Using Calculus to Calculate Elasticities

Using Calculus to Calculate Elasticities

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By
Mike Moffatt
Mike Moffatt
Professor of Business, Economics, and Public Policy
  • Ph.D., Business Administration, Richard Ivey School of Business
  • M.A., Economics, University of Rochester
  • B.A., Economics and Political Science, University of Western Ontario
Mike Moffatt, Ph.D., is an economist and professor. He teaches at the Richard Ivey School of Business and serves as a research fellow at the Lawrence National Centre for Policy and Management.
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Updated on March 28, 2017

[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:

  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

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Your Citation
Moffatt, Mike. "Using Calculus to Calculate Elasticities." ThoughtCo, Aug. 27, 2020, thoughtco.com/using-calculus-to-calculate-elasticities-1146248. Moffatt, Mike. (2020, August 27). Using Calculus to Calculate Elasticities. Retrieved from https://www.thoughtco.com/using-calculus-to-calculate-elasticities-1146248 Moffatt, Mike. "Using Calculus to Calculate Elasticities." ThoughtCo. https://www.thoughtco.com/using-calculus-to-calculate-elasticities-1146248 (accessed May 29, 2023).