# Overlapping generations model

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The overlapping generations (OLG) model is one of the dominating frameworks of analysis in the study of macroeconomic dynamics and economic growth. In contrast, to the  Ramsey–Cass–Koopmans neoclassical growth model in which individuals are infinitely-lived, in the OLG model individuals live a finite length of time, long enough to overlap with at least one period of another agent's life.

Economic growth is the increase in the inflation-adjusted market value of the goods and services produced by an economy over time. It is conventionally measured as the percent rate of increase in real gross domestic product, or real GDP.

The Ramsey–Cass–Koopmans model, or Ramsey growth model, is a neoclassical model of economic growth based primarily on the work of Frank P. Ramsey, with significant extensions by David Cass and Tjalling Koopmans. The Ramsey–Cass–Koopmans model differs from the Solow–Swan model in that the choice of consumption is explicitly microfounded at a point in time and so endogenizes the savings rate. As a result, unlike in the Solow–Swan model, the saving rate may not be constant along the transition to the long run steady state. Another implication of the model is that the outcome is Pareto optimal or Pareto efficient.

## Contents

The OLG model is the natural framework for the study of: (a) the life-cycle behavior (investment in human capital, work and saving for retirement), (b) the implications of the allocation of resources across the generations, such as Social Security, on the income per capita in the long-run, [1] (c) the determinates of economic growth in the course of human history, and (d) the factors that triggered the fertility transition.

Human capital is the stock of habits, knowledge, social and personality attributes embodied in the ability to perform labour so as to produce economic value.

Retirement is the withdrawal from one's position or occupation or from one's active working life. A person may also semi-retire by reducing work hours.

In economics, resource allocation is the assignment of available resources to various uses. In the context of an entire economy, resources can be allocated by various means, such as markets or planning.

## History

The construction of the OLG model was inspired by Irving Fisher's monograph The Theory of Interest. [2] It was first formulated in 1947, in the context of a pure-exchange economy, by Maurice Allais, and more rigorously by Paul Samuelson in 1958. [3] In 1965, Peter Diamond [4]  incorporated an aggregate neoclassical production into the model. This OLG model with production was further augmented with the development of the two-sector OLG model by Oded Galor, [5] and the introduction of OLG models with endogenous fertility. [6] [7]

Irving Fisher was an American economist, statistician, inventor, and Progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt deflation has been embraced by the post-Keynesian school. Joseph Schumpeter described him as "the greatest economist the United States has ever produced", an assessment later repeated by James Tobin and Milton Friedman.

Maurice Félix Charles Allais was a French physicist and economist, the 1988 winner of the Nobel Memorial Prize in Economic Sciences "for his pioneering contributions to the theory of markets and efficient utilization of resources", for Maurice Allais contribution, along with John Hicks and Paul Samuelson, to neoclassical synthesis. They formalize the self-regulation of markets, that Keynes refuted, while reiterating some of his ideas.

Paul Anthony Samuelson was an American economist. The first American to win the Nobel Memorial Prize in Economic Sciences, the Swedish Royal Academies stated, when awarding the prize in 1970, that he "has done more than any other contemporary economist to raise the level of scientific analysis in economic theory". Economic historian Randall E. Parker has called him the "Father of Modern Economics", and The New York Times considered him to be the "foremost academic economist of the 20th century".

Books devoted to the use of the OLG model include Azariadis' Intertemporal Macroeconomics [8] and de la Croix and Michel's Theory of Economic Growth. [9]

Constantine Christos "Costas" Azariadis is a macroeconomist born in Athens, Greece. He has worked on numerous topics, such as labor markets, business cycles, and economic growth and development. Azariadis originated and developed implicit contract theory.

David de la Croix is a Belgian scholar and author in the field of economic growth and demographic economics. He is professor at the University of Louvain (UCLouvain).

Philippe Michel was a French mathematical economist.

## The Pure-Exchange OLG Model

The most basic OLG model has the following characteristics: [10]

• 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 are born in every period. ${\displaystyle N_{t}^{t}}$ denotes the number of individuals born in period t.
• ${\displaystyle N_{t}^{t-1}}$ 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 ${\displaystyle N_{0}}$ .
• The size of the initial old generation is normalized to 1: ${\displaystyle N_{0}^{0}=1}$.
• People do not die early, so ${\displaystyle N_{t}^{t}=N_{t+1}^{t}}$.
• 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.

## The OLG Model with Production

### The Basic One-Sector OLG Model

The pure-exchange OLG model was augmented with the introduction of an aggregate neoclassical production by Peter Diamond. [4]  In contrast, to Ramsey–Cass–Koopmans neoclassical growth model in which individuals are infinitely-lived and the economy is characterized by a unique steady-state equilibrium, as was established by Oded Galor and Harl Ryder, [11] the OLG economy may be characterized by multiple steady-state equilibria, and initial conditions may therefore affect the long-run evolution of the long-run level of income per capita.

Peter Arthur Diamond is an American economist known for his analysis of U.S. Social Security policy and his work as an advisor to the Advisory Council on Social Security in the late 1980s and 1990s. He was awarded the Nobel Memorial Prize in Economic Sciences in 2010, along with Dale T. Mortensen and Christopher A. Pissarides. He is an Institute Professor at the Massachusetts Institute of Technology. On June 6, 2011, he withdrew his nomination to serve on the Federal Reserve's board of governors, citing intractable Republican opposition for 14 months.

Since initial conditions in the OLG model may affect economic growth in long-run, the model was useful for the exploration of the convergence hypothesis. [12]

The economy has the following characteristics: [13]

• 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).

### The Two-Sector OLG Model

The one-sector OLG model was further augmented with the introduction of a two-sector OLG model by Oded Galor. [5] The two-sector model provides a framework of analysis for the study of the sectoral adjustments to aggregate shocks and implications of international trade for the dynamics of comparative advantage. In contrast to the Uzawa two-sector neoclassical growth model, [14] the two-sector OLG model may be characterized by multiple steady-state equilibria, and initial conditions may therefore affect the long-run position of an economy.

### The OLG Model with Endogenous Fertility

Oded Galor and his co-authors develop OLG models where population growth is endogenously determined to explore: (a) the importance the narrowing of the gender wage gap for the fertility decline, [6] (b) the contribution of the rise in the return to human capital and the decline in fertility to the transition from stagnation to growth, [7] [15] and (c) the importance of population adjustment to technological progress for the emergence of the Malthusian trap. [16]

## Dynamic Inefficiency

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. [17]

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. [4] 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 ]

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 [18] and whether government programs to transfer wealth from young to poor do reduce dynamic inefficiency[ citation needed ].

Another 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. [10]

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