The Gray Rhino: Dealing with the Aging Population in China

The Gray Rhino: Dealing with the Aging Population in China 

Education Research

In this thesis, I examine the implications of the potential reforms in retirement age and birth control policy in China using the OLG model. It is shown that without reforms, the capital stock, labor supply and aggregate output will experience dramatic declines in the coming years of this century. The impact on return to capital and wage rate differs due to the changes in K/L ratio. The postponement of retirement age proves effective in delaying the rise of old-age dependency ratio, while birth control policy reform produces better results in coping with the aging population in the long run. 


Since the implementation of the one-child policy in 1982, China’s population growth rate has experienced a decreasing trend. During the same period, the life expectancy of Chinese people has increased from 67.63 years in 1982 to 76.91 years in 20191. These two factors have resulted in the rapid aging demographics in China. 

According to the United Nations’ population projection (DESA, 2019), the old age dependency ratio (population 65 and older divided by population 15-64) in China will increase from 12% in 2010 to 58% in 21002. The growing elderly population has posed significant challenges to the government’s ability to deliver social welfare policy as well as the long-term development of the Chinese economy. 

Panel A: Total population and growth rate Panel B: Old-age dependency ratio 

Figure 1: UN demographic projections in China 

Source: United Nations, Department of Economic and Social Affairs, Population Division (2019). World Population Prospects 2019, Online Edition. Rev. 1. 

Compared with the aging problem in other countries, there are some special characteristics in China’s problem. Firstly, the decline in population growth rate of China is not a natural process which is often the case in other countries. Its steady decrease is accompanied by the implementation of the one-child policy. By relaxing the 

1 Data is taken from the World Bank. See: 


2The official retirement age in China currently is 60 for men, 55 for female civil servants and 50 for female workers. Suppose both men and women retire at 60, the old-age dependency ratio could be as high as 78% in 2100. 

strict one-child policy, the long-run birth rate in China could rebound and ease the aging problem. Secondly, the official retirement age in China (women in particular) is smaller than other aging countries3 and this leaves room for the Chinese government to delay retirement age in order to lower the old-age dependency ratio. 

In 2016, Chinese policy makers abolished the one-child policy and introduced the two-child policy which allows two children for each couple. Besides, there are also calls for delaying retirement age and abolishing birth control policy in recent years. This thesis aims to explore the implications of the aging population in China and evaluate the potential policy measures which focus on birth control and retirement age. 

The remainder of this thesis is organized as follows. Section 2 provides the literature review on aging demographics and Section 3 presents the model used in this thesis. Section 4 and 5 introduce the implementation of this model in MATLAB and its calibration. The quantitative findings are presented in Section 6 and Section 7 concludes this thesis. Appendix A gives detailed derivation of the Euler Equation and the second-order difference equation used in the backward shooting algorithm. 

3In 2018, OECD average retirement age is 64.2 for men and 63.5 for women(OECD, 2019). 5

Literature Review 

The process of aging is not exclusive to China, many countries in the world have already experienced it for many years. As pointed out by Conesa and Kehoe (2018), the rapid increase of old-age dependency ratios is not only observed in developed countries, but some emerging countries are also projected to encounter this process. In this context, a large amount of literature has focused on the macroeconomic implications of aging and most studies seem to agree that there exists a negative association between aging population and economic growth. 

Bernheim et al. (2001) use US data to demonstrate that there is significantly neg ative relationship between consumption and retirement, and they infer the increase of retirees would pose a negative impact on consumption and economic growth. Lee and Mason (2007) reveal that the increase of old-age dependency would raise the tax burden of the declining working-age group, which tends to have higher consumption than the old-age group. This would reduce the overall consumption of the economy and hinder its growth. Apart from consumption, the savings pattern would also be negatively impacted by the aging population. Davies and Reed (2006) show that overall saving tends to decline due to the increase of the old-age population since saving becomes the main source of spending for pensioners. Besides, there are also other routes of influencing economic growth by an aging population, such as public expenditures (Lee et al., 2011) and human capital (Bell and Rutherford, 2013). 

