Forward and backward dynamics in implicitly defined overlapping generations models

In dynamic economic models derived from optimization principles, the forward equilibrium dynamics may not be uniquely defined, while the backward dynamics is well defined. We derive properties of the global forward equilibrium paths based on properties of the backward dynamics. We propose the framework of iterated function systems (IFSs) to describe the set of forward equilibria and apply the IFS framework to a one- and a two-dimensional version of the overlapping generations (OLG)-model. We show that if the backward dynamics is chaotic and has a homoclinic orbit (a “snap-back repeller”), the set of forward equilibrium paths converges to a fractal attractor. Forward equilibria may be interpreted as sunspot equilibria, where a random sunspot sequence determines equilibrium selection at each date.     

The dynamics of capital accumulation in an overlapping generations model

This paper studies capital accumulation in a slightly altered, explicit OLG model. The fundamental difference with the standard model lies in the initial conditions. If a portion of the initial allocation of the capital stock is not assigned to retirees, the framework allowing for a genuine accumulation of capital is provided. Dynamic aspects of the resulting model are analyzed, including dynamics of the quotas of capital stock, shares of total output, partial influences of parameters, the connection between the depreciation rate and capital's contribution to production, and the relation of the latter to saddle-node bifurcations and the existence of real-valued equilibria. It is demonstrated that the model is able to describe the initial phase of accumulation.     

Competitive equilibria in overlapping generations Experiments

Summary The paper presents the results of four overlapping generations experiments performed at the California Institute of Technology. Overlapping generations markets were created in which each agent had a two period life span. With the exception of the first period, there were eight agents trading in each period; four buyers (two young and two old) and four sellers (two young and two old). Parameters were selected so that a small set of equilibria existed. The markets were open for twenty-nine periods with a demand shift occurring at the fifteenth and sixteenth periods.This work provides a method of computing all competitive equilibria for a class of environments—called the opposing shift environments. The main conclusion of the experiments is that the experimental price data converge to near the stationary portions of the competitive equilibria. Demographic dynamics are also explored as part of the price adjustment process. Dynamic features found in non-overlapping environments are also observed in the overlapping generations case.We thank E. J. Nanale, a Caltech undergraduate student, who helped design and conduct the experiments. Special thanks are due to Hsing Yang Lee for his help in adapting the Multiple Unit Double Auction Program to accomodate the overlapping generations environment and for his help with the data processing. We also thank David Grether who found a serious problem with the statistical analysis in an early draft. The research assistance of Kay-Yut Chen was also very valuable to the completion of this work. The comments and suggestions of P. Rangazas were very helpful. The research of the second author was supported in part by a National Science Foundation Grant and the California Institute of Technology Laboratory for Experimental Economics and Political Science.     

On stationary overlapping generations models

Consider a stationary consumption–loan economy with fixed amount of fiat money. We show that although the economy is stationary if the number of goods and the number of ‘types’ of consumers are larger than one, there might be no Pareto optimal stationary competitive equilibrium.     

Rational expectations in the overlapping generations model

This paper reports results on the character of the rational expectations equilibria of a stochastic overlapping generations model with heterogenous markets. The model considered is a stationary overlapping generations model in which the endowments of young agents are subject to i.i.d. random shocks. The main result shown is that if there are l > 1 commodities traded in every period, then for most preferences, the rational expectations equilibrium stochastic process of prices and allocations necessarily exhibits serial correlation. This is in marked contrast to the one commodity model in which there always exists an equilibrium which is measure isomorphic to the endowment process.     

Signal-extracting education in an overlapping generations model

Summary. In order to get good positions in companies, people try to enter highly-ranked universities. However, abilities vary greatly between individuals. High-ability individuals have an incentive to send signals to firms by obtaining a higher level of education in order to distinguish themselves from low-ability individuals. This paper constructs an overlapping generations model in order to examine the macroeconomic consequences of such sorting behavior of individuals. There are two kinds of possible equilibria in our model. In one equilibrium, only the high-ability agent can obtain higher education and thus an elite society emerges. In the other equilibrium, all ability types have the chance to obtain higher education and thus a society with mass higher education emerges. We also discuss the possibility of multiple equilibria of these different steady states and the dynamic change in wage differentials.Received: 9 October 2002, Revised: 15 July 2003, JEL Classification Numbers: D82, J31, O10.The authors acknowledge Osamu Hayashida, Noriyoshi Hemmi, Hideshi Itoh, Michihiro Kandori, Toshihiro Matsumura, Takuya Nakaizumi, Osamu Nishimura, Ryoji Ohdoi, Tadashi Yagi, Noriyuki Yanagawa, and seminar participants at Doshisha University, the University of Tokyo, and Contract Theory Workshop at Kyoto University for helpful comments and suggestions. We would also like to thank an anonymous referee for valuable comments. This paper is part of the academic Project on Intergenerational Equity (PIE), funded by a scientific grant from Japans Ministry of Education, Culture, Sports, Science and Technology (grant number 603).     

