Abstract
This paper studies how the upcoming demographic transition will affect the returns to risky capital and safe government debt. Using a neoclassical two-generations-overlapping model, we show that the entrance of smaller cohorts into the labor market will lower both interest rates. The risky rate, however, will react more sensitive than the risk-free rate. Consequently, the risk premium declines when an economy transitions from high fertility to low fertility.
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Notes
- 1.
See Poterba (2001) for a survey.
- 2.
Similarly, even if agents can anticipate how given changes in labor supply change interest rates, they may not be able to anticipate these changes in labor supply correctly. That is, agents born in the first half of the twentieth century may have been surprised by the drastic increase in female labor force participation.
- 3.
There are obviously many different debt policies perceivable. However, the following results will be valid for all perceivable debt policies provided that taxes τ t and the amount of debt B t which is issued at time t do not depend on variables that are not yet realized in period t, e.g., the future capital intensity k t+1.
- 4.
The individual receives his wage w t after the realization of z t is known. Note also that taxes are known once z t is known.
- 5.
The rate of return R t+1 = z t+1 f′(k t+1) inherits its log-normal distribution from the technology shock z t+1. Thus, ln(R t+1) follows a normal distribution.
- 6.
Given our specification of the production sector, second-period wage income and capital are perfectly correlated (perfect substitutes), i.e., \(corr(w_{t+1},R_{t+1}) = \frac{cov(w_{t+1},R_{t+1})} {\sigma _{w}\sigma _{R}} = 1\).
- 7.
From (26), we have \(\frac{df'(k_{t+1})} {dn_{t+1}}> 0\); in turn, we find that \(\frac{d\gamma } {dn_{t+1}} <0\), together with γ r < 0 and γ f′ > 0, implies that \(\frac{dr_{t+1}} {dn_{t+1}}> 0\).
- 8.
Note that there are two ways to think about the savings decision:
$$\displaystyle\begin{array}{rcl} \max _{\gamma _{t},s_{t}}U(\Omega _{t} - s_{t}) +\beta E_{t}[U((r_{t+1} +\gamma _{t}(R_{t+1} - r_{t+1}))s_{t})].& & {}\\ \end{array}$$Expanding (30) at the point γ = 0 and \(s =\bar{ s}\), we have:
$$\displaystyle\begin{array}{rcl} & & U(\Omega _{t} -\bar{ s}_{t}) +\beta U(\bar{s}_{t}r_{t+1}) \\ & & +\beta U'(\bar{s}_{t}r_{t+1})E[R_{t+1} - r_{t+1}]\bar{s}_{t}\gamma _{t} + \frac{1} {2}\beta U''(\bar{s}_{t}r_{t+1})\Big(E[R_{t+1} - r_{t+1}]\bar{s}_{t}\gamma _{t}\Big)^{2} \\ & & +\Big(-U'(\Omega _{t} -\bar{ s}_{t}) +\beta U'(\bar{s}_{t}r_{t+1})r_{t+1}\Big)(s_{t} -\bar{ s}_{t}) \\ & & +\frac{1} {2}\Big(U''(\Omega -\bar{ s}_{t}) +\beta U''(\bar{s}_{t}r_{t+1})r_{t+1}^{2}\Big)(s_{t} -\bar{ s}_{t})^{2} \\ & & +\gamma _{t}E[R_{t+1} - r_{t+1}]\beta U'(\bar{s}_{t}r_{t+1})\Big(\frac{U''(\bar{s}_{t}r_{t+1})} {U'(\bar{s}_{t}r_{t+1})}r_{t+1}\bar{s}_{t} + 1\Big)(s_{t} -\bar{ s}_{t}). {}\end{array}$$(29)The last term in (29) indicates the interaction between the size and the composition of the portfolio (cross derivative). There are now two ways to think of our approximation in the main text: (i) The household chooses savings according to the usual Euler equation and chooses the portfolio shares according to the Taylor approximation (31). Put differently, the household chooses savings according to a precise rule. The portfolio shares, however, rely on an approximation. (ii) The household chooses both s and γ according to (29). In this (less appealing) case, there would be an additional component in the FOC for γ of ambiguous sign.
- 9.
The respective partial derivatives are \(\gamma _{f'} = \frac{\sigma _{R}^{2}-E[\Pi ]^{2}-E[\Pi ]2f'\sigma _{z}^{2}} {C}\) and \(\gamma _{r} = \frac{-\sigma _{R}^{2}+E[\Pi ]^{2}} {C}\) where C is a positive constant. Hence, if γ f′ > 0, then γ r < 0.
- 10.
For large equity premia, the term E t [(R t+1 − r t+1)]2 in the denominator grows very large compared to E t [(R t+1 − r t+1)] in the numerator. Increases in f′(k) may now in principle decrease the portfolio share in the risky asset.
- 11.
Apparently, for small rates of return, there is not much difference between the two formulas. For small x, we have the first-order Taylor-series log(1 + x) = 0 + 1 ⋅ x = x. Thus, \(log(E[R_{t+1}]) - log(r_{t+1})\thickapprox E[R_{t+1} - r_{t+1}]\). Moreover, in the denominator, \(var(zf')\thickapprox var(log(z))\) if the variance of z is small and the (net) rate of return close to zero, i.e., \(f'\thickapprox 1\). With regard to our analysis of the equity premium, however, the expression in (34) has the disadvantage that the relative risk aversion is now a function of savings and changes during the transition. In this case, it is not possible to derive appealing conditions for the relation between the risk premium and the growth rate for population.
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Acknowledgements
I thank the anonymous referee, Axel Boersch-Supan, Brian Cooper, Nataliya Demchenko, Alexander Ludwig, Edgar Vogel, Matthias Weiss, and seminar participants in Mannheim for comments.
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Kuhle, W. (2017). Demographic Change and the Rates of Return to Risky Capital and Safe Debt. In: Bökemeier, B., Greiner, A. (eds) Inequality and Finance in Macrodynamics. Dynamic Modeling and Econometrics in Economics and Finance, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-54690-2_8
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