Endogenous human capital formation, distance to frontier and growth

Research in Economics 68 (2014) 117–132

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Endogenous human capital formation, distance to frontier and growth Sujata Basu a,n, Meeta Keswani Mehra b a Center for International Trade and Development (CITD), School of International Studies (SIS), Jawaharlal Nehru University (JNU), New Delhi, India b Professor of Economics, CITD, SIS, JNU, New Delhi, India

a r t i c l e i n f o

abstract

Article history: Received 11 January 2013 Accepted 12 November 2013 Available online 26 November 2013

We examine human capital's contribution to economy-wide technological progress through two channels – imitation and innovation – innovation being more skillintensive than imitation. We develop a growth model based on the endogenous abilitydriven skill acquisition decision of an individual. It is shown that skilled human capital is growth enhancing in the “imitation-innovation” regime and in the “innovation-only” regime whereas unskilled human capital is growth enhancing in the “imitation-only” regime. Steady state exists and, in the long run, the economy converges to the world technology frontier. In the diversified regime, technological progress raises the return to ability and generates an increase in wage inequality between and within groups – consistent with the pattern observed across countries. & 2013 University of Venice. Published by Elsevier Ltd. All rights reserved.

Keywords: Economic growth Endogenous labor composition Imitation–innovation Convergence Wage inequality

1. Introduction 1.1. Background The relationship between level of education and economic growth has been much debated upon. Krueger and Lindahl (2001) find that education is statistically significant and positively associated with subsequent growth only for countries with lowest levels of education, and is insignificant and negative for countries with middle and high levels of education. That is, an inverted U-shaped relationship is found to exist between the level of education and the growth rate of an economy.1 The relationship is insignificant when the regression is run for the Organization of Economic Cooperation and Development (OECD) countries alone. This finding is apparently puzzling – basic education helps growth but the relationship between the two dies down as the country progresses in terms of education. A possible explanation for this phenomenon could be that education favors the adoption of new technologies but the aggregate level of human capital does not matter, or its impact gets weaker as the economy progresses toward the frontier, as noticed by Nelson and Phelps (1966). On the other hand, according to Romer (1990), education favors the innovation of new technologies. But Nelson and Phelps (1966) seem to have ignored the fact that a country can also improve its technology level through innovation, while Romer (1990) overlooks the fact that technology improvement can be possible through imitation from the world technology frontier, especially for a technologically backward economy. Benhabib and

n Correspondence to: Room No 209, Centre for International Trade and Development, School of Social Science, Jawaharlal Nehru University, New Mehrauli Road, New Delhi 110067, India. Mobile: þ 91 9717675108. E-mail addresses: [email protected], [email protected] (S. Basu). 1 This finding is similar to Durlauf and Johnson (1995) and Kalaitzidakis et al. (2001).

1090-9443/$ - see front matter & 2013 University of Venice. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.rie.2013.11.002

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Spiegel (1994) and Barro (1998) find that the catch-up component of growth is the dominant factor for the technologically backward economy. To find a possible explanation to the puzzle posed by Krueger and Lindahl (2001), we need to focus on both – economy's distance to the world technology frontier and on the composition of its human capital (as much as on its level). The cross-country analysis of Vandenbussche et al. (2006) and the cross US-state analysis of Aghion et al. (2009) bring out the role of appropriate institutions in analyzing the relationship between the growth and the composition of human capital in the diversified regime (that is, where an economy performs both imitation and innovation for further technology improvement). Using a theoretical model, they show that skilled human capital has a higher growth-enhancing effect closer to the technology frontier but unskilled human capital is the main source of growth for a technologically backward economy.2 Furthermore, Vandenbussche et al. (2006) provide evidence in favor of this prediction using a panel data set covering 19 OECD countries covering the time period 1960 and 2000. Unlike earlier research, they not only take into account the stock of human capital but also its composition, which makes a significant difference to their analysis. They are able to solve the puzzle posed by Krueger and Lindahl (2001) by showing that education has a positive and significant impact on economic growth even for an economy with high level of education. Aghion et al. (2009) confirm the same prediction in respect of cross-US-state panel data. The main drawback of both these studies is that they assume that there exists an exogenously given composition of skilled– unskilled human capital. They only seem to consider the benefit of skilled human capital and ignore the fact that there is a cost associated with skill acquisition. Also, the growth enhancing composition of skilled human capital cannot be the same for an economy, irrespective of its distance to the frontier. We attempt to fill this important gap by endogenizing skill composition, based on individual's decision to acquire education or not. Di Maria and Stryszowski (2009) also endogenize the composition of human capital in the above discussed setup but the purpose of their analysis is entirely different. They want to examine the impact of migration on the process of growth. They do not characterize the importance of different compositions of human capital on economic growth as we do in this paper. Moreover, the modeling setup of endogenizing the education decision of our model is completely different from Di Maria and Stryszowski (2009). Unlike them, we assume that education decision is not only affected by the cognitive ability of an individual but also by the opportunity cost of education, that is, the outside scope of income of an individual. Furthermore, Aghion et al. (2009) and Vandenbussche et al. (2006) only characterize the intermediate economies, which perform both types of research activities – imitation and innovation. Unlike our research, they are entirely silent about the growth enhancing education policy of sufficiently advanced and backward economies, which may be completely specializing in either imitation or innovation activities alone. The distinguishing features of our research are 1. Unlike Vandenbussche et al. (2006) and Aghion et al. (2009), we consider heterogeneous agents, and by endogenizing individual's schooling decision, the composition of skilled and unskilled human capital in any time period is ascertained. Further, we characterize the specialized economies – which perform either only-imitation or only-innovation activities. 2. Under the assumption that innovation is skilled human capital-intensive, in the diversified regime, the level of skilled human capital increases and that of unskilled human capital decreases as the economy progresses, ascribable to the rising importance of innovation. However, this increase in the share of skilled human capital in the overall human capital composition is missing in Aghion et al. (2009) and Vandenbussche et al. (2006) works. As a consequence, the shift of both skilled and unskilled human capital from imitation to innovation activity is relatively more pronounced in our case than in their work. Later in Section 3.5.3, we elaborate on how this changes the growth enhancing policy of the diversified regime as prescribed by Aghion et al. (2009) and Vandenbussche et al. (2006). On the other hand, in our research, there exists a fixed composition of skilled and unskilled human capital in the only-imitation and only-innovation regimes, since further technology improvement has a similar impact on both types of human capital. 3. By considering both the cost and the benefit associated with choosing to be skilled, we show that skilled human capital is growth enhancing for an economy which performs both imitation and innovation activities and only-innovation activity, while unskilled human capital is growth enhancing for an economy which performs only-imitation activity. Our findings contradict the policy prescription of Vandenbussche et al. (2006) and Aghion et al. (2009), since according to them, unskilled human capital is growth enhancing for a relatively technologically backward economy, which performs both imitation and innovation activities. They did not characterize the specialized economies – which perform either onlyimitation or only-innovation activities. Also, since the motivation of the study is entirely different for Di Maria and Stryszowski (2009), they also do not say anything about the growth enhancing education policy of an economy irrespective of its distance to the frontier. Furthermore, we contradict the theory of Grossman and Helpman (1991) according to whom skilled human capital is growth enhancing whereas unskilled human capital is growth depressing irrespective of the economy's distance to frontier. 4. The dynamics of the stylized economy show that by implementing an appropriate education policy, in the long run an economy will convergence to the world technology frontier irrespective of its distance to the frontier. 5. Moreover, in the cone of diversified region, as an economy progresses technologically, the average income of skilled human capital rises while that of unskilled human capital falls. Consequently, this raises the income inequality between skilled and unskilled human capital groups. Since technology improvement has a heterogeneous effect on individuals'

2 Here unskilled human capital implies those individuals who have technical skill but do not acquire the general purpose technology. That is individuals with primary/secondary education are considered as an unskilled worker and people with tertiary education as a skilled worker.

