From Here to There: Achieving Fiscal Sustainability Under Alternative Demographic Contingencies

Abstract

This paper develops a model with overlapping generations of households, productive public and private capital, and a golden rule of fiscal policy aimed at maximizing economic growth. Studying the transitional dynamics between steady states triggered by different exogenous shocks, we find that a waning fertility rate, coming through a weaker preference for having children, increased longevity, a decrease in subjective discounting or lower financial support for child rearing, will require a policy intervention to ensure convergence to the growth maximizing debt level. A simple calibration exercise shows that when faced with projected demographic aging the adjustments in public debt and public investment required for keeping the economy at its maximum steady state growth rate may be small.

Appendices

Appendix 1: Jacobian, Hessian, and Policy Function Partial Derivative

The Jacobian for the motion equation in (16) is:

$$ J\left( {X_{t} ,\mathop {\dot{G}_{t} }\limits } \right) \equiv \frac{{\partial X_{t + 1} \left( {X_{t} ,\mathop {\dot{G}_{t} }\limits } \right)}}{{\partial X_{t} }} = \frac{{\overline{C}\mathop {\dot{G}_{t} }\limits X_{t}^{\omega - 1} \left[ {\alpha X_{t} + \omega \left( {1 + C + \alpha X_{t} } \right)} \right]}}{{\left[ {\overline{C} - \mathop {\dot{G}_{t} }\limits X_{t}^{\omega } \left( {1 + C + \alpha X_{t} } \right)} \right]^{2} }} $$

The Hessian is:

$$ \begin{aligned} H\left( {X_{t} ,\mathop {\dot{G}_{t} }\limits } \right) & \equiv \frac{{\partial^{2} X_{t + 1} \left( {X_{t} ,\mathop {\dot{G}_{t} }\limits } \right)}}{{\partial X_{t}^{2} }} = \frac{{\overline{C}\mathop {\dot{G}_{t} }\limits X^{ - 2 + \omega } }}{{\left[ { - \overline{C} + \mathop {\dot{G}_{t} }\limits X^{\omega } \left( {1 + C + \alpha X} \right)} \right]^{3} }} \\ & \quad \left\{ \begin{gathered} - \overline{C}\omega \left[ {\left( {1 + C} \right)\left( { - 1 + \omega } \right) + \alpha \left( {1 + \omega } \right)X} \right] \hfill \\ - \mathop {\dot{G}_{t} }\limits X^{\omega } \left[ {\left( {1 + C} \right)^{2} \omega \left( {1 + \omega } \right) + 2\alpha \left( {1 + C} \right)\omega \left( {2 + \omega } \right)X + \alpha^{2} \left( {1 + \omega } \right)\left( {2 + \omega } \right)X^{2} } \right] \hfill \\ \end{gathered} \right\} \\ \end{aligned} $$

The policy function partial derivative becomes:

$$ \frac{{\partial X_{t + 1} \left( {X_{t} ,\mathop {\dot{G}_{t} }\limits } \right)}}{{\partial \mathop {\dot{G}_{t} }\limits }} = \frac{{\overline{C}X_{t}^{\omega } \left( {1 + C + \alpha X_{t} } \right)}}{{\left[ {\overline{C} - \mathop {\dot{G}_{t} }\limits X_{t}^{\omega } \left( {1 + C + \alpha X_{t} } \right)} \right]^{2} }} $$

Appendix 2: Evaluations of Jacobian, Hessian and Policy Function Partial at Maximum Growth Steady State and Under Steady State Condition

Here the evaluation of the stability implicit function criteria for the maximum growth steady state to be a tangent bifurcation is shown. As seen in de la Croix and Michel (2002), the conditions are:

$$ \left\{ {\begin{array}{*{20}c} {\varphi_{X}^{^{\prime}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) = 0} \\ {\varphi_{{\dot{G}}}^{^{\prime}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) \ne 0} \\ {\varphi_{X}^{^{\prime\prime}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) \ne 0} \\ \end{array} } \right. $$

where $\varphi \left( {X_{t} ,\mathop {\dot{G}_{t} }\limits } \right) \equiv X_{t + 1} \left( {X_{t} ,\mathop {\dot{G}_{t} }\limits } \right) - X_{t}$.

