A Game-Theoretic Analysis of Fiscal Policy under Economic Growth

from the Perspective of MMT: Toward a Neoclassical Basis of MMT

YASUHITO TANAKA

Faculty of Economics

Doshisha University

Kamigyo-ku, Kyoto, 602-8580

JAPAN

Abstract: - We present a game-theoretic analysis of fiscal policy under economic growth from the perspective

of MMT using a simple two-periods overlapping generations (OLG) model. We show the following results. 1)

Sustained budget deficits are necessary to maintain full-employment under economic growth driven by popula-

tion growth. 2) An excessive budget deficit triggers inflation, and after one period inflation full-employment is

maintained by sustained budget deficits with constant price. 3) Insufficient government deficit causes involuntary

unemployment, and we need extra budget deficit over its steady state value to recover full-employment. These

budget deficits need not be, and must not be redeemed. Therefore, if it is institutionally and legally possible, they

should be financed by seigniorage not by public debt.

Key-Words: - Overlapping generations model, Full-employment, Budget deficit, Growth, MMT

Received: June 15, 2021. Revised: January 26, 2022. Accepted: February 17, 2022. Published: March 3, 2022.

1 Introduction

This paper will present a game theoretic analysis of

a simple two-periods overlapping generations (OLG)

model with pay-as-you-go pensions in which goods

are produced solely by labor, and mainly show the

following three results.

1. Sustained budget deficits are necessary to main-

tain full-employment under economic growth

driven by population growth. Budget deficit is

necessary under growth because of deficiency

of the consumptions by the older generation

consumers. Since the budget deficits to main-

tain full-employment must be sustained, if it is

institutionally and legally possible, the budget

deficits should be financed by seigniorage not by

public debt. This budget deficit need not to be,

and must not be redeemed.

2. An excessive budget deficit triggers inflation.

About this excessive budget deficit that has

caused inflation, only the excess portion should

be reduced. There is no need to make up for

past excesses by creating surpluses or reducing

deficits.

3. Insufficient government deficit causes involun-

tary unemployment, and we need extra budget

deficit over its steady state value to recover full-

employment. After recovery of full-employment

we can maintain it with sustained budget deficits.

Thus, the extra budget deficit should not be re-

deemed.

In the next section we present a model of this pa-

per. In Section 3 we consider budget deficit for eco-

nomic growth. In Section 4 we analyze inflation by

excessive budget deficit, and in Section 5 we consider

involuntary unemployment due to insufficient gov-

ernment deficit, and necessity of extra budget deficit

to recover full-employment. In Section 6 we will

consider a case of economic growth by technological

progress instead of population growth.

This paper is one of the attempts to give a theo-

retical basis to the so-called functional finance theory

by [3] and [4]. It also present a theoretical foundation

to MMT (Modern Monetary Theory, [5]). In partic-

ular, we provide a rationale for the following claims

([2]). We refer to the summary of Kelton’s book by

[1]. In fact, Hogan argues that Kelton is wrong, but

he summarizes Kelton’s argument to the point.

1. The treasury creates new money.

The money supply equals the net savings (the

savings minus the pay-as-you-go pensions). An

increase in the money supply equals the increase

in the net savings. As expressed in the equation

(4) in Section 3.1, an increase in the net savings

from a period to the next period equals the bud-

get deficit in the latter period. The rate of an in-

crease in the net savings, which equals the rate

of in increase in the money supply, equals the

rate of economic growth, and therefore the bud-

get deficit and an increase in the money supply

does not cause inflation in this case.

2. Inflation is caused by federal government deficit

spending, not by Fed policy.

As we will show in Section 4.1, if the actual

budget deficit is larger than the budget deficit

that is necessary and sufficient to maintain full-

employment under economic growth with con-

stant price, the price of the good will rise.

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3. Federal government spending is not related to

taxes or borrowing.

As summarized above, sustained budget deficits

are necessary to maintain full-employment under

economic growth, and these budget deficits make

it possible to maintain full-employment. It is im-

possible to maintain full-employment in a grow-

ing economy with a balanced budget. There-

fore, even if the budget deficit to maintain full-

employment is financed by the national debt, it

does not need to be repaid or redeemed, and must

not be repaid or redeemed. Future budget sur-

pluses need not and must not make up the deficits

for growth.

Recently, we have published some studies about

theoretical or mathematical bases of MMT using a

model with or without microeconomic foundations

of behaviors of consumers and firms under perfect

or monopolistic completion (for example, [10], [11],

[12]). This paper is an attempt to interprete these re-

sults in the framework of Game Theory.

2 The model

We consider the following model. It is a two gener-

ations overlapping generations (OLG) model, and a

simplified version of the model used by [6], [7], [8]

and [9] with pay-as-you-go pensions.

1. There are one good and one firm. The firm pro-

duces one unit of the good by one unit of labor

at present (Period t). Although there is only one

firm, it behaves competitively. Let wbe the nom-

inal wage rate and pbe the price of the good.

Then, p=w, and the real wage rate is one.

2. Consumers live over two periods, Period 1 and

Period 2. In Period 1 (younger period, working

period) they can work and earn the wages. They

buy and consume the good by wages. In Period

2 (older period, retired period) they buy and con-

sume the good by their savings and the pay-as-

you-go pensions. The good can be consumed by

any unit, that is, a consumer can consume, for ex-

ample, 0.1 units of the good. The population of

consumers grows from a period to the next period

at the rate γ−1>0.

3. There are four younger generation consumers in

Period t. If a consumer is employed, he sup-

plies one unit of labor (labor supply is indivisi-

ble), receives the wage, consumes the good, pays

tax for pay-as-you-go pensions for consumers

of the older generation, lends money to unem-

ployed younger generation consumers (if invol-

untary unemployment exists) and leaves the re-

maining income.

If a consumer is not employed, his income is

zero. However, he can receive a pay-as-you-

go pension after retirement, so he consumes the

good in his Period 1 by borrowing money from

employed younger generation consumers using

his pension as collateral. We assume that utility

of an employed consumer is larger than utility of

an unemployed consumer.

Each consumer determines his consumption and

labor supply at the beginning of Period 1 depend-

ing on the situation that he is employed or not

employed. He spends half of his income, respec-

tively, on consumption in Period 1 and that in Pe-

riod 2.

4. There are 4

γolder consumers in Period t(four

older generation consumers in Period t+ 1). If a

consumer was employed in his Period 1, he con-

sumes the good by his savings and the pay-as-

you-go pension. If he is not employed, he con-

sumes the good only by the pay-as-you-go pen-

sion, and repays his debt due to consumption in

Period 1.

We assume zero interest rate. Repayment by a con-

sumer who was unemployed in the younger period

is assured. An employed consumer is indifferent be-

tween lending money to unemployed consumers and

leaving money.

We assume that the tax other than that for the pay-

as-you-go pension system is zero.

3 Sustained full-employment under

growth with constant price

3.1 Period t

Let us consider the following pair of strategies of

the firm and the younger generation consumers at a

steady state with full-employment and constant price

under economic growth due to population growth at

the rate γ−1. The strategies of the consumers are

derived from their utility maximization over two pe-

riods. Denote the pay-as-you go pension received by

each older generation consumer in each period is γτ.

On the other hand, the tax for the pay-as-you-go pen-

sion paid by each younger generation consumer is τ

because of the population growth. The nominal wage

rate and the price are 1.

1. Firm: employs four younger generation con-

sumers and produces four units of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage 1, con-

sumes 1

2[1 + (γ−1)τ]units of the good, pays τ

units of tax for the pay-as-you-go pension system

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for the older generation consumers, and leaves

the rest of the income;

2[1 −(1 + γ)τ].

In the next period he consumes 1

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension γτ .

Each older generation consumer consumes 1

2[1+(γ−

1)τ]units of the good by his savings and the pay-as-

you-go pension (γτ ). Let Gtbe the government ex-

penditure other than the pay-as-you-go pension. The

total demand for the good is

21 + 1

γ[1 + (γ−1)τ] + Gt.

Note that there are four younger and 4

γolder genera-

tion consumers. This must be equal to the total sup-

ply, 4. Thus,

Gt=2(γ−1)

γ[1 −(1 + γ)τ]>0.(1)

The total net savings of the older generation con-

sumers, which is the difference between their con-

sumptions and the pensions, is

γ[1 + (γ−1)τ]−

γγτ =2

γ[1 −(1 + γ)τ].(2)

On the other hand, the total net savings of the older

generation consumers in the next period, who are the

younger generation consumers in Period t, is

2[1 + (γ−1)τ]−4γτ = 2[1 −(1 + γ)τ].(3)

The difference between (3)and (2) is

γ(γ−1)[1 −(1 + γ)τ] = Gt.(4)

Therefore, the government expenditure Gtin (1)

equals the increase in the net savings of the consumers

from Period t−1to Period t.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of four units of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, sup-

ply of one unit of labor and 1

2[1 + (γ−1)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

Since we assume zero tax other than the tax for the

pay-as-you-go pension system, the government ex-

penditure equals the budget deficit, and the existence

of the positive government expenditure means the ex-

istence of the budget deficit. Since the budget deficits

to maintain full-employment must be sustained, if it is

institutionally and legally possible, the budget deficits

should be financed by seigniorage not by public debt.