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incomes (depending on their cognitive ability), it tends to raise within group inequality among skilled human capital. On the other hand, in the imitation-only region, due to diminishing marginal return to the imitation activity, as an economy progresses, average income and, consequently, average consumption of skilled and unskilled human capital decreases. Further, aggregate consumption of the economy decreases as an economy progresses. There exists a constant income inequality between skilled and unskilled human capital and within skilled human capital groups. In the innovation-only region, as the technology level increases, the average income and consumption of skilled and unskilled human capital increase and so does aggregate consumption of the economy. In this region also there exists a constant income inequality between and within groups of skilled and unskilled human capital. 1.2. Structure of the paper The paper is organized as follows. In Section 2, the basic structure of the model is discussed, and the economic intuition for the behavior of the economy over time is provided. Section 3 contains the main analytical results for a decentralized market economy. Here, we derive the labor market clearing condition and the equilibrium growth rate of the economy. We also characterize the equilibrium wage rate and the income of skilled and unskilled human capital and between and within group inequality levels of the economy. We derive the balanced growth path for the economy in the decentralized equilibrium. The equilibrium consumption basket for both skilled and unskilled human capital and the consumption growth rate of the overall economy are also characterized. Section 4, shows the importance of the composition of human capital on growth with some degree of complementarity between imitation and innovation activities. Section 5 concludes our work and discusses some policy implications of this research. Moreover, the direction for future research is provided. In what follows immediately, we describe the structure of the economy. 2. The economic environment 2.1. Production There are a finite number of small open economies. In each economy there is an entrepreneur who is engaged in the production of a final output, and there is a continuum of intermediate input producers. Each intermediate input producer also invests in the R&D sector for productivity improvement, where the latter is characterized by free entry and exit. In any time period, each intermediate input producer possesses the locally available highest technology level; otherwise, he/she will be thrown out of the market. There is size 1 population of workers. Workers have heterogeneous cognitive ability. We also assume that once an individual becomes educated (or else decides not to go for education), the marginal return from education (or the lack of it), in terms of the wage rate, would be the same for all skilled (or unskilled) workers with different cognitive abilities. That is, all individuals working as skilled (resp. unskilled) workers are equally productive in the R&D activity irrespective of their cognitive ability.3 We assume that among skilled workers any one organizes the production of final output, implying that he/she is the entrepreneur. All the other skilled workers engage in the production of intermediate inputs. A small open economy implies that an individual can lend or borrow any amount from the world economy, and the economy is a price taker. Therefore, the rental rate for borrowing/lending is fixed at r t in any period t. We consider discrete time intervals, where all the agents live for two time periods in an overlapping generations framework. In every time period, and in each country, the final output is produced competitively by using land and a continuum of mass 1 of intermediate inputs. For simplicity, we normalize the total supply of land to 1. We consider Cobb–Douglas production function of 1α R 1 1α α the form: Y t ¼ lt xit di; 0 o αo 1, where Yt is the final output in period t, lt is the total amount of land (lt ¼ 1, 8 t), Ait is the 0 Ait level of technology in sector i in period t, and xit is the amount of intermediate input i in period t. Since the final good sector produces under perfect competition, the price of the ith intermediate input in period t is equal to its marginal product in the final good sector. That is, pit ¼ ∂Y t =∂xit ¼ αA1it  α xαit  1 , where pit denotes the price of the intermediate input in sector i in period t. Each unit of the intermediate good is produced by using one unit of the final good as capital. The monopolist produces the following amount of the intermediate good in sector i in period t: xit ¼ α2=1  α Ait . Accordingly, the profit of the intermediate input producer in sector i in period t is π it ¼ ðpit  1Þxit ¼ ð1=α 1Þα2=1  α Ait ¼ δ1 Ait , where δ1 ¼ ð1=α 1Þα2=1  α . Note that both the equilibrium production and the profit in the ith intermediate input sector in period t are linearly dependent on the local (national) technology level in sector i in that period. That is, the technology adjusted intermediate inputs and profit are the same for all the sectors in every time period. 2.2. Dynamics of productivity A country can improve its technology level through two channels – imitating from the world technology frontier or innovating upon the local technology level. The technology improvement function is postulated as s ðA t  1  At  1 Þ þ γuϕnit s1nit ϕ At  1 ; Ait ¼ Ai;t  1 þ λ½usmit s1mit

3

λ 40; γ 4 0;

This assumption will become clearer once we introduce technology improvement process in Eq. (1).

ð1Þ

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where A t  1 is the world technology frontier at time t  1, At  1 is country's local technology frontier at time t  1, um;i;t (resp. sm;i;t ) is the amount of unskilled (resp. skilled) human capital input used in imitation at time t, un;i;t (resp. sn;i;t ) is the amount of unskilled (resp. skilled) human capital used in innovation at time t, s (resp. ϕ) is the elasticity of unskilled human capital in imitation (resp. innovation), γ measures the relative efficiency of innovation compared to imitation, and λ measures the efficiency of the overall process of technological progress. Since imitation and innovation are assumed to substitute for each other, one cannot ignore the possibility of a corner solution. Therefore, the specific technology improvement function for a sufficiently technologically backward economy can be expressed as Ait ¼ Ai;t  1 þ λusit s1it  s ðA t  1 At  1 Þ. We call this the “imitation-only” regime. Similarly, the specific technology improvement function for a sufficiently technologically advanced economy can be written as Ait ¼ Ai;t  1 þ λγuϕnit s1nit ϕ At  1 . Henceforth, this is referred to as the “innovation-only” regime. To satisfy our basic assumption that innovation is relatively more skilled human capital intensive than imitation, we make the following assumption4: A1. The elasticity with respect to skilled human capital is higher in innovation activity than in imitation activity or s 4 ϕ. In other words, the elasticity of unskilled human capital is higher in imitation than in innovation. In the same vein, under the imitation-only regime, imitation is unskilled human capital intensive, that is, s 4 1=2 and in the innovation-only regime, innovation is skilled human capital intensive, that is, ϕo 1=2.5

2.3. Consumption We assume that an individual devotes τ fraction of his/her first period time on education and the remaining (1  τ) fraction to work as a skilled worker in the R&D sector. Accordingly, to acquire education an individual has to sacrifice his/her income as an unskilled worker during the time spent on education. That is, the opportunity cost of education is the foregone wage rate of unskilled worker in that period. Furthermore, we assume that getting educated is easier for the more talented individual (that is, individual with higher cognitive ability). It is assumed that cognitive ability is uniformly distributed over the interval [0, 1]. On the other hand, without bearing any additional cost, an individual can work as an unskilled worker. A skilled human capital works and studies simultaneously. The entrepreneur, the intermediate input producers and both the types of human capital retire in the second period of their lives. Moreover, an individual consumes in both the periods of his/her life. The cost of education of an individual with cognitive ability level, θ, is expressed as Eðθ; At  1 Þ ¼ τwut =θ; 0 o τ o1. Both skilled and unskilled human capital maximize their lifetime utility subject to the budget constraint. Each individual maximizes the following utility function Wk ¼ log ct;t þ ð1 þ ρÞ  1 log ct;t þ 1 ; where k ¼ s; u, where Wk is the lifetime utility of the kth individual, ct;t is the consumption of an individual in period t who is born in period t, ct;t þ 1 is the consumption of an individual in period tþ 1 who is born in period t, s denotes skilled worker and u denotes unskilled workers, ρ is the discount rate. Along with his/her consumption in the first and second period of his/her life, skilled human capital has to bear the cost of education. Accordingly, the budget constraint for skilled human capital is ct;t þτwut =θ þ ct;t þ 1 =1 þ r t þ 1 ¼ ð1  τÞwst ; where wst is the wage rate of skilled human capital in period t. Since unskilled human capital spends his/her entire income on consumption, the budget constraint of unskilled human capital is ct;t þ ct;t þ 1 =1 þ r t þ 1 ¼ wut . We assume a perfectly competitive labor market. Individuals have perfect foresight. For more mathematical tractability, we assume that there is no population growth. Each parent bears one child. At the end of the first period of the tth generation, a new (tþ1)th generation appears. Before moving on to the analytical results, we provide an intuitive description of how the economy behaves over time. In period t þ1 a new generation appears. Depending on the available knowledge in period t, and his/her cognitive ability, each individual decides whether he/she would opt for education or not. This determines the composition of skilled and unskilled human capital in period tþ1. A perfectly competitive labor market implies equalization of the wage rates of both skilled and unskilled human capital between imitation and innovation activities. Depending on the efficiency of both – imitation and innovation – activities in period tþ1, the market allocates the total stock of skilled and unskilled human capital to these two activities. For a sufficiently technologically backward (resp. advanced) economy, the market allocates neither skilled nor unskilled human capital to innovation (resp. imitation) activity; instead, these are committed entirely to the imitation (resp. innovation) activity. This constitutes our “imitation-only” (resp. “innovation-only”) regime. In comparison, the intermediate economies perform both imitation and innovation activities. This is known as the “imitation–innovation” regime. The efficiency of imitation and innovation activities yields the next period's technology level, At þ 1 . The technology level in period tþ 1 determines the production of the intermediate inputs in sector i, which in turn, determine the final output in this period. This generates the income and the consumption for both skilled and unskilled human capital in period t þ1. Next, the technology level in period tþ1 ascertains the next period's technology level. This process continues and, in the long run, the economy tends to converge to the world technology frontier, as will be seen below. 4 Here efficiency or productivity of skilled or unskilled human capital is measured in terms of the elasticity of skilled or unskilled human capital in imitation or innovation activity. 5 In the diversified regime our work does not require any assumption on the absolute intensity of skilled or unskilled human capital in the imitation or innovation activities. Hence, these parametric restrictions pertain only to the specialized economies.

