Demographic Transition, Human Capital and Economic Growth in China

Abstract

We assess the impact of demographic changes on human capital accumulation and aggregate output using an overlapping generations model with endogenous savings and human capital investment decisions. We focus on China as it has experienced rapid changes in demographics as well as human capital levels between 1970 and 2010. Additionally, further variations in demographics are expected due to the recently introduced two-child policy. Model simulations indicate that education shares and income per capita will be lower with a fertility rebound as compared to status quo fertility. We find education policy to be effective in mitigating these adverse outcomes associated with higher fertility. While long-run declines in output per capita can be offset by a 4.7% increase in the government education budget, it requires a 28% increase to achieve the same outcome in the short-run.

1. Introduction

Population aging and an associated slowdown in economic growth is a major concern in many countries. Rising old age dependency ratios may increase the private burden of caring for elderly parents and threaten the fiscal sustainability of pay-as-you-go pension and public healthcare systems. This is particularly true in China, which has recently expanded its partially funded pension and health insurance systems into rural areas and to migrants (Bairoliya and Miller, 2020). While such social insurance programs may overcome market failures and improve welfare (Bairoliya et al., 2017), they may not be as sustainable as a fully funded personal account system (Feldstein and Liebman, 2008). As population aging is driven more by below replacement fertility than longer life spans (Bloom et al., 2010), it seems natural to propose higher fertility rates as one of the potential remedies (Banister et al., 2010; Turner, 2009).

China’s fertility decline has been hastened by its one child policy and fertility is now well below replacement at a fairly low level of income, raising the prospect that China will get old, slowing economic growth, before it gets rich. It may be, therefore, that relaxing fertility restrictions in China improves individual welfare, by allowing families to have the number of children they want, while also improving macroeconomic performance. In 2015, China moved to a universal two child policy which has been forecast to raise the total fertility rate (TFR) from a level near 1.5 children per women to 1.8 by 2030 (Zeng and Hesketh, 2016). This is bound to cause interesting variations in demographics in the near future. We analyze the effects of these expected changes in demographics on human capital levels and macroeconomic outcomes relative to a counterfactual of a continuation of fertility at the level of 1.5 children per woman.

Bloom et al. (2010) show in a theoretical framework that a reduction in fertility below replacement levels can result in a sharp decline in the working age share of the population and potential slow down of economic growth. Aging could also substantially increase the tax burden of health care and pension programs due to declining support ratios and increased health expenditures per capita (Christiansen et al., 2006; Bloom et al., 2011; Seshamani and Gray, 2004). However, declining fertility can induce higher investments in health and human capital which can offset some of the negative effects of aging by raising average effective labor supply (Fougère and Mérette, 1999; Lee and Mason, 2010b,a; Prettner et al., 2013). It can also induce higher physical capital accumulation by encouraging workers to save for retirement rather than rely on their children for old-age support (İmrohoroğlu and Zhao, 2018). In the light of these potential countervailing mechanisms, the macroeconomic effects of the recent relaxation of fertility controls by the Chinese government are unclear. Moreover, the macroeconomic outcomes may differ in the short run versus the long run.

In order to quantitatively assess the impact of these demographic variations in China in a general equilibrium framework, we use an overlapping generations (OLG) model featuring inter-generational altruism to mimic the important role of family in China in providing social insurance. The unit of analysis in the model is a household composed of several generations living together and engaging in various economic activities. While we treat fertility as exogenous, we allow for endogenous human capital accumulation to capture the quality-quantity trade-off as it is an important mechanism to determine the effect of fertility changes on macroeconomic outcomes. We allow for public subsidies on education, health insurance, social security and private savings and model uncertainty in survival, labor productivity and medical expenditures. The government operates public pension and health insurance programs and subsidizes primary, secondary and college education in the model. While pension payments are financed through labor income taxes, public spending on health insurance and education is jointly financed through consumption taxes.

Our quantitative exercise yields four main insights. First, we find that the impact of fertility on output in the long run crucially depends on how fertility affects saving, education, and female labor supply. If savings and physical capital accumulation is the only behavioral mechanism (i.e. education and female labor supply are held constant), an increase in fertility can increase income per capita in the long run by reducing the old age dependency rate and the taxes needed to finance public pensions and health care. However, if we allow for even modest effects on female labor supply and investment in children’s education, income per capita is lower in the long run with higher fertility. An important point is that increasing the working age share of the population is not the same as maximizing income per capita. Income is affected not just by labor supply but by human and physical capital investments that may move in opposite directions to labor supply.

Second, the short run effects of higher fertility along the transition path to the long run involve a lower level of income per capita than with no fertility increase. In the short run, the higher fertility rate increases the youth dependency ratio and these children require consumption, child care, and education, while not producing any output until they reach working age. The surprising point here however is how long the short run lasts. In our benchmark experiment, it takes approximately sixty years for the working age share to increase with higher fertility. While the first cohort of children from the fertility shock enters the workforce at around age 15, higher fertility means more children coming after them and it takes a considerable period of time for the age structure to reach a new steady state.

Third, through alternate fertility experiments, we find that the effect of fertility changes on long run income per capita is not monotonic. While moving from a total fertility rate of 1.5 to 1.8 lowers income per capita in the long run, reducing fertility to 1.2 also lowers income per capita. At this very low fertility level the increased female labor supply and education effects from lower fertility are insufficient to counterbalance the negative effect of population aging. This is consistent with Lee et al. (2014) who find that the total fertility rate that maximizes consumption for China is slightly lower than the replacement fertility. Whereas both very low or very high fertility rates have adverse economic effects.

Finally, we find that there are significant externalities (both positive and negative) associated with higher fertility through taxes and subsidies. Higher fertility on one hand reduces the fiscal burden of financing old-age pensions and medical expenditures by increasing the fiscal support ratio. On the other hand, it also lowers the education subsidies per child under the assumption of a fixed government budget (as a share of output) for education. We conduct two policy experiments to further elucidate the importance of these general equilibrium effects. Specifically, we search over the space of public education subsidies to find the required permanent increase in the education budget with a TFR of 1.8 that results in the same per capita output as the 1.5 TFR benchmark. We find that while a modest 4.7% increase in the government education budget can offset the negative long run effects of higher fertility on output, it requires an almost 28% increase to achieve this in the short run (within the next forty years).

Our quantitative analysis is not without limitations, but one that warrants mention at the outset is the assumption of exogenous fertility. One way to address the issue studied in this paper would be to allow households in the model to endogenously make decisions on both their fertility and human capital investments once the policy constraints are lifted. However, we abstract away from this in favor of a richer exogenous demographic structure for two reasons. First, it not clear what the long-run total fertility rate will be in China due to the recent policy changes. Hence, it would be difficult to interpret the model predictions for both TFR and economic outcomes. Moreover, the focus of the current analysis is on the macroeconomic changes that might occur due to the potential variations in future demographics. Second, endogenizing fertility decisions is computationally feasible only in a model with a much simpler demographic structure. However, we find that realistically introducing multiple generations (both child and elderly dependents) generates important transition dynamics in the model.

Our paper contributes to a growing body of related literature on demography and economic growth in China. First, demographics have been shown to have important implications for savings in China. There is empirical evidence that fertility has a negative effect on savings at the household level (e.g. Banerjee et al. 2010; Ge et al. 2012; Choukhmane et al. 2013; Banerjee et al. 2014). At the aggregate level, Modigliani and Cao (2004) use time series data from China to argue that fertility influenced savings over the past several decades through changes in demographic structure. Structural OLG models have since been used to analyze and quantify the link between demographics and the observed increases in aggregate savings in China (e.g. Curtis et al. 2015; Banerjee et al. 2014; He et al. 2015; Choukhmane et al. 2013). With a two-way altruism model most closely related to ours, İmrohoroğlu and Zhao (2018) find the interaction of demographics, productivity growth, and uncertain long-term care of elderly parents to be an important driver of Chinese savings rates. This is consistent with Chamon and Prasad (2010) who find evidence in support of rising average savings rates due to rising private burden of both health care spending and education. However, İmrohoroğlu and Zhao (2018) abstract from human capital considerations and the role of children more broadly. Using a general equilibrium model of endogenous fertility decisions, Liao (2013) looks at the welfare effects of relaxing fertility constraints in China but abstracts away from some key modeling details. For instance, this paper is not able to match the evolution of age-structure over time due to a simple demographic structure. Matching the precise evolution of age-distribution is crucial in pinning down the demographic dividend, hence the short run and the long run effects of fertility changes. It also abstracts away from the government programs on education, pensions and healthcare which have assumed a significant role in China in the recent times. Our general equilibrium effects indicate that the tax externality associated with these public transfer programs is significant.

