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{{Short description|Framework in macroeconomics}}

{{For|the topic in population genetics|Overlapping generations}}

{{Macroeconomics sidebar}}

An '''overlapping generations model''', abbreviated to '''OLG model''', is a type of representative agent [[economics|economic]] model in which [[Agent (economics)|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

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 model|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.

across the different generations.

The OLG model is the natural framework for the study of: (a) the life-cycle behavior (investment in [[human capital]], work and [[Retirement plan|saving]] for [[retirement]]), (b) the implications of the [[resource allocation|allocation of resources]] across the generations, such as [[Social security|Social Security]], on the [[income per capita]] in the long-run,<ref>{{cite journal|last1=Imrohoroglu|first1=Selahattin|last2=Imrohoroglu|first2=Ayse|last3=Joines|first3=Douglas|year=1999|title=Social Security in an Overlapping Generations Economy with Land|journal=Review of Economic Dynamics|volume=2|issue=3|pages=638–665|doi=10.1006/redy.1999.0066}}</ref> (c) the determinants of economic growth in the course of human history, and (d) the factors that triggered the [[Demographic transition|fertility transition]].

== History ==

The construction of the OLG model was inspired by [[Irving Fisher]]'s monograph ''The Theory of Interest''.<ref name="ABB29">{{harvtxt|Aliprantis|Brown|Burkinshaw|1988|p=229}}:

The concept of an OLG model was inspired by [[Irving Fisher]]'s monograph ''The Theory of Interest''.<ref name="ABB29">{{harvtxt|Aliprantis|Brown|Burkinshaw|1988|p=229}}:<p>{{cite book|last1=Aliprantis |first1=Charalambos&nbsp;D.|authorlink1=Charalambos D. Aliprantis| last2=Brown|first2=Donald&nbsp;J.|last3=Burkinshaw|first3=Owen|title=Existence and optimality of competitive equilibria|chapter=5 The overlapping generations model (pp.&nbsp;229–271)|publisher=Springer-Verlag|___location=Berlin|date=April 1988|edition=1990 student|pages=xii+284|isbn=3-540-52866-0|mr=1075992}}</ref> 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 [[Costas Azariadis|Azariadis]]' Intertemporal Macroeconomics<ref>{{Cite web|title = Wiley: Intertemporal Macroeconomics - Costas Azariadis|url = http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1557863660.html|website = eu.wiley.com|accessdate = 2015-10-24}}</ref> and [[David de la Croix|de la Croix]] and [[Philippe Michel (economist)|Michel]]'s Theory of Economic Growth.<ref>{{Cite web|title = A Theory of Economic Growth - 9780521001151 - Cambridge University Press|url = https://www.cambridge.org/asia/catalogue/catalogue.asp?isbn=9780521001151|website = www.cambridge.org|accessdate = 2015-10-24}}</ref>

{{cite book|title=Existence and optimality of competitive equilibria|last1=Aliprantis|first1=Charalambos&nbsp;D.|last2=Brown|first2=Donald&nbsp;J.|last3=Burkinshaw|first3=Owen|date=April 1988|publisher=Springer-Verlag|isbn=978-3-540-52866-1|edition=1990 student|___location=Berlin|pages=xii+284|chapter=5 The overlapping generations model (pp.&nbsp;229–271)|mr=1075992|author-link1=Charalambos D. Aliprantis}}</ref> 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.<ref>{{Cite journal|last=Samuelson|first=Paul A.|date=1958|title=An exact consumption-loan model of interest with or without the social contrivance of money|journal=Journal of Political Economy|volume=66|issue=6|pages=467–482|doi=10.1086/258100|s2cid=153586213 }}</ref> In 1965, [[Peter Diamond]]<ref name="Diamond65" /> 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]],<ref name=":0">{{cite journal|last1=Galor|first1=Oded|author-link=Oded Galor|year=1992|title=A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System|journal=[[Econometrica]]|volume=60|issue=6|pages=1351–1386|jstor=2951525|doi=10.2307/2951525}}</ref> and the introduction of OLG models with endogenous fertility.<ref name=":1">{{Cite journal|last1=Galor|first1=Oded|last2=Weil|first2=David N.|date=1996|title=The gender gap, fertility, and growth|journal=American Economic Review|volume=86|issue=3|pages=374–387}}</ref><ref name=":2">{{Cite journal|last1=Galor|first1=Oded|last2=Weil|first2=David N.|date=2000|title=Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyond|journal=American Economic Review|volume=90|issue=4|pages=806–828|doi=10.1257/aer.90.4.806|citeseerx=10.1.1.195.5342}}</ref>