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3. Analytical results We now formally derive the key results of our analysis.

3.1. Labor supply The total supply of human capital in period t is determined by individual's lifetime utility maximization exercise. From a comparison of costs and returns associated with education, the cut-off or threshold value of the parameter θ(  θ^ t ) above in which an individual chooses the status of being skilled is given by θ^t Z

τwut : ð1  τÞwst  wut

ð2Þ

For simplicity we assume that individuals who are indifferent between working as skilled or unskilled choose to work as skilled.

3.2. Imitation-only regime We first consider the case where an economy performs only-imitation activity.6

3.2.1. Demand for labor An intermediate input producer chooses skilled and unskilled human capital by maximizing its profit in the R&D sector. Since an intermediate input producer operates the production process for one period only, he/she maximizes current profit net of labor costs. Profit maximizing exercise of the intermediate input producer is max λδ1 U sit S1it  s ðA t  1  At  1 Þ  wut U it wst Sit :

ð3Þ

U it ;Sit ;

Given that all intermediate input producers face the same maximization problem, in equilibrium we have umit ¼ umt ; smit ¼ smt ; unit ¼ unt ; snit ¼ snt . There is mass 1 of intermediate firms, so that labor market clearing condition is St ¼ smt þsnt ; U t ¼ umt þ unt . 3.2.2. Labor market in equilibrium Under perfect competition, labor market always clears. By equating relative demand for skilled and unskilled human capital with its respective relative supply we get the equilibrium level of skilled and unskilled human capital in a sufficiently technologically backward economy.7 Due to diminishing marginal productivity to technology improvement in the imitationonly regime the returns to both skilled and unskilled workers falls as an economy progresses. From the first-order condition in Eq. (3) it can be inferred that this decrement has a similar effect on both types of human capital, implying a constant relative wage rate of skilled–unskilled human capital. As a result, there exists a fixed composition of skilled and unskilled human capital in the imitation-only equilibrium, and there is absence of any transitional dynamics in this case. One can now comment on the growth enhancing education policy of an economy, in terms of whether to invest in primary, secondary or tertiary education based on the varying cost of education. Our main instrument for this is the cost of education, denoted by τ. We, therefore, attempt comparative dynamics with respect to τ. From Eq. (31) it is clear that with an increase in τ there is an increase in the level of unskilled human capital and a decrease in the level of skilled human capital. Intuitively, an increment in τ reduces the income of a skilled individual, but does not impact the income of unskilled worker. As a result, individuals who earlier marginally benefited from education will now prefer to work as unskilled workers. This reduces the level of skilled human capital and raises that of unskilled human capital. As unskilled worker works for one period alone and bears no additional cost of education, his/her absolute income is identical to his/her absolute wage rate. This leads us to conclude that the absolute income and the consumption of unskilled worker behaves in the same manner as the wage rate of unskilled worker. On the other hand, the average income of skilled M

human capital, which is an increasing function of its cognitive ability is found to be I s;t ¼ I st jθ ¼ 1 þ I st jθ ¼ θ^ t =2 ¼ M I s;t

measures the average income of skilled ðλδ1 =2ÞðSn =U n Þ  s ðA t  1  At  1 Þ½2ð1  τÞð1  sÞ τsðSn =U n Þð1 þ U n =U n Þ, where human capital in the imitation-only regime and Ikt denotes the income of kth type of human capital in time period t, where k ¼ s; u. As discussed, in Eq. (32) the wage rate of skilled human capital decreases in the imitation-only regime. Similarly, the average income as well as consumption of skilled human capital decreases as an economy moves to the technology frontier. 6 7

Later in Section 3.4.4 we explicitly characterize the parametric restrictions under which the economy will perform only-imitation activity. Detailed mathematical derivations are provided in Appendix A.

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3.3. Innovation-only regime Now, let us consider an economy which performs only-innovation.8 Similar to the imitation-only regime, here too we get a fixed composition as well as a constant relative wage rate for both types of human capital.9 Specifically, now the marginal productivity of skilled and unskilled human capital increases, as the technology level increases and so does their wage rate and the average income. But, from the profit maximizing exercise of the R&D sector, in the innovation-only regime, it is clear that the relative gain of higher knowledge is identical for both the factors. As a result, the relative wage rate of skilled and unskilled human capital is constant and this induces a fixed composition of skilled and unskilled human capital in the innovation-only regime. From the comparative dynamics exercise, we understand that as the cost of education increases (that is, there is a rise in τ) there is an increase in the level of unskilled human capital and a decrease in that of skilled human capital. The logic for the above argument is the same as that provided for the imitation-only regime. 3.4. Combined imitation and innovation regime Finally, we consider those economies which perform both imitation and innovation activities, or are characterized by diversified R&D sector. We now formally analyze this case. 3.4.1. Demand for labor We begin by determining the demand for skilled and unskilled human capital in innovation and imitation activities. Each intermediate producer chooses umit ; smit ; unit ; snit by solving the following maximization problem: max

umit ;unit ;smit ;snit

s λδ1 ½usmit s1mit ðA t  1  At  1 Þ þ γuϕnit s1nit ϕ At  1   ½wut ðumit þ unit Þ þ wst ðsmit þsnit Þ:

ð4Þ

At any time t, the allocation of skilled and unskilled human capital in the diversified regime is solved to be hðat  1 ÞU t St ψSt  hðat  1 ÞU t ; snt ¼ ; ψ 1 ψ 1 ψ½hðat  1 ÞU t  St  ψSt  hðat  1 ÞU t ; unt ¼ ; umt ¼ ðψ  1Þhðat  1 Þ ðψ 1Þhðat  1 Þ

smt ¼

ð5Þ

where ψ ¼ sð1  ϕÞ=ϕð1  sÞ 4 1, at  1 ¼ At  1 =A t  1 , and hðat  1 Þ ¼ ½ð1 sÞψ s ð1 at  1 Þ=γð1 ϕÞat  1 1=s  ϕ . That is, at  1 is the inverse measure of country's distance to the world technology frontier. As a country moves to the world technology frontier, the value of at  1 increases and approaches 1. The ratio ð1  at  1 Þ=at  1 measures the relative distance of the economy from the world technology frontier. As an economy progresses to the frontier, the difference between the local/national technology level and the world frontier decreases, thus reducing the relative distance of the economy from the frontier. Since hðat  1 Þ is a monotonic function of the relative distance of the economy from the frontier, as an economy progresses technologically, the value of hðat  1 Þ decreases. That is, hðat  1 Þ is a decreasing function of at  1 . Eqs. (5) imply smt hðat  1 Þ ; ¼ ψ umt

snt ¼ hðat  1 Þ: unt

ð6Þ

which is used later to determine the growth enhancing level of skilled and unskilled human capital and its allocation to both imitation and innovation activities. 3.4.2. Labor market equilibrium By substituting Eq. (6) and the first-order condition associated with Eq. (4) into Eq. (2) we get the growth enhancing composition of human capital in period t as10 Ut ¼

