## Money 101 — part 2

### The Equation of Exchange and the Price Level[3.5]

Imagine that for each specific sort of good or of service in the economy over some period (say a year), we multiply the price of the good or service by the quantity sold, and sum the results.[4] p1 · c1 + p2 · c2 + … = Σ (pi · ci) = pT · c where ci is the quantity of the good or of the service and pi is the unit-price. (If the price of a good or service changes during the period, we can do a separate product for each price, with the quantity sold at that price.)

We might be tempted to think that this sum would be equal to the total amount of money in use in the economy; however, consider that a dollar (or whatever the unit of money) is typically going to be used in more than one transaction. What we can actually say is that M · n = Σ (pi · ci) where M is the total amount of money in use, and where n is the average number of times that a unit of monetary unit is exchanged in these transactions. (One might determine n itself from no more than a record of each occasion on which money changed hands in the sale of commodities, the record completely disregarding what commodities were sold, their prices, and their quantities.) In practice, economists refer not to n but to the average time-rate at which a unit is exchanged, ν = n/t . Although ν is a pure number over time and essentially a frequency, it has been given the unfortunate name velocity, and is usually symbolized with a V.[5] Given that we are using this rate, technically the equation must be recast as M · ν = Σ (pi · qi) where qi is the time-rate ci/t at which the i-th good or service is traded. (In verbal discourse about the relationship, economists tend to ignore the distinctions between n and ν and between ci and qi without actual confusion.)

Note that this equation is necessarily true. If any variable in this equation changes, then some other variable must also change. For example, if M is increased, and none of the qi increases, then ν must drop, or some of the pi must increase, or both.

Now, let's define nY = Σ (pi · qi) So nY (which just one variable, not n·Y) is the summary rate of expenditure spent across transactions. (Many people, including perhaps most economists, would claim that this were the aggregate value of all transactions; but there is a terrible conceptual error lurking thereïn, which I will not labor here.[6]) Also, let's pretend that, every d_mn'd year, people make and buy the same d_mn'd things, in the same d_mn'd quantities. If nY is different from one year to another, then what must have changed are the prices. If we pick some year to be our reference, and its nY as our nY0, then we can talk about prices of year τ in terms of Pτ = nYτ / nY0 which P we'll call the price level.

We can also go on to define rYτ = nYτ / Pτ

In that case Mτ · ντ = Pτ · rYτ And the subscripts are mostly needless clutter here, so we can write the equation as M · ν = P · rY This formula is called the equation of exchange.

Of course, in the real world, the qi (the rates at which various quantities of things are bought-and-sold) change from year to year. At this point, the macrœconomists do something … er, questionable. They propose that we compute a price level from a weighted average, using as weights either qi from the reference year or from the examined year, or from both.

The Laspeyres Index uses the qi from the reference year: PL = Σ(pi,τ · qi,0) / Σ(pi,0 · qi,0)

The Paasche Index uses the qi from the examined year: PP = Σ(pi,τ · qi,τ) / Σ(pi,0 · qi,τ)

(Only a jerk would expect you to remember which is the Laspeyres index and which is the Paasche index, but there are lots of jerks teaching economics.[7])

The Fisher index uses the harmonic average of the Laspeyres and Paasche indices: PF = 2 / (1 / PL + 1 / PP)

None of these indices, nor any other that we might construct, is really perfect. The macrœconomists tell us that if the qi don't change much, then the difference won't matter much; but it's just their guts telling them what much is. Oh well.

Anyway, now the definition is effectively switched around: nYτ = Pτ · rYτ The idea is that nY is just a nominal Y, while rY is the real Y, with the illusions of price removed. (The problem is that the illusions are not truly removeable by a simple act of division. Oh well.)[8]

BTW, more than two decades ago, we really lost the ability to track M. There's still some useful insight in at least some of these equations, but they no longer have as much practical application as once they did.

#### Demand for Cash and Velocity[8.5]

Above, I identified M as the total amount of money in use. If we concern ourselves instead with the total money stock, including money not in use, the mathematics do not change in any essential way, because money not being exchanged may be mathematically regarded as exchanged zero times.

Let's now use M for the total money stock, ME for money exchanged in transactions, MI for money not exchanged in transactions, ν for the average frequency in which a unit of money in the whole stock is exchanged, νE for the frequency associated just with ME, and νI for the frequency associated just with MI. ME · νE = Σ (pi · qi) Since MI isn't in use, it has no velocity. νI = 0 The average velocity of the whole stock of money is ν = (ME · νE + MI · νI) / M = (ME · νE + MI · 0) / M = ME · νE / M So M · ν = ME · νE = Σ (pi · qi) But what we may note now is that, for a given M and νE, an increase in idle cash MI causes a decrease in the average velocity of the money supply ν, and a decrease in idle cash causes an increase in the average velocity of money. Positive demand for holdings of cash are inversely related to velocity. Many economists, in discussing velocity, emphasize this relationship though the underlying importance of the determinants of νE is still greater because, if nothing braked it, ν would rise indefinitely and with it prices.

### The Inflationary Process

At one time, the word inflation referred to a general increase in the money supply. People were so utterly convinced that an increase in the money supply would lead to an over-all increase in prices that any use of the word inflation was taken to imply a general increase in prices, and this became another meaning of the word itself. When an effort was made to convince people that a general increase in prices would often or typically be caused by something else, the word inflation ended up mostly being applied to price increases, instead of to money supply increases.

