14.2 Complements and Substitutes
The way the demand curve shifts in response to the price of another good depends on the relationship between those two goods:
- Goods like peanut butter and grape jelly are complements: they are generally consumed together, for example in PB&J sandwiches.
- Goods like strawberry jam and grape jelly are substitutes: they generally serve the same purpose.
- Goods like t-shirts and jelly are independent goods: there’s no obvious relationship between them.
Let’s look at each of these in turn, by examining the effect of a price increase in one flavor of jelly on the quantity demanded of peanut butter, strawberry jam, and t-shirts.
Complements
If two goods are complements, an increase in the price of either good will result in a decrease in the quantity bought of both goods. For example, if you enjoy sandwiches made with peanut butter (good 1) and grape jelly (good 2), an increase in the price of jelly increases the price of making a PB&J sandwich; so you might have fewer such sandwiches and do something else for lunch.
- In Good 1 - Good 2 space, this means that an increase in the price of grape jelly (good 2) leads to a decrease in the quantity demanded of both goods: that is, the optimal bundle moves down and to the left.
- If we look at the demand curve for peanut butter (good 1), we can see that the quantity of peanut butter demanded at every price of peanut butter decreases, shifting the demand curve to the left.
Substitutes
If two goods are substitutes, an increase in the price of one good will result in a decrease in the quantity bought of that good, and an increase in the quantity of the other. For example, if you view strawberry jam (good 1) and grape jelly (good 2) as substitutes, then an increase in the price of grape jelly will cause you to use more of the relatively cheaper strawberry jam in recipes which could use either.
- In Good 1 - Good 2 space, this means that an increase in the price of grape jelly (good 2) leads to a decrease in the quantity demanded of grape jelly, but an increase in the quantity demanded of strawberry jam: that is, the optimal bundle moves down and to the right.
- If we look at the demand curve for strawberry jam (good 1), we can see that the quantity of strawberry jam demanded at every price of strawberry jam increases, shifting the demand curve to the right.
Independent Goods
Finally, let’s think about goods like t-shirts (good 1) and grape jelly (good 2), which have no obvious connection. For such goods, we would not expect a change in grape jelly to affect the quantity of t-shirts bought at all:
- In Good 1 - Good 2 space, this means that an increase in the price of grape jelly (good 2) leads to a decrease in the quantity demanded of grape jelly, but no change in the quantity demanded of t-shirts: that is, the optimal bundle moves straight down.
- If we look at the demand curve for t-shirts (good 1), we can see that the demand curve is unaffected by the price of grape jelly.
One utility function we’ve seen that exhibits this behavior is Cobb-Douglas, in which the consumer will spend a given fraction of their income on each good. For example, with the Cobb-Douglas utility function $u(x_1,x_2) = x_1x_2$, the consumer’s demand for good 1 is $x_1^\star(p_1,p_2,m) = m/2p_1$, which doesn’t depend at all on $p_2$.
The CES Function
One useful family of utility functions for analyzing complements and substitutes is a CES utility function, of the form \(u(x_1,x_2) = (ax_1^r + bx_2^r)^{1 \over r}\) In this case the $r$ parameter measures how complementary or substitutable goods are.
Negative values of $r$ correspond to complements. As $r$ approaches $-\infty$, the preferences approach the “perfect complements” or Leontief functional form we’ve become familiar with. However, for any negative value of $r$, the goods are complementary; we sometimes call these “weak complements” to distinguish them from “perfect complements,” but just “complements” works too.
Likewise, positive values of $r$ correspond to substitutes. When $r = 1$, the function is just linear; so the goods are “perfect substitutes.” However, a value of $r$ between 0 and 1 represents goods which are substitutes but not perfect substitutes — again, we sometimes call these “weak substitutes” or just “substitutes.”
Finally, when $r = 0$, the goods are neither substitutes nor complements, but independent goods.
The graph below allows you to play with changes in $p_1$ and $p_2$ for different values of $r$, for the simple CES utility function \(u(x_1,x_2) = (x_1^r + x_2^r)^{1 \over r}\) The initial value of $p_2$ is 4; try changing that to a smaller or larger value to see how the demand curve shifts, and the optimal bundle shifts: