Measuring Price Elasticity of Demand
Measuring Price Elasticity from a Linear Demand Function
Suppose a linear demand function is given as-
Q = a − b · P
Given the demand function, the formula for measuring price elasticity of demand (ep) through a demand function can be expressed as follows:
ep = − b · (P/Q)
(where b = ΔQ/ΔP).
The derivation of this is as follows:
Given the demand function, the total demand at a given price, say P1, can be estimated as-
Q1 = a − b · P1
When the price changes from P1 to P2, the total demand can be worked out as-
Q2 = a − b · P2
Given the formula for measuring the price elasticity, we need two ratios− ΔP/ΔQ and P/Q. Given the demand at two different prices, P1 and P2, the ratio ΔP/ΔQ can be obtained as follows:
By substituting −b for ΔQ/ΔP in the elasticity formula, we get-
ep = − b · (P/Q)
Alternatively, ΔQ/ΔP can be obtained (especially in the case of point elasticity) by differentiating the demand function Q = a − bP.
Here, −b denotes the decrease in quantity demanded when the price increases by Rs 1. Given a demand function, price elasticity can be expressed as ep = − b · (P/Q).
Given this formula, the price elasticity can be measured by substituting the numerical values for a, b, P and Q from an estimated demand function.
For a numerical example, suppose a demand function is given as-
Q = 100 − 5P
In this demand function, −5 denotes ΔQ/ΔP. This can be proved as follows. Now, the price elasticity for P = 10 can be obtained as follows-
At P = 10,
Q = [100 − 5 (10)] = 50
By substituting these values into the elasticity formula, we get-
ep = (−5) (10/50) = -1
Similarly, at P = 8, Q = 100 − 5(8) = 60, and
ep = (−5) (8/60) = −0.67
Measuring Price Elasticity from a Non-Linear Demand Function
Suppose a non-linear demand function of multiplicative form is given as-
Q = a · P-b
By differentiating the demand function, we get-
δQ/δP = −b · a · P-b-1
By substitution, the price elasticity formula given it can be written as-
Since Q = aP−b, by substitution, it can be written as-
This shows that the price elasticity coefficient in the case of a power demand function equals the power of price (P) and remains constant: it does not change with a change in price.
For a numerical example, suppose a non-linear demand function is given as-
Q = 5P−2
By differentiating the demand function, we get-
By multiplying it by Q/P, we get price elasticity as-
Since Q = 5P−2, by substitution, we get-
Thus, price elasticity in the case of a non-linear demand function equals the power of the variable price, P. In our example, price elasticity equals −2.
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