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From Citizendium, the Citizens' Compendium

Elasticity of Demand (an example of the algebra of elasticity)

Supposing that Q is the quantity of a product that would be bought by consumers when its price is P, and that Q is related to P by the equation:

Q = − AP + B

- then the elasticity of demand, E, for the product is given by:

E = (dQ / Q) / (dP / P), or

E = (dQ / dP)(P / Q),

- where dQ and dP are small changes in the values of Q and P.

It can be shown that, for the simplified linear example,:

dQ / dP = − A so that E = − A(P / Q)

- and E will vary in value with different values of P and Q because as P increases the fraction P/Q will increase.

The terms "elastic" and "inelastic" are applied to commodities for which E is respectively numerically (ie ignoring the sign) greater or less than 1. If the elasticity of demand for a product is greater than 1, a price increase will lead to a fall in the amount PQ spent on the product, because demand Q will fall more than the rise in P. Conversely, if its elasticity is numerically less than 1, a price rise will result in a rise in the amount spent on it.

Cross elasticity of demand is defined as above in algebraic terms, except that Q is the quantity of one of the products that will be bought when P is the price of the other.