Checking the conditions we see that they all hold:

$$ \varphi_{X}^{^{\prime}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) = \left. {\frac{{\left( {1 + X} \right)\left( {\alpha X + \omega \left( {1 + C + \alpha X} \right)} \right)}}{1 + C + \alpha X}} \right|_{{X = X^{*} }} - 1 = 0 $$

$$ \varphi_{{\dot{G}}}^{^{\prime}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) = \frac{{X^{{*}{\upomega }} \left( {1 + X^{*} } \right)^{2} \left( {1 + {\text{C}} + {\upalpha }X^{*} } \right)}}{{\overline{C}}} > 0 $$

$\varphi_{X}^{^{\prime\prime}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) = \frac{1}{2}\left\{ {\frac{{\alpha \omega^{2} }}{1 + C} - \frac{4\alpha }{{1 - \alpha + C}} - \omega^{2} + \frac{{\sqrt {4\alpha \left( {1 + C} \right) + \left( {1 - \alpha + C} \right)^{2} \omega^{2} } }}{{\left( {1 + C} \right)\left( {1 - \alpha + C} \right)}}\left[ {2 + \omega - \alpha \omega + C\left( {2 + \omega } \right)} \right]} \right\} > 0$ Overall, these characteristics prove that the maximum growth steady state is a tangent bifurcation.

Note that here it is also shown that $\varphi_{{\dot{G}}}^{^{\prime}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) > 0$ and $\varphi_{X}^{{^{\prime\prime}}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) > 0$. The sign of these two derivatives is important to assert the existence and stability of steady states in the vicinity of the maximum growth steady state, in that the motion equation is convex and next period’s capital ratio, $X_{t + 1}$, decreases as the policy function ${\dot{G}}_{t}$ is reduced. Furthermore, since $\varphi_{X}^{^{\prime}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) = 0$ and $\varphi_{X}^{{^{\prime\prime}}} \left( {X^{*} ,\gamma^{A} \left( {X^{*} } \right)} \right) > 0$, the Jacobian of the motion Eq. (16) near the point ${X}^{*}$ has a value lower than 1 for values below the maximum growth point ${X}^{*}$ and higher than 1 for $X$ values bigger than ${X}^{*}$.

Appendix 3: Derivation of Demography-Related Partial Derivatives of ${{\varvec{X}}}^{\mathbf{*}}$

The demography-related derivatives of ${X}^{*}$ are formally evaluated in Hughes Hallett et al. (2019). Summarizing their results:

Parameter	Derivative of $X^{*}$
$\rho$	$\frac{{\partial X^{*} }}{\partial \rho } = \underbrace {{X_{n}^{*} }}_{ > 0}\underbrace {{n_{\rho } }}_{ < 0} < 0$
$\lambda$	$\frac{{\partial X^{*} }}{\partial \lambda } = \underbrace {{X_{n}^{*} }}_{ > 0}\underbrace {{n_{\lambda } }}_{ > 0} > 0$
$\varepsilon$	$\frac{{\partial X^{*} }}{\partial \varepsilon } = \underbrace {{X_{n}^{*} }}_{ > 0}\underbrace {{n_{\varepsilon } }}_{ > 0} > 0$
$\rho_{w}$	$\frac{{\partial X^{*} }}{{\partial \rho_{w} }} = \underbrace {{X_{{\rho_{w} }}^{*} }}_{ > 0} + \underbrace {{X_{n}^{*} n_{{\rho_{w} }} }}_{ > 0} > 0$

Using as short ${\Psi } = \frac{1}{2}\left( {\rho_{w} \frac{{\left( {1 - \alpha } \right)zn}}{{\alpha \left( {1 - zn} \right)}} + \frac{1 + \alpha }{\alpha }} \right) + \left( {\frac{1}{{\omega^{2} }} - 1} \right)$, the derivatives are:

$$ X_{n}^{*} = \frac{\omega }{{\left( {1 + \omega } \right)}}\rho_{w} \frac{{\left( {1 - \alpha } \right)z}}{{2\alpha \left( {1 - zn} \right)^{2} }}\left[ {\frac{{\Psi }}{{\sqrt {{\Psi }^{2} - \left( {\frac{1}{{\omega^{2} }} - 1} \right)^{2} - \left( {\frac{1}{{\omega^{2} }} - 1} \right)} }} - 1} \right] > 0 $$

