Mathematical Foundations for Mosler's Full Employment-Zero Inflation Model
by
Pavlina R. Tcherneva
Gettysburg College
Conceptual Framework
This paper will begin to develop a mathematical model that shows the basic relationship between prices paid by the government and the quantity received by the government from the private sector*, assuming the following cause and effect, as outlined in Mosler's analysis "Soft Currency Economics":
- The government needs real goods and services (g&s).
- The government imposes taxes, in dollars, in order to create sellers who offer real g&s in return for the needed dollars.
- The government purchases the desired g&s.
- The government is the monopoly issuer of its currency. Therefore, it has the ultimate power to determine the price it pays for g&s; i.e. prices are an exogenous variable.
Model
The terms and conditions are defined as follows:
Population--10 people.
T--tax liability of the economy.
PL--wage paid by the government for one fire fighter.
QL--quantity of labor that works for the government as fire fighters.
O<QL<M, where QLM is the maximum capacity of the economy (10 people).
G--government spending.
Gd--government deficit.
I--aggregate investment.
S--aggregate savings.
Sn--net nominal savings in the economy.
Assumptions
- T is fixed (a fixed property tax is a good example).
- There is only one tax period.
*The private sector includes all participants holding dollars, domestically or abroad; i.e. everyone except the monopoly issuer of the currency.
The model will start with the accounting identity:
G + I = T + S
The equation above can be restated:
G = T + (S - I), where
(S - I) = Sn = Gd.
Case 1
Supplementary Assumptions:
1. The economy produces one service only, fighting fires.
2. The only expenditure of the government is paying wages to the fire fighters.
3. Sn = 0 (assume that in our model there is no provision for a government deficit).
4. T= $10 --total tax bill for the entire community.
5. Labor is not divisible (only 1, 2, 3 fire fighters can be hired).
6. Labor hours are not divisible. Every worker is hired full-time for the entire tax period.
7. The lowest unit of account is $1.00.
8. PL-the price of labor is set by the government.
The following is a general formula for the level of government spending for a given tax period:
n
å (Qi x Pi) = T
i=1
The expression indicates that, in order to calculate the amount of dollars necessary to pay taxes, we need to sum the revenue of the sale of quantity Q real goods and services offered by the private sector, where each good Qi is sold for some corresponding price Pi. Since in this model the private sector offers only one service--working as fire fighters--and government's only expense is paying wages, the expression is simplified to the following:
PL X QL = T.
From this equation we can extrapolate that, in order to obtain T dollars for taxes, the private sector must sell QL = T/PL services to the government. In Figure 1, points A, B, C, and D indicate the quantity of labor (QL) that is transferred from the private sector to the government at a corresponding wage level (PL); i.e. how many people sell their labor as fire fighters at different wages, given the aggregate taxobligation (T) of $10.
Figure 1.
Since labor and wages are not divisible, the graph in Figure 1. consists of specific points as opposed to a continuous curve. For example, the government cannot hire 1.5 fire fighters and, under the initial assumption, it cannot offer 2.5 dollars as a price for labor.
Point A indicates that if the government offers $10 per fire fighter, only one will be hired. Points B, C, and D show that a wage of $5, $2, and $1 will result in 2, 5, and 10 fire fighters, respectively. No other points exist, since Sn = 0.
The example illustrated in Case 1 establishes the following inverse relationship: an increase in prices paid by the government corresponds a decrease in the quantity of goods and services it receives. Conversely, higher prices result in fewer goods and services purchased.
Case 2
Supplementary Assumptions:
Sn = 0.
T = $1 0.
This case removes supplementary assumptions 4 and 5 form Case 1, allowinglabor-hours and wages to be divisible; i.e. labor can work less than full time for a given tax period and the smallest unit of account is no longer $1.00. The inverse relationship between the price the government pays for labor and the amount it receives from the private sector continues, and is now represented by a function, instead of discrete points:
PL = T/QL PL(min) = $1 (minimum wage)
Figure 2 demonstrates the following:
- With Sn = 0, there is only one unique curve.
- As QLM = 10 defines the limit of the capacity of the economy, it also defines the minimum wage PL(min) the government can offer. In Case 2 this wage is denoted by point A.
- The curve is asymptotic to the y-axis. It never crosses it because the existence of any tax means that some labor will always be attracted.
Case 2a
Supplementary Assumptions:
- T Sn = 0
- T=$10.
- T.he government chooses to set the price of labor (PL is exogenously determined).
This case will be described by the same continuous graph as in Figure 2. The graph demonstrates that at any given PL the government knows the amount of QL it will receive. PL does not change via market forces.