τϕhðat  1 Þ τshðat  1 Þ ¼ : ½ð1  τÞð1 ϕÞ  ϕhðat  1 Þ ½ψð1 τÞð1  sÞ  shðat  1 Þ

ð7Þ

For the existence of positive quantities of both skilled and unskilled human capital in equilibrium, the following condition needs to be satisfied U t 4 0 ) ½ð1  τÞð1 ϕÞ  ϕhðat  1 Þ 4 0:

ð8Þ

Now, we attempt some comparative dynamics. We first discuss the impact of an increase (resp. a decrease) in the cost of education on the composition of the human capital. As can be seen from Eqs. (7) and (8), dSt =dτ o 0; and dU t =dτ 40. Intuitively, due to reasons similar to those discussed earlier in Section 3.2.2, as the cost of education increases, the economy-wide equilibrium level of skilled human capital decreases and that of unskilled human capital increases. 8 9 10

Later in Section 3.4.4 we characterize the parametric restrictions under which an economy performs only-innovation. Detailed mathematical derivations are provided in Appendix B. Detailed derivation of the first order condition of Eq. (4) is provided in Eq. (33) in Appendix C.

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Next, we study the transition path of skilled and unskilled human capital as the economy moves to the world technology frontier. We have dU t =dat  1 o 0 and dSt =dat  1 40. From Eq. (1), the catch-up effect is high for a technologically backward economy (that is, at  1 is low and hðat  1 Þ is high). Under A1, the demand for unskilled human capital is relatively high in a technologically backward economy. Moreover, as a country moves toward the world technology frontier, imitation becomes relatively less important than innovation. Under A1, the demand for skilled (resp. unskilled) human capital increases (resp. decreases) and associated with this, the marginal productivity of skilled (resp. unskilled) human capital increases (resp. decreases) with economic progress. Therefore, the growth enhancing level of skilled human capital increases and unskilled human capital decreases as a country gets closer to the world technology frontier. This indicates that, keeping a fixed amount of skilled and unskilled human capital for any distance from the world technology frontier cannot be an optimum decision, an assumption made by Aghion et al. (2009), and Vandenbussche et al. (2006). Our above discussion casts a doubt on their assumptions. Thus, we have Lemma 1. Under A1, (i) The stock of skilled human capital increases (resp. decreases) and unskilled human capital decreases (resp. increases) as a country moves to the world technology frontier, even as the aggregate stock of human capital remains unchanged for the country which is performing both imitation and innovation activities. (ii) Any increment (resp. decline) in the cost of education decreases (resp. increases) the total stock of skilled human capital and increases (resp. decreases) the stock of unskilled human capital.

Further, we attempt to characterize the growth enhancing allocation of human capital into both – imitation and innovation – activities. Substituting the value of Ut from Eq. (7) into Eq. (5), the solution to the growth enhancing allocation of skilled and unskilled human capital in imitation and innovation activities to be: ψτϕ½1 þ hðat  1 Þ ψ  ; ðψ  1Þ½ð1  τÞð1  ϕÞ  ϕhðat  1 Þ ðψ 1Þhðat  1 Þ ψ τϕ½ψ þhðat  1 Þ  ; unt ¼ ðψ  1Þhðat  1 Þ ðψ  1Þ½ð1 τÞð1  ϕÞ  ϕhðat  1 Þ τϕhðat  1 Þ½1 þ hðat  1 Þ 1 ;  smt ¼ ðψ 1Þ½ð1  τÞð1 ϕÞ  ϕhðat  1 Þ ψ 1 ψ τϕhðat  1 Þ½ψ þ hðat  1 Þ  : snt ¼ ψ  1 ðψ 1Þ½ð1  τÞð1 ϕÞ  ϕhðat  1 Þ umt ¼

ð9Þ

We again carry out a comparative dynamics exercise. We first analyze the impact of a relatively higher cost of education on the allocation of skilled and unskilled human capital in imitation and innovation activities. From Eq. (9), we get dumt =dτ 4 0; dsmt =dτ 4 0; dunt =dτ o 0; dsnt =dτ o 0. Intuitively, under Lemma 1, as the cost of education increases, the level of unskilled human capital increases. From A1, imitation attracts more unskilled human capital and, due to the complementarity effect, it also raises the demand for skilled human capital in the imitation activity. This again raises the marginal productivity of unskilled human capital in imitation activity as a second round effect. This process continues entailing an increase in the employment of both skilled and unskilled human capital in imitation activity and thus fall in innovation activity. We now turn toward an analysis of the transition path of the allocation of skilled and unskilled human capital as an economy progresses to the world technology frontier. From Eq. (9), we have dumt =dat  1 o 0; dsmt =dat  1 o 0; dunt =dat  1 4 0; dsnt =dat  1 40. First, from Lemma 1, as an economy progresses, the level of skilled human capital increases and by A1, both skilled and unskilled human capital shift from imitation to innovation activity. Secondly, as the economy moves to the world technology frontier, that is, at  1 increases and hðat  1 Þ decreases, the catch-up effect tends to get weaker and the economy relies more on innovation activity for further technological progress. Under A1, innovation requires more skilled human capital. Due to the complementarity effect, both skilled and unskilled human capital move away from imitation into innovation activity. Both these factors induce a decline in the employment of skilled and unskilled human capital in imitation and an increase of it in innovation activities, as the economy closes its gap with the world technology frontier. Hence, Lemma 2. Under A1, (i) The optimal amount of skilled and unskilled human capital employment increases (esp. decreases) in innovation (resp. imitation) activity, as a country moves to the world technology frontier. (ii) A relatively more (resp. less) expensive education system shifts skilled and unskilled human capital from innovation (resp. imitation) to imitation (resp. innovation) activity.

3.4.3. Wage rate and income of skilled and unskilled human capital We now try to derive the market clearing wage rate and, consequently, the income of both skilled and unskilled human capital inside the cone of diversification. In this region, we also analyze the behavior of the competitively-determined wage rate and income as the economy moves toward the frontier.

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First, we derive the wage rates of skilled and unskilled human capital from the first-order conditions associated with Eq. (4), together with Eq. (6), to be11: wst ¼ λδ1 γð1  ϕÞh 1ϕ

wut ¼ λδ1 γϕh

ϕ

ðat  1 Þat  1 A t  1 ¼ λδ1 ð1 sÞψ s h

ðat  1 Þat  1 A t  1 ¼ λδ1 sψ

 ð1  sÞ 1  s

h

s

ðat  1 Þð1 at  1 ÞA t  1 ;

ðat  1 Þð1  at  1 ÞA t  1 :

