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:

e_p = − 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-

e_p = − 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 e_p = − 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-

e_p = (−5) (10/50) = -1

Similarly, at P = 8, Q = 100 − 5(8) = 60, and

e_p = (−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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