Saturday, November 7, 2009

Taylor Rule - Equation Discussion

Sometimes I discuss these topics with my son Jerry. This is one of those times. He got back to me after taking a look at the Taylor Rule. He makes very good sense of it...

Jerry writes:

Roughly (leaving out things like logarithms for now [1])

(target interest rate) = (inflation rate) + (equilibrium real interest
rate) + 0.5 * (actual inflation - desired inflation) + 0.5 * (GDP -
potential output)

So, to help clarify what it is saying (without commenting on if it is
correct or reasonable or anything) -

1. If the economy is "in equilibrium" and you don't need to do
anything to it... actual inflation equals "desired inflation" and the
GDP is up to its potential ... then the equation looks like:
(target interest rate) = (inflation rate) + (equilibrium real interest rate)
I think that makes sense... if you have a 3% interest rate, but
inflation is 5%, you are losing you have to set your
"interest rate" at 8% to have a 3% "real interest rate" (i.e. - "how
much money you make").

2. now say that it comes, by a small amount, out of equilibrium
because there is "too much inflation"
(target interest rate) = (inflation rate) + (equilibrium real
interest rate) + 0.5 * (actual inflation - desired inflation)
If you "desire" 3% inflation, and you have 4% instead... you should
increase your interest rate to try to cut down on the inflation:
(target interest rate) = (whatever it was before in step 1) PLUS 0.5%
Conversely, if your inflation rate is LOWER than you want it... you
should REDUCE your interest rate.

3. Now say that the inflation goes back to the "desired" rate. But the
economy slumps and starts underproducing, and GDP falls below its
(target interest rate) = (inflation rate) + (equilibrium real
interest rate) + 0.5 * (GDP - potential output)
So... if GDP is below it's potential, that term on the right is
negative, and so you should lower your interest rates to "stimulate"
Conversely, I guess that if GDP were ever above it's potential, you
should raise interest rates. (hahaha)

4. then you just put that all together.

So basically, the "taylor rule" is what you get if you start from step
1 (which i think makes sense, such as it is), and add the statements:
- if inflation goes up, you should increase your interest rates
- if output goes down, you should cut your interest rates
and then assume, mathematically, that it is a linear function of those
two variables, and no other variables.

That last part isn't quite as bad as it sounds... it is a common
mathematical trick. basically, if the deviation from equilibrium is
SMALL ENOUGH, it is valid to treat everything as linear (i mean,
assuming that your model is close to correct to begin with...if those
variables don't belong there or he has the + sign where he should have
a - sign or something, then it won't be correct)...
but basically like - if you are sitting on top of a circle...if you
are SOOOO SMALL and don't walk around very much, you will think you
are sitting on a flat surface. Everything looks flat if you get close
enough to it.

Of course, even in cases where that is valid, it goes out the window
when you have a BIG change. (if you take a BIG STEP you will just fall
off the edge of the circle... and the model already breaks down if you
take a "sort of medium step" and start slowly sliding down off the

And then you can make this assumption about the form of the equation,
and fit for the two "a" parameters (i called them both 0.5 ... but if
you wanted to you could go figure out what value they have
"empirically" by matching them against the data and picking whatever
values give you the closest fit).

But yeah... it is not derived from any general theory of economics or
anything like that... it is just someone's opinion on what variables
come into it and how, written down in basically the simplest
mathematical form possible, and then fitted to past policy decisions
to find the "a"s that make it work the best.

There is not even an argument made as to why the opinion should be
valid. It's kind of funny.


[0] - it seems to me that what it means for the economy to be "in
equilibrium", the "real interest rate", the "equilibrium real interest
rate", the "desired inflation", and the "potential output" are all
more or less matters of almost every "number" in this
equation is just pulled out of someone's ass, and I don't really have
much faith in any of this. But, be that as it may...

[1] - i say "GDP - potential output" instead of log(GDP) -
log(potential output). But it's sort of the same thing. taking the log
of something is a way to get a number that is like "percentage
difference" instead of "dollar difference". let's say - i am less
uncomfortable with that log-taking than with almost anything else in
the equation.


Anonymous said...

Pretty interesting!

How do you konow the "equilibrium real interest rate" for a given country?

Let's say: Which is the equilibrium real interest rate for Brasil, Chile o Argentina today.

The Arthurian said...

Hello, Anon.

"Pretty interesting!"

Yeah, I thought my son did a great job of making sense of that ugly equation.

"How do you know the 'equilibrium real interest rate' for a given country?"

I think what happens is this: They make up a value that that gives them the result they want, and then they hold up that result as proof they were right!

Jerry said...

Basically, yes, i think it is made up.

But that's not necessarily as silly as it sounds. I think it's the "desired rate" or "target rate".

I think the Taylor Rule is just a sort of approximate assertion about how the economy works. A theory about what effects inflation has. Something like that.

It's "a mathematical way of saying" this:
"We think that low interest rates cause GDP growth, but also cause inflation. So if inflation (or growth) is higher than we'd like, we raise interest rates. But if growth (or inflation) is lower than we want, then we lower the interest rate."

That's all it is. It doesn't mean that the "theory" about what the effects of inflation are is right or wrong (it's just a statement, not a proof or a derivation or an argument).

But if they are right, then this formula would probably be a reasonable approximation of how it works. (probably the actual reality would be more complicated, but this "linear approximation" would be about correct for small percentages. e.g. as long as inflation, growth, interest, etc are under 10% (or something), this form would be OK.)

If the "theory" is not right, then of course it wouldn't help you at all to have an equation that says the same wrong thing.