feasiblea higher proportion of high-tech workers among the labor force with humancapital. The implication of the degree of complementarity in human capital isnow clear from (A6). As two kinds of human capital are more complementary,i.e., a higher positive value of ρ, a smaller value of θt is compatible with a givenlevel of technology than when they are less complementary.

Appendix 2. The lower bound of the level of technology

Substitution of (4) into the LHS of (19) yields:

λt ≥(

s

1− γ − wu

)(1− α)−1zαβ−1θ

β−1t . (A7)

Substitution of (5) into (20) yields:

λt ≥(

h

1− γ − wu

)(1− α)−1zα(1− β)−1θ

βt . (A8)



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Equating the RHS of (A7) with that of (A8), we obtain the critical value of θ, θ ′:

θ ′ ≡ 1− ββ

s(1− γ )−1− wu

h(1− γ )−1− wu.

The lower bound of λt for a given θ depends on these parameters. That is,

if θ ≤ [≥] θ ′, then the RHS of (A8) [(A7)] is the lower bound. (A9)

Appendix 3. The lower bound of θ : Case II

If the wage rate for skilled human capital starts to satisfy (27), then it increasesfurther until it satisfies (28). All steady-state values λ and θ should satisfy (27)and (28) simultaneously. That is,

(h− s)(1+ r )

(1− α)z−αθβ

(1− β)− βθ ≥{

h

1− γ + (h− s)(1+ r )− wu

}× (1− α)−1zαβ−1θ β−1.

After rearranging and simplifying, we obtain the lower bound of the steady-stateratio of high-tech human capital to skilled human capital, θ :

θ ≥ 1− ββ

h(1− γ )−1− (h− s)(1+ r )− wu

h(1− γ )−1− wu≡ θ . (A10)

The steady-state value of θ cannot be less than θ .It can be shown that if θ ≥ θ , then every possible steady-state values, λ,

satisfying (27) and (28), also satisfies (A9). Figures 2(a) and (b) illustrate thelocation of curves in our system, ζ(θ) and ω(θ). The steady-state pairs of(θ , λ) should be on ω(θ) with θ ≥ θ . Both ζ(θ) and ω(θ), are above the lowerbound.

Appendix 4. Proof of Proposition 1

Substitution of steady-state values by using (8) and (29) into (28) yields:

(Hh)φ−β(Ls+h− Hh)β[(1−β)−β

(Hh

Ls+h− Hh

)]− (h− s)(1+ r )

(1−α)z−αλleader

Hh,leader= 0. (A11)




Letting the LHS of (A11) be S(H h, Ls+h) and utilizing the implicit function theo-rem, we can obtain the sign of dHh/dLs+h = −(∂S/∂Ls+h)/(∂S/∂ Hh).

∂S

∂Ls+h= β(1− β)(Hh)φ−β(Ls+h − Hh)β−1 > 0. (A12)

∂S

∂ Hh= (Hh)φ−β−1(Ls+h − Hh)β−1

{(φ − β)[Ls+h(1− β)− H h] − H h

}(A13)

Using (A12) and (A13),

dHh

dLs+h= −β(1− β)H h

(φ − β)[Ls+h(1− β)− H h] − H h. (A14)

If φ ≤ β, then ∂ H h/∂Ls+h < 0 since (1− β)Ls+h > H h which comes from that(1− β)Ls+h is the upper bound of Hh. Thus, φ ≤ β is the sufficient conditionfor dHh/dLs+h > 0. Then, noting (9), dλ/dLs+h > 0. That is, the steady-statelevels of high-tech human capital and technology are higher as the number ofindividuals with human capital is larger if φ ≤ β.

If φ > β, it might be possible that (φ − β)[Ls+h(1− β) − H h] − H h ≥ 0 andconsequently dHh/dLs+h ≤ 0. However, as can be seen in (A13), if technol-ogy increases sufficiently high, which is more likely as Ls+h is larger, then(1− β)Ls+h − H h approaches 0, which makes dHh/dLs+h > 0.

Next, we will show the condition for dHh/dLs+h > 0even if φ > β. Substitutionof θ = Hh/(Ls+h − H h) into (A10) and some rearrangement yields:

H h ≥ (1− β)Ls+h h(1− γ )−1− (h− s)(1+ r )− wu

h(1− γ )−1− (1− β)(h− s)(1+ r )− wu. (A15)

From (A14) we know that in order to have dHh/dLs+h > 0 even if φ > β, we needthat:

H h >φ − β

1+ φ − β Ls+h(1− β). (A16)

If the RHS of (A15) is greater than the RHS of (A16), then any Hh that doesnot satisfy (A16) cannot be a steady-state value since that fails to satisfy (A10),the lower bound for θ . That is, if

φ − β1+ φ − β Ls+h(1− β) < h(1− γ )−1− (h− s)(1+ r )− wu

h(1− γ )−1− (1− β)(h− s)(1+ r )− wu, (A17)

then we will have dHh/dLs+h > 0 even if φ > β.By simple rearrangement of (A17) we get:

β(φ − β) < h(1− γ )−1− (h− s)(1+ r )− wu

(h− s)(1+ r ). (A18)



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Thus, we need to assume only (A18) to guarantee that dHh/dLs+h > 0. Sine1> φ > β > 0, the maximum possible value of the β(φ−β) is at most 0.25 whenφ = 1 and β = 0.5.

If the world technology grows steadily without bound, then the technologylevel of an imitating country also grows steadily to allow more high-tech workersgiven the size of human capital pool, Ls+h, as given in (28). This makes (1− β)Ls+h− Hh to approach 0 for all levels of Ls+h in the long-run even if φ > β. Thus,if there is no limit to the achievable level of world technology, dHh/dLs+h > 0for all parameter values.


Since H = (H s)β(Hh)1−β and H s = Ls+h − Hh,

H = (Ls+h − Hh)β(Hh)1−β. (A19)

Using (A19),

dH

dLs+h= (Ls+h − H h)β−1(Hh)−β

{β Hh + [Ls+h(1− β)− H h]

dHh

dLs+h

}. (A20)

Substituting (A14) into (A20) yields:

dH

dLs+h= β(Ls+h − H h)β−1(Hh)−β

(φ − 1)[Ls+h(1− β)− H h] − H h

(φ − β)[Ls+h(1− β)− H h] − H h. (A21)

Since 0<φ<1 and (1−β)Ls+h> Hh, the numerator of (A21) is always negative.Thus, the sign of dH/dLs+h is the same as the negative of the sign of the denom-inator of (A21). The conditions are exactly the same as those that determinethe sign of (A14). The sufficient condition for dH/dLs+h > 0 is φ ≤ β, while thecondition (A18) is just enough.


Substitution of (28) into (4) and (5) yields:

ws = (h− s)(1+ r )βθ

(1− β)− βθ ; wh = (h− s)(1+ r )(1− β)(1− β)− βθ . (A22)

Using Proposition 4 and noting (1−β)−βθ > 0, dws/dLs+h = d(ws/dθ )(dθ/dLs+h)

> 0. Similarly, dwh/dLs+h = d(wh/dθ )(dθ/dLs+h) > 0.Noting (26), we can easily show that ∂ yII/∂Ls+h > 0 when population L is

constant since dlu/dLs+h < 0 and dwh/dLs+h (or dwh/dLs+h) as shown above.




Appendix 7. The evolution of income inequality

We need to show that the measure of income inequality for our Case II, JII , de-fined in (32) first increases and then decreases as per-capita income increases.Since the steady-state per-capita income is negatively related to the proportionof unskilled workers in the labor force in Case II (Section 3.2), it is sufficientto show that d JII/dlu first takes a negative sign and then changes its sign topositive.

d JII

dlu= (1− l u)

∂bs+h

∂l u

[1

l ubu + (1− l u)bs+h− 1

bs+h

]+ ln

(bs+h

bu

)+ bu − bs+h

l ubu + (1− l u)bs+h. (A23)

Since ∂bs+h/∂l u < 0 and bu < bs+h noting (21) and since bs+h = bh = bs, thefirst two terms in (A23) are positive and the last term is negative. As l u is smaller,bs+h/bu is larger. Thus, as per-capita income is higher or as l u is smaller, themore likely d JII/dlu is positive. Thus, d JII/dlu should have a negative sign andthen a positive one. This completes the proof.

Acknowledgments

An anonymous referee’s comments are gratefully appreciated. The author wouldalso like to thank the Alfred P. Sloan Foundation for financial support.

Notes

1. In closely related contributions, Banerjee and Newman (1991, 1993) analyzed the implicationsof income distribution on growth in a stochastic environment where capital market imperfec-tions affect investment behavior. Alesina and Perotti (1994, 1996) and Perotti (1992, 1993,1996) investigated theoretically and empirically the implications of income distribution utilizingpolitico-economic equilibrium approaches. Chou and Talmain (1996) demonstrated the impli-cations of income distribution via the labor supply on growth. See also additional references inRodrik (1994, 1996) and Galor and Tsiddon (1997b).