The concept of an OLG model was inspired by [[Irving Fisher]]'s monograph ''The Theory of Interest''.<ref name="ABB29">{{harvtxt|Aliprantis|Brown|Burkinshaw|1988|p=229}}:<p>{{cite book|last1=Aliprantis |first1=Charalambos&nbsp;D.|authorlink1=Charalambos D. Aliprantis| last2=Brown|first2=Donald&nbsp;J.|last3=Burkinshaw|first3=Owen|title=Existence and optimality of competitive equilibria|chapter=5 The overlapping generations model (pp.&nbsp;229–271)|publisher=Springer-Verlag|___location=Berlin|date=April 1988|edition=1990 student|pages=xii+284|isbn=3-540-52866-0|mr=1075992}}</ref> 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 [[Costas Azariadis|Azariadis]]' Intertemporal Macroeconomics<ref>{{Cite web|title = Wiley: Intertemporal Macroeconomics - Costas Azariadis|url = http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1557863660.html|website = eu.wiley.com|accessdateaccess-date = 2015-10-24}}</ref> and [[David de la Croix|de la Croix]] and [[Philippe Michel (economist)|Michel]]'s Theory of Economic Growth.<ref>{{Cite web|title = A Theory of Economic Growth - 9780521001151 - Cambridge University Press|url = https://www.cambridge.org/asia/catalogue/catalogue.asp?isbn=9780521001151|website = www.cambridge.org|accessdateaccess-date = 2015-10-24}}</ref>

==Basic Pure-exchange OLG model ==

[[File:OLG model- Generation.png|thumb|Generational Shifts in OLG Models]]

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*<math>N^{t-1}_t </math> 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 <math>N_0 </math> .

*The size of the initial old generation is normalized to 1: <math>N^{0}_0 =1</math>.

*People do not die early, so <math>N^{t}_t </math> = <math>N^{t}_{t+1} </math>.

*Population grows at a constant rate n:

::<math> N_t^t = (1+n)^t </math>

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:where <math> \beta </math> is the rate of time preference.

== OLG model with production ==

==Attributes==

=== Basic one-sector OLG model ===

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 condition]]s for when an OLG competitive equilibrium allocation is inefficient.<ref name="Cass72">{{cite journal | last1 = Cass| first1 = David | authorlink=David Cass| year=1972 |title= On capital overaccumulation in the aggregative neoclassical model of economic growth: a complete characterization | journal =[[Journal of Economic Theory]] | volume = 4| pages = 200–223 | doi = 10.1016/0022-0531(72)90149-4 | issue = 2}}</ref>

The pure-exchange OLG model was augmented with the introduction of an aggregate neoclassical production by [[Peter Diamond]].<ref name="Diamond65" />  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,<ref>{{cite journal|last1=Galor|first1=Oded|author-link=Oded Galor|last2=Ryder|first2=Harl E.|year=1989|title=Existence, uniqueness, and stability of equilibrium in an overlapping-generations model with productive capital|journal=[[Journal of Economic Theory]]|volume=49|issue=2|pages=360–375|doi=10.1016/0022-0531(89)90088-4}}</ref> 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.

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]].<ref>{{Cite journal|last=Galor|first=Oded|date=1996|title=Convergence? Inferences from theoretical models|url=https://www.brown.edu/academics/economics/sites/brown.edu.academics.economics/files/uploads/1996-3.pdf|journal=The Economic Journal|volume=106|issue=437|pages=1056–1069|doi=10.2307/2235378|jstor=2235378}}</ref>[[File:OLG Model - Diamond.png|thumb|Convergence of OLG Economy to Steady State]]

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.<ref name="Diamond65">{{cite journal | last1 = Diamond| first1 = Peter | authorlink=Peter Diamond| year=1965 |title= National debt in a neoclassical growth model | journal =[[American Economic Review]] | volume = 55| pages = 1126–1150 | issue = 5}}</ref> 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|date=May 2012}}

The economy has the following characteristics:<ref>{{cite book|title=OLG Model|last=Carrol|first=Christopher|title=OLG Model}}</ref>

*Two generations are alive at any point in time, the young (age 1) and old (age 2).

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.<ref name="LjungqvistSargent2004"/>

*The size of the young generation in period t is given by N_t = N₀ E^t.

*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 K_t+1 = N_t_,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 N_t-1_,a₁_,_t-1 = K_t.