We include an endogenous schooling decision in our model as fertility has been theoretically and empirically linked to human capital investments in China. Using Chinese twin births for identification, Li et al. (2008) find that higher fertility significantly reduces educational attainment and enrollment while Rosenzweig and Zhang (2009) also find reductions in schooling progress, expected college enrollment, and school grades. Compared to savings, the impact of demographics on human capital accumulation in China has received far less attention in the structural macro literature. An exception is Choukhmane et al. (2013) whose partial equilibrium model predicts that changing demographics lead children of the one-child policy generation to have at least 20% higher human capital compared to their parents.¹ Meng (2003) and Chamon and Prasad (2010) also highlight the potential role of underdeveloped financial markets in amplifying savings motives under demographic change, particularly in terms of education spending. Importantly, we restrict the borrowing capacity of families and allow for an interaction between demographics and public spending on education through a government budget constraint. Finally, as previous studies have established important connections between demographics and the macroeconomic fluctuations since the end of China’s centralized economy, we turn our eye to the future. In the wake of renewed interest in relaxing the restrictive fertility policies in China, we examine the implications of changes in demographic structure moving forward under alternate fertility paths. An important point in our paper is that we assess the economic effects of these upcoming demographic variations, not the welfare effects. Families may enjoy having additional children, and these children may improve their parent’s utility level, even if it lowers income per capita and their economic circumstances. In addition, a welfare analysis would have to take into account the utility of the children born due to the policy change which raises difficult ethical questions of measuring welfare with different population sizes (Blackorby et al., 2005).

2. Background

In this section, we provide some background information on demographics and human capital investments in China. We begin with a brief discussion on the mechanics linking fertility to an economy’s demographic structure and provide some time-series data and future projections for China under section 2.1. Next, we discuss an important behavioral response to changing demographics — changes in human capital investments. This behavioral mechanism is briefly detailed along with supporting data in section 2.2.

2.1. Demographics

One of our primary focuses in this paper is how fertility changes affect the age dependency structure across an economy over time. Fertility changes affect the working age share² of the population by altering both the old-age and child dependency ratios. For instance, an increase in fertility lowers the working age share by increasing the number of child dependents per worker. On the other hand, higher fertility in the long run also reduces the number of retirees or older dependents per worker, thereby increasing the working age share of the population. On net, the long run effect of fertility on working age share depends on the relative strengths of these two opposing forces.

Figure 1 provides an illustrative example of the theoretical relationship between total fertility rate and working age share of the population in the long run steady state. When there is a marginal decrease in fertility from a very high rate, the reduction in the number of child dependents outweighs the increased share of retirees in the long run, resulting in an overall increase in working age share. This is the case typical in many developing countries (including China prior to the 1990s). However, indefinite declines in fertility ultimately result in a lower working age share in the long run as the relative number of retirees increases. Moreover, when fertility rates are very low, there is a potential for substantial increases in working share from relatively small increases in fertility in the long run.

Figure 1:

Stable Long run Relationship Between Fertility Rate and working age Share

Note: For this example, we hold age-specific survival rates constant at current levels in China. Survival probabilities are taken from UN life table estimates for 2010–15. To obtain an analytic solution, we assume mothers give birth to all children at age 29.

Figure 2 shows the U.N. estimates (1960–2010) and high, low and medium variant projections (2010–2100) for total fertility rate and working age share in China. Between 1970 and 2000 there was a sharp decline in fertility largely attributed to China’s one-child policy. Correspondingly, over this time-frame there was a steep rise in the working age share driven by the drop in child dependents. However, now that the one-child generation is moving into the workforce, there is a projected decline in working age share in the coming decades due to a sharp rise in the number of old-age dependents per worker.³

Figure 2:

Demographic Transition in China

Note: Data is taken from the 2012 revision of the World Population Prospects. Projections are shown under three different fertility variant assumed by the U.N. Dased line indicates the introduction of the two child policy in 2016.

2.2. Human Capital

Our framework allows fertility changes to operate through multiple channels to impact individual and aggregate human capital investments. To illustrate this point, figure 3 shows the estimated share of 25–29 year-olds in China with completed secondary or college education from 1960–2010. Secondary completion rose substantially over the entire time-frame but experienced the sharpest growth during the 1980s and 90s. College completion also started to increase in the 90s and is largely believed to continue to rise substantially into the foreseeable future. The primary channel of influence we focus on is through the child expenditure channel. For example, some households may not be wealthy enough to send a large number of children to school. A decline in fertility would thus promote investments in human capital by relaxing the household budget constraint. The fertility rate may also influence individuals indirectly by altering the budgets of government programs. For example, if government outlays on education are fixed, the decreased share of school-aged individuals accompanying lower fertility results in a decrease in the per student private cost of education. Lastly, there are general equilibrium effects that will influence the strength of the above mechanisms. For example, a decline in after-tax wage relative to interest rate reduces the returns to schooling and has a dampening effect on average educational attainment.

Figure 3:

Education shares in China Over Time

Data Source: Barro and Lee (2013)

In the next section, we build our dynamic general equilibrium model which captures all these important mechanisms linking fertility with demographics and human capital accumulation decisions.

3. Model

Consider an economy populated by a large number of households that each consist of overlapping generations of a family. Household members are altruistic towards each other and make decisions as a single economic unit. Over time, children in the family grow up, have children of their own, and eventually replace their parents in the household. In this way each household in the economy is an infinitely lived dynasty. As children grow up, they accumulate human capital by attending school. As individuals age, they eventually face medical expenditure and mortality risk. Time is discrete and in each period a new generation of individuals is born.

3.1. Technology

Aggregate output in the economy (Y) is assumed to be produced by a representative firm using the technology:

$$Y_t=A_t{K}_t^{\alpha }{N}_t^{1-\alpha }\hspace{1em}\alpha \in \left(0,1\right),$$ (1)

where K_t and N_t are the aggregate capital stock and labor inputs (measured in efficiency units) in period t, A_t is total factor productivity, and α is the capital share. Output can be consumed (C), invested in physical capital (I), expended on education (E), or expended on medical care (M):

$$Y_t=C_t+I_t+E_t+M_t.$$

Finally, letting δ equal per-period depreciation, the law of motion of capital is given by:

$$K_{t+1}=(1-\delta )K_t+I_t.$$

3.2. Households

Demographic Structure

The economy is populated by overlapping generations of individuals residing in family households. Each individual lives through four stages of life—child, young adult, old adult, and elderly. As a child, an individual simply consumes and spends a fraction of childhood in primary school. Young adulthood begins by either continuing in school (secondary school and eventually college) or entering the labor market. Regardless of educational choice, after schooling is complete, the remainder of the young adult stage is spent working. As an old adult, an individual continues to work in the labor market and consume. Old adults eventually begin to receive public pension benefits as well. Finally, the elderly continue to receive pension payments but are assumed retired from the labor force. It is upon becoming elderly that individuals begin to face medical expenditure and mortality risk.

The economic decision making unit is a household. Each household in the economy is indexed by “household age” j = 1, …, J. This index completely defines the age structure of the entire household. At each age j, all members of the household pool their resources to maximize a joint objective function. Following Laitner (1992), individuals derive utility from their own lifetime consumption and from the utility of other household members and descendants.

Regardless of age j, each household always consists of one old adult and their n young adult children which is the implied model fertility rate.⁴ At household age j = J_c < J, each young adult has n offsprings of their own, who live with the household as children until age J. After age J, the young adult siblings become old adults and split into n separate households with their own young adult children. The siblings are assumed to evenly split household assets and share the continuing financial burden of their now elderly parent. This implies that $\frac{1}{n}$ elderly parents are included in each household, conditional on survival. Moreover, any pension income received by the elderly parent is distributed evenly to old adult children to help finance consumption and medical care for the elderly.

Note that due to the birth of children and mortality risk faced by the elderly members of the family, a household can have four different compositions. One where all four generations are present; where only young and old adults are present (children are not born as yet and elderly have deceased); where children, young, and old adults are present and elderly have deceased; and finally where young adults, old adults, and elderly are present. Figure 4 summarizes the timeline for both households and individuals in the model. The figure highlights the rich demographic structure available in this framework—an individual’s life may overlap with that of his children, parents, grandchildren, grandparents, great-grandchildren, and great-grandparents.

Figure 4:

Evolution of Generations and Timeline

Labor Earnings

In each model period, every young and old adult is endowed with one unit of productive time. Young adults either spend this time in school, taking care of children, or supply it inelastically to the labor market. More specifically, a young adult of age j < J_c earns the following pre-tax labor income:

$$w\left(1-{\kappa }_{e_{y}j}\right)e_y{\epsilon }_j{\eta }_y,$$

where w is the competitive wage rate, ε_j is age-specific life-cycle productivity, and η_y is a permanent idiosyncratic shock realized by an individual upon becoming a young adult (j = 1). Labor productivity is also conditional on the young adult’s level of education e_y. Moreover, ${\kappa }_{e_{y}j}$ indicates whether a young adult of education type e_y is enrolled in school at age j. Specifically, ${\kappa }_{e_{y}j}=1$ if the young adult is enrolled in school and ${\kappa }_{e_{y}j}=0$ otherwise.

Schooling is assumed complete by the time young adults have their own children at age J_c. As such, a young adult of age j ≥ J_c earns the following pre-tax labor income:

$$w{e}_y{\epsilon }_j{\eta }_y\left(1-n{\theta }_f\right),$$

where n is the number of children they have and θ_f is the time-cost of raising their children. Even though our model is gender-neutral, θ_f captures in a simple way, the effect of fertility on female labor supply.

