Income equals expense. This simple accounting identity underlays all economic trade. It expresses in mathematical form the concept that property is traded for property. The identity is true for both traders, only the direction is reversed.
It is important to this discussion to be aware that income and expense are considered to have occurred over some time period. The length of the time period is unspecified but is usually on a year-to-year basis. Notice that the equation stands alone without interaction between the time periods, a fact that we will overcome later in the discussion.
We can use the 'income equals expense' identity in macroeconomics by writing M*V = P*Q.  The left side of the equation can be considered as a general expression of income. The right side is the data driven value of all prices and quantities traded, considered as expenses.
On the income side, the term M (Money supply) represents a master supply of money available during the data period. The master supply of money can be measured with M1, M2, and MZM, terms often associated with the quantity theory of money . We will look for another way to measure money supply, a measure linked to past prices and quantities.
Establish Government Borrowing as a Stable Feature of the Macro-economy. 
We begin by using M*V = P*Q to take a better look at the private economy, keeping in mind that we are looking for another measure of money supply.
Assume that M*V = P*Q = GDP (Gross Domestic Product) as measured by Federal Reserve statistics. We can then write for the macro-economy
GDP = GDP
which gives us an expression for macro-trade where income equals expenses.
We will find the private sector GDP (PGDP) by subtracting the government sector. If we used the government budget constraint components Expense (Exp) = Taxes (T) + Borrowing (B), the result of the subtraction would be
GDP - Exp = GDP - T - B (1)
giving us an expression of private economy activity defined in two ways. It would also be correct to write
PGDP = GDP - Exp = GDP - T - B. (2)
This identity becomes troubling because many economist will deny that the B term is money. They would be right. The term B is borrowing recorded in amount equal to an identical amount spent to pay government expenses. Despite having a non-money status, the B term must represent new money created in a manner similar to money creation by bank lending (A loan note is traded for money, the money is spent to form new bank deposits, and only the note remains as evidence of the trade).
Equation (2) reveals that term B must carry between measuring periods. Stated another way, borrowing not paid in one period would continue into the next period. This would be an important consideration in any model of the economy that spanned between time periods. This delayed purchasing power is in addition to the purchasing power represented by all the deposits created by government spending. Stated another way, government borrowing measures BOTH outstanding money and delayed purchasing power.
When we look at the historical record for the United States, we see that the national budget has been in deficit for nearly all the years since the 1930s. The sum of all the borrowing has reached a number that nearly matches the size of annual GDP. It is reasonable to consider that the debt can be used as a measure of monetary base, with some fraction of the base constantly available in a form of the commonly exchanged property money.
Government borrowing has been described as a stable record of money creation which would make the record a suitable monetary base. We will label the government borrowing monetary base as MGB.
This base recognition would agree with concepts from Modern Monetary Theory (MMT) as expressed by Randall Wray, Joseph Laliberte and many others. (Laliberte's post delves into the use of M*V = P*Q as a vehicle for monetary policy.)
Comparing M2 and the Sum of Government Borrowing MGB
As we go boldly into the realm of a new money supply measure, we need to compare the well accepted measures with the new. We will look at M2 as an example remembering that M1 and MZM have similar underpinnings.
The most notable characteristic of all three measures of money supply is that they are all bank based. That is, the amount of money on deposit in banks is an important component of each measure. This has an inherent problem because bank loans create the illusion of increased money supply as described previously. The central bank can reduce the visible amount by exchanging bonds for currency (some would say 'reserves'). M2 is one measure of the remaining money supply left in banks.
Chart 1 is a comparison of the actual values of M2 and government borrowing MGB.  The reader can see that the two values are roughly the same and follow roughly the same trajectory. Why would the two measurements follow roughly the same trajectory? Because they both attempt to measure a base money supply.
|Chart 1. M2 and MGB. MGB is "Federal Debt Held by the Public" FYGFDPUN .
We will compare private sector GDP (PGDP) vigor with the ratio of pricing to money supply, which is commonly called velocity. We use two measures of money supply, M2 and MGB as the reference scale. Velocity is the balancing term plotted in Chart 2.
|Chart 2. Private GDP velocity vs M2 and MGB. MGB is the longer trace. 
Chart 2 illustrates the difficulties a theorist would have when using either money supply measurement as a stable base. MGB has an advantage that the value of borrowing is driven by fiscal policy and is recorded as a matter of accounting. M2 would need an adjustment term to account for delayed purchasing power if it was used in a model.
MGB can be as stable as the government desires. It will increase or decrease at the rate of change in government deficits.
In Chart 2, the rate of annual money turnover (MGB) is seen to have declined as the supply has grown. This is an indication that prices are not directly related to money supply.
A trend line can be drawn along the bottom of the peaks. The peaks seem to correspond to periods of high construction activity which can be seen to have lasted for several years. This may be the result of higher rates of bank borrowing activity, a possibility that would be worthy of further study.
Note . M*V = P*Q is a widely used equation associated with the quantity theory of money . Money Supply (M) times Velocity (V) equals Price (P) times Quantity (Q). Velocity and Quantity both are scales measured in number of transactions. Money Supply and Price both are scales measured in price per transaction. The quantity theory postulates that a change in the money scale will drive a change in the price scale. In this post, the equation is used in a different way.
Note . A relationship between government borrowing and money supply is found in a different way in the post "Suggestions for Enhancement of Modern Monetary Theory (MMT)".
Note . The Federal Reserve has provided the data series "Federal Debt Held by the Public" FYGFDPUN which seems to capture the sum of all Federal borrowing. FYGFDPUN seems to be the sum of Federal Reserve series FDHBFRBN and FDHBPIN, identical except for scale.
(c) Roger Sparks 2014