Mathematical Model of MMT with Profit Return under Monopolistic Competition

Even under constant returns to scale technology there is a positive profit return if the goods are produced in monopolistic competition. By a two-periods overlapping generations (OLG) model with production in monopolistic competition under constant returns to scale in which the economy grows by technological progress and the older generation consumers receive the profits, we consider the problem of budget deficit. The following results will be proved. 1) A budget deficit is necessary to realize full employment with constant price when the economy grows. 2) If the budget deficit exceeds the level necessary and sufficient to maintain full employment in a growing economy with constant price, inflation will occur. A stable budget deficit is necessary to prevent further inflation. 3) If the budget deficit is insufficient to maintain full employment, a recession with involuntary unemployment occurs. We can overcome a recession and restore full employment making a budget deficit larger than the one necessary and sufficient to maintain full employment without a recession. Since we can maintain full employment by constant budget deficits, we should not offset the deficit created for overcoming the recession by budget surpluses. Also, we show that the budget deficit equals the difference between the net savings of the younger generation consumers and that of the older generation consumers in each case.


Introduction
In previous studies we examined the arguments for fiscal policy by Lerner's (1943Lerner's ( , 1944 Functional Finance Theory and MMT (Modern Monetary Theory, Wray (2015), Mitchell, Wray and Watts (2019), Kelton (2020)) 1 by a static model or an overlapping generations model of perfect competition with constant returns to scale technology. Under perfect competition with constant returns to scale there is no corporate profit. In order to include the existence of corporate profits in the analysis, it is possible to consider the case of production using not only labor but also capital, but this paper deals with a simpler model of monopolistic competition with constant returns to scale in which labor is the only production factor. Maintaining the basics of the neoclassical microeconomic framework, such as consumers' utility maximization under budget constraints, and equilibrium of supply and demand of goods in monopolistic competition with constant returns to scale technology, the idea of the effect of fiscal policy in Functional Finance Theory and MMT is discussed by a simple mathematical model.
Using a two-periods (two-generations) overlapping generations model with production in monopolistic competition under constant returns to scale technology in which the economy grows by technological progress and the older generation consumers receive the corporate profits, the following results will be proved. 1) A budget deficit is necessary to achieve full employment under constant price when the economy grows by technological progress. (Section 3) 2) If the budget deficit exceeds the level necessary and sufficient to maintain full employment in a growing economy under constant price, inflation will occur. A stable budget deficit is necessary to prevent further inflation. (Section 4) 3) If the budget deficit is insufficient to maintain full employment, a recession with involuntary unemployment will occur. (Section 5) We can overcome a recession and restore full employment making a budget deficit larger than the one necessary and sufficient to maintain full employment without a recession.
Since we can maintain full employment through constant budget deficits, we should not offset the deficit created for overcoming the recession by budget surpluses.
We consider restoration of full employment by fiscal spending (Section 6) and tax reduction (Sections 7).
Also, we show that the budget deficit equals the difference between the net savings of the younger generation consumers and that of the older generation consumers in each case. In the overlapping generations model the net savings means the consumptions of consumers in their older period (the second period) net of the profit returns which are received in their older 3 period.
We think that the essence of MMT with respect to fiscal policy lies in the following two points.
1. "Financial resources is not necessary for fiscal spending" or "Taxes are not a source of revenue for fiscal spending" From a macroeconomic perspective, fiscal spending has a role to increase the demand for goods, while taxes have a role to reduce demand by reducing disposable income of consumers. To achieve full employment and stable growth without inflation, an adequate balance between the size of fiscal spending and taxes is necessary, not those taxes are necessary for fiscal spending.
2. There is no need to pay off government debt with taxes.
Although not discussed specifically in this paper, all government bonds could be redeemed tomorrow without raising taxes if it wanted to (although it does not have to). The central bank could just buy them all up. This may have the effect of lowering interest rates, but it will not directly increase demand for goods and will not cause high-rate inflation because people's assets will not increase (unless the central bank buys them at prices above face value) and it will not generate income. If it is done during a recession, even low-rate inflation will not occur. Since government bonds are also money in the same broad sense as bank time deposits ("liquidity in the broad sense"), and the central bank's purchase of government bonds does not increase the money supply in that sense, inflation does not occur from the perspective of the Quantity Theory of Money.
In the next section we consider behavior of consumers and firms. In Appendix we will show that if the profit returns are received by the younger generation consumers, the budget deficit equals the difference between the savings of the younger generation consumers and that of the older generation consumers in a case of economic growth with full employment under constant price.

