Human-capital spillover, population and R&D-based growth

Bharat Diwakar; Gilad Sorek

doi:10.1515/bejm-2016-0029

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Human-capital spillover, population and R&D-based growth

Bharat Diwakar and Gilad Sorek

Published/Copyright: October 12, 2016

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From the journal The B.E. Journal of Macroeconomics Volume 17 Issue 1

Abstract

We study two-sector R&D model with endogenous human capital accumulation. Allowing for fractional human capital spillover from parents to their offspring, which are subject to congestion in fertility rate, we establish non-monotonic relations between population growth and economic growth. These non-monotonic relations, which are polynomial in general, are determined by the base level of human capital spillover and the magnitude of the congestion effect: a U shape relation can arise under low congestion factor, whereas a hump shape may present for high congestion factor. Hence this work contributes to the recent line of modified R&D-based growth models, aimed to align theory with the mixed empirical data on the relation between population growth and economic prosperity.

Keywords: human-capital spillover; innovation-based growth; population growth

JEL Classification: O-31; O-40

1 Introduction

A recent line of research proposed several modifications to modern growth theory, aimed to remove the counterfactual “weak scale-effect” that presented in second and third generation models of R&D-based growth: that is a positive effect of population growth on economic prosperity. ^[1] A common element in this literature is the introduction of human capital as productive input in the R&D sector. This modification induces substitution between the quantity and quality of workers, which enables an increase in overall effective labor supply and thereby enhanced economic growth, even with a constant or declining population of workers.

Within this line of research the assumed process of human capital formation is crucial to the relation between population growth and economic growth. More specifically, the potential diluting effect of population growth on the average human capital level was emphasized in the literature as hindering economic growth; See for example Dalgaard and Kreiner (2001), Strulik (2005), Bucci (2013), and Chu et al. (2013). In these papers, young agents are assumed to enter the labor force with zero human capital. However, other related studies abstracted from this diluting effect entirely, following Lucas (1988) exact formulation, assuming that newborns are inherited with the same level of human capital as their parents; See for example Tournemaine and Luangaram (2012) and Bucci (2015).

When discussing his own formulation of human capital accumulation, Lucas (1988) emphasized the plausibility of fractional transmission of human capital within dynasties, from parents to their offspring: “…One needs to assume… that the initial level each new member begins with is proportional to (not equal to!) the level already attained by older members of the family. This is simply one instance of a general fact that I will emphasize…: that human capital accumulation is a social activity, involving groups of people in a way that has no counterpart in the accumulation of physical capital” (p. 19).

The present work pursues and elaborates such intermediate approach regarding the diluting effect of newborns on human capital accumulation. First, we allow for fractional human capital spillover from parents to their offspring. That is, we consider the entire range between the two extreme cases presented in the aforementioned literature. This kind of spillover was widely considered in other strands of the literature on growth and human capital accumulation, without R&D-based innovations (See for example Becker et al. 1990, Galor and Tsiddon 1997, De La Croix and Doepke 2004, and Fioroni 2010), as well as in recent R&D-driven growth models written in the Overlapping Generations framework; See for example Strulik et al. (2013) and Prettner (2014).

Second, we consider congestion effects in the transmission of human capital within dynasties. That is, the degree of human capital spillover from parents to their offspring is decreasing with the number of kids. The intuition that motivates this analysis is the following: parental human capital spillover is transmitted through direct interaction between parents and their offspring in the household, where parenting time is not a pure public good.

This is consistent with the essential notion of trade-off between the quantity and quality of children, as phrased by Hanushek (1992, p. 86): “The trade-off between child quantity and quality enters essentially because parents’ time and resources must be spread thinner with more children.”

Our framework is an extended version of Young’s (1998) model of two-sector R&D, that incorporates population growth and human-capital accumulation. The analysis yields a rich set of possible relations between population growth and economic growth, including non-monotonic ones, depending on the assumed type of spillover. The welfare analysis shows that the rates of human capital accumulation and technological progress in the decentralized economy may deviate from the efficient ones, in various ways.

Several theoretical papers have established ambiguous effect of population growth on technological progress, which depends on the strength of dynastic altruism toward future generations (Dalgaard and Kreiner 2001, Strulik 2005, Bucci 2013), the (potentially adverse) effect of specialization on the production complexity (Bucci 2015), and the effect of technological progress on the stock of human capital – appreciation vs. depreciation (Bucci 2008). In these studies however, unlike in the present work, the effect of population growth on technological progress depends on the values of model parameters, and are monotonic given the parameters set.

Our work is closely related to the recent contribution by Boikos et al. (2013), which studies the effect of fertility on human capital accumulation, in a model with no R&D-based innovation and endogenous fertility. In their theoretical analysis the effect of population growth on human capital accumulation is allowed to be positive, negative, and non-monotonic. Hence,they allow for “negative dilution” effect of population growth on human capital accumulation, under which population growth enhances per-capita human-capital accumulation.

The theoretical part of their work shows that the overall effect of fertility on per-capita human capital accumulation and income growth depends crucially on the sign of the dilution effect, which is left unspecified (see discussion on p. 50 and footnote there). Similar approach was taken also by Boucekkine and Fabbri (2013) and Marsiglio (2014), who assume unspecified and quadratic diluting effect, respectively, in models of physical capital accumulation and endogenous fertility. ^[2]

All these studies were able to establish a hump shape relation between fertility growth and economic growth, consistent with the empirical findings reported by Boikos et al. (2013), ^[3] and Kelley and Schmidt (1995). ^[4] In comparison with these studies, our results are derived in a full fledged R&D-based growth model, ^[5] based on a simple specification of the diluting effect, which has an intuitive economic interpretation. Namely, the diluting effect here is defined by fractional human-capital spillover from parent to their offspring, which is subject to congestion in the number of offspring.

That is, we assume population growth always dilutes per-capita human capital accumulation, but not necessarily in a linear fashion. Yet, we establish non-monotonic polynomial relation between population growth and economic growth, which varies – from U shape to Hump shape, depending on the congestion factor in parental human capital spillover.

In another recent relevant paper by Prettner (2014), human capital is formed through public education system where higher fertility rate decreases schooling-intensity – i.e. per-student public spending. Prettner (2014) shows that for economies with developed public education system – in terms of spending level and teachers’ productivity – there is a non-monotonic relation between fertility rate and economic growth: for initially low (high) rates increase in fertility has negative (positive) effect on economic growth. For economies with under-developed public education system the effect of fertility on population growth is definite-positive. ^[6] Prettner’s (2014) results, derived in the overlapping generations framework, are similar to ours in the special case of no congestion in parental human-capital spillover, which we derive for infinitely living agents and decentralized human capital accumulation.

Our specification of human capital formation could yield similar qualitative relations between population growth and economic growth in models where human capital accumulation is the sole growth engine, like the one studies by Boikos et al. (2013). ^[7] Nonetheless, in R&D-based growth models, like the one studied here, it is the rate of increase in over effective labor supply that affects innovation rate and economic growth, ^[8] whereas in typical models with human capital accumulation only economic growth requires an increase in average (per-capita) human capital level, as in Boikos et al. (2013). Hence, different parameter values will characterize the alternative relations between population growth and economic growth in these two prototype models.

The paper is organized as follows. Section 2 presents the detailed model. Section 3 analyzes the dynamic equilibrium and the effect of population growth on technological progress. Section 4 presents welfare analysis for the model economy, and Section 5 concludes this study.

2 The model

We extend Young’s (1998) two-sector R&D model by adding population growth and human capital accumulation. Time is discrete, and population grows at exogenous rate n≥0. Population size in each period is denoted L_t=L₀(1+n)^t, where L₀ is normalized to one. In each period, each worker is endowed with one unit of time. To enhance exposition clarity, the analysis focuses first on exogenous human capital accumulation, and then human capital accumulation is endogenized through education choice.

2.1 Preferences

Consumer’s lifetime utility is given by

(1)U=∑t = 0∞ρtln(ct)

where ρ∈(0, 1) is the subjective discount factor, and c is the per-capita instantaneous utility from consuming N differentiated products, i.e. “varieties,” subject to a CES utility function

(2)ct=(∑i = 1Ntci,t1ε)ε

with ε=ss−1, and s is the elasticity of substitution across all varieties. The consumption level of each variety c_i is defined as c_i=q_ix_i, where x_i and q_i, denote the consumed quantity and product quality, respectively. The assumed preferences imply the instantaneous demand for each variety

(3)xi,td=qi,ts − 1(λpi,t)−s(∑i = 1Ntci,t1ε)ε

where λ is the Lagrange multiplier from the instantaneous utility maximization (i.e. the shadow value of the given periodic spending level). The logarithmic specification in (1) implies the standard Euler condition for optimal consumption smoothing, written in terms of aggregate spending, denoted E

(4)Et + 1Et=ρ(1+rt + 1)

where 1+r_t+1 is the (gross) interest rate earned between periods t and t+1.

2.2 Production and innovation

We will start by analyzing a model with exogenous rate of human capital accumulation, subject to the following aggregate growth rate

(5)1+gh≡htht − 1=(1+ωn)(1+g˜h)1+n

where h_t is the per-capita human capital, and the parameter ω∈(0, 1) measures parental human capital spillover. With constant population, i.e. n=0, per-capita human capital grows at the rate g˜h. For positive population growth and ω=1, parental human capital spillover is complete, and thus population growth has no effect on the per-capita human capital level. For ω=0, population growth rate works as a full dilution factor over h_t. Our analysis focuses on the intermediate cases with fractional transmission of human capital from parents to their offspring. Furthermore, we consider nonlinear spillover, due to congestion in the number of offspring, that is, ω=ω(n) and ω′(n)<0. To enhance tractability, we focus on the following specification

(6)ω(n)=ω0exp(−μn)

where ω₀∈(0, 1), and μ≥0 is the congestion factor. With μ=0 there is no congestion in human capital spillover. Notice that (5)–(6) imply that population growth slows down the accumulation of per-capita human capital, that is δghδn<0,^[9] hence we assume effective diluting. Aggregate human capital, denoted H, is defined as the product of population size and per-capita human capital

(7)Ht=Ltht

Effective labor supply is the sole input for production and innovation, and the wage rate is normalized to one. One unit of labor produces one unit of consumption good (regardless of its quality).

We follow Young’s (1998) specification of the innovation cost function

(8)f(qi,t + 1, q¯t)={exp(ϕqi,t + 1q¯t)if qi,t + 1>qi,texp(ϕ)if qi,t + 1<qi,t

Innovation cost in sector i is increasing with the rate of improving its product quality, over the highest quality that was already developed in the economy – denoted q̅_t. This specification implies vertical knowledge-spillovers, as the present cost of providing a certain quality today is decreasing in the highest quality level that was already developed in the past. As innovation is assumed to be certain, vertical innovation (i.e. quality improvements) implies that the effective lifetime of each product is one period. Hence, each firm maximizes the profit

(9)Πi,t=(pi,t + 1−1)xi,t + 1dLt + 11+rt + 1−f(qi,t + 1, q¯t)

Maximizing (9) for price p_i,t+1 yields the standard optimal monopolistic price p^*=ε, ∀t, i. The first order for optimal quality choice is derived after plugging the optimal price and the demand function (3) into (9)

(10)1qi,t + 1∗(ε−1)(s−1)(λε)−s(∑i = 1Ntci,t + 11ε)εLt + 11+rt + 1=ϕq¯tf(qi,t + 1∗, q¯t)

The asterisk superscript denotes optimally chosen values for the variables in the decentralized economy. Assuming free entry to the R&D sector implies that in equilibrium the profit in (9) equals zero. Combining this assumption with the optimality condition (10) we obtain the equilibrium rate of quality improvement

(11)∀i:1+gq≡qt + 1∗q¯t=s−1ϕ

We assume the cost parameter ϕ is low enough to guarantee g_q>0 and to make vertical competition between successive product generations redundant, i.e. p∗<1+gq⇒ε<s−1ϕ.

As the rate of quality improvement is time invariant, so is equilibrium innovation cost f(q_i,t+1, q̅_t)=e^s−1, ∀t, i. Notice that under symmetric equilibrium demand for each variety is xtd=EtεNt∀i, and thus the free entry condition can be also written as

(12)(ε−1)Et + 1εNt + 1f=(1+rt + 1)

3 Equilibrium and growth dynamics

3.1 Exogenous human capital accumulation

Combining (4) and (12) we have,

(13)Et=fNt + 1(1−1ε)ρ

and plugging (13) back into (12) yields the interest rate for the assumed stationary equilibrium

(14)1+gNρ=(1+rt + 1)

where 1+gN≡Nt + 1Nt. The aggregate resources-uses constraint for the economy is defined by the allocation of labor between production and R&D investment

(15)Ht=Etε+fNt + 1

Plugging (13) into (15) yields

(16)Ht=fNt + 1(ε−1)ρ+fNt + 1⇒Nt + 1=Htf(1(ε−1)ρ+1)

Hence, variety expansion rate equals to the exogenous growth rate of effective labor supply (1+g_N)=(1+g_H), which, following (5)–(6), implies:

(17)(1+gN)=(1+g˜h)[1+ω(n)n]

Observe that under symmetric equilibrium, equation (2) can be written as

ct=(∑i = 1Nt(qi,txi,t)1ε)ε=Ntεqtxt=NtεqtEtεLtNt

After plugging (13) into c_t, the above expression implies that in the stationary equilibrium per-capita consumption grows at a constant rate

(18)1+gc≡ctct − 1=Lt − 1Ntε − 1qtNt + 1LtNt − 1ε−1qt − 1Nt=(1+gq)(1+gN)ε1+n

Then we substitute (17) and (6) into (18) to rewrite

(19)1+gc=(1+gq)(1+g˜h)ε[1+ω0exp(−μn)n]ε1+n

Equation (19) reveals the two opposing effect induced by population growth on per-capita consumption growth. The positive effect is due to the increase in aggregate human capital supply, which accelerates variety expansion – according to equation (16). This positive effect is generated through the spillover parameter and is then amplified by the preference parameter ε, which is decreasing with the elasticity of substitution across varieties – s. With lower s gains from faster variety expansion, driven by faster human capital accumulation are higher. The negative effect of population growth on per-capita consumption growth, which presents in the denominator of (19), is the regular pure dilution effect.

Differentiating (19) for n shows that δgcδn is positive (negative) if the following (reverse) inequality holds

(20)ε(1+n)(1−μn)−n>exp(μn)ω0

Proposition 1:With exogenous human capital accumulation, for sufficiently high ω₀and μ, the function g_c(n) is hump shape. That is for sufficiently strong base spillover and congestion effect economic growth first accelerates with population growth rate and then slows down.

Proof: For μ>0, the right hand side of (20) is increasing with n. For ε(1−μ)<1, the left hand side of (20) is monotonically decreasing with n, and hence, for sufficiently high n it is guaranteed that (20) does not hold, that is ∂gc∂n<0. If ω0>1ε, condition (20) holds for n=0. Hence under these conditions, ∂gc∂n is positive (negative) under sufficiently low (high) population growth rates■

For ε(1−μ)<1 and ω0<1ε, the function g_c(n) is monotonically decreasing. Under lower values of μ, for which ε(1−μ)>1, the left-hand side of (20) is increasing with n up to n=ε(1−μ)−12εμ, and then starts decreasing (for high n values). Then, g_c(n) still follows a hump shape for ω0>1ε. However, for ω0<1ε, condition (20) holds only for intermediate values of n, implying that g_c(n) is first decreasing with n – to a local minimum, and then it is increasing to a local maximum from where it is monotonically decreasing. Hence, within this parameters set the shape of g_c(n) follows a co-sine shape, which combines U shape with Hump shape. As the value of μ decreases the range of the U shape is expanding.

Proposition 2:With exogenous human capital accumulation, for sufficiently low ω₀and μ, the function g_c(n) is U shape. That is for sufficiently weak base spillover and congestion effect economic growth first slows down with population growth rate and then accelerates.

Proof: For the limit case μ=0, (20) is modified to ε+n(ε−1)>1ω0. The right-hand side of this condition is increasing with n. For ω0>1ε the latter inequality does (not) hold for sufficiently high (low) n, implying that ∂gc∂n is negative (positive) for low (high) values of n■

For μ=0 and ω0>1ε we have ∀n>0:∂(1+gc)∂n>0, that is g_c(n) is monotonically increasing.

3.2 Endogenous human capital accumulation

We turn now to incorporate endogenous human capital accumulation in the model, subject to the conventional specification

(21)ht + 1=(1+ωn)(ξet+1−δ)ht1+n

(22)Δht + 1≡ht + 1−ht=[(1+ωn)(ξet+1−δ)1+n−1]ht

where e∈(0, 1) is the time invested in human capital formation, δ is a depreciation rate, and ξ captures the productivity of the human capital formation technology. ^[10] Equation (21) implies that 1+gh≡ht + 1ht=(1+ωn)(ξet+1−δ)(1+n), and following (7) we obtain

(23)1+gH≡Ht + 1Ht=(1+gh)(1+n)=(1+ωn)(ξet+1−δ)

The return on investment in human capital should equal the return on R&D investment defined in (9)

