Using Calculus To Calculate Cross-Price Elasticity of Demand

Using Calculus To Calculate Income Elasticity of Demand

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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:

  • 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 firms 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')))


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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


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

Next: Using Calculus To Calculate Price Elasticity of Supply

Other Price Elasticity Equations

  1. Using Calculus To Calculate Price Elasticity of Demand
  2. Using Calculus To Calculate Income Elasticity of Demand
  1. Using Calculus To Calculate Cross-Price Elasticity of Demand
  2. Using Calculus To Calculate Price Elasticity of Supply
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Moffatt, Mike. "Using Calculus To Calculate Cross-Price Elasticity of Demand." ThoughtCo, Mar. 27, 2017, thoughtco.com/calculate-cross-price-elasticity-of-demand-1146246. Moffatt, Mike. (2017, March 27). Using Calculus To Calculate Cross-Price Elasticity of Demand. Retrieved from https://www.thoughtco.com/calculate-cross-price-elasticity-of-demand-1146246 Moffatt, Mike. "Using Calculus To Calculate Cross-Price Elasticity of Demand." ThoughtCo. https://www.thoughtco.com/calculate-cross-price-elasticity-of-demand-1146246 (accessed January 21, 2018).