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Population dynamics and monetary policy

Abstract

This paper shows that in a general equilibrium model with interest-rate feedback rules of the Taylor-type population dynamics give rise to multiple steady states. Under an active monetary policy, real determinacy occurs only around the steady state with zero net financial wealth, where aggregate consumption is equally distributed among agents of different generations. By contrast, in a neighborhood of the steady state displaying a positive stock of financial wealth and intergenerational inequality, real determinacy requires monetary policy to be passive. Changes in the demographic profile of the economy are shown to have relevant implications for the aggregate accumulation of wealth.

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Notes

  1. The inverse of the instantaneous probability of death is equal to the life expectancy and is constant throughout life as in Blanchard (1985).

  2. This assumption is necessary to make money always essential. See Obstfeld and Rogoff (1983).

  3. See Deaton and Muellbauer (1980), and for a similar application in the context of the Yaari–Blanchard model, see Marini and van der Ploeg (1988).

  4. The condition Γ′(R t )>0 follows from Ω ccΩ cm Ω c/Ω m<0 and Ω mmΩ cm Ω m/Ω c<0, whereas the sign of [Γ(R t )−R t Γ′(R t )] depends on the elasticity of substitution between real money balances and consumption, σ=Γ′(R t ) R t /Γ(R t ), which is assumed to be less than one.

  5. For this reason, the fiscal rule (Eq. 18) can be defined as “passive” in the spirit of Leeper (1991) or “locally Ricardian” as in Woodford (2003). Allowing the fiscal parameter θ to be time variant would open up to multiple cases. For instance, if θ were equal to the nominal interest rate R t at each instant of time, we would have a balance budget rule à la Benhabib et al. (2002).

  6. If R(π) is a nonlinear function of π, as in Benhabib at el. (2001b), to ensure that the nominal interest rate is always above a certain lower bound for any π, the economy will display four steady states.

  7. Under a budget balance rule, where θ t =R t at each instant of time, the unintended steady state would exhibit an inflation rate \(\widehat{\pi } = - n\) and a level of per capita government liabilities â< (>)0 if R′> (<)1. See Appendix B for details.

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Acknowledgments

We would like to thank Alessandro Cigno, the editor, two anonymous referees, and Giancarlo Marini for their valuable comments and suggestions. We are also grateful to Andrea Costa and Pasquale Scaramozzino for useful discussions. We acknowledge financial support from Il Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR). All remaining errors are ours. A detailed technical appendix is available from the authors upon request.

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Correspondence to Barbara Annicchiarico.

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Appendices

Appendix A

Linearizing Eqs. 21 and 22 around the intended steady state (π*, a*), we obtain

$${\left( {\begin{array}{*{20}l} {{{\mathop \pi \limits^ \bullet }_{t} } \hfill} \\ {{{\mathop a\limits^ \bullet }_{t} } \hfill} \\ \end{array} } \right)} = J*{\left( {\begin{array}{*{20}l} {{\pi _{t} - \pi *} \hfill} \\ {{a_{t} - a*} \hfill} \\ \end{array} } \right)}$$
(24)

where

$$J* = {\left( {\begin{array}{*{20}c} {{{\left[ {R\prime {\left( {\pi *} \right)} - 1} \right]}\frac{{L{\left[ {R{\left( {\pi *} \right)}} \right]}}}{{L\prime {\left[ {R{\left( {\pi *} \right)}} \right]}R\prime {\left( {\pi *} \right)}}} - }} \\ {0} \\ \end{array} \begin{array}{*{20}c} {{\frac{{\beta {\left( {\rho + \mu } \right)}}}{{L\prime {\left[ {R{\left( {\pi *} \right)}} \right]}R\prime {\left( {\pi *} \right)}}}}} \\ {{\rho - n - \theta }} \\ \end{array} } \right)}$$
(25)