Short-memory equilibrium in stochastic overlapping generations economies

In stochastic OLG exchange economies, we show that short-memory equilibria—the natural extension from deterministic economies of steady states, low-order cycles, or finite state-space stationary sunspots equilibria—fail to exist generically in utilities. As a result, even with independent and identically distributed exogenous shocks there is serial correlation in endogenous economic variables in equilibrium. This arises even if utilities are time-separable, some goods inferior, and there are no technological lags. Hence, the origins of economic fluctuations can be traced only to the demographic structure of a heterogeneous agent, multiple-good economy.     

Gains from trade with overlapping generations

Summary This paper examines the welfare effects of international trade in a context of overlapping generations. It shows that, for a single trading country, uncompensated free trade may be Pareto inferior to autarky. However, for each government there are compensation schemes which guarantee welfare improvements for all local individuals when free trade is allowed, or when for a small open economy the terms of trade improve or the number of tradable goods increases, or when a customs union is formed.We acknowledge with gratitude the probing comments of Henry Y. Wan, Jr. and two anonymous referees.     

Stationary Markov equilibria for overlapping generations

Summary.  At a stationary Markov equilibrium of a Markovian economy of overlapping generations, prices at a date-event are determined by the realization of the shock, the distribution of wealth and, with production, the stock of capital. Stationary Markov equilibria may not exist; this is the case with intra-generational heterogeneity and multiple commodities or long life spans. Generalized Markov equilibria exist if prices are allowed to vary also with the realization of the shock, prices and the allocation of consumption and production at the predecessor date-event. (Stationary) Markov -equilibria always exist; as allocations and prices converge to equilibrium prices and allocations that, however, need not be stationary.Received: 2 March 2004, Revised: 2 April 2004, JEL Classification Numbers:   D50, D52, D60, D80, D90.Correspondence to: Felix KublerWe thank participants in seminars in Athens and Lund, at Penn, at IMPA and at Stanford, the 2002 CEME (NBER) General Equilibrium Conference and the 2002 SED meetings, and especially Martin Hellwig, George Mailath and an anonymous referee for very helpful comments.     

Two-period cycles in a three-period overlapping generations model

We study the properties of two-period monetary cycles in simple pure exchange overlapping generations economies in which the households live for three periods. We demonstrate that these economies can support cycles under a much broader—and, arguably, more plausible—range of assumptions than the analogous two-period economies. We show that economies that fail the well-known Grandmont (Econometrica 53 (1985) 995) condition can have cycles, and that economies that satisfy the condition can fail to have cycles. In addition, we show that economies can have monetary cycles when they do not have conventional monetary steady states, and when aggregate demand for assets is not decreasing in the real return rate at a gross real rate of unity.     

Efficiency and prices in economies of overlapping generations

In a general economy of overlapping generations, I introduce a notion of uniform inefficiency, corresponding to the occurrence of a Pareto improvement with a small uniform destruction of resources [G. Debreu, The coefficient of resource utilization, Econometrica 19 (1951) 273-292]. I provide a necessary and sufficient condition for uniform inefficiency in terms of prices at a competitive equilibrium: an allocation is uniformly inefficient if and only if the relative price of the aggregate endowment in a given period into the aggregate endowment up to that period does not vanish over periods of trade, a sort of Modified Cass Criterion [D. Cass, On capital overaccumulation in the aggregative neoclassical model of economic growth: a complete characterization, J. Econ. Theory 4 (1972) 200-223]. Minimal assumptions on fundamentals are needed for such a complete characterization. Furthermore, proofs reduce to simple and short direct arguments. Finally, I verify that uniform inefficiency is preserved under perturbations, a property that might fail for the canonical notion of inefficiency. Remarkably, an allocation is uniformly inefficient if and only if a non-vanishing redistribution, like a social security mechanism, is welfare improving.     

Habit formation and indeterminacy in overlapping generations models

I introduce habit formation into an otherwise standard overlapping generations economy with pure exchange populated by three-period-lived agents. Habits are modeled in such a way that current consumption increases the marginal utility of future consumption. With logarithmic utility functions, I demonstrate that habit formation gives rise to stable monetary steady states in economies with hump-shaped endowment profiles and reasonably high discount factors. Intuitively, habits imply adjacent complementarity in consumption, which in turn explains why income effects are sufficiently strong in spite of the logarithmic utility. The three-period horizon further strengthens the income effect.     

Multiple equilibrium in the overlapping generations model revisited

Bliss [Bliss, C., 2008. Multiple equilibrium in the Diamond capital model. Economics Letters 100, 143-145] finds numerically that the Diamond OLG model can have uncountably many steady states. We use log preferences and show analytically that a continuum of steady states can still exist.     