However, whether we accept the idea that M · ν = P · rY is somehow close enough, or insist on M · ν = Σ (pi · qi) we can see that, if the economy is generally growing, then an over-all increase in prices requires an increase in M or an increase in ν or both.

Velocity ν is determined by the costs and benefits of spending money more or less quickly. Spending money involves transactions costs, which begin to climb as the speed with which money is spent is increased. For any given gain to be got by spending money incrementally faster, there is an associated cost. When the costs of incrementally greater speed would exceed the benefit of that greater speed, the speed of spending money is at its rational level. (If people aren't perfectly rational, still they don't deviate from rationality in unlimited ways for indefinite spans of time.) If prices are expected to be generally stable, then that's all there is to it. Velocity will fall within some natural band, changing only as technology changes the cost of spending money still faster.

But if over-all prices are not expected to be generally stable, then things become more complicated.[9] If prices are generally decreasing, then one forgoes the gains in the value of money by spending it, so there is then downward pressure on ν; but a decrease in ν will itself creäte a downward pressure on prices! If prices are generally increasing, then one incurs losses in the value of money by holding it, so there is then upward pressure on ν; but an increase in ν will itself creäte an upward pressure on prices! In either case, there is thus a feedback loop! However, one may see from the equation above that the effect of ν on prices is, roughly speaking, linear, while the technologies of moving money will generally have costs that increase at an increasing rate; so increases in prices and increases in velocity will not feed each on the other indefinitely.

As a practical matter, an increase in M is necessary for there to be a sustained general increase in prices. There is, however, meaningful controversy over shorter-run fluctuations in ν. (Unfortunately, in practice ν is computed as nY/M; so, without a reliable measure of M, we do not have a reliable measure of ν.)

The theory that a proportionate relationship holds between the money supply M and prices pi is a naïve version of the quantity theory of money. The simplest way that this might be true would be if ν were perfectly constant, and the various qi were simply unresponsive to changes in M (though a more complicated system of relationships could support such a proportionality). A weaker version of the theory is that an increase in M will not merely enable prices to be higher than they otherwise would be, but will ultimately cause them to be higher than they otherwise would be, roughly proportional that increase. The equations above don't really tell us how an increase in the money supply would have this effect, but it's not hard to see.

If the money supply is increased then, even if people don't believe that prices will generally increase, none-the-less prices will, because whoever gets the new, extra money is going to try to spend some or all of it. (And, if they saved some of it by deposting it with a financial intermediary that invests it, then it would be borrowed by or otherwise assigned to others who would spend some of it.) Their increased spending will provoke sellers to increase prices, both in attempts to increase profits and also to meet the rising costs of increased production. As the new money goes from person to person, it drives up prices in the markets in which they are buyers. Of course, as these increases ripple through, those who get the new money later-and-later in the process get less-and-less value from it. By the time that the new money reaches some people, the prices that they face have increased more than have their holdings of money; the people at the end of the process are left poorer; and, as they adjust their spending, so are some of the markets in which they were buyers. Further, some of the markets where quantities were initially increased now face increased costs which push up their prices to where quantities are at or below previous levels. The economy will have been disturbed, and the qi will never get back to where they were.

If people do believe that prices will generally increase, then people will attempt to push up their own wages and prices in anticipation of this. If they rationally anticipate the effects of increasing the money supply, then over-all prices will increase sooner. If people were really good at anticipation, then the qi wouldn't change nearly so much.

(If people believe that prices will generally increase but there is no increase in the money supply, then ν must be pushed up to allow the reälized increases in prices, which will be less than the anticipated increases. And when the anticipated increases are not fully reälized, ν will slow — because the cost of spending money faster will be less than the benefit of spending it before its value is diminished — and prices will receed.)

In the event that the money supply is reduced one way or another, those who have less money will reduce their spending. Sellers will cut prices in an attempt not to lose more business than optimal, and so forth.[10]

### The Effects of Inequality in the Accumulation of Money

[3.5 (2021:10/25 & 27)] I've made a few technical tweaks to this section.

[4] The expression pT · c is in vector notation. You may just ignore it if you wish.

[5] When last I knew, Wikipedia was grotesquely bobbling the definition of ν. (More generally, Wikipedia articles on economics tend to be rife with error, for a variety of reasons.)

[6] The error lies in using what I have called a quasi-measure on a scale where its deficiencies are utterly out-of-hand.

[7] Some people say that I am a jerk, but those people are jerks!

[8] The Wikipedia bobbling of the definition of velocity, mentioned in foot-note [5], can now be specifically described. Editors there have a repeated tendency to define velocity as ν = P · rY / M so that it is simply a ratio involving three quantities, two of which do not even correspond to anything real (not-withstanding that the name of one of those two things contains the word real). But while, in practice, velocity is computed in just such a way, and velocity would be equal to ν = Σ (pi · qi) / M = nY / M there is no intrinsic need to know p1, q1, p2, q2, &c, nor certainly to compute P to determine ν. As far as the definition of velocity is concerned, nY is a black box.

[8.5] This sub-section was added on 14 November 2021.

[9] Note that we're not discussing a one-shot increase or decrease, but a situation in which over-all prices are, for some interval, continually rising or falling.

[10] Some people (most notably some followers of Keynes) believe that market prices are downwardly sticky, resisting adjustment to decreases in the money supply. If so, then quantities bought-and-sold would have to decrease. (An increase in velocity would not make sense with the cost of holding money not increasing!) I think that the real problem is that prior contractual commitments are confused with current market prices by some would-be sellers. In any case, even if the decrease in prices is not as proportional as would be an increase, one still expects a decline explained by individual responses to decreased income.