Quality of schooling, fertility and economic growth

Saini, Swati; Mehra, Meeta Article Quality of schooling, fertility and economic growth Review of Economic Analysis (REA) Provided in Cooperation with: International Centre for Economic Analysis (ICEA), Waterloo, Ontario Suggested Citation: Saini, Swati; Mehra, Meeta (2024) : Quality of schooling, fertility and economic growth, Review of Economic Analysis (REA), ISSN 1973-3909, International Centre for Economic Analysis (ICEA), Waterloo (Ontario), Vol. 16, Iss. 2, pp. 175-220, https://doi.org/10.15353/rea.v16i2.5180 This Version is available at: https://hdl.handle.net/10419/328164 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Review of Economic Analysis 16 (2024) 175–220 1973-3909/2024175 Quality of Schooling, Fertility and Economic Growth SWATI SAINI*† Department of Economics, Delhi School of Economics, University of Delhi MEETA KESWANI MEHRA Centre for International Trade and Development, Jawaharlal Nehru University The existing body of literature underscores the crucial role of technology, driven by both innovation and imitation, in fostering economic growth. Human capital emerges as a key factor influencing technology adoption and innovation.We consider a R&D-based growth model to analyze how improvement in schooling quality impacts technical progress (via the twin channels of imitation and innovation) and therefore, longrun economic growth of an economy by working through the influence of fertility rates and education decisions at household level. The results indicate that improvement in schooling quality triggers a child quantity-quality trade-offat the household level when quality of schooling exceeds an endogenously determined threshold. At household level, parents invest more in the education of their children and have lesser number of children. This micro-level trade-off has two opposing effects on aggregate human capital accumulation at the macroeconomy wide level. A higher investment in education of a child stimulates the accumulation of human capital, which fosters technical progress, but the simultaneous decline in fertility rate reduces total factor productivity growth by contracting the stock of human capital. The former effect prevails over the latter only when quality of schooling is higher than the threshold and therefore, economic growth is driven by rate of aggregate human capital accumulation under both innovation and imitation regimes. However, when the quality of schooling is lower than this threshold, parents do not invest in the education and focus on maximizing fertility. Therefore, the economy grows at the rate of population growth at the macro level under the two regimes. Also, it is advantageous for an economy to innovate upon the local technology frontier instead of imitating from the world technology frontier if the rate of human capital accumulation is higher than the growth rate of world technology frontier in the presence of constant or diminishing returns to R&D sector. *This paper is based on one of the essays from the PhD thesis of Swati Saini. We gratefully acknowledge the invaluable comments and suggestions received from Debasis Mondal and Mausumi Das on the earlier version of the paper. We thank the participants of the 4th International Conference on South Asian Economic Development, South Asian University, New Delhi, CoRe: IGIDR PhD Colloquium, Indira Gandhi Institute of Development Research, Mumbai, DAAD workshop on “Comparative Economic Development Issues”, Centre for International Trade and Development, SIS, JNU, New Delhi, IMR Doctoral Conference, Indian Institute of Management, Bengaluru for their valuable comments and suggestions on earlier version of the paper. We are grateful to the editor, Jerzy (Jurek) Konieczny and the anonymous referee for their comments and suggestions. †Corresponding author: [email protected] ©2024 Swati Saini and Meeta Keswani Mehra. Licenced under the Creative Commons AttributionNoncommercial 3.0 Licence (http://creativecommons.org/licenses/by-nc/3.0/). Available at http://rofea.org. 175 Review of Economic Analysis 16 (2024) 175–220 Keywords: fertility, quality of schooling, economic growth, innovation, imitation JEL Classifications: J13, J11, O11, O30 1 Introduction According to Barro & Lee (2013) estimates, the share of population without any formal schooling in developing countries has declined from 54.6 percent in 1960 to approximately 17.4 percent in 2010. However, merely expanding access to education does not ensure that children actually learn in schools. The learning outcomes in schools closely hinge upon the quality of schooling, which has been given inadequate attention in the development policy paradigms of most developing countries until now. But more recently, development policies of most countries are making a shift towards improving learning quality in schools than merely expanding access to education. This policy paradigm shift in education policy is also reflected in the post-2015 development agenda. Imparting quality education features as the fourth Sustainable Development Goal set by the United Nations. This shift is motivated by two factors. First, there is growing empirical evidence that quality of schooling matters more for economic growth. Several studies have found that human capital quality has a significant positive impact on growth (Hanushek & Kimko 2000, Hanushek & Woessmann 2012, Ciccone & Papaioannou 2009, Islam et al. 2014, Neycheva 2016, Altinok & Aydemir 2017, Campbell & ¨ Ung¨ or 2020). Second, poor quality of schooling remains a dismal reality in developing countries. According to the Annual Status of Education Report (ASER) (2019) survey titled ’Early Years’, at least 25 percent of Indian school children in the four-eight age group do not have age-appropriate cognitive and numeracy skills, making for a massive learning deficit at an early age. Similarly, Glewwe et al. (2010) report that teachers from rural schools in Kenya were absent 20 percent of the time; while, in Zambia and Pakistan, teachers were absent, respectively, 18 percent and 10 percent of the time (Das et al. 2004, Reimers 1993). In addition to these observations about quality of schooling, certain demographic changes have been observed in the world since past few decades. The total fertility rate has declined markedly in many regions of the world over the last few decades.1The average fertility has declined in sub-Saharan Africa from 6.3 births per woman in 1990 to 4.6 in 2019. Other regions have also witnessed a fertility decline over the same period - Northern Africa and Western Asia (from 4.4 to 2.9), Central and Southern Asia (4.3 to 2.4), Eastern and South-Eastern Asia (2.5 to 1.8), Latin American and the Carribean (3.3 to 2.0) and the Oceania (4.5 to 3.4)2(United Nations, Department of Economic and Social Affairs, Population Division 2019). Around 40 percent of the world’s population lives in intermediate-fertility countries and close to 50 percent 1Total fertility rate is defined as the average number of children born to women over a lifetime. 2The region of Oceania excludes New Zealand and Australia in the UN Report. 