16.2 Demand Effects: Complements and Substitutes

When we introduced the notion of complements and substitutes, we posited that grape jelly and peanut butter might be complements (because they are used together to make peanut butter and jelly sandwiches), while grape jelly and strawberry jam might be substitutes.

Now suppose the price of grapes goes up, causing the supply of grape jelly to shift to the left. How does this affect the equilibrium price and quantity of peanut butter? What about strawberry jam?

Because grape jelly and peanut butter are complements, an increase in the price of grape jelly would cause the demand for peanut butter to decrease. The following diagram shows the market for grape jelly and the demand for peanut butter. Use the slider to increase the cost of grapes in the diagram below; as you can see, this raises the price of grape jelly, which in turn shifts the demand for peanut butter to the left:

However, the story doesn’t end there! After all, we would expect a decrease in demand for peanut butter to reduce the price of peanut butter. Again because the two goods are complements, this would cause an increase in the demand for grape jelly, (slightly) raising the price of grape jelly even further. These effects would continue to ricochet between the two markets until they read a new equilibrium. Use the slider in the diagram below to see the equilibrium effect of the same change in the cost of grapes:

Note that the net effect on the market for grape jelly is still a decrease in quantity and an increase in price, as we would expect from a leftward shift in the supply. The direct effect of the shift in the supply curve is sometimes called a “first-order” effect; the fact that demand increases slightly due to the change in the price of peanut butter is likewise called a “second-“ or “third-order” effect.

Solving for equilibrium in two markets

How do we solve for the overall equilibrium in the two markets? Let’s model this situation “as if” demand came from a single consumer with preferences \(u(x_1,x_2) = \min\{x_1,x_2\}\) so their demand for goods 1 and 2 is \(D_1(p_1,p_2) = {m \over p_1 + p_2}\) \(D_2(p_1,p_2) = {m \over p_1 + p_2}\) the supply of good 1 comes from a single price-taking firm with cost function \(c_1(q_1) = \frac{a}{2}q_1^2\) so its supply curve is given by \(S_1(p_1) = {p_1 \over a}\) and the supply of good 2 comes from a single price-taking firm with cost function \(c_2(q_2) = \frac{b}{2}q_2^2\) so its supply curve is given by \(S_2(p_2) = {p_2 \over b}\) Notice that this is a little easier to solve for, because in equilibrium $Q_1 = Q_2$. Let’s call this quantity $Q^E$; this gives us three equations in three unknowns: \(\begin{aligned} Q^E &= {m \over p_1 + p_2}\\ Q^E &= {p_1 \over a}\\ Q^E &= {p_2 \over b} \end{aligned}\) A bit of algebra yields \(\begin{aligned} Q^E &= \sqrt{m \over a + b}\\ p_1^E &= a\sqrt{m \over a + b}\\ p_2^E &= b\sqrt{m \over a + b} \end{aligned}\) As you can see, an increase in $a$ (the cost of producing good 1) reduces the equilibrium quantity, increases the price of good 1, and decreases the price of good 2. If you’d like to try to re-create the diagrams above, try starting with $m = 144$ and $a = b = 2$, and look at the effect of an increase in $a$ from 2 to 7…