Old adults supply their unit of time inelastically to the labor market. As such, an old adult of age j earns the following pre-tax labor income:

$$w{e}_o{\epsilon }_j{+}_J{\eta }_o,$$

where η_o is the permanent idiosyncratic shock that they received when they were a young adult. In this way, the productivity shock η remains constant throughout an individual’s life. Moreover, we assume the productivity shock of a young adult (η′) is correlated with their parent’s shock (η) through a finite-state Markov chain with stationary transitions over time:

$${\mathrm{\Gamma }}_t^{\eta }\left({\eta }^{\prime },\mathcal{E}\right)=Prob\left({\eta }^{\prime }\in \mathcal{E}|\eta \right)={\mathrm{\Gamma }}^{\eta }\left({\eta }^{\prime },\mathcal{E}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall t.$$

Education

We model three discrete choices for educational attainment—primary school, secondary school, and college. All children exogenously enter primary school at age J_p and are in school for the remainder of childhood (i.e. through household age J). However, at age J, households decide if children will drop out of school and enter the labor market the following period as young adults, will continue their education through secondary school, or will continue through college. Primary school requires an annual tuition cost of θ_p that is entirely subsidized by the government. Continuing education beyond primary school incurs an annual cost of θ_e which may be fully or partially subsidized. It is important to note that education level is chosen prior to realization of an individual’s productivity shock η. This implies that idiosyncratic returns to education are uncertain at the time when schooling decisions are made.

Medical Expenditures and Mortality

Elderly individuals of age j survive to age j + 1 with probability ψ_j. At the end of period J, they die with probability one. Conditional on being alive, the elderly are characterized by a medical expenditure state $x\in X$. Conditional on expenditure state, households are required to finance medical expenditure m_x for the care of their elderly parent. The elderly are assumed to start in the lowest medical expenditure state $\overline{x}$. The medical expenditure state then evolves stochastically over the remaining life-cycle. The stochastic process follows a finite-state Markov chain with stationary transitions over time. The Markov process is assumed to be identical and independent across individuals:

$${\mathrm{\Gamma }}_t^x\left(x^{\prime },\mathcal{X}\right)=Prob\left(x^{\prime }\in \mathcal{X}|x\right)={\mathrm{\Gamma }}^x\left(x^{\prime },\mathcal{X}\right),\hspace{0.17em}\hspace{0.17em}\forall t,$$

where x is the current medical expenditure state and x′ is that of the following period.

3.3. Government

The government operates three programs in the model. First, a pay-as-you-go social security system which is defined by pension benefits SS for each old adult above age J_ss and for all surviving elderly. Pension benefits are determined by a replacement rate b_s of national average earnings. Second, the government subsidizes the health care of the elderly by covering a fraction b_h of their medical expenditure bill. Finally, the government provides a subsidy for education. The cost of primary school is fully covered by the government. For secondary school and college, the government covers a fraction λ_e of the total annual tuition cost θ_e. As a majority of public revenues in China are collected through direct or indirect consumption taxes, we assume public spending on education and health care is financed with a proportional tax on individual consumption τ.⁵ However, as the Chinese pension system is primarily financed with labor income taxes, we assume the social security budget is balanced through a proportional tax on labor income τ_ss.

3.4. Decision Problem

At any given time, a household can be characterized by a vector of state variables ζ = (a, x, d, e_y, e_o, η_y, η_o, j), where a denotes current holdings of one-period, risk-free assets, x is elderly member’s medical expenditure state, d is an indicator for whether the elderly is deceased, e_y and e_o are education levels of the young and old adults respectively, η_y and η_o are productivity levels of young and old adult respectively, and j is the age of the household. Given this state vector, a household chooses total consumption c, and next period assets a′, to maximize the present utility of the household plus the expected discounted utility of all future periods of the family dynasty. In period J, the education level of the next generation of adults $e_y^{\text{'}}$ is also chosen. The decision problem facing a household of age j < J may be written:

$$\begin{gathered} \nu \left(a,x,d,e_y,e_o,{\eta }_y,{\eta }_o,j\right)= \\ \underset{c,a^{\prime }}{\max }\left\{\tilde{n}u\left(\frac{c}{\tilde{n}}\right)+\beta E_{x^{\prime }{d}^{\prime }}\left[\nu \left(a^{\prime },x^{\prime },d^{\prime },e_y,e_o,{\eta }_y,{\eta }_o,j+1\right)\right]\right\} \end{gathered}$$

subject to:

$$\begin{gathered} c\left(1+\tau \right)+a^{\prime }=y\left(1-{\tau }_{ss}\right)+a\left(1+r\right)+\frac{\left(1-d\right)}{n}\left(SS-\left(1-b_n\right)m_x\right) \\ +SS\left(j\ge J_{ss}\right)-n{\kappa }_{e_{y}j}\left(1-{\lambda }_e\right){\theta }_e \\ {a}^{\prime }\ge 0,c>0 \end{gathered}$$

where y refers to total household labor income given by:

$$y=\{\begin{array}{cc}w{e}_o{\epsilon }_{j+J}{\eta }_o+n\left(1-{\kappa }_{e_{y}j}\right)w{e}_y{\epsilon }_j{\eta }_y& \text{if}\hspace{0.17em}j<J_c\\ w{e}_o{\epsilon }_{j+J}{\eta }_o+n\left[w{e}_y{\epsilon }_j{\eta }_y\left(1-n{\theta }_f\right)\right]& \text{if}\hspace{0.17em}j\ge J_c,\end{array}$$

and $\tilde{n}$ is the number of adult equivalents in the household:

$$\tilde{n}=n+1+\frac{\left(1-d\right)}{n}+\gamma nn\left(j\ge J_c\right)$$

where γ is the consumption requirement of a child relative to an adult.

The current period utility of an individual is given by u (.) and value function V (.) is the total expected discounted utility of arriving in a period of time with a given state vector. Note that expectations are taken with respect to the stochastic process for the medical expenditure state and the survival risk of the elderly. The first constraint is the household budget constraint. Note that the total annual private cost of education for each of the n young adults for education level e_y is given by (1 − λ_e)θ_e. Also note the role of the elderly in the decision problem. Conditional on being alive (d = 0), households have access to $\frac{1}{n}$ of the elder’s pension income SS but are also responsible for the same fraction of the elder’s unsubsidized medical care (1 − b_h) m_x. Finally, note that households also face a no borrowing constraint (a′ ≥ 0).

In period J, the decision problem facing a household may be written:

$$\begin{gathered} \nu \left(a,x,d,e_y,e_o,{\eta }_y,{\eta }_o,j\right)= \\ \underset{c,a^{\prime },e_y^{\text{'}}}{\max }\left\{\tilde{n}u\left(\frac{c}{\tilde{n}}\right)+\eta \beta E_{{\eta }_y^{\text{'}}}\left[\nu \left(\frac{a^{\prime }}{n},x^{\prime },d^{\prime },e_y^{\text{'}},e_o^{\text{'}},{\eta }_y^{\text{'}},{\eta }_o^{\text{'}},1\right)\right]\right\} \end{gathered}$$

subject to:

$$c\left(1+\tau \right)+a^{\prime }=y\left(1-{\tau }_{ss}\right)+a\left(1+r\right)+\frac{\left(1-d\right)}{n}\left(SS-\left(1-b_h\right)m_x\right)+SS$$

$$a^{\prime }\ge 0,c>0,$$

and y and $\tilde{n}$ are defined as above. Expectations over next period’s value function are now taken with respect to the productivity shock of the children η, who will become young adults. Moreover, in the following period young adults become old adults so we have ${\eta }_y={\eta }_o^{\text{'}}$ and $e_y=e_o^{\text{'}}$.

3.5. Definition of Stationary Competitive Equilibrium

Let $a\in {ℝ}_{+}$, $x\in \mathcal{X}=\left\{x_1,x_2,\dots ,x_n\right\}$, $d\in \mathcal{D}=\left\{0,1\right\}$, $e_y,e_o\in {\mathcal{E}}_d=\left\{e_1,e_2,\dots ,e_n\right\}$, ${\eta }_y,{\eta }_o\in \mathcal{E}=\left\{{\eta }_1,{\eta }_2,\dots ,{\eta }_n\right\}$, $j\in \mathcal{J}=\left\{1,2,\dots ,J\right\}$ and $\mathcal{R}={ℝ}_{+}\times \mathcal{X}\times \mathcal{D}\times {\mathcal{E}}_d\times {\mathcal{E}}_d\times \mathcal{E}\times \mathcal{E}\times \mathcal{J}$. Let $B\left({ℝ}_{+}\right)$ be the Borel σ-algebra of ${ℝ}_{+}$ and $P\left(\mathcal{X}\right)$, $P\left(\mathcal{D}\right)$, $P\left({\mathcal{E}}_d\right)$, $P\left(\mathcal{E}\right)$, $P\left(\mathcal{J}\right)$ the power sets of X, $\mathcal{D}$, ${\mathcal{E}}_d$, $\mathcal{E}$, $\mathcal{J}$ respectively. Let ${\text{Σ}}_{\mathcal{R}}\equiv B\left({ℝ}_{+}\right)\times P\left(\mathcal{X}\right)\times P\left(\mathcal{D}\right)\times P\left({\mathcal{E}}_d\right)\times P\left({\mathcal{E}}_d\right)\times P\left(\mathcal{X}\right)\times P\left(\mathcal{X}\right)\times P\left(\mathcal{J}\right)$. Let $\mathcal{M}$ be the set of all finite measures over the measurable space $\left(\mathcal{R},{\text{Σ}}_{\mathcal{R}}\right)$.