176 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth of the global population lives in low-fertility countries.3It is expected that slightly less than 30 percent of the world population will live in intermediate-fertility countries and 70 percent of the total population will live in low-fertility countries in 2050. Human capital is a direct factor of production, which is positively related to output growth just like other factors, such as physical capital and labor (Lucas Jr 1988, Rebelo 1991, Mankiw et al. 1992). Additionally, human capital facilitates the adoption and development of technology (sometimes differentiated by imitation and innovation activity as two distinct routes for technological progress) (Nelson & Phelps 1966, Benhabib & Spiegel 1994). Using a horizontal R&D-based growth model ´ a la Romer (1990) with endogenous human capital supply, Boikos et al. (2022) model leisure externalities in R&D activity and examine their impact on innovation rate when interacted with technological spillovers. They show that ceteris paribus, leisure has a positive influence on innovation activity if there are weak intertemporal spillover effects arising from existing ideas. However, if there exists strong intertemporal spillover effects, then, leisure has a negative influence on innovation activity. As it is evident from this endogenous growth literature, human capital is a major determinant of economic growth. A decline in population implies a decline in human capital which can lead to a decline in economic growth as human capital is the driving force for R&D activities. Thus, declining fertility rate can strangle economic growth. However, empirical literature finds that population growth and economic growth are negatively correlated (Kelley & Schmidt 1994, Brander & Dowrick 1994, Li & Zhang 2007, Herzer et al. 2012, Prettner et al. 2013). Boikos et al. (2013) investigates the relationship between the fertility rate and per capita human capital investment. Using a theoretical framework, they show that birth rate has a monotonically negative impact on economic growth rate when birth rate is exogenously given and it is assumed that its impact on per capita human capital investment is linear and monotonically negative. In the absence of any specific assumption regarding impact on per capita human capital investment and after endogenizing birth rate, they show that total impact of population growth on economic growth is non-monotonic which conforms with the possibility that impact of population growth on economic growth may differ (in sign and magnitude) across countries characterized by different birth rates. There exists a strand of theoretical literature linking R&D based growth with endogenous fertility and education decisions (Strulik 2005, Strulik et al. 2013, Hashimoto & Tabata 2016, Bucci et al. 2020) that provides an explanation of why empirical literature finds no supportive evidence of the pessimistic prediction that declining fertility can strangle economic growth. The child quantity-quality trade-offat the micro level is posited as one of the plausible explanations. 3Intermediate-fertility countries are countries (such as India, Indonesia, Pakistan, the Philippines and Egypt) where women have an average lifetime fertility that ranges between 2.1 and 4 live births. Lowfertility countries are those countries where fertility is below 2.1 live births per woman. It includes almost all of Europe, Northern America, Australia and New Zealand. 177 Review of Economic Analysis 16 (2024) 175–220 This micro level trade-offensures that declining fertility is accompanied by higher human capital endowment per person in terms of education and health. As a result, although, declining fertility reduces workforce size but, also, leads to higher human capital endowment per person. This higher human capital accumulation averts the negative economic impact of declining fertility. Given that human capital is the driving force for R&D, this entails a higher R&D output and higher R&D-based growth. This strand of literature uses discrete-time overlapping generations framework to analyze the effect of child quantity-quality trade-offon economic growth. Rise in life expectancy of parents (Hashimoto & Tabata 2016),increase in health investment of children (Baldanzi et al. 2021), rising wages (Strulik et al. 2013) are some of the common mechanisms used to explain child quantity-quality trade-offat the household level and its consequent impact on R&D-based economic growth.4In a slight deviation from the usual mechanism of child quantity-quality trade-offadopted by most of the studies in this area, Cervellati et al. (2023) utilize a twosector framework to show how complementarities between human and physical capital yield endogenous dynamics of population, physical capital, and human capital. Our work is related to this strand of literature linking R&D based growth with endogenous fertility and education decisions and attempts at showing quality of schooling can be another mechanism that can trigger child quantity-quality trade-offat the household level. There exists another strand of theoretical literature that analyzes the linkages between quality of schooling and economic growth. Many existing studies (Tamura 2001, Gilpin & Kaganovich 2012) on quality of schooling and economic growth focus on explaining how determinants of quality of schooling, such as teacher-student ratio and teacher quality together, impact the learning process, and the consequent human capital formation and, therefore, economic growth. However, most of these studies assume exogenously determined population growth and also do not consider technical progress in their models. Consequently, these studies are unable to analyze the impact of schooling quality and the resulting demographic change on R&D activities, which are major determinants of technological development in the real world. We improve upon these papers by endogenizing both - population growth and technical change. The concept that the technology diffusion plays a crucial role in economic growth has a rich historical background. Nelson & Phelps (1966) was one of the first studies to highlight the 4In general, this child quantity-quality trade-offbetween fertility and education is a crucial component of unified growth theory, which models the transition from Malthusian state to sustained growth of an economy. The genesis of this strand of literature can be traced back to the seminal work of Becker (1960). Galor & Weil (2000) postulate that a higher rate of technical progress triggers a child quantity-quality trade-offwhich induces a demographic transition in which fertility rates decline and investments in human capital of children increase. This, eventually, paves way to the period of sustained economic growth of an economy. Besides technical progress, declining child mortality (Soares 2005), rise in life expectancy of parents (Boucekkine et al. 2002, 2003, Kalemli-Ozcan 2002, 2003), and decline in gender wage gap (Galor & Weil 1996) are other mechanisms to explain child quantity-quality trade-offat the household level and the consequent long-run development from stagnation to modern growth of economies. 178 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth role of human capital in facilitating technology diffusion. Jovanovic & Rob (1989) formulate an aggregate model featuring diverse agents, where interactions generated both novel ideas and the replication of ideas—a process of diffusion from initially more productive agents to less productive ones. Jovanovic & MacDonald (1994) delved into innovation and diffusion within a competitive industry, exploring how firm-level incentives were influenced by the distribution of technologies among their competitors. Kortum (1997) introduced the concept of a technology frontier to describe the knowledge state within a society. Lucas Jr (2009, 2015) developed models where this frontier evolved through interactions among agents, leading to the transfer of knowledge. Lucas Jr & Moll (2014) expanded on these models, allowing agents to allocate their time between production and acquiring knowledge. Perla & Tonetti (2014) show that growth is generated as a positive externality from risk-taking by less productive firms imitating more productive firms, leading to technology diffusion and sustained growth. Benhabib et al. (2014) develop frameworks in which agents optimally choose the amount to invest in improving growth through innovation as well as through technology diffusion. Benhabib et al. (2021) extend the analysis further and show how innovation and technology diffusion interact to endogenously determine the shape of the productivity distribution and generate aggregate growth. Stokey (2021) look at technology-skill complementarity and explain how the interplay between skill acquisition and technology drives economic growth in a stylized economy. The empirical literature on technology diffusion employs various methodologies, examining geographic diffusion within a country, diffusion among firms