A Brief Rationale for MMT Arguments

In this note we examine MMT (Modern Monetary Theory) arguments by a simple macroeconomic model without microeconomic foundation. Mainly we will show the following results. 1) In the underemployment case the national income is determined by the budget deficit. 2) In the full employment case we can define the budget deficit which is necessary and sufficient to achieve full employment. 3) The excessive budget deficit causes inflation. 4) We need budget deficit to achieve and maintain full employment under economic growth. 5) We can recover recession by the budget deficit which is larger than that when full employment is maintained. Also, we show that the budget deficit equals the increase in the savings between generations.


Introduction
In some previous studies (Note 1) we have examined the arguments for fiscal policy from the perspectives of Lerner's (1943Lerner's ( , 1944 Functional Finance Theory and MMT (Modern Monetary Theory, Wray (2015), Mitchell, Wray and Watts (2019), Kelton (2020) (Note 2) by several models, a static model, an overlapping generations model (according to Otaki, 2007Otaki, , 2009Otaki, , 2015 of perfect competition under constant returns to scale technology, an overlapping generations model of perfect competition under decreasing returns to scale technology with positive profits, an overlapping generations model of monopolistic competition (oligopoly with differentiated goods) with positive profits; with or without pay-as-you-go pensions for the older generation consumers, with or without unemployment insurances, with or without consumption in the childhood period before the younger (working) period. In these models, we studied the problems caused by budget deficits from the perspectives of Functional Finance Theory and MMT based on the microeconomic foundations of consumer and firm behavior: utility maximization by consumers under budget constraint and profit maximization by firms. In this note, on the other hand, we will provide a discussion of the effects of budget deficits using a very simple macroeconomic model without the microeconomic foundations.
Mainly we will prove the following results.
1) Underemployment case  The real national income is determined by the budget deficit given the marginal propensity to consume, the constant part of consumption function and tax. (Proposition 1)  The real national income is determined by the budget deficit given the marginal propensity to consume, the constant part of consumption function and the fiscal spending. (Proposition 2) If full employment is not achieved, the independent variable is the budget deficit, and the national income and the savings are determined by the budget deficit not that the savings of consumers determine the budget deficit.
2) Full employment case  We can define the budget deficit which is necessary and sufficient to achieve full employment given the propensity to consume, the constant part of the consumption function and the tax (or the fiscal spending) by the full employment real national income and the investment. (Proposition 3 and 4)  If we achieve full employment under balanced budget, the constant part of the consumption function must take a specific value. (Proposition 5) The results in 1) and 2) mean that insufficient budget deficit (the budget deficit which is smaller than the budget deficit necessary and sufficient to achieve full employment) causes recession with involuntary unemployment.

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ISSN 2329-521X 2021, Vol. 9, No. 2 www.macrothink.org/iss 3 The excessive budget deficit (the budget deficit which is larger than the budget deficit necessary and sufficient to achieve full employment) causes inflation. (Proposition 6)

4) Economic growth with full employment
We need budget deficit to achieve and maintain full employment under economic growth. (Proposition 7)

5) Recovery from recession
In order to recover full employment from recession we need budget deficit larger than that when full employment is maintained. (Proposition 9) Also, we will show that the budget deficit equals the increase in the savings between generations (Proposition 8). Although our model is not an overlapping generations model, one generation of consumers will consequently leave savings for the next generation.
We think that the essence of MMT with respect to fiscal policy lies in the following two points.
 "Financial resources is not necessary for fiscal spending" or "Taxes are not a source of revenue for fiscal spending" From a macroeconomic point of view, fiscal spending has the role of increasing the demand for goods, while taxes have the role of reducing disposable income of consumers and thus reducing demand. In order to achieve full employment and stable growth without inflation, an appropriate balance between the size of fiscal spending and taxes is necessary. We do not need taxes for fiscal spending  There is no need to pay off government debt with taxes.
The central bank could just buy up all the government bonds. This might have the effect of lowering interest rates, but it would not directly increase demand for goods and would not cause high rates of inflation, because people would not have more assets or income (unless the central bank bought them at a price above face value). If it is implemented during a recession, it will not cause low inflation either. Government bonds are money in the same broad sense as bank term deposits ("liquidity in the broad sense"), and the central bank's purchase of government bonds does not increase the money supply in that sense, so from the point of view of the quantity theory of money there is no inflation.