*Labor and capital markets are perfectly competitive and the aggregate production technology is CRS, Y = F(K,L).

=== Two-sector OLG model ===

OLG models allow us to look at intergenerational redistribution and systems such as [[Social Security (United States)|Social Security]].<ref>{{cite journal | last1= Imrohoroglu | first1= Selahattin | last2=Imrohoroglu | first2 =Ayse | last3= Joines | first3= Douglas | year=1999 | title=Social Security in an Overlapping Generations Economy with Land | journal = Review of Economic Dynamics | volume =2 | issue=3 }}</ref>

The one-sector OLG model was further augmented with the introduction of a two-sector OLG model by [[Oded Galor]].<ref name=":0" /> 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,<ref>{{Cite journal|last=Uzawa|first=Hirofumi|date=1964|title=Optimal growth in a two-sector model of capital accumulation|journal=The Review of Economic Studies|volume=31|issue=1|pages=1–24|doi=10.2307/2295932|jstor=2295932}}</ref> 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.

=== OLG model with endogenous fertility ===

==Production==

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,<ref name=":1" /> (b) the contribution of the rise in the return to human capital and the decline in fertility to the transition from stagnation to growth,<ref name=":2" /><ref>{{Cite journal|last1=Galor|first1=Oded|last2=Moav|first2=Omer|date=2002|title=Natural selection and the origin of economic growth|journal=The Quarterly Journal of Economics|volume=117|issue=4|pages=1133–1191|doi=10.1162/003355302320935007|citeseerx=10.1.1.199.2634}}</ref> and (c) the importance of population adjustment to technological progress for the emergence of the [[Malthusian trap]].<ref>{{Cite journal|last1=Ashraf|first1=Quamrul|last2=Galor|first2=Oded|date=2011|title=Dynamics and stagnation in the Malthusian epoch|journal=American Economic Review|volume=101|issue=5|pages=2003–2041|doi=10.1257/aer.101.5.2003|pmid=25506082|pmc=4262154}}</ref>

== Dynamic inefficiency ==

A OLG model with an aggregate neoclassical production was constructed by [[Peter Diamond]].<ref name="Diamond65"/> A two-sector OLG model was developed by [[Oded Galor]].<ref>{{cite journal | last1 = Galor| first1 = Oded | authorlink=Oded Galor |year=1992 |title= A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System| journal =[[Econometrica]] | volume = 60| pages = 1351–1386 | jstor=2951525 | issue = 6 }}</ref>

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 [[FirstFundamental Welfaretheorems Theoremof welfare economics|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,<ref>{{Cite book |last=Acemoglu |first=Daron |title=Introduction to modern economic growth |date=2009 |publisher=Princeton University Press |isbn=978-0-691-13292-1 |___location=Princeton, New Jersey Oxford}}</ref> similar to the logic described in the [[Hilbert Hotel]]. 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 condition]]s for when an OLG competitive equilibrium allocation is inefficient.<ref name="Cass72">{{cite journal | last1 = Cass| first1 = David | authorlinkauthor-link=David Cass| year=1972 |title= On capital overaccumulation in the aggregative neoclassical model of economic growth: a complete characterization | journal =[[Journal of Economic Theory]] | volume = 4| pages = 200–223 | doi = 10.1016/0022-0531(72)90149-4 | issue = 2}}</ref>

Another attribute of OLG type models is that it is possible that '[[Global saving glut|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.<ref name="Diamond65">{{cite journal | last1 = Diamond| first1 = Peter | authorlinkauthor-link=Peter Diamond| year=1965 |title= National debt in a neoclassical growth model | journal =[[American Economic Review]] | volume = 55| pages = 1126–1150 | issue = 5}}</ref> 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|date=May 2012}}

Unlike the [[Ramsey–Cass–Koopmans model]] the steady state level of capital need not be unique.<ref>{{cite journal | last1 = Galor| first1 = Oded | authorlink=Oded Galor | last2 = Ryder| first2 = Harl E. |year=1989 |title= Existence, uniqueness, and stability of equilibrium in an overlapping-generations model with productive capital| journal =[[Journal of Economic Theory]] | volume = 49| pages = 360–375 | doi = 10.1016/0022-0531(89)90088-4 | issue = 2 }}</ref> Moreover, as demonstrated by Diamond (1965), the steady-state level of the capital labor ratio need not be efficient which is termed as "[[dynamic efficiency|dynamic inefficiency]]".