Definition 1. Given fiscal policies of the government {λ_e, b_s, b_h, τ, τ_ss} and a fertility rate n, a stationary competitive equilibrium is a set of value functions v(ζ), households’ decision rules {c(ζ), a′(ζ), e_y(ζ)}, prices {r, w}, tax rates {τ, τ_ss}, pension benefits {SS}, and time-invariant measure of households $\mathrm{\Phi }\left(\zeta \right)\in \mathcal{M}$ such that:

1. Given fiscal policies and prices, household’s decision rules solve household’s decision problem.

2. Prices w and r satisfy:

   $$r=A\alpha {\left(\frac{N}{K}\right)}^{1-\alpha }-\delta$$

   $$w=A\left(1-\alpha \right){\left(\frac{K}{N}\right)}^{\alpha }.$$
3. Individual and aggregate behavior are consistent:

   $$K=\int a^{\prime }\left(\zeta \right)\mathrm{\Phi }\left(d\zeta \right)$$

   $$N=\int \left(e_o{\epsilon }_{j+\mathcal{J}}{\eta }_o+n\left(\left(1-{\kappa }_{e_{y}j}\right)e_y{\epsilon }_j{\eta }_y\right)\left(1-n{\theta }_f\left(j\ge {\mathcal{J}}_c\right)\right)\right)\mathrm{\Phi }\left(d\zeta \right).$$
4. Goods market clears⁶:

   $$\begin{gathered} \int \left(c\left(\zeta \right)+\frac{\left(1-d\right)}{n}{m}_x+n{\kappa }_{e_{y}j}{\theta }_e\right)\mathrm{\Phi }\left(d\zeta \right)+\int n^2{\theta }_p\mathrm{\Phi }\left(d\zeta \left(j\ge J_p\right)\right) \\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=A{K}^{\alpha }{N}^{1-\alpha }-\delta K. \end{gathered}$$
5. Measure of households satisfy:

   $$\begin{gathered} \mathrm{\Phi }\left(a^{\prime },\overline{x},0,e_y^{\text{'}},e_o^{\text{'}},{\eta }_y^{\text{'}},{\eta }_o^{\text{'}},1\right) \\ =n{\sum }_{\left\{\zeta :a^{\prime }=a\left(\zeta \right)/n,e_y^{\text{'}}=e_y\left(\zeta \right),e_o^{\text{'}}=e_y,{\eta }_o^{\text{'}}={\eta }_y\right\}}{\mathrm{\Gamma }}_t^{{\eta }_o}\left({\eta }_y^{\text{'}},\mathcal{E}\right)\mathrm{\Phi }\left(\zeta \right)\hspace{1em}for\hspace{1em}j=J. \end{gathered}$$

   $$\begin{gathered} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\mathrm{\Phi }\left(a^{\prime },x^{\prime },d^{\prime },e_y,e_o,{\eta }_y,{\eta }_o,j+1\right) \\ =\frac{1}{n^{1/J}}{\sum }_{\left\{a,x,d:a^{\prime }=a\left(\zeta \right)\right\}}{\mathrm{\Gamma }}^x\left(x^{\prime },\mathcal{X}\right)\mathrm{\Psi }\left(d^{\prime },d_j\right)\mathrm{\Phi }\left(a,x,d,e_y,e_o,{\eta }_y,{\eta }_o,j\right)\hspace{1em}for\hspace{1em}j<J \end{gathered}$$

   where $\mathrm{\Psi }\left(d^{\prime },d_j\right)$ is the probability of transitioning from state d at age j to state d′ with Ψ(1, 1) = 1 and Ψ(0, 0) represents the survival probability ψ_j for age j.
6. Government budget for education and medical expenses balances:

   $$\int \left(b_h\frac{\left(1-d\right)}{n}{m}_x+{\lambda }_{e}n{\kappa }_{e_{y}j}{\theta }_e\right)\mathrm{\Phi }\left(d\zeta \right)+\int n^2{\theta }_p\mathrm{\Phi }\left(d\zeta \left(j\ge J_p\right)\right)=\tau \int c\left(\zeta \right)\mathrm{\Phi }\left(d\zeta \right).$$
7. Social security budget balances⁷:

   $${\tau }_{ss}wN=\int SS\mathrm{\Phi }\left(d\zeta \left(j\ge J_{SS}\right)\right)+\int dSS\mathrm{\Phi }\left(d\zeta \right),$$

   $$SS=\frac{b_{s}wN}{\int \left(1+n\left(1-{\kappa }_{e_{y}j}\right)\right)\mathrm{\Phi }\left(d\zeta \right)}.$$

4. Calibration

We use a calibrated version of the model to understand the effect of demographic transition and education policies on macroeconomic variables in China in both the short run and the long run. Key to our quantitative analysis is matching the human capital and demographic structure of the Chinese economy circa 2010, prior to the relaxation of fertility restrictions. However, since the Chinese economy has undergone massive changes in the last five decades, we calibrate our model economy in two stages. First, we calibrate an initial steady state to match some key features of the Chinese economy and demographics circa the 1960s. We chose to go back to the 1960s as fertility and mortality started declining dramatically in the 1970s (UN, 2015) and steady states assume a stable demographic structure.⁸ Second, we calibrate the transition economy to capture some key changes in the Chinese economy between 1960 and 2010. We later show that this approach yields a good approximation of important features of the economy in 2010.

In our calibration exercise, we take some parameter values directly from the literature or estimate them using micro level survey data. For instance, we use data from the China Health and Retirement Longitudinal Study (CHARLS) to estimate the stochastic process for elderly medical expenditures. Other parameters we estimate jointly using our general equilibrium model by minimizing the distance between certain data and model moments. The following subsections lay out the details of our calibration exercise.

4.1. Demographics

Each model period is assumed to represent one calendar year. In order to capture the demographics accurately, we model the entire life cycle of an individual from ages 0 to 98. The final “household age” index is set to J = 29. We set J_c = 17 implying young adults have children at real age 29. Table 1 gives the number⁹ and age-structure of different generations living in a household. Finally, we set set J_ss = 19 implying that old adults begin receiving pension payments at real age 60.

Table 1:

Household Composition

Stage	Individual’s age (yrs.)	Number	Household ages lived
Children	0–11	nn	17…J
Young Adults	12–40	n	1…J
Old Adults	41–69	1	1…J
Elderly	70–98	$\frac{1}{n}$	1…Death

In our initial steady state we set n = 2.7 so that the implied fertility rate matches the UN estimate for China in 1965–70 of 5.4.¹⁰ Fertility along the transition from 1960–2010 is shown in figure 5. We maintain the initial fertility rate for 10 years then gradually reduce the rate to 1.5 between 1970 and 2000 to approximate the declines estimated by the UN.¹¹

Figure 5:

Fertility Rates: 1960–2010

4.2. Preferences

Individual’s preferences over consumption are defined as follows:

$$u\left(c\right)=\frac{c^{1-\sigma }}{1-\sigma }.$$

The parameter σ controls risk aversion and is set to a value of 2, implying an intertemporal elasticity of substitution of 0.5. As children consume fewer resources than adults, we set the child consumption weight γ = 0.3 following the OECD-modified consumption equivalent scale (Hagenaars et al., 1994).

4.3. Medical Expenditures and Mortality

We assume there are two possible realizations of the elderly medical expenditure state x—high and low. We estimate transition probabilities between states and associated medical expenditures m_x using data from the 2011 and 2013 waves of the CHARLS, a nationally representative survey of Chinese residents ages 45 and older. We first divide surveyed individuals over the age of 70 into percentiles based on reported annual total medical expenditures.¹² As has been documented in other countries, the expenditure distribution is highly skewed with a thin right tail driven by a limited number of catastrophic events. As such, we categorize those in the bottom 90 percentiles of the expenditure distribution as our low expenditure state. Analogously those in the top 10 percentiles are categorized into the high expenditure state. Annualized transition probabilities between states across the two waves of the CHARLS are shown in the first columns of table 2.¹³ We next compute the mean expenditures among those categorized into the high/low expenditure state using the 2013 wave. We set low/high medical expenditures m_x to be a constant share of output per capita in every model period. The last column of table 2 shows the estimated average expenditures as a share of output per capita from the CHARLS.

Table 2:

Medical Expenditures & Transition Probabilities

x	Transition probability		Mean expenditures (% GDP per capita)
	Low	High
Low	0.94	0.06	2.4
High	0.53	0.47	156.6

Survival probabilities are taken from UN life table estimates (UN, 2015). As the initial steady state is calibrated to match key features of the Chinese economy circa the 1960s, we use age-specific survival probability estimates for 1965–70 as a starting point in the model. However, age-specific mortality rates have significantly improved in China over the past decades and are projected to continue to improve into future. As a simple means of capturing this improvement in the model, we linearly decrease the mortality risk along our transition path from 1970 to 2100 to reach the levels projected by UN for 2095–2100. Figure 6a shows the corresponding set of UN survival probability estimates used in our initial and final steady states.