Budget deficit is necessary under growth because of

deficiency of the savings of the older generation. This

budget deficit is not debt and does not need to be, and

must not be redeemed.

Since consumers leave the savings by money, the

money supply equals the net savings (the savings mi-

nus the pay-as-you-go pensions). An increase in the

money supply equals the increase in the net savings.

As expressed in the equation (4), an increase in the net

savings from Period t−1to Period tequals the bud-

get deficit in Period t. The rate of an increase in the

net savings, which equals the rate of in increase in the

money supply, equals the rate of economic growth,

and therefore the budget deficit and an increase in the

money supply does not cause inflation in this case.

3.2 Period t+ 1

Next we consider Period t+1. Let us consider the fol-

lowing pair of strategies of the firm and the younger

generation consumers in Period t+ 1.

1. Firm: employs 4γyounger generation con-

sumers and produces 4γunits of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage 1, con-

sumes 1

2[1 + (γ−1)τ]units of the good, pays τ

units of tax for the pay-as-you-go pension system

for the older generation consumers, who are the

younger generation consumers in Period t, and

leaves the rest of the income;

2[1 −(1 + γ)τ].

In the next period he consumes 1

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension γτ .

Each older generation consumer consumes 1

2[1+(γ−

1)τ]units of the good by his savings and the pay-as-

you-go pension (γτ ). Let Gt+1 be the government ex-

penditure other than the pay-as-you-go pension. The

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total demand for the good is

2(1 + γ)[1 + (γ−1)τ] + Gt+1.

This must be equal to the total supply, 4γ. Thus,

Gt+1 = 2(γ−1)[1 −(1 + γ)τ] = γGt.(5)

The government expenditure Gt+1 in (5) equals the

increase in the net savings of the consumers from Pe-

riod tto Period t+1. This relationship holds true even

after Period t+ 1.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of 4γunits of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, sup-

ply of one unit of labor and 1

2[1 + (γ−1)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

From these results we get the following proposi-

tion.

Proposition 1. In order to maintain economic growth

we need sustained budget deficits. The budget deficit

in each period equals the increase in the total net sav-

ings of the older generation consumers (the difference

between their consumptions and the pensions) from

the previous period to this period.

4 Inflation by excessive budget deficit

4.1 Period t

In this section we suppose that the government expen-

diture in Period tis larger than Gtin the previous case.

Denote it by ˆ

Gt. The actual price of the good in Period

tis ρ > 1, and the actual nominal wage rate is also

ρ. We assume that the inflation is not anticipated by

the older generation consumers in the previous period

(when they are young in their Period 1). We consider

the value of ˆ

Gtwhich triggers inflation at the rate ρ

under full-employment. Let us consider the following

pair of strategies of the firm and the younger genera-

tion consumers.

1. Firm: employs four younger generation con-

sumers and produces four units of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage ρ, con-

sumes 1

2[1 + (γ−1)τ]units of the good at the

price ρ, pays ρτ units of tax for the pay-as-you-

go pensions for the older generation consumers,

and leaves the rest of the income;

ρ−ρτ −

2[1 + (γ−1)τ] = ρ

2[1 −(1 + γ)τ].

In the next period he consumes 1

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension ργτ at the price ρif the price of

the good is ρin Period t+ 1.

Each older generation consumer consumes 1

2ρ[1 +

(ργ −1)τ]units of the good by his savings and the

pay-as-you-go pension (ργτ ). The nominal total de-

mand for the good is

2ρ[1 + (γ−1)τ] + 2

γ[1 + (ργ −1)τ] + ˆ

γ1 + ργ + (ργ2

−1)τ+ˆ

Gt.

If this is equal to the nominal total supply, 4ρ, we have

Gt=2

γργ −1−(ργ2

−1)τ.

Comparing this with Gtin (1),

Gt−Gt(6)

γργ −1−(ργ2

−1)τ−(γ−1)[1 −(1 + γ)τ]

=2(ρ−1)(1 −γτ )>0.

This means that an excessive budget deficit expressed

in (6) causes inflation under full-employment.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of four units of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm and consumptions of the

other consumers, supply of one unit of labor and

2[1 + (γ−1)τ]units of consumption are optimal

under the assumption that each consumer spends

half of his income, respectively, on consumption

in Period 1 and that in Period 2.

From these results we get the following proposi-

tion.

Proposition 2. The excessive budget deficit ex-

pressed in (6) causes inflation under full-employment.

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4.2 Period t+ 1

Next we consider Period t+1. Let us consider the fol-

lowing pair of strategies of the firm and the younger

generation consumers in Period t+ 1. The pay-as-

you-go pension for each older generation consumer is

ργτ . Assume that the price of the good in this period

is ρ.

1. Firm: employs 4γyounger generation con-

sumers and produces 4γunits of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage ρ, con-

sumes 1

2[1 + (γ−1)τ]units of the good at the

price ρ, pays ργτ units of tax for pay-as-you-go

pension to the older generation consumers, who

are the younger generation consumers in Period

t, and leaves the rest of the income;

ρ−ρτ −

2[1 + (γ−1)τ] = ρ

2[1 −(1 + γ)τ].

In the next period he consumes 1

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension ργτ at the price ρ.

Each older generation consumer consumes 1

2[1+(γ−

1)τ]units of the good by his savings and the pay-as-

you-go pension (ργτ ) at the price ρ. Let ˆ

Gt+1 be the

government expenditure other than the pay-as-you-go

pension. The nominal total demand for the good is

2ρ(1 + γ)[1 + (γ−1)τ] + ˆ

Gt+1.

Note that there are 4γyounger generation consumers.

This must be equal to the nominal total supply, 4ργ.

Thus,

Gt+1 = 2ρ(γ−1)[1 −(1 + γ)τ].

From (5) we find

Gt+1 =ρGt+1.

This relationship holds true even after Period t+ 1.

Therefore, after one period inflation full-employment

under constant price is maintained by sustained bud-

get deficits. Only the excess portion should be re-

duced. There is no need to make up for past excesses

by creating surpluses or reducing deficits.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of 4γunits of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, sup-

ply of one unit of labor and 1

2[1 + (γ−1)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

From these results we obtain the following propo-

sition.

Proposition 3. After one period inflation full-

employment under constant price is maintained by

sustained budget deficits. Only the excess portion

should be reduced. There is no need to make up

for past excesses by creating surpluses or reducing

deficits.

5 Insufficient budget deficit and

recovery from recession

5.1 Period t

Now we assume that the government expenditure in

Period tis insufficient. Let the actual value of the

government expenditure be

Gt=2

γ(γ−1)[1 −(1 + γ)τ]−

It is smaller than Gtin (1). We assume that it is non-

negative. The price of the good and the nominal wage

rate in Period tis 1. We also assume that upto the

previous period, Period t−1, full-employment has

been realized. Let us consider the following pair of

strategies of the firm and the younger generation con-

sumers.

1. Firm: employs three younger generation con-

sumers and produces three units of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage 1, con-

sumes 1

2[1+(γ−

3)τ]units of the good, pays 4

3τ

units of tax for pay-as-you-go pensions for the

older generation consumers, and leaves the rest

of the income:

1−

3τ−

21 + γ−

3τ=1

21−γ+4

3τ.

In the next period he consumes 1

2[1 + γ−

3τ]

units of the good by his savings and the pay-as-

you-go pension γτ .

3. One unemployed consumer: consumes 1

2γτ units

of the good by borrowing money from employed

younger generation consumers using his pension

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as collateral. In the next period he consumes 1

2γτ

units of the good by the pay-as-you-go pension

γτ , and repays his debt due to consumption in

Period 1

Each older generation consumer consumes 1

2[1 +

(γ−1)τ]units of the good by his savings and the

pay-as-you-go pension (γτ ). The total demand for the

good is

21 + γ−

3τ+1

2γτ (7)

γ[1 + (γ−1)τ] + ˜

21 + γ−

3τ+1

2γτ +2

γ[1 + (γ−1)τ]

γ(γ−1)[1 −(1 + γ)τ]−

2= 3.

Note that there are 4

γolder generation consumers,

three employed and one unemployed younger con-

sumers. (7) means that the total demand equals the

total supply, and there exists one involuntary unem-

ployment.

