An integral equation representation for overlapping generations in continuous time

doi:10.1016/j.jet.2008.03.006

Journal of Economic Theory

Volume 143, Issue 1, November 2008, Pages 596-609

https://doi.org/10.1016/j.jet.2008.03.006 Get rights and content

Abstract

This paper develops a method for solving for the dynamic general equilibrium of a deterministic continuous time overlapping generations model with a finite-horizon life-cycle. The model has isoelastic preferences and allows for general assumptions about individual endowments and demographics. Solving for an equilibrium reduces to solving a nonlinear integral equation. In the special case of log utility, the integral equation is linear and global approximations to a solution are easily computed with linear algebra.

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Demichelis (2002) and Demichelis and Polemarchakis (2007) study the limit of the discrete-time counterpart of the continuous-time model. Boucekkine et al. (2002) and Edmond (2008) perform numerical simulations. Mierau and Turnovsky (2014a,b) consider an approximation of the solution with a system of ordinary differential equations. d’Albis and Augeraud-Véron (2009)
Stability and determinacy conditions for mixed-type functional differential equations
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This paper analyzes the solution of linear mixed-type functional differential equations with either predetermined or non-predetermined variables. Conditions characterizing the existence and uniqueness of a solution are given and related to the local stability and determinacy properties of the steady state. In particular, it is shown that the relationship between the uniqueness of the solution and the stability of the steady-state is more subtle than the one that holds for ordinary differential equations, and gives rise to new dynamic configurations.
Spectral collocation method for solving Fredholm integral equations on the half-line
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Among this methods, a direct and commonly used approach is based on certain orthogonal approximations on infinite intervals, i.e., the Laguerre and Hermite spectral methods (see for example, [11,12,18]). With the same importance occupied in practice by differential equations defined on unbounded domains, integral equations on unbounded intervals are also one of most class of integral equations along with increased interest in the application of these models to significant scientific and technological problems (see for example, [7,8,16]). Therefore, we attempt to invest such techniques developed for solving differential equations to integral equations, of course, with the difference in approximation under the integral sign instead of differential.
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In the general equilibrium model of Cass and Yaari (1967), agents have certain and finite lifetimes. Their approach has been extended to the infinite lifetime case by Weil (1989) and, recently, to several other directions by Boucekkine et al. (2002), d’Albis and Augeraud-Veron (2009), Demichelis and Polemarchakis (2007) and Edmond (2008). In these models, the representative agent is assumed to have a fixed lifetime and her behaviour is specified independently of any demographic variable.
An extension to the Yaari (1965)–Blanchard (1985) continuous time overlapping generations model for an endowment Arrow–Debreu economy with an age-structured population is presented. It is proved that Arrow–Debreu equilibrium prices are represented by a double linear integral equation, and depend on the age-distributions of population and endowments. For an economy with a balanced growth, and logarithmic utility, we prove that bubbles may exist if endowments are distributed earlier than some critical age.
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NoteAn integral equation representation for overlapping generations in continuous time

Abstract

J. Public Econ.

Uncertain lifetime, life insurance, and the theory of the consumer

Rev. Econ. Stud.

Debt, deficits, and finite horizons

J. Polit. Economy

Dynamic Fiscal Policy

Life-cycle economies and aggregate fluctuations

Rev. Econ. Stud.

The Numerical Solution of Integral Equations of the Second Kind

The Numerical Treatment of Integral Equations

Numerical Methods in Economics

Numerical Recipes in C: The Art of Scientific Computing

Note
An integral equation representation for overlapping generations in continuous time