Figure 6:

Survival and Labor Productivity

4.4. Labor Productivity

We use data from the China Family Panel Studies (CFPS) to estimate age-specific labor productivity ε_j over potential working years of the life-cycle (ages 12 to 69).¹⁴ We use the 2010 and 2012 waves and regress log of hourly income on age, age-squared and an individual fixed effect to obtain our life-cycle productivity estimates. Due to lack of observations, we assume productivity is constant prior to age eighteen. Figure 6b plots the estimated life-cycle profile of labor productivity. Productivity increases steadily until age fifty, at which point it begins to decline throughout the remainder of an individual’s working life.

We estimate the Markov chain for the stochastic component of productivity η by assuming an underlying AR(1) process in logs:

$$\ln \left({\eta }^{\prime }\right)=\rho \ln \left(\eta \right)+{\epsilon }_{\eta },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\epsilon }_{\eta }~N\left(0,{\sigma }_{\eta }^2\right).$$

We then use the Tauchen method to approximate this process with a Markov chain over eight discrete states. Parameters governing the stochastic process ρ and σ_η are jointly estimated using the predictions of the model (see section 4.8 for details).

Finally, recall that children in the model (ages 0–11) cost their parents a fraction of their labor time endowment θ_f. Following Bloom et al. (2009) we set θ_f = 0.16.¹⁵

4.5. Education

We assume primary education lasts for six years (age 6–11), secondary for six years (age 12–17), and college for four years (age 18–21).¹⁶ Empirical estimates suggest very low returns to education in China in the 1970s—in the range of 0–3% (Yang, 2005; Fleisher and Wang, 2005). As such, after normalizing education-specific productivity e for primary education to one, we use a 3% annual return to secondary school and college for our initial (1960) steady state. The compression of wages is largely attributed to equalization policies carried out under the centrally planned economy, with particular downward pressure on more educated workers. Following economic reforms of the early 1980s, there was a steep rise in the returns to schooling.¹⁷ Most recent estimates have found overall annual returns in the 10–20% range (e.g. Li (2003); Li and Luo (2004); Zhang et al. (2005); Fang et al. (2012)). Heckman and Li (2004) estimated annual returns to college close to 10%. Moreover, the average return globally across countries is estimated at around 10% (Psacharopoulos and Patrinos, 2004). As such, to capture the decompression of wages in China over time, we assume annual returns over primary school increase to 10% by 2010 for each year of secondary schooling and college. The initial and final education-specific productivity estimates are given in table 3 (refer to appendix B for more details).

Table 3:

Educational Productivity Returns and Costs

Level	Ages	Productivity (e)		Annual total tuition (θ)
		1960	2010+	1960	2010+
Primary	6–11	1.00	1.00	0.08y	0.16y
Secondary	12–17	1.19	1.77	0.37y	0.23y
College	18–21	1.34	2.59	3.79y	0.84y

Note: y refers to output per capita.

Total college tuition is expressed as a fraction of output per capita and the estimates of each education category (θ_p, θ_s, θ_c) are taken directly from the data (China Education Statistical Yearbook from various years). The last two columns of table 3 show the total tuition costs for the years 1960 and 2010 used in the model (refer to appendix B for more details). Note that even though college tuition levels went up between 1960 and 2010, the cost as percentage of output per capita went down from 379% in 1960 to 84% in 2010. On the other hand, cost of primary schooling rose sharply from 8% to 16% of output per capita during the same time period.

4.6. Technology

We set α to match the long run average capital share of income (1970–2010) for China while the depreciation rate δ is set to 10%. Total factor productivity A is normalized to one, even though we allow for labor augmenting TFP change by increasing returns to education over time as suggested in the literature.

4.7. Government Policies

The government operates the pension and health insurance programs and subsidizes primary, secondary and college education in the model economy. The medical expenditure reimbursement rate is set at b_h = 0.7. This is the estimated rate for urban workers in China and the target rate for rural workers as well (Yip et al., 2012). The pension replacement rate is set as 35% of national average earnings (b_s = 0.35), the target rate for the pay-as-you-go component of the current urban system (OECD, 2010).

Finally, government subsidizes primary, secondary and college education at the rates of λ_p, λ_s and λ_c respectively. Primary school has been historically funded through various levels of government in China, as such we maintain λ_p = 1 throughout our analyses. College and secondary education is only partially subsidized by the government and we allow these subsidies in the model to change over time to reflect the changes that took place in China between 1960–2010. Specifically, in the initial steady state, no public subsidies are available for college and secondary education. Starting in 2010, we fix government expenditures on education at 4.3% of GDP. This matches the latest empirical estimates and is near the government’s stated long-term goal of 4% (China, 2014; Tsang, 1996). After subtracting the entire cost of primary school from the government’s budget, the remainder is split in a 60/40 ratio between secondary school and college in order to determine the respective subsidy rates (refer to appendix B for more details).¹⁸

4.8. Estimation of Other Parameters

We use the model to jointly estimate three remaining parameters—(β, ρ, σ_η )—by targeting relevant empirical data moments in the initial steady state. We estimate the discount factor β by targeting the average capital-output ratio (1960–70) of 3.23 in the Penn World Tables 8.1 (Feenstra et al., 2015). For estimating the persistence and standard deviation of the labor productivity shock, we target the inter-generational income mobility and the income Gini coefficient, respectively. Gong et al. (2012) provide estimates of inter-generational income mobility in urban China for father-son, father-daughter, mother-son and mother-daughter. We use a simple average of these for our targeted moment. The income Gini coefficient for China in 1981 is taken from the World Bank Development Indicators.¹⁹ Table 5 provides a summary of all parameter estimates along with data and model moments. The model does an excellent job of matching the data moments.

Table 5:

Estimation Results

Parameter	Value	Moment	Data	Model
Discount factor β	0.944	K/Y	3.23	3.16
Persistence of prod. shock ρ	0.62	Inter-generational income mobility	0.60	0.60
S.D. of prod. shock ση	0.43	Income Gini	0.29	0.29

5. Model-Fit

While our benchmark model is calibrated to match the data in terms of mortality, income mobility, etc., it is useful to compare predictions along some other dimensions not targeted during calibration. First, the model is able to match the changing demographic structure rather well. This is evident from figure 7 showing the old-age and child dependency ratios both in the model and the data (UN, 2015).²⁰ These dependency ratios were not targeted directly in our calibration exercise. As fertility and mortality rates were not completely stable prior to 1960, the initial steady state somewhat over-predicts these ratios. However, as our demographic structure evolves over time, it become quite similar to the data. This is perhaps unsurprising as we feed in fertility rates and age-specific mortality rates that approximate the data along the transition. Nonetheless, it is reassuring that estimates seem reasonable.

Figure 7:

Model Fit: Demographics

Table 6 shows the primary, secondary and college share for the year 2010 both in the model and the data.²¹ The model slightly over-predicts the secondary share and under-predicts the college and primary shares. However, it is worth noting that the model is able to capture the the fact that college educational attainment levels in China have been very low even in the recent times. The most notable changes in educational attainment in China has been a sharp rise in secondary education share and a corresponding decline in primary education share over the period 1960–2010 and the model is able to capture that well (refer to appendix figure C.1).

Table 6:

Model Fit: Education Shares in 2010

	Primary	Secondary	College
Model	73.7	24.6	1.8
Data	77.7	20.0	2.4

Note: columns may not sum to 100 due to rounding.

The model also does well in matching the average age-specific consumption, labor earnings and transfer profiles from the National Transfer Accounts (NTA) data (Lee and Mason, 2011).²² Figure 8 shows these average age-specific profiles for the year 2002 in both the data and the model.²³ In this figure, consumption is expressed as a percentage of age 45 consumption, and earnings and transfers as a percentage of age 45 earnings.

Figure 8:

Model Fit: Consumption, Earnings and Transfers

Similar to the data, we have an increase in consumption through early years of life as education expenditures and consumption requirements rise. This is followed by a sustained level of consumption throughout much of the individual’s working life before there is an increase due to health expenditures later in life. In contrast to the data, our model maintains the elevated consumption throughout old-age as health expenditures remain high. Moreover, our model over-predicts the gap between mid and late-life consumption as we do not incorporate health spending prior to becoming elderly. In terms of the labor earnings profile, the model matches the data well, though the rise in income is somewhat less steep and the peak a few years later.

The life-cycle deficit is defined as consumption less labor income and gives a sense of how the model does in predicting intra-household transfers. Life-cycle deficits become positive only a few years later in the model compared to the data, while they return negative a few years later as well. Compared to the data, the deficit is somewhat lager in the model during the school-aged years and for the elderly. This is primarily due to the over-prediction of education and health spending during the respective age ranges.