Most of the literature focuses on research in developed countries (Teixeira et al., 2017) and as the largest developing country in the world, China has not received enough attention yet. Existing literature differs in the economic impact of the aging population in China. Modigliani and Cao (2004) argue that population aging has a negative impact on saving rates and economic growth in China. However, a recent study by Hsu et al. (2018) finds that the aging population in China would promote more savings and accelerate physical and human capital accumulation. 

To deal with an aging society, many scholars have recommended two approaches: (i) the official retirement age must be delayed (Heijdra and Romp, 2009) and (ii) the fertility rate must be improved above the replacement level fertility4.  

Replacement fertility is the total fertility rate at which women give birth to enough babies to sustain population levels. 

The Institute of Medicine (US) Committee on the Long-Run Macroeconomic Effects of the Aging U.S. Population (Council, 2013) points out that retirement age delay is the short term solution to aging problem, and fertility rate improvement would be the long term remedy after conducting a large-scale study on the implications of demographic shift in the US. 

Lastly, to evaluate the impact of policy in dealing with an aging population, many researchers employ models with overlapping generations. These models range from stylised 2-3 period representations of the life cycle that build on Samuelson (1958) and Diamond (1965) work, to detailed multi-period models that build on Auerbach and Kotlikoff (1987) 65-period model. 


The model of this thesis is based on Problem Set 3. I choose an overlapping generation (OLG) model to evaluate the impact of the potential policy reforms the government could implement. 

3.1 Firm 

There is a representative firm that produces a single good using a Cobb-Douglas production function: 

Yt = AtKθt L1−θ 

where θ is the output share of capital, Kt and Lt are the capital and labor input at time t, and Atis the total factor productivity at time t. 

Capital depreciation rate is δ ∈ (0, 1). The representative firm maximizes its profits such that the rental rate of capital, rt, and the wage rate wt, are given by: 

rt = θAt 

 Kt Lt 


− δ (1) 



3.2 Demographics 

wt = (1 − θ)At 

 Kt Lt 

This model treats one period as a five-year interval and individuals enter the economy at the age of 20. After entering the economy, they will face mortality risk and work up until the mandatory retirement age of 60 5, after which individuals can live up to 100 years old. Therefore, the maximum periods T individuals live up to is 16 periods 5) in this model, during which the working life R lasts 8 periods ( 6020 


5) and 

The rest periods are their retirement life. The population of each generation is given 


Nt =XT 


 Y i−



Ni,t (3) 

where ψjis the conditional probability of surviving to period j + 1, and Ni,t is the number of individuals at the age of i who are born in time t

5Here I ignore the differences in official retirement age between men and women in China and assume everyone retires at the age of 60 in the benchmark scenario.

The size of the population evolves over time exogenously at the rate gt, so Nt+1 = (1 + gt)Nt (4) 

3.3 Government 

Government taxes labor income at rates τw, and uses the revenues to finance an exogenously given amount of government consumption expenditures G, which I assume is a fixed percentage of total output Yt. Both tax rates are determined endogenously 

in this model. 

Gt = τwXR i=1

 Y i−



wi,t+i−1Ni,t (5) 

The government also runs a pay-as-you-go social security program that is financed entirely by a payroll tax τss



 Y i−



SStNi,t = τssXR i=1

 Y i−



wi,t+i−1Ni,t (6) 

3.4 Households 

In this model economy, the labor income of an individual is divided into two parts. Before retirement, the income of an individual at time t is given by the economy wide wage ratio wt. Once retired, he receives constant social security income SSt every year, which is equal to a fixed share of his wage ratio at the age of retirement, µ · wR,t+R−1. The after-tax income, ei,t+i−1, of an individual is given by: 



(1 − τw − τss)wi,t+i−1 i ≤ R SSt i > R 


where i = 1, 2, …16, R = 8 is the official retirement age, ei,t+i−1 is the after-tax income of an individual of age i in time period t + i − 1, and wi,t+i−1 is the labor income of an individual of age i in time period t + i − 1. 

The intertemporal budget constraint of an individual is given by: ki+1,t+i + ci,t+i−1 = (1 + rt+i−1)ki,t+i−1 + ei,t+i−1 (8) 

where ki,t+i−1 is the individual wealth of an individual of age i in time period t+i−1, ci,t+i−1 is the consumption of an individual of age i in time period t + i − 1, and rti−1 is the interest rate in time period t + i − 1. 

