Financial Mathematics Text

Thursday, January 9, 2014

My Failed Inflation Prediction

In August 2009 I made some tongue-in-cheek type predictions. To be clear, I roughly believed them but at the same time I knew they were some gut feelings based on some rough models I had in my head.

My inflation prediction was that that CPI would hit 250 by Fall 2011. It was about 215 at the time which put inflation to be close 8% a year. Apparently we're still not there.

By Fall 2011, CPI was only about 226 which would put inflation at about 2.5% annualized for those two years. So how could I be so wrong?

My prediction was roughly based on at least two of the following beliefs:

1) An increase in money supply results in price increase.

2) An increase in base money results in the increase of money supply.

By modus ponens, one can deduce that an increase in the base money will result in price increase.

In answer to my question of why I'm wrong, one possibility is that I'm wrong because one of my premises is false. 

Alternatively, it could be that I was right, but my timing is off.

Another possibility is that somehow I was right but reality is wrong (aka we live in Fairyland).

Which is it? Let's take a closer look at each belief.

Does an Increase in Money Supply result in an Increase in Prices?

Many people hold to this belief based on the equation of exchange. The equation of exchange is intended as a sort of identity that describes GDP in terms of two separate equations.

The first equation expresses GDP as the amount of money in the economy multiplied by the velocity of money:
$$GDP = mv$$
The velocity of money is a measure of how much money is changing hands every year (e.g. how fast money is traveling). In some sense, we are implicitly defining velocity of money as:
$$v = \frac{GDP}{m}$$
The second equation expresses GDP as sum of all transactions. Suppose that in a given year, there are $n$ transactions. For each transaction, a given quantity of goods are sold at a specific price. If you add up all of these transactions you get GDP:
$$GDP = \sum_{i=1}^n p_i q_i$$
We can treat each set of prices and each set of quantities as vectors and express it the dot product as follows:
$$GDP = \vec{p} \cdot \vec{q}$$
We can then combine the two equations to get the equation of exchange:
$$GDP = mv = \vec{p} \cdot \vec{q}$$
The interesting observation about this is that, even though the price level is a vector, we measure it with a scalar (CPI). So to all of those who complain that CPI is a bad measure of the price level, they are correct. Unfortunately the alternatives that they propose are bad for the same reason.

Now at this point, introductory economics textbooks sometimes make a simplification that amounts to "economic students are really bad at math". (Some would claim that economists are as well.) The price level and the quantity vectors get transformed to scalar by a process we call magic. The result is this equation:
$$GDP = mv = pq$$
Now this equation we can do some cool stuff with. In particular, we can rewrite it in terms of the price level:
$$p = \frac{mv}{q}$$
Or by taking the total differential and dividing by $p$, we can get a more explicit relationship:
$$\frac{dp}{p} = \frac{dm}{m} + \frac{dv}{v} - \frac{dq}{q}$$
This tells us that the change in the price level will increase by an increase in the money supply, an increase in the velocity of money or a decrease in the quantity of output.

Now if you assume that velocity and quantity are constant, then you get you get the following result:
$$\frac{dp}{p} = \frac{dm}{m}$$
And that's what we're hoping to derive. Inflation is driven by changes in the money supply.

But this result is a consequence of two assumptions (which may end up being false) and a piece of mathematics that I referred to as "magic". 

While there appears to be a good relationship between the money supply and prices, it's not what the theory says it should be (if the above theory was correct, we'd expect to see a power law or a linear relationship with zero slope):

So let's turn now to the next premise. . .

Does an increase in the monetary base result in an increase in the money supply?

Recall, that when the Federal Reserve made a massive increase in its balance sheet, it did so by increasing the monetary base and purchasing securities. But does that result in an actual increase in the overall money supply?

The way the textbooks talk about what is referred as "fractional reserve banking" goes something like this.

Fractional Reserve Banking

When a bank receives a deposit it doesn't just put it in a safe. It actually loans a portion (fraction) of that money out. In the US, banks are required to keep 10% of their deposits as reserves. The other 90% may be loaned out.

So suppose there is only one bank and it receives a deposit of \$100. It loans out 90% or \$90. That \$90 lent out typically ends back in the bank but it shows up as someone else's deposit. With this new deposit of \$90, the bank may loan out another 90% or \$81. As you can imagine this process can continue indefinitely.

The result is that this initial \$100 deposit gets multiplied as follows:
$$ \$100 + \$90 + \$81 + \$72 + . . . + \$100 \times 0.9^n + . . . $$
This turns out to be a geometric series with $a=\$100$ and $r=0.9$. If we allow this to process to go to infinity, the series sums to:
\frac{a}{1-r} = \frac{\$100}{1-0.9} = \$1000
In other words, we started out with \$100 and the banking system multiplied this figure by a factor of 10. This is the money multiplier principle.

While the math may be roughly correct, there's a causality that's assumed in all of this.

Multiplier Myth in Fairyville

Imagine there's a bank in a little town called Fairyville (which so happens to be where all economists live). They have all of the money lent out which they are permitted to lend out. A customer walks in and requests a loan and the banker says, "sorry, we're all out of loans today."

The bank manager remarks, "gee, I really wish someone would make a deposit soon. Otherwise, we'll keep having to turn customers away as we haven't any money to lend out."

The causality assumes that banks are waiting on deposits and once they receive them that will cause them to start lending out money (as they can't do anything else.) The myth in all of this is that this is the exact opposite of how banks operate in the real world.

For a more thorough look at all of this, I recommend Steve Keen's The Myth of the Money Multiplier.

But roughly speaking banks don't worry much about reserves. And there's a good reason for that. They don't have to!

You see if a bank lends out too much money, they can borrow reserves from another bank. If the bank can find no other banks to borrow reserves from, there is always the lender of last resort (here in the US: The Fed). The Fed loans the bank money to cover the reserves. If it fails to do so, would be a failure in the financial system (and all that goes with it.)

And the primary purpose of the Fed (which is not the dual mandate) is to make sure the financial system remains functioning.

So the way it actually works is that banks lend money first and the Fed plays catchup later. The causality is reversed. 

And there's actually empirical evidence to support this narrative:

Business Cycles: Real Facts and a Monetary Myth

The reality is that bank credit expansion leads the business cycle while the monetary base may even lag it a tad.

Was My Timing Off or Was I Just Plain Wrong?

So I think this is a relevant question. There are still plenty of hyperinflationists out there claiming that the collapse is coming. They may not have gotten the timing right, but ultimately they're right?

At this point I'm largely convinced that my simplistic models were not only too simplistic but in some respects were just plain wrong. And the evidence seems to be supporting the view that I was wrong. After all, the Fed has massively increased its balance sheet and we haven't seen prices budge much at all.

A Quick Word on Hyperinflation

Now I've seen plenty of people predict hyperinflation in the last few years and some of them stubbornly hold onto this view. And like a stopped clock that's correct twice a day, they may turn out right. But I think there's a lot that's lacking in their models assuming they're working with models at all.

The argument seems to be nothing more than this:

"OMGZ, the Fed's printing lots of moneyz and the US govt's running huge deficits. Therefore hyperinflation."

I'm inclined to think that hyperinflation is a much more complex issue than such simplifications. I certainly have no model that would enable me to predict it. But I think there are things well beyond "money printing" involved in hyperinflation episodes.

For some interesting reads on the subject, here are two articles:

Hyperinflations, Hysteria, and False Memories, James Montier via GMO

Hyperinflation - It's More than Just a Monetary Phenomenon - Cullen Roche (see more here at pragcap).

That's about all I have to say on that topic.

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