9.5 Complements, Substitutions, and Independent Goods - One Last Time

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; if one of the goods is inferior, they’ll buy more of the inferior good and less of the normal good.

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

The net effect on good 1 is unambiguous: the consumer will buy less good 1, both because of the income and substitution effects.

The effect on good 2 is ambiguous: the substitution effect causes them to buy more good 2, but the income effect causes them to buy less. Which one wins?

• If the substitution effect dominates, the consumer ends up buying more good 2 when the price of good 1 increases. This is our definition of substitutes!
• If the income effect dominates, the consumer ends up buying less good 2 when the price of good 1 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 2 when the price of good 1 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 doubling of the price of good 1 — 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:

You can also see the way that the complementarity or substitutability affects the degree of travel of the IOC and the slope of the POC:

• If goods are complements ($r < 0$), the IOC shifts very little, so the substitution effect is small, and the net effect is a decrease in the quantity of good 2.
• If goods are substitutes ($r > 0$), the IOC shifts dramatically, so the substitution effect is big, and the net effect is an increase in the quantity of good 2.