Debt to GDP Ratio from the perspective of Functional Finance Theory

and MMT

YASUHITO TANAKA

Faculty of Economics

Doshisha University

Kamigyo-ku, Kyoto, 602-8580

JAPAN

Abstract: - This paper will argue that since the ratio of government debt to GDP cannot diverge to infinity, fiscal

collapse is not possible. Using a basic macroeconomic model in which the interest rate of government bonds

is endogenously determined, with overlapping generations model in mind, we show the following results: 1)

The budget deficit including interest payments on the government bonds equals an increase in the savings from a

period to the next period. 2) If the savings in the first period is positive (unless the savings are made solely through

stocks), we need budget deficit to maintain full employment under constant prices or inflation in the later periods.

3) Excess budget deficit induces inflation under full employment. 4) Under an appropriate assumption about the

proportion of the savings consumed, the debt to GDP ratio converges to a finite value. It does not diverge to

infinity.

Key-Words: - Budget deficit, Debt to GDP ratio, MMT, Functional Finance Theory

1 Introduction

One of the most commonly used conditions for ex-

amining fiscal stability is the Domar condition ([1],

[18]). The Domar condition compares the interest rate

with the economic growth rate under balanced bud-

get (excluding interest payments on the government

bonds), and if the former is greater than the latter, pub-

lic finance will become unstable, and the government

debt to GDP ratio will continue to grow. Yoshino and

Miyamoto (2020) try to modify the Domar condition

by focusing not only on the supply side of govern-

ment bonds but also on the demand side, while keep-

ing the idea of fiscal instability indicated by the Do-

mar condition. However, our interest is different from

that. We consider a problem of the debt to GDP ra-

tio from the perspective of Functional Finance Theory

([3], [4]) and MMT (Modern Money Theory or Mod-

ern Monetary Theory, [2], [13], [17]1) using a simple

macroeconomic model, and we will show that the Do-

mar condition is meaningless.

In the next section, we examine the relation be-

tween the budget deficit and the debt to GDP ratio,

and will show the following results.

1. The budget deficit including interest payments on

the government bonds equals an increase in the

savings from a period to the next period. (Propo-

sition 1)

2. If the savings in the first period (Period 0) is pos-

itive (unless the savings are made solely through

stocks), we need budget deficit to maintain full

employment under constant prices or inflation in

the later periods. (Proposition 2)

1Japanese references of MMT are [5], [6], [7], [11], [12].

3. Excess budget deficit induces inflation under full

employment. (Proposition 3)

4. Under an appropriate assumption about the pro-

portion of the savings consumed, the debt to GDP

ratio converges to a finite value. It does not di-

verge to infinity. (Proposition 4)

In Section 3 we consider endogenous determina-

tion of the interest rate on the government bonds by

the monetary policy of the government.

In Section 4 we examine the so-called Domar con-

dition that under balanced budget excluding interest

payments on the government bonds the interest rate

should be smaller than the growth rate to prevent the

debt to GDP ratio diverging infinity, and we will show

that it is meaningless.

2 Budget deficit and debt to GDP

ratio

Using a simple macroeconomic model we analyze

budget deficit and the debt to GDP ratio. In a

broad sense, savings are made by government bonds,

money, and stocks, of which those made by govern-

ment bonds and money are analyzed as savings in

this paper. The amounts of government bonds and

money supply are determined by the government. Al-

though money does not earn interest and government

bonds earn interest, consumers are willing to hold a

certain amount of money for reasons such as the liq-

uidity of money. The holding of money is consid-

ered to be a decreasing function of the interest rate of

the government bonds (while the holding of govern-

ment bonds is an increasing function of the interest

rate). The reasons for this are as follows. This part

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2022.2.9

Yasuhito Tanaka

E-ISSN: 2769-2477

Volume 2, 2022

implicitly assumes an overlapping generations model

in which people live for two periods2. People de-

cide how much money and government bonds to hold

so that the marginal utility of holding one more unit

of money and the marginal utility of interest income

from holding government bonds are equalized. Since

the marginal utility of money decreases as the amount

of money held increases, the amount of money held is

a decreasing function of the interest rate of the gov-

ernment bonds.

The share of government bonds in savings is de-

noted by b(r),0< b(r)≦1.ris the interest rate

of the government bonds. The share of money in sav-

ings is 1−b(r). The investment is financed by sav-

ings in the form of stocks, and it may be a decreasing

function of the interest rate of the government bonds.

However, for simplicity we assume that the invest-

ment is constant in each period. The interest rate of

the government bonds is endogenously determined by

the monetary policy of the government.

2.1 Period 0

First consider Period 0 at which the world starts. All

variables represent nominal values. Let Y0,C0,I0,

T0and G0be the GDP, consumption, investment, tax

and fiscal spending in Period 0. Then,

Y0=C0+I0+G0.

The consumption is written as

C0=¯

C0+α(Y0−T0).

C0is the constant part of consumption in Period 0. It

is financed by the savings carried over from the previ-

ous period. Since there is no previous period of Period

0, ¯

C0= 0.

Then,

C0=α(Y0−T0),

and

Y0=α(Y0−T0) + I0+G0.

From this

(1 −α)(Y0−T0) = I0+G0−T0.

The savings in Period 0, which is carried over to the

next period, is

S0= (1 −α)(Y0−T0)−I0.

Therefore, we have

G0−T0= (1 −α)(Y0−T0)−I0=S0.

2In other studies, for example, [14], [15] and [16], which are according

to the model by M. Otaki such as [8], [9], [10], we use an overlapping

generations model to analyze the problem of budget deficit in a growing

economy

Let us assume full employment in Period 0, and de-

note the full employment GDP by Yf, that is,

Y0=Yf.

Then, we obtain

G0−T0= (1 −α)(Yf−T0)−I0=S0.(1)

This is the budget deficit we need to achieve full em-

ployment in Period 0. It is determined by Yf,I0and

T0. From this we get the following equation.

G0= (1 −α)(Yf−T0) + T0−I0.

This is the fiscal spending needed to achieve full em-

ployment given T0and I0. If the budget deficit is

larger than this value, then Yfincreases and (1) still

holds.

Unless the savings are made solely through stocks,

(1) is positive.

Note that as we said at the beginning of this sub-

section, all variables represent nominal values.

2.2 Period 1

Next, consider Period 1. Again all variables represent

nominal values. Let Y1,C1,I1,T1and G1be the GDP,

consumption, investment, tax and fiscal spending in

Period 1. Then,

Y1=C1+I1+G1.

The consumption is written as

C1=¯

C1+α(Y1−T1).

C1is the constant part of consumption in Period 1. It

is financed by the savings carried over from Period 0.

Let r0be the interest rate of the government bonds,

which is carried over from Period 0 to Period 1. Let

δbe the proportion of the savings consumed. Then,

C1=δ(1 + b(r0)r0)S0,0< δ ≦1,

and

C1=δ(1 + b(r0)r0)S0+α(Y1−T1),

and so

Y1=δ(1 + b(r0)r0)S0+α(Y1−T1) + I1+G1.

From this

(1−α)(Y1−T1) = δ(1+b(r0)r0)S0+I1+G1−T1.

Therefore,

G1−T1= (1−α)(Y1−T1)−I1−δ(1+b(r0)r0)S0.

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2022.2.9

Yasuhito Tanaka

E-ISSN: 2769-2477

Volume 2, 2022

The savings in Period 1, which is carried over to Pe-

riod 2, is

S1= (1−α)(Y1−T1)−I1+(1−δ)(1+b(r0)r0)S0.

This means

G1−T1=S1−(1 + b(r0)r0)S0.

Alternatively,

G1−T1+b(r0)r0S0=S1−S0.(2)

We assume that the economy grows by technological

progress. The real growth rate is g > 0. Also the

prices may rise from Period 0 to Period 1, that is, there

may be inflation. Let pbe the inflation rate. Then,

(1 + g)(1 + p)−1 = g+p+gp

is the nominal growth rate.

Under nominal growth at the rate of g+p+gp,

Y1= (1 + g)(1 + p)Yf.

Tax and investment also increase at the same rate as

follows under the assumption that inflation is pre-

dicted,

T1= (1 + g)(1 + p)T0, I1= (1 + g)(1 + p)I0.

