Models of overlapping generations

Fig. 1: Basic structure of an OLG model: Each generation lives for a certain time (here: two periods); the young from the new generation ( orange ) coexist with the old from the previous generation ( blue ).

In economics, models of overlapping generations or overlapping generation models (OLG models) are a specific class of theoretical models that describe the long-term development of an economy. OLG models are characterized by the fact that individuals are grouped into generations , with each generation always living for a certain (finite) time over an infinite time horizon and going through stages of life (for example “young” and “old”); It is named after the fact that at least two generations are always alive at the same time (each in different phases of life), which amounts to an "overlapping" of the generations.

Basic model in a pure exchange economy (Samuelson model)

Fig. 2: Effect of increasing population growth.

Samuelson (1958) starts from a simple scenario in which individuals have a certain initial configuration of a good that they cannot take over from one period to the next; In other words, the shelf life of the equipment is exactly one period or, in Samuelson's formulation, the interest rate is −1. Time is assumed to be discrete and infinite; the observed periods are given by. Now assume, as usual, that an individual lives for two periods. Every generation consists of individuals and the intertemporal equipment vector of each individual of generation t is given through , whereby stands for equipment in the first phase of life (“young”) and for that in the second phase of life (“old”); The intertemporal consumption vector of every individual of generation t is defined analogously . The intertemporal utility function is loud and, according to common assumptions, is strictly monotonically increasing , strictly concave and twice continuously differentiable (in each case in both arguments). The OLG economy described in this way is subject to the (aggregated) budget constraint in period t ${\displaystyle t=0,1,\ldots }$ ${\displaystyle N_{t}}$ ${\displaystyle (\omega _{t}^{j},\omega _{t}^{a})}$ ${\displaystyle \omega _{t}^{j}}$ ${\displaystyle \omega _{t}^{a}}$ ${\displaystyle (c_{t}^{j},c_{t}^{a})}$ ${\displaystyle U_{t}(c_{t}^{j},c_{t}^{a})}$

${\displaystyle N_{t-1}c_{t-1}^{a}+N_{t}c_{t}^{j}=N_{t-1}\omega _{t-1}^{a}+N_{t}\omega _{t}^{j}}$,

which is to be understood as follows: The consumption of all "old" from the previous generation plus the consumption of all "young" from the current generation (hence the total consumption of all living individuals) must correspond to the current equipment of all living individuals.

If the equation is set with the aid of the population growth rate ,, the following equivalent, but more easily accessible form for a meaningful graphical representation, results: ${\displaystyle n_{t}\equiv (N_{t}-N_{t-1})/N_{t-1}}$

${\displaystyle c_{t-1}^{a}-\omega _{t-1}^{a}=(1+n_{t})\cdot \left(\omega _{t}^{j}-c_{t}^{j}\right)}$

It illustrates the inter-generational interdependence that defines the OLG model: all consumption by the old population that exceeds their own resources is financed in full from the entire savings made by the younger population. Fig. 2 illustrates this. If every generation living in t consumes exactly the amount of their equipment, one is in the so-called equipment point . An exogenously induced redistribution of per capita consumption of the younger generation by one unit in favor of the old generation (movement to the top left) would result in an increase in per capita consumption for them. Changes in population growth lead to a rotation of the budget line by . ${\displaystyle E_{0}}$ ${\displaystyle 1+n_{t}}$ ${\displaystyle E_{0}}$

Model with production

Diamond (1965) extends the basic model to include companies and a production technology and thus analyzes the effects of national debt. Later literature often relies on this expanded model, which is referred to in literature as the Diamond model after its creator, Nobel Prize winner Peter A. Diamond .

Utility function in the OLG model

In the OLG model with subsistence consumption, the lifetime utility of households can be described by the following utility function

${\displaystyle U_{t}={\begin{cases}\operatorname {log} (c_{t}-{\bar {c}})+\beta \operatorname {log} (c_{t+1})&{\text{ falls }}c_{t}>{\bar {c}}\\-\infty &{\text{sonst.}}\end{cases}}}$

Representations:

${\displaystyle c_{t}}$ denotes the consumption of households in the period ${\displaystyle t}$

${\displaystyle c_{t+1}}$ denotes the consumption of households in the period ${\displaystyle t+1}$

${\displaystyle {\bar {c}}}$ denotes subsistence consumption

Web links

• John Geanakoplos : Financial Theory. Lecture 12 - Overlapping Generations Models of the Economy. Lecture recording and script. Open Yale Courses, Yale University, 2009. [English]

literature

• Charalambos D. Aliprantis, Donald J. Brown, and Owen Burkishaw: Existence and Optimality of Competitive Equilibria. Springer, Berlin a. a. 1989, ISBN 3-540-50811-2 . [Chapter 5, pp. 229-271]
• Truman F. Bewley: General Equilibrium, Overlapping Generations Models, and Optimal Growth Theory. Harvard University Press, Cambridge and London 2007, ISBN 978-0-674-02288-1 . [Chapters 9 and 10, pp. 360–589]
• Karl Farmer and Matthias Schelnast: Growth and International Trade. An Introduction to the Overlapping Generations Approach. Springer, Heidelberg a. a. 2013, ISBN 978-3-642-33668-3 (e-book: doi : 10.1007 / 978-3-642-33669-0 ).
• John Geanakoplos: Overlapping generations model of general equilibrium. In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, doi : 10.1057 / 9780230226203.1236 (online edition).
• Lars Ljungqvist and Thomas J. Sargent : Recursive Macroeconomic Theory. 3. Edition. MIT Press, Cambridge and London 2012, ISBN 978-0-262-01874-6 . [Chapter 9, pp. 315–362]
• Mich Tvede: Overlapping Generations Economies. Palgrave Macmillan, New York 2010, ISBN 978-0-230-24334-7 .
• Philippe Weil: Overlapping Generations: The First Jubilee. In: Journal of Economic Perspectives. 22, No. 4, 2008, pp. 115–143, doi : 10.1257 / jep.22.4.115 (freely accessible at http://www.philippeweil.com/research/OLGjubilee.pdf , accessed on August 13, 2013) . [Basic model and literature review]

Remarks

1. ^ Paul A. Samuelson : An Exact Consumption-Loan Model of Interest With or Without the Social Contrivance of Money. In: Journal of Political Economy. 66, No. 6, 1958, pp. 467-482 ( JSTOR 1826989 ).
2. ↑ The following presentation differs from Samuelson's model setting in several respects. Samuelson assumes, for example, that individuals live three periods and assumes that they have zero equipment in their last phase of life.
3. ↑ If this does not seem intuitive, consider that after defining the growth rate. ${\displaystyle (1+n_{t})=N_{t}/N_{t-1}}$
4. ^ Peter A. Diamond: National Debt in a Neoclassical Growth Model. In: The American Economic Review. 55, No. 5, 1965, pp. 1126–1150 ( JSTOR 1809231 ) (also online free of charge: PDF file ( memento of the original from March 5, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this note. , 0.7 MB).