Underemployment Case
We consider a simple macroeconomic model without the microeconomic foundations of consumers and firms. Let be the full employment real national income, and be the actual real national income. We assume underemployment, that is, < . The price of the goods is one. The consumption function is = 0 + ( − ).
is the tax. is the marginal propensity to consume. 0 < < 1. The national income is = + = 0 + ( − ) + + . (1) is the real government expenditure. is an investment of firms. No detailed analysis of investment is given. We assume that it is constant. From this

+G,
These mean the following results.
Proposition 1. The real national income is determined by the budget deficit − given , 0 and .

Proposition 2. The real national income
is determined by the budget deficit − given , 0 and .

Full Employment Case
is not a variable, but constant. If full employment is achieved, (1) is rewritten as = 0 + ( − ) + + .

This means
Thus, we find Proposition 3. The budget deficit which is necessary and sufficient to achieve full employment given , 0 and is The results in Proposition 1, 2, 3 mean that insufficient budget deficit (the budget deficit which is smaller than the budget deficit necessary and sufficient to achieve full employment) causes recession with involuntary unemployment.
(2) is rewritten as follows, ISSN 2329-521X 2021 www.macrothink.org/iss 5 This means Proposition 4. The budget deficit which is necessary and sufficient to achieve full employment given , 0 and is

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Also we find Proposition 5. If full employment is achieved under balanced budget ( − = 0), 0 must be If the budget deficit is smaller than the value in (2) and (3), there will be underemployment, or the recession with involuntary unemployment.

Inflation by Excessive Budget Deficit
Let be the price of the good. Suppose that the nominal national income is , the nominal government expenditure is ′. 0 and T do not change, and full employment is achieved. Then, we have = 0 + ( − ) + + ′.
Alternatively we assume that the tax is decreased to ′, and G does not change. Then, we have = 0 + ( − ′) + + . This means Comparing this with (3)  > 1 when − ′ > − . Therefore, we have shown the following result.
Proposition 6. The excessive budget deficit causes inflation.
As for the inflation process, we can think of a story in which excess demand for goods generates excess demand for labor, which raises the nominal wage rate, which in turn raises the prices of the goods. ISSN 2329-521X 2021 www.macrothink.org/iss 6

Economic Growth with Full Employment
We can assume that 0 is financed by the savings of the previous generation consumers. If the economy grows between generations at the rate of − 1 > 0 with full employment, Investment and tax also increase at the rate of − 1. Then, from (2) This means Proposition 7. We need budget deficit to achieve and maintain full employment under economic growth.

The Budget Deficit and Savings by Consumers
(6) implies the following results.

Proposition 8. The budget deficit equals the increase of the savings between generations.
Let us consider generally. Let ̅ be the real national income in the previous period (Period 0), and be the real national income and the price in this period (Period 1). The savings in Period 0 is equal to Therefore, Proposition 8 holds whether we have full employment or not, whether we have inflation or not.

Recovery from Recession
Suppose that a recession is caused by an insufficient budget deficit in Period 0, and full ISSN 2329-521X 2021 www.macrothink.org/iss 7 employment is restored in Period 1. The price is constant at 1. ̅ < , and 0 is the same as that in the previous case. Let ′ and ′ be the fiscal spending and the tax in Period 1.

From this
If full employment is achieved in Period 0, and the fiscal spending and the tax in Period 1 are and , we have If = ′, (8) and (9) mean On the other hand, if = ′, (8) and (9) (10) implies that the budget deficit when recession is recovered by an increase in the fiscal spending is larger than the budget deficit when full employment is maintained. (11) implies that the budget deficit when recession is recovered by the tax reduction is larger than the budget deficit when full employment is maintained. We have shown Proposition 9 In order to recover full employment from recession we need budget deficit larger than that when full employment is maintained.

Conclusion
We have shown the main statements of MMT (Modern Monetary Theory) and Functional Finance Theory by a simple macroeconomic model. The purpose of this note was to show that the basic propositions of MMT can be established by simple models without the use of complicated ones including microeconomic foundations.