within an industry, and crosscountry diffusion respectively. Griliches (1957) is an early study on hybrid corn adoption in U.S. that finds hybrid adoption across geographic regions is well explained by their relative profitability across regions. Foster & Rosenzweig (1996) and Foster & Rosenzweig (2010) looks at introduction of high-yielding varieties in India and across multiple countries respectively and find that schooling played a crucial role in explaining variations. Along similar lines, Manuelli & Seshadri (2014) studied the gradual diffusion of tractors in U.S. agriculture. Comin & Hobijn (2004) explore the cross-country diffusion of specific technologies over two centuries, analyzing 23 industrial economies and Comin & Hobijn (2010) expand their study to 166 countries encompassing the entire spectrum of income levels.In the former investigation, they identify human capital as a crucial factor influencing the speed of adoption. Building upon this, Comin & Mestieri (2018) delve into adoption lags and the intensity of technology use. The findings suggest that variations in adoption lags played a significant role in the cross-country income disparities during the nineteenth century, while distinctions in intensity of use further contributed to divergence in the twentieth century. As it is evident from the literature, human capital plays a pivotal role in facilitating technology diffusion. In an influential empirical study, Krueger & Lindahl (2001) observe that human capital enhances growth only for the countries with the lowest level of education. That is, education matters only for catching up but not for innovation at the frontier. In an attempt 179 Review of Economic Analysis 16 (2024) 175–220 to resolve this Krueger-Lindahl puzzle, Vandenbussche et al. (2006) argue that human capital does not affect innovation and imitation uniformly. They develop an endogenous growth model, where innovation makes relatively more intensive use of skilled labor and imitative activities make relatively intensive use of unskilled labor and show that skilled labor has a higher growthenhancing effect closer to the technological frontier. Using a panel dataset covering 19 OECD countries for period 1960-2000, they find evidence in support of their theoretical findings.5Ang et al. (2011) empirically investigate the predictions of the theoretical model of Vandenbussche et al. (2006) for developing countries. Their results show that the growth enhancing effects of tertiary education attainment or skilled human capital increase when high and medium income countries move closer to the technology frontier. Human capital is not contributing to growth in low income countries, suggesting that they neither innovate nor imitate. Thus, it can be concluded from the extant literature that technology (via the twin channels of innovation and imitation) is a pivotal driver of economic growth and human capital plays a fundamental role in technology adoption and innovation in the process of technical progress. The endogenous fertility and education decisions at the household level can influence human capital accumulation at the aggregate level, which in turn, can affect the economic growth via its impact on technical progress. However, impact on economic growth can differ depending upon whether innovation or technology adoption is driving technical progress. We try to integrate these different strands of literature by building an overlapping generations version of an R&D-based growth model ´ a la Diamond (1965) and Jones (1995) to examine the impact of a child quantity-quality trade-offtriggered by improvement in quality of schooling on technical progress and economic growth. We focus on characterizing two cases of high and low quality of schooling for an economy and examine the corresponding drivers of economic growth - rate of human capital accumulation and population growth in these two cases under two distinct regimes of technological improvement - innovation and imitation. Under the innovation regime, technological improvements occur by innovating upon local technology frontier, whereas technological progress occurs by imitating existing foreign technologies under imitation regime. We derive the condition under which it will be advantageous for an economy to innovate on local technology frontier. In this respect as well, this work is an improvement over existing research in this area. We find that the quality of schooling triggers a child quantity-quality trade-offat the micro 5They show that human capital affects the rate of technical progress via a level effect and a composition effect. Holding the composition of human capital constant, an increase in the stock of human capital is always growth-enhancing. However, holding its level constant, the growth-enhancing properties of human capital depend on both its composition and the distance to the technological frontier. The growthenhancing impact of skilled labor increases with a country’s proximity to the world technology frontier, where proximity is measured by the ratio between the total factor productivity in the country and the corresponding variable for a frontier economy such as US. Conversely, the growth-enhancing impact of unskilled labor decreases with the proximity to the world technology frontier. 180 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth level when quality of schooling surpasses an endogenously determined threshold under both the technology regimes. When quality of schooling surpasses the threshold, parents invest in education of their children and bear fewer number of children. This micro-level trade-offhas two opposing effects on aggregate human capital accumulation at the macro level. A higher investment in the education of a child stimulates the accumulation of human capital, which fosters technical progress, but the simultaneous decline in fertility rate reduces the total factor productivity growth and economic growth by contracting the pool of available researchers. The first effect prevails over latter only when quality of schooling is higher than the threshold. When quality of schooling is less than the threshold, parents do not educate their children and focus on maximizing fertility. In such a scenario, economic growth is solely driven by population growth under both innovation and imitation regimes.Distance from technology frontier is another driver of growth under imitation regime. Our results show that it is advantageous for an economy to innovate upon the local technology frontier instead of imitating from the world technology frontier if the rate of human capital accumulation is higher than the growth rate of world technology frontier in the presence of constant or diminishing returns to R&D sector. Furthermore, a mere surpassing of the threshold level of quality schooling is not sufficient enough for an economy to experience a higher economic growth rate as compared to an economy with quality of schooling lower than the threshold level. Under the two technology regimes, quality of schooling should be high enough such that it leads to high enough investments in education of children, entailing that the growth-stimulating effect dominates the growth-impeding effect of quality of schooling. The rest of the paper is organized as follows. Section 2 discusses the basic structure of the model. Sections 3 and 4 discuss the market clearing condition and the key analytical results for a decentralized economy, which provide the key propositions of this study. Section 5 concludes. 