The economics of indeterminacy in overlapping generations models

Recent research indicates that there are robust examples of overlapping generations economies in which there are indeterminate equilibria without fiat money and equilibria with more than one dimension of indeterminacy. This paper presents simple examples of a stationary, pure exchange overlapping generations economy with one good in each period and a representative consumer, who lives for three periods, in each generation. These examples exhibit every possible form of indeterminacy and instability. Furthermore, the parameters of the principal example agree with empirical evidence. We use our examples as case studies for analyzing the problems involved in computing the equilibria of such economies.     

Taxation of capital income in overlapping generations economies

Whether capital income should be taxed in overlapping generations economies is vividly discussed. It is shown that intergenerational lump‐sum taxes cannot implement the Golden Rule allocation when agents have private information on their earnings potential. Hence, the seminal Atkinson–Stiglitz result that optimal income taxation pre‐empts any role for indirect taxation cannot be interpreted to imply that capital income taxation (affecting intertemporal relative prices) should not be taxed. Specifically, capital income should unambiguously be taxed in small open economies, and the optimal tax rate depends inversely on the elasticity of total savings to disposable income and the after‐tax rate of return.     

Capital asset pricing in an overlapping generations model

This paper attempts to contribute to two rapidly growing branches in economic theory: asset pricing and “overlapping generations” models. The model is formulated and it is shown that equilibrium prices exist, and some of their properties are discussed. Then the model is applied to an asymmetric information environment to see if randomness in the number of informed agents could confuse the uninformed. Surprisingly, it could not.     

Optimal growth and cycles in overlapping generations models

Summary This paper investigates the dynamical properties of optimal paths in one-sector overlapping generations models without assuming that the utility function of the representative agent is separable. When the utility function is separable, the optimal growth paths monotonically converges toward the modified golden rule steady state. In the non-separable case, we show that the optimal growth path may be oscillating and optimal two-period cycles may exist. Applying these results to the model with altruism, we show that the condition of operative bequest is fully compatible with endogeneous fluctuations provided that the discount factor is close enough to one. All our results are illustrated using Cobb-Douglas utility and production functions.We thank C. Blackorby, J. Blot, P. Cartigny and one anonymous referee for helpful comments and suggestions which generally improved the exposition of the paper. We would also like to thank the participants of the Population and demography session of the European Economic Association 10th Annual Congress (Prague, The Czech Republic, September 1–4, 1995).     

Socially responsible investment in an environmental overlapping generations model

One of the problems associated with the conservation of the environment is that short-lived individuals fail to account for the long-term effects of pollution, which implies that future generations bear the costs imposed by the current generation. Such inter-generational externalities are usually tackled by (Pigovian) taxes, fiscal policy or environmental regulation. Alternatively, we propose that socially responsible investment funds create a role for the stock market to deal with intergenerational environmental externalities. We analyze the role of the stock market in an environmental overlapping generations model of the Diamond-type, in which agents choose between investing in “clean” government bonds or “polluting” firm equity. We show that although socially responsible investors are short-lived, the forward-looking nature of stock prices can help to resolve the conflict between current and future generations.     

Short run core equivalence in an overlapping generations model

This essay establishes the equality between the set of competitive equilibrium allocations and the set of allocations which belong to the cores of a sequence of short run economies for an overlapping generations model. A short run economy is a finite period economy obtained by truncating the original economy so that the truncated economy is irreducible.     

The determinacy of equilibrium in economies of overlapping generations

We study the problem of uniqueness of equilibrium paths in the overlapping generations model. We show that, despite local calculation based on counting equations and unknowns, the equilibrium path may be unique. We do this by constucting an example of an economy of overlapping generations with just one equilibrium up to time shift, beside the steady states. Time is either discrete or continuous; in either case, it extend into the infinite future and, possibly, the infinite past. There is one, non-storable commodity at each date. The economy is stationary; intertemporal preferences are logarithmic; the endowments and discount factors of individuals need not depend continuously on time. With continuous time, equilibrium paths of prices are smooth; this, even for endowments and discount factors of individuals that do not depend continuously on time. With discrete time, as the number of periods in the life-span of individuals increases, equilibrium paths converge to the continuous time solutions. If time extends infinitely into the infinite past as well as into the infinite future, in continuous time, all non-stationary equilibrium paths of prices are time-shifts of a single path; in addition, there are two stationary solutions; in discrete time, there is a one dimensional family of non-stationary solutions, up to time-shift; however, the indeterminacy vanishes as the number of periods in the life span of individuals tends to infinity. If, alternatively, time has a finite starting point, in discrete time the degree of indeterminacy increases with the life-span of individuals, and, in continuous time, it is infinite; however, these are families of exponentially decreasing oscillations which, although they may exhibit pseudo-chaotic behaviour for a while, as time tends to infinity they all get damped, and asymptotic behaviour is that of the economy that originates in the infinite past. Stefano Lovo made interesting comments.