Financial Mathematics Text

Monday, November 3, 2014

Some Comments on the Quantity Theory of Money

As I mentioned in my previous post - Inflation - Why the official numbers are wrong! - I pointed out that the general theory for which inflation is based upon implies that the "price level" is a vector but measures of inflation represent this as a scalar. Today I want to explore how that complicates the picture for the Quantity Theory of Money.

The Equation of Exchange

The Quantity Theory of Money is based on the Equation of Exchange discussed in the previous post. As I pointed out, both sides of the equation are equal to GDP. On the left side we have this:
$$GDP = MV$$
where $M$ is the "money supply" and $V$ is the "velocity of money". It's worth noting that this equation implicitly defines what the velocity of money is (or at least how it is to be measured.) In that sense it's an operational definition.

On the right side of the equation, we have this:
GDP &= \sum_{i=1}^n p_i q_i \\
&= \vec{p} \cdot \vec{q}
This equation is a definition of GDP which is the sum of all (value added) transactions. This can be mathematically represented as the dot product of two vectors: a price vector and a quantity vector. The components of these vectors will be each price and each quantity of good for the price and quantity vectors respectively.

The end result is the Equation of Exchange which looks like this:
MV &= \sum_{i=1}^n p_i q_i \\
MV &= \vec{p} \cdot \vec{q}

Economists sometimes ignore the fact that price and quantity are vectors and then proceed to reason from that false assumptions. I'm going to discuss that and a few other problems this presents.

Rates of Change

One really nice thing one can do if one makes the assumption that $P$ and $Q$ are not vectors is something which is conveniently shown on the wikipedia article which involves taking a total differential. This allows one to see how changing one variable affects changing others.

We'll begin by expressing the equation of exchange in terms of the price and assume that both quantities are scalars:
$$P = \frac{MV}{Q}$$
From this we can take the total differential (I'm going to handwave the math; take a course in differential equations if you're bored):
$$dP = \frac{V}{Q}dM + \frac{M}{Q}dV - \frac{MV}{Q^2}dQ$$
The $dP$, $dM$, etc, express the absolute change in price, money supply, etc. For example, if the money supply goes from $\$100\text{ billion}$ to $\$101\text{ billion}$ then $dM= \$1\text{ billion}$.

And last but not least, we'll divide both sides by the original price equation (we'll divide the left side by $P$ and the right side by $MV/Q$.) Then end result looks like this:
$$\frac{dP}{P} = \frac{dM}{M} + \frac{dV}{V} - \frac{dQ}{Q}$$
What this says is that the percentage change in the price level (the "inflation rate") is equal to the percentage change in the money supply plus the change in velocity minus the change in productivity.

The Quantity Theory of Money says if $V$ and $Q$ are constant (e.g. $dV/V = dQ/Q = 0$) then the rate of inflation is simply going to be equal to the rate of change in money supply. In other words:
\[ \begin{align*}

\frac{dP}{P} &= \frac{dM}{M}
The problem, of course, is that $P$ and $Q$ aren't scalars. So all of that math was invalid.

Now perhaps one can argue that even though the above equations are mathematically incorrect, they may still be useful approximations. I can't argue with that (although one would have to actually show evidence that that is the case.) 

So how should it really look?

Well, for starters, we can't really reformulate this in terms of price as the right side of the equation is taking the dot product of two vectors. I don't know how to "divide" by a vector. But we can still look at rates of change. Let's consider this in terms of the money supply.
M &= \frac{1}{V}\sum_{i=1}^n p_i q_i \\
&= \frac{1}{V} \left[p_1 q_1 +  p_2 q_2 + . . . p_n q_n\right]
Now we can take the total differential of that, but it's not going to look pretty:
dM &=  -\frac{1}{V^2} \left[p_1 q_1 +  p_2 q_2 + . . . p_n q_n\right]dV + \frac{1}{V} \left[p_1 dq_1 + q_1 dp_1+  p_2 dq_2 + q_2 dp_2 + . . . p_n dq_n + q_n dp_n\right] \\
dM &=  -\frac{1}{V^2} \left[\sum_{i=1}^n p_i q_i\right]dV + \frac{1}{V} \left[\sum_{i=1}^n (p_i dq_i + q_i dp_i)\right]
And that's about as pretty as it's going to get. So a change in the quantity of money (not percentage change as we haven't yet divided by $M$) is going to be equal to all of the changes in each price and each quantity of good.

Now we can look at the percentage change but it's not going to look much prettier. This is as pretty as I can make it (dividing both sides by "$M$"):
$$\frac{dM}{M}= \frac{\sum_{i=1}^n (p_i dq_i + q_i dp_i)}{\sum_{i=1}^n p_i q_i} - \frac{dV}{V}$$
And this is where you'll get things like Cantillon effects in which some prices go up some, some go up a lot and some even go down.

Of course there's another ambiguity here. One of the assumptions made by economists is that the quantity of goods sold is constant. So does that mean we sell exactly the same amount of each good (e.g. all of the $dq_i=0$)? With a total of $n$ transactions, it seems unlikely that all the quantities will be the same (perhaps due to another sort of Cantillon effect).