Are the strategies of the firm and the younger gen-

eration consumers above in a Nash equilibrium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expen-

diture, employment of three consumers and pro-

duction of three units of the good are optimal.

2. Each younger employed consumer: given the be-

havior of the firm, consumptions of the other con-

sumers and the government expenditure, supply

of one unit of labor and 1

2[1 + (γ−

3)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

3. One younger unemployed consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, 1

2γτ

units of consumption is optimal given the situa-

tion that he is unemployed, and spends half of his

income, respectively, on consumption in Period

1 and that in Period 2.

From these results we get the following proposi-

tion.

Proposition 4. The insufficient government expendi-

ture (or budget deficit) causes involuntary unemploy-

ment.

5.2 Period t+ 1

Next we consider Period t+ 1. Suppose that we re-

cover full-employment in Period t+ 1 by increasing

the government expenditure (or the budget deficit).

Let ˜

Gt+1 be the government expenditure other than

the pay-as-you-go pension which is sufficient to re-

cover full-employment.

Let us consider the following pair of strategies of

the firm and the younger generation consumers in Pe-

riod t+ 1. The pay-as-you-go pension for each older

generation consumer is γτ . Assume that the price of

the good in this period is 1.

1. Firm: employs 4γyounger generation con-

sumers and produces 4γunits of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage 1, con-

sumes 1

2[1 + (γ−1)τ]units of the good, pays

τunits of tax for pay-as-you-go pension for the

older generation consumers, who are the younger

generation consumers in Period t, and leaves the

rest of the income;

1−τ−

2[1 + (γ−1)τ] = 1

2[1 −(1 + γ)τ].

In the next period he consumes 1

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension γτ .

Each older generation consumer, who was employed

in the previous period, consumes 1

21 + γ−

3τ

units of the good by his savings and the pay-as-you-

go pension (γτ ), and one older generation consumer,

who was unemployed in the previous period, con-

sumes 1

2γτ units of the good by the pay-as-you-go

pension. He also repay the debt for consumption in

the previous period.

The total demand for the good is

2γ[1+(γ−1)τ]+ 3

21 + γ−

3τ+1

2γτ +˜

Gt+1.

Note that there are three older generation consumers

who was employed, and one of them was unem-

ployed. Suppose that this is equal to the total supply,

4γ. Then,

Gt+1 = 2(γ−1)[1 −(1 + γ)τ] + 1

Comparing this with (5)

Gt+1

−Gt+1 =1

2>0.

Therefore, to recover full-employment from a state

with involuntary unemployment we need extra bud-

get deficit. After recovery of full-employment we can

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maintain it with sustained budget deficit. Thus, the

extra budget deficit should not be redeemed.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of 4γunits of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, sup-

ply of one unit of labor and 1

2[1 + (γ−1)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

From these results we obtain the following propo-

sition.

Proposition 5. In order to recover full-employment

from a state with involuntary unemployment we

need extra budget deficit. After recovery of full-

employment we can maintain it with sustained budget

deficit. Thus, the extra budget deficit should not be

redeemed.

6 Growth by technological progress

6.1 Sustained full-employment under

growth with constant price

6.1.1 Period t

Let us consider the following pair of strategies of

the firm and the younger generation consumers at a

steady state with full-employment and constant price

under economic growth at the rate γ−1. The strate-

gies of the consumers are derived from their utility

maximization over two periods. We suppose that the

labor productivity is 1 in Period t, and it increases at

the rate γ−1. Also we assume that the nominal wage

rate is 1 in Period t. Denote the pay-as-you go pension

in this period by τ. We assume

(1 + γ)τ < 1.

1. Firm: employs four younger generation con-

sumers and produces four units of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage 1, con-

sumes 1

2[1 + (γ−1)τ]units of the good, pays τ

units of tax for the pay-as-you-go pension system

for the older generation consumers, and leaves

the rest of the income;

2[1 −(1 + γ)τ].

In the next period he consumes 1

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension γτ .

Since we consider a steady state under growth at the

rate γ−1, each older generation consumer consumes

2γ[1 + (γ−1)τ]units of the good by his savings and

the pay-as-you-go pension (τ). Let Gtbe the govern-

ment expenditure other than the pay-as-you-go pen-

sion. The total demand for the good is

21 + 1

γ[1 + (γ−1)τ] + Gt.

