An Integral Equation Representation for Overlapping Generations in Continuous Time

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.Keywords: overlapping generations, continuous time, life-cycle, intertemporal prices.The overlapping generations model is one of the workhorses of modern macroeconomics. This paper outlines a continuous time OLG model where agents live for a given finite interval of time so that genuine life-cycle behavior is possible. 1In applied macroeconomics, OLG models are set in discrete time and individuals have many decision periods per lifetime (Auerbach and Kotlikoff, 1987;Ríos-Rull, 1996). Long-lived models of this kind are attractive because they allow for realistic timing conventions and an intuitive calibration to annual or quarterly data. In practice, however, long-lived discrete time models are difficult to work with computationally. The main advantage of the continuous time model outlined below is that it simultaneously permits agents to have many decisions per lifetime but is computationally simple.The model is presented in Section 1 and allows for isoelastic utility and general assumptions about individual endowments (and demographics). The main result of this paper is given in Section 2 which shows how to represent the equilibrium fixed point problem as a specific, generally nonlinear, integral equation which needs to be solved for an unknown intertemporal price function.

Continuous-Time Overlapping Generations Models

“…The distribution At first sight, these figures are similar to those obtained by Edmond (2008). The distribution of initial assets barely affects the intertemporal prices profile.…”

“…Function f (t) Prices p (t) Figure 1 Asset distribution Function f (t) Prices p (t) Figure 2 At first sight, these figures are similar to those obtained by Edmond (2008).…”

Convergence analysis of Galerkin and multi-Galerkin methods for linear integral equations on half-line using Laguerre polynomials

“…Integral equations on unbounded intervals are important class of integral equations along with increased interest in the applications to the scientific and technological problems (Atkinson 1969, Anselone and Sloan 1985, 1988, Chandler and Graham 1987, Amini and Sloan 1989, Guo 1998, Edmond 2008, Parand and Babolghani 2012. Numerical methods for integral equation of type (1.1) have attracted a lot of attention.…”

Local determinacy of prices in an overlapping generations model with continuous trading

“…Demichelis [13] and Demichelis and Polemarchakis [14] study the limit of the discrete-time counterpart of the continuoustime model. Boucekkine et al [9] and Edmond [15] perform numerical simulations. Mierau and Turnovsky [30], [31] consider an approximation of the solution with a system of ordinary differential equations.…”