3.1. Cointegrating polynomial regression

Our empirical analysis aims to demonstrate that the U-shaped relationship between the US current account balance with China and the life expectancy gap shown in Figure 2 is a statistically valid result. For this purpose, we use the polynomial cointegration technique developed by Wagner (Reference Wagner2015, Reference Wagner2023) and Wagner and Hong (Reference Wagner and Hong2016), which enables us to examine nonlinear cointegrating relationships involving integrated regressors and their powers and is particularly useful when economic theories predict that underlying variables have nonlinear relationships and/or linear models have poor predictive accuracy.

We estimate the polynomial regression model based on time series data as

(1) \begin{align} z_{t}=\beta _{0}+\beta _{11}x_{1t}+\beta _{12}x_{1t}^{2}+\beta _{2}x_{2t}+\cdots +\beta _{m}x_{mt}+u_{t}, \end{align}

where $\beta _{0},\cdots, \beta _{m}$ are coefficients to be estimated. An explained variable $z_{t}$ and explanatory variables in linear form $x_{1t},\cdots, x_{mt}$ are integrated of order one. Only $x_{1t}$ is allowed to have a polynomial relationship with $z_{t}$ , which represents the most empirically relevant case, as noted by Wagner (Reference Wagner2023). (1) is regarded as a cointegrating polynomial regression if an error term $u_{t}$ is stationary.

The existence of polynomial cointegration is a prerequisite for the successful modeling of the U-shaped relationship between the US current account balance with China ( $z_{t}$ ) and the life expectancy gap ( $x_{1t}$ ) because the rejection of the stationarity of $u_{t}$ means that (1) is a spurious regression. Note that standard tests for linear cointegration cannot be applied to (1) because powers of integrated variables are not integrated variables (Wagner, Reference Wagner2012); thus, $x_{1t}$ and $x_{1t}^{2}$ cannot be separately treated as integrated regressors. This problem is resolved by using the polynomial cointegration technique.

The testing procedure for polynomial cointegration involves two steps. First, the fully modified ordinary least squares (FMOLS) method developed by Phillips and Hansen (Reference Phillips and Hansen1990) is used to estimate (1), and the FMOLS residual $\hat{u}_{t}^{+}$ is obtained.Footnote ⁴ This estimation method is designed to nonparametrically correct for endogeneity and serial correlation biases. Next, we calculate the test statistic for the null hypothesis that $u_{t}$ is stationary as

(2) \begin{align} CT=\frac{1}{T\hat{\omega }}\sum _{t=1}^{T}\left(\frac{1}{\sqrt{T}}\sum _{j=1}^{t}\hat{u}_{j}^{+}\right)^{2},\end{align}

where $T$ is the sample size and $\hat{\omega }$ is a consistent estimator of the long-run variance of $\hat{u}_{t}^{+}$ . Wagner and Hong (Reference Wagner and Hong2016) showed that the $CT$ test statistic has a nuisance parameter-free limiting distribution and, thus, that the corresponding critical values can be calculated in the special case where only one integrated regressor has a polynomial relationship with an explained variable, such as in (1). The critical values are tabulated in Wagner (Reference Wagner2023). If the $CT$ test statistic is not significant, then (1) is regarded as a cointegrating relationship with a nonlinear impact of $x_{1t}$ .

3.2. Data

The sample period is from the first quarter of 2003 to the fourth quarter of 2019 because the bilateral US current account balance with China is only available beginning in 2003. In addition, it is likely that the pandemic crisis affected macroeconomic variables after 2019, especially life expectancy; thus, the sample period has the advantage of eliminating possible biases due to the pandemic crisis. The estimation results for the periods 2003–2016 and 2003–2019 are very similar. Therefore, the impact of the life expectancy gap is robust to the recent trade tensions between China and the US. The details of the data are reported in Appendix A. The empirical analysis incorporates the US current account balance, the life expectancy gap, and six control variables, as discussed below.

4.1. Unit root test

Before estimating (1), it is necessary to examine the stationarity of each of the variables introduced in Section 3 because the polynomial cointegration technique is applicable when the variables are integrated of order one. This paper uses the unit root test developed by Elliott et al. (Reference Elliott, Rothenberg and Stock1996), which is a modification of the Dickey–Fuller test (Dickey and Fuller, Reference Dickey and Fuller1979).

The results are reported in Table 1. The test does not reject the null hypothesis of a unit root for the variables in levels. However, the null hypothesis of a unit root is rejected for the variables in first differences. Therefore, all the variables are integrated of order one.Footnote ⁷ This finding is consistent with those of previous studies (e.g., Arghyrou and Chortareas, Reference Arghyrou and Chortareas2008).

Table 1. Unit root test

The Dickey–Fuller test with GLS detrending developed by Elliott et al. (Reference Elliott, Rothenberg and Stock1996) is used. The lag length of the regression is selected by the modified Akaike information criterion developed by Ng and Perron (Reference Ng and Perron2001). The test includes a linear trend and a constant term for variables in levels and a constant term for variables in first differences. The results remain unchanged if the test includes only a constant term for variables in levels. ** and * indicate significance at the 1% and 5% levels, respectively.

