(Untitled)

[jcreed]

A thing that's bugged me for a long time is the apparent arbitrariness with which we have to define inflation. The CPI, for instance, picks some ol' basket of goods, and measures how its price changes. Depending on how that representative basket of goods is chosen, you get a different answer. In particular, if the basket of goods has twice as many

Comments

[altamira16]

Do they count things twice? I thought that the items occurred in the basket just once, but the trick was picking what went into the basket because things become obsolete. For example, it may have made sense to have buggy whips in there at some point. Or it may have made sense to have a first generation Nintendo in there at some point.

[jcreed]

Well, but do you include a Nintendo and a Sega, or just one gaming console? Ten apples per week, or five?

[platypuslord]

If you're looking at a given point in time, the decision is easy: include every item on the market, in proportion to how many dollars were spent on that item.

As the above commentor wrote, the trouble starts when you get products that can become obsolete. Cars are a good example.

Suppose I put a 2000 Honda Accord in my "basket". I quickly notice that the price of a 2000 Honda Accord decays at about 10% per year. I add some other models -- 2001 Honda Accord, 2002 Honda Accord -- and discover to my horror that every car on the market has the same yearly price decay! Does this mean inflation is at -10%?

[platypuslord]

IANAE, but I assume what economists actually do is they ignore any item for which supply or demand changed drastically during the period surveyed. For such items, the matrix method you describe might work fine -- but I think it would also be perfectly reasonable to simply weight each item by the volume sold per year.

[platypuslord]

It seems very strange that you're representing your items as vectors in Rn. I have difficulty imagining any useful vector space which could represent both (for example) an Xbox and a watermelon.

[jcreed]

Sorry if that was not at all clear - the vectors would be their prices over time.

[platypuslord]

Oho.

[r_transpose_p]

Wait, is Σiμivi a basket of goods, or something you divide by its previous value to get the inflation rate?

Also, is Σiμivi in Rn? That doesn't look like an average to me!

Which quantities in here are goods and which are prices?

[jcreed]

Sorry, it's just kind of a suggestive setup. I imagine the vectors in Rn are something like the price histories of a good, or better yet, the history of the log ratio of price from one year to the next.

Why is Σiμivi being in Rn make it seem not like an average? It would be the mean if μi were constantly 1/m.

[r_transpose_p]

Hadn't read your comment to platypuslord.

Now it all makes sense.

(my initial confusion was that, while I'm perfectly okay with the average of a bunch of velocities being a velocity, my mind wants the "inflation rate" to be in R. I assume you're not talking about making inflatin rates vectors in Rn, but if you are, that is kindof interesting [and might require a bit more explanation of the economics involved])

[bhudson]

why is this robustness desirable? It seems like you are saying that if potato prices go up 50%, it should have the same effect on the inflation rate whether we are yuppies who buy one potato per month but usually buy lattes, or if we are Irish and famined and spend half our income on potatoes.

[r_transpose_p]

Oh yeah, something thats actually robust probably has to deal with substitute goods.

Good point.

[demoness101]

An excellent point.

[r_transpose_p]

Wait, why is rotation invariance desirable?

What does rotation even mean when rotating bundles of time-series of good prices? Or does it become some other sort of "rotation" when you pick a different norm?

Speaking of which, which norm are you using for distance?

[jcreed]

I can't really justify it that well. My thought process wandered from the initial intuitive problem to a more abstract formulation, and now that I think about it, it's not totally appropriate.

It probably came out of thinking through the impossibility theorem that you can't have something that is invariant under all linear transformations, and also duplication of samples: for consider first the endpoints of an equilateral triangle. Rotation-invariance (a special case of linear transformation obviously) that their "average" is in the ordinary geometric center of the triangle. But if I linearly squish the triangle so the two points on the base come together just below the top point, then that ordinary center is carried by that linear transformation to a height just 1/sqrt(3) above the base, which it shouldn't; the "average" of two points --- since we're supposed to be agnostic about the fact that there's a redundant point sitting on top of the bottom one --- should obviously be half-way between them.