The Equation of Exchange and The Requirement Of Debt

The equation of exchange is one on the few physics equation in economics.  It is typically presented as


Where M is the stock of money in the flow, V is the rate of turnover for that stock.  p is the price of goods, and q is the quantity of goods sold.

It says that, when all of the price times quantities of all goods are added together, the sum is equal to the stock of money in the flow times the number of times the money is used in the transactions.

V is not really velocity in the physics sense of distance per unit time. It is frequency of exchange.  Depending on whether the time period is explicit or assumed, it can be in two forms.  It is in units of $ used per $ in the stock, $/$. As $/$ cancels, it is unitless.  If it includes the time period, it is in units of $ used per $ in the stock per time period, or $/$*time. The $s cancel and it is in /time, like /year.

This equation, and the relationships, are valid for all economic systems, it is simply a description, as long as the definition of M being that of the money in the flow is adhered to.

For the economy that we are most interested in, our US economy of end user consumption, GDP =  M·V = Σ(p·q).

And, the stock of money, M, in the equation of exchange is strictly the money in the flow.  It is not the stock of money in reserve accounts at the bank. It isn’t even the stock of money in checking and savings accounts.  It is called the equation of exchange for this reason.

For instance, a checking account may have $1000 in it at the beginning of the month, with two paychecks of $500 deposited during the month, $900 spent during the monthly, for a final balance of $1100.  By measures of M1, this represents $1000 and $1100 for each successive measure while the actual money that flowed is $900.

This equation of exchange is related to inflation.  If the stock of money in the flow is increased while the velocity remains constant and the quantity of good is steady, then the price of goods increases.  This is what is being referred to when people say that “printing” money causes inflation.

An increase in GDP is not necessarily the result of inflation.  Inflation refers to the process by which Q remains constant while P and GDP increase.  This is the difference between real and nominal GDP.  Nominal GDP may increase with no increase in output.  As such, it is price inflation and real GDP has not changed.

On the other hand, if Q increases while P remains the same, GDP will increase if, in fact, either M or V increases.  This is an increase in real GDP.  And, of course, if Q increases but M and V remain constant, the P must decline which is deflation.

One thing to note is that if population increases without an increase in GDP due to Q, then standard of living has fallen as the quantity of goods consumed per person has declined.

As well, for the standard of living to increase, an increase in Q per person, without deflation occurring, then M must increase or the velocity of money must increase.

There are measures of V published by the Federal Reserve of St. Louise. These are VM1, VM2 and VMzM.  They differ in terms of the measure of money used, M1, M2, and MzM.  Each captures an ever increasing set of money, from coins and currency in circulation, to checkable deposits, to savings and large time deposits.  Another measure, the monetary base, included notes and coins in circulation along with and commercial banks’ reserves with the central bank.

The problem with these measure of the money supply is that they do not represent the stock of money in the flow, only the stock of money that is available to flow. Being available doesn’t necessitate that it will.

So, the basic equation of exchange, GDP =  M·V = Σ(p·q), does not necessarily lead directly to the conclusion that inflation will occur when the monetary base is increased.  The monetary base includes money in the reserve accounts at the Fed.  For inflation to occur, the reserve accounts have to somehow get into the flow.

Arguments surrounding this equation of exchange have to do with whether the velocity is constant and what the stock of money in the equation is. Is it the stock of money in the monetary base or M1?  It is, in fact, neither.  It is the money in the flow.

Other criticisms involve the interpretation of what is affected by a change in M.  A change in M, as noted, can be reflected by a change in P or Q.  It simply depends on other factors.  Still, if either p or q change, M must also change if V is limited.

The current monetary policies of the Federal Reserve are basically three forms, increasing reserves of private banks, open market operations where securities are purchased and sold on the open market, and changes in the interest rate at the discount window.

There seems to be no other methods by which the money supply is affected.  There does not appear to be any injection by “printing” a new quantity of money and using that to purchase t-bills, otherwise spend it or hand it out.

One affects the money supply by putting more money in the hands of the owners of securities, the others affects the availability of credit.  None of them put money directly into the majority of the flow of money at a level of consumption that affects GDP.

In order for t-bills to be purchased by the fed, they have to have been previously purchased by the public.  Open market operations serve as a buffer, to adjust the stock of money.  It does not directly increase the stock in the flow.

One has to consider how the money supply in the flow has been increased, if and how the stock of money in the flow has increased due to an increase in M1 or the monetary base.

A finer issue is with regard to credit.  There is no doubt that, in terms of Σ(p·q), charging $200 on a credit card adds to the stock of money that flows.  For the example above, if this $400 is also part of the months purchases, it adds to the $900 in cash, for a total of $1300 in the flow of Σ(p·q).  And, again, M1 at $1000 and $1100 doesn’t represent the flow of money.

This addition of credit can occur if the demand for credit is up relative to the level of reserve accounts and the discount rate.

Given the affect of credit and drawing from savings (or taking the money out from under the mattress), the equation of exchange is better written as;

GDP(2) = Σ(p·q)(2) = (Mf(1) + ΔC(2) + ΔS(2))*V

The GDP at time 2 is equal to the price times quantity of all goods at time 2.  These are equal to the monetary flow stock at time 1 (previous GDP) plus the additional credit, or that drawn from savings, added at time 2.

This form illuminates the effect of credit on the GDP.  If the amount of money in the flow is otherwise constant, demand is high, and saving have been depleted, then given low interest rates, M may be increased through consumer borrowing on revolving credit.

This addition of credit can then increase the money supply.  And, an increase in the money supply allows for either inflation, increase in population and an increase in the standard of living.

The problem is that this increase is a one shot deal.  In order for the money in the flow to continue to increase, incurred credit must continue to increase.  At some point, servicing of credit becomes a limiting factor.  When servicing of credit becomes a limiting factor, borrowing stops.  And, as people are reactionary, when the stop borrowing, the tend towards paying credit off. As such, the stock of money in the flow drops by both the amount of missing credit increase plus the amount that is being used to pay off credit.

There are two issues.  The primary one is that it essentially requires a debtor class.  There is little doubt that the people able and motivated to accumulate this debt is the middle class.  The second one is that it is simply limited and eventually leads to a crash.

If we look at the level of consumer revolving and non-revolving credit, available at the Federal Reserve of St Louise, we find that consumer credit has continued to accumulate since the 1960’s.  In this recent recession, it reversed direction at the beginning of the recession.  As the recession eased, the level of consumer credit leveled off.

Thankfully, the federal government has been kind enough to continue to accumulate debt.  There is every reason to conclude that this has helped offset the requirement that the public accumulate it.

Still, even this is limited.

And lacking any method by which the stock of money in the flow of exchanges is increased without the necessity of debt, then debt must be accumulated.

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