Note that there are four younger and four older gen-

eration consumers. This must be equal to the total

supply, 4. Thus,

Gt=2(γ−1)

γ[1 −(1 + γ)τ]>0.(8)

The total net savings of the older generation con-

sumers, which is the difference between their con-

sumptions and the pensions, is

γ[1 + (γ−1)τ]−4τ=2

γ[1 −(1 + γ)τ].(9)

On the other hand, the total net savings of the older

generation consumers in the next period, who are the

younger generation consumers in Period t, is

2[1 + (γ−1)τ]−4γτ = 2[1 −(1 + γ)τ].(10)

The difference between (10)and (9) is

γ(γ−1)[1 −(1 + γ)τ] = Gt.(11)

Therefore, the government expenditure Gtin (8)

equals the increase in the net savings of the consumers

from Period t−1to Period t.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of four units of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, sup-

ply of one unit of labor and 1

2[1 + (γ−1)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

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6.1.2 Period t+ 1

Next we consider Period t+1. Suppose that the labor

productivity in Period t+ 1 is γ−1, and the nominal

wage rate is also γ−1. Let us consider the following

pair of strategies of the firm and the younger gener-

ation consumers in Period t+ 1. The pay-as-you-go

pension for each older generation consumer is γτ.

1. Firm: employs four younger generation con-

sumers and produces 4γunits of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage γ, con-

sumes γ

2[1 + (γ−1)τ]units of the good, pays γτ

units of tax for the pay-as-you-go pension system

for the older generation consumers, who are the

younger generation consumers in Period t, and

leaves the rest of the income;

2[1 −(1 + γ)τ].

In the next period he consumes γ

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension γ2τ.

Each older generation consumer consumes 1

2[1+(γ−

1)τ]units of the good by his savings and the pay-as-

you-go pension (γτ ). Let Gt+1 be the government ex-

penditure other than the pay-as-you-go pension. The

total demand for the good is

2(1 + γ)[1 + (γ−1)τ] + Gt+1.

This must be equal to the total supply, 4γ. Thus,

Gt+1 = 2(γ−1)[1 −(1 + γ)τ] = γGt.(12)

The government expenditure Gt+1 in (12) equals the

increase in the net savings of the consumers from Pe-

riod tto Period t+1. This relationship holds true even

after Period t+ 1.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of 4γunits of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, sup-

ply of one unit of labor and γ

2[1 + (γ−1)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

From these results we get the following proposi-

tion.

Proposition 6. In order to maintain economic growth

we need sustained budget deficits. The budget deficit

in each period equals the increase in the total net sav-

ings of the older generation consumers (the difference

between their consumptions and the pensions) from

the previous period to this period.

6.2 Inflation by excessive budget deficit

6.2.1 Period t

In this subsection we suppose that the government ex-

penditure in Period tis larger than Gtin the previous

case. Denote it by ˆ

Gt. The actual price of the good in

Period tis ρ > 1, and the actual nominal wage rate is

also ρ. We assume that the inflation is not anticipated

by the older generation consumers in the previous pe-

riod (when they are young in their Period 1). We con-

sider the value of ˆ

Gtwhich triggers inflation at the

rate ρunder full-employment. Let us consider the fol-

lowing pair of strategies of the firm and the younger

generation consumers.

1. Firm: employs four younger generation con-

sumers and produces four units of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage ρ, con-

sumes 1

2[1 + (γ−1)τ]units of the good at the

price ρ, pays ρτ units of tax for the pay-as-you-

go pensions for the older generation consumers,

and leaves the rest of the income;

ρ−ρτ −

2[1 + (γ−1)τ] = ρ

2[1 −(1 + γ)τ].

In the next period he consumes 1

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension ργτ at the price ρif the price of

the good is ρin Period t+ 1.

Each older generation consumer consumes 1

2ργ [1 +

(ργ −1)τ]units of the good by his savings and the

pay-as-you-go pension (ρτ). The nominal total de-

mand for the good is

2ρ[1 + (γ−1)τ] + 2

γ[1 + (ργ −1)τ] + ˆ

γ1 + ργ + (ργ2

−1)τ+ˆ

Gt.

If this is equal to the nominal total supply, 4ρ, we have

Gt=2

γργ −1−(ργ2

−1)τ.

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Comparing this with Gtin (8),

Gt−Gt(13)

γργ −1−(ργ2

−1)τ−(γ−1)[1 −(1 + γ)τ]

= 2(ρ−1)(1 −γτ )>0.

