1. Introduction
People are born, grow, work, retire and die. This continues to be true forever.
But many formal economic models don’t reflect this truth of human being at all.
The representative individual or household will live or exist forever. It will make an optimal plan within infinite time horizon.
This unrealistic hypothesis may be eased considering parental concern for descendants and inheritance to children. But, of course, such economic customs aren’t complete.
So economists sometimes suppose an utterly opposite assumption. People work for one period after they grew up and spend retirement life for the next period. Their planning time horizons are restricted within their own life time.
This concept of overlapping generations model is normally used for considering the tendency of economic growth or the character of endogenous economic fluctuations. It, however, was proposed from another purpose at all and then
The Formation of Overlapping Generations Model and Its Implications for Japan and Korea as Aging Countries
Yoshihiro Yamazaki
＊＊Faculty of Economics, Fukuoka University, Fukuoka, Japan
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（ １ ）
ሺଵǡ ଶሻ ൌ ሺͳ െ ଵǡ Ͳ െ ଶሻǡ Ǥ Ǥ ଵ ୲ଶൌ Ͳ.
ଵሺ୲ሻ ୲ଶሺ୲ሻ ؠ Ͳ୲. introduced into macroeconomics.
This paper will follow up the history of overlapping generations briefly first and we will deduce some implication for Japan and Korea as aging countries from this model.
2. The Origin of Overlapping Generations Model−Samuelson
It is said to be Samuelson [1958] that proposed the concept of overlapping generations model originally.1 Samuelson built this model so as to show his biological origin of interest in an economy.
Samuelson supposes that people work for only one period after grown up and then enjoy retirement for the next period. People will solve the maximizing problem as follows ;
Here the utility function U depends on consumptions of each period C1, C2. People work in their young day and earn income 1. They save a part of income S1 for his old day. If we use symbolRtas the exchange ratio of the future goods to the present goods, the restricting condition of the above equation will be held.
This restricting condition, however, is actually the budget identity. It means
To clear the market, the condition
1 Some say that Maurice Allais devised in his 1947’s work in advance to Samuelsson. But I have not still checked it.
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Ͳ ൌ ୲ଵሺ୲ሻ ୲ିଵଶሺ୲ିଵሻ ൌ Ͳǡ േͳǡ േʹǡ ǥ
Ͳ ൌ ቂଵሺሻ ଵ
ଵା୫ଶሺሻቃ.
ൌଵା୫ଵ ൌ Ͳ ൏ ଵሺሻ ൌ െଶሺሻ.
is required. HereBi is the population that newly enter the labor market at timei.
IfB＝B(1＋m)tandRt＝Rt＋1＝…＝R, the last equation can become
Comparing this with the identity above, we can assure that this equation has a solution like
Here i is the interest rate of this community and Samuelson called this the biological rate of interest.
Samuelson didn’t assume any time preference and productivity effect at all.2 So zero or negative rates of interest are also possible. If the population grows, the biological rate of interest will be positive. But if it is stable or decreasing, it can be zero or negative.
But remember now that the consumptions and savings under this biological rate of interest may not be the optimal solution of behavior of an economic agent.
Actually old people will have nothing to exchange a part of goods that young people produce. Because the goods will go bad by the next period, they cannot stock them for themselves. But even if young people share the goods with old
2 These are corresponding to Boehm-Bawerk’s second and third reasons respectively.
As his first reason is economic growth, a negative interest rate is possible when the economy is decaying.
The Formation of Overlapping Generations Model and
Its Implications for Japan and Korea as Aging Countries（Yamazaki） −４０１−
（ ３ ）
people, they will find the elder gone in the next period. No one will give the goods back for their retirement.
The other option is that young people lend their goods to younger generation.
But this is utterly impossible because younger generation aren’t born yet. Because of this situation, the optimum solution is notS1＞0butS1＝S2＝0. As a result, R must become infinity and the interest rate minus 100 ％. And we can easily check this solution satisfies the last equation, too.
Samuelson introduced money as human being’s contrivance to avoid this paradoxical case. If there exists money and any young people willingly accept it, people can makeS1of money and but goods by the money in their old day.
When population grows at the rate m, output also grows at the same rate. If the amount of money fixed, the monetary price of goods will go down at the same rate as population. Because of this relation, the real interest rate is positive or negative corresponding to population growth or decay even when monetary rate of interest is zero.
3. The Introduction of Overlapping Generations Concept into Macroeconomics
−Modigliani
As we saw in the previous chapter, the original concept of overlapping generations by Samuelson was an abstract and microeconomic one. But five years later than Samuelson’s, Ando＝Modigliani [1963] introduced this concept into macroeconomic field.3
3 Modigliani first started this research with Brumberg. But his sudden death compelled Modigliani to continue the research with Ando and publish under the last two names.