ð10Þ

We analyze the transition path of the wage rate of both kinds of labor force as an economy progresses to the world technology frontier. Differentiating Eq. (10) w.r.t. at  1 we get that dwst =dat  1 4 0 and dwut =dat  1 o 0. From Eq. (1) this can be explained as follows: the contribution of imitation is lower in the case of a relatively advanced economy. From A1, the marginal productivity of skilled human capital increases and unskilled human capital decreases and, consequently, the wage rate of skilled human capital also rises and unskilled human capital falls as an economy moves to the world technology frontier. While Di Maria and Stryszowski (2009) only concentrate on the relative movement of the wage rate, our analysis not only characterizes the relative wage rate but also the absolute levels of wages of skilled and unskilled human capital. Observe that, in the diversified regime as an economy progresses to the frontier, the total stock of skilled human capital rises, and so does its wage rate. That is, as the supply of skilled human capital rises, the absolute wages are not lowered; rather these are raised. This might appear contradictory to the traditional demand-supply theory. Intuitively what is at work is as follows. An increase in the stock of skilled human capital raises the next period's technology level. Since we consider knowledge-driven technological progress, by A1, this further raises the demand for skilled human capital. That is, the relative demand for skilled human capital is now proportionately higher than its relative supply. So, our finding that there will be an increment in the wage rate of skilled human capital, following an increase in the supply of skilled human capital, in the transition, is in conformity with standard theory. This finding is also in line with Galor and Moav (2000), Acemoglu (2002) and Acemoglu (1998).12 In the standard literature this is known as “skilled-biased technological progress”. We now analyze the average income and consumption paths of skilled and unskilled human capital. As discussed earlier, the absolute income and the consumption of unskilled human capital behave in the same manner as their wage rate. The average income of skilled human capital in the cone of diversified region is captured by I s;t ¼ I st jθ ¼ 1 þ ϕ I st jθ ¼ θ^ t =2 ¼ λδ1 γð1  τÞh ðat  1 Þ½ð1  ϕÞ þ ϕhðat  1 Þat  1 A t  1 =2, where I s;t measures the average income of skilled human capital in period t. A decrement in the distance of an economy from the world technology frontier not only decreases the cost of education but also raises the return from education. As a result, the average income as well as the consumption of skilled human capital increases as an economy moves to the world technology frontier. 3.4.4. Existence possibility of imitation or innovation activities At this point, we examine the private decision of the intermediate input producers to invest in imitation or innovation activity. The existence of an interior solution requires that smt rSt , snt r St ; smt Z0; and snt Z0. Lemma 3. For further technology improvement an economy performs both imitation and innovation activities if and only if ½ð1  ϕÞ  ϕhðat  1 Þ ψ½ð1  ϕÞ  ϕhðat  1 Þ oτo ; 2 ½ð1 ϕÞ þϕhðat  1 Þð1 þ hðat  1 ÞÞ ϕh ðat  1 Þ þψ½ð1 ϕÞ þϕhðat  1 Þ accordingly, an economy performs only-imitation if and only if τZ

ψ½ð1 ϕÞ ϕhðat  1 Þ 2

ϕh ðat  1 Þ þ ψ½ð1  ϕÞ þ ϕhðat  1 Þ

;

and an economy performs only-innovation if and only if τr

½ð1  ϕÞ  ϕhðat  1 Þ : ½ð1  ϕÞ þϕhðat  1 Þð1 þ hðat  1 ÞÞ

ð11Þ

For the existence of both imitation and innovation activities, the condition in (11) is both necessary and sufficient. This condition is similar to those derived by Aghion et al. (2009), Vandenbussche et al. (2006) and Di Maria and Stryszowski (2009) for an interior solution. We endogenize human capital allocation based on individual's private optimization decision and the cost of education, which determine the growth enhancing composition of human capital depending on its distance to frontier. One can interpret this condition in one of the following two ways – either based on economy's distance to the frontier (that is, at  1 or hðat  1 Þ), or related to the cost of education (that is, τ), both of which determines the relative composition of skilled–unskilled human capital. Given the relative factor endowments of skilled–unskilled human capital in a period, a country never opts for innovation (resp. imitation) if it is too far away from (resp. close to) the frontier. Instead, it specializes in imitation (resp. innovation) activity since the catch-up effect is high (resp. low), implied by a high (resp. low) value of hðat  1 Þ. We can also interpret this condition in another way. Under A1, for any value of a, a relatively skilled (resp. 11

Detailed derivation of the first order condition of Eq. (6) is provided in Eq. (33) in Appendix C. This finding is also empirically supported by Autor et al. (1998) and Katz et al. (1992) for the United States, Machin (1998) for the OECD countries, and Himanshu (2007), Sarkar and Mehta (2010) for India. 12

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unskilled) human capital abundant economy (that is, when τ is low (resp. high)) completely specializes in innovation (resp. imitation) activity. 3.4.5. Existence of skilled human capital An individual will not opt for education if the benefit from acquiring education is lower than that from not having it. Since the net income of skilled human capital is an increasing function of his/her cognitive ability, if the individual with maximum cognitive ability is not induced to go for education, then it is not optimal for anyone to opt for education. Therefore, a sufficient condition for no individual opting for education will be: I ut jθ ¼ 1 4 I st jθ ¼ 1 )

ð1  ϕÞ ϕhðat  1 Þ o τ: ð1  ϕÞ þϕhðat  1 Þ

ð12Þ

Under (12), there is no skilled human capital in the economy. Thus, the converse of this condition is necessary for the existence of skilled human capital. The condition required for the existence of both imitation and innovation activities ensures the existence of skilled human capital in an economy. That is, the converse of condition (12) is already ensured by condition (11). 3.5. Growth rate of an economy at the decentralized equilibrium In this subsection we characterize the growth rate of an economy in period t depending on its distance to frontier. We also characterize the growth enhancing education policy of an economy depending on its distance to the frontier. Rearranging the technological progress in Eq. (1) and summing over all i, one can define the growth rate of a R1 decentralized economy in period t as g t ¼ 0 Ait  At  1 =At  1 di. 3.5.1. Imitation-only regime The growth rate of the economy in the imitation-only regime is captured by13   Z 1 Ait  At  1 s 1  s 1 at  1 gM di ¼ λU n Sn ; t ¼ At  1 at  1 0 M

ð13Þ M

where gt denotes the growth rate of an economy in the imitation-only regime. From Eq. (13), we have dgt =dat  1 o 0. Due to diminishing marginal returns to the imitation activity, the scope of further technology improvement decreases as an economy progresses. As a result, in the imitation-only regime, the growth rate of an economy decreases as the distance to the frontier decreases. One can now find out the growth enhancing education policy of an economy which is in the imitation-only regime. For this we carry out a comparative dynamics analysis. We get    n s 2 d gM U Un 2 1 s n ð1  sÞ 40: t n  ¼ λ n Sn 2 at  1 U S S dat  1 d n S with, wn 41, it is implied that 0 o sðSn =U n Þ oð1  sÞ o1. Under A1, the increment in technology level is higher when the relative endowment of unskilled human capital is larger. That is, unskilled human capital is growth enhancing in the imitation-only regime. Therefore, by raising higher cost of education, τ, an economy in the imitation-only regime will tilt its labor composition in favor of unskilled human capital and, consequently, attain a higher growth rate. That is, investing in the primary/secondary education or in vocational training in comparison with tertiary education would be more growth enhancing in the imitation-only regime. 3.5.2. Innovation-only regime The growth rate of an economy which is in the innovation-only regime can be expressed as ϕ

n gN Sn t ¼ λγU

1ϕ

;

ð14Þ

N gt

denotes the growth rate of an economy in the innovation-only regime. As discussed earlier, there exists a fixed where composition of human capital in the innovation-only regime. Consequently, a marginal increment in the technology level is linearly dependent on the country's own technology level. This induces a constant growth rate in the innovation-only regime. Thus, all the economies which perform only-innovation are growing at the same rate, and, hence, this would be identical to the growth at the world technology frontier. We denote it as g. Moreover, in this regime the growth enhancing composition of skilled–unskilled human capital and the absolute and relative wage rates of skilled and unskilled human capital are identical across all the economies. This also points to the fact that all the economies which are in the innovationonly regime will be at the frontier. Thus, at a moment in time when an economy moves from an incompletely specialized 13 As already mentioned in Section 2.1, output level is linearly dependent on the local technology level, so the growth rate of an economy is the same as with the relative change in the technology level.

