## Sunday, April 28, 2013

### Other Things Equal

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

I first ran into the phrase ceteris paribus many years ago in Maynard's General Theory. It means "other things equal" or "everything else unchanged". It lets you consider the effect of a change in one factor, uncomplicated by the fact that everything in the economy is related to everything else.

Ceteris paribus

A ceteris paribus assumption is often fundamental to the predictive purpose of scientific inquiry. In order to formulate scientific laws, it is usually necessary to rule out factors which interfere with examining a specific causal relationship...

One of the disciplines in which ceteris paribus clauses are most widely used is economics, in which they are employed to simplify the formulation and description of economic outcomes. When using ceteris paribus in economics, assume all other variables except those under immediate consideration are held constant.

(Hesitantly) Okay...

According to the Wikipedia article, the concept is used "to consider the effect of some causes in isolation, by assuming that other influences are absent."

But you have to remember it's an assumption. The purpose of "other things equal" is to help to focus on the one topic under consideration, and for that it is most effective. But it's only an assumption, not the reality. If you happen to be arguing (PDF) that "average growth falls considerably" when government debt reaches "a threshold of 90 percent of GDP", it helps to assume that nothing else is changing. So then, the results that you see can only have been caused by the cause you describe.

If we are looking at increasing debt as the cause of slowing growth, and you want to focus on the Federal debt as the cause, then you have to assume that debt other than Federal debt didn't change at all. Because if "other" debt increased a little, it may share a little of the blame for slowing growth. And if other debt increased as much as Federal debt, it may share equally in the blame. But if other debt increased more than Federal debt, perhaps it is "other" debt that should get most of the blame for slow growth.

In the graph below, the blue line shows the gross Federal debt as a percent of GDP. The red line shows "other" debt as a percent of GDP. The green line is a benchmark showing 90% of GDP.

 Graph #1: Debt as the Cause of Slow Growth

Assume there was no increase in debt other than Federal, and we can say the Federal debt must be responsible for slow growth. But I just can't bring myself to say it.

Jerry said...

I think:
Something (Unemployment? whatever) is a function of several variables.

Let's pretend it's two variables. Then if you graph the function it is a 2-dimensional surface. So, like a landscape with hills and valleys. The two variables are "north/south" and "east/west", and the function gives you height. (Maybe the east/west is interest rates and north/south is TCMDO/GDP, and height is unemployment rate. If you go north all the way the ground slopes up.)

The partial derivative is a real thing. It means: at any given location, if you just move straight east, how much do you go up or down? Then, if you go back where you started, and just move straight north, how much do you move up or down?

Of course, the answer can be fairly complicated, depending on what the landscape looks like. Sometimes going north will bring you up, and sometimes it'll bring you down, depending on where you are. It's not likely to be a simple "north is always up". In general, the partial derivative is also a function of both variables. But it's a real thing and it makes sense.

I think that what's hard in economics is that you can't really see the landscape. You just sort of wander around blindly on a specific path and record your coordinates. And then you are trying to reconstruct the landscape from those measurements. So that might be hard to do -- but, it should be possible to do it, eventually, as long as you know what you're doing.

So, it can be pretty complicated. But it's still a valid thing to do. If you're doing it right.

I guess that I'm not convinced that economists are measuring the relevant coordinates. (maybe they are making a plot of temperature and humidity vs height, instead of east/west and north/south). And sometimes the landscape might change over time (another coordinate) due to earthquakes or something. And there is certainly a lot of "pretend this and that" going on in economics.

I don't know. I think my point is that there may be no connection between "partial derivatives" and "ceteris paribus as it is used in economics".

Min said...

Ceteris are seldom paribus.