(24)1+rt + 1=(ξet − 1+1−δ)htetht

Plugging the interest rate (14) into (24) and rearranging yields

(25)∀t:e∗=1−δ1+gNρ−ξ

and plugging (25) back into (23) yields

(26)1+gH=(1+ωn)(1−δ)1−ξρ1+gN

Modifying the resources-uses constraint (16) for the time invested in human capital formation yields

(27)(1−e∗)Ht=fNt + 1(ε−1)ρ+fNt + 1

Plugging the interest rate (14) into (27) and rearranging yields

(28)Nt + 1=(1−e∗)Htf[1(ε−1)ρ+1]

Equation (28) shows that the aggregate human-capital stock and the varieties span share the same growth rate, as in Section 3. Imposing (1+g_H)=(1+g_N) in (26) and simplifying we obtain

(29)1+gN=(1−δ)(1+ωn)+ξρ

Hence, following (18)–(19), per-capita consumption growth rate remains 1+gc=(1+gq)(1+gh)ε(1+ωn)ε1+n which can be written explicitly as ^[11]

(30)1+gc=(1+gq)[(1−δ)(1+ω0exp(−μn)⋅n)+ξρ]ε1+n

Equation (30) shows that the effect of population growth on per-capita consumption growth under endogenous human capital accumulation is very similar to the one presented in equation (19), for exogenous rate of human capital accumulation. Nonetheless, here, the effect of population growth rate on per-capita consumption growth depends also on the technological parameters of human capital formation, and the time preference parameter.

Following (30), ∂gc∂n is positive (negative) if the following (reverse) inequality holds

(31)ε>1+ω0exp(−μn)⋅n+ξρ(1−δ)ω0exp(−μn)⋅(1−μn)(1+n)

Proposition 3:With endogenous human capital accumulation, for μ>0 andε>1ω0(1+ξρ(1−δ)),the relation between population growth and per-capita consumption growth follows a hump shape.

Proof: Condition (31) does not hold for n>1μ, as the denominator turns negative, but it does hold for sufficiently low n if ε>1ω0(1+ξρ(1−δ)). Hence, under these conditions g_c(n) is non-monotonic and follows a hump shape■

Proposition 4:With endogenous human capital accumulation, for sufficiently low congestion effect andε<1ω(1+ξρ(1−δ)),the relation between g_cand n follows non-monotonic U shape.

Proof: For the limit case μ=0, condition (31) becomes ε>1+ω0n+ξρ(1−δ)ω0(1+n)⇒ε+n(ε−1)>1+ξρ(1−δ)ω. The latter condition holds for sufficiently high n, but it does not hold for sufficiently low (yet non-negative) n if ω0<1+ξρ(1−δ)ε. Hence, under these conditions ∂gc∂n is negative (positive) for sufficiently low (high) values of n, implying that g_c(n) is U shaped

Having ε>1+ξρ(1−δ)ω implies that ∀n>0:∂gc∂n>0, that is positive monotonic relation between population growth and economic growth.

The relation between g_c and n established in Proposition 4 is similar to the one presented in Prettner (2014) for an overlapping generations economy with public education system. In Prettner’s work, as in the present study, high productivity in human capital formation (interpreted there as teachers’ productivity and schooling efficiency) is needed to obtain such non-monotonic relation. In addition, his result also requires a high level of public spending on schooling, which is set exogenously, whereas Proposition 4 above is derived for decentralized investment in education, chosen by the households.

4 Welfare analysis

We turn now to evaluate the welfare performance of our extended version of Young’s model. In Young’s (1998) original work, growth is driven solely by vertical (quality improving) innovation. There, the rate of quality improvements is slower than the social optimum (see p. 59 there) because investors do not internalize the vertical knowledge spillover defined in the innovation process (8). In our extended framework there are additional knowledge spillover through the process of human capital accumulation, which determines the rate of variety expansion. These spillover are not internalized either in the decentralized economy, as agents are choosing investment level according to the private rate of return, according to equation (24). Therefore, the introduction of human capital accumulation in our model generates a second source of efficiency distortion.