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regime to a specialized regime, it will attain a new steady state and will be at the frontier. So, there is a concurrent move from diversified regime to innovation-only regime as well as to the new steady state and the world technology frontier. Now let us try to figure out the growth enhancing education policy of an economy which is in the innovation only regime. For this, we again rely on a comparative dynamic exercise. We get !ϕ " # N 2 dgt Sn Sn n ! ¼λ U ð1 ϕÞ ϕ 40: Un Un Sn d Un Now, wn 41, implies 0 o ϕðSn =U n Þ o ð1 ϕÞ o1. One can now derive the basis for policies toward promotion of higher education in the innovation-only regime. Under A1, a reduction in the cost of education (implied by a reduction in τ) leads to higher relative endowment of skilled human capital. This raises the increment in the technology level and, consequently, the growth rate of the economy. This implies that skilled human capital is growth enhancing in the innovation-only regime. 3.5.3. Diversified regime Here, we characterize the growth rate of an economy in the cone of diversified region in period t to be g t ¼ λγ ð1  ϕÞh

ϕ

ðat  1 Þ 

1ϕ

λγϕτh ðat  1 Þ½ð1  ϕÞ  ϕhðat  1 Þ : ½ð1  τÞð1 ϕÞ  ϕhðat  1 Þ

ð15Þ

2

Comparative dynamics imply that dg t =dτ o0 and d g t =dat  1 ∂τ 40. The intensity of the effect of higher cost of education can be understood separately for a technologically advanced economy relative to a technologically backward economy. With an increase in the cost of education, the growth rate of the economy is lowered, irrespective of its distance from the world technology frontier.14 Moreover, this decrease is more pronounced for a relatively more technologically advanced economy. That is, the loss in the growth rate of an economy on account of a higher cost of education is greater for a technologically advanced economy than for a backward economy. Let us look at the intuition of this result. First, consider the case for a relatively less advanced economy. By Lemma 2, with an increase in the cost of education in period t, an economy shifts its relatively costly input (that is, skilled human capital) into imitation activity. But, by A1, skilled human capital is more efficient in innovation activity. Hence, the use of a relatively costly input (that is, skilled human capital), in an activity where its use is relatively less efficient, cannot be growth enhancing. Now, consider a relatively technologically advanced economy, for which innovation is the main source of technological progress. Here, by increasing the cost of education, an economy not only shifts its costly input (that is, skilled human capital) into an activity where it is less productive, but also shifts its labor force into an activity where its comparative advantage is lower. This reduces the increment to technology gain and, in turn, reduces the growth rate of an advanced economy in period tþ1 in the decentralized equilibrium. Proposition 1. Under A1 (i) Skilled human capital is growth enhancing in the innovation-only regime. (ii) Skilled human capital is growth enhancing and unskilled human capital is growth depressing in the cone of diversified regime. Furthermore, in the diversified regime unskilled human capital is more growth depressing for the relatively advanced economy than the relatively backward economy. (iii) Unskilled human capital is growth enhancing in the imitation-only regime.

Proposition 1 contradicts the findings of Aghion et al. (2009) and Vandenbussche et al. (2006). Under A1, both these papers show that in the diversified region a marginal increase in the total number of unskilled (resp. skilled) human capital is growth enhancing for the relatively technologically backward (resp. advanced) economy. A key assumption of their research is that the composition of labor force is exogenously specified. They only focus on the gains from acquisition of skills/education but ignore the costs associated with it. Therefore, if a country is farther away from the world technology frontier, shifting skilled human capital from innovation to imitation is costless in their model. This assumption is restrictive. We improve upon this by explicitly characterizing the cost of education of an individual and endogenously determining the composition of human capital. Therefore, if an economy shifts its costly input into a relatively less efficient activity, there is a reduction in the growth rate of the economy, irrespective of its distance from the world technology frontier. Now, let us try to define the path or transition in the growth rate of an economy as it moves to the frontier. To study the behavior of the economy, we need to ascertain the change in the growth rate as it progresses. From, Eq. (15) it is clear that the growth rate of an economy increases as it moves to the world technology frontier, that is, ∂g t =∂at  1 4 0. Intuitively, from Lemma 1, as an economy closes its gap with the world technology frontier it raises the stock of skilled human capital; and from Proposition 1, skilled human capital is growth enhancing for an economy in the diversified regime, whatever may be 14 As already discussed in Section 1.1, our policy prescription for education in the diversified regime is entirely different from the idea provided by Vandenbussche et al. (2006) and Aghion et al. (2009).

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the distance of the economy from the world technology frontier. Consequently, as an economy progresses toward the frontier it attains increasingly higher growth rate. Further, since we consider a knowledge driven model of growth, an economy innovates upon technology depending on its previously available local/national technology frontier. This concept is similar to the “building on the shoulders of the giant”. The level of knowledge is also higher for a technologically advanced economy, which generates a higher growth rate for it. This result is contrary to Di Maria and Stryszowski (2009), where growth rate is falling as an economy progresses while in the diversified regime. They measure the level of labor force in terms of their working hour. As a result, there is no change in the total payment to skilled–unskilled human capital as an economy progresses. In their framework, producers grab the entire additional benefit of higher technology. That is, the benefit of higher technology level accruing to skilled human capital is relatively low in their analysis; consequently, the increment in skilled human capital is getting adversely affected. Also, from Proposition 1, skilled human capital is growth enhancing in the diversified regime irrespective of its distance to frontier. So, low accumulation of skilled human capital in Di Maria and Stryszowski (2009) leads them to the conclude that growth rate is decreasing in the diversified regime. 3.6. Steady state analysis for the decentralized economy We now analyze the behavior of the economy in steady state. We try to find answers to the following questions. Is there absolute or conditional convergence? What is the growth rate of an economy in steady state? The absolute convergence of an economy implies that, in the long run, the economy will be at the frontier. That is ð1  at  1 Þ-0 at a sufficiently large value of t. If there is conditional convergence then the economy may not reach the world technology frontier, but instead it will converge to some value of a characterized by its own technology and preference parameters. In view of this, we define the growth rate of an economy in terms of its distance to the world technology frontier, which helps us to examine whether the economy will converge to the frontier or to some in-between value of a in the long run. From the definition of gt (Eq. (13)) one can express gt ¼

at ð1 þgÞ  at  1 ð1 þ g t Þ at  1 : ) at ¼ ð1 þgÞ at  1

ð16Þ

Let us define the transition path of the economy as its moves to the world technology frontier. First consider the imitationonly regime. By using Eqs. (16) and (13), we get i dat 1 h s 1s 1  λU n Sn ¼ 4 0: ð1 þ gÞ dat  1 Next, we characterize the transition path of the economy which performs both imitation and innovation. By using Eq. (13) we get that dat =dat  1 4 0, for all such economies. Therefore, once an economy is in the imitation-only regime it will reduce its gap with the world technology frontier and shift to imitation–innovation regime and, then onward, with certainty, it will in the long run shift to the innovation-only regime. Further as already discussed in Section 3.5.2, all the economies which are in the innovation-only regime are at the frontier. Therefore, in the long run, all the economies will converge to the world technology frontier. Hence, by optimally choosing the time cost of education an economy can converge to the world technology frontier. While the growth enhancing policy for a technologically backward economy, which is in the imitation only regime is to raise the cost of education, that for the diversified and innovation-only regime economies'education subsidization is the optimal policy prescription. An economy which is in the diversified regime or in the innovation-only regime should subsidize the cost of education not only to attain a higher growth rate but also to reach the frontier. Analytically as well, it is proven below that in steady state all the economies will grow at the same rate, and their growth rate will be identical to that of the frontier economy. We know that in steady state at will converge to an, that is, at -an and the growth rate of the economy will converge to gn, that is, g t -g n . Therefore, from Eq. (16) it is evident that in steady state either g n ¼ g or an ¼ 0. Thus, in the long run, all the economies will grow at the same rate. Proposition 2. Under A1, (i) In the long run all the economies will converge to the world technology frontier in steady state, and will grow at the same rate. (ii) An economy which is performing only-innovation is at the technology frontier.

