3.2 Supply and Demand Changes

When the price of a complement changes—what happens to the equilibrium price and quantity of the good? Such questions are answered by comparative statics, which are the changes in equilibrium variables when other things change. The use of the term “static” suggests that such changes are considered without respect to dynamic adjustment; instead, one just focuses on the changes in the equilibrium level. Elasticities will help us quantify these changes.

How much do the price and quantity traded change in response to a change in demand? We begin by considering the constant elasticity case, which allows us to draw conclusions for small changes for general demand functions. We will denote the demand function by q_d(p) = a * p^−ε and supply function by q_s(p) = bp^η. The equilibrium price p* is determined at the point where the quantity supplied equals to the quantity demanded, or by the solution to the following equation:

$$q_d(p*)=q_s(p*).$$

Substituting the constant elasticity formulae,

$$ap{*}^{(-\mathrm{\epsilon )}}=q_d(p*)=q_s(p*)=bp{*}^{\mathrm{(\eta )}}.$$

Thus,

$$\frac{a}{b}=p{*}^{\epsilon +\eta },$$

or

$$p*={\left(\frac{a}{b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\epsilon +\eta $}\right.}.$$

The quantity traded, q*, can be obtained from either supply or demand, and the price:

$$q*=q_s(p*)=bp{*}^{\eta }=b{\left(\frac{a}{b}\right)}^{\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\epsilon +\eta $}\right.}=a^{\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\epsilon +\eta $}\right.}{b}^{\raisebox{1ex}{$\epsilon $}\!\left/ \!\raisebox{-1ex}{$\epsilon +\eta $}\right.}.$$

There is one sense in which this gives an answer to the question of what happens when demand increases. An increase in demand, holding the elasticity constant, corresponds to an increase in the parameter a. Suppose we increase a by a fixed percentage, replacing a by a(1 + ∆). Then price goes up by the multiplicative factor ${\left(1+\Delta \right)}^{{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\epsilon +\eta $}\right.}}$ and the change in price, as a proportion of the price, is $\frac{\Delta p*}{p*}={\left(1+\Delta \right)}^{{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\epsilon +\eta $}\right.}}-1\text{.}$ Similarly, quantity rises by $\frac{\Delta q*}{q*}={\left(1+\Delta \right)}^{{}^{\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\epsilon +\eta $}\right.}}-1\text{.}$

These formulae are problematic for two reasons. First, they are specific to the case of constant elasticity. Second, they are moderately complicated. Both of these issues can be addressed by considering small changes—that is, a small value of ∆. We make use of a trick to simplify the formula. The trick is that, for small ∆,

$${\left(1+\Delta \right)}^{{}^r}\approx 1+r\Delta .$$

The squiggly equals sign ≅ should be read, “approximately equal to.”The more precise meaning of ≅ is that, as ∆ gets small, the size of the error of the formula is small even relative to δ. That is, ${\left(1+\Delta \right)}^{{}^r}\approx 1+r\Delta$ means $\frac{{\left(1+\Delta \right)}^{{}^r}-\left(1+r\Delta \right)}{\Delta }\underset{\Delta \to 0}{\to }0.$ Applying this insight, we have the following:

For a small percentage increase ∆ in demand, quantity rises by approximately $\raisebox{1ex}{$\eta \Delta $}\!\left/ \!\raisebox{-1ex}{$\epsilon +\eta $}\right.$ percent and price rises by approximately $\raisebox{1ex}{$\Delta $}\!\left/ \!\raisebox{-1ex}{$\epsilon +\eta $}\right.$ percent.

The beauty of this claim is that it holds even when demand and supply do not have constant elasticities because the effect considered is local and, locally, the elasticity is approximately constant if the demand is “smooth.”