6. Results

In order to understand the short and long run effects of demographic changes on macroeconomic outcomes in China, we analyze two different future fertility scenarios. First, we assume that the 2010 fertility of 1.5 is maintained indefinitely into the future. Second, beginning in 2010, we assume an unexpected gradual increase to a long-term fertility rate of 1.8, which is somewhat higher than the current average in China.²⁴ Both fertility paths are identical along all other exogenous dimensions.²⁵ The fertility rates fed along both transition paths are given in figure 9 along with a comparison to the UN projections (medium variant). These two paths — 1.5 and 1.8 — are henceforth referred to as the “low” and “high” fertility scenarios.

Figure 9:

Fertility Rates Along Transition

As our theory outlined above has several countervailing mechanisms, we begin our quantitative analysis by discussing the long run and short run macroeconomic outcomes in our endogenous human capital model (benchmark) as outlined above. We then compare the differences in macroeconomic outcomes across high and low fertility scenarios for two other models — 1) a simplified model featuring exogenous human capital and no time cost of raising children, 2) a model with exogenous human capital but a time cost of raising children as in our benchmark. Calibration details of the alternate model specifications are available in Appendix A. It should be noted that, for a given fertility rate, the old-age and child-dependency ratios and the working-age share are the same across all three models as the demographic structure is identical in all three frameworks. The following sub-sections discuss our benchmark results and then compare across the three models how the long run and short run macroeconomic outcomes respond to the changes in the fertility rate.²⁶ This is followed by analyses of alternate fertility scenarios and education policy experiments.

6.1. Benchmark Model

Figure 10 shows transition paths for select macroeconomic variables for both fertility scenarios for the benchmark model. All per capita variables have been normalized to output in the low fertility final steady state. First, notice there are macroeconomic increases along both fertility paths going forward from 2010. However, comparing across fertility paths, we find overall macroeconomic declines when moving from a long-term fertility rate of 1.5 to 1.8. Table 7 gives percentage change in select variables across fertility scenarios in the long run steady state. Note that the working age share is initially lower under the higher fertility path due to an increase in the number of child dependents. However, it starts increasing over the 1.5 fertility path around the year 2070 when enough of these child dependents start entering the workforce as productive agents. Finally in the long run, working age share is 5.4% higher under the 1.8 fertility path. Despite a higher working age share in the long run, there is a decline of 0.8% in per capita effective labor supply and 1.3% in capital, resulting in lower long term per capita output (1.0%) and consumption (3%). However, in the short run (year 2070) these declines under the higher fertility path are more severe where output is lower by 3.8%, consumption by 5%, capital by 4.2%, and effective labor by 3.4%. It is interesting to note that the short-run declines under higher fertility is driven by a lower working age share of the population, despite college shares being somewhat higher, whereas in the long-run, lower output under the higher fertility path is driven by lower levels of human capital.

Figure 10:

Transition Dynamics: Benchmark

Note: All per capita variables have been normalized to output in the 1.5 fertility final steady state.

Table 7:

Final Steady State Results

Variables	% Change low to high fertility
Working share	5.40
Output per capita	−1.01
Consumption per capita	−2.99
Capital per capita	−1.25
Effective labor per capita	−0.78
Public medical expenditures	−18.54
SS payments	−17.20
College share	−5.40
After-tax wage	6.15
Investment per capita	5.61

In our model, demographic changes and exogenous improvements in returns to education since the 1960’s result in an increase in both college and secondary education shares and a drop in primary share over the coming decades under both fertility scenarios (panels (e) and (f) of figures 10). After 2070, when the primary education share goes to zero, further improvements in educational attainment are brought about by a decline in secondary share and a further increase in college share. However, the gains in college share are smaller under the higher fertility scenario.

The behavioral response to higher fertility is influenced by several additional factors when human capital is endogenous in addition to savings. First, some poor households may not be willing to send additional children to school without the ability to borrow funds for interim consumption, leading them to reduce their choice of educational attainment due to the household borrowing constraint. Moreover, recall that educational returns are uncertain at the time schooling decisions are made, making human capital a riskier investment than physical capital. As our functional form for preferences exhibits decreasing absolute risk aversion, this implies that human capital investment will weakly increase with household wealth, even if the household borrowing constraint does not bind. Thus, long run declines in wealth reduce education both by placing additional budgetary pressure on families and making them less willing to invest in risky human capital. Fertility changes also influence human capital investment decisions by altering the education budget of the government. In our model, government outlays on education are fixed as a share of output so an increase in the share of school-aged children leads to an increase in the per student private cost of secondary school and college (refer to figure C.2d in appendix C). If families are unable or unwilling to invest in lumpy school tuition despite relatively high returns, they may shift resources to physical capital as an alternate means of investment.

With higher fertility, there is a higher transfer requirement for future generations of children. There is also a shift from human capital to physical capital investments due to the aforementioned reason. Together these lead to a long run 5.6% increase in average annual investments in physical capital. However, despite the increase in investments, there is a small decline in average capital stock due to lost income from the long run decline in human capital.

Finally, there are declines in fiscal outlays for elderly social insurance programs with higher fertility. Specifically, there is an 18.5% reduction in public outlays on medical expenditures as share of output, which places downward pressure on the consumption tax. There is also a 17.2% decline in social security pension payments as a share of output, and correspondingly a drop in social security labor income tax. Combining this reduction in distortionary income tax with a small increase in the ratio of physical to human capital results in a net increase in the after-tax wage of more than 6%. Nonetheless, this increase in the private returns to schooling is not able to counteract the negative aggregate effects of higher fertility on education previously detailed. All together, we find that the favorable macroeconomic effects of higher fertility—larger working age share, additional savings for future children, and reduced public spending on pensions and health care—are outweighed by reductions in average educational attainment and capital stock in the long run steady state.

6.2. Model Comparisons

We now discuss key differences in our benchmark model results with two alternate models of exogenous human capital. In the first model, we shut down both the human capital investment channel available to the households as well as the effect of fertility on female labor supply through a time cost of raising children (labeled “Exg. HC”). In the second model, we only shut down the aforementioned investment channel while still allowing for time cost of raising children (labeled “Exg. HC + TC”). Since we are primarily interested in comparing the differences between high and low fertility scenarios, figure 11 shows the percentage difference in key macroeconomic variables between high and low fertility paths $\left(x_t=\frac{x_t^{1.8}-x_t^{1.5}}{x_t^{1.5}}\ast 100\right)$ for the three models.²⁷

Figure 11:

Transition Dynamics: Model Comparison

Due to identical demographic structure, the effect of increased fertility on working-age share is the same across the three models — an immediate decline due to increased child dependents followed by an increase in the long run (refer to appendix table C.1). However, there is divergence in the impact of fertility on other macroeconomic outcomes. In the model with exogenous human capital and no time cost of children, the effect of the fertility change on effective labor supply, capital, and output is driven by the effects on working-age share over the transition. About 70 years from the start of the fertility transition, the effective labor supply in the 1.8 fertility path exceeds that of the 1.5 fertility path, with capital and output following suit a few years later. Adding a time cost of children exogenously lowers labor supply — and hence capital and output — throughout the transition, but the pattern of results remain unchanged. The addition of endogenous human capital further lowers effective labor supply with higher fertility in the long run, by reducing average educational levels.

In the long run, higher fertility maintains roughly 2% higher effective labor and 1.6% higher output in the exogenous human capital model without time cost. Adding a time cost lowers this to around 1.3% and 0.8% but maintains the positive long run impact of higher fertility. In contrast, in our benchmark model, the long run increase in working-age share in the higher fertility path is not able to compensate for the loss in education shares. As a result, effective labor, capital, and output are around 1% lower in the long run.

6.3. Alternate Fertility Scenarios

Our benchmark experiment found lower short and long run macroeconomic outcomes under a high (1.8) compared to a low (1.5) fertility path. In order to test the generalizability of these results, we feed two alternate long-term fertility paths (1.2 and 2.0) into our benchmark model.²⁸ Figure 12 shows results for percentage changes from the 1.5 fertility path for key macroeconomic variables, for these two alternate fertility scenarios, along the transition. In the short run (2010–2070), there is a negative monotonic relationship between fertility and outcomes — lower fertility results in higher levels of average effective labor (despite small short run declines in college share as seen in figure 12e), capital stock, and output. This is primarily driven by the short run increase in working-age share accompanying lower fertility. In the long run, higher fertility affects macroeconomic variables through two countervailing mechanisms — an increase in working-age share and a decline in human capital. When moving from 1.5 to 2.0 fertility, the latter mechanism dominates the increase in working-age share, resulting in an overall negative effect of higher fertility in the long run (just as in the 1.8 fertility case). However, with a very low fertility of 1.2, the increase in college share is unable to make up for the large decline in working-age share. As a result the long run output per capita is lower than in the 1.5 case. These results indicate that while a fertility rate that maximizes the long run working-age share (2.0) may not maximize long run output, it is also true that a fertility rate that maximizes average human capital (1.2) may not maximize output either. Rather, it is the delicate balance between the working-age share and human capital levels that determine long run output.

Figure 12:

Transition Dynamics: Fertility Comparisons

6.4. Policy Experiments

Our benchmark model indicates that a bump in fertility can result in macroeconomic declines for China, both in the short-run and the long-run. While the short-run decline is triggered by an increase in the childcare burden, in the longer run, lower levels of economic output under higher fertility is driven mostly by lower levels of human capital. We next conduct a set of experiments to gauge the potential of education policy to influence output, both in the short-run and long-run, through the human capital channel in our benchmark model. Specifically, we would like to use the quantitative model developed here, to understand how much of an increase in aggregate government budget on education can completely offset the negative effects of higher fertility. Towards this goal, we search over the space of public education subsidies to find the required permanent increase in the education budget with high fertility that results in the same per capita output as the low fertility benchmark.²⁹ We identify the required increase in education budget to equate output both in the long-run steady state (policy 1) and in the year 2060, forty years from the start of the reform (policy 2). Figure 13 provides percentage change in key macroeconomic variables from the benchmark 1.5 fertility case for three different scenarios — a) benchmark 1.8 fertility case, b) 1.8 fertility under policy 1 and finally c) 1.8 fertility under policy 2.

Figure 13:

Transition Dynamics: Policy Experiments

Note: Policy 1 refers to a permanent increase in government education budget to 4.5% of GDP and policy 2 to 5.5% of GDP, starting year 2020, in the 1.8 fertility case.

Recall that in the benchmark calibration, the public education budget is fixed at 4.3% of GDP starting from the year 2010. We find that a small increase to 4.5% starting in 2020 can wash out the long-run negative effects of higher fertility on output (refer to figure 13, panel d). This underscores the importance of the human capital channel in our model, especially in mitigating the long-run effects of demographic transition. An increase in per child educational subsidy due to the increase in education budget lowers the long-run private cost of college by 5.5% as compared to the benchmark high fertility case. While this only has a small impact on effective labor (college share goes up by 4.4%), it frees up household resources for investments in physical capital. Additionally, a long run increase in earnings, due to improvements in human capital levels, increases the average capital stock in the economy. Together, these effects prevent the long run declines in output observed when fertility increases and the education budget is held fixed at 4.3%. However, note that these are very long run effects. In the short run, as seen in figure 13, the positive effects of this policy regime are unable to counteract other dominant channels including a declining working age share and higher transfer requirements for future generations of children. For example, with an education budget of 4.5%, it takes more than eighty years for the output per capita to reach the low fertility scenario with the benchmark budget of 4.3%.

Our analysis reveals that changes in human capital in the model economy take place over a long period of time due to the realistic nature of demographics modeled here. As a result, education policy is somewhat less effective in undoing the negative effects of higher fertility in the short-run, where other dominant channels are at play. In fact, equating output between the two fertility scenarios, forty years from the reform, requires a roughly 28% permanent increase in the education budget (from 4.3% to 5.5% of GDP) in the 1.8 fertility case starting in 2020. The mechanisms underlying the human capital channel operate in the same way. For instance, as expected, a further increase in the educational subsidy lowers private tuition, thereby raising human capital levels and average physical capital stock under the high fertility scenario. However, note that the college share under the benchmark high fertility is already somewhat higher than the low fertility case in the short-run. As a result, it takes a much larger increase in college shares to compensate for the lower working age share and equate output in the short-run (refer to panel c of figure 13). In the long-run, due to the large increases in both college share and physical capital investments, output remains substantially higher compared to the lower fertility case.

7. Robustness

In this section, we test the robustness of our key results to some of the assumptions made in the benchmark model. Specifically, we test how our final steady state results would change if we allowed for (1) cross-sectional heterogeneous fertility or (2) medical expenditures (as a share of output per capita) to grow over time.

7.1. Heterogeneous Fertility

In our benchmark model we focused on changing aggregate fertility over time but assumed fertility was the same across all households at a given point in time. However, it is quite plausible that there will be heterogeneous response to the relaxation of the fertility restrictions in China. While a fully endogenous dynamic response is beyond the scope of this study, we examine if implementing limited exogenous fertility heterogeneity has substantial quantitative implications for our aggregate results. Specifically, we solve for the final steady state (for both 1.5 and 1.8 fertility scenarios) under the assumption of an exogenous “high” and “low” fertility type. It is well-known that fertility is negatively correlated with parental characteristics such as income and education. As such, we assume fertility is an exogenous function of the parent’s productivity type η —a parent below/above some productivity threshold is assigned to the high/low fertility type.³⁰ In the baseline, high fertility types are assigned a fertility rate of 1.68 and low fertility types a rate of 1.25. This yields an aggregate fertility rate of 1.5 as in our benchmark status quo scenario. We compare this baseline to a steady state in which each fertility rate is increased by 20%—yielding an aggregate fertility rate of 1.8.

Column 2 of table 8 provides some key comparisons between the 1.5 and 1.8 scenarios with heterogeneous fertility. Aggregate results are similar to the benchmark model (reproduced in column 1 for convenience), though per capita output falls a little more with heterogeneous fertility. This is due to a larger drop in college share and, to a lesser extent, per capita capital stock. However, the brunt of the cost of higher fertility is borne by high fertility households. For example, college share and consumption per capita fall by nearly 15% and 6% for high fertility households compared to less than 1% for low fertility households. While low fertility households experienced a proportionally equivalent increase in fertility as high types, losses were almost entirely offset by positive general equilibrium effects—primarily lower taxes. Further experiments reveal that an increase in the government education budget from 4.3% to 4.5% of GDP can mostly offset the negative aggregate effects on per capita output by increasing human capital investments (results available on request). This is the same increase in the government budget required to offset effects in the benchmark experiments. Thus overall, while results suggest there are potentially interesting and important distributional consequences, the aggregate results are similar to the benchmark.

Table 8:

Final Steady State Results: Sensitivity

Variables	Benchmark	Heterogeneity	Med. Exp.
Output per capita	−1.01	−1.62	−1.77
Consumption per capita	−2.99	−3.31	−1.79
Type 1	-	−5.97	-
Type 2	-	−0.03	-
Capital per capita	−1.25	−1.91	−2.87
Type 1	-	−3.02	-
Type 2	-	−0.24	-
College share (%)	−5.40	−9.19	−5.76
Type 1	-	−14.90	-
Type 2	-	−0.85	-

Notes: Percentage change from (aggregate) low to high fertility shown. Type 1 refers to households with a high fertility old adult and Type 2 refers to a low fertility old adult. College share by type refers to the college share of young adults in the household.

7.2. Medical Expenditures

In our benchmark specification we have assumed that individual medical expenditures, as a percent of income per capita, remain fixed at the initial steady state levels. Note that despite this assumption, the model generates a large increase in overall expenditures as a percent of GDP due to population aging. For instance, medical expenditures rise from 0.9% of GDP in 2010 to roughly 5% of GDP by 2100 (refer to appendix figure C.5). However, it is conceivable that average expenditures could also rise in the future due to medical inflation or increased utilization. For example, health expenditures grew 1.7% faster than the overall Chinese economy between 1993 and 2012, largely due to increased real expenditures per case of disease (Zhai et al., 2017). Medical expenditures as a share of GDP in China are also projected to increase by 70% between 2015 and 2035 (Zhai et al., 2019). In light of this, we conduct an experiment where we double both high and low medical expenditures per person (e.g. we increase low expenditures from 2.4% to 4.8% of GDP per capita).

Column 3 of table 8 provides some key steady state results, comparing the 1.5 and 1.8 fertility scenarios with higher medical expenditures. Overall, results are similar to the benchmark, though the decline in output per capita due to higher fertility (1.77%) is somewhat higher. This is mostly driven by a larger decline in capital, likely due to increased budgetary pressure and a larger decline in precautionary savings due to the higher medical expenditures. Further experiments reveal that an increase in the government education budget from 4.3% to 4.75% of GDP can fully offset these negative effects on per capita output by increasing human capital investments (results available on request). This is somewhat higher than the 4.5% government budget required in the benchmark.

8. Conclusion

After having implemented one of the most stringent fertility policies in the late 1970’s to slow down population growth, China recently relaxed its one child policy in hopes of promoting economic growth and social welfare. In this paper, we develop a dynamic general equilibrium model for understanding the short and long run macroeconomic effects of feasibly induced changes in aggregate fertility rates. We incorporate endogenous education decisions and intra-household transfers to capture the effect of changing demographics on human and physical capital accumulation.

Our results indicate that a fertility bump results in lower short run income per capita in our benchmark model economy. These results may persist in the long run as well where a small bump in fertility results in declines in human capital investments. More generally, we find that maximizing the working age share of the population is not the same as maximizing income per capita. As such, even though our focus in this paper has been on China, our findings are generalizable to other countries which are experiencing population aging. Faster aging economies with already high levels of human capital, such as Japan, Germany, Italy, Finland and Portugal (projected to have old-age dependency ratios greater than 40% by 2030), might benefit from a bump in fertility in the long run.

Our study is not without limitations. Most notable being the treatment of fertility as an exogenous change. However, our robustness exercise indicates that while a heterogenous response to fertility relaxations may have interesting distributional consequences, substantial deviations from the aggregate prediction is unlikely. We also make several simplifying assumptions about human capital investments and the evolution of labor productivity. We abstract away from labor supply decisions, health dynamics and model medical expenditure uncertainty only for the very old. We are also limited to analyzing the economic impact of fertility changes and stress that a welfare analysis is outside the scope of our current framework. Addressing some these limitations leaves room for important future research in this direction.