This means that an excessive budget deficit expressed

in (13) causes inflation under full-employment.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of four units of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm and consumptions of the

other consumers, supply of one unit of labor and

2[1 + (γ−1)τ]units of consumption are optimal

under the assumption that each consumer spends

half of his income, respectively, on consumption

in Period 1 and that in Period 2.

From these results we get the following proposi-

tion.

Proposition 7. The excessive budget deficit

expressed in (13) causes inflation under full-

employment.

6.2.2 Period t+ 1

Next we consider Period t+ 1. Suppose that the la-

bor productivity in Period t+ 1 is γ, and the nominal

wage rate is ργ. Let us consider the following pair of

strategies of the firm and the younger generation con-

sumers in Period t+ 1. The pay-as-you-go pension

for each older generation consumer is ργτ . Assume

that the price of the good in this period is ρ.

1. Firm: employs four younger generation con-

sumers and produces 4γunits of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage ργ, con-

sumes γ

2[1 + (γ−1)τ]units of the good at the

price ρ, pays ργτ units of tax for pay-as-you-go

pension to the older generation consumers, who

are the younger generation consumers in Period

t, and leaves the rest of the income;

ργ −ργτ −

ργ

2[1+(γ−1)τ] = ργ

2[1−(1+γ)τ].

In the next period he consumes γ

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension ργ2τat the price ρ.

Each older generation consumer consumes 1

2[1+(γ−

1)τ]units of the good by his savings and the pay-as-

you-go pension (ργτ ) at the price ρ. Let ˆ

Gt+1 be the

government expenditure other than the pay-as-you-go

pension. The nominal total demand for the good is

2ρ(1 + γ)[1 + (γ−1)τ] + ˆ

Gt+1.

This must be equal to the nominal total supply, 4ργ.

Thus,

Gt+1 = 2ρ(γ−1)[1 −(1 + γ)τ].

From (12) we find

Gt+1 =ρGt+1.

This relationship holds true even after Period t+ 1.

Therefore, after one period inflation full-employment

under constant price is maintained by sustained bud-

get deficits. Only the excess portion should be re-

duced. There is no need to make up for past excesses

by creating surpluses or reducing deficits.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of 4γunits of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, sup-

ply of one unit of labor and γ

2[1 + (γ−1)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

From these results we obtain the following propo-

sition.

Proposition 8. After one period inflation full-

employment under constant price is maintained by

sustained budget deficits. Only the excess portion

should be reduced. There is no need to make up

for past excesses by creating surpluses or reducing

deficits.

6.3 Insufficient budget deficit and recovery

from recession

6.3.1 Period t

Now we assume that the government expenditure in

Period tis insufficient. Let the actual value of the

government expenditure be

Gt=2

γ(γ−1)[1 −(1 + γ)τ]−

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It is smaller than Gtin (8). We assume that it is non-

negative. The price of the good and the nominal wage

rate in Period tis 1. We also assume that upto the

previous period, Period t−1, full-employment has

been realized. Let us consider the following pair of

strategies of the firm and the younger generation con-

sumers.

1. Firm: employs three younger generation con-

sumers and produces three units of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage 1, con-

sumes 1

2[1+(γ−

3)τ]units of the good, pays 4

3τ

units of tax for pay-as-you-go pensions for the

older generation consumers, and leaves the rest

of the income:

1−

3τ−

21 + γ−

3τ=1

21−γ+4

3τ.

In the next period he consumes 1

2[1 + γ−

3τ]

units of the good by his savings and the pay-as-

you-go pension γτ .

3. One unemployed consumer: consumes 1

2γτ units

of the good by borrowing money from employed

younger generation consumers using his pension

as collateral. In the next period he consumes 1

2γτ

units of the good by the pay-as-you-go pension

γτ , and repays his debt due to consumption in

Period 1

Each older generation consumer consumes 1

2γ[1 +

(γ−1)τ]units of the good by his savings and the pay-

as-you-go pension (τ). The total demand for the good

21 + γ−

3τ+1

2γτ (14)

γ[1 + (γ−1)τ] + ˜

21 + γ−

3τ+1

2γτ +2

γ[1 + (γ−1)τ]

γ(γ−1)[1 −(1 + γ)τ]−

2= 3.

Note that there are four older generation consumers,

three employed and one unemployed younger con-

sumers. (14) means that the total demand equals the

total supply, and there exists one involuntary unem-

ployment.

