Using Calculus To Calculate Cross-Price Elasticity of Demand

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')))

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.

Next: Using Calculus To Calculate Price Elasticity of Supply

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