## Inverse Demand and Supply Curves

We can look at market equilibrium in a slightly different way that is often useful. As indicated earlier, individual demand curves are normally viewed as giving the optimal quantities demanded as a function of the price charged. But we can also view them as inverse demand functions that measure the price that someone is willing to pay in order to acquire some given amount of a good. The same thing holds for supply curves. They can be viewed as measuring the quantity supplied as a function of the price. But we can also view them as measuring the price that must prevail in order to generate a given amount of supply.

These same constructions can be used with market demand and market supply curves, and the interpretations are just those given above. In this framework an equilibrium price is determined by finding that quantity at which the amount the demanders are willing to pay to consume that quantity is the same as the price that suppliers must receive in order to supply that quantity.

Thus, if we let Ps(q) be the inverse supply function and Pd(q) be the inverse demand function, equilibrium is determined by the condition

EXAMPLE: Equilibrium with Linear Curves

Suppose that both the demand and the supply curves are linear:

The coefficients (a, c, d) are the parameters that determine the intercepts and slopes of these linear curves. The equilibrium price can be found by solving the following equation:

The equilibrium quantity demanded (and supplied) is

We can also solve this problem by using the inverse demand and supply curves. First we need to find the inverse demand curve. At what price is some quantity q demanded? Simply substitute q for D(p) and solve for p. We have q — a — bp, so

In the same manner we find

Setting the demand price equal to the supply price and solving for the equilibrium quantity we have

Note that this gives the same answer as in the original problem for both the equilibrium price and the equilibrium quantity.

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