$$ X_{{\rho_{w} }}^{*} = \frac{\omega }{{\left( {1 + \omega } \right)}}\frac{{\left( {1 - \alpha } \right)zn}}{{2\alpha \left( {1 - zn} \right)}}\left[ {\frac{{\Psi }}{{\sqrt {\left[ {\Psi } \right]^{2} - \left( {\frac{1}{{\omega^{2} }} - 1} \right)^{2} - \left( {\frac{1}{{\omega^{2} }} - 1} \right)} }} - 1} \right] > 0 $$

$$ n_{\rho } = \frac{{ - \varepsilon \left( {1 - \lambda } \right)}}{{z\left( {1 - \rho_{w} } \right)\left[ {1 + \left( {1 - \lambda } \right)\rho + \varepsilon } \right]^{2} }} < 0 $$

$$ n_{\lambda } = \frac{\varepsilon \rho }{{z\left( {1 - \rho_{w} } \right)\left[ {1 + \left( {1 - \lambda } \right)\rho + \varepsilon } \right]^{2} }} > 0 $$

$$ n_{\varepsilon } = \frac{{1 + \left( {1 - \lambda } \right)\rho }}{{z\left( {1 - \rho_{w} } \right)\left[ {1 + \left( {1 - \lambda } \right)\rho + \varepsilon } \right]^{2} }} > 0 $$

$$ n_{{\rho_{w} }} = \frac{\varepsilon }{{z\left( {1 + \left( {1 - \lambda } \right)\rho + \varepsilon } \right)\left( {1 - \rho_{w} } \right)^{2} }} > 0 $$

Appendix 4: Calibration of Hughes Hallett et al. (2019)

Below are four tables from the calibration exercise in Hughes Hallett et al. (2019)

Parameter values of the baseline simulation
$\alpha$	Private capital's share in national income	0.37
$\beta$	Private capital’s importance in productivity factor	0.55
$\omega$	Private capital elasticity in production	0.716
$A$	TFP scale factor	34
$\rho$	Subjective discount factor	0.69
$z$	Rearing time per child	0.281
$\rho_{w}$	Child-rearing subsidy rate	0.193
$\lambda$	Death rate	0.7
$\varepsilon$	Importance of children in utility	0.539

1. Source: Hughes Hallett et al. (2019)

Results of steady state baseline simulation
$d^{*}$	Optimal debt to GDP ratio in steady state (no TFP)	49.67%
$X$	Public private capital ratio	0.2546
$\gamma^{A}$	Annual economic growth rate	1.44%
$\gamma$	Annual economic growth rate per capita	0.75%
$\theta$	Tax rate	14.5%
$\tilde{C}$	Saving rate (out of disposable income)	11.86%
$r$	Annualized interest rate	5.14%
$n$	Number of children per person	1.36
	Old-age dependency ratio	22%
	Life expectancy at birth	78.5

1. Source: Hughes Hallett et al. (2019)

Sensitivities to demographic parameters, OECD economies
$X_{n}^{*}$	$\partial {\text{X}}^{*} /\partial {\text{n}}$	0.0067
${\text{n}}_{{\uplambda }}$	$\partial {\text{n}}/\partial {\uplambda }$	0.538
${\text{n}}_{{\upvarepsilon }}$	$\partial {\text{n}}/\partial {\upvarepsilon }$	1.746
${\text{n}}_{{\uprho }}$	$\partial {\text{n}}/\partial {\uprho }$	−0.234
${\text{d}}_{{\uplambda }}^{*}$	$\partial {\text{d}}^{*} /\partial {\uplambda }$	0.005
${\text{d}}_{{\upvarepsilon }}^{*}$	$\partial {\text{d}}^{*} /\partial {\upvarepsilon }$	0.016
${\text{d}}_{{\uprho }}^{*}$	$\partial {\text{d}}^{*} /\partial {\uprho }$	−0.002

1. Source: Hughes Hallett et al. (2019)

OECD population growth, old-age dependency ratios and life expectancy (45 year averages)
	1970–2015	2015–2060	2060–2100
45-year population growth rate	36.1%	6.3%	−5.2%
Old-age dependency ratio	21.9%	44.0%	60.1%
Life expectancy at birth	75.2	83.9	89.2
Parameter $\lambda$	0.7	0.532	0.43
Parameter $\varepsilon$	0.539	0.42	0.382
${X}^{*}$	25.5%	25.3%	25.2%
${d}^{*}$	49.7%	49.4%	49.3%

1. Source: Hughes Hallett et al. (2019)