Individuals do not care about their descendants or predecessors, which means they do not leave bequests to their children or provide financial support to their parents. Therefore, individuals face addition constraints given by: 

k1,t = kT +1,t = 0 (9) 

Individuals obtain utility from their own life-time consumption only and their utility function is characterized by constant relative risk aversion (CRRA). Therefore, the household’s maximization problem can be described as: 



XT i=1 


 Y i−





1 − γ 

subject to Equation 7, 8 and 9, where β is the subject discount factor, and γ is the degree of relative risk aversion. 

3.5 Characteristics in equilibrium 

In equilibrium, the households’ optimization problem is solved and all markets are cleared. 

For labor market, assume one generation of individuals are endowed with one unit of labor, the labor supply in each period is given by: 

Lt =XR i=1

 Y i−



Ni,t−i+1 (10) 

which means that the aggregate labor supply is equivalent to the sum of the working-age population. 

For capital market, the aggregate capital in the economy is simply the sum of the individuals’ wealth: 

Kt =XT i=1

 Y i−



ki,t+i−1Ni,t (11) 

For the goods market, the aggregate demand should be equal to the supply: Yt = Ct + It + Gt (12) 

where the aggregate consumption Ct =PTi=1  Qi−1 

j=0 ψj 

gate investment It = Kt (1 − δ)Kt−1

ci,t+i−1Ni,t, and the aggre 

MATLAB Implementation 

The procedures of solving for steady state basically follow hints on Problem Set 3. Step 1 

Based on the individual’s optimization problem, I set up the Lagrangian equation to solve for the Euler Equation and derive the optimal path of individual wealth from it6. The optimal path of individual wealth (Equation 16) is suitable for backward shooting algorithms. 

Step 2 

As is shown in Equation 17, the coefficients α1, α2, α3, α4in the optimal path need to be decided in addition to solving for the individual’s problem. The steady state interest rate and tax rates are determined endogenously in this model, and all of them depend on the aggregate capital in the equilibrium. Thus, I provide a guess of aggregate capital Kt to obtain the coefficients and prepare for the backward shooting. 

Step 3 

The transversality condition in Equation 9 states that individuals leave no wealth upon their deaths (kT +1,t = 0). By providing another guess of the second to last individual wealth kT,t, together with the wage ratio one period earlier, I get the value of third to last individual wealth kT −1,t. Repeat this process until the first period of individual wealth k1,t. If k1,t is not approximately equal to zero7, I will update the initial guess of kT,t and repeat the above process so that eventually k1,t is close to zero8. These iterations can be achieved using while and for loops in MATLAB. 

Step 4 

The capital market clearing condition (Equation 11) requires that the sum of individual wealth should equal the aggregate capital in the economy. If the 

6The detailed derivation is shown in Appendix A. 

7Throughout this model, I choose 1 × 104 as the tolerance level. 

8The update algorithm follows the hint on Problem Set 3. 


T ,t = kiT ,t 0.01 · ki1,t 

where i is the number of iterations. If the initial guess of kiT ,t results in negative capital ki1,t in period 1, this rule will automatically choose ki+1 

T ,t > kiT ,t on the i + 1 iteration. Similarly, if ki1,t is 

positive, this rule will give a smaller initial guess of ki+1 

T ,t .

The difference between the guess of Ktin Step 2 and the sum of individual wealth obtained in Step 3 is greater than the tolerance level, the guess of aggregate capital Ktis updated and Step 2 and 3 is repeated. The iteration will not stop until these two values converge. I use the fsolve function in this step to solve for the equilibrium aggregate capital. 


I calibrate the model so that the initial steady state matches the survival probability and old-age dependency ratio 9in China in 2010s and the final steady state in the end of this century (around Year 2100). Most of the parameters are calibrated according to ˙Imrohoroglu and Zhao (2018) work. Table 1 summarizes the main calibration used in the model. 

Parameter Description Value 

β Subject discount factor 1.01 

γ Relative risk aversion 3 

δ Capital depreciation rate 0.1 

θ Capital output share 0.3 

µ Benchmark pension dependency ratio 0.15 

T Time horizon 16 

R Benchmark retirement period 8 

A TFP Factor 1 

q Government expenditure to total output 0.16 

g1 Benchmark population growth rate at Stage 1 0.4% 

g2 Benchmark population growth rate at Stage 2 -0.5% 


As is mentioned in Section 3, the model period is 5 years and there are T = 16 periods in this economy. Individuals enter the economy at the age of 20 (t = 1) and retire at the age of 60 (t = R) in benchmark scenarios. 