Case 2b
Supplementary Assumptions:
- Sn = 0.
- T=10.
- The government forfeits its option to set PL by choosing to pay a market price for labor (PL is market determined).
This case, where G=T = PL X QL is described by Figure 3.
Without the government setting PL, as in Case 2a, this model shows that the possible outcomes for the time period include all points on the line.
In Figure 3, the market price of labor will not fall bellow $ 1, because then there will be no solutions for PL X QL = T. The market price also can never exceed $10, because the government is spending $10 only. In order to fulfill its tax obligation, the private sector will have to provide only one unit of labor. Therefore, in this case, the market price of labor has the following boundaries:
G/QL(min) > PL > G/QL(max) or 10/1 > PL > 10/10 or $10 > PL > $1.
In Case 2a the government sets the price of labor, making it an exogenous variable. In Case 2b the government pays a market determined (endogenous) price of labor and thus it does not know the price it will pay and the corresponding number of fire fighters it will receive. By allowing the taxpayers to set prices, the government also gives them the power to determine quantity. This could pose an interesting dilemma. For example, an organized labor pool could see to it that PL would be $10 at all times, and that any tax increase would be answered with a corresponding price increase. Therefore, any attempt by the government to get more than the minimum QL would fail. Please note that it is the government's (exogenous) choice whether to implement policy 2a or 2b.
Case 3
Case 3 incorporates an additional government expenditure--the purchase of park benches from the private sector. In Case 3a, the government chooses to set the wage of the fire fighters, and to pay market determined prices for a bench. In Case 3b, the government decides to determine both the wage it offers for a fire fighter and the price it pays for a bench. And in Case 3c, it does not set any prices. (Case 3a is a basic outline for the full employment model discussed later in this paper.
Case 3a
Supplementary Assumptions:
The government sets the price of labor (PL), and purchases benches at a market determined price (PB).
Sn = 0
Case 3a illustrates how a change in the price of benches affects the number of people who will work for the government as fire fighters. The graphs below show that, as the government increases the number of benches it purchases from the private sector, the number of fire fighters it gets will decrease. The general relationship between the two variables is given by the formula:
PB = T/QB - PL/QB X QL,
where T, PL, and PB are given.
As the graphs bellow demonstrate, when QB increases, QL decreases.
T = $10 PL = $1
PB = $3 T=PB x QB + PLX x QL
Graph A
Graph B
Graph C
QB = 1 bench QB = 2 benches QB = 3 benches 10 = 3x1 + 1xQL 10 = 3x3 + 1xQL 10 = 3x3 + 1xQL Ql = 7 Ql = 4 Ql = 1 Pb = 10-Ql Pb = 10/2-1/2 * Ql Pb = 10/3-1/3 * Ql
Figure 4.
Consider prices PB1 and PB2 at every given level of purchase of benches by the government in Figure 5, which is identical to Figure 4. Notice that the government will attract fewer fire fighters with a high market price of benches than with a low market price. In other words, when market prices are high, the number of people working for the government is low. When market prices fall, more public service jobs are taken to obtain the needed dollars.
Figure 5.
In Figure 5, point K is defined by the ratio T/PL. If Sn > 0 then point K will shift to the right and will be given by G/PL.
Case 3b
Supplementary Assumptions:
- Government spending includes the purchase of labor and park benches.
- Government sets both the wage of a fire fighter and the price of a bench.
Conditions:
- PB--government determined price of a bench.
- QB--the number of benches sold to the government.
- Sn = 0.
The amount of goods and services the government will be able to purchase is given by the following budget constraint:
G = PB x QB + PL x QL,
restated:
QB = (G / PB) - (PL / PB) QL.
Changing PB alters the slope of the graph but not the x-intercept:
Graph A
Graph B
Graph C
government price= PB government price=PB + 1 government price=PB - 1
We have incorporated the purchase of another good by the government. The relationship between how much it pays and how much it gets continues to be inverse. If the price of benches (PB) is raised, all else equal, the quantity sold to the government (QB) declines:
Each point on the line represents a different basket of goods and services. The y-intercepts on all 3 graphs are the points where the government gets only benches and no fire fighters. The x-intercept is the point where the government only employs fire fighters and does not purchase any benches. Of course, the private sector may want to sell some combination of the two; pt E is an example. The function does not give any specific equilibrium level of goods and services transferred from the private to the public sector, resulting in an uncertain quantity of labor and/or benches.
Case 3c
Supplementary-Assumptions:
- Government spending includes the purchase of labor and park benches.