2 The Model and Equilibrium Solutions 2.1 The Economic Environment We consider a model economy populated by overlapping generations of people, each of whom lives for two periods: adulthood and old age. Time is discrete and spans from 0 to ∞. During childhood, which is not modeled explicitly, individuals are reared and educated by their parents. All the decisions are made at the beginning of adulthood. Adults are identical in all aspects.They inelastically supply their skills in the labor market. Adults care about the consumption of a homogeneous final good, number and human capital level of their children. During old age, individuals consume their savings plus interest earned on these. Abstracting from gender differences, each household has a single parent. For avoiding the indivisibility problem, we assume that children are borne in continuous number. All individuals survive up to adulthood. The education of current period’s children (though childhood is not modeled 181 Review of Economic Analysis 16 (2024) 175–220 explicitly) determines human capital endowment of next period’s adult generation. Akin to Castell´ o-Climent & Hidalgo-Cabrillana (2012), human capital accumulation function depends on an exogenously given quality of education, parental investment in education and human capital of parent. Parental investment in education is a fraction of income spent on education of each child. The production structure of the economy closely follows Romer (1990) and Jones (1995). The economy consists of three sectors: final goods, intermediate goods and R&D. Perfect competition prevails in the final goods and R&D sectors whereas monopoly prevails in the intermediate goods sector. The intermediate goods are horizontally differentiated, each produced with the help of a blueprint design developed in the R&D sector. The entire range of intermediate goods constitutes as an input in the final goods sector. 2.2 Individuals Individuals derive utility from c1,t, their own consumption of the final good during adulthood; c2,t+1, their own consumption during old age; nt, number of children borne and ht+1, human capital per child. Parents’ motivation to invest in human capital of children by spending on children’s education is driven by a ‘warm glow’ of giving (Andreoni 1989) or preference for having ‘higher-quality’ children (Becker 1960). The lifetime expected utility of individuals in generation tis given by: ut=log c1,t+β1log c2,t+1+β2log(ht+1nt),(1) where positive weights β1and β2measure the weights on future consumption, c2,t+1, child quantity, ntand quality, ht+1relative to current consumption, ct, in the utility function. Alternatively, following De la Croix & Doepke (2004), β2can be interpreted as an “altruism” factor. An individual’s embodied human capital is denoted by htand the wage per unit of human capital is wt. Young adults spend their income on current consumption, savings for old-age consumption and child’s education expenditure. Rearing a child necessarily takes fraction τ∈ (0,1) of an adult’s time, which is given exogenously. Accordingly, the budget constraints for the adults and old individuals are given by: wtht(1 −τnt)=c1,t+st+et(wtht)nt; (2) c2,t+1=(1 +rt+1)st,(3) where etis the fraction of income per child spent on education, stis savings and rt+1is interest rate. Non-negativity constraints apply to all the variables. The human capital of children, ht+1, depends on human capital of parents, ht, parental investment in education per child, et, and quality of education system, θ, which is exogenously given. ht+1=(µ+θet)ht.(4) 182 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth competition among multiple R&D firms to become the first to succeed in creating and patenting a new blueprint and/or process. 7 If all other factors are held constant, an increase in R&D activity will induce increased duplication of research effort leading to stepping-on-toes effect. Additionally, R&D productivity depends on a catch-up term, ¯ At At under the imitation regime. Akin to Nelson & Phelps (1966), ¯ At At is the catch-up term, which signifies the fact that greater the technological gap between leader and follower economy, higher the potential of the follower economy to catch up through imitation of existing technologies. Since all R&D firms end up in a symmetric equilibrium, the production function of technology under innovation regime at the aggregate level reduces to: At+1−At=¯ δ(HA t)λAφ t.(25) Under the imitation regime, the aggregate production function reduces to: At+1−At=¯ δ(HA t)λAφ t"¯ At At#.(26) The catch-up effect is specific to the imitation regime only. Each firm in the R&D sector maximizes profits, given by: πt,A=pA t(At+1−At)−wAHA t, where pA tis price of a blueprint, At+1−Atare number of new blueprints discovered and wAis the wage rate. Under both imitation and innovation regimes, using eq. (23), the profit function of R&D firm can be expressed as: πt,A=pA tδt(HA t)−wAHA t.(27) In case of both the technology regimes, maximization of profits would lead to the following optimality condition: wA=pA tδt.(28) Substituting for δtfrom eq. (24), the wage rate under innovation regime is now given by: win A=pA t¯ δ(HA t)λ−1Aφ t= pA t¯ δ(HA t)λAφ t HA t,(29) Similarly, wage rate under imitation regime is expressed as: wim A= pA t¯ δ(HA t)λAφ t ¯ At At HA t ,(30) 7The term, (HA t)λ−1in eq (24) captures the stepping-on-effect. There exists diminishing returns to R&D effort as 0 <λ<1. The standing-on-shoulders effect is captured by Aφ tin eq. (24). 189 Review of Economic Analysis 16 (2024) 175–220 where superscripts ’in’ and ’im’ refer to variables under the innovation and imitation regimes. Using eqs. (25) and (26), the wage rate under both the technology regimes simplifies to: wA="pA t(At+1−At) HA t#.(31) where wages of researchers are increasing in the price of blueprint (price of patent) and number of blueprints discovered. We, next, consider the research arbitrage condition. Shareholders of intermediate firms face two options. First, they can make an investment of pA tin a risk-free asset and earn the market rate of interest, rtwhich is exogenously given. Alternatively, they can purchase shares of intermediate firms in period t and receive πt+1as dividend in period t+1 and can sell these shares to next generation to earn capital gain/loss resulting from the change in price of patents over time. In equilibrium, the rate of return from both these investments should be the same. That is, rt+1pA t=πt+1+(pA t+1−pA t). The left hand side of this equation is the interest earned from investing in a risk-free asset. The right hand side is the sum of the dividend earned and the capital gain/loss. 3 Market clearing condition and Dynamics of the System Human capital is used for production of final good, intermediate inputs and R&D activities. Now, in equilibrium, the demand for human capital in R&D sector, intermediate inputs sector and final good sector should add up to: HA t+HY t+HI t=Ht.(32) This gives the labor market clearing condition. Next, we consider the equilibrium condition for final goods market. Final good is used for consumption and for incurring education expenditure on children. The final goods market clearing condition is given by: Yt=c1,tNt+c2,tNt−1+Et(33) where Et=etwthtntNtis the total education expenditure incurred on next generation by the present generation. Furthermore, the aggregate savings of young adults in period tmust be used for net investment in R&D activities i.e. pA t(At+1−At)=stNt−pA tAt 190 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth The term on left-hand side is the net investment in R&D activities and the first term on righthand side is aggregate savings of young adults and second term is the dissavings of the old and the two terms together on the right-hand side represent the net savings. Eliminating pA tAtfrom both the sides yields the following asset market clearing condition8: pA tAt+1=stNt(34) In equilibrium, wages in final good, intermediates and R&D sectors should equalize, that is, wY=wA=wI. Let that equalized wage rate be wt. Substituting for wYand wIfrom eqs. (14) and (22), we get that: HI t HY t =α2 1−α(35) Now, HY t+HI t=Ht−HA t.Substituting for HY tfrom eq.(35), we derive the human capital stock engaged in intermediates sector as: HI t=α2 1−α+α2[Ht−HA t] (36) Next, we substitute for stfrom eq.