Using Calculus to Calculate Elasticities

Using Calculus to Calculate Elasticities

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[Q:] I understand the equations you have on your site regarding the 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:]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 calculus, 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 to four different situations: