Utility maximization with perfect complements

Perfect complements

Perfect complements are goods which only provide utility or happiness when they are consumed together. They are an extreme case of complementary goods, which are goods which complement other goods (fries and ketchup are an example of complements). Perfect complements are different from normal complementary goods in that they only provide utility if consumed together whereas you can consume complements individually. The best way to think of perfect complement is "has to be consumed together, otherwise, they provide you with no happiness".

Examples of perfect complements

  • Left and right shoes
  • Portable gaming devices and batteries
  • Computers and operating systems

Why are these perfect complements?

In terms of left and right shoes, typically nobody wants just a left or right shoe. Therefore, they must be consumed in bundles. For gaming devices, they will not operate without batteries. Computers are unable to run without an operating system and having an operating system and no computer is unlikely to provide any utility.

A graphical representation of perfect complements

Consider our left and right shoe example. We typically want to consume shoes in a 1:1 ratio.
perfect complements

In the above case, we see that U = min(Good 1, Good 2) which says "our utility equals whichever value is smaller between good 1 or good 2". The blue curve represents the indifference curve and is an "L" shape. It takes on the "L" shape because given we consume only one unit of good 2, it does not matter how many units of good 1, we will still only receive one unit of utility. We can see that from the diagram. If we consume 1 unit of both, U = min(1,1) and the minimum value of 1 and 1 is 1. Now suppose that we increase our consumption of good 1 to 2 units whilst leaving consumption of good 2 at 1 unit. The utility now equals min(1,2) and the minimum value of 1 or 2 is still 1, so our utility does not change and remains at 1.

Maximizing utility with perfect complements

Suppose that we are looking at our shoe example again so our utility function is U = min(L, R) where L = left shoe and R = right shoe. Recall, that our values for utility do not matter as long as they display the following characteristic:

More is better, so having 2 left shoes and 2 right shoes is better than having 1 left shoe and 1 right shoe, which we can see is true since:

min(2,2) = 2 - which is the utility from having 2 left and right shoes

min(1,1) = 1 - which is the utility from having 1 left and 1 right shoe

So this characteristic is satisfied.

And let's suppose that our consumer has an income of $40 and each shoe cost $10 each. Thus our budget constraint looks like:

$40 = $10L + $10R

Unlike most utility maximization problems for which you are familiar, you cannot solve this by taking derivatives. There are relatively few possibly combinations of shoes so we could solve this by brute force to see which combination of goods gives us the most utility. For example, we could consume the following combinations

Left | Right | Utility
4           0            0
3           1            1
2           2            2
1           3            1
0           4            0

As we can see, the maximum utility is when we consume 2 units of both left and right shoes which should have hopefully been obvious by now. This process could be pretty tedious if we had a larger set of possible combinations. Fortunately, there is an easier way to do this. If we realize that we need to have the same proportion of shoes, we could simple set L = R. Then we could substitute this into our budget constraint s.t

$40 = $10L + $10L

$40 = $20L

L = 2

since L = R

R = 2

This is a lot more convenient way of solving such a question and as we see this yields the same result as above.

Now consider another example. Suppose that we have income of $200, a gameboy costs $40 and batteries cost $5 each have a gameboy which requires 2 batteries. If B = batteries and G = gameboy, then our utility function is U = min(G, .5B). It is as such since we need 2 batteries for each gameboy. We can see what happens if we plug in a few values.

Suppose we have 1 gameboy and 1 battery then U = min(1, .5) = .5 (this assumes that we can consume half a gameboy).

Suppose we have 1 gameboy and 2 batteries then U = min(1, .5(2)) = 1

This is the same as saying 1 G = .5 B. Therefore, if we substitute G = 2B into the budget constraint then we get:

$100 = $40G + $5B

$200 = $40(.5B)+ $5B

$200 = $25B

B = 8

G = .5B

G = 4

There you go. That's how maximize utility when you have perfect complements.

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