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（ ４ ）
୲ൌ Ƚଵ୲ Ƚଶ୲ୣ Ƚଷ୲ିଵ.
୲ൌ ሺȽଵ Ƚଶሻ୲ Ƚଷ୲ൌ Ƚ୲ Ƚଷ୲ିଵ.
୲ൌ ୲ ୲ିଵെ ୲ൌ ሺͳ െ Ƚሻሾ୲ ୲ିଵሿ െ Ƚଷ୲ିଵ Ƚ୲ିଵ
ൌ ሺͳ െ Ƚሻሾ୲ ୲ିଵሿ െ ሺȽଷെ Ƚሻ୲ିଵ.
౪ି౪షభ
౪షభ ൌ ሺͳ െ Ƚሻ ቂଢ଼౪ା୰ ౪షభ
౪షభ െ୬ାଵିయି୰ቃ.
In Modigliani’s model, people work in their young day and save a part of income as assets. After retirement, they spend all of the saved income in their old day and leave no inheritance. Following the style of those days’ macroeconomics, people don’t solve the optimization problem to decide the consumption in each period. In his model, the community will make consumption expenditure according to the consumption function as follows ;
Here Ct, Yt, Yte and At−1 are aggregate consumption, labor income, expected future labor income and asset of the period t respectively. For simplicity, if we assume that people expect their future income the same as this period’s, the function above will become
If people earn asset income in addition to labor income and they save all of the former, saving functionStcan be shown as
Here r is the return on asset. Also asSt＝At−At−1, dividing the equation above byAt−1, we can obtain the relation like
Even though any ncan satisfy this equation, it can be interpreted as natural rate of economic growth.
The Formation of Overlapping Generations Model and
Its Implications for Japan and Korea as Aging Countries（Yamazaki） −４０３−
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ൌ୬ାଵିయି୰.
ୗ౪
ଢ଼౪ା୰౪షభൌ౪ି౪షభ
౪షభ
౪షభ ଢ଼౪ ൌ୬୦.
୲
୲ିଵ
ൌ െ
ൌଢ଼౪ିୗ౪
౪షభ ൌ୦ି୬୦ି୰
େ౪ ଢ଼౪
౪షభ
ଢ଼౪ .
C
Y C＝（h-n）/（h-r）Y
C＝αY＋α3A
We shall introduce the symbolh like
When the income-asset ratio achieves to h, the asset will also grow at the same rate asn. And then the saving-income ratio will be constant as
Similarly we can find the relations like
and
In the equations above, Yt-rAt−1 means the labor income. When the interest rate r is Samuelson’s biological rate of interest n, this consumption-income ratio must be 1. Thus we can show the relations using the figure below.
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ሺ ୲ଵǡ ୲ଶሻǤ Ǥ ୲ଵ ୲ൌ ୲ǡ ୲ଶൌ ሺͳ ୲ାଵሻ୲ ୲ଵଵା୰ୡ౪మ
౪శభൌ ୲. This figure means that an economy moves up along the long term consumption function C＝(h−n)/(h−r)Y as it accumulates asset A. But in economic fluctuation, though the short term Y will change temporally, asset A will not change. So the economy will deviate from the long term consumption function along the short term consumption functionC＝αY＋α3A.
4. Economic Growth in Overlapping Generations Model − Diamond and Others
Because Modigliani was a Keynesian, he concentrated the analysis to the demand side of an economy. But some economists started to use the overlapping generations model into the analysis of economic growth in relatively earlier stage of development of economics. Diamond [1965] was seemingly one of the first trials.
In their models, household has unit labor and earn wt in its young day. Its young days consumption isc1tand savingst. Saving goes to buy bonds issued by firms and comes back with real interest rate rt＋1. This money will be paid as its old days consumption c2t. In this situation, the household will solve this maximization problem.
By solving this problem, we can obtain such a saving function as st＝s(wt, rt＋1). If both of today’s consumption and tomorrow’s are normal goods, the first derivative by wage must be between 0 and 1. But because the income effect and substitute effect work in opposite directions, the signature of the first derivative by
The Formation of Overlapping Generations Model and
Its Implications for Japan and Korea as Aging Countries（Yamazaki） −４０５−
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ሺ୲ሻ െ ୲ᇱሺ୲ሻ ൌ ୲ǡ ᇱሺ୲ሻ ൌ ୲.
୲ାଵൌୱሾሺ୩౪ሻି୩౪ᇲሺ୩౪ሻǡᇲሺ୩౪ሻሿ
ଵା୬ .
kt＋1
kt
interest rate is undetermined. If today’s consumption and tomorrow’s are substitute, the household increases saving according to interest rate. But in the inverse case, it decreases saving.
When firm’s technology can be shown by production function per capita f(k), we can obtain the relation of marginal productivity like
These firms have totally Kt capital and Kt＝Lt−1st−1. Population Lt increases at the rate ofn. From these equations, we can deduce the dynamic equation like
If there exists a solutionk＊＝kt＋1＝kt, it is the stationary one. This solution must locate on the 45 degree line of the figure as below.
Overlapping generations model’s stable growth path, though, are very complicated. For example, the complicated curve in the figure above shows the case in which the economy shows a chaotic move. But if both production function and utility function are Cobb＝Douglas type, the model has only one
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（ ８ ）
ൌ ୟଵିୟൌ ଵିୟ.
ൌ ଵିୟǡ ൌଵିୟ୩.
ൌ ଵଵିୠଶୠ.
ଵൌ ሺͳ െ ሻǡ ଶൌ ሺͳ ሻ.
ሺͳ െ ሻ ሺͳ െ ሻ ൌ . non-zero solution and it is stable like the simple curve in the figure.
5. Some Simulations on Aging Economies
In this chapter, we specify the equations in the model. Firstly consider the closed economy. Production function is
From this, wage rate and interest rate are
And utility function is also Cobb＝Douglas type.
Maximizing this utility under the budget constraint w＝C1＋C2/(1＋r), we get the relation
Young people earn wL＝aY and spend (1−b)wL＝(1−b)aY totally. And old people earn asset income rK＝(1−a)Y and spend with their saving. So their expenditure is (1−a)Y＋K.
Then derive the stationary solution. Because the growth rate of population isn, the capital K also has to grow at the rate of n. This economy needs nK investment every year. So equilibrium of goods requires
The Formation of Overlapping Generations Model and
Its Implications for Japan and Korea as Aging Countries（Yamazaki） −４０７−
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ൌ
ͳ ൨
ଵ ୟ
ൌ ቂଵା୬ୟୠቃ
భష
ǡ ൌ ሺͳ െ ሻଵା୬ୟୠ.
୬
ଢ଼ ൌୟୠ୬
ଵା୬.
כൌୠ୵ଵା୬ൌ ቂଵିୟ୧ ቃ
భష
ଵା୬ൌ ଵା୬ଢ଼ . From this,
follows. This means that in aging society like Japan and Korea, capital per capita must increase.
On the other hand, wage rate and interest rate in equilibrium are
Because of these, we can say that in aging society, wage rate is high and interest rate low. But the distribution between wage and interest keeps constant in the process of aging. This character comes from the Cobb＝Douglas production function.
The saving rate is
This goes down in aging economies.
Finally we suppose an open economy. There the interest rate is given as i.
And the capital K＊that domestic people own and the capital K that is used in domestic production are different.
Because of this, the distribution ratio between capital and labor is
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୧כ
୵ൌଵା୬୧ୠ.
ሺכെ ሻ ൌ ቂଵିୟ୧ ቃ
భ
ቂሺଵିୟሻሺଵା୬ሻୟୠ୧ െ ͳቃ .
−600
−500
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2
−400
−300
−200
−100 0 100 200 300 400
From this relation, we can say that in aging society, the weight of capital income will increase.
Current surplus or the difference between investment and saving is
We can confirm this will decrease when population growth rate goes down. For example, ifa＝0.8,b＝0.6,4i＝0.95andL＝10000, then this current account will move as below when the raten goes down 0.96to−0.2.
4 This means that people’s subjective rate of discount is minus. So they want to consume more after their retirement than in their young days.
5 Suppose that the interval between generations is 40 year. Because this interest rate is 40 years’ one, the annual rate corresponding this is 0.016.
6 As we explain in the previous footnote, the annual rate of population growth is 0.01 when 40 years’ rate of the growth is 0.5.
The Formation of Overlapping Generations Model and
Its Implications for Japan and Korea as Aging Countries（Yamazaki） −４０９−
（ １１ ）
6. Conclusion
We have surveyed the history of overlapping generations model briefly and checked the implication to aging economy like Japan and Korea. It is a very abstract model and if you try to add some realistic assumption, it becomes almost too complicated to treat. But we can check it gives some useful implication even by using the simplest one.
References
Ando, A. and F. Modigliani, F., 1963, ‘The “Life Cycle” Hypothesis of Saving : Aggregate Implications and Tests,’American Economic Review, Vol.55, pp.55‐84.
Diamond, P. A., 1965, ‘National Debt in a Neoclassical Growth Model,’American Economic Review, Vol.55, pp.1126‐50.
Samuelson, P. A., 1958, ‘An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money,’The Journal of Political Economy, Vol. LXVI, pp.467‐82.
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