Overlapping generations model

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]

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]

Generational Shifts in OLG Models

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.

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.

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.

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]

• Peter A. Diamond
• Karl Shell
• Macroeconomic model
• First welfare theorem
• Walrasian equilibrium

• Acemoğlu, Daron (2008). "Growth with Overlapping Generations". Introduction to Modern Economic Growth. Princeton University Press. pp. 327–358. ISBN 978-0-691-13292-1.
• Barro, Robert J.; Sala-i-Martin, Xavier (2004). "Appendix: Overlapping-Generations Models". Economic Growth (Second ed.). New York: McGraw-Hill. pp. 190–200. ISBN 978-0-262-02553-9.
• Blanchard, Olivier Jean; Fischer, Stanley (1989). "The Overlapping Generations Model". Lectures on Macroeconomics. Cambridge: MIT Press. pp. 91–152. ISBN 978-0-262-02283-5.
• Romer, David (2006). "Infinite-Horizon and Overlapping-Generations Models". Advanced Macroeconomics (3rd ed.). New York: McGraw Hill. pp. 47–97. ISBN 978-0-07-287730-4.
• Weil, Philippe (2008). "Overlapping Generations: The First Jubilee". Journal of Economic Perspectives. 22 (4): 115–34. CiteSeerX 10.1.1.513.4087. doi:10.1257/jep.22.4.115.
• Azariadis, Costas (1993), "Intertemporal Macroeconomics", Wiley-Blackwell, ISBN 978-1-55786-366-9.
• de la Croix, David; Michel, Philippe (2002), "A Theory of Economic Growth - Dynamics and Policy in Overlapping Generations", Cambridge University Press, ISBN 9780521001151.