In introductory economics courses, students are taught that elasticities are calculated as ratios of percent changes. Specifically, they are told that price elasticity of supply is equal to the percent change in quantity supposed divided by the percent change in price. While this is a helpful measure, it is an approximation to some degree, and it calculates what can (roughly) be thought of as an average elasticity over a range of prices and quantities.

To calculate a more exact measure of elasticity at a particular point on a supply or demand curve, we need to think about infinitesimally small changes in price and, as a result, incorporate mathematical derivatives into our elasticity formulas. to see how this is done, let's take a look at an example.

## An Example

Suppose you're given the following question:

Demand is Q = 100 - 3C - 4C^{2}, where Q is the amount of the good supplied, and C is the production cost of the good. What is the price elasticity of supply when our per unit cost is $2?

We saw that we can calculate any elasticity by the formula:

- Elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)

In the case of price elasticity of supply, we are interested in the elasticity of quantity supplied with respect to our unit cost C. Thus we can use the following equation:

- Price elasticity of supply = (dQ / dC)*(C/Q)

In order to use this equation, we must have quantity alone on the left-hand side, and the right-hand side be some function of cost. That is the case in our demand equation of Q = 400 - 3C - 2C^{2}. Thus we differentiate with respect to C and get:

- dQ/dC = -3-4C

So we substitute dQ/dC = -3-4C and Q = 400 - 3C - 2C^{2} into our price elasticity of supply equation:

- Price elasticity of supply = (dQ / dC)*(C/Q)

Price elasticity of supply = (-3-4C)*(C/(400 - 3C - 2C^{2}))

We're interested in finding what the price elasticity of supply is at C = 2, so we substitute these into our price elasticity of supply equation:

- Price elasticity of supply = (-3-4C)*(C/(100 - 3C - 2C
^{2}))

Price elasticity of supply = (-3-8)*(2/(100 - 6 - 8))

Price elasticity of supply = (-11)*(2/(100 - 6 - 8))

Price elasticity of supply = (-11)*(2/86)

Price elasticity of supply = -0.256

Thus our price elasticity of supply is -0.256. Since it is less than 1 in absolute terms, we say that goods are substitutes.