4.2. Test for polynomial cointegration

We can now examine the presence of polynomial cointegration. Following Wagner (Reference Wagner2015, Reference Wagner2023), the $CT$ test statistic for the null hypothesis of polynomial cointegration is calculated from the model with $m=1$ .Footnote ⁸

The test results are shown in Table 2. The $CT$ test statistic for the null hypothesis of quadratic polynomial cointegration between the life expectancy gap and the US current account balance with China is well below the 5% critical value of 0.2927, indicating that they have a nonlinear cointegrating relationship. It is also found that the other explanatory variables do not have nonlinear impacts on the US current account balance with China. Therefore, there is evidence in favor of our model in (1), in which only the life expectancy gap ( $x_{1t}$ ) has a quadratic polynomial relationship to the US current account balance with China ( $z_{t}$ ).

Table 2. Test for quadratic polynomial cointegration

** and * indicate that the null hypothesis of quadratic polynomial cointegration is rejected at the 1% and 5% significance levels, respectively. The critical values are tabulated in Wagner (Reference Wagner2023).

4.3. Estimation of polynomial cointegrating regression

Using the FMOLS method, we estimate (1) with all possible combinations of the control variables to check for the robustness of the nonlinear impact of the life expectancy gap. By doing so, we obtain 64 estimation results for this impact.Footnote ⁹ Similarly, (1) with $m=1$ (basic model) is estimated using both annual and quarterly data to check for robustness, while the larger models are not estimated using annual data because of their small sample size. Furthermore, following Wagner (Reference Wagner2015), we use the dynamic ordinary least squares (DOLS) method developed by Saikkonen (Reference Saikkonen1991) and Stock and Watson (Reference Stock and Watson1993) to estimate the same models for reference purposes.Footnote ¹⁰ Consequently, we obtain 130 estimation results in total, and the nonlinear impact of the life expectancy gap is robust for all cases. These results do not suffer from endogeneity and serial correlation biases because the FMOLS and DOLS methods are designed to correct for these biases.

For space reasons, Table 3 reports the complete estimation results of only the basic and full models, and the results for the other models are summarized in Table 4.Footnote ¹¹ For all cases, the FMOLS estimation results show that the coefficients on the life expectancy gap and its square are significantly negative and positive, respectively. Therefore, the life expectancy gap has a U-shaped relationship with the US current account balance with China. A similar U-shaped relationship holds if we use the DOLS method; the estimation results are reported in Supplementary Material B.

Table 3. Cointegrating polynomial regression model for the US current account balance with China

The estimation method is the FMOLS. Numbers within parentheses are heteroskedasticity and autocorrelation consistent (HAC) standard errors. The long-run variance is estimated by the quadratic spectral (QS) kernel, and the bandwidth parameter is selected by the procedure described in Andrews (Reference Andrews1991). ** and * indicate significance at the 1% and 5% levels, respectively.

Table 4. Robustness checks for the U-shaped impact of the life expectancy gap

$x_{2}$ is the growth rate difference in real domestic demand, $x_{3}$ is the real exchange rate, $x_{4}$ is the VIX, $x_{5}$ is the relative SMC-to-GDP ratio, $x_{6}$ is the growth rate difference in the working-age population, and $x_{7}$ is the difference in the fiscal balance-to-GDP ratio. The FMOLS method is used. Numbers within parentheses are HAC standard errors. ** indicates significance at the 1% level.

The fitted line of the basic model is presented in Figure 2.Footnote ¹² The threshold level of the life expectancy gap is calculated as $-\beta _{11}/2\beta _{12}$ based on (1) and is approximately 3.2, as shown in Table 3. Hence, the sign of the impact of the life expectancy gap is reversed when the gap narrows by 3.2 years or less, and this case is applied to the period after 2013. In other words, the catching-up of Chinese life expectancy first worsens the US deficit, but eventually, the increased US deficit is improved by the further catching-up of Chinese life expectancy after 2013. A more detailed discussion on this threshold level is provided in Section 5.

These findings are obtained after controlling for the major determinants of current account balances. The estimation results for the full model show that the US current account balance is improved by weak domestic demand, a real depreciation of the US dollar, higher market risk, a higher degree of financial market development, an increased working-age population, and an improved fiscal balance. These results are consistent with those in the existing empirical literature (e.g., Belke and Dreger, Reference Belke and Dreger2013; Chinn and Prasad, Reference Chinn and Prasad2003; De Santis and Lührmann, Reference De Santis and Lührmann2009; Gervais et al. Reference Gervais, Schembri and Suchanek2016; Gruber and Kamin, Reference Gruber and Kamin2007, Reference Gruber and Kamin2009; Lane and Milesi-Ferretti, Reference Lane and Milesi-Ferretti2012; Unger, Reference Unger2017).

4.4. Other analyses

For further robustness checks, this paper conducts three analyses. First, the fertility gap (defined as the US total fertility rate minus the Chinese total fertility rate) is added to the cointegrating polynomial regression model to control for fertility policies and related household behaviors. The U-shaped impact of the life expectancy gap is robust if the model includes the fertility gap. Second, we compare the performances of the linear and nonlinear models. The model performance improves remarkably by allowing for nonlinearity in the impact of the life expectancy gap. Third, we estimate the error correction model that includes the error correction term calculated as the residual in (1). The adjustment coefficient is significantly negative, and thus, the error correction mechanism works appropriately. These results are reported in Supplementary Materials D, E, and F.

6.1. Motivation and background

There is empirical evidence that the catching-up of Chinese life expectancy has a U-shaped impact on the US current account balance. Therefore, we address the next question: What is its underlying mechanism? Because our findings are new in the empirical literature, theoretical support is useful for better understanding the U-shaped impact. Our theoretical analysis uses an open-economy OLG model that is consistent with the literature on the macroeconomic impact of life expectancy (e.g., Aísa et al. Reference Aísa, Pueyo and Sanso2012; Backus et al. Reference Backus, Cooley and Henriksen2014; Hirazawa and Yakita, Reference Hirazawa and Yakita2017; Ito and Tabata, Reference Ito and Tabata2010; Li et al. Reference Li, Zhang and Zhang2007).

This analysis aims to provide a possible theoretical explanation for the U-shaped impact of the life expectancy gap. Therefore, we examine only the general equilibrium effect of foreign life expectancy in an open economy. Although a replication of the full dynamics of the US current account balance with China is also of great interest, our empirical analysis assesses the impact of the life expectancy gap separately from the other impacts by using many control variables. Hence, the theoretical analysis focuses on the mechanism of the U-shaped impact, and our model consists of variables closely related to life expectancy. This model is analytically tractable and serves to provide a clear understanding of the mechanism of this U-shaped impact.

The existence of such a U-shaped impact suggests that life expectancy has both positive and negative impacts on savings, and these impacts were discussed by Ito and Tabata (Reference Ito and Tabata2010). Specifically, agents who live longer have an incentive to save more for their retirement, while this effect is offset when population aging increases the social security burden and reduces disposable income. To extend their model, this paper uses model elements developed by Aísa et al. (Reference Aísa, Pueyo and Sanso2012) and Ludwig and Vogel (Reference Ludwig and Vogel2010) and incorporates elderly labor supply into the model. We show that this extension is essential for theoretically describing the U-shaped impact.

A principal feature of the elderly is lower productivity because declines in physical health and cognitive ability generally occur in old age (Fries, Reference Fries1980). As Aísa et al. (Reference Aísa, Pueyo and Sanso2012) pointed out, given that better health mitigates a decline in elderly productivity, life expectancy has yet another negative impact on savings because agents who earn more in old age save less over their working lives. Furthermore, improved elderly productivity increases the aggregate labor supply. Consequently, life expectancy affects both capital and labor inputs in our model, and this composite effect lowers the capital–labor ratio, which determines the rate of return on capital and thus international capital flows. This effect is expected to be greater in China and the US, where elderly people work longer, compared to other countries. For example, in 2010, the labor force participation rate of those aged 65 years and over was 21.1% for China and 17.4% for the US, both values of which were higher than the OECD average of 12.5% (source: OECD).Footnote ¹⁸

Krueger and Ludwig (Reference Krueger and Ludwig2007) also used an open-economy OLG model with an elderly labor supply and presented simulation results for the welfare effects of population aging. Although our model is consistent with their model, a welfare analysis is not conducted in this paper because our focus is on the U-shaped impact of the life expectancy gap. Instead, to better understand our empirical results, we derive an analytical expression for the general equilibrium effect of foreign life expectancy on the home current account balance.

A theoretical analysis of the US current account balance with China was conducted by Eugeni (Reference Eugeni2015). The analytical results based on a two-country OLG model showed that China’s social security reform reduces the US deficit. We extend this study by examining the impacts of life expectancy and the elderly labor supply, which are also important from a social security perspective because these variables affect the social security burden in China.

Coeurdacier et al. (Reference Coeurdacier, Guibaud and Jin2015) used an open-economy OLG model with household credit constraints and indicated that the divergence in savings (and thus current account balances) between China and the US is strongly explained by heterogeneous household credit constraints and their interaction with growth differentials. Our analysis is complementary to that of Coeurdacier et al. (Reference Coeurdacier, Guibaud and Jin2015). Although our model does not include financial frictions, it provides a theoretical foundation for the U-shaped impact of the life expectancy gap, which is also an important determinant of the US current account balance with China, as shown in Section 4.

6.2. Basic setup

We use a two-country, one-good, two-period OLG model with lifetime uncertainty. The world consists of a home country and a foreign country, and they are linked through an integrated commodity market and an international capital market. However, the domestic labor market is closed. Although the structure of the home country is explained in this section, the structure of the foreign country can be discussed in the same way as that of the home country. The foreign variables and parameters are denoted by an asterisk.

Our model assumes that a new generation, called generation $t$ , is born in each period $t=1,2,3,\ldots$ . Generation $t$ is composed of a continuum of $N_{t}\gt 0$ units of identical agents. In period 1, there are $N_{0}$ units of initial old agents who live only in period 1. Let $n\gt 0$ be the population growth rate, and thus $N_{t}=(1+n)N_{t-1}$ . The length of lifetime of agents is uncertain, and they live for a maximum of two periods, young and old age. An agent in the home country survives to old age with a probability of $p\in [0,1]$ and dies at the beginning of old age with a probability of $1-p$ . Therefore, in each period, there are $N_{t}$ units of young agents and $pN_{t-1}$ units of old agents. A rise in the survival probability $p$ leads to population aging in the home country because it increases the share of old agents. Hence, the survival probability corresponds to life expectancy (e.g., Aísa et al. Reference Aísa, Pueyo and Sanso2012; Ferrero, Reference Ferrero2010; Hirazawa and Yakita, Reference Hirazawa and Yakita2017; Li et al. Reference Li, Zhang and Zhang2007). Indeed, the correlation coefficient between the survival rate to age 65 years and life expectancy at birth for the period 1960–2018 is 0.999 for China and 0.992 for the US (source: World Development Indicators of the World Bank).

Under these assumptions, the old-age dependency ratio is given by

(3) \begin{align} \frac{pN_{t-1}}{N_{t}}=\frac{p}{1+n}. \end{align}

Therefore, the effect of the old-age dependency ratio can be decomposed into the effects of life expectancy and the growth rate of the working-age population.Footnote ¹⁹ For this reason, our empirical analysis uses these two variables instead of the old-age dependency ratio.

The theoretical model is used to examine how the home current account balance responds to the reduction in the life expectancy gap $p-p^{*}$ caused by an increase in $p^{*}$ . To describe the catching-up process of foreign life expectancy, we restrict our analysis to the case where the parameter condition $p^{*}\lt p$ holds. We also assume that the home and foreign countries have the same population growth rate. This assumption is helpful for obtaining clear analytical results for the impact of the life expectancy gap.Footnote ²⁰

6.3. Households

Households derive utility from their own consumption in young and old age. In this paper, the expected lifetime utility of the representative agent in generation $t$ is defined as

(4) \begin{align} u_{t}=\ln \left(c_{1,t}\right)+p\ln \left(c_{2,t+1}\right), \end{align}

where $c_{1,t}$ and $c_{2,t+1}$ denote consumption in young and old age, respectively.

Young agents in generation $t$ have one unit of time endowment. They inelastically supply one unit of effective labor and obtain wage income $w_{t}$ . Part of the wage income is devoted to current consumption $c_{1,t}$ and the payment of taxes $\tau _{t}w_{t}$ , where $\tau _{t}$ is the rate of tax levied on wage income, and the rest is devoted to savings $s_{t}$ .

Following Ito and Tabata (Reference Ito and Tabata2010), we assume the existence of an actuarially fair insurance company developed by Yaari (Reference Yaari1965). This company collects funds and invests them in firms or foreign countries. Because individuals do not migrate, the insurance contracts between the two countries differ. Returns on investment are distributed uniformly among insured surviving old agents. Therefore, those in the home and foreign countries receive payments $R_{t+1}s_{t}/p$ and $R_{t+1}s_{t}^{*}/p^{*}$ in exchange for $s_{t}$ and $s_{t}^{*}$ , respectively.Footnote ²¹ $R_{t+1}$ is the gross rate of interest, and an insurance contract is more attractive to agents as long as the survival probability is less than 1.

Old agents in generation $t$ also have one unit of time endowment and consume their entire wealth. Following Ludwig and Vogel (Reference Ludwig and Vogel2010), this paper assumes that old agents work a given fraction $\omega$ of their time. For the rest of their time $(1-\omega )$ , they are retired and receive pension benefits $b_{t+1}$ . Hence, $\omega$ can be regarded as a measure of retirement age, and the length of the working period in the lifetime is given by $1+\omega$ . Another interpretation of $\omega$ is that it is the proportion of old agents who work (i.e., the labor force participation rate of the elderly) because of the assumption of identical agents. Either way, $\omega$ operates as a policy variable (Ludwig and Vogel, Reference Ludwig and Vogel2010), and a similar assumption was used by Echevarría (Reference Echevarría2004), Inagaki (Reference Inagaki2021), and Krueger and Ludwig (Reference Krueger and Ludwig2007).

Let $\delta$ be the fraction of young productivity maintained in the second period of life. Old agents in generation $t$ inelastically supply $\delta \omega$ units of effective labor, obtain wage income $\delta \omega w_{t+1}$ , and pay taxes ${\tau _{t+1}}\delta \omega w_{t+1}$ . In this paper, $\delta$ is specified as

(5) \begin{align} \delta =p^{\gamma }, \end{align}

where $\gamma$ is a positive parameter. This specification is essentially the same as that developed by Aísa et al. (Reference Aísa, Pueyo and Sanso2012) and Hirazawa and Yakita (Reference Hirazawa and Yakita2017). Because $\delta \leq 1$ , productivity in old age is lower than that in young age. However, the productivity decline in old age is mitigated by an increase in life expectancy $p$ (i.e., improvement in health). This phenomenon is consistent with the theory of the compression of morbidity developed in the field of health and medical sciences (e.g., Fries, Reference Fries1980). Furthermore, we show that this assumption holds empirically (Appendix B).

Based on these settings, the lifetime budget constraints of the representative agent in generation $t$ are expressed as

(6) \begin{align} c_{1,t}+s_{t}&=\left(1-\tau _{t}\right)w_{t}, \end{align}

(7) \begin{align} c_{2,t+1}&=\frac{R_{t+1}}{p}s_{t}+\left(1-\tau _{t+1}\right)\delta \omega w_{t+1}+\left(1-\omega \right)b_{t+1}. \end{align}

Maximizing (4) subject to (6) and (7) gives the following savings equation:

(8) \begin{align} s_{t}=\frac{p}{1+p}\left[\left(1-\tau _{t}\right)w_{t}-\left\{\frac{\left(1-\tau _{t+1}\right)\delta \omega w_{t+1}+\left(1-\omega \right)b_{t+1}}{R_{t+1}}\right\}\right]. \end{align}

This equation shows that a higher life expectancy $p$ increases savings because it means a higher probability of enjoying consumption in old age. However, a greater earning ability in old age $\delta$ decreases savings, which suggests that young agents who expect to earn more in old age do not have incentives to save more.

6.4. Government

The government runs a social security system. The government budget is assumed to be balanced in each period. Therefore, total contributions by workers are equal to total pension payments as

(9) \begin{align} \tau _{t}w_{t}N_{t}+\tau _{t}w_{t}\delta \omega pN_{t-1}=b_{t}\left(1-\omega \right)pN_{t-1}.\end{align}

Following Bárány et al. (Reference Bárány, Coeurdacier and Guibaud2018), Ito and Tabata (Reference Ito and Tabata2010), Krueger and Ludwig (Reference Krueger and Ludwig2007), and Ludwig and Vogel (Reference Ludwig and Vogel2010), our model incorporates a pension scheme where the calculation of payments uses a replacement rate. Let us assume that $b_{t}=\phi (1-\tau _{t})w_{t}$ , where $\phi \in [0,1)$ is the replacement rate.Footnote ²² Then, the tax rate is given by

(10) \begin{align} \tau _{t}=\frac{\left(1-\omega \right)\phi p}{1+n+\delta \omega p+\left(1-\omega \right)\phi p}\equiv \tau. \end{align}

From this equation, we have that

(11) \begin{align} \frac{d\tau }{dp}=\underset{+}{\underbrace{\frac{\partial \tau }{\partial p}}}+\underset{-}{\underbrace{\overset{-}{\overbrace{\frac{\partial \tau }{\partial \delta }}}\overset{+}{\overbrace{\frac{\partial \delta }{\partial p}}}}}. \end{align}

An increase in life expectancy increases the tax rate because it increases the number of pensioners, while an improvement in elderly productivity eases the tax burden per worker and decreases the tax rate. Using (5) and (10), we obtain the following:

(12) \begin{align} \frac{d\tau }{dp}=\frac{\left(1+n-\gamma \delta \omega p\right)\left(1-\omega \right)\phi }{\left[1+n+\delta \omega p+\left(1-\omega \right)\phi p\right]^{2}}\gt 0. \end{align}

Life expectancy is positively associated with the tax rate unless $\gamma$ in (5) is large enough. The numerical analysis in Section 7 shows that the positive sign in (12) holds.

6.5. Firms

Firms produce output using standard Cobb–Douglas technology as

(13) \begin{align} Y_{t}=\overline{A}K_{t}^{\alpha }L_{t}^{1-\alpha }, \end{align}

where $Y_{t}$ is aggregate output, $\overline{A}$ is a technology parameter, $K_{t}$ is aggregate capital, $L_{t}$ is aggregate labor, and $\alpha \in (0,1)$ . Following Ludwig and Vogel (Reference Ludwig and Vogel2010), $L_{t}$ is given by the sum of effective labor inputs supplied by young and old agents as

(14) \begin{align} L_{t}=N_{t}+\delta \omega pN_{t-1}=\left(\frac{\varepsilon }{1+n}\right)N_{t}, \end{align}

where $\varepsilon \equiv 1+n+\delta \omega p$ . Therefore, we have that $L_{t}=(1+n)L_{t-1}$ .

The existing capital stock is fully used for old-age consumption, and the gross interest rate $R_{t}$ contains a 100% depreciation rate. Assuming that firms operate in perfect competition, the first-order conditions for profit maximization are

(15) \begin{align} w_{t}=\left(1-\alpha \right)\overline{A}k_{t}^{\alpha }, \end{align}

(16) \begin{align} R_{t}=\alpha \overline{A}k_{t}^{\alpha -1}, \end{align}

where $k_{t}\equiv K_{t}/L_{t}$ .

6.6. Equilibrium

Assuming the free mobility of capital, the rates of return on capital in the home and foreign countries converge to one world price $R_{t}^{W}$ , where $R_{t}^{W}=R_{t}=R_{t}^{*}$ . This means that $k_{t}=k_{t}^{*}$ and $w_{t}=w_{t}^{*}$ from (15) and (16).

Let $N_{t}^{W}\equiv N_{t}+N_{t}^{*}$ and $L_{t}^{W}\equiv L_{t}+L_{t}^{*}$ . Because $N_{t}$ , $N_{t}^{*}$ , $L_{t}$ , and $L_{t}^{*}$ grow at the same rate $n$ , we have that $N_{t}^{W}=(1+n)N_{t-1}^{W}$ and $L_{t}^{W}=(1+n)L_{t-1}^{W}$ . The market clearing condition for world capital is $K_{t+1}^{W}=s_{t}N_{t}+s_{t}^{*}N_{t}^{*}$ , where $K_{t+1}^{W}\equiv K_{t+1}+K_{t+1}^{*}$ . Dividing both sides of this condition by $N_{t}^{W}$ gives

(17) \begin{align} k_{t+1}^{W}\left[\varepsilon \theta +\varepsilon ^{*}\left(1-\theta \right)\right]=s_{t}\theta +s_{t}^{*}\left(1-\theta \right)\equiv s_{t}^{W}, \end{align}

where $k_{t}^{W}\equiv K_{t}^{W}/L_{t}^{W}$ and $\theta \equiv N_{t}/N_{t}^{W}=N_{0}/N_{0}^{W}$ . The world share of the home population $\theta$ is derived using the facts that $N_{t}=(1+n)^{t}N_{0}$ and $N_{t}^{W}=(1+n)^{t}N_{0}^{W}$ . Furthermore, from the definition of $K_{t}^{W}$ , we have that $k_{t}^{W}=\theta k_{t}+(1-\theta )k_{t}^{*}$ . Therefore, $k_{t}=k_{t}^{*}=k_{t}^{W}$ holds when $k_{t}=k_{t}^{*}$ .

Let us assume that firms in the home and foreign countries have the same production technology. Then, by substituting (8), (10), (15), (16), $b_{t}=\phi (1-\tau _{t})w_{t}$ , and those of the foreign country counterparts into (17) and taking $k_{t}=k_{t}^{*}=k_{t}^{W}$ into account, we can describe the dynamics of $k_{t}^{W}$ as

(18) \begin{align} k_{t+1}^{W}={\Gamma} \left(k_{t}^{W}\right)^{\alpha }, \end{align}

where

\begin{equation*} {\Gamma} \equiv \frac{\theta \lambda +\left(1-\theta \right)\lambda ^{*}}{\theta \mu +\left(1-\theta \right)\mu ^{*}}, \end{equation*}

\begin{equation*} \lambda \equiv \frac{p}{1+p}\left(1-\tau \right)\left(1-\alpha \right)\overline{A}, \end{equation*}

\begin{equation*} \mu \equiv \frac{p}{1+p}\frac{1-\alpha }{\alpha }\left(1-\tau \right)\left[\delta \omega +\left(1-\omega \right)\phi \right]+\varepsilon, \end{equation*}

\begin{equation*} \lambda ^{*}\equiv \frac{p^{*}}{1+p^{*}}\left(1-\tau ^{*}\right)\left(1-\alpha \right)\overline{A}, \end{equation*}

\begin{equation*} \mu ^{*}\equiv \frac{p^{*}}{1+p^{*}}\frac{1-\alpha }{\alpha }\left(1-\tau ^{*}\right)\left[\delta ^{*}\omega ^{*}+\left(1-\omega ^{*}\right)\phi ^{*}\right]+\varepsilon ^{*} . \end{equation*}

Therefore, the steady-state capital–labor ratio in an open economy $k^{W}$ is given by

(19) \begin{align} k^{W}={\Gamma} ^{\frac{1}{1-\alpha }}. \end{align}

If $\theta =1$ , the world economy consists of only the agents of the home country. In this case, (19) is interpreted as the steady-state capital–labor ratio in the closed home economy $k_{c}$ , which is expressed as

(20) \begin{align} k_{c}={\Psi} ^{\frac{1}{1-\alpha }},\end{align}

where ${\Psi} \equiv \lambda /\mu$ . Similarly, when $\theta =0$ , the steady-state capital–labor ratio in the closed foreign economy $k_{c}^{*}$ is determined as

(21) \begin{align} k_{c}^{*}={{\Psi} ^{*}}^{\frac{1}{1-\alpha }},\end{align}

where ${\Psi} ^{*}\equiv \lambda ^{*}/\mu ^{*}$ .

6.7. Current account balances

When the economies are opened to trade, the current account balance in the home country equals the change in net foreign assets as

(22) \begin{align} G_{t}=\left(A_{t+1}-K_{t+1}\right)-\left(A_{t}-K_{t}\right), \end{align}

where $A_{t}\equiv s_{t-1}N_{t-1}$ denotes the wealth in the home country, which includes opportunities to acquire claims on foreign capital (Geide-Stevenson, Reference Geide-Stevenson1998). Using (6), (7), and (9) and the assumption of perfect competition, (22) can be rewritten as

(23) \begin{align} G_{t}=Z_{t}+\left(R_{t}-1\right)\left(A_{t}-K_{t}\right), \end{align}

where $Z_{t}\equiv Y_{t}-c_{1t}N_{t}-c_{2t}pN_{t-1}-K_{t+1}$ is the trade balance in the home country. Hence, the current account balance in this model consists of the trade balance and net primary income.

Let $g_{t}\equiv G_{t}/L_{t}$ and $a_{t}\equiv A_{t}/L_{t}$ . Then, the steady-state value of the current account balance per unit of effective labor $g$ is given by

(24) \begin{align} g=n\left(a-k^{W}\right), \end{align}

where $a=s/\varepsilon$ is the steady-state value of $a_{t}$ , and $s$ is the steady-state value of $s_{t}$ .Footnote ²³

7.1. Analytical results

The comparative static analysis based on the steady-state equilibrium provides useful information on the properties of our model, which can be effectively used to understand the mechanism of the U-shaped impact of the life expectancy gap on the home current account balance.Footnote ²⁴ In the model, the pattern of international capital flows is determined by the difference in the capital–labor ratio in the closed economies. For example, $k_{c}^{*}\gt k_{c}$ means that $R^{*}\lt R$ from (16). Thus, after the economies are opened to trade, young agents in the foreign country invest part of their savings in the home country to obtain a higher rate of return. The home country imports capital from the foreign country and runs a current account deficit in its equilibrium (i.e., $g\lt 0$ ). Similarly, $g\gt 0$ holds when $k_{c}^{*}\lt k_{c}$ . Therefore, to examine the impact of $p^{*}$ on $g$ , we first calculate the change in $k_{c}^{*}$ based on the derivative of $k_{c}^{*}$ with respect to $p^{*}$ . Then, the resulting level of $k_{c}^{*}$ is compared with the level of $k_{c}$ . This procedure is the same as that used in Ito and Tabata (Reference Ito and Tabata2010).

