# Overlapping generations model

For the topic in population genetics, see Overlapping generations.

An overlapping generations model, abbreviated to OLG model, is a type of representative agent economic model in which agents live a finite length of time long enough to overlap with at least one period of another agent's life. As it models explicitly the different periods of life, - such as schooling, working or retirement periods --, it is the natural framework to study the allocation of resources across the different generations.

## History

The concept of an OLG model was inspired by Irving Fisher's monograph The Theory of Interest.[1] Notable improvements were published by Maurice Allais in 1947, Paul Samuelson in 1958, and Peter Diamond in 1965. Books devoted to the use of the OLG model include Azariadis' Intertemporal Macroeconomics[2] and de la Croix and Michel's Theory of Economic Growth.[3]

## Basic model

Generational Shifts in OLG Models

The most basic OLG model has the following characteristics:[4]

• Individuals live for two periods; in the first period of life, they are referred to as the Young. In the second period of life, they are referred to as the Old.
• A number of individuals is born in every period.N tt denotes the number of individuals born in period t.
• N t-1t denotes the number of old people in period t. Since the economy begins in period 1, in period 1 there is a group of people who are already old. They are referred to as the initial old. The number of them can be denoted as N0 .
• The size of the initial old generation is normalized to 1: N 00 = 1.
• People do not die early, so N tt = N tt+1.
• Population grows at a constant rate n:
${\displaystyle N_{t}^{t}=(1+n)^{t}}$
• In the "pure exchange economy" version of the model, there is only one physical good and it cannot endure for more than one period. Each individual receives a fixed endowment of this good at birth. This endowment is denoted as y.
• In the "production economy" version of the model (see Diamond OLG model below), the physical good can be either consumed or invested to build physical capital. Output is produced from labor and physical capital. Each household is endowed with one unit of time which is inelastically supply on the labor market.
• Preferences over consumption streams are given by
${\displaystyle u(c_{t}^{t},c_{t}^{t+1})=U(c_{t}^{t})+\beta U(c_{t}^{t+1}),}$
where ${\displaystyle \beta }$ is the rate of time preference.

## Attributes

One important aspect of the OLG model is that the steady state equilibrium need not be efficient, in contrast to general equilibrium models where the First Welfare Theorem guarantees Pareto efficiency. Because there are an infinite number of agents in the economy (summing over future time), the total value of resources is infinite, so Pareto improvements can be made by transferring resources from each young generation to the current old generation. Not every equilibrium is inefficient; the efficiency of an equilibrium is strongly linked to the interest rate and the Cass Criterion gives necessary and sufficient conditions for when an OLG competitive equilibrium allocation is inefficient.[5]

Another attribute of OLG type models is that it is possible that 'over saving' can occur when capital accumulation is added to the model—a situation which could be improved upon by a social planner by forcing households to draw down their capital stocks.[6] However, certain restrictions on the underlying technology of production and consumer tastes can ensure that the steady state level of saving corresponds to the Golden Rule savings rate of the Solow growth model and thus guarantee intertemporal efficiency. Along the same lines, most empirical research on the subject has noted that oversaving does not seem to be a major problem in the real world.[citation needed]

A third fundamental contribution of OLG models is that they justify existence of money as a medium of exchange. A system of expectations exists as an equilibrium in which each new young generation accepts money from the previous old generation in exchange for consumption. They do this because they expect to be able to use that money to purchase consumption when they are the old generation.[4]

OLG models allow us to look at intergenerational redistribution and systems such as Social Security.[7]

## Production

A OLG model with an aggregate neoclassical production was constructed by Peter Diamond.[6] A two-sector OLG model was developed by Oded Galor.[8]

Unlike the Ramsey–Cass–Koopmans model the steady state level of capital need not be unique.[9] Moreover, as demonstrated by Diamond (1965), the steady-state level of the capital labor ratio need not be efficient which is termed as "dynamic inefficiency".

### Diamond OLG Model

Convergence of OLG Economy to Steady State

The economy has the following characteristics:[10]

• Two generations are alive at any point in time, the young (age 1) and old (age 2).
• The size of the young generation in period t is given by Nt = N0 Et.
• Households work only in the first period of their life and earn Y1,t income. They earn no income in the second period of their life (Y2,t+1 = 0)
• They consume part of their first period income and save the rest to finance their consumption when old.
• At the end of period t, the assets of the young are the source of the capital used for aggregate production in period t+1.So Kt+1 = Nt,a1,t where a1,t is the assets per young household after their consumption in period 1. In addition to this there is no depreciation.
• The old in period t own the entire capital stock and consume it entirely, so dissaving by the old in period t is given by Nt-1,a1,t-1 = Kt.
• Labor and capital markets are perfectly competitive and the aggregate production technology is CRS, Y = F(K,L).

In Diamond's version of the model, individuals tend to save more than is socially optimal, leading to dynamic inefficiency. Subsequent work has investigated whether dynamic inefficiency is a characteristic in some economies[11] and whether government programs to transfer wealth from young to poor do reduce dynamic inefficiency[citation needed].

## References

1. ^ Aliprantis, Brown & Burkinshaw (1988, p. 229):

Aliprantis, Charalambos D.; Brown, Donald J.; Burkinshaw, Owen (April 1988). "5 The overlapping generations model (pp. 229–271)". Existence and optimality of competitive equilibria (1990 student ed.). Berlin: Springer-Verlag. pp. xii+284. ISBN 3-540-52866-0. MR 1075992.

2. ^ "Wiley: Intertemporal Macroeconomics - Costas Azariadis". eu.wiley.com. Retrieved 2015-10-24.
3. ^ "A Theory of Economic Growth - 9780521001151 - Cambridge University Press". www.cambridge.org. Retrieved 2015-10-24.
4. ^ a b Lars Ljungqvist; Thomas J. Sargent (1 September 2004). Recursive Macroeconomic Theory. MIT Press. pp. 264–267. ISBN 978-0-262-12274-0.
5. ^ Cass, David (1972). "On capital overaccumulation in the aggregative neoclassical model of economic growth: a complete characterization". Journal of Economic Theory. 4 (2): 200–223. doi:10.1016/0022-0531(72)90149-4.
6. ^ a b Diamond, Peter (1965). "National debt in a neoclassical growth model". American Economic Review. 55 (5): 1126–1150.
7. ^ Imrohoroglu, Selahattin; Imrohoroglu, Ayse; Joines, Douglas (1999). "Social Security in an Overlapping Generations Economy with Land". Review of Economic Dynamics. 2 (3).
8. ^ Galor, Oded (1992). "A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System". Econometrica. 60 (6): 1351–1386. JSTOR 2951525.
9. ^ Galor, Oded; Ryder, Harl E. (1989). "Existence, uniqueness, and stability of equilibrium in an overlapping-generations model with productive capital". Journal of Economic Theory. 49 (2): 360–375. doi:10.1016/0022-0531(89)90088-4.
10. ^ Carrol, Christopher. OLG Model.
11. ^ N. Gregory Mankiw; Lawrence H. Summers; Richard J. Zeckhauser (1 May 1989). "Assessing Dynamic Efficiency: Theory and Evidence". Review of Economic Studies. 56 (1). pp. 1–19. doi:10.2307/2297746.