with ρnθ<0 being a sufficient condition to ensure the respect of the fiscal solvency (Eq. 19). Linearizing Eqs. 21 and 22 around the unintended steady state \({\left( {\widehat{\pi },\,\widehat{a}} \right)}\), we obtain

$${\left( {\begin{array}{*{20}c} {{{\mathop \pi \limits^ \bullet }_{t} }} \\ {{{\mathop a\limits^ \bullet }_{t} }} \\ \end{array} } \right)} = \widehat{J}{\left( {\begin{array}{*{20}c} {{\pi _{t} - \widehat{\pi }}} \\ {{a_{t} - \widehat{a}}} \\ \end{array} } \right)},$$
(26)

where

$$\widehat{J} = {\left( {\begin{array}{*{20}c} {{\widehat{J}_{{11}} }} & {{ - \frac{{\beta {\left( {\rho + \mu } \right)}}}{{L\prime {\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}R\prime {\left( {\widehat{\pi }} \right)}}}}} \\ {{{\left[ {R\prime {\left( {\widehat{\pi }} \right)} - 1} \right]}\frac{{{\left[ {R{\left( {\widehat{\pi }} \right)} - \widehat{\pi } - \rho } \right]}L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}}}{{{\left( {\rho + \mu } \right)}\beta }}}} & {0} \\ \end{array} } \right)},$$
(27)

with \(\widehat{J}_{{11}} = \frac{{{\left[ {R\prime {\left( {\widehat{\pi }} \right)} - 1} \right]}L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}}}{{L\prime {\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}R\prime {\left( {\widehat{\pi }} \right)}}} + {\left[ {R{\left( {\widehat{\pi }} \right)} - \widehat{\pi } - \rho } \right]} > 0\).

Since π is a jump variable and a is predetermined (state variable), the systems Eqs. 24 and 26 have a unique convergent solution if the real part of the eigenvalues are of opposite sign (real determinacy); there are multiple convergent solutions if the real part of both eigenvalues are negative (real indeterminacy) and unstable if the real part of both eigenvalues are positive.

Proof of Proposition 1

Under an active monetary policy, the model displays real determinacy around (π*, a*) and instability around \({\left( {\widehat{\pi },\,\widehat{a}} \right)}\) because det (J*)<0, det \({\left( {\overline{J} } \right)} > 0\), and tr \({\left( {\overline{J} } \right)} > 0\).▪

Proof of Proposition 2

Under passive monetary policy, the model displays real indeterminacy around (π*, a*) and real determinacy around \({\left( {\widehat{\pi },\,\widehat{a}} \right)}\) because det (J*)>0, tr (J*)<0 and det \({\left( {\overline{J} } \right)} < 0\).▪

Appendix B

We now explore the properties of the model under the assumption that tax revenues, adjusted for interest savings on the monetary base, are set equal to interest payments. This case results when the fiscal parameter θ is time varying and is set equal to the nominal interest rate R t at each instant of time.

Under this balanced budget, fiscal rule Eq. 22 collapses to

$${\mathop a\limits^ \bullet }_{t} = - {\left( {\pi _{t} + n} \right)}a_{t} ,$$
(28)

being θ t =R t . Let θ* and \(\widehat{\theta }\) denote the fiscal parameter at the intended and at the unintended steady state, respectively.

At the intended steady state, R(π*)=ρ+π*=θ*. Fiscal solvency requires that π*+n>0, where π*=θ*−ρ. It follows that θ*+nρ>0.