Table 4:

Technology Parameters

Parameter	Value	Target/Source
Capital share α	0.48	Feenstra et al. (2015)
Period depreciation δ	0.10	Chow and Li (2002)
Factor productivity A	1	Normalization

A. Alternate Model Specifications

In terms of calibration, we simply set the parameter θ_f = 0 for eliminating the time cost of children. For removing the endogenous human capital margin, we eliminate education choice from our benchmark model (i.e. e = 1 for all individuals). In both cases, we re-estimate the parameters β, ρ and σ_η to match the same three data targets as our benchmark model. However, we keep the first stage calibration (parameters set/estimated outside the model) unchanged across all three models. Table A.1 provides the calibration results for both models without the human capital investment channel. Exogenous human capital increases income inequality and reduces inter-generational persistence in labor earnings compared to our benchmark model. As a result, we find that the calibration of these alternate models without the human capital choice results in a higher persistence and a lower variance in the labor productivity shock when matching the same targeted moments. It should be noted that consumption taxes only finance health care expenditures in the first two models but also finance education expenditures in the benchmark model.

Table A.1:

Calibration Results for Alternate Models

Parameters	Value		Moment	Data	Exg. HC	Exg. HC + TC
	Exg. HC	Exg. HC + TC
β	0.945	0.945	K/Y	3.23	3.16	3.18
ρ	0.63	0.64	Income mobility	0.60	0.60	0.60
ση	0.39	0.42	Income Gini	0.29	0.29	0.29

Notes: Exg. HC refers to the exogenous human capital model and Exg. HC + TC is the model with exogenous human capital and time cost of children

B. Calibration: Human Capital

Due to the lack of empirical estimates on returns to education between 1960 and 2010, we allow them to increase linearly from 3% (in 1960) to 10% (in 2010) and then hold them fixed at the 2010 level. Table B.1 below provides these numbers.

The total tuition costs are taken from the China Statistical Yearbooks from various years and expressed as a fraction of output per capita. Even though the yearbooks report total tuition costs for several years in between 1960 and 2010, we assume a linear increase in these costs between 1960 and 2010 for two reasons. First, the data is missing for many years as shown in figure B.1 (e.g. we have only one data point prior to 1978). Second, these costs have been fluctuating quite a bit during the time period of interest which causes numerical convergence issues while simulating the transition dynamics in our model.

Table B.1:

Returns to Education

Year	Primary	Secondary	College	Year	Primary	Secondary	College
1960	1	1.19	1.34	1987	1	1.51	2.02
1961	1	1.21	1.37	1988	1	1.52	2.04
1962	1	1.22	1.39	1989	1	1.53	2.07
1963	1	1.23	1.42	1990	1	1.54	2.09
1964	1	1.24	1.44	1991	1	1.55	2.12
1965	1	1.25	1.47	1992	1	1.56	2.14
1966	1	1.26	1.49	1993	1	1.58	2.17
1967	1	1.27	1.52	1994	1	1.59	2.19
1968	1	1.29	1.54	1995	1	1.6	2.22
1969	1	1.3	1.57	1996	1	1.61	2.24
1970	1	1.31	1.59	1997	1	1.62	2.27
1971	1	1.32	1.62	1998	1	1.63	2.29
1972	1	1.33	1.64	1999	1	1.64	2.32
1973	1	1.34	1.67	2000	1	1.66	2.34
1974	1	1.36	1.69	2001	1	1.67	2.37
1975	1	1.37	1.72	2002	1	1.68	2.39
1976	1	1.38	1.74	2003	1	1.69	2.42
1977	1	1.39	1.77	2004	1	1.7	2.44
1978	1	1.4	1.79	2005	1	1.71	2.47
1979	1	1.41	1.82	2006	1	1.73	2.49
1980	1	1.43	1.84	2007	1	1.74	2.52
1981	1	1.44	1.87	2008	1	1.75	2.54
1982	1	1.45	1.89	2009	1	1.76	2.57
1983	1	1.46	1.92	2010	1	1.77	2.59
1984	1	1.47	1.94	.	.	.	.
1985	1	1.48	1.97	.	.	.	.
1986	1	1.49	1.99	2100+	1	1.77	2.59

While information is available on the total public education budget of the Chinese government from 1960–2000, it is not clear how resources were allocated across different levels of education during this time period. As such we assume that between 1960–2000, no public subsidies are available for college and secondary school and tuition costs are fully borne by households. This implies the government budget is endogenously determined in the model pre-2000 based on the number of enrolled primary students each year (recall primary school is always free). Starting in 2000, we assume that the government budget on education (as a share of output) increases linearly until it reaches 4.3% in 2010 and remains fixed thereafter. This strategy gives a reasonable approximation of the total education budget as seen in the data (see figure below). Starting in 2000, after subtracting the entire cost of primary school from the government’s budget, the remainder is split in a 60/40 ratio between secondary school and college in order to determine the respective subsidy rates (until secondary becomes free, at which point a higher share starts to accrue to college). This split is consistent with more recent data that shows a relatively stable 60/40 ratio post-2000 (China, 2014).

Figure B.1:

Total Tution Costs

Figure B.2:

Government Education Budget

It is also important to acknowledge that in the mid-1980s, the central Chinese government introduced a law requiring local governments to establish procedures and deadlines for attaining free nine-year compulsory education (six years of primary and three years of secondary education). This may partially explain the rising government education budget observed in the data, which we also approximate in the model (along with free primary education throughout our analysis). However, the fact that the law aims to make basic education compulsory may have also contributed to the subsequent rise in education in 1990s and 2000s. As we do not enforce any penalty for choosing to stop after primary schooling, we would not be capturing any compulsory effect in our model. We also model secondary school as a full six years, without the option of choosing only three years. These abstractions may imply we are underestimating the rise in education above primary school during the transition. However, we are still able to reasonably match the steep decline in primary share between 1960 and 2010 through a combination of increasing returns to schooling, lower fertility rates, and increased public subsidies to education (see figure C.1). Moreover, while compulsory basic education could alter our transitions prior to 2010, as almost all of the human capital implications from a rebound in fertility post-2010 is on the secondary/college margin, these abstractions are unlikely to significantly alter our main (post-2010) results.

C. Additional Figures and Tables

Figure C.1:

Model Fit: Education Shares

Figure C.2:

Benchmark Transition Dynamics: Prices

Table C.1:

Final Steady State Results

Variables	Exg. HC		Exg. HC + TC		Benchmark
	1.5	1.8	1.5	1.8	1.5	1.8
Working-age share (%)	51.89	54.69	51.89	54.69	51.89	54.69
Output per capita	1.00	1.02	1.00	1.01	1.00	0.99
% change	-	1.56	-	0.78	-	-1.01
Consumption per capita	0.66	0.67	0.66	0.66	0.61	0.60
% change	-	0.23	-	-0.59	-	-2.99
Consumption per ad equ	0.71	0.72	0.71	0.72	0.65	0.65
% change	-	2.35	-	1.51	-	-0.94
Capital per capita	3.16	3.19	3.16	3.17	3.12	3.08
% change	-	1.03	-	0.24	-	-1.25
Average labor supply	0.61	0.64	0.60	0.62	0.52	0.52
% change	-	4.42	-	3.78	-	1.27
Effective labor per capita	0.35	0.35	0.35	0.35	0.35	0.35
% change	-	2.05	-	1.29	-	-0.78
Investment per capita	0.29	0.31	0.29	0.31	0.28	0.30
% change	-	8.01	-	7.29	-	5.61
Interest rate (%)	5.21	5.30	5.18	5.27	5.40	5.44
Wage rate	1.50	1.50	1.51	1.50	1.49	1.48
SS tax (%)	24.88	20.16	24.92	20.16	28.53	23.62
Consumption tax (%)	5.37	4.43	5.38	4.44	12.80	11.97
Private Medical expenditures (% of GDP)	1.53	1.24	1.53	1.24	1.53	1.24
Public Medical expenditures (% of GDP)	3.56	2.90	3.56	2.90	3.56	2.90
SS payments (% of GDP)	12.94	10.47	12.94	10.48	14.83	12.28
Primary share (%)	-	-	-	-	0.00	0.00
Secondary share (%)	-	-	-	-	12.01	16.76
College share (%)	-	-	-	-	87.99	83.24
Private secondary cost	-	-	-	-	0.00	0.00
Private college cost	-	-	-	-	0.32	0.36

Notes: Per capita outcomes have been normalized to output in the 1.5 fertility case for each model specification.

Figure C.3:

Transition Dynamics: Exogenous Human Capital

Note: All per capita variables have been normalized to output in the 1.5 fertility final steady state.

Figure C.4:

Transition Dynamics: Exogenous Human Capital & Time Cost of Children

Note: All per capita variables have been normalized to output in the 1.5 fertility final steady state.

Figure C.5:

Medical Expenditures in the Benchmark (1.5 Fertility Case)