14.4 Income and Substitution Effects for Complements and Substitutes

So, how can we use income and substitution effects to explain complements and substitutes? Well, the signs and magnitudes of the income and substitution effects on each good can tell us a lot about the way the consumer sees the two goods.

First, note that the substitution effect reflects the change in the relative prices of the goods. If one price changes and the other remains the same, one will become relatively more expensive and the other relatively cheaper. As long as the consumer’s indifference curve is downward sloping, the substitution effect will always represent a shift away from the (newly) relatively more expensive good and toward the (newly) relatively cheaper good. So, if the price of good 1 increases or the price of good 2 decreases, the substitution effect will involve buying less good 1 and more good 2; and if the price of good 1 decreases or the price of good 2 increases, the substitution effect will involve buying more good 1 and less good 2.

Second, note that the income effect reflects the change in the real income of the consumer. Any price increase will decrease a consumer’s purchasing power, reducing their real income. If both goods are normal, the consumer will buy less of both goods; again, we’ll learn more about normal goods in Wednesday’s lecture.

Let’s assume that both goods are normal goods, and let’s think about the effect an increase in the price of good 2. The substitution effect means that the consumer will buy less good 2 and more good 1. The income effect means that the consumer will buy less of both goods.

The net effect on good 2 is unambiguous: the consumer will buy less good 2, both because of the income and substitution effects.

The effect on good 1 is ambiguous: the substitution effect causes them to buy more good 1, but the income effect causes them to buy less. Which one wins?

• If the substitution effect dominates, the consumer ends up buying more good 1 when the price of good 2 increases. This is our definition of substitutes!
• If the income effect dominates, the consumer ends up buying less good 1 when the price of good 2 increases. This is our definition of complements!
• If neither effect dominates — if they exactly offset each other — the consumer ends up not changing their consumption of good 1 when the price of good 2 increases. This is our definition of independent goods!

To see how this plays out, let’s look at a fixed price change — in particular, a quadrupling of the price of good 2, from 2 to 8 — and vary whether the goods are complements or substitutes. Recall that for the CES utility function \(u(x_1,x_2) = (x_1^{r}+x_2^{r})^{1/r}\) the parameter $r$ determines whether the goods are complements ($r < 0$), independent/Cobb-Douglas ($r = 0$), or substitutes ($0 < r \le 1$).

In the diagram below, change $r$ to see how the relative magnitude of the income and substitution effect on good 2 changes:

Why this relationship?

Note what’s going on here. The substitution effect is a movement along an indifference curve from the point where the MRS is equal to the initial price ratio, to the point where the MRS is equal to the new price ratio. If the indifference curve is very shallow (as it is when $r > 0$), this requires traveling a long distance along the indifference curve, so there is a large substitution effect. If the indifference curve bends rapidly (as it does when $r < 0$), the same price change results in a much shorter distance traveleed along the indifference curve, so there is a small substitution effect. Indeed, for a very negative value of $r$, there is almost no substitution effect!

In short, the shape of the indifference curve directly translates into observable behavior. In real life, what economists generally do is the opposite of this exercise: they observe behavior in the real world, and then try to find a utility function which captures that behavior mathematically. In other words, rather than starting with some arbitrary parameter $r$ and solving for a consumer’s optimal bundle, they observe purchase decisions that consumers make, and then try to estimate the value of $r$ which could have given rise to the data they see in the real world.