China is expected to experience dramatic demographic changes over the next several decades. In this thesis, I choose to model these changes in two stages, which corresponds to DESA (2019) projection. The first stage ranges from Year 2020 to 2030, when China’s population reaches its summit. The second stage is from Year 2031 to 2100, when the population declines gradually. 

During 2010-2020, the population growth rate of China remains 0.5% on av 9Hereafter the old-age dependency ratio is defined as the retired population (t = R + 1, R + 2, …, T) over the working population (t = 1, 2, …, R). Erage10. Therefore, I choose Year 2020 as the initial steady state and the population growth rate during this period is calibrated to 0.4%, which reflects the mild growth momentum of China’s population. The population growth rate in the second stage is calibrated to -0.5%, which follows DESA (2019) projection and the final steady state is set in Year 2100. 

The benchmark population projection in this model is shown in Figure 2. Both the total population and the share of the working population (Age 20-60) shrink over time. 

10 Data is taken from the Seventh National Population Census results released on May 11, 2021. 

See: 1817185.html 11Data is computed based on the Life Table published by DESA (2019). See: 

Figure 2: Population projection during Year 2020-2100 

5.2 Preference and technology 

The value of relative risk aversion γ is assumed to be 3, which is commonly used in literature. The subject discount factor β for individuals is set to 1.01, which I follow ˙Imrohoroglu and Zhao (2018) finding to account for the mortality risk faced by individuals. 

The capital depreciation ratio δ is set to 0.1 according to Bai et al. (2006), and the capital output share is set to 0.3. For simplicity, I assume there is no technological progress and assign the TFP factor A to 1, which is constant across the whole time horizon. 

5.3 Government 

The Chinese government expenditure is, on average, 16% of the GDP from 2000 to 201912. Based on this information, I calibrate the government expenditure to total output (q) to 0.16. 

I assume there is no government deficit in the model and the labor income tax 12 Data is taken from the World Development Indicators (WDI) database published by the World Bank. See: 

revenues exactly balance the government expenditure. The labor income tax rate in steady state is determined endogenously in the model. 

The pension replacement rate13 is calibrated to be 0.15, which follows the same value in ˙Imrohoroglu and Zhao (2018). The pay-as-you-go social welfare program is also assumed to be self-financing. That is to say, the payroll tax revenue levied on working population balances the pensions received by the retirees in the economy. Hence, the payroll tax rate in steady state is also determined endogenously. 

Policy reforms 

As is discussed in Section 1, there are two potential policies the Chinese government could consider to reform: postpone retirement age and relax birth control policy. 

Figure 3: Population projection with Three-child policy during Year 2020-2100 

In this thesis, I will present results for three reforms which aims to deal with the aging population: (1) an increase of retirement age from 60 to 65; (2) a relaxation of birth control policy from ”two-child” to ”three-child”, which I assume would improve the population growth rate in both stages; and (3) a combination of reforms (1) and (2). 

13The pension replacement rate is defined as the pension income over the average wage ratio. 16

The calibration of retirement age reform is plain, I just changed the retirement period R from 8 to 9. To account for the impact of birth control policy reform, there are two adjustments in the benchmark demographics prediction. Firstly, I increase the horizon of the first stage from Year 2020 to 2040, and decrease the length of the second period from Year 2041 to 2100. Then I change the benchmark population growth rate in both stages to reflect the impact of the three-child policy. The population growth rate is set to 0.6% at stage 1 and -0.2% at stage 2. The population projection with this reform is shown in Figure 3. 


6.1 Time paths of individual wealth and consumption 

In this subsection, I present the time paths of individual wealth and consumption in two steady states under different policy reforms. The results from these reforms are shown in Figure 4. 

Panel A: Benchmark without policy reforms Panel B: Retirement age reform

Panel C: Birth control policy reform Panel D: Both policy reforms 

Figure 4: Time paths of individual capital and consumption 

As is shown in Figure 4, individuals accumulate wealth during their working periods and consume it until death in steady states. To maximize their utilities, they choose to smooth consumption over their life. Their consumption is hump shaped with the maximum consumption appearing at their retirement period. 

