# Using Calculus To Calculate Price Elasticity of Demand

## Price Elasticity with Calculus

Suppose you're given the following question:

Demand is Q = 110 - 4P. What is price elasticity (point version at \$5?

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 price elasticity of demand, we are interested in the elasticity of quantity demand with respect to price. Thus we can use the following equation:

• 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 price. That is the case in our demand equation of Q = 110 - 4P. Thus we differentiate with respect to P and get:

• dQ/dP = -4

So we substitute dQ/dP = -4 and Q = 110 - 4P into our price elasticity of demand equation:

• Price elasticity of demand: = (dQ / dP)*(P/Q)
Price elasticity of demand: = (-4)*(P/(110-4P)
Price elasticity of demand: = -4P/(110-4P)

•

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

• Price elasticity of demand: = -4P/(110-4P)
Price elasticity of demand: = -20/90
Price elasticity of demand: = -2/9

•

Note that you could have plugged in values for P and Q earlier rather than carrying out the algebra in terms of P.  In addition, this value, because it only considers one point on the demand curve rather than a region of the demand curve, is classified as point elasticity rather than arc elasticity.

Thus our price elasticity of demand is -2/9. Since it is less than 1 in absolute terms, we say that Demand is Price Inelastic

Next: Using Calculus To Calculate Income Elasticity of Demand.

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