Longevity-induced vertical innovation and the tradeoff between life and growth

Abstract

We analyze the economic growth effects of rising longevity in a framework of endogenous growth driven by quality-improving innovations. A rise in longevity increases savings and thereby places downward pressure on the market interest rate. Since the monopoly profits generated by a successful innovation are discounted by the endogenous market interest rate, this raises the net present value of innovations, which, in turn, fosters R&D investments. The associated increase in the employment of scientists leads to faster technological progress and a higher long-run economic growth rate. From a welfare perspective, the direct effect of an increase in life expectancy tends to be larger than the indirect effect of the induced higher consumption due to faster economic growth. Consequently, the debate on rising health care expenditures should not be predominantly based on the growth effects of health care.

Appendix: The model solution for a continuum of firms in the intermediate goods sector

The model solution in the main text refers to a setting in which output is produced by using only one intermediate good as input. In this section, we show that the central results remain valid when assuming a continuum of intermediate goods sectors instead (Aghion and Howitt 1999). To be able to derive analytical solutions for this case, however, we have to use continuous time approximations in the derivation of the long-run economic growth rate.

Assume, in contrast to the baseline model, that there is one R&D sector for each intermediate good, with the firms in each research sector competing to discover the next generation of that particular good. One necessary additional assumption is that, although the arrival rates in different sectors are independent of each other, innovations are drawn from the same pool of knowledge, i.e., each new innovation increases the technological frontier available to all research firms. This ensures that innovations during one period arrive gradually. Using a continuous time approximation to be able to derive the analytical solution of the long-run economic growth rate, the labor market clearing condition and the research arbitrage condition are given by

$$\begin{array}{@{}rcl@{}} L &=&n+ \frac{(1-\alpha) \left( \frac{\alpha^{2}}{\omega }\right)^{\frac{1}{1-\alpha }}}{1+\ln (\gamma )-\alpha}, \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} \left( \frac{\alpha^{2}}{\omega}\right)^{\frac{1}{\alpha-1}} &=& \frac{\lambda \frac{1-\alpha}{\alpha}}{r+\lambda n + \frac{\alpha}{1-\alpha}\lambda n \ln(\gamma)}. \end{array} $$

(28)

The main difference to the baseline model is the crowding out effect, represented by $\lambda n$ in the denominator of Eq. 28. This is a consequence of the competition between the sectors because lower monopoly profits reduce the incentives to invest in R&D. The number of researchers across all sectors is the same because the expected payoffs in all research sectors are identical. The flow of innovations can, therefore, still be expressed as

$$ g_{t}=\lambda \cdot n_{t} \cdot \ln\gamma. $$

(29)

Since the new production structure does not affect the consumption-savings behavior of the households, the long-run growth rate of the economy is given by

$$ g=\max\left\{\frac{(1-\alpha) \log (\gamma ) \{\beta \phi [(1-\alpha) \lambda L+\alpha ]+\beta \lambda L \phi \log (\gamma )-\alpha\}}{\log (\gamma ) [(1-\alpha) \alpha (1-\beta \phi)+\beta \phi ]+(1-\alpha) \beta \phi },0\right\}. $$

(30)

The effect of population aging, as represented by an increase in the survival probability (ϕ), is still unambiguously positive. This is formulated in the following proposition.

Proposition 2

If there is a continuum of intermediate goods sectors instead of a single sector in our vertical innovation economic growth model with overlapping generations, the long-run growth rate (g) still increases in response to a higher survival probability ( ϕ ).

Proof

The partial derivative of the growth rate with respect to the survival probability is given by

$$ \frac{\partial g}{ \partial \phi}=\frac{(1-\alpha) \alpha \beta \log (\gamma ) [1+\log (\gamma )-\alpha] [(1-\alpha) \lambda L \log (\gamma )+ 1]}{\{(\alpha -1) \beta \phi -\log (\gamma ) [(1-\alpha) \alpha (1-\beta \phi)+\beta \phi ]\}^{2}}. $$

(31)

The denominator of this expression is always positive. Since $0<\alpha <1$, the numerator is also always positive such that the survival probability has a strictly positive effect on the long-run growth rate of the economy. □

The economic intuition is identical to the one in the baseline case with only one intermediate good. An increase in the probability to survive raises aggregate savings. Higher savings induce higher investments into shares of intermediate goods companies. This, in turn, raises the demand for innovation and, thus, for scientists. Having a larger number of scientists in the economy raises the frequency at which new innovations occur, increasing the long-run growth rate of aggregate consumption and of output. As apparent from Table 3, the growth effect of increasing life expectancy is still positive for different values of the EIS.

Table 3 Sensitivity of the growth rate to changes in the elasticity of intertemporal substitution (EIS)

The welfare effects of increasing longevity are reported in Table 4. The slightly larger productivity shares, as compared to the one-sector model, are a result of the crowding out effect mentioned above, which reduces the incentives to invest in research. This, in turn, increases the productivity share in the welfare analysis.

Table 4 Decomposition of additional utility for an increase in life expectancy of 1 year, multi-sector approach

To summarize, the introduction of a continuum of intermediate goods does not change our baseline results from a qualitative perspective and only slightly from a quantitative perspective. While the multi-sector case is arguably more realistic, it comes at the cost of using a continuous time approximation to be able to calculate analytical results.