And what does it mean to say that the "price level" goes "up"? It's important to think about. Because we're not saying that "all prices go up". Some will go up and some will not. But "in aggregate" do they go up? What does that mean?

Typically what is meant is we can take some weighted basket of prices and see that this basket goes up (such as CPI). But depending on how one constructs that weighted basket, you could end up with very different price levels and price increases.

The Money Multiplier Myth, Exogenous and Endogenous Money

One question that should concern us is which measure of money is relevant? There are a number of ways to measure money as there are different types of money. There are coin and currency. There are deposits at the Fed. There are checking and savings accounts. There are certificates of deposits (CDs). There are money market funds (many of which one can write checks on). All of these are different types of money and they get aggregated in different ways.

Now if we are to believe standard economic theory, this shouldn't matter much. This comes down to what is called the "money multiplier". The story goes like this.

A deposit is made to a bank (say, cash comes from a Central bank). This is external money. For concreteness, let's say \$100 is deposited.

Once the bank has this money it now has reserves. The bank is required to hold a portion of its cash as reserves while it can lend out the rest. In the U.S., that percentage is 10%. So it holds \$10 as reserves and lends out \$90.

Now if that 90% that gets lent out gets back into the banking system, that's another \$90 in deposits for a grand total of \$190. So now the banking system still needs to hold 10% of that (\$9) but can lend out the other 90% (\$81). And this process can repeat itself indefinitely.

If you work out the math on this, it turns out to be a geometric series. In particular, if $R$ is the reserve requirement (10% in our example), then the money multiplier is given by:
$$M = \frac{1}{R}$$
So for our example, the \$100 initial deposit could increase by a factor of 10 to \$1000.

Now economists recognize some problems with this. For example, it's assumed that the money enters back into the banking system; one could hold that cash underneath a mattress. But overall there is a belief in a causal mechanism. Money is increased externally and then the banks do their thing via a money multiplier.

The way the story looks its as if the bankers are sitting around thinking to themselves, "boy I hope someone deposits some more money so that we can make another loan".

The Multiplier Myth

Reality is a bit more complicated than this. An entirely different story (and one that is better supported by actual data) is that banks extend credit when they see there's a good opportunity and worry about reserves later.  If later on they find that lack sufficient reserves they use the interbank lending system. If the entire banking sector is short on reserves they call up the Federal Reserve as "lender of last resort". The Fed basically has no choice but to supply reserves.

Steve Keen goes into more detail on this works with a simple model: The Myth of Money Multiplier. Keen also notes that reserve data is lagged by 30 days. This further corroborates the story that banks don't lend reserves.

Kydland and Prescott in their paper, Business Cycles: Real Facts and a Monetary Myth, look at business cycles and various economic variables and their relations. The authors note that:
There is no evidence that either the monetary base or M1 leads the cycle, although some economists still believe this monetary myth. Both the monetary base and M1 series are generally procyclical and, if anything, the monetary base lags the cycle slightly. [. . .]

The difference of M2-M1 leads the cycle by even more than M2, with the lead being about three quarters.
This suggests a picture in which banks extend credit leading into the business cycle and the Federal Reserves shores up reserves (M0) in the midst of the cycle (or perhaps toward the end of the cycle). 

Endogenous Money?

So that raises an interesting question on what causes what (if anything.) Instead of the picture that exogenous money causes an expansion of bank credit (endogenous money) via some money multiplier effect, the exact opposite appears to be the case. As a result, the process by which banks extend credit to businesses and consumers drives monetary expansion.

The Velocity of Money

Now, what are we to make of this velocity of money concept? First off, the theory holds that money velocity is stable. This isn't exactly true. Money velocity varies quite a bit. Here is the velocity of money (depending on whether the money supplied is measured by M1, M2 or MZM):

The stable velocity part of the the theory is clearly counterfactual. Beyond that, I'm not sure to what extent it has an effect on prices. I would imagine, especially in a hyperinflation scenario in which there is currency rejection, it play some role.

What causes what?

From an epistemological standpoint, the equation of exchange appears to be an identity. After all, the left side of the equation defines money velocity and the right side of the equation defines GDP. The Quantity Theory of Money, however, makes at least three postulates:
  1. Velocity is stable.
  2. Quantity is stable.
  3.  A change in money causes an increase in prices.
Now if the equation of exchange were just scalars:
$$GDP = MV = PQ$$
then fixing $V$ and $Q$ would imply that changes in $M$ would imply changes in $P$. That's fairly trivial. What isn't trivial, however, is the causality imposed upon it. All the equation says is how 4 different variables relate to one another; it says nothing about causality.

The supposition is that an increase in the money supply has an effect on prices. But as we noted earlier with the money multiplier myth, increases in the money supply is primarily the result of banks extending credit. In that sense, one could argue that businesses and consumers requiring loans for purchases ($Q$) causes an increase in the money supply, a position discussed here:

Why is the Quantity Theory of Money Wrong and can Anything be Salvaged from it?

I'm inclined to think that since commercial banks are independent profit seeking entities, it stands to reason that there is a sort of two way street. Businesses and consumers demand loans for purchases ($Q$) while banks create loans for profit ($M$).

Regardless of how it's interpreted, it should be clear that reality is a bit different from the standard view.

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