In Diamond's version of the model, individuals tend to save more than is socially optimal, leading to [[Dynamic efficiency|dynamic inefficiency]]. Subsequent work has investigated whether dynamic inefficiency is a characteristic in some economies<ref name="Mankiw89">{{cite news|author1=N. Gregory Mankiw|author2=Lawrence H. Summers|author3=Richard J. Zeckhauserjournal|title=Assessing Dynamic Efficiency: Theory and Evidence|author1=N. Gregory Mankiw|date=1 May 1989|journal =[[Review of Economic Studies]] |author2=Lawrence volumeH. Summers|issue=1|volume= 56| pages = 1–19 | doi = 10.2307/2297746 |author3=Richard issueJ. Zeckhauser|jstor= 12297746}}</ref> and whether government programs to transfer wealth from young to poor do reduce dynamic inefficiency{{Citation needed|date=November 2014}}.

===Diamond OLG Model===

[[File:OLG Model - Diamond.png|thumb|Convergence of OLG Economy to Steady State]]

The economy has the following characteristics:<ref>{{cite book|last=Carrol|first=Christopher|title=OLG Model}}</ref>

*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 N_t = N₀ E^t.

*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 K_t+1 = N_t_,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 N_t-1_,a₁_,_t-1 = K_t.

*Labor and capital markets are perfectly competitive and the aggregate production technology is CRS, Y = F(K,L).

A thirdAnother 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.<ref name="LjungqvistSargent2004" />

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<ref name="Mankiw89">{{cite news|author1=N. Gregory Mankiw|author2=Lawrence H. Summers|author3=Richard J. Zeckhauser|title=Assessing Dynamic Efficiency: Theory and Evidence|date=1 May 1989|journal =[[Review of Economic Studies]] | volume = 56| pages = 1–19 | doi = 10.2307/2297746 | issue = 1}}</ref> and whether government programs to transfer wealth from young to poor do reduce dynamic inefficiency{{Citation needed|date=November 2014}}.

==See also==

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* [[Karl Shell]]

* [[Macroeconomic model]]

* [[Fundamental theorems of welfare economics|First welfare theorem]]

* [[First Welfare Theorem]]

* [[Walrasian equilibrium]]

Line 65 ⟶ 73:

==Further reading==

* {{cite book |first=Daron |last=Acemoğlu |chapter=Growth with Overlapping Generations |title=Introduction to Modern Economic Growth |___location= |publisher=Princeton University Press |year=2008 |isbn=978-0-691-13292-1 |pages=327–358 }}

* {{cite book |first1=Robert J. |last1=Barro |authorlink1author-link1=Robert J. Barro |first2=Xavier |authorlink2author-link2=Xavier Sala-i-Martin |last2=Sala-i-Martin |chapter=Appendix: Overlapping-Generations Models |title=Economic Growth |___location=New York |publisher=McGraw-Hill |year=2004 |edition=Second |isbn=978-0-262-02553-19 |chapterurlchapter-url={{Google books |plainurl=yes |id=jD3ASoSQJ-AC |page=190 }} |pages=190–200 }}

* {{cite book |firstfirst1=Olivier Jean |lastlast1=Blanchard |first2=Stanley |last2=Fischer |chapter=The Overlapping Generations Model |title=Lectures on Macroeconomics |___location=Cambridge |publisher=MIT Press |year=1989 |isbn=978-0-262-02283-45 |pages=91–152 |chapterurlchapter-url=https://books.google.com/books?id=j_zs7htz9moC&pg=PA91 }}

* {{cite book |last=Romer |first=David |year=2006 |edition=3rd |chapter=Infinite-Horizon and Overlapping-Generations Models |title=Advanced Macroeconomics |___location=New York |publisher=McGraw Hill |pages=47–97 |isbn=978-0-07-287730-84 }}

* {{cite journal |last=Weil |first=Philippe |year=2008 |title=Overlapping Generations: The First Jubilee |journal=[[Journal of Economic Perspectives]] |volume=22 |issue=4 |pages=115–34 |doi=10.1257/jep.22.4.115 |citeseerx=10.1.1.513.4087 }}

* [[Costas Azariadis|Azariadis, Costas]] (1993), "Intertemporal Macroeconomics", Wiley-Blackwell, {{ISBN |978-1-55786-366-9}}.

* [[David de la Croix|de la Croix, David]]; [[Philippe Michel (economist)|Michel, Philippe]] (2002), "A Theory of Economic Growth - Dynamics and Policy in Overlapping Generations", Cambridge University Press, {{ISBN |9780521001151}}.

{{MacroeconomicsEconomics}}

[[Category:Economics models]]