Business Mathematics and Statistics · Ch 6 — Applications of Differentiation (incl. Business/Economics applications, Maxima/Minima, Partial Derivatives)

Elasticity of Demand

A demand function relates the quantity demanded $x$ to