(6) and pA tfrom eq. (28) in the asset market clearing condition and use the fact that wY=wA=wI=wtat equilibirum and that the size of the workforce or supply of labor is given by Lt=(1 −τnt)Ntto get9: At+1=ζδtHt(37) where ζ=β1 1+β1+β2(1−τnt). Now, we get the following expression for At+1under the two technology regimes after substituting for δtfrom eq. (24): Innovation regime : At+1=ζ¯ δ(HA t)λAφ t Ht HA t ; (38) Imitation regime : At+1=ζ¯ δ(HA t)λAφ t"¯ At At#Ht HA t . which after substituting from eqs. (25) and (26), simplifies to a single expression for both the regimes: HA t Ht At+1=ζ(At+1−At) 8Detailed derivation of asset market clearing condition is provided in Appendix B 9We can treat ntas a constant as we know from eq. (8) that fertility rate is constant over time 191 Review of Economic Analysis 16 (2024) 175–220 We get the employment share in R&D sector for both technology regimes after dividing both sides by Atas HA t Ht =ζgA,t 1+gA,t (39) Accordingly, the equilibrium wage rate in the three sectors is given by: wA=wY=wI=(1 −α)Yt HY t .(40) We, next, examine the dynamic properties of our stylized economy. First, we discuss the dynamics of physical factors of production. The aggregate population, Nt, grows at the fertility rate, ntas follows: Nt+1=ntNt,(41) where ntis constant and endogenously given by eq. (8). Taking child rearing time into account, the size of the workforce or supply of labor is given by Lt=(1 −τnt)Nt. Since child rearing costs are constant over time, and from eq. (8) we know that fertility rate is also constant over time, the workforce grows at the fertility rate as: Lt+1=ntLt.(42) Next, we discuss the dynamics of aggregate human capital, Ht≡htLt. The dynamics of per capita human capital are given by eq. (9). Using eqs. (9) and (42), the equation for aggregate human capital accumulation can be written as: Ht+1 Ht =           µnt,if θ≤µ τ ; (τθ −µ) (1 −) nt,otherwise. (43) The dynamics of total factor productivity are given by eq.(38) as: Innovation regime: At+1=ζ¯ δ(HA t)λAφ t Ht HA t .(44) Imitation regime: At+1=ζ¯ δ(HA t)λAφ t"¯ At At#Ht HA t .(45) Using eqs.(19),(35) and (36), the dynamics of aggregate output can be written as: Yt=(1 −α)1−αα2 α2(1−α)(1 −α+α2)A1−α t"1−HA t Ht#Ht(46) 192 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth The system of equations,(eqs. (41)-(46)) fully describes the equilibrium dynamics of our model economy for all the plausible cases. The next section characterizes the balanced growth paths of an economy for two cases - a) when the economy’s quality of education system is sufficiently high, that is, θ > µ τ , and b) when quality of schooling is less than the threshold, or, θ≤µ τ . 4 Balanced Growth Path and Steady-State Properties of the Stylized Economy 4.1 Characterizing the Balanced Growth Path We denote the growth rate of xalong the balanced growth path by gx, that is, by omitting the time index for brevity. 10 We, first, consider the growth rate of human capital accumulation. The proportion of workforce employed in final goods, intermediates and R&D sectors, (given by eqs. (35),(36), (39) respectively) are constant as gA,tis constant along the balanced growth path. Therefore, along the balanced growth path, we have: HY t+1 HY t =HA t+1 HA t =HI t+1 HI t =Ht+1 Ht =(1 +gH).(47) Thus, the human capital stocks in the final good, intermediates and the R&D sectors grow at the rate of total human capital accumulation along the balanced growth path. Next, we consider the growth rate of total factor productivity. Under innovation regime, we observe from eq. (44) that: (1 +gA,t)=ζ¯ δHt(HA t)λ−1 A1−φ t ; Since along balanced growth path, the left hand side is constant, therefore, right hand side must also be constant and this holds true when (1 +gA)=[(1 +gh)n]λ 1−φ.(48) The right hand side follows from the definition of aggregate human capital Ht=htLtand from eq. (47). Further, we observe from eq. (45) that the rate of technical progress under the imitation regime can be written as: (1 +gA,t)=ζ¯ δHt¯ At(HA t)λ−1 A2−φ t . 10A balanced growth path is a long run equilibrium of the economy, also defined as the steady state, along which growth rate of variables is either zero or constant over time. For any variable x, the growth rate is denoted by gx,t=(xt+1−xt)/xt, and its rate of change by ˜gx,t=(gx,t+1−gx,t)/gx,t. The balanced growth, thus, requires ˜gx,t=0. 193 Review of Economic Analysis 16 (2024) 175–220 Similarly, using the definition of the balanced growth path, we derive the long-run rate of technological progress under imitation regime as: (1 +gA)=(1 +gH)λ 2−φ(1 +g¯ A)1 2−φ=[(1 +gh)n]λ 2−φ(1 +g¯ A)1 2−φ.(49) Thus, intuitively, under both imitation and innovation regimes, technological progress is driven by growth in aggregate human capital. Human capital accumulation improves productivity of researchers, which fosters technological progress. Besides aggregate human capital, the growth of world technology frontier is also a driver of growth, but only in case of imitation regime. The follower economy takes advantage of existing technologies through technology adoption. Therefore, as the world technology frontier grows, it enhances the potential of the follower country to catch up through imitation. Next, we ascertain the growth rates of aggregate output and per capita consumption along the balanced growth path. From eq. (46) we observe that: (1 +gY)=(1 +gA)1−α(1 +gH).(50) Along the balanced growth path, growth rate of aggregate output depends on growth rate of human capital and total factor productivity. From eqs. (48), (49) and (50), we derive the balanced growth path of the stylized economy under the two technology regimes as: Innovation regime: (1 +gY)=[(1 +gh)n]1−φ+λ(1−α) 1−φ; (51) Imitation regime: (1 +gY)=[(1 +gh)n]2−φ+λ(1−α) 2−φ(1 +g¯ A)1 2−φ,(52) where [(1 +gh)n]=(1 +gH)=                      β2θ (1 +β1+β2)µ1−,if θ < µ τ ; β2µ (1 +β1+β2)τ,if θ=µ τ ; β2θ(1 −)1− (1 +β1+β2)(τθ −µ)1−,otherwise. This follows after substituting the value of nfrom eq. (8) into eq. (43). Thus, along the steady state, the growth rates of aggregate output are determined by the rate of human capital accumulation under both the regimes of technical improvement. 194 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth Furthermore, we observe from the consumer’s optimization exercise that the Euler equation has the following standard form: ct+1 ct =β1(1 +rt+1).(53) The right hand side follows from substituting values of ctand stfrom eqs. (5) and (6) in eq. (3). Along the balanced growth path, per capita consumption grows at a constant rate under both the technology regimes as rtis exogenously given. We next compare the economic growth rates under the two technology regimes. We denote the economic growth rate under innovation regime by ginn Yand under imitation regime by gimi Y. The innovation economy will grow at a higher rate if ginn Y>gimi Y. Substituting for ginn Yand gimi Yfrom eqs. (51) and (52), we get: (1 +gH)1−φ+λ(1−α) 1−φ>(1 +gH)2−φ+λ(1−α) 2−φ(1 +g¯ A)1 2−φ which on simplification yields:11 (1 +gH)λ(1−α) 1−φ>(1 +g¯ A). Thus, we have, Proposition 4.1 (i) Under innovation regime, total factor productivity, aggregate output and per capita consumption grow at a constant rate along the balanced growth path characterized by eqs. (48), (51) and (53). (ii) Under imitation regime, total factor productivity, aggregate output and per capita consumption grow at a constant rate along the balanced growth path characterized by eqs. (49), (52) and (53). (iii) Along the balanced growth path, the economy under innovation regime exhibits a higher economic growth rate as compared to imitation regime if gH>g¯ A, λ +φ≤1 Intuitively, under both the technology regimes, the self-sustaining endogenous growth path is driven by human capital accumulation when quality of schooling exceeds the threshold, θ > µ τ . In this case, at the micro level, parents decide to have fewer number of children and invest more 11We get the same condition if we compare per capita economic growth rates under the two technology regimes. 