Taking the derivative of $k_{c}^{*}$ with respect to $p^{*}$ gives

(25) \begin{align} \frac{dk_{c}^{*}}{dp^{*}}=\underset{+}{\underbrace{\frac{\partial k_{c}^{*}}{\partial p^{*}}}}+\underset{-}{\underbrace{\overset{-}{\overbrace{\frac{\partial k_{c}^{*}}{\partial \tau ^{*}}}}\overset{+}{\overbrace{\frac{d\tau ^{*}}{dp^{*}}}}}}+\underset{-}{\underbrace{\overset{-}{\overbrace{\frac{\partial k_{c}^{*}}{\partial \varepsilon ^{*}}}}\overset{+}{\overbrace{\frac{\partial \varepsilon ^{*}}{\partial p^{*}}}}}}+\underset{-}{\underbrace{\overset{-}{\overbrace{\frac{\partial k_{c}^{*}}{\partial \delta ^{*}}}}\overset{+}{\overbrace{\frac{d\delta ^{*}}{dp^{*}}}}+\overset{-}{\overbrace{\frac{\partial k_{c}^{*}}{\partial \varepsilon ^{*}}}}\overset{+}{\overbrace{\frac{\partial \varepsilon ^{*}}{\partial \delta ^{*}}}}\overset{+}{\overbrace{\frac{d\delta ^{*}}{dp^{*}}}}}}.\end{align}

The proof is provided in Appendix C. This equation shows that foreign life expectancy affects the home current account balance through four channels. First, from (8), higher savings promote capital accumulation and increase the capital–labor ratio (first term: savings effect). Second, from (12), an increased tax rate ( $\tau ^{*}$ ) leads to lower savings and thus a lower capital–labor ratio (second term: tax effect). Third, from (14), aggregate labor (a linear function of $\varepsilon ^{*}$ ) increases as the number of old workers ( $\omega ^{*}p^{*}N_{t-1}^{*}$ ) increases, which decreases the capital–labor ratio (third term: labor quantity effect). Fourth, from (5), (8), and (14), an improvement in elderly productivity ( $\delta ^{*}$ ) decreases savings (fourth term) and increases aggregate labor (fifth term), and this composite effect decreases the capital–labor ratio (sum of the fourth and fifth terms: productivity effect). The total effect $dk_{c}^{*}/dp^{*}$ varies in accordance with the sizes of these effects.

7.2. Numerical analysis based on the analytical results

Because it is very difficult to analytically specify the sign of the total effect in (25), this paper adopts a numerical comparative static analysis. The parameters based on the data for China and the US are reported in Table 6.Footnote ²⁵ To describe the catching-up process of foreign life expectancy, the initial conditions are given by $p=0.87$ and $p^{*}=0.81$ , and then, only $p^{*}$ is increased to 0.86. This assumption is based on the fact that the survival rate to age 65 years in China was 80.37% in 2003 and increased to 86.16% in 2018 (source: World Development Indicators of the World Bank). Using these parameters, we obtain the numerical results for the analytical expressions of the four effects in (25).

Table 6. Parameters for numerical analysis

To solve the model analytically, the capital share, technology parameter, and population growth rate are assumed to be the same between the home and foreign countries. To calculate the capital share, the average of the US labor share of income ( $1-\alpha$ ) for the period 2000–2016 is used, and the source is Giandrea and Sprague (Reference Giandrea and Sprague2017). Population share $\theta$ is defined as the ratio of the US population aged 25–64 years to the sum of the Chinese and US populations aged 25–64 years, and the average value over the period 2003–2019 is used. The average of the annual growth rates of the population aged 25–64 years for China and the US over the period 2003–2019 is 0.0091, and the parameter is calculated as $1.0091^{40}-1$ . The source of these data is the World Development Indicators of the World Bank. The technology parameter is the same as that used in Ito and Tabata (Reference Ito and Tabata2010). For the labor force participation rate of people aged 65 years and over, the data for China (foreign country) in 2010 and the US (home country) in 2019 are used because of data availability. The source of these data is the OECD. The US pension replacement rate in 2018 is used because of data availability, and the source is the OECD. Similarly, the pension replacement rate for urban employees in China in 2015 is used, and the source is Gongcheng (Reference Gongcheng2017). Because Chinese data for estimating $\gamma ^{*}$ are not available, the numerical analysis assumes that $\gamma =\gamma ^{*}$ . The estimation procedure for this parameter is explained in Appendix B.

The results of the numerical comparative static analysis are presented in Table 7. The total effect is positive for $p^{*}\leq 0.84$ but negative for $p^{*}\gt 0.84$ . Hence, $k_{c}^{*}$ increases at first but then decreases when the life expectancy gap $p-p^{*}$ is narrowed by an increase in $p^{*}$ .Footnote ²⁶ Corresponding changes in the home current account balance (per unit of effective labor) $g$ are described in Figure 5, in which $p-p^{*}$ has a U-shaped relationship with $g$ .Footnote ²⁷ Using these results, we can provide a theoretical foundation for the impact of the life expectancy gap as follows.

Table 7. Numerical comparative statics: effect of $p^{*}$ on $k_c^{*}$ based on (25)

The parameters reported in Table 6 are used. The initial condition is given by $p^{*}=0.81$ , and then, only $p^{*}$ is increased to describe the catching-up process of foreign life expectancy.

Figure 5. Catching-up of foreign life expectancy and its impact on the home current account balance. The parameters reported in Table 6 are used. The initial conditions are given by $p=0.87$ and $p^{*}=0.81$ , after which only $p^{*}$ increases.

First, the savings effect is dominant (Table 7); thus, $k_{c}^{*}$ increases as $p^{*}$ increases, which means that an increase in $p^{*}$ leads to an increase in $s^{*}$ from (8) and promotes capital accumulation. A subsequent decrease in $R^{*}$ leads to larger capital flows to the home country, and thus, $g$ decreases as $p^{*}$ starts to catch up with $p$ (point A in Figure 5). Next, $g$ reaches the turning point when $p^{*}$ is approximately 0.842 (point B in Figure 5). Finally, $k_{c}^{*}$ decreases in the middle of the convergence process of $p^{*}$ because the sum of the tax, labor quantity, and productivity effects exceeds the savings effect (Table 7). Consequently, further narrowing of the life expectancy gap $p-p^{*}$ improves $g$ (point C in Figure 5). Importantly, the survival rate to age 65 years in China exceeded the threshold level of 0.842 in 2013. This result is consistent with the empirical evidence that the life expectancy gap has been below the threshold level since 2013 (see Section 4.3).

It is also found that the elderly labor supply is an essential factor for the reversal of $g$ because the total effect in (25) is always positive if the labor quantity and productivity effects are excluded from the model (see Table 7). To examine the importance of the elderly labor supply, we calculate $g$ under the assumption that $\omega =\omega ^{*}=0$ (i.e., zero participation of old workers). The results are presented in Figure 6. We find that $g$ does not improve even though the life expectancy gap $p-p^{*}$ is narrowed by an increase in $p^{*}$ . Therefore, the threshold level of the life expectancy gap depends on the labor force participation of the elderly. The empirical analysis for the impacts of $\omega$ and $\omega ^{*}$ is of interest but is very difficult in the case of this study because the data on the labor force participation rate of the elderly in China are available per 10-year period, and the sample size is 1 for the period 2003–2019.

Figure 6. Relationship between $g$ and $p-p^{*}$ under the assumption that $\omega =\omega ^{*}=0$ . The labor force participation rates of the elderly in the home and foreign countries ( $\omega$ and $\omega ^{*}$ ) are assumed to be zero. The other parameters are the same as those in Table 6. The initial conditions are given by $p=0.87$ and $p^{*}=0.81$ , after which only $p^{*}$ increases.

7.3. Discussion on savings and productivity effects

The numerical comparative static analysis in Table 7 indicates that the sizes of the savings and productivity effects are particularly large among the four effects of life expectancy in (25). Because the signs of the savings and productivity effects are opposite, the U-shaped relationship between $p-p^{*}$ and $g$ suggests that these effects have different timings of their peaks. Specifically, given a continuous increase in $p^{*}$ , the savings effect works immediately, and $g$ increases at first. However, in the late stage of the catching-up process, the productivity effect becomes apparent, and $g$ starts to decrease. The difference between the two effects is shown in Figure 7 (Panel A). A discussion on this difference is helpful for better understanding the mechanism of the U-shaped relationship between $p-p^{*}$ and $g$ .

Figure 7. Savings and productivity effects. The parameters reported in Table 6 are used.

From Table 7, it can be seen that the productivity effect is a major cause of the reversal of $g$ . Elderly productivity is determined by (5), and we obtain an estimate of 3.394 for $\gamma$ (see Appendix B). Therefore, the relationship between elderly productivity and life expectancy is described in Figure 7 (Panel B). Because elderly productivity remains low for a while after life expectancy starts to rise, the productivity effect does not work immediately. Conversely, elderly productivity improves remarkably when life expectancy becomes sufficiently high. This situation holds in the late stage of the catching-up process of life expectancy, and improved elderly productivity amplifies the productivity effect. Intuitively, this means that good health is required for elderly people to work well.

In contrast to the productivity effect, the savings effect peaks when life expectancy is sufficiently low (Figure 7, Panel A). Young agents have little savings if their life expectancy is extremely low. Therefore, the results in Figure 7 suggest that young agents who save less have incentives to save more in response to an increase in life expectancy. Given a continuous increase in life expectancy, the savings effect is positive but decreases, which implies that savings increase at a decreasing rate, with other things being equal. Therefore, in the late stage of the catching-up process of life expectancy, young agents have more savings, and the savings effect becomes much smaller than its peak. This characteristic is opposite that of the abovementioned productivity effect, and this difference promotes the reversal of $g$ in the case of our model.