Behavior of consumers and firms
We use a two-periods (generations) overlapping generations (OLG) model according to the model used by Otaki (2007Otaki ( , 2009Otaki ( , 2015. Consumers live over two-periods, the younger period (the first period) and the older period (the second period). They work only in the younger period. They consume goods in their older period by their savings carried over from their younger period and the profit returns received from firms. There is a continuum of goods indexed by ∈ [0,1]. Each good is monopolistically produced by Firm with only labor as a production factor under constant returns to scale technology. The number (or the size) of the firms is normalized to one. Let ( ) be the consumption of good z in Period , = 1,2 of a consumer, ( ) be the price of good in Period , = 1,2. Period 1 is the 4 younger period, and Period 2 is the older period for consumers. Consumers choose their consumptions over two periods to maximize their utility subject to the budget constraint. Let L be the employment and be the labor supply or the employment in the full employment state. 0<* < 1. It is the propensity to consume of consumers. Denote the disposable income over two periods of employed and unemployed consumers by ,-. The budget constraint for a consumer is

Let
The Lagrange function is The first order conditions for maximization of ℒ are From them They mean From (1) and (2) They mean By (3)  From them and (1) This means ( ) = (56 This is the demand function for good of a consumer in his younger period. The demand of an older generation consumer in the same period is ( ) = ( ()56 : . ,-; is the disposable income of an older generation consumer. The total disposable income is written as [*,-+ (1 − *),-; ] . Then, ,and ,-; mean the average income of the younger and the older generation consumers. Now let us consider profit maximization of a firm in Period 1. The profit of the firm producing good is . @ is the nominal wage rate and A is the labor productivity. The condition for profit maximization with respect to ( ) given is in the equilibrium. Hereafter we call and the prices in Period 1 and 2. The government determines the demand for good , F( ) , given its expenditure G to maximize The condition for maximization of H is

This means
. This is the demand for good z of the government. In the equilibrium The total demand for the goods is *,-+ (1 − *),-; + G .
In the equilibrium the total demand equals the total supply which is y . is the employment. Under full employment = .
The economy grows by technological progress. The labor productivity increases at the rate L − 1 > 0 from period to period. The nominal wage rate @ also increases at the rate L − 1 under constant price. In a period, Period 1, it is y. If @ = A, Let Π be the profit return of a firm for a consumer in Period 1. Then, For simplicity we assume that every older generation consumer receives the profit return including consumers who were unemployed in their younger period. Of course, we can assume that only the older generation consumers who were employed receive the profit returns. Essentially, the result is the same. In the next period (Period 2) the labor productivity increases at the rate of L − 1. Thus, the profit return of a firm for a consumer in that period is L@ N .
Note that L@ is the nominal wage rate (without inflation) in Period 2. Denote the profit return which is received by an older generation consumer in Period 2 by Π . Then, Π = L@ N . 8

Economic Growth and Budget Deficit
We suppose full employment in this section. Thus, Π = L@.
Let G and O be the government expenditure and the (total) tax in Period 1. The younger generation consumers pay the taxes. Suppose @ = A. The disposable income of younger generation consumers in total is The (total) savings of the younger generation consumers in Period 1 is This is equal to the consumption of the older generation consumers in Period 2. Since they receive the profit returns in the next period, their net savings is From the equilibrium between the total supply and the total demand, we get (2) implies that if L > 1, the budget deficit should be positive, and that the budget deficit equals the difference between the net savings of the younger generation consumers and the net savings of the older generation consumers.
Summarizing the results, we obtain Proposition 1 We need a budget deficit to achieve and maintain full employment under economic growth with constant price even if there are profit returns for the older generation consumers. The budget deficit equals the difference between the net savings of the younger generation consumers and that of the older generation consumers.
This budget deficit should not be offset by future surplus so long as the economy grows with constant price.

Inflation by Excess Budget Deficit
Let S = S 1− 1 $ be the price of the goods in Period 2, the nominal wage rate be SLy, the government expenditure be TG. The profit return for an older generation consumer is We assume that the nominal value of the tax is LO. The nominal total supply in Period 2 is S LA .
Also, we assume that the profit return in the next period for a younger generation consumer is Π R = LΠ .
Their net savings is The consumption of the younger generation consumers is Their net savings is This is the difference between the net savings of the younger generation consumers and that of the older generation consumers. Comparing this with (7), Proposition 2 A budget deficit (given tax 2 ) that exceeds the level necessary and sufficient to maintain full employment in a growing economy under constant price will cause inflation.
(8) means that even in this case the budget deficit equals the difference between the net savings of the younger generation consumers and that of the older generation consumers.
We need a stable budget deficit discussed in the previous section to prevent further inflation.

