# Trouble with the equation of exchange.

Discussion in 'Economy' started by itfitzme, Feb 1, 2012.

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### itfitzmeVIP Member

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

MV=&#931;PQ

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, of course, not really velocity in the physics sense of distance per unit time. It is in units of \$ used per \$ in the stock, \$/\$. Never the less, it has the sense of flow and the term velocity suffices.

This equation of exchange is related to inflation. If the stock of money in the flow is increased while the velocity remains constant, then the price of goods increases, thus, inflation. 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.

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 = MV = &#931;PQ.

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.

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.

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 = MV = &#931;PQ, 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 and modifications of 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.

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.

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 &#931;PQ, 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 &#931;PQ. 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) = &#931;PQ(2) <= (Mf(1) + dC(2) + dS(2) ) V

Where the GDP at time 2 is less than the monetary flow stock at time 1 plus the additional credit and 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.

This addition of credit can then increase the money supply. And, an increase in the money supply allows for either inflation or 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. And, 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.

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### DSGEVIP Member

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I'm super tired and need to go to bed right now, but I'll read the whole post tomorrow and respond. For now, I disagree with this much. The right hand side of the equation of exchange is NGDP which is not a \$/\$ unit, it is a \$/time unit. I submit that velocity is in fact of dimension "per unit time" and the money stock of dimension "dollars" in order for NGDP to be M*V = \$ * time^(-1) = \$/time.

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### itfitzmeVIP Member

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Nice. Yes, this is true. By \$/\$, I mean V specifically

I've gotta be strict about the units and they have to make sense physically per quantity and cancel out properly, of course. It's the physics training. I know, in econ, we "assume" the /time as often just understood. And, in many cases, it doesn't matter. Like in supply and demand curves, the axis can be \$ and units, \$/year and units/year, or \$/month and units/month. It really doesn't matter as the whole thing just scales.

So, let's see how I got there.

Okay, how about P is \$/unit and Q is units/year. So PQ is \$/year. Of course, NGDP \$/year, which is great.

I went back to wiki and get "The velocity of money (also called velocity of circulation) is the average frequency with which a unit of money is spent in a specific period of time. Velocity has to do with the amount of economic activity associated with a given money supply. When the period is understood, the velocity may be presented as a pure number; otherwise it should be given as a pure number over time. "

Wiki also says, "That \$100 level is possible because each dollar was spent an average of twice a year, which is to say that the velocity was 2 / yr."

I really want to do Q in just units which gives

NGDP = \$/year = MV = PQ=(\$/unit)*(units)

which isn't going to fly unless NGDP is taken as \$ or Q is in units/year.

If we can go with the wiki "When the period is understood", we can do this thus, NGDP in just \$, where the period is understood.

Sticking with Q in units then,

NGDP = \$ = MV = PQ=(\$/unit)*(units)

Then with M in \$, V is unitless. Still, \$/\$ is also unitless, and interprets at (\$ used)/(\$ existing).

Alternatively, I can go with

NGDP = \$/year = MV = PQ=(\$/unit)*(units/year)

So MV is in \$/year and, unless I'm missing something, M is in \$. That puts V in /year which is just as easily (\$ used)/(( existing*year). M is in \$ existing.

I do like, though, now that you mention it, V in \$/(\$*year). There is the velocity in terms of time. Of course, it could be in year, months, or simply time.

"/year" is more like velocity, as in feet/second, than that \$/\$ is.

So, I stand corrected.

Or do I?

Still, it's not feet per second. Perhaps if I take it as (\$/\$)/year, it is more appealing. Still, no matter how I do it, I am just not happy with it as strictly "velocity".

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### expat_panamaGold Member

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Many people use a morass of big words to hide that fact they don't know anything and they just want to sink others into a pit of muddled thought. Is there a chance you could state your point in 25 or less small words?

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### itfitzmeVIP Member

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So I reread that wiki, "The velocity of money (also called velocity of circulation) is the average frequency", and there you have it. Frequency is cycles per second. And this makes sense, with a one dollar used ( cycling ) twice (or whatever) for every dollar existing in a time period.

Frequency is a valid physical measure. It's typical in electrical and electronic engineering. It is also in mechanics of rotation.

So that is great. Thanks. It really bugged me.

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### samjonesMember

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Heh heh heh - he said "morass".

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### expat_panamaGold Member

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OK, so anyone could have said "chaotimyre" but at this juncture the operative specate is to always eschew obfuscation.

I think.

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### itfitzmeVIP Member

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Yeah, I knew you were around somewhere. Really, is that why you post? To prove something? I realize that you don't get it. That's okay, then it's not for you. I really don't know what to tell you.

There aren't any big words in there. Unless you want to count "modifications" as a big word, it's got 13 letters.

It's just the detail that I have to think it through. And it doesn't even got no calculus. It is what it takes to say it.

It's just a bunch of statements about what MV=PQ means or implies. Each statement, itself, should be a "Duh!". There is nothing hidden, nothing complicated.

It really is simply a description of what is. Each should follow, in a manner, from something that comes before. And as it is based on what is a physical condition, then each that follows must be true. You won't find it much clearer.

And it leads to some interesting observations of how the economy must function. For instance it demonstrates that increasing the stock of money in the flow can lead to growth or inflation, depending on something else. And that is the key question, what the something else is, or the conditions.

And while I mentioned per population, I didn't play around with that. If you do it on a per capita basis, it reveals some other things.

I originally considered that MV>=PQ, representing an upper bound for GDP. That is, GDP=PQ<=MV. Instead, I went with sticking to the condition that M is restricted to the flow. As an upper bound, the M in GDP=PQ<=MV can then be M1, M2, MzM, or the monetary base.

Or you can go with "shit happens".

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### expat_panamaGold Member

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LOL --neither of us knows what the other is saying, talk about a disconnect!!

My mistake was thinking you had some point you were making, that some particular policy was good or bad maybe. That's what me and most folk I run into here are always doing. OK, you weren't and you're just posting stuff for the heck of it. Sounds good, we can always use more diversity in approaches here --enjoy and cheers!

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### DSGEVIP Member

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Cool beans. I guess if you want to treat Q as just units, so NGDP as just \$, you can treat V as being dimensionless, just a constant that multiplies the money stock. Kind of like angles in physics.

W is in joules (N*m), F is in newtons, d is in metres, which leaves &#952; dimensionless.

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