Calculate Cross-Price Elasticity of Demand (Calculus)

man writing notes
Portra Images / Getty Images
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.
Learn about our Editorial Process
Updated on February 10, 2019

Suppose you're given the following question:

Demand is Q = 3000 - 4P + 5ln(P'), where P is the price for good Q, and P' is the price of the competitors good. What is the cross-price elasticity of demand when our price is $5 and our competitor is charging $10?

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 cross-price elasticity of demand, we are interested in the elasticity of quantity demand with respect to the other firm's price P'. Thus we can use the following equation:

Read More
Using Calculus To Calculate Income Elasticity of Demand
By Mike Moffatt
  • Cross-price elasticity of demand = (dQ / dP')*(P'/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 the other firm's price. That is the case in our demand equation of Q = 3000 - 4P + 5ln(P'). Thus we differentiate with respect to P' and get:

  • dQ/dP' = 5/P'

So we substitute dQ/dP' = 5/P' and Q = 3000 - 4P + 5ln(P') into our cross-price elasticity of demand equation:

  • Cross-price elasticity of demand = (dQ / dP')*(P'/Q)
    Cross-price elasticity of demand = (5/P')*(P'/(3000 -4P + 5ln(P')))

We're interested in finding what the cross-price elasticity of demand is at P = 5 and P' = 10, so we substitute these into our cross-price elasticity of demand equation:

  • Cross-price elasticity of demand = (5/P')*(P'/(3000 -4P + 5ln(P')))
    Cross-price elasticity of demand = (5/10)*(5/(3000 - 20 + 5ln(10)))
    Cross-price elasticity of demand = 0.5 * (5 / 3000 - 20 + 11.51)
    Cross-price elasticity of demand: = 0.5 * (5 / 2991.51)
    Cross-price elasticity of demand: = 0.5 * 0.00167
    Cross-price elasticity of demand: = 0.5 * 0.000835

Thus our cross-price elasticity of demand is 0.000835. Since it is greater than 0, we say that goods are substitutes.

Other Price Elasticity Equations

  1. Using Calculus To Calculate Price Elasticity of Demand
  2. Using Calculus To Calculate Income Elasticity of Demand
  3. Using Calculus To Calculate Price Elasticity of Supply
Format
mla apa chicago
Your Citation
Moffatt, Mike. "Calculate Cross-Price Elasticity of Demand (Calculus)." ThoughtCo, Aug. 27, 2020, thoughtco.com/calculate-cross-price-elasticity-of-demand-1146246. Moffatt, Mike. (2020, August 27). Calculate Cross-Price Elasticity of Demand (Calculus). Retrieved from https://www.thoughtco.com/calculate-cross-price-elasticity-of-demand-1146246 Moffatt, Mike. "Calculate Cross-Price Elasticity of Demand (Calculus)." ThoughtCo. https://www.thoughtco.com/calculate-cross-price-elasticity-of-demand-1146246 (accessed April 19, 2024).