At the unintended steady state, \(R{\left( {\widehat{\pi }} \right)} = \widehat{r} + \widehat{\pi } = \widehat{\theta }\) and \(\widehat{a} = {\left( {\widehat{\theta } - \widehat{\pi } - \rho } \right)}{L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}} \mathord{\left/ {\vphantom {{L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}} {\beta {\left( {\mu + \rho } \right)}}}} \right. \kern-\nulldelimiterspace} {\beta {\left( {\mu + \rho } \right)}}\), with \(\widehat{\pi } = - n\). If monetary policy is active, then \(\widehat{\pi } < \pi *\) will imply that the real interest rate \(\widehat{r}\) prevailing at this steady state is lower than in the intended steady state. It follows that \(\widehat{a} = {\left( {\widehat{\theta } + n - \rho } \right)}{L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}} \mathord{\left/ {\vphantom {{L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}} {\beta {\left( {\mu + \rho } \right)}}}} \right. \kern-\nulldelimiterspace} {\beta {\left( {\mu + \rho } \right)}} < 0\), since \(\widehat{\theta } + n - \rho < 0\). By contrast, if monetary policy is passive, the real interest rate \(\widehat{r}\) will be larger than in the intended steady state. It follows that \(\widehat{a} = {\left( {\widehat{\theta } + n - \rho } \right)}{L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}} \mathord{\left/ {\vphantom {{L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}} {\beta {\left( {\mu + \rho } \right)}}}} \right. \kern-\nulldelimiterspace} {\beta {\left( {\mu + \rho } \right)}} > 0\), since \(\widehat{\theta } + n - \rho > 0\).

The Jacobian of the system of Eqs. 21 and 28 at (π*, a*) and at \({\left( {\widehat{\pi },\widehat{a}} \right)}\) is given, respectively, by

$$J* = {\left( {\begin{array}{*{20}c} {{{\left[ {R\prime {\left( {\pi *} \right)} - 1} \right]}\frac{{L{\left[ {R{\left( {\pi *} \right)}} \right]}}}{{L\prime {\left[ {R{\left( {\pi *} \right)}} \right]}R\prime {\left( {\pi *} \right)}}}}} & {{ - \frac{{\beta {\left( {\rho + \mu } \right)}}}{{L\prime {\left[ {R{\left( {\pi *} \right)}} \right]}R\prime {\left( {\pi *} \right)}}}}} \\ {0} & {{ - \pi * - n}} \\ \end{array} } \right)},$$
(29)
$$\widehat{J} = {\left( {\begin{array}{*{20}c} {{\widehat{J}_{{11}} }} & {{ - \frac{{\beta {\left( {\rho + \mu } \right)}}}{{L\prime {\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}R\prime {\left( {\widehat{\pi }} \right)}}}}} \\ {{ - \frac{{{\left( {\widehat{\theta } + n - \rho } \right)}L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}}}{{\beta {\left( {\rho + \mu } \right)}}}}} & {0} \\ \end{array} } \right)},$$
(30)

with \(\widehat{J}_{{11}} = \frac{{{\left[ {R\prime {\left( {\widehat{\pi }} \right)} - 1} \right]}L{\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}}}{{L\prime {\left[ {R{\left( {\widehat{\pi }} \right)}} \right]}R\prime {\left( {\widehat{\pi }} \right)}}} + {\left( {\widehat{\theta } + n - p} \right)}.\).

An inspection of Eqs. 29 and 30 reveals that under an active monetary policy, the model exhibits real determinacy around (π*, a*), since det (J*)<0, and instability or real indeterminacy around \({\left( {\widehat{\pi },\widehat{a}} \right)}\) since det \({\left( {\overline{J} } \right)} > 0\) and the sign of the trace is ambiguous. Under a passive monetary policy, the model displays real indeterminacy around (π*, a*) and real determinacy around \({\left( {\widehat{\pi },\widehat{a}} \right)}\) since det (J*)>0, tr(J*)<0, and det \({\left( {\overline{J} } \right)} < 0\).

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Annicchiarico, B., Piergallini, A. Population dynamics and monetary policy. J Popul Econ 19, 627–641 (2006). https://doi.org/10.1007/s00148-005-0035-x

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Keywords

  • Population dynamics
  • OLG
  • Interest rate rules

JEL Classification

  • E31
  • E52
  • J10