- Government accepts market determined prices for both the wage of a fire fighter and the price of a bench.
On Figure 6, the price and quantity of benches change but the x-intercept (point K) remained constant, since it is defined as G/PL. In this case, however, PL also fluctuates due to changing market conditions. Hence Case 3c is similar to Case 2b, where there were infinite possible outcomes. Both Case 3c and 2b are characterized by the absence of exogenous price control by the government, with price and, therefore, quantity determined by the taxpayer.
Case 4
All cases so far have assumed that Gd = Sn = 0. Case 4 introduces different levels of Sn to Case 2a.
Supplementary Assumptions:
- T=$10.
- QLM = 10.
- The government purchases fire fighters only.
This case graphs 3 conditions:
- Sn = $0
- Sn = $2
- Sn = $-2, where Sn = $-2 means that the private sector dis-saves as it borrows form the government.
Here is the data for the three cases and the graphs that describe the number of fire fighters that will work for the government at different wage-levels.
Case 4a
Case 4b
Case 4c
Sn=Gd=$0
Sn=Gd=$2
Sn=Gd=$2
G=10 + 0 = $10
G=10 + 2 = $12
G=$8
PL= G/QL = 10/Q
PL= 12/QL
PL= 8/QL
Figure 7.
All three curves in Figure 7 are downward sloping: -the more the government pays, the less it receives from the private sector (at a given tax level).
The three situations are characterized by different minimum wages that the government can offer. With the full capacity confined to 10 people, in the first situation (curve PL = 10/QL), the government is not running a deficit (Sn = 0); hence $1 is be the minimum that the government can pay for a fire fighter. At that point (point A) it will attract 100% of the capacity and everyone will be working in the public sector, fighting fires. In the second situation (curve PL=12/QL) when the private sector net saves $2, the minimum wage is given by point B, $1.20. Similarly for the third curve that wage is $0.80.
Changing Sn does not change the basic relationship, it only shifts the curve.
This shift can be expressed mathematically in general terms as opposed to usingparticular values for our variables:
Let G = PL x QL = T + Sn
PL = (T + Sn)/QL = T/QL + Sn/QL
Let net savings increase from Sn to Sn *, by a change of Sn = Sn *-Sn
G*=PL* x QL = T+Sn*
PL* = (T + Sn*)/QL = T/QL + Sn*/QL
= > PL* = T/QL + (Sn + Sn/QL = T/QL + Sn/QL + Sn/QL
= > PL* = PL + Sn/QL
therefore, the curve has shifted by Sn/QL. We have found a general expression for the shift of the curve.
Figure 8.
Conclusion
The mathematical framework presented in this paper outlines the inverse relationship between the price government pays for goods and services and the quantity of real goods and services it receives, given the following assumptions:
- The government is the monopoly issuer of its fiat currency.
- The government exogenously sets taxes, and creates sellers of real goods and services.
- The government has the ultimate power to set exogenously the prices it pays for real good and services.
The inverse relationship is maintained regardless of the fact that the private sector may or may not have a desire to net save. Net savings equals the government deficit by definition, which can be incorporated into a fiscal policy that provides for full employment and price stability. Let the government offer a job to anyone that is willing and able to work at a fixed wage rate, allowing for the deficit to float. As a result, full employment will immediately result.
The government employees will be a supplementary labor force that will then act as a stabilizer of the economy. When the economy slows down and the private sector starts laying off workers, the government as an employer of last resort will provide jobs for those that need them. The resulting increase in the government deficit has an expansionary bias.
As the economy picks up again, new demand for those supplementary government workers in the private sector will result. In a growing economy, the supplementary workers will get hired away form the public sector. This will cause the government deficit to shrink, which is a contractionary fiscal bias. It is important to keep in mind that government should maintain a fixed rate of pay to the public sector employees that does not compete with the rate of pay offered by the private sector. As long as the supplementary wage remains constant, it will act as a stabilizing and a defining factor for the currency.
To recapitulate, the math model outlines some basic relationships which must be considered in the enforcement of the proposed fiscal policy option. Considering that taxation is the driving force behind fiat currency, that the government has the ultimate power to exogenously set taxes and the prices, this fiscal policy can eliminate the problems of unemployment and inflation altogether.*
References:
Mosler, Warren B., Sep. 1995 Soft Currency Economics, West Palm Beach, Florida: III Finance
*For a more elaborate discussion on Warren Mosler's analysis of fiat currency, refer to "A Critical Review of Soft Currency Economics" by Pavlina R. Tcherneva.