Then, the savings in Period 1 is

S1=(1 −α)(1 + g)(1 + p)(Yf−T0)

−(1 + g)(1 + p)I0+ (1 −δ)(1 + b(r0)r0)S0.

It is rewritten as

S1= (1+g)(1+p)S0+(1−δ)(1+b(r0)r0)S0.(3)

If δ < 1, we have

S1>(1 + g)(1 + p)S0.(4)

From (2) and (3) we obtain

G1−T1+b(r0)r0S0=(1 + g)(1 + p)S0+ [b(r0)r0

−δ(1 + b(r0)r0)]S0,

and

G1−T1= (1 + g)(1 + p)S0−δ(1 + b(r0)r0)S0

<(1 + g)(1 + p)(G0−T0).

They are budget deficits, with or without interest pay-

ments on the government bonds, we need to achieve

full employment in Period 1 under nominal growth at

the rate of g+p+gp.

G1−T1+b(r0)r0S0= (1 + g)S0+ [b(r0)r0

−δ(1 + b(r0)r0)]S0

<(1 + g)(1 + p)S0+ [b(r0)r0

−δ(1 + b(r0)r0)]S0,

the economy grows at the real growth rate gwithout

inflation. Therefore, we can say that excess budget

deficit induces inflation.

2.3 Period 2

Next, consider Period 2. Also in this subsection all

variables represent nominal values. Let Y2,C2,I2,

T2and G2be the GDP, consumption, investment, tax

and fiscal spending in Period 2. Then,

Y2=C2+I2+G2.

The consumption is

C2=¯

C2+α(Y2−T2).

C2is the constant part of consumption in Period 2. It

is financed by the savings carried over from Period 1.

Let r1be the interest rate of the government bonds,

which is carried over from Period 1 to Period 2. Sim-

ilarly to the case of Period 1,

C2=δ(1 + b(r1)r1)S1,

and

C2=δ(1 + b(r1)r1)S1+α(Y2−T2),

and so

Y2=δ(1 + b(r1)r1)S1+α(Y2−T2) + I2+G2.

From this

(1−α)(Y2−T2) = δ(1+b(r1)r1)S1+I2+G2−T2.

Therefore,

G2−T2= (1−α)(Y2−T2)−I2−δ(1+b(r1)r1)S1.

The savings in Period 2, which is carried over to Pe-

riod 3, is

S2= (1−α)(Y2−T2)−I2+(1−δ)(1 +b(r1)r1)S1.

This means

G2−T2=S2−(1 + b(r1)r1)S1.

Alternatively,

G2−T2+b(r1)r1S1=S2−S1.(5)

Again we suppose that the economy nominally grows

by technological progress and inflation at the rate of

g+p+gp, then

Y2= (1 + g)2(1 + p)2Yf.

We assume that, for simplicity, the inflation rate p

is constant. Tax and investment also increase at the

same rate as follows,

T2= (1 + g)2(1 + p)2T0, I1= (1 + g)2(1 + p)2I0.

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2022.2.9

Yasuhito Tanaka

E-ISSN: 2769-2477

Volume 2, 2022

Then, the savings in Period 2 is

S2=(1 −α)(1 + g)2(1 + p)2(Yf−T0)

−(1 + g)2(1 + p)2I0+ (1 −δ)(1 + b(r1)r1)S1.

It is rewritten as

S2= (1+g)2(1+p)2S0+(1−δ)(1+b(r1)r1)S1.(6)

If δ < 1, since

S1>(1 + g)(1 + p)S0,

assuming

(1 + b(r1)r1)(1 + g)>1 + b(r0)r0,(7)

we have

S2>(1 + g)(1 + p)S1.(8)

If the interest rate is constant, and the nominal growth

rate is positive, (7) is satisfied. From (5) and we ob-

tain

G2−T2+b(r1)r1S1= (1 + g)2(1 + p)2S0(9)

+ [b(r1)r1−δ(1 + b(r1)r1)]S1.

and

G2−T2=(1 + g)2(1 + p)2S0(10)

−δ(1 + b(r1)r1)S1.

Since

G1−T1= (1 + g)(1 + p)S0−δ(1 + b(r0)r0)S0,

by the assumption in (7), we obtain

G2−T2<(1 + g)(1 + p)(G1−T1).

