Using Calculus to Calculate Elasticities

[Q:] I understand the equations you have on your site regarding change in quantity demanded and change in price to calculate the elasticity. How would I convert this equation into those types? I don't quite understand what this equation means. No other info was given.

Demand is Qx = 110 - 4Px. What is price (point) elasticity at $5?

[A:]Thanks for your question!

In the article A Beginner's Guide to Elasticity we learned that elasticity is given by the formula:

Elasticity = (percentage change in Z) / (percentage change in Y)

We saw how to calculate various elasticities when we're given numerical examples. But how do we calculate an elasticity when we're given a formula such as Z = f(X)?

Use Calculus to Find the Elasticity!

Using some fairly basic calcululus, we can show that

(percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z)

where dZ/dY is the partial derivative of Z with respect to Y. Thus we can calculate any elasticity through the formula:

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

We'll look at how to apply this for four different situations:

1. Using Calculus To Calculate Price Elasticity of Demand
2. Using Calculus To Calculate Income Elasticity of Demand
3. Using Calculus To Calculate Cross-Price Elasticity of Demand
4. Using Calculus To Calculate Price Elasticity of Supply

Next: Using Calculus To Calculate Price Elasticity of Demand