In the benchmark scenario without policy reforms (Panel A), Compared with initial steady states in 2020, the individual wealth level in 2100 is higher in any 


period of life, and the consumption level in 2100 is lower in the early stage but exceeds the 2020 level in later periods of life. With an older population, individuals in 2100 choose a lower consumption level and accumulate more wealth in their early periods of life to support their consumption level in retirement. This is in line with the view that an aging population would promote more savings and accelerate capital accumulation. 

Panel B demonstrates the effects of retirement age reform on steady state in 2100. The postponement of the retirement age from 60 to 65 has allowed individuals to have longer working periods and shorter retirement horizons during their life. Since individuals can earn a higher labor income during working periods, this reform enables them to accumulate wealth at a slower pace. Besides, the longevity risk faced by individuals decreases due to the shorter retirement horizon. These effects may explain the fact that the consumption level after reform dominates benchmark scenarios throughout the entire horizon. 

The time paths of individual wealth and consumption with birth control policy reform do not deviate a lot from benchmark scenarios (Panel C). Although the total population after reform stays at a higher level under three-child policy, the distribution of population across various age groups remains relatively stable. Individuals choose to follow the same pattern of saving and consumption once the effect of reform fades away in the long run. 

The impact of both reforms on individual wealth and consumption in steady states is simply the combination of each reform (Panel D). As discussed above, the dominant impact is from the retirement age reform. 

Properties of steady states 

Column Bench mark presents the properties of initial steady state in 2020, and Column No Reform represents the final steady state without policy reforms but in 2100 demographics. The results for retirement age reform, birth control policy reform and a combination of these reforms are presented in Columns Reform 1, Reform 2, and Reform 3

In the model, the Chinese demographics has undergone dramatic changes from 2020 to 2100. The total population shrinks by about 30% (from 1 to 0.7238) under current two-child policy, and the old-age dependency ratio more than doubles in the No Reform scenario. The increase in retirement age of Reform 1 partially alleviates the rising old-age dependency ratio (from 60.82% to 43.39%). Compared with retirement age reform, the three-child policy succeeds in preventing the decline of the total population in the long run, but it fails to deal with the aging population. The old-age dependency ratio in Reform 2 is as high as the No Reform scenario. Reform 3 presents the best outcome in both maintaining population scale and deferring the rapid growth of old-age dependency ratio. 

The aging population has imposed a great challenge to the pension system. Under the No Reform scenario, the payroll tax rate τSS increases from 4.39% to 9.12% to balance the pension expenditures. The rise of the payroll tax rate will impact individuals’ disposable income negatively. As is shown in Equation 6, holding other parameters fixed, the payroll tax rate is mainly influenced by old-age dependency ratio. Therefore, postponing retirement age will be more effective in curbing the rising payroll tax rate than birth control policy reform (6.51% and 6.37% under different birth control policies), as it results in a lower old-age dependency ratio. 

Due to the negative demographic trend discussed above, the aggregate labor in 20

No Reform scenario witnesses dramatic decline in 2100. Without reforms on birth control policy, it is shown in the table that retirement age postponement alone has limited effect in offsetting the negative impact of the aging population on the labor force, as the aggregate labor in Reform 1 (0.5048) is slightly higher than the No Reform case (0.4500). In comparison, the introduction of a three-child policy is relatively effective to make up the labor force gap between two steady states. As Council (2013) discusses, fertility rate improvement is the long-term solution to the aging problem. This argument is supported in this model. 

Compared with aggregate labor, the impact of the aging problem on capital stock is milder in 2100. As is shown in Figure 4 Panel A, the individual wealth level in 2100 dominates that in 2020 under No Reform scenario. People tend to accumulate more wealth to maintain a higher consumption level after retirement. From the macroeconomic perspective, this increase in individual capital stock offsets part of the negative impact of changing demographics, resulting in a mild decrease in aggregate capital under the No Reform scenario. However, with the implementation of the three-child policy, this positive feedback from individuals outweighs the aging population, and the aggregate capital in the economy (1.0140 and 0.9834) exceeds the benchmark scenario (0.8825). 

