1. Introduction
The fact that Japanese government keeps much national debt and issues more and more every year is well-known all over the world nowadays. Even in the country, people are discussing the burden of this national debt loudly. Of course, some follows the wrong belief and the others make their opinion on any economic logic. One purpose of this paper is to make some suggestions to this situation of Japan’s from the viewpoint of macroeconomic growth theory.
Though even the economists including classic ones had also discussed the burden of national debt till the 1950s, it was Modigliani who introduced the problem into the field of macroeconomic growth theory for the first time. His model was so simple that it could not derive a full implication from the framework.
The model, however, gave the perspective into the following development.
Diamond developed the idea of Modigliani and reached the complete model with which an economist treated the questions of national debt. This was overlapping generations framework.
National Debt in the History of Macroeconomic Growth Theory
Yoshihiro Yamazaki ＊
＊
Faculty of Economics, Fukuoka University, Fukuoka, Japan
−７７−
（ １ ）
Although overlapping generations framework was proposed by Modigliani
1, it did not happen in his paper referred above. The paper was issued in 1961. And his overlapping generations paper was published 2 years later than that. This is also the reason why Modigliani could not propose the complete framework for the question.
Diamond completed the overlapping generations model as the analytical tool in macroeconomics in 1965’s paper of his. This paper was actually oriented to the clarification of national debt’s burden in the macroeconomic meaning. He referred to the optimized situation and investigated in what meaning national debt gave burden on an economy.
Here did I talk about optimization. If we understand this term concerning economic welfare, debt issuing can possibly be the optimizer of it. Barro proved what is called neutrality theorem of national debt in his paper published 10 years later than Diamond’s. We can make it clear in what case national debt helps optimize the economic situation by re-reading his paper.
To follow the development of macroeconomic growth theory in the light of national debt is another purpose of this paper.
2. Emergence of National Debt : Modigliani’s Model
Modigliani [1961] shed new light on the question of national debt. He thought the burden of national debt as the effect on the economic growth. The path of
1
When Modigliani proposed the model, Samuelson had already published the paper using overlapping generations framework. And he explained the reason of interest payment. Because of this, it was not connected with macroeconomic research on economic growth.
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（ ２ ）
economic growth depends on accumulation of capital when population increases constantly and there is no technological development.
He set a very simple model of an economy. In his model, income Y is constant for simplicity. Then the identity below holds among the income, consumption C and savings S .
Y＝C＋S
When we represent propensity to consume as c, this identity follows.
Y ＝ cY ＋ S
If government needs some money and collects a lump-sum tax from the generation, this identity among the differences will hold.
0 ＝− cT ＋ dS ＋ T dS＝−sT
Here s means propensity of savings. The last identity shows that the lump- sum tax will reduce the savings in the generation by the amount sT .
Of course, the tax will also reduce economic welfare of this generation through decreasing their disposable income and consumption by T and cT . In addition, however, another negative effect on the economy will remain for a while because the savings also decrease and the capital accumulation slows down temporarily.
The slowing-down of capital accumulation turned out to reduce the income for several years
2.
National Debt in the History of Macroeconomic Growth Theory（Yamazaki） −７９−
（ ３ ）
We shall then treat the case of national debt. The government raises money only by issuing national debt.
In this case, the identity among the differences will be as follows.
0＝dS＋B dS ＝− B
Here B shows treasury bonds issuing.
When the government issues bonds, the generation’s disposable income does not decrease. This is because people in the generation buy the bonds with their savings. So bonds issuing does not affect economic welfare of the generation.
Bonds issuing, however, reduce capital accumulation by the amount B . Because the amount in the case of taxation is sT , the slowing-down of capital accumulation is much larger in bonds issuing
3.
In addition to that, while the amount of capital will recover after several years in taxation, the amount of capital will eternally continue smaller than normal case in bonds issuing. This is, of course, because one-to-one replacement happens between physical capital and treasury bonds in the case of bonds issuing
4.
2
Diamond also pointed this inconsistency. As he referred to the fact, however, Modigliani knew the inconsistency and neglected it for simplicity. It is because Modigliani essentially treated only the case of long-term stationary state.
3
Of course, we assume B is equal to T here.
4
Because Modigliani assumes the constant income, economic welfare of the nation is not reduced in the long run. Actually, however, the slowing-down of capital accumulation will discourage economic growth and seemingly reduce the nation’s welfare.
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（ ４ ）
3. Full Development of the Theory : Diamond’s Model
Modigliani’s model is clearly old Keynesian and demand-sided. It treats only the demand of capital market. It lacks production function and has to assume that the income is constant.
Diamond [1965] completed overlapping generations model and analyzed the equilibrium path of economic growth.
Government issues treasury bonds b per person to the younger generation. The government keeps the level b into the future. It pays interest rb per person to the older generations. The rate of interest r is the same as the return to capital. The government collects money for interest payment by imposing a lump-sum tax t on the younger generation
5.
The budget constraint for the government is as follows when we assume the rate of population growth as n.
1＋r
1＋n b−b＝t r−n 1＋n b＝t The budget constraint for a generation is as follows.
c
1＝w−s−t c
2＝(1＋r) s
5
This premise of bonds issuing is utterly the same as pension system through imposing premiums on the younger generation.
National Debt in the History of Macroeconomic Growth Theory（Yamazaki） −８１−
（ ５ ）
Here c
1and c
2are consumptions of the young days and the old days respectively. And w is the wage rate.
Diminishing s from both the equations, we obtain one budget constraint.
c
1＋ c
21 ＋ r ＝w−t
Also using n, we can represent the accumulation of capital like this.
w−c
1−b−t＝(1＋n) k
By substituting the government’s budget constraint, we will have this equation.
w − c
1− b − t ＝ (1 ＋ n) k s＝(1＋n) k＋b
Here k is capital per person. The right side of the last equation shows the supply of loanable fund and the left is the demand for it.
Differentiating both sides respectively, we obtain the equation below.
∂ s
∂ k dk ＋ ∂ s
∂ b db＝(1＋n) dk＋db (1＋n− ∂ s
∂ k ) dk＝( ∂ s
∂ b −1) db Finally we have gotten the derivative of k with b.
−８２−
（ ６ ）
dk db ＝
∂ s
∂ b − 1 1 ＋ n − ∂ s
∂ k
Here the partial derivative of s with respect to b must be smaller than unity because the consumptions are normal goods. Therefore it follows that the numerator of the right side is negative
6.
Then we assume that the partial derivative of s with respect to k is negative or slightly positive. When it is the case, the denominator of the left hand is positive.
If k increases, the rate of capital return r must go down. The reduction will probably decrease the savings s
7.
From these discussions, we can conclude that bond issuing per person will reduce the level of capital accumulation in the long-run equilibrium condition because the derivative of k with respect to b is negative.
4. Neutrality again ? : Barro’s Model
Here we introduce a simple production function f (k). Then the rate of interest and the wage rate can be expressed as follows.
r＝f ′ (k), w＝f (k)−f′ (k) k
6
When b increases, the present value of total tax burden also increases. Because of this, people transfer more of income into their old days as savings s. This is not, however, less than unity since they care about their young days’ consumption, too.
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