Few discussions of monetary policy take place without at least implicitly invoking the famous Equation of Exchange, a popular economic model attributed to Irving Fisher, but with roots dating back to David Hume and John Locke. The Equation of Exchange can be presented as follows:
M x V = P x O
where M is the money supply, V is the velocity of circulation of this money, P is the average price level, and O is the quantity of output, or actual goods produced.
At first glance, the Equation of Exchange seems to reveal some compelling causal relationships to help explain what’s really happening in the economy. Being a simple formula with four interdependent variables, we should be able to solve for each in terms of the others, thereby generating a variety of insights to guide our analyses of monetary policy as well as the various elements of economic growth.
But as we explore its implications more carefully, I think we have to conclude that its descriptive and prescriptive powers are largely illusory.
Central to any discussion of monetary policy is the variable P, the average price level, whose rate of change is popularly known as inflation when positive and deflation when negative. Measuring and monitoring P allegedly allows the Federal Reserve to manage M, the money supply, whose rate of change is understood by economists, via the Quantity Theory of Money, a derivative of the Equation of Exchange, to determine the average price level.
For example, if we want to know the rate of price inflation, we can try to solve for P over time and then compute the percentage change in P from one year to the next:
P = M x V / O
Thus, the average level of prices, P, is equal to the money supply, M, times the velocity of circulation, V, divided by the total output of the economy, O.
It sounds so logical and precise.
Now, the only thing we have to do is develop measures of M, V, and O.
We know that we can get the first variable M, the money supply, from the Federal Reserve’s published reports.
But what about V?
Well, the Fed also publishes an estimate of V based on the following equation:
V = GDP / M
Given that we can observe and measure GDP, the gross domestic product, as well as M, the money supply, then we can solve for V, the velocity of circulation, thereby giving us the second variable we need to determine P, the average price level.
Now if we insert this new model of V into the equation for P, we get the following:
P = (M x (GDP / M)) / O
This cumbersome equation can then be simplified because the Ms in the numerator cancel each other out. Thus, by striking out the two Ms, we’re left with:
P = GDP / O
So the average price level equals the GDP, which we know we can get from the government's published reports, divided by O, the total output of the economy.
In our effort to solve for P, this leaves us with one remaining variable: O, the total output of the economy. So what is this total output, O?
The answer suggests itself when we recognize that the GDP used by the Fed in its velocity equation is what the government refers to as Nominal GDP, or the GDP statistic before adjusting to remove the effects of price inflation, which we have been referring to as changes in P.
So, O must be associated with the Real GDP that, according to the government, indicates the level of economic output independent of the effects of any price inflation that might be manifesting in the economy.
Thus, we can clarify the above equation as follows:
P = NGDP / RGDP
where NGDP is Nominal GDP and RGDP is Real GDP.
I won’t digress to prove the mathematics, but this equation can be transformed into another powerful equation:
P^ = NGDP^ - RGDP^
where the change in the average price level, P^, is roughly equal to the change in NGDP, denoted as NGDP^, minus the change in RGDP, denoted as RGDP^.
Simple translation: Price inflation(deflation) is roughly equal to the difference between the growth in Nominal GDP and the growth in Real GDP.
Makes intuitive sense, right?
But is the government's Real GDP statistic a valid measure of O, the level of economic output independent of the effects of any price inflation that might be manifesting in the economy?
Stepping back from the reported statistics, recall that in the Equation of Exchange, O refers to the actual, that is nonmonetary, output of the economy. In other words, the total amount of shoes, cars, computers, etc. produced in a particular period of time.
This is the real wealth that we create each year with all our hard work. After all, the only value in money is in its capacity to buy real goods that help satisfy whatever real demands we have. O is an expression of all these real goods created in any given year (or any other chosen period of time).
Theoretically, we could catalog everything that was produced each year, but we could never add up all these different goods to generate a meaningful aggregate statistic. After all, what is the sum of 1 billion pairs of shoes plus 16 million cars plus 58 million computers? A big pile of stuff, that's what. Clearly, they are incommensurable without some common denominator.
Therefore, because O cannot be quantified, it is not a valid, independent variable for this mathematical Equation of Exchange. It cannot be observed or measured all by itself, the way we can observe and measure M and NGDP.
What to do?
Perhaps we can rearrange the original equation to solve for O, as follows:
O = M x V / P
We’ve already determined that M and V are available in the Fed’s published reports, and we’ve seen that V = NGDP / M. So we can insert this new ratio into the equation to yield this:
O = (M x (NGDP / M)) / P
Look familiar? Canceling the Ms in the numerator, we’re left with:
O = NGDP / P
Now that’s no better than P = NGDP / O.
We’re basically chasing our tails here, trying in vain to calculate both P and O from a single known variable, NGDP, which is the only observable, measurable, valid indicator of total output.
In our mind’s eye, we can appreciate that NGDP might be the mathematical product of some tangible yet incommensurable mountain of goods, O, multiplied by some average price level, P. But we cannot observe either of these in the natural world. They simply do not exist.
Now at this point someone must be thinking that we can estimate P more directly via the statistical analysis that yields the various inflation rates published by the government, whether it's the familiar Consumer Price Index or the GDP Deflator (which, incidentally, is NGDP / RGDP).
But this approach side-steps a very important problem: If we cannot observe or measure or deduce in any quantitative way one of the two variables that is supposed to be a component of Nominal GDP, then we cannot, by definition, observe, measure, or deduce the other variable. We may convince ourselves that we can, but we really cannot.
It may sound a bit too definitive for some, but if NGDP is supposed to be the product of P x O, and O is a completely fictitious aggregate statistic, never to be observed in nature other than as a catalog of distinct products that cannot be added together, then P must also be a fictitious aggregate statistic, never to be observed in nature other than as a catalog of distinct prices that cannot be validly separated from the distinct products to which they have been associated through unique acts of market exchange. Simply put: if no independent O, then no independent P.
Thus, with the help of the deceptively rigorous Equation of Exchange, I come to the logical, if somewhat disappointing, conclusion that there really are no valid, independent measures of price inflation and real output, despite the very great importance of these economic constructs.
Perhaps I am missing something.
But it seems as though the only real GDP is what the government calls Nominal GDP and what the government calls Real GDP is not a real statistic at all, but rather something conjured up in order to convey the deceptive notion that we can still enjoy robust growth in output despite even more robust growth in money supply by virtue of the logical impression that low velocity of circulation can somehow "absorb" excess monetary inflation, resulting in low price inflation whose reasonably accurate independent measurement can be deducted from a reasonably accurate measure of Nominal GDP growth to yield a reasonably accurate measure of Real GDP growth.
And most people fall for this story when journalists and economists simply parrot the government's press releases that summarize statistics for carefully manipulated price inflation and carefully conjured real output. And then the Fed adds to the confusion by highlighting the relatively low inflation statistics together with the relatively strong growth in Real GDP as if this were proof positive of the benign nature of its shockingly high monetary inflation.
In terms of the Economic Philosopher's Stone, the Equation of Exchange, it seems that the only robust aggregate statistics we can use are the money supply, which is tracked very carefully by the Fed, and nominal output, which is reported as Nominal GDP. And when money supply has been increasing at a higher rate than nominal output over the course of many years, we need to be very concerned about what that implies for both the level and the sustainability of real output, the real wealth of nations.
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