The impact on aggregate output can be obtained by Cobb-Douglas production equation once the aggregate capital and labor supply are fixed. Since the capital output share is 0.3 in this model, labor supply carries more weight in determining the output. This is reflected in the result, as the aggregate output under two-child policy witnesses a large drop relative to the benchmark scenario, and the relaxation of birth control policy keeps the output from declining further. It should be noted that the TFP factor is assumed to be constant over time in the model. When considering technological progress, the aggregate output should be larger compared with current results. 

According to the results, the implications of an aging population in return to capital and wage rate are dramatically different: the return to capital in 2100 experiences considerable drop under all scenarios, whereas the wage rate witnesses an increase regardless of reforms. As is shown in Equation 1 and 2, the value of return to capital and wage rate is mainly governed by K/L ratio. The relative changes in aggregate capital and labor supply in different scenarios result in the variations in return to capital and wage rate. The result has demonstrated that the capital stock is more abundant relative to labor supply in 2100. Therefore, the return to capital decreases due to oversupply and wage rate increases thanks to the aging population. 


It is widely anticipated that China will experience dramatic demographic changes in the coming decades of this century. However, the rapid aging process far exceeds people’s anticipation14. At the end of this thesis-writing period, the Chinese government unexpectedly relaxed its birth control policy and allows couples to have up to three children15, which underlines the gravity of the aging problem. It becomes even more important to understand the consequences of these changes. 

In this thesis, I examine the implications of the potential reforms in retirement age and birth control policy in China using the OLG model. It is shown that without reforms, the capital stock, labor supply and aggregate output will experience dramatic declines in the coming years of this century. The impact on return to capital and wage rate differs due to the changes in K/L ratio. The postponement of retirement age proves effective in delaying the rise of old-age dependency ratio, while birth control policy reform produces better results in coping with the aging population in the long run. 

14The latest Population Census shows that there are only 12 million babies born in 2020, continuing the descent to a near six-decade low. See: 

Press Release/202105/t20210510 1817185.html 

15On 31 May, 2021, the Chinese government lifted the cap on births and married Chinese couples are allowed to have up to three children. See: ministries/202106/01/content WS60b61ab7c6d0df57f98da86e.html 

The Gray Rhino: Dealing with the Aging Population in China Written by Zixuan Ma


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Conesa, J. C. and Kehoe, T. J. (2018). An introduction to the macroeconomics of aging. The Journal of the Economics of Aging, 11:1–5. 

Council, N. R. (2013). Aging and the Macroeconomy : Long-Term Implications of an Older Population. National Academies Press, Washington, D.C. 

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A Appendix 

To solve the household optimization problem, Lagrangian method can be used: 

L =XT i=1 


 Y i−




1 − γ+λt+i−1[ki+1,t+i+ci,t+i−1(1+rt+i−1)ki,t+i−1−ei,t+i−1] ψj 

where λ is the Lagrange multiplier. The first-order conditions can be obtained by: 


∂ci,t+i−1= βi−1 ∂L 

 Y i−




i,t+i−1 + λt+i−1 = 0 (13) 

∂ki+1,t+i= λt+i−1 − λt+i(1 + rt+i) = 0 (14) From Equations 13 and 14, Euler Equation can be derived as: 

ci+1,t+i = [βψi(1 + rt+i)] 1γ ci,t+i−1 (15) 

Using intertemporal budget constraint 8 to solve for the consumption and substitute into the Euler Equation 15. This will give the law of motion for individual’s wealth: 

ki,t+i−1 = α1t+i−1ki+1,t+i + α2t+i−1ki+2,t+i+1 + α3t+i−1ei,t+i−1 + α4t+i−1ei+1,t+i (16) 


α1t+i−1 =

1 + rt+i−1+

i(1 + rt+i)(1−γ)(1 + rt+i−1)(17) 


α2t+i−1 =

i(1 + rt+i)1(1 + rt+i−1)(18) 


α3t+i−1 = 1 − τw − τss 

1 + rt+i−1(19) 

α4t+i−1 =1 − τw − τss 

i(1 + rt+i)1(1 + rt+i−1)(20) β1ψ1

The Gray Rhino: Dealing with the Aging Population in China Written by Zixuan Ma

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The Gray Rhino: Dealing with the Aging Population in China