Previously on the New Arthurian Blog...

Last time, we looked at a typical "money supply relative to output" graph. Then I drew the graph again with the deflator inverted. Inverting the deflator flipped the blue line. The line moved with the deflator. I said this shows that the deflator -- the "correction" to output -- is the reason the typical graph mimics the CPI so closely.

People say it is the relation between money and output that mimics the CPI. But when I inverted the deflator, the blue line inverted. In other words, the line shows what the deflator is doing. Not what "money relative to output" is doing. Money and output play an insignificant role in the graph, despite what people say.

What would happen if we left the deflator alone, and flipped money and output?

What If...

Messin' with the numbers.

Okay, let's do a typical "money supply relative to output" graph. We take the GDP as it comes, purchased at actual prices, and total it up. (That's nominal output.) Then we divide the GDP number by the price level (to "deflate" the GDP down to what it would have been if there was no inflation). This calculation gives us "real output", which is often just called "output."

Next, we take the quantity of money (M2) and divide it by the "real output" number. This gives us the famous "money supply relative to output."

Then we "index" this number: Figure the average value for the whole series, and divide each number in the series by the average. The new values we get are the ones we will display on a graph.

We also grab the CPI, which shows the level of prices. And we index this series of numbers just as we indexed our "money supply relative to output."

We index both sets of numbers, as Milton Friedman said, "To make the two series comparable."

Important Stuff

(Important to me, anyhow.)

The Secret Economist recently posted an evaluation of the Taylor rule.

What's the Taylor rule? According to the Wik, the Taylor rule "stipulates how much the central bank ... should change the nominal interest rate" when inflation and economic growth wander from their targets.

Long story short, the Secret Economist became "suspicious" of the good match between the Taylor numbers and the "Federal Funds" interest rate, and decided to investigate. The bold conclusion: "The Taylor rule fits because it is an identity."

Now, Wikipedia says, "An identity is an equality that remains true regardless of the values of any variables that appear within it." So SE's conclusion is that the Taylor calculation will always give a good match to the interest rate. Suspicions justified.

But -- as Arlo Guthrie said -- that's not what I came to tell you about.

Fractions

"To divide by a fraction, invert and multiply."

Suppose we have a calculation like . But we know that is a fraction, and that . In order to understand our calculation, we can replace the in the calculation with the thing it equals, thus: . It is evident now that in our calculation, we are dividing by a fraction.

To divide by a fraction, use the rule they teach in elementary school: invert and multiply. Inverting the fraction gives us . Now, multiply by the inverted fraction: .

In this form our calculation is simpler because there is only one division. It is okay to remove the parentheses: , and to rearrange terms: . We can add parentheses to show we do the division first: . All of this is valid arithmetic.

Our new formula will produce the same result as , our original calculation.


is the quantity of money.
is real output.
is output in actual prices.
is the price level (the "deflator").

In its final version, our calculation divides the quantity of money by output in actual prices, and multiplies the result by the price level.

The calculation we started with was used by Milton Friedman in Money Mischief to produce all those graphs that show how "money relative to output" follows the same trend-line as prices.

His numbers follow the price trend because he multiplies by the price level.

You can mimic the price trend in many ways. You don't have to use the quantity of money and output. You can use any number you want, pretty much. Just multiply by the price level, and your answer takes on the shape of the price trend.

The Three Little Graphs

Wow! This was easy to do!

Below is a Google Docs spreadsheet. It contains three pages. Each page has calculations and a graph. The graphs are off to the right, near the top, but you won't see them unless you go looking.

This post is mostly in the nature of a test. Probably not much here of interest. This is the spreadsheet used to develop graphs for my comments at The Secret Economist.