Are the strategies of the firm and the younger gen-

eration consumers above in a Nash equilibrium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expen-

diture, employment of three consumers and pro-

duction of three units of the good are optimal.

2. Each younger employed consumer: given the be-

havior of the firm, consumptions of the other con-

sumers and the government expenditure, supply

of one unit of labor and 1

2[1 + (γ−

3)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

3. One younger unemployed consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, 1

2γτ

units of consumption is optimal given the situa-

tion that he is unemployed, and spends half of his

income, respectively, on consumption in Period

1 and that in Period 2.

From these results we get the following proposi-

tion.

Proposition 9. The insufficient government expendi-

ture (or budget deficit) causes involuntary unemploy-

ment.

6.3.2 Period t+ 1

Next we consider Period t+ 1. Suppose that we re-

cover full-employment in Period t+ 1 by increasing

the government expenditure (or the budget deficit).

Let ˜

Gt+1 be the government expenditure other than

the pay-as-you-go pension which is sufficient to re-

cover full-employment.

Suppose that the labor productivity in Period t+ 1

is γ, and the nominal wage rate is γ. Let us consider

the following pair of strategies of the firm and the

younger generation consumers in Period t+ 1. The

pay-as-you-go pension for each older generation con-

sumer is γτ . Assume that the price of the good in this

period is 1.

1. Firm: employs four younger generation con-

sumers and produces 4γunits of the good.

2. Each employed younger generation consumer:

supplies one unit of labor, receives wage γ, con-

sumes γ

2[1 + (γ−1)τ]units of the good, pays

γτ units of tax for pay-as-you-go pension for the

older generation consumers, who are the younger

generation consumers in Period t, and leaves the

rest of the income;

γ−γτ −

2[1 + (γ−1)τ] = γ

2[1 −(1 + γ)τ].

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In the next period he consumes γ

2[1 + (γ−1)τ]

units of the good by his savings and the pay-as-

you-go pension γ2τ.

Each older generation consumer, who was employed

in the previous period, consumes 1

21 + γ−

3τ

units of the good by his savings and the pay-as-you-

go pension (γτ ), and one older generation consumer,

who was unemployed in the previous period, con-

sumes 1

2γτ units of the good by the pay-as-you-go

pension. He also repay the debt for consumption in

the previous period.

The total demand for the good is

2γ[1+(γ−1)τ]+ 3

21 + γ−

3τ+1

2γτ +˜

Gt+1.

Note that there are three older generation consumers

who was employed, and one of them was unem-

ployed. Suppose that this is equal to the total supply,

4γ. Then,

Gt+1 = 2(γ−1)[1 −(1 + γ)τ] + 1

Comparing this with (12)

Gt+1

−Gt+1 =1

2>0.

Therefore, to recover full-employment from a state

with involuntary unemployment we need extra bud-

get deficit. After recovery of full-employment we can

maintain it with sustained budget deficit. Thus, the

extra budget deficit should not be redeemed.

Is the pair of strategies above in a Nash equilib-

rium?

1. Firm: given the consumption behavior of the

younger generation consumers and the older gen-

eration consumers, and the government expendi-

ture, employment of four consumers and produc-

tion of 4γunits of the good are optimal.

2. Each younger generation consumer: given the

behavior of the firm, consumptions of the other

consumers and the government expenditure, sup-

ply of one unit of labor and γ

2[1 + (γ−1)τ]units

of consumption are optimal under the assumption

that each consumer spends half of his income, re-

spectively, on consumption in Period 1 and that

in Period 2.

From these results we obtain the following propo-

sition.

Proposition 10. In order to recover full-employment

from a state with involuntary unemployment we

need extra budget deficit. After recovery of full-

employment we can maintain it with sustained budget

deficit. Thus, the extra budget deficit should not be

redeemed.

7 Conclusion

This paper attempts to provide a game-theoretic ba-

sis to functional finance theory and MMT (Modern

Monetary Theory). We have shown that to achieve

and maintain full-employment in a growing economy

we need sustained budget deficits even without infla-

tion, and this budget deficit must not be redeemed.

The model of this paper is very simple. For exam-

ple there is no capital and investment. In the future

research we would like to consider a model in which

firms make investments.

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Sources of funding for research

presented in a scientific article or

scientific article itself

This work was supported by JSPS KAKENHI Grant

Number 18K01594 in Japan.

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(Attribution 4.0 International, CC BY 4.0)

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Commons Attribution License 4.0

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