The social planner that maximizes (1) along the balanced growth path is facing the following objective function

(32)U=11−ρ(lnc0+ρln(1+gc)1−ρ)

The welfare function (32) should be maximized by allocating labor efficiency over production and the two investment activities – quality improvements and human capital formation. This maximization problem is still subject to the resources-uses constraint (27), and also to the implied explicit expression for (1+g_c) in (30). Imposing these restrictions on (32) we obtain the constrained objective function ^[12]

(33)U=11−ρ(ln[N0εq0[(1−e)h0−fN0(1+ωn)(ξe+1−δ)]εN0]++ρln[1ϕlnf((1+ωn)(ξe+1−δ))ε1+n]1−ρ)

After normalizing all initial values to unity, we derive the first order conditions with respect to investment in education and quality improvements

(34)∂U∂e:1+ξf(1+ωn)((1−e∗∗)−f(1+ωn)(ξe∗∗+1−δ))=ρ1−ρ[ξεξe∗∗+1−δ]

(35)∂U∂f:(1+ωn)(ξe+1−δ)((1−e)−f∗∗(1+ωn)(ξe+1−δ))=ρ1−ρ1f∗∗lnf∗∗

The superscript with double asterisk denotes the solution values for the maximization of (33). Combining conditions (34)–(35) yields the efficient investment in quality

(36)ξ(1+ωn)=1f∗∗[ε(lnf∗∗)−1]

Note that the efficient investment in quality improvement decreases with the productivity of human capital formation ξ and the degree of human capital spillover ω, and increases with the elasticity across varieties, s. By comparison, the decentralized investment in quality implied by equation (11) depends only on the preferences parameter s and it does not account for the human capital formation and spillover parameters. The first optimization condition can be written as

(37)(1ρ−1)lnf∗∗+1=ξ(1−e∗∗)ξe∗∗+1−δ⇒e∗∗=εlnf−1−1−δξ[(1ρ−1)(lnf)+1](1ρ−1)lnf+εlnf

The efficient investment in education implies the following rate of human capital accumulation

(38)1+gH=(1+ωn)(εlnf−1)(ξ+1−δ)(1ρ−1)(lnf)+εlnf=1+1−δξflnf[(1ρ−1)+ε]

Proposition 5:The growth rates of human capital accumulation and products’ quality improvements may deviate from the efficient one in various ways. Overall efficiency is achieved iff1+1−δξf∗∗lnf∗∗[(1ρ−1)+ε]=(1−δ)(1+ωn)+ξρ

Proof: Comparing (36) with (10) shows that the market will provide efficient rate of quality improvements only if, ξ(1+ωn)=1exps − 1(s−1). Hence, generally, the rate of quality improvements in the decentralized economy can be higher or lower than the efficient one. Comparing (38) with (26) implies that the rate of human capital accumulation in the market is efficient only if

1+1−δξf∗∗lnf∗∗[1ρ−1+ε]=(1−δ)(1+ωn)+ξρ

Clearly, this condition may hold only for a very specific set of parameter■

As explained above, in the present model there are externalities in the both the vertical and horizontal dimensions of innovation, due to knowledge externalities in quality improvements and in the process of human capital which determines the rate of variety expansion.

Both types of spillover are not internalized on the decentralized economy. Hence, the deviation of the decentralized economy from the efficient performance depend not only on the overall level of positive externalities but also on their relative strength.

5 Conclusion

In this work we have established a polynomial relation between population growth and economic growth, building on the notion of human-capital spillover from parents to their o springs. We have shown that the shape of the non-monotonic relation between population growth and economic growth can be altered and even inverted in the presence of congestion in human-capital spillover. Our findings contribute to the recent literature that is aimed to modify R&D-based model to remove the counterfactual definite positive effect of population growth on technological progress, and economic growth (“weak scale effect”).

In particular, this work adds to the few recent studies that established non-monotonic relation between population growth and economic growth. We have shown that under sufficient congestion impact, the effect of population growth on economic growth may follow a hump shape, that is consistent with the empirical finding of Boikos et al. (2013) and Kelley and Schmidt (1995). Finally, we have shown that the rates of human capital accumulation and products’ quality improvements in the decentralized economy may deviate in various ways from the welfare maximizers.

Subsequent research is called to explore the implications of endogenous fertility rates to the results derived in this work, including the potential for equilibria multiplicity that was pointed out but not fully explored by Boikos et al. (2013, p. 49).

Acknowledgments

We thank Pietro Peretto for insightful comments. We also thank the two anonymous referees of the journal for helpful and constructive comments.

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Published Online: 2016-10-12

Articles in the same Issue

https://doi.org/10.1515/bejm-2016-0029

Keywords for this article

human-capital spillover; innovation-based growth; population growth