3.7. Income inequality In this subsection we try to analyze the implications for the level of inequality (in terms of between group and within group inequality), and the changes therein (rise or fall) in inequality as an economy progresses toward the frontier. We define the extent of income inequality between the groups of skilled and unskilled human capital by comparing the average income of skilled and unskilled workers. From our previous discussion it is clear that in the imitation-only regime

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the average income of both skilled and unskilled workers decreases by the same proportion as an economy moves to the frontier. Therefore, there exists a constant income inequality between the two as long as the economy is in the imitationonly regime. Using the same logic, in the innovation-only regime as well, there will exist a constant level of between group income inequality. Furthermore, from our earlier analysis, it is clear that, in the diversified economy, the average income of skilled workers increases and that of unskilled workers falls as an economy moves to the frontier. This leads to a rise in income inequality between skilled and unskilled human capital as the economy progresses. Next, we analyze the within group inequality. Due to heterogeneity of income of skilled human capital, an economy always faces some degree of income inequality within the skilled human capital category. Since we assume that individual's cognitive ability is uniformly distributed over the interval [0, 1], one can define the income inequality within skilled human capital by comparing the incomes of skilled human capital with highest and lowest cognitive ability. We know that, the return from education is much higher for individuals with high cognitive ability. From Lemma 1, the total stock of skilled human capital increases in the diversified economy as an economy moves to the world technology frontier. That is, the diversity (in terms of cognitive ability) among skilled human capital increases. This leads to a higher income gap within the skilled human capital group. Consequently, the within group inequality rises in the diversified regime. The pattern of between and within group income inequality in the diversified regime is also similar to Galor and Moav (2000). Galor and Moav (2000) consider that in the short run technology improvement has a direct negative impact on the returns to both skilled and unskilled human capital and that negative impact is more pronounced for the low ability individuals. Consequently, there is a rise in income inequality both between and within skilled and unskilled human capital. The income distribution implications in our paper are seemingly close to Galor and Moav (2000) but the channel of transmission is entirely different. In our paper as the economy progresses, its dependence on skilled-biased technological improvement tends to rise. This is combined with an inverse relationship between the cost of acquiring skills (education) and cognitive ability, which yields that low ability workers will tend to have a smaller gain in income relative to high ability workers. In our analysis there is no direct depreciation of human capital (skilled or unskilled) due to technological progress. In spite of these effects, we are able to show that in equilibrium, there are negative effects of technological progress on the stock of unskilled human capital. Moreover, in moving to the frontier, there is rising income inequality between skilled and unskilled human capital and within skilled human capital. Thus, despite ignoring the fact that a technological progress has direct negative impact on human capital, we are able to show a similar income distributional outcome in equilibrium. In this sense our result is somewhat stronger. On the contrary, in the imitation-only regime, the total stock of skilled human capital remains unchanged during the transition to the world technology frontier, and the additional benefit from a higher technology level is the same for all the skilled workers. This leads to a constant income gap among skilled human capital. Consequently, there exists a constant within group inequality in the imitation-only regime. Again, by the same logic as in the imitation-only regime, there is a constant level of income inequality among skilled human capital in the innovation-only regime.15 Proposition 3. Under A1, (i) In the imitation-only regime, the wage rate, average income and consumption of both skilled and unskilled human capital decrease and consequently, the aggregate consumption of the economy decreases as distance to the frontier falls. (ii) In the imitation–innovation regime, the wage rate, income and average consumption of skilled human capital increase and unskilled human capital decreases as an economy moves to the world technology frontier. (iii) In the innovation-only regime, the wage rate, average income and average consumption of both skilled and unskilled human capital increase and also the aggregate consumption of the economy increases as technology level rises. (iv) In both the imitation-only and the innovation-only regimes there exists a constant income inequality between skilled and unskilled human capital and within skilled human capital group. (v) The income inequality between skilled and unskilled human capital as well as that within skilled human capital increases as an economy moves to the world technology frontier. (vi) As an economy progresses to the world technology frontier, the production of the final output and the intermediate inputs also rises; this increase occurs at the same rate as the growth in the technology level.

4. Complementarity between imitation and innovation activities So far, we have assumed that there exists perfect substitutability between imitation and innovation activities. This allows an economy to completely specialize in imitation or innovation activity. But, it is possible that both imitation and innovation activities are essential for technology improvement. That is, corner solutions do not exist. We therefore, relax our earlier assumption of perfect substitutability. It is shown that, with some degree of complementarity between imitation and innovation activities, skilled labor is growth enhancing even for an economy which has a very low level of skilled human capital, and which is relatively more dependent on imitation activity. That is, the economy engages a very low level of skilled 15 In Proposition 3, we also state the results related to the nature of the consumption of the skilled and unskilled human capital as well as that of the aggregate economy. Detailed proofs can be obtained from the corresponding author.

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human capital in innovation activity. We consider a more generalized form of technology improvement process, by taking a constant elasticity of substitution (CES) production function:  sÞ ρ  ϕÞ ρ ρ Ait ¼ Ai;t  1 þ λ½ðusmit sð1 Þ ðA t  1  At  1 Þρ þ γðuϕnit sð1 Þ At  1 1=ρ ; mit nit

ð17Þ

where 0 o ρo 1 measures the complementarity between imitation and innovation activities. Each intermediate input producer chooses umit ; unit ; smit and snit by maximizing the profit function as it is given in (3.4.1). From the first order conditions of the maximization exercise we get that:  1 ð1  sÞρ  1 ð1  ϕÞρ ρ susρ smt ðA t  1  At  1 Þρ ¼ γϕuϕρ snt At  1 mt nt

ð1  sÞð1  ρÞ  1 ð1  ϕÞρ  1 ρ ð1 sÞusρ ðA t  1 At  1 Þρ ¼ γð1 ϕÞuϕρ At  1 ; mt smt nt snt

ð18Þ

Dividing across equations we get the same relative demand function as in Eqs. (5) and (6). Replacing these in Eq. (18) we get, kðsmt ; St ; at  1 Þ ¼ vðat  1 ÞU t  wðsmt ; St Þ½St þ ðψ  1Þsmt   0;    1=ðs  ϕÞρ ð1 sÞψ sρ 1  at  1 ρ ; where vðat  1 Þ ¼ γð1 ϕÞ at  1  ρ  1=ρðs  ϕÞ St  smt : and wðsmt ; St Þ ¼ smt

ð19Þ

Now, we want to show that there exists a unique solution of smt which satisfies Eq. (19). We get kð0; S; U; aÞ ¼ vðat  1 ÞU t 4 0; kðS; S; U; aÞ ¼  1 o0;   ∂k 1 ρ St ½St þ ðψ  1Þsmt  o 0: ¼ wðsmt ; St Þ ðψ  1Þ þ ∂smt ρðs ϕÞ St smt

ð20Þ

Therefore, an implicitly defined equilibrium demand for skilled and unskilled workers in period t is kðsnmt ; Snt ; at  1 Þ ¼ vðat  1 ÞU nt  wðsnmt ; Snt Þ½Snt þ ðψ 1Þsnmt   0:

ð21Þ

Now, let us look at the implicitly defined equilibrium supply of skilled and unskilled human capital in period t. From Eq. (2) and the first order conditions of Eq. (17) ensure that Snt ¼ 1 

τs½Snt þ ðψ  1Þsnmt  : ð1  τÞð1  sÞψð1 Snt Þ  τs½Snt þ ðψ  1Þsnmt 

ð22Þ

We want to know how the equilibrium levels of skilled and unskilled human capital change on account of a parametric change in the cost of education, captured by τ. From Eq. (21),   ∂wðsnmt ; St Þ  n St þ ðψ 1Þsnmt þ ðψ  1Þw snmt ; Snt n dSt dsmt ∂snmt ¼ n n    dτ dτ ∂wðsmt ; St Þ n  vðat  1 Þ  St þ ðψ  1Þsnmt  w snmt ; Snt ∂Snt n

n

dSt ) dτn 4 0; dsmt dτ

if smt -St ;

ð23Þ

which is a sufficiency condition but not a necessary condition for the same sign of the change in the total level of skilled human capital and the change in skilled human capital in imitation activity due to a change in the cost of education. From Eq. (22),   Un  ðψ 1ÞU nt 1 þ t n n n n dSt U Ut dsmt τ ¼  t 1 þ U nt þ : ð24Þ  n n n n dτ τ dτ s½St þ ðψ 1Þsmt  ½St þðψ  1Þsmt  Therefore, from Eqs. (23) and (24) n

dSt o0 dτ

n

and

dsmt o 0: dτ

ð25Þ

Eq. (25) ensures that even for an economy which starts out with very low level of skilled human capital in innovation activity, with an increase in the cost of education, the equilibrium level of skilled human capital decreases. Alongside, skilled human capital shifts from imitation to innovation activities. That is, our Lemma 1 and Lemma 2 still hold even if there is complementarity between imitation and innovation activities. We next analyze the growth rate of an economy in period t. We have, Z 1 At  At  1 gt ¼ di At  1 0

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¼ λδ1

    ψ s ws ðsmt ; St Þ 1  at  1 ð1  sÞ ðSt smt Þ 1=ρ 1 þ : s mt vs ðat  1 Þ ð1  ϕÞ smt at  1

