Using Calculus To Calculate Price Elasticity of Supply

Suppose you're given the following question:

Demand is Q = 100 - 3C - 4C², where Q is the amount of the good supplied, and C is the production cost of the good. What is the price elasticity of supply when our per unit cost is $2?

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 supply, we are interested in the elasticity of quantity supplied with respect to our unit cost C. Thus we can use the following equation:

Price elasticity of supply = (dQ / dC)*(C/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 cost. That is the case in our demand equation of Q = 400 - 3C - 2C². Thus we differentiate with respect to C and get:

dQ/dC = -3-4C

So we substitute dQ/dC = -3-4C and Q = 400 - 3C - 2C² into our price elasticity of supply equation:

Price elasticity of supply = (dQ / dC)*(C/Q)
Price elasticity of supply = (-3-4C)*(C/(400 - 3C - 2C²))

We're interested in finding what the price elasticity of supply is at C = 2, so we substitute these into our price elasticity of supply equation:

Price elasticity of supply = (-3-4C)*(C/(100 - 3C - 2C²))
Price elasticity of supply = (-3-8)*(2/(100 - 6 - 8))
Price elasticity of supply = (-11)*(2/(100 - 6 - 8))
Price elasticity of supply = (-11)*(2/86)
Price elasticity of supply = -0.256

Thus our price elasticity of supply is -0.256. Since it is less than 1 in absolute terms, we say that goods are substitutes.

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