# Economists Do It With Models

## Reader Question: Why Only Slippers?

Reader and friend Lilei emailed me the following a few weeks ago:

An individual has a constant marginal rate of substitution of shoes for slippers of 3/4 (i.e. he or she is always willing to give up 3 pairs of slippers to get 4 pairs of shoes). If slippers and shoes are equally costly, the individual will
C) spend his or her income equally on slippers and shoes
D) wear slippers only 3/4 of the time (ha)

I thought all the answers were rather stupid. But I contest A. It seems illogical to me that you would conclude that a “rational” consumer would buy only slippers. It’s like saying seamless bras and underwires are the same. (Editor’s note: hee hee.)

My response:

In my view, not only are some of the answers rather stupid, but the question itself is kind of stupid…or, if not specifically stupid, not particularly realistic and tricky for the sake of being tricky, which is annoying. (I get the notion of giving a tricky question to test mastery of material. But why not give something realistic so that students aren’t left with a “well this is all irrelevant to my life” taste in their mouths?) But let’s go through this…

The marginal rate of substitution is a measure of a person’s preferences, or how much they like different things. First off, I think your exam should have said “marginal rate of substitution of slippers for shoes” rather than the other way around, since my way measures how many slippers the consumer would be willing to give up to get one more pair of shoes. To say that the MRS of slippers for shoes is 3/4 implies that the consumer likes slippers more than shoes, since he would have to be compensated with more than 3 pairs of shoes in order to offset a happiness loss from giving up 3 pairs of slippers. (You could also be cute and say that he would have to be compensated with 4/3 of a pair of shoes to be willing to give up a pair of slippers, but what would the guy do with 1/3 of a pair of shoes? And would that be 1/3 of each foot, or 2/3 of the left and none of the right? 2/3 of two pairs even? Or…well, you get the idea.)

The crux of this problem is the word always. Normally, the MRS depends on how much of each thing we are currently consuming. This is best illustrated by an example. Each year during the consumer choice unit, I go to CVS and buy a big bag of Starburst and a big bag of Reese’s peanut butter cups. In class, I ask for a student volunteer, and I make sure that said volunteer is not allergic to chocolate. I then dump the entire bag of Starburst out on the student’s desk (mainly because I appreciate the visual that it creates). I hold one Reese’s peanut butter cup and ask how many Starburst the student is willing to trade for a peanut butter cup. The number is usually pretty high, and the trade is executed. (The yellow Starburst are almost always the first to get traded. Go figure.) I then ask the question again, and the number gets a little lower. This repeats for a while, and we get to a point where the student isn’t willing to give up any more Sturburst for a single peanut butter cup. I then have to offer up 2 peanut butter cups, or 3, 4, etc. to get any more Sturburst from the student. In the end, I give the student all of the Sturburst and peanut butter cups so that I don’t eat them myself (and the students usually share with the class).

The point is that, when you have a lot of something, you are generally willing to give up a relatively large amount to get one more of something that you don’t already have a lot of. When you have only a little bit of something, you hold onto it for dear life (compared to when you had a lot of it at least) and need a much better trade to get you to part with any of it. Put in terms of MRS and shoes and slippers, this would mean that the MRS of slippers for shoes is higher when the consumer has a lot of slippers but few shoes and lower when the consumer has few slippers and a closet full of shoes. Therefore, this notion of the MRS ALWAYS being 3/4 is, well, contrived, since it means that the consumer’s preferences are not affected by how many shoes and slippers he currently has lying around. Taking the likely unrealistic preference as given, the answer to the question is A. This just boils down to the fact that if the consumer always likes slippers more than shoes and the two items cost the same, then this person is going to enjoy a slipper-filled Hugh Hefner type of lifestyle.

To put it in more nerdy terms (translation: you can skip this if you haven’t taken an econ course), consumer choice, at least in a textbook sense, is governed by the utility maximization framework. This framework represents mathematically both the consumer’s preferences and his budget in order to compute optimal consumption. Preferences are expressed by indifference curves, which are basically level sets that show different levels of happiness. (You can think of indifference curves as a topographic map of pleasure.) The general goal is to be on the highest indifference curve that is still affordable, since that is where the consumer will be the happiest. Normally, this situation looks something like this:

The straight line represents the consumer’s budget, and it has a slope of -1, since the consumer HAS to give up one slipper to afford one more shoe. (This is merely due to the fact that slippers and shoes are the same price. The slope is technically the price of shoes divided by the price of slippers, at least in an absolute value sense.) The curved lines represent the consumer’s indifference curves, and they are bowed in the way that they are because of the Starburst/peanut butter cup concept described above. I’ve arbitrarily labeled them happiness (or utility) of 1, 2 and 3- all that is important to know is that higher numbers are better, and higher-numbered curves are up and to the right on the graph. (You may also know that the magnitude of the slope of the indifference curves at any point is equal to the MRS described above.) I’ve labeled the optimal consumption point above, and generally the consumer is purchasing some of each of the goods at that point.

Now let’s consider the specific situation in the problem above. Because were told that the MRS is ALWAYS 3/4, this means that the slope of the indifference curves should be constant at -3/4. (Indifference curves have negative slope because you are giving up one good and getting more of the other.) That picture looks like this:

Again, you will notice that a happiness level of 3 is the best that the consumer can do, and he can only get to that level by consuming only slippers, so that is the optimal consumption. The general lesson to be learned is that if you make a wacky assumption you can get a strange answer, even if your model itself is perfectly reasonable.

Tags: Econ 101 · Reader Questions

### 71 responses so far ↓

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This might make a good part a) of a problem of which part b) is “Do you think this is realistic? If not, what assumption do you think should be changed, and why?”

• 2 the weakonomist // Dec 9, 2009 at 9:41 pm

My mother-in-law loves slippers. Unfortunately for her my dog does too. His q* seems to be somewhere around 4 pairs annually, but the marginal substitution I’ve found is about 4 treats for each pair. Let’s see the graph!

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