ð26Þ

We now look at the change in the growth rate of an economy in period t as the cost of education rises. We get that, dg t ∂g dSt ∂g t dsmt ¼ t þ ; dτ ∂St dτ ∂smt dτ

ð27Þ

which implies, ∂g t 40 ∂St

if ρ4

ϕ s

and

St -0

and

smt -St :

Moreover, ∂g t 40 ∂smt

if ρ 4

1 ð1 þ s ϕÞ

and

smt -St :

ð28Þ

Therefore, Eqs. (27), (25) and (28), imply that dgt =dτ o 0. This proves that skilled human capital is growth enhancing even for an economy which has a very low level of skilled human capital (that is, St -0), and which engages very low level of skilled human capital in innovation activities (that is smt -St ) given ρ4 ϕ=s. That is, even with complementarity between imitation and innovation activities, skilled human is growth enhancing for a technologically backward economy – which has a very low level of skilled human capital and high dependence on imitation activity. That is, Proposition 1 still holds with complementarity between imitation and innovation activities. 5. Conclusions Technological progress is a dual phenomenon. A country can attain technological progress by imitating from the technology frontier or by innovating. An economy which is lagging far behind the world technology frontier can improve its technology level by allocating its labor force mainly into imitation. Similarly, an advanced economy can progress technologically by innovating new knowledge. Under the assumption that different types of human capital are efficient in different activities, Vandenbussche et al. (2006) and Aghion et al. (2009) show that unskilled human capital is the main source of growth for the technologically backward economy and skilled human capital is the key source of growth for the technologically advanced economy among the economies which perform both imitation and innovation activities. By relaxing their assumption of exogenous composition of the labor force, and by utilizing an endogenous growth model, we show that skilled human capital is the engine of growth in the diversified economy. We also characterize the economies which are performing only-imitation or only-innovation activity and show that unskilled human capital is growth enhancing in the imitation-only regime and skilled human capital is growth enhancing in the innovation-only regime. There exists a growth enhancing composition of skilled and unskilled human capital and by implementing that composition an economy can converge to the world technology frontier. That is, a growth enhancing education policy is crucial for both – technologically developing and developed economies. But, as has been derived, this might raise the income inequality between skilled and unskilled human capital and also within the group of skilled human capital, since due to skilled biased technological change, the reward to skilled human capital is much higher in a technologically advanced economy. Our work can be extended in several directions. First, one can allow the possibility of outsourcing of the R&D activity by a developed economy to a less developed economy. Wage rate of skilled human capital is relatively lower and the average cognitive ability of skilled human capital is higher in the less developed economy. This increases the profit of the R&D producer in the developed economy and also raises the income of skilled and unskilled human capital in the underdeveloped economy. Second, in this entire work, we assume that new knowledge is freely available to all the economies. Instead, one can characterize the growth path and the convergence condition of the economy by ruling out the assumption that world technology level is freely accessible. Third, till now all the research in this area has abstracted from international trade in commodities. One can develop a dynamic Ricardian model of international trade around the core idea of our paper and study cross-sectoral allocation of skilled and unskilled human capital in the context of international specialization in goods production and trade. Fourth, one can analyze the consequences of heterogeneous cost of education depending on individual's parental education level and study the impact of that on the growth rate, inequality and intergenerational mobility of an economy depending on its distance to the world technology frontier. This would yield deeper insights into the interface between distance to frontier, composition of human capital and economic growth.

Acknowledgments We are grateful to Mausumi Das, Ashok Guha, Manas Ranjan Gupta and Debasis Mondal for their comments and suggestions which have helped this paper take its present form. We are also thankful for the comments received at the seminars/conferences at the Indian Statistical Institute, Kolkata (August 2011), Centre for Development Economics, Winter

S. Basu, M. Keswani Mehra / Research in Economics 68 (2014) 117–132

131

School 2011, Delhi School of Economics (December 2011), XXIst Annual Conference, Economics, Jadavpur University (December 2011), Centre for Economic Studies and Planning Young Scholars' Seminar (March 2012), 3rd Bread-IGC-ISI Summer School (July 2012), Engines for More and Better Jobs in Europe, Seek Conference, Centre for European Economic Research (ZEW), Mainheim, Germany (April 2013) and University of Applied Sciences (HTW), Berlin, Germany (May 2013).

Appendix A From Eq. (3), in any time period t the demand for skilled and unskilled human capital in the imitation-only regime is wDD t ¼

wst 1  s Ut ¼ s wut St

ð29Þ

From Eqs. (29) and (2), we get the equilibrium level of skilled and unskilled human capital in any period t in the imitationonly regime SS wDD t ¼ wt

)

1  s Ut τ þU t ¼ s St U t ð1  τÞ

) U 2t ð1 þ Σ Þ  ð1  τÞU t  τ ¼ 0;

) Ut ¼

ð1  τÞ 7

where Σ ¼ ð1 τÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  τÞ2 þ 4τð1 þ ΣÞ

1s s

2ð1 þ ΣÞ

ð30Þ

where wDD and wSS t t respectively measure the relative demand and supply curve of skilled and unskilled human capital in the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi imitation-only regime in any period t. Now, U t ¼ ð1  τÞ þ ð1  τÞ2 þ4τð1 þ ΣÞ=2ð1 þ ΣÞ is always positive. But, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U t ¼ ð1  τÞ  ð1  τÞ2 þ4τð1 þ ΣÞ=2ð1 þ ΣÞ o0, since ð1 τÞ2 þ 4τð1 þ ΣÞ 4 ð1 τÞ. Therefore, the equilibrium level of unskilled human capital in period t in imitation-only regime is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð1  sÞ ð1  τ Þ ð1 τÞ þ ð1 τÞ2 þ 4τ 1 þ s τ   : ð31Þ ¼ Un ¼ ð1 sÞ ½ð1  τÞwn 1 ð1 τÞ 2 1þ s The absolute and relative wage rates of skilled–unskilled human capital are respectively given by 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð1  sÞ 2 6 7 ð1  τ Þ ð1  τÞ þ ð1  τÞ þ4τ 1 þ 7 s wst ð1  sÞ 6 6 7 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wn ¼ ¼ ; 6    7 7 s 6 wut 42 1 þ ð1  sÞ ð1  τÞ  ð1 τÞ  ð1 τÞ2 þ 4τ 1 þ ð1  sÞ ð1  τÞ 5 s s wut ¼ λδ1 sU n

s  1 n1  s

S

s

wst ¼ λδ1 ð1  sÞU n Sn

ðA t  1  At  1 Þ;

s

ðA t  1  At  1 Þ:

ð32Þ

Appendix B In the similar fashion as mentioned in Appendix A, one can derive the equilibrium level of skilled and unskilled human capital and their respective wage rates in period t in the innovation-only regime. The level of unskilled human capital in the innovation-only regime is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi ð1 ϕÞ ð1  τ Þ ð1  τÞ þ ð1  τÞ2 þ4τ 1 þ ϕ τ n   : U ¼ ¼ ð1  ϕÞ ð1  τÞwn 1 ð1 τÞ 2 1þ ϕ

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The relative wage rate of skilled–unskilled human capital in innovation-only regime is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3  ffi ð1 ϕÞ 2 ð1  τ Þ ð1 τÞ þ ð1 τÞ þ 4τ 1 þ 6 7 ϕ 7 wst ð1  ϕÞ 6 6  7: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r wn ¼ ¼  h i 7 ϕ 6 wut 42 1 þ ð1  ϕÞ ð1  τÞ  ð1  τÞ  ð1  τÞ2 þ 4τ 1 þ ð1  ϕð ð1  τÞ 5 ϕ ϕ

Appendix C The first-order conditions of the maximization exercise in Eq. (4) are  ∂L1t 1 1s ¼ λδ1 susmit smit A t  1  At  1  wut ¼ 0; ∂umit ∂L1t ¼ λδ1 γuϕnit 1 s1nit ϕ At  1 wut ¼ 0; ∂unit  ∂L1t s ¼ λδ1 ð1  sÞusmit smit A t  1  At  1  wst ¼ 0; ∂smit ∂L1t ϕ ¼ λδ1 γ ð1  ϕÞuϕnit snit At  1  wst ¼ 0: ∂snit

ð33Þ

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