(9) and (10) are budget deficits, with or without in-

terest payments on the government bonds, we need to

achieve full employment in Period 2.

By (3) and (6),

S2=[(1 + g)2(1 + p)2(11)

+ (1 −δ)(1 + b(r1)r1)(1 + g)(1 + p)

+ (1 −δ)2(1 + b(r1)r1)2]S0.

G2−T2=(1 + g)2S0−δ(1 + b(r1)r1)S1

<(1 + g)2(1 + p)2S0−δ(1 + b(r1)r1)S1,

the economy grows at the real growth rate gwithout

inflation. Therefore, we can say that excess budget

deficit induces inflation.

2.4 Period 3 and beyond

From now on, for simplicity, the interest rates in all

periods are equal. Also in this subsection all variables

represent nominal values, and the inflation rate is con-

stant. It may be zero. Denote the interest rate by r. By

similar reasoning for Period 3, we get

G3−T3=S3−(1 + b(r)r)S2.

and

G3−T3+b(r)rS2=S3−S2.(12)

The savings in Period 3 is

S3= (1+g)3(1+p)3S0+(1−δ)(1+b(r)r)S2.(13)

Thus,

G3−T3+b(r)rS2(14)

= (1 + g)3(1 + p)3S0+ [b(r)r−δ(1 + b(r)r)]S2,

and

G3−T3= (1+g)3(1+p)3S0−δ(1+b(r)r)S2.(15)

(14) and (15) are budget deficits, with or without in-

terest payments on the government bonds, we need to

achieve full employment in Period 3.

From (11) and (13), we get

S3=[(1 + g)3(1 + p)3+ (1 −δ)(1 + b(r)r)(1 + g)2(1 + p)2

+ (1 −δ)2(1 + b(r)r)2(1 + g)(1 + p)

+ (1 −δ)3(1 + b(r)r)3]S0.

Proceeding with this argument, we obtain the fol-

lowing result for Period n,n≧1.

Gn−Tn+b(r)rSn−1=Sn−Sn−1.(16)

With or without inflation in Period n, we have

Sn=[(1 + g)n(1 + p)n(17)

+ (1 −δ)(1 + b(r)r)(1 + g)n−1(1 + p)n−1+· · ·

+ (1 −δ)n−1(1 + b(r)r)n−1(1 + g)(1 + p)

+ (1 −δ)n(1 + b(r)r)n]S0.

Similarly, for Period n−1,

Sn−1=[(1 + g)n−1(1 + p)n−1(18)

+ (1 −δ)(1 + b(r)r)(1 + g)n−2(1 + p)n−2+· · ·

+ (1 −δ)n−2(1 + b(r)r)n−2(1 + g)(1 + p)

+ (1 −δ)n−1(1 + b(r)r)n−1]S0.

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2022.2.9

Yasuhito Tanaka

E-ISSN: 2769-2477

Volume 2, 2022

2.5 Some propositions

From (2), (5), (12) and (16) we obtain the following

proposition.

Proposition 1. The budget deficit including inter-

est payments on the government bonds equals an in-

crease in the savings from a period to the next period.

(2), (4), (5), (8) and (12) mean the following result.

Proposition 2. If the savings in the first period (Pe-

riod 0) S0is positive (unless the savings are made

solely through stocks), we need budget deficit to main-

tain full employment under constant prices or infla-

tion in the later periods.

About inflation we found

Proposition 3. Excess budget deficit induces infla-

tion under full employment.

2.6 Debt to GDP ratio

Since

Yn= (1 + g)(1 + p)Yn−1,

(17) and (18) mean

−Sn−1

Yn−1

=(1 −δ)(1 + b(r)r)

(1 + g)(1 + p)nS0

.(19)

Since δis the proportion of the savings consumed, we

can assume

δ > 1

Then, for



(1 −δ)(1 + b(r)r)

(1 + g)(1 + p)



<1,

it is sufficient that

b(r)r < 1 + 2(g+p+gp).(20)

Note that g+p+gp is the nominal growth rate. Since

0< b(r)≦1and ris the interest rate of the govern-

ment bonds, (20) is definitely satisfied. Therefore,

When n→ ∞,Sn

−Sn−1

Yn−1

→0.

From (17), we obtain

=1 + (1 −δ)(1 + b(r)r)

(1 + g)(1 + p)+· · ·

+(1 −δ)(1 + b(r)r)

(1 + g)(1 + p)n−1

+(1 −δ)(1 + b(r)r)

(1 + g)(1 + p)nS0

Then, if



(1 −δ)(1 + b(r)r)

((1 + g)(1 + p)) 



<1,

we get

→1

1−(1−δ)(1+b(r)r)

(1+g)(1+p)

Therefore, the debt to GDP ratio Sn

converges to a

finite value. It does not diverge to infinity.

Summarizing the result,

Proposition 4. Under an appropriate assumption

about the proportion of the savings consumed (δ >

2), the debt to GDP ratio converges to a finite value.

It does not diverge to infinity.

3 Determination of interest rate

The demand for money in Period 0 is

(1 −b(r0))S0.

Denote the money supply by M0. Then, r0is deter-

mined so that

(1 −b(r0))S0=M0

is satisfied. Similarly, let Mnbe the money supply in

Period n. Then, the interest rate in Period nis deter-

mined so that

(1 −b(rn))Sn=Mn

is satisfied. As the money supply Mnincreases, rn

and b(rn)must be smaller. Therefore, an increase in

the money supply lowers the interest rate, and also in-

terest payment b(rn)rnSnin Period 1 decreases. This

is the effect of monetary policy.

4 About Domar condition

From the above discussion, the interest rate can be

changed by monetary policy so that the so-called Do-

mar condition ([1], [18]), that the interest rate must

be less than the economic growth rate to prevent the

ratio of government debt to GDP from becoming in-

finitely large (in particular, if a balanced budget can

be achieved excluding interest payments on govern-

ment bonds), can be satisfied, but even if this condi-

tion is not satisfied, the ratio of government debt to

GDP will not become infinitely large. When savings

are made in both government bonds and money, the

issue is not the interest rate on government bonds it-

self, but the product of the share of savings held in

government bonds and the interest rate on govern-

ment bonds b(r)rand the proportion of the savings

consumed δ. We call

δ(1 + b(r)r)−1(21)

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2022.2.9

Yasuhito Tanaka

E-ISSN: 2769-2477

Volume 2, 2022

the adjusted interest rate. Since δ≦1and b(r)≦1,

It is not larger than r.

Let us assume balanced budget excluding interest

payments on the government bonds in Period 1 as fol-

lows.

G1−T1= 0.

Then, (2) means that the following equation must

hold.

(1 + g)(1 + p) = δ(1 + b(r0)r0).

1 + g < δ(1 + b(r0)r0),

there is excess demand for goods. Then, the prices

rise and the nominal growth rate g+p+gp equals

δ(1 + b(r0)r0)−1.

For the periods after Period 1 we obtain similar re-

sults.

Then, under the assumption that δ > 1

2, (19) and

1−δ

δ<1

mean Sn

−Sn−1

Yn−1

<S0

and

when n→ ∞,Sn

−Sn−1

Yn−1

→0.

Therefore, the debt to GDP ratio can not diverge to

infinity even if

1 + g < δ(1 + b(r0)r0),

g < δ(1 + b(r0)r0)−1,

Summarizing the result,

Proposition 5. Even if the adjusted interest rate (21)

is larger than the real growth rate, the debt to GDP

ratio can not diverge to infinity.

5 Conclusion

We have argued that fiscal collapse is impossible be-

cause the ratio of the government debt to GDP can

not diverge to infinity. Using a simple macroeco-

nomic model including peoples’ money holding we

have shown the following results.

1. The budget deficit including interest payments on

the government bonds equals an increase in the

savings from a period to the next period.

2. If the savings in the first period (Period 0) is pos-

itive (unless the savings are made solely through

stocks), we need budget deficit to maintain full

employment under constant prices or inflation in

the later periods.

3. Excess budget deficit induces inflation under full

employment.

4. Under an appropriate assumption about the pro-

portion of the savings consumed, the debt to GDP

ratio converges to a finite value. It does not di-

verge to infinity.

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E-ISSN: 2769-2477

Volume 2, 2022

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DOI: 10.37394/232028.2022.2.9

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E-ISSN: 2769-2477

Volume 2, 2022