We have already used the terms substitutes and complements, but it is now appropriate to give a formal definition. Since we have seen perfect substitutes and perfect complements several times already, it seems reasonable to look at the imperfect case.
Let’s think about substitutes first. We said that two brands of sparkling water might be thought of as perfect substitutes for a consumer who can’st taste any difference between the two. But what about sparkling water and flat water? This is a case of “imperfect” substitutes. That is, sparkling and flat water are, to some degree, substitutes for each other, but they don’st substitute as perfectly as two different brands of sparkling water do.
Similarly, we said that right shoes and left shoes were perfect complements. But what about a pair of shoes and a pair of socks? Right shoes and left shoes are nearly always consumed together, and shoes and socks are usually consumed together. Complementary goods are those like shoes and socks that tend to be consumed together, albeit not always.
Now that we’ve discussed the basic idea of complements and substitutes, we can give a precise economic definition. Recall that the demand function for good 1, say, will typically be a function of the price of both good 1 and good 2, so we write . We can ask how the demand for good 1 changes as the price of good 2 changes: does it go up or down?
If the demand for good 1 goes up when the price of good 2 goes up, then we say that good 1 is a substitute for good 2. In terms of rates of change, good 1 is a substitute for good 2 if
The idea is that when good 2 gets more expensive the consumer switches to consuming good 1: the consumer substitutes away from the more expensive good to the less expensive good.
On the other hand, if the demand for good 1 goes down when the price of good 2 goes up, we say that good 1 is a complement to good 2. This means that
Complements are goods that are consumed together, like coffee and sugar, so when the price of one good rises, the consumption of both goods will tend to decrease.
The cases of perfect substitutes and perfect complements illustrate these points nicely. Note that is positive (or zero) in the case of perfect substitutes, and that is negative in the case of perfect complements.
When it comes to substitutes and complements, the two-good case is rather special. Since income is being held fixed, if you spend more money on good 1, you’ll have to spend less on good 2. This puts some restrictions on the kinds of behavior that are possible.
When we expand our models to include more than two goods at a time, richer possibilities emerge for substitute and complement patterns across goods. Consider, for example, electric cars and gasoline. They are neither substitutes nor complements—not directly, anyway. However, electric cars are substitutes for gas-powered cars, and gas-powered cars are complements with gasoline. So it’s not surprising that electric vehicles and gasoline behave like substitutes: when gas prices go up, so does demand for electric cars.
Higher demand for electric cars is good for the bottom line of producers who specialize in making those vehicles. One recent study examined how the stock prices of the world’s biggest car companies responded to fluctuations in the price of oil (which also determines the price of gas).2 That relationship was weakly negative for almost all companies (which by now all make both gas-powered and electric vehicles)—but one company stood out: Tesla. It was the only company whose value exhibited a strong positive correlation with the price of oil. Not surprisingly, this was the only large company focused exclusively on electric car production.
Although our definitions of substitutes and complements in terms of consumer demand behavior seem reasonable, those definitions run into trouble in more general environments. For example, in a situation involving more than two goods, good 1 may end up being a substitute for good 3, while good 3 is a complement for good 1. To avoid peculiar outcomes like this, more advanced treatments typically use somewhat different definitions of substitutes and complements. The concepts we have defined in terms of consumer behavior are known as gross substitutes and gross complements. Those concepts will be sufficient for our needs.