Financial Mathematics Text

Monday, August 11, 2014

Inflation - Why the official numbers are wrong!

I'm certainly not the first to claim this and I won't be the last. But I think I'll say this differently than most who happen to say it.

For starters, when I refer to inflation, I'm preferring to price inflation.  Some (particularly Austrians), by inflation, mean monetary inflation so I think it's important to be clear on what's being discussed.

Many feel that the current inflation figures (whether we're talking CPI or CPI minus food and energy) understate the actual rate of inflation. Other methods are sometimes used which in many cases are not any better than the official numbers (and in many ways worse.)

Howerever, what I'd like to suggest is that Any method of coming up with an inflation number is flawed. To understand this we'll have to take a brief detour.

The Equation of Exchange

Economists often talk about the equation of exchange.While this equation is not without criticism, many claims regarding money supply and inflation are dependent on this equation. The problem is that the way it's often presented (at least in introductory courses) is not technically correct. Fortunately Wikipedia's entry is actually correct. This probably has to do with the fact that most introductory economics students aren't that familiar with linear algebra. Typically, they see something like this:
where $M$ is the money supply, $V$ is the velocity of money, $P$ is the price level and $Q$ is the quantity of goods.

We'll get back to this in a moment.

The Right Side of the Equation

Imagine you have an economy and you want to measure the total income of a nation. This is often referred to as GDP. To do this in detail, we would have to treat each transaction. Each transaction results in a quantity of units sold multiplied by a price per unit. If we sum up all of these transactions we get total GDP.

Now there are some details I'll leave out. For example, what counts as a transaction? If I sell my used car is that a transaction? We'll leave those questions aside and assume we've worked out a satisfactory answer.

So what does this all look like? Suppose that in a given period of time (typically a year), there were $n$ transactions. For each transaction there is a price (per unit) and there is a quantity (of units) sold. Then GDP looks like this:
GDP &= p_1 q_1 + p_2 q_2 + . . . + p_n q_n \\
&= \sum_{i=1}^{n} p_i q_i
Now it's also worthwhile to consider units here. For GPD, the units will be something like dollars per unit of time (e.g. \$/yr). So the right side will have to be the same. However, each individual price figure will be quoted for particular units. For example, we might have \$5 per pound of beef or \$500 per iPad. The "per year" comes in because we're only considering quantities sold within a particular time frame.

But there's another way to rewrite this side of the equation. If you're familiar with some basic vector analysis, and in particular the dot product, then we can think of "p" and "q" as vectors:
$$GDP = \vec{p}\cdot \vec{q}$$
where $\vec{p}$ is $[p_1, p_2, . . ., p_n]$ and $\vec{q}$ is $[q_1, q_2, . . ., q_n]$. We'll get back to this in a moment.

The Left Side of the Equation

Now I don't want to focus too much on the left side of the equation. It's certainly not without its problems. But the idea is that there is a quantity of money $M$ and that quantity of money changes hands $V$ times per year. This is how we get to the dollars per year units of GDP.

But what is the money supply? Do we consider the monetary base? What about M1 or MZM? In fact, FRED provides data on money velocity for M1, M2 and MZM.

Obviously there's a problem of measuring the money supply as its a moving target which changes frequently throughout the year. And we'd have to agree upon which measure of money to use.

But what about measuring the "velocity of money"? As far as I'm aware, we don't have an independent means of measuring money velocity. What that means is that this equation - $GPD = MV$ - is a definition of velocity of money.

Now you might ask whether or not we could, say, track a particular dollar and see how many times it changes hands. But the vast majority of the money in our economy is not a physical piece of paper; it's an accounting creation. So tracking all of those dollars won't exactly work well.

But even if we did, what would that give us? For each dollar in our economy there would be a unique velocity for that dollar. So would we average them? Take the median?

The Inflation Numbers are Wrong

The problem with the inflation numbers, however, comes from the fact that we measure inflation as a scalar. But as I noted above, it's more accurate to think of the "price level" as a vector. So any attempt to classify the "price level" as a scalar is simply wrong!

But wait! Aren't there ways of characterizing a vector by a scalar?

One standard approach is measuring Euclidean length. If we have a vector in Euclidean n-space we can measure the length of that vector by taking the dot product with itself and then taking the square root of the resultant. For example, if $\vec{v}$ is a vector then the length of $\vec{v}$ is given by:
$$||\vec{v}|| = \sqrt{\vec{v}\cdot \vec{v}}$$
Can't we do something like that?

Here's where paying attention to units is important. Imagine an economy with 2 prices: beef selling at \$5 per lb and iPads selling at \$500 per unit. What would our vector look like?
$$\vec{p} = [\$5/\text{lb beef}, \$500/ \text{iPad}]$$
If we were to measure the "length" of this vector, that would give us:
$$||\vec{p}|| = \sqrt{\$25/ \text{lb beef}^2 + \$250,000/\text{iPad}^2}$$
Now I have no idea how to add up these entirely different units.

The fact remains, however, is that the price level is a vector while price indices are scalars. This implies that all price indices fail to correctly measure inflation.

What's the Solution?

What price inflation indices actually do is take some sort of weighted average of all of the prices they measure. This is probably the most sensible approach. But there are decisions that need to be made regarding:
  1. What goods it measures prices for.
  2. What weights to assign to said prices.
This is where there are disagreements. There are some standards we might employ but they are not without problems. For example, we might choose weights that reflect the real weights in our economy. So if we spend 1% of our money on beef then 1% of our price index should be influenced by beef prices.

But then you would not have a terribly consistent price series. If beef prices rise (or fall) that will influence next year's weights and quantities since some consumers may switch from beef to chicken. Do we keep the weights the same to have a consistent price series or do we adapt the weights to reflect the new weights?

If you're curious about some of the details on how the Bureau of Labor Statistics does this, you might consider taking a look at Addressing misconceptions about the Consumer Price Index. Starting on page 5, the authors discuss substitution with a candy bar example.

Concluding Remarks

All measures of inflation or price indices (that I've seen) characterize the price level as a scalar. The equation which many economists use to understand prices - the equation of exchange - however, treats the price level as a vector. So as a matter of technicality, all price indices fail to "reflect" the actual "price level".

Solutions that attempt to treat the price level as a scalar require that we adopt a variety of standards and conventions. I am doubtful that there is a "correct" procedure for doing so. If the theory is correct, then no such procedure is correct.

However, I do suspect that measuring price levels by scalars is still a useful activity. Some conventions ought to be adopted that aid us in creating a useful scalar measure of the price level. But I suspect there's a considerable amount of freedom in how we make such decisions.

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