Elasticity is always a ratio between two percentage changes. For a general expression, y = f(x), the elasticity of y with respect to x is:
Note that the term ?y/?x shows the marginal effect, or slope of the function in the x-direction, and (y0, x0) is the starting point. For a linear function the slope is always constant, but the elasticity is not constant because it varies with the starting point.
As an example, consider the following demand function for beer:
QD = 100 − 2. p + 0.05. ps − 0.25 . pc − 0.00125 . Y
QD is here expressed in a volume measure (liters or bottles or cans etc) per month and prices and income in money (kr, €, $, £ etc). The substitute good maybe wine and the complement good sausage. If p = 10kr/bottle, ps = 50kr/bottle, pc = 30kr/kilo and Y = 20000kr/month. Inserting these numbers:
QD = 100 – 2 x 10 + 0.05 x 50 − 0.25 x 30 − 0.00125 x 20000 = 50,
i.e., this household, or consumer, demands 50liters of beer per month. Now, the demand curve is a graph which shows the relationship between the own price and quantity demanded, so if we aggregate all the numbers above except for the price we get:
Q = (100 + 0.05 x 50 − 0.25x 30 − 0.00125 x 20000) − 2 x pQ = 70 − 2. p
QD = 70 − 2. p
Given that the assumed value of the all other variables except the price. The marginal effect is −2, and if we start at the equilibrium we calculated initially,
i.e. p0 = 10, Q0 = 50, the price elasticity is,
This means that if the price changes by one percent, the quantity demanded changes by 0.4 percent. If the own price elasticity is less than 1 (after changing sign, since the marginal effect is always negative), we say that demand is inelastic.
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