Complements and Substitutes - EconGraphs (2024)

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. In other words, the price offer curve has a positive slope.
• 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.

If peanut butter and jelly are perfect complements with the utility function $u(x_1,x_2) = \min{x_1,x_2}$, we could illustrate the effect of this price change in the following graph:

Note, however, that goods might be complements without being perfect complements. That is, we can observe complementary behavior even if you don’t only consume goods in a strict ratio. For example, you might have other uses for peanut butter than just making PB&J sandwiches!

One useful family of utility functions 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. 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. For example, the graph below shows the same price change for $r = -1$:

Even though these goods aren’t perfect complements, an increase in the price of good 2 still causes the consumer to buy less good 1 and less good 2, and causes the demand curve for good 1 to shift in (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. In other words, the price offer curve has a negative slope.
• 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.

For a CES function, a value of $r$ between 0 and 1 represents goods which are substitutes but not perfect substitutes — again, we sometimes call these “weak substitutes.” (When $r = 1$, you can check to see that they’re perfect substitutes.) For example, with $r = {1 \over 2}$, we can illustrate the effect of this price change in the following graph:

As you can see, an increase in the price of good 1 causes the consumer to buy less good 1 and more good 2, meaning the POC has a negative slope. Likewise, an increase in the price of good 2 causes the consumer to buy more good 1 at any price, meaning an increase in the price of good 2 causes the demand curve to shift to the right.

As you can see, the case of perfect substitutes is an extreme example of this kind of behavior: when the price of one good increases beyond a certain threshold, the optimal bundle jumps from buying only one good to buying only the other. But that really is an extreme case: in general, changes in the prices of imperfect substitutes like strawberry jam and grape jelly will affect the demand for the other good, but not necessarily drive it down to zero. After all, sometimes you crave grape jelly, even if it’s pricier!

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. In other words, the price offer curve for good 2 is a vertical line. (The price offer curve for good 1 will be a horizontal line.)
• 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.

The 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 $p_2$, and is therefore a horizontal line, as we saw previously.

In the next chapter, we’ll go into much more detail about what makes goods complements, substitutes, or independent goods.