195 Review of Economic Analysis 16 (2024) 175–220 in their education. This follows from Lemma 1. At the macro level, this trade-offraises the rate of human capital accumulation, which encourages faster technological progress and, therefore, economic growth. Besides human capital, growth of world technology frontier is an additional driver of growth under the imitation regime via the catch-up effect. Alternatively, when quality of schooling is less than the threshold, θ≤µ τ , parents do not invest in the education of children and instead, maximize fertility. In this case, the balanced growth path of the economy is driven only by population growth, which in turn, is determined by the fertility rate. This result is similar to the findings of neo-classical models of SolowSwan and Cass-Koopmans-Ramsey. The only difference is that population growth and technical progress are endogenously determined in our model structure. Thus, the drivers of economic growth differ depending upon the level of quality of schooling. When quality of schooling surpasses the threshold level, economic growth is driven by human capital accumulation whereas it is driven by population growth when quality of schooling is less than the threshold. In a similar context, while Eckstein & Zilcha (1994) do not specifically model the quality of schooling in their OLG framework, they show the effect of compulsory education on economic growth and distribution. Their analysis reveals that compulsory schooling financed by proportional taxes on income increases economic growth, and makes income distribution more equitable in the long-run. Similarly, Tamura (2001) explicitly model quality of schooling in their model of human capital accumulation but does not consider technical progress. He shows that quality of education fosters human capital formation and, therefore, economic growth. Also, it is advantageous for an economy to innovate upon the local technology frontier instead of imitating from the world technology frontier if the rate of human capital accumulation is higher than the growth rate of world technology frontier in the presence of constant or diminishing returns to R&D sector (i.e. λ+φ≤1). This implies that an economy has the potential to become the new world technology frontier if quality of schooling should be sufficiently high such that it leads to high enough investment in the education of children so that human capital accumulation emerges as a driver of economic growth. We next turn to characterizing the evolution of wage rate along the steady state. It is known from eq. (40) that wage rate can be expressed as: wA=wY=wI=(1 −α)Yt HY t . Further, from eq. (51), under the innovation regime: (1 +gY)=[(1 +gh)n]1−φ+λ(1−α) 1−φ,and from eq. (52), under imitation regime, we have: (1 +gY)=[(1 +gh)n]2−φ+λ(1−α) 2−φ(1 +g¯ A)1 2−φ. 196 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth Eqs. (40), (47), (51) and (52) together imply that, along balanced growth path, wage rate under innovation regime grows as follows: (1 +gw)=[(1 +gh)n]λ(1−α) 1−φ; (54) And the wage rate under imitation regime grows at the rate: (1 +gw)=[(1 +gh)n]λ(1−α) 2−φ(1 +g¯ A)1 2−φ,(55) where [(1 +gh)n]=(1 +gH)=                      β2θ (1 +β1+β2)µ1−,if θ < µ τ ; β2µ (1 +β1+β2)τ,if θ=µ τ ; β2θ(1 −)1− (1 +β1+β2)(τθ −µ)1−,otherwise . Intuitively, the wage rate depends on aggregate output and human capital as expressed by eq. (40). It is known from eq. (50) that growth rate of aggregate output or GDP depends on growth rate of human capital and total factor productivity along the balanced growth path. Therefore, the wage rate grows at the rate of technical progress along the balanced growth path under both the regimes of technological improvement. Next, we compare the two cases of high and low quality of schooling and determine the condition under which the economy exhibits higher growth rate of per capita income under the case of higher quality of schooling, θ > µ τ , as compared to the case of low quality of schooling, θ≤µ τ , under the two technology regimes. 4.2 Comparative Analysis of Per Capita Economic Growth Rates of Economies with Higher and Lower Quality of Schooling We assume that when θ > µ τ , quality of schooling is denoted by θhfor that particular economy whereas quality of schooling is denoted by θlfor an economy with quality of schooling less than the threshold, θ≤µ τ . Derivations provided in Appendix C show that an economy with higher quality of schooling, θh, will experience a higher per capita income growth rate as compared to an economy with a lower quality of schooling, θl, if the following conditions hold under the individual technology regimes. Innovation regime: θh> θl"(τθh−µ) µ(1 −)#λ(1−α)−(1−φ+λ(1−α)) λ(1−α) ; (56) 197 Review of Economic Analysis 16 (2024) 175–220 Imitation regime: θh> θl"(τθh−µ) µ(1 −)#λ(1−α)−(2−φ+λ(1−α)) λ(1−α) .(57) Thus, we have, Proposition 4.2 An economy with higher quality of schooling, θh>µ τ , will experience a higher per capita income growth rate as compared to an economy with lower quality of schooling, θl≤µ τ if the quality of schooling, θh, is sufficiently high captured by the following parametric restriction: Innovation regime: θh> θl"(τθh−µ) µ(1 −)#λ(1−α)−(1−φ+λ(1−α)) λ(1−α) > θl; (58) Imitation regime: θh> θl"(τθh−µ) µ(1 −)#λ(1−α)−(2−φ+λ(1−α)) λ(1−α) > θl. Intuitively, a mere surpassing of the threshold level of quality schooling is not sufficient enough for an economy to experience a higher growth rate of per capita output as compared to an economy with quality of schooling lower than the threshold level. Under the two technology regimes, quality of schooling should be sufficiently high as indicated by the eq. (58) such that it leads to high enough investment in the education of children, entailing that the growthstimulating effect overpowers the growth-impeding effect of quality of schooling. Otherwise, the possibility that an economy with lower quality of schooling experiences a higher per capita economic growth rate than an economy with higher quality of schooling is not ruled out, especially for large enough values of child rearing costs, τor for small enough value of inter-generational human capital spillovers, µand returns to education, respectively. This follows directly from eq. (58). It can be observed that the expression, (τθh−µ) µ(1 −)is increasing in τ. In this particular case, when the value of τis sufficiently high, population growth rate may turn out to be a more effective driver of economic growth as the threshold value of quality of schooling for higher economic growth is so high that an economy may find not investing at all in education of the future generation as a relatively more beneficial outcome. Similarly, it can be shown that: ∂ ∂µ (τθh−µ) µ=−τθh µ2<0; ∂ ∂  1−=−1 (1 −)2<0. These imply that the threshold value of quality of schooling for higher per capita economic growth is decreasing in the value of µand respectively. Thus, this threshold value of quality of schooling can be high enough for sufficiently small µand such that population growth rate 198 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth is decreasing in the value of . This implies that, ceteris paribus, this critical threshold value decreases as returns to schooling increase when quality of schooling exceeds the threshold. Therefore, ceteris paribus, parents educate their children and bear lesser number of children in response to an increase in returns to education. Similar to the impact of quality of schooling, this micro level trade-offgenerates a growth-stimulating effect and a growth-impeding effect at the macro level. ∂gH ∂ =(1 +gh)n "1 (1 −)+log(τθ −µ) 1−# | {z } Growth-stimulating effect −1 1− |{z} Growth-impeding effect  . The growth-stimulating effect overpowers the growth-impeding effect of a change in returns to education when quality of schooling exceeds the threshold as shown by eq. (70). Resultantly, an increase in returns to education yields higher rate of technical progress under both innovation and imitation regimes. Also, the growth rate of per capita income rises as returns to education increase under both the technology regimes when quality of schooling exceeds the threshold. The intuitive explanantion for this effect is similar as the explanantion for a rise in growth rate of per capita income as quality of schooling improves under the two regimes. The economic growth rate, gY, is increasing in as the growth-stimulating effect overpowers the growth-impeding effect of a change in returns to education. Also, it is known from Lemma 2 that the population growth rate is decreasing in returns to education, . Therefore, the growth rate of per capita income rises as returns to education increase. When quality of schooling is strictly lower than the threshold, parents make zero investment in education of their children and focus on having more children. Thus, higher population growth drives economic growth in this case. Additionally, it can be observed from eq. (9) that inter-generational human capital spillovers become more productive and spur growth rate of per capita human capital as returns to education increase whereas when θ=µ τ , an increase in returns to education yield higher rate of technical progress and, therefore, economic growth under both the regimes due to higher inter-generational human capital spillovers as fertility rate and therefore, population growth rate does not depend on returns to education in this case. Similar to the effect of quality of schooling, it can be observed from eqs. (63) and (64) that returns to education raise population growth rate by a lesser proportion as compared to the proportionate rise in growth rate of aggregate output under both the technology regimes. Therefore, growth rate of per capita income rises as returns to education rises. This completes the characterization of the balanced growth path of our decentralized economy. 205 Review of Economic Analysis 16 (2024) 175–220 5 Discussion This paper formulates an analytical framework to examine the impact of quality of schooling on technical progress and therefore, economic growth of an economy. Since technical advancements can happen through innovation and technology adoption (imitation), one can observe varying impact on economic growth depending upon whether innovation or technology adoption is driving technical progress. Therefore, we characterize two types of economies. The first is an innovation economy where technological improvements occur by innovating upon the local technology frontier. The second is an imitation economy where technological progress occurs by imitating existing foreign technologies. We examine how the endogenous fertility and education decisions at the household level (triggered by schooling quality) influence human capital accumulation at the aggregate level, which in turn, affect the economic growth under the two technology regimes. We find that the quality of schooling triggers a child quantity-quality trade-offat the micro level when quality of schooling surpasses an endogenously determined threshold under both the technology regimes. When quality of schooling surpasses the threshold, parents invest in the education of their children and bear lesser number of children. However, parents focus on maximizing fertility and do not educate their children when quality of schooling is less than the threshold. This micro-level trade-offgenerates two types of effects on economic growth at the macro level - a growth-stimulating effect and a growth-impeding effect. Our results show that the former effect dominates over latter only when the quality of schooling is higher than the threshold, and the economy is on a self-sustaining growth path. Alternatively, when the quality of schooling is less than the threshold, parents do not educate their children and focus, instead on maximizing fertility. Higher fertility rate leads to higher population growth, which propels economic growth rate under both innovation and imitation regimes. Furthermore, it is advantageous for an economy to innovate upon the local technology frontier instead of imitating from the world technology frontier if the rate of human capital accumulation is higher than the growth rate of world technology frontier in the presence of constant or diminishing returns to R&D sector. Also, a mere surpassing of the threshold level of quality schooling is not sufficient enough for an economy to experience a higher growth rate of per capita output as compared to an economy with quality of schooling lower than the threshold level. Under the two technology regimes, quality of schooling should be high enough such that it leads to high enough investments in education of children, entailing that the growthstimulating effect dominates the growth-impeding effect of quality of schooling. This research can be extended in several directions in future. First, it is assumed that quality of schooling is exogenous in our analytical framework. One possible extension can be endogenizing quality of schooling. Quality of schooling can be endogenized by introducing an education sector in which teacher-pupil ratio and teacher quality determine the quality of education system of the economy. It will be interesting to examine how the dynamics of the economy 206 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth change when an additional education sector is introduced. Introduction of education sector can lead to competition between R&D and education sectors for hiring skilled labor, which may in turn, influence the growth dynamics of the economy. Second, we have focused only on skilled labor. Unskilled labor can also be introduced in the present theoretical structure to determine the impact of quality of schooling on the distribution of income between skilled and unskilled workers in the long-run. Third, there exists a possibility that human capital accumulation can be influenced by technological progress as shown by Bucci (2008). This possibility can be explored in the future by including technological progress in the human capital accumulation function. Fourth, we have characterized innovation-only and imitation-only regimes. Akin to Vandenbussche et al. (2006) and Basu & Mehra (2014), a diversified regime can be introduced where both innovation and imitation activities (with unskilled and skilled labor force) lead to technological improvements. Appendix A Solution to Household’s Optimization Exercise The utility function is described as follows: Maximize ut=log c1,t+β1log c2,t+1+β2log(ht+1nt) subject to wtht(1 −τnt)=c1,t+st+et(wtht)nt c2,t+1=(1 +rt+1)st ht+1=(µ+θet)ht,  < 1 After substituting for c2,t+1and ht+1, the lagrangian for this problem is formulated as : L=log c1,t+β1log[(1 +rt+1)st]+β2log nt+β2log(µ+θet)+β2log ht +ψ[wtht(1 −τnt)−c1,t−st−etnt(wtht)] The choice variables are c1,t,st,etand nt.The first-order conditions are: ∂L ∂c1,t =0⇔1 c1,t −ψ=0⇔c1,t=1 ψ.(A.1) ∂L ∂st =0⇔β1 st −ψ=0⇔st=β1 ψ.(A.2) ∂L ∂nt =0⇔β2 nt −ψτwtht−ψetwtht=0⇔β2 nt =ψ[τ+et]wtht⇔nt=β2 ψ[τ+et]wtht .(A.3) 207 Review of Economic Analysis 16 (2024) 175–220 ∂L ∂et =0⇔β2θ µ+θet −ψntwtht=0⇔nt=β2θ ψ[µ+θet]wtht .(A.4) From eqs. (A.3) and (A.4), the left hand side can be equated to yield: µ+θet=θ[τ+et]⇔µ−θτ =etθ[−1] et=µ−θτ θ(−1) =θτ −µ θ(1 −) Hence, we have: et=         0,if θ≤µ τ ; τθ −µ θ(1 −),otherwise.(A.5) Next, we know that the budget constraint is given by: wtht(1 −τnt)=c1,t+st+et(wtht)nt. From eq. (A.3), etnt(wtht) can be expressed as: etnt(wtht)=β2 ψ−τntwtht.(A.6) Substituting from eqs. (A.1), (A.2) and (A.6), the budget constraint can be expressed as: wtht−τntwtht=1 ψ+β1 ψ+β2 ψ−τntwtht which on simplifying leads to: ψ=1+β1+β2 wtht (A.7) whose substitution into eqs. (A.1) and (A.2) yields: c1,t=wtht 1+β1+β2 ; st=β1wtht 1+β1+β2 , Substituting for etfrom eq. (A.5) and for ψfrom eq. (A.7) in eq. (A.4), yields: nt=                   β2θ (1 +β1+β2)µ,if θ < µ τ ; β2 (1 +β1+β2)τ,if θ=µ τ ; β2θ(1 −) (1 +β1+β2)(τθ −µ),otherwise. This completes the solution to the utility maximization exercise of households. 208 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth Appendix B Derivation of Asset market clearing condition It is known from eq.(33) that final goods market clearing condition is given by: Yt=c1,tNt+c2,tNt−1+Et We know from national accounting that GNP can be calculated using the final use approach or the income approach. Let GNP be denoted by Zt. From the final use side, it is known that Ztis used for consumption, investment in R&D activities and for incurring education expenditure on children, i.e., Zt=c1,tNt+c2,tNt−1+Et+pA t(At+1−At)=Yt+pA t(At+1−At).