Web Notes > Microeconomics

Working with Perfect Complements Preferences

If a person likes to consume goods in rigidly fixed proportions, such as one left shoe for each right shoe, he or she is said to regard the goods as perfect complements. Expressing that proportion mathematically can be confusing at first.  Here's a frequently asked question about that.


"I had a question regarding a couple of the perfect-complements examples on your website. It appears that the variables used for the goods and the proportions that they consumed in relation to one another are mixed up.

Here's what I mean: For the movies and popcorn example it says the person eats exactly 2 serving of popcorn for each movie she sees. Let x be the quantity of movies she sees and let y be the servings of popcorn she eats. This says that x=2y.

However, in the solution page you have y=2x which, when substitued back into the demand equations along with prices, yield different results (quantities are not integers).  The same seems to be true for the beer and pizza problem. Please advise."


This is a confusing point with perfect complements preferences. It comes up because the way preferences are expressed in English makes it tempting to write the equation relating the goods incorrectly.

If a statement says: "She eats 2 servings of popcorn for each movie she sees" order of the words makes it tempting to write "2*Qp = Qm" because the "2" occurs near "popcorn". That's what the person asking the question did.  However, that's falling into a trap laid by English syntax: the equation you'll get is wrong.  To see that, imagine you wanted to use "2*Qp = Qm" to find out how many movies go with Qp=2.  Given her preferences, the answer should be 1. However, the equation would give 4 movies instead.

A better way to approach the problem is not to trust your ear, so to speak, and instead to start by writing the person's preferred ratio as just that, a ratio.  In the popcorn and movies case it would be:

`(Qp)/(Qm) = 2/1`

Then multiply both sides by Qm to get a more convenient equation:

`Qp = 2*Qm`

This equation works correctly.  If Qm=1 it predicts that she'll want Qp=2, which is exactly right.

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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 08/22/2018