Population age structure and consumption growth: evidence from National Transfer Accounts

Abstract

We assess the effects of the population age structure and the population dynamics on economic growth. Following recent research, we focus on the generational turnover effect to characterize the influence of birth and death rates, depending on the age profile of individual consumption, the extent of annuity market imperfections, and the willingness of households to shift consumption over time. Using data from the National Transfer Accounts on age profiles of consumption for a number of different countries, we assess—in a comparative way—the sign and the magnitude of the generational turnover effect and its impact on economic growth. We find considerable cross-country differences and trace them back to the underlying variation in demography and in the age structure of consumption.

Appendices

Appendix A: Derivations

1.1 A.1 The aggregate Euler equation

Maximizing Eq. 1 subject to Eq. 2 implies that the optimal consumption path of an individual is characterized by

$$ \frac{\dot{c}(t_{0},t)}{c(t_{0},t)}=\frac{r(t)-\rho -\left( 1-\theta \right) \mu \left( t-t_{0}\right) }{\sigma }, $$

(5)

which corresponds to the standard Euler equation in case of complete annuity markets (𝜃 = 1). By contrast, uninsured mortality within an incomplete annuity market (𝜃 < 1) implies a downward drag on consumption growth. This is the more realistic case on which we focus our attention: as μ(t − t ₀) becomes large for age a = t − t ₀ approaching D, it follows that consumption declines toward the end of life. Given that the interest rate exceeds the rate of time preference, this will generate the hump-shaped age profiles of consumption that are typically observed in the data (see Hansen and Imrohoroglu 2008; Heijdra and Mierau 2012).

We can write aggregate consumption as

$$C\left( t\right) =N\left( t\right) {\int}_{t-D}^{t}n\left( t_{0},t\right) c\left( t_{0},t\right) dt_{0}, $$

where $n\left (t_{0},t\right ) =\beta e^{-\pi \left (t-t_{0}\right ) -M\left (t-t_{0}\right ) }$ is the population share at time t of the cohort born at time t ₀, given the survival function $e^{-M\left (t-t_{0}\right ) }$. Differentiating with respect to time, we obtain

$$\begin{array}{@{}rcl@{}} \overset{\cdot }{C}\left( t\right) & \,=\,&\pi C\left( t\right) \,+\,N\left( t\right) \left\{ \beta c\left( t,t\right) \,+\,{\int}_{t-D}^{t}\left[ \,-\,\left( \pi \,+\,\mu \left( t\!-t_{0}\right) \right) \,+\,\frac{\overset{\cdot }{c}(t_{0},t)}{c(t_{0},t) }\right] n\left( t_{0},t\right) c(t_{0},t)dt_{0}\right\} \\ & =&\frac{r-\rho }{\sigma}C(t)+N(t)\left\{ \beta c(t,t)-\left[ 1+\frac {1-\theta }{\sigma}\right] {\int}_{t-D}^{t}\mu (t-t_{0})n(t_{0},t)c(t_{0},t)dt_{0}\right\} , \end{array} $$

where we use $\dot {N}\left (t\right ) =\pi N\left (t\right )$. Defining average consumption across the deceased by

$$c^{\dag }\left( t\right) :=\frac{1}{\overline{\mu }}{\int}_{t-D}^{t}\mu \left( t-t_{0}\right) c(t_{0},t)n\left( t_{0},t\right) dt_{0} $$

allows us to write the aggregate Euler equation as

$$\frac{\dot{C}(t)}{C(t)}=\frac{r-\rho }{\sigma}+{\Omega} \left( t\right) , $$

where

$$ {\Omega} \left( t\right) :=-\overline{\mu }\left\{ \left[ 1+\frac{1-\theta}{\sigma } \right] \frac{c^{\dag }\left( t\right) }{c\left( t\right) }-\frac{ \beta }{\overline{\mu }}\frac{c\left( t,t\right) }{c\left( t\right) }\right \} $$

(6)

with c(t) = C(t)/N(t) being per capita consumption.

1.2 A.2 The balanced growth path for an AK production structure

Following Mierau and Turnovsky (2014a), we assume a production function

$$Y_{i}(t)=A(t)K_{i}(t)^{\alpha }L_{i}(t)^{1-\alpha }, $$

where Y _i is output of firm i, K _i (t) is its capital stock, L _i (t) is its employment level, α ∈ [0, 1] denotes the elasticity of output with respect to capital, and A(t) = A k(t)^1−α refers to the productivity index as a function of learning-by-doing spillovers. In this case, the aggregate production function is of the AK type. Denoting the rate of depreciation by δ ∈ [0, 1] , we obtain r(t) = α A − δ and, thus, a balanced growth path of

$$g=\frac{\dot{k}\left( t\right) }{k\left( t\right) }=A-\delta -\pi -\frac{ c\left( t\right) }{k\left( t\right) }=\frac{1}{\sigma }\left[ \alpha A-\delta -\rho \right] +{\Omega} \left( t\right) -\pi =\frac{\dot{c}\left( t\right) }{c\left( t\right) }. $$

In this case, an increase in the generational turnover effect, Ω(t), would also predict an increase in the growth rate of capital accumulation and, thus, an increase in the gross savings rate and in the per capita growth rate of the economy. Solving the interior equation yields

$$\frac{c\left( t\right) }{k\left( t\right) }=\left( 1-\frac{\alpha }{\sigma } \right) A-\left( 1-\frac{1}{\sigma }\right) \delta +\frac{\rho }{\sigma } -{\Omega} \left( t\right)\!, $$

implying a negative relationship between the consumption/capital ratio and the generational turnover effect, Ω(t).

Appendix B: Data

The National Transfer Accounts (NTA) project consists of research teams in more than 40 countries. The teams are measuring how people at each age consume and save. The particular advantage of NTA data over other data is that they include the evolution of intergenerational transfers. Hence, the data allow to keep track of the true underlying consumption-savings profiles over the life cycle. For a detailed description of the NTA project and the data, see: http://www.ntaccounts.org/web/nta/show/.

Figure 1 contains the private consumption level of each age group (from birth to 89 years in 1-year steps and for the group 90 + ) divided by average labor income at ages 30 to 49 for the following countries for which the necessary data of both sources (the National Transfer Accounts and the Human Mortality Database) were available: Austria, Chile, Finland, Germany, Japan, Slovenia, Spain, Sweden, Taiwan, and the USA. We clearly observe a steep consumption-age profile for the USA, while the profile is rather flat for Taiwan.

Fig. 1

Age-specific consumption profiles of the countries included in our analysis

Appendix C: Robustness

To assess the sensitivity of the results with respect to the elasticity of intertemporal substitution, we present the results regarding the generational turnover effect and the growth drag for the cases of σ = 1, σ = 2, and σ = 4 in Tables 3, 4, and 5. The cases σ = 1 and σ = 4 can be considered as the lower and upper bounds of reasonable estimates of σ (cf. Chetty 2006; Guvenen 2006). We see that neither the generational turnover effect nor the growth drag change their signs. However, we see that the larger is σ (which implies a lower elasticity of intertemporal substitution), the smaller is the negative effect of incomplete annuitization on economic growth. Note that this is intuitively clear because, ceteris paribus, a higher elasticity of intertemporal substitution implies a stronger drag of mortality on individual consumption change.

Table 3 Generational turnover and growth drag for $\protect \sigma =1$

Table 4 Generational turnover and growth drag for $\protect \sigma =2$

Table 5 Generational turnover and growth drag for $\protect \sigma =4$