Recession and Involuntary Unemployment by Insufficient Budget Deficit
Let be the employment in Period 2. Assume that the government expenditure is TG, the tax is LO . The price of the goods is constant. The profit return for an older generation consumer is If full employment will be restored in Period 3, the profit return in the next period for a younger generation consumer is The nominal total supply in Period 2 is L 2 A .
Their net savings is The consumption of the younger generation consumers is 2 A change in fiscal spending and that in tax have different effects on the national income. An increase in fiscal spending and tax while keeping the budget deficit constant will lead to inflation, while a decrease in fiscal spending and tax while keeping the budget deficit constant will lead to a recession. It is because the multiplier of a change in tax is smaller than that of a change in fiscal spending. On the other hand, a change in tax influences on consumption of the older generation consumers in the next period.
From the equilibrium between the total supply and the total demand we obtain This is the difference between the net savings of the younger generation consumers and that of the older generation consumers. Assume that Π R in (9) equals LΠ in (7). Comparing (9) with (7), Proposition 3 A budget deficit (given tax 4 ) that is insufficient to maintain full employment will cause a recession with involuntary unemployment.
(4) means that even in this case the budget deficit equals the difference between the net savings of the younger generation consumers and that of the older generation consumers.

Recovery from Recession by the Government Expenditure
We will recover full employment in Period 3. Suppose that the taxes in Periods 2 and 3 are LO and L O. The price is constant at . The total supply is L A .
The profit return for an older generation consumer is denoted by The profit return for a younger generation consumer is The consumption (or the savings) of the older generation consumers is By (10) and (11) When < , we have T > L . Therefore, a larger budget deficit is required than would be required to maintain full employment in the absence of a recession. From (10) Summarizing the results, we obtain Proposition 4 A budget deficit larger than the one necessary and sufficient to maintain full employment without a recession can overcome a recession caused by insufficient budget deficit and restore full employment.
(12) means that even in this case the budget deficit equals the difference between the net savings of the younger generation consumers and that of the older generation consumers.
We should not offset the deficit created to overcome the recession by subsequent surpluses because we need a stable budget deficit to maintain full employment with economic growth under constant price.

Recovery from Recession by the Tax Reduction
Again, we will recover full employment in Period 3. The price is constant at . Suppose that the government expenditures in Periods 2 and 3 are LG and L G. As in the previous case the profit return for an older generation consumer is denoted by The profit return for a younger generation consumer is The nominal total supply is L A .
Their net savings is Let $O be the tax in Period 3. Then, the consumption of the younger generation consumers is Their net savings is The total demand is Note that is employment in Period 2. From the equilibrium between the total supply and the total demand, we have L A = *PL A + LΠ R − $OQ + (1 − *)PLA + Π R − LOQ + L G.
Again, if there is no recession, (11) holds in Period 3. By (11) and (13), When < , we get $ < L . Therefore, a larger budget deficit by tax reduction is required than would be required to maintain full employment in the absence of a recession. Since *<1, we can see that the tax cut in this case must be larger than the additional fiscal spending in the previous case. This is because the marginal propensity to consume is smaller than one. However, as discussed in footnote 2, the tax cut will affect consumption in the next period and will have the same effect as an increase in fiscal spending in the long run.
Summarizing the results, we obtain Proposition 5 A budget deficit larger than the one necessary and sufficient to maintain full employment without a recession can overcome a recession caused by insufficient budget deficit and restore full employment by tax reduction.
(15) means that even in this case the budget deficit equals the difference between the net savings of the younger generation consumers and that of the older generation consumers.

Concluding Remark
Using a two-periods overlapping generations (OLG) model with production of goods in monopolistic competition with constant returns to scale technology and profit returns to older generation consumers, we have shown that excessive budget deficits lead to inflation, but budget deficits are necessary and useful for full employment under economic growth and overcoming recessions. Although in this paper we assume monopolistic competition to derive the existence of profits, similar conclusions may be drawn by considering an economy in which goods are produced by both capital and labor as factors of production. Assuming that old-age consumers, who hold capital through savings, receive profits, the sum of which is included in aggregate supply because it is distributed from firm revenues along with wages paid to the younger generation. On the other hand, the older generation will probably allocate some of the profits to consumption in their youth in anticipation of receiving them. Or, by the same token, the younger generation will calculate the future profits they will receive and spend part of it on consumption in their younger years. The same conclusion holds because the net savings in youth will be less than the amount of consumption spent from the future profits received.

Appendix: When the profit returns are received by the younger generation consumers
Suppose that the profit returns are received by the younger generation consumers. We consider a case of economic growth with full employment under constant price. Let Π be the profit for a younger generation consumer in Period 1. Then,
The profit for a younger generation consumer in Period 2 is Π = LA = LΠ .