2. Greenwood and Jovanovic (1990) analyzed the implications of financial development and growthon income distribution. Their model may also generate the Kuznets inverted-U relation.

3. For excellent surveys on the Kuznets curve and related test results, see Adelman and Robinson(1989), Sundrum (1990), and Fields (1991). Brenner et al. (1991) covered experiences of sevenindustrialized countries that conform with the Kuznets hypothesis.

4. Schmitz (1989) emphasized the imitative activities of entrepreneures in the growth process.Rustichini and Schmitz (1991) studied the implications of both the development of knowledge(research) and the acquisition of knowledge (imitation) on long-run growth.

5. Abramovitz (1986), Lucas (1990), Edwards (1992), and Parente and Prescott (1992) pointed outthat technological spillovers may not be effective unless a receiving country has technical andsocial capabilities to absorb and implement advanced technologies. Nelson and Phelps (1966)and Benhabib and Spiegel (1992) presented an idea that human capital stock affects the speedof adoption and implementation of technology from abroad.

6. Among others, Barro (1991), Mankiw et al. (1992), and Durlauf and Johnson (1992).



268 ZANG

7. There are some exceptions: For instance, Perotti (1993) utilized a political-economy argumentto arrive at similar conclusions as ours.

8. Griliches (1969), Dougherty (1972), Fallon and Layard (1975), Mincer (1992, 1996), Brito andIntriligator (1991), and Galor and Weil (1996).

9. A CES specification for the production of human capital would be better since it could showthe implications of the degree of complementarity in human capital. See Appendix 1.

10. Our result is not sensitive to this assumption. Parente and Prescott (1992) presented a modelwhere some minimum average level of education is necessary for technology adoption. Thethreshold externalities were first introduced formally by Azariadis and Drazen (1990).

11. Since the technology level of an imitating country depends on the absolute level of high-techhuman capital in our model, countries with more people would have an advantage in supplyinghigh-tech human capital, ceteris paribus. However, by the construction of a sufficiently small φ,the absolute difference in the level of technology would be relatively small between countrieswith very different sizes of population. Suppose that, country 1 has 5 times as many as high-tech human capital as country 2, but the country 1 has high-tech human capital just half asmany as the leader country. If φ = 0.2, the steady-state level of technology of the larger country(country 1) will be 87% of that of the leader country, while the smaller one (country 2) attains asmuch as 63%. If we assume that technology transfer depends on the average level of high-techhuman capital, the size of population would not matter.

12. If population growth would be introduced in the model, we would have some interesting resultsat the cost of complexity, although our main conclusion would not be affected. If populationis allowed to change, then per-child human capital investment or bequest determines the dis-tribution of workers in the labor force. An increase in family size will affect adversely per-childhuman capital investment (see Galor and Zang, 1997). However, the increase in the size of pop-ulation may have a positive effect on the absolute number of high-tech human capital whichincreases the level of technology of the country. Thus, population growth has two potentialopposing effects on economic growth.

13. Allowing consumers to borrow with higher rate than firms would not alter our results (see Galorand Zeira, 1993). There are several papers studying the implications of income distribution ongrowth that are based on borrowing constraints. Notable contributions are Galor and Zeira(1993), Banerjee and Newman (1991, 1993), Aghion and Bolton (1991, 1992), and Perotti (1993).

14. Thus there is a room for government intervention. For an insightful discussion about privateversus public education, see Glomm and Ravikumar (1992).

15. An alternative way to model the bequest motive is to assume that individuals draw utility fromtheir offspring’s utility, not directly from the bequest itself. Both specifications lead to similarresults.

16. Appendix 2 shows the lower bound for feasible values of λt for a given θ . The upper bound forλt to satisfy (19) will be discussed later and it turns out to be the same as the RHS of (27) or thecurve ζ(θ) in figures 2(a) and (b). If λt is within the upper bound for a given θ , we are in Case I;otherwise, we are in Case II. We do not draw the lower bound in figures 2(a) and (b).

17. We draw the case where the wage rate for high-tech human capital increases as more individualsbecome high-tech workers.

18. Actually, a weaker restriction on parameters than φ ≤ β is sufficient. See Appendix 4. It canbe also shown that if the feasible level of the world technology is unbounded, no parameterrestrictions are needed to obtain this result.

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