(B.1) From the income side, gross national income (GNI) is given by GNI =wYHY t+wAHA t+wIHI t+Atπt(B.2) Substituting from eqs. (14), (20), (22) and (31), it can be observed that GNI =Yt+pA t(At+1−At)=Zt Thus, the value of GNI is equivalent to value of GNP. Equating eqs.(B.1) and (B.2), we have c1,tNt+c2,tNt−1+Et+pA t(At+1−At)=wYHY t+wAHA t+wIHI t+Atπt(B.3) =wtHt+Atπt where the last term on the right-hand side is derived using eq.(32) and wY=wA=wI=wt at equilibirum. Now, it is known from the household’s budget constraint that wtht(1 −τnt)= c1,t+st+et(wtht)nt. Multiplying both sides of the household’s budget constraint by Nt, we get wtHt=c1,tNt+stNt+Et where Et=et(wtht)ntNt.Substituting back in eq.(B.3), we deduce pA t(At+1−At)=stNt+Atπt−c2,tNt−1(B.4) It is known from the research arbitrage condition that Atπt=(1 +rt)pA t−1At−pA tAt.Substituting for Atπtand using that c2,tNt−1=(1 +rt)st−1Nt−1, we get that pA tAt+1=stNt+(1 +rt)pA t−1At−c2,tNt−1=stNt+(1 +rt)(pA t−1At−st−1Nt−1).(B.5) Since initial assets are given by pA −1A0−s−1N−1at t=0, we get the following asset market clearing condition for any period t>0, pA tAt+1=stNt. 209 Review of Economic Analysis 16 (2024) 175–220 Appendix C Derivations of Eqs. (56) and (57) We derive the conditions when an economy with high quality of schooling θ > µ τ exhibits a higher per capita output growth rate as compared to an economy with lower quality of schooling θ≤µ τ . We assume that when θ > µ τ , quality of schooling is denoted by θhfor that particular economy whereas quality of schooling is denoted by θlfor an economy with quality of schooling less than the threshold θ≤µ τ . An economy with higher schooling quality (θh) will grow at a higher rate as compared to an economy with lower quality of schooling (θl) when the following condition holds true under both the technology regimes: gy,θh>gy,θl(C.1) We first, derive the condition for innovation economy. At steady state, growth rate of per capita output under innovation regime is given by: gy=(1 +gh)1−φ+λ(1−α) 1−φnλ(1−α) 1−φ−1. Thus, we have (1 +gh,θh)1−φ+λ(1−α) 1−φ(nθh)λ(1−α) 1−φ>(1 +gh,θl)1−φ+λ(1−α) 1−φ(nθl)λ(1−α) 1−φ,(C.2) where gh,θhand gh,θlare per capita human capital accumulation rates when θ > µ τ and θ≤ µ τ respectively and nθhand nθlare the fertility rates when θ > µ τ and θ≤µ τ respectively. Substituting from eqs. (9) and (8) for (1 +gh,θh), (1 +gh,θl), nθhand nθlto get: "(τθh−µ) 1−#(1−φ+λ(1−α)) "θh(1 −) (τθh−µ)#λ(1−α) > µ(1−φ+λ(1−α)"θl µ#λ(1−α) which on simplification, yields: θh> θl"(τθh−µ) µ(1 −)#λ(1−α)−(1−φ+λ(1−α)) λ(1−α) .(C.3) Similarly, we derive the condition for imitation economy. Under imitation regime, the growth rate of per capita output given by: gy=(1 +g¯ A)1 2−φ(1 +gh)2−φ+λ(1−α) 2−φnλ(1−α) 2−φ−1. Thus, we have (1 +g¯ A)1 2−φ(1 +gh,θh)2−φ+λ(1−α) 2−φ(nθh)λ(1−α) 2−φ>(1 +g¯ A)1 2−φ(1 +gh,θl)2−φ+λ(1−α) 1−φ(nθl)λ(1−α) 2−φ. 210 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth Substituting from eqs. (9) and (8) for (1 +gh,θh), (1 +gh,θl), nθhand nθl, we derive: "(τθh−µ) 1−#(2−φ+λ(1−α)) "θh(1 −) (τθh−µ)#λ(1−α) > µ(2−φ+λ(1−α)) "θl µ#λ(1−α) , which simplifies to: θh> θl"(τθh−µ) µ(1 −)#λ(1−α)−(2−φ+λ(1−α)) λ(1−α) .(C.4) This completes derivations of eqs. (56) and (57). To prove that these two conditions hold true under the two technology regimes, we postulate that, λ(1 −α)−(1 −φ+λ(1 −α)) λ(1 −α)>1,and λ(1 −α)−(2 −φ+λ(1 −α)) λ(1 −α)>1. which can be simplified to yield the following expressions: −(1 −φ+λ(1 −α)) λ(1 −α)>0,and −(2 −φ+λ(1 −α)) λ(1 −α)>0. This is a contradiction as φ < 1, λ < 1, α < 1 and  < 1. Therefore, λ(1 −α)−(1 −φ+λ(1 −α)) λ(1 −α)<1,and λ(1 −α)−(2 −φ+λ(1 −α)) λ(1 −α)<1.(C.5) Further, we know that θ > µ τ . Multiplying both sides by τand then, subtracting µfrom both sides yields (τθ −µ) 1−> µ, Since µ≥1, we have: (τθ −µ) 1−>1.(C.6) Thus, "(τθh−µ) µ(1 −)#>1. Therefore, eqs. (C.5) and (C.6) together imply that "(τθh−µ) µ(1 −)#λ(1−α)−(1−φ+λ(1−α)) λ(1−α) >1,and "(τθh−µ) µ(1 −)#λ(1−α)−(2−φ+λ(1−α)) λ(1−α) >1.(C.7) Appendix D Derivations of Eq. (60) We know that (1 +gH)=(1 +gh).n.Differentiating both the sides w.r.t θyields: ∂gH ∂θ =(1 +gh)∂n ∂θ +n∂gh ∂θ .(D.1) 211 Review of Economic Analysis 16 (2024) 175–220 When θ > µ τ , we know from Lemma 1 that ∂nt ∂θ =−µβ2(1−) (1+β1+β2)(τθ−µ)2and it is given that (1 +gh)= (τθ −µ) (1 −) from eq (9). Differentiating ghw.r.t θ, we get that: ∂gh ∂θ ="(τθ −) 1−# .τ τθ −µ=(1 +gh)τ τθ −µ.(D.2) Substituting this into eq. (D.1), we get that: ∂gH ∂θ =(1 +gh)−µβ2(1 −) (1 +β1+β2)(τθ −µ)2+(1 +gh)τ τθ −µ∗n, Substituting for nfrom eq. (8), ∂gH ∂θ =(1 +gh)"τβ2θ(1 −) (1 +β1+β2)(τθ −µ)2−µβ2(1 −) (1 +β1+β2)(τθ −µ)2# =(1 +gh)[τθ −µ]β2(1 −) (1 +β1+β2)(τθ −µ)2. This completes derivation of eq. (60). Appendix E Derivations of Eqs. (65), (66), (67) and (68) The growth rate of per capita income can be expressed as: (1 +gy)=(1 +gY) n.(E.1) Under innovation regime, substituting for (1 +gY) from eq. (51) and simplifying, we get: (1 +gy)=(1 +gh)1−φ+λ(1−α) 1−φnλ(1−α) 1−φ.(E.2) Taking log on both sides and differentiating w.r.t θ, yields, 1 1+gy ∂gy ∂θ =1−φ+λ(1 −α) (1 +gh)1 −φ ∂gh ∂θ +λ(1 −α) (1 −φ)n ∂n ∂θ ,(E.3) When θ > µ τ , substituting for n from eq. (8), ∂n ∂θ from Lemma 1 and ∂gh ∂θ from eq. (D.2) yields: ∂gy ∂θ =1+gy 1−φ"(1 −φ+λ(1 −α))τ τθ −µ−µλ(1 −α) θ(τθ −µ)#. It can be observed that ∂gy ∂θ >0, if (1 −φ+λ(1 −α))τ (τθ −µ)>µλ(1 −α) θ(τθ −µ)⇔θ > µλ(1 −α) τ(1 −φ+λ(1 −α)). 212 SAINI, MEHRA Quality of Schooling, Fertility and Economic Growth This holds true as θ > µ τ ,λ < 1 and φ < 1. Thus, ∂gy ∂θ >0. Next, we consider the case when θ < µ τ . Substituting for n from eq. (8), ∂gh ∂θ from eq.(61) and ∂n ∂θ from Lemma 1 in eq. (E.3), we deduce: ∂gy ∂θ ="λ(1 −α)(1 +gy) θ(1 −φ)#. We now, consider the imitation regime. Substituting for (1 +gY) from eq. (52) and simplifying, we get: (1 +gy)=(1 +g¯ A)1 2−φ(1 +gh)2−φ+λ(1−α) 2−φnλ(1−α) 2−φ.(E.4) Taking log on both sides and differentiating w.r.t θ, we get: 1 1+gy ∂gy ∂θ =2−φ+λ(1 −α) 2−φ(1 +gh) ∂gh ∂θ +λ(1 −α) (2 −φ)n ∂n ∂θ .(E.5) When θ > µ τ , substituting for n from eq. (8), ∂n ∂θ from Lemma 1 and ∂gh ∂θ from eq. (D.2), we have: ∂gy ∂θ =1+gy 2−φ"(2 −φ+λ(1 −α))τ τθ −µ−µλ(1 −α) θ(τθ −µ)#. Now, ∂gy ∂θ >0 if (2 −φ+λ(1 −α))τ τθ −µ>µλ(1 −α) θ(τθ −µ)⇔θ > µλ(1 −α) τ(2 −φ+λ(1 −α)). This also holds true as θ > µ τ ,λ < 1 and φ < 2. Therefore, ∂gy ∂θ >0. We next, consider the case when θ < µ τ . When θ < µ τ , substituting for n from eq. (8), ∂gh ∂θ from eq.(61) and ∂n ∂θ from Lemma 1 in eq. (E.5) yields: ∂gy ∂θ ="λ(1 −α)(1 +gy) θ(2 −φ)#. This completes derivations of eqs. (65), (66), (67) and (68). Appendix F Derivations of Eqs. (70) and (71) We know that:(1 +gH)=(1 +gh).n. Differentiating both the sides w.r.t , we get that: ∂gH ∂ =(1 +gh)∂n ∂ +n∂gh ∂ (F.1) 213 Review of Economic Analysis 16 (2024) 175–220 When θ > µ τ , we know from the interior solution of eq. (9) that (1+gh)=(τθ −µ) (1 −) . Taking log on both sides and differentiating w.r.t , we get the following expression: 1 1+gh ∂gh ∂ =1+log+log(τθ −µ)+ 1−−log(1 −) =(1 +gh)"1 (1 −)+log "(τθ −µ) 1−##.(F.2) Also, from Lemma 2, we have ∂nt ∂ =−β2θ (1 +β1+β2)(τθ −µ). Substituting for ∂n ∂ from Lemma 2 and ∂gh ∂ from eq. (F.2) into eq. (F.1), we derive that: ∂gH ∂ =−β2θ(1 +gh) (1 +β1+β2)(τθ −µ)+(1 +gh)n"1 (1 −)+log(τθ −µ) 1−# Substituting for β2θ (1 +β1+β2)(τθ −µ)from eq. (8), yields: =(1 +gh)"n"1 (1 −)+log(τθ −µ) 1−#−n 1−#=(1 +gH)log "(τθ −µ) 1−#. We next derive the expression for ∂gH ∂ when θ≤µ τ . When θ≤µ τ , it is known from eq. (9) that:(1 +gh)=µ. Taking log on both sides and differentiating ghw.r.t yields: ∂gh ∂ =(1 +gh)logµ. (F.3) Substituting for ∂n ∂ and ∂gh ∂ from Lemma 2 and eq. (F.3) into eq. (F.1), we deduce that: ∂gH ∂ =         (1 +gH)h1 +logµi>0,if θ < µ τ , (1 +gH)logµ>0, θ =µ τ . (F.4) This completes the derivations of eqs. (70) and (71). Appendix G Derivations of Eqs. (72), (73), (74) and (75) It is known from eq. (E.2) that the growth rate of per capita income under innovation regime is given by: (1 +gy)=(1 +gh)1−φ+λ(1−α) 1−φnλ(1−α) 1−φ.(G.1) Taking log on both sides and differentiating w.r.t yields: 1 1+gy ∂gy ∂ =1−φ+λ(1 −α) (1 +gh)1 −φ ∂gh ∂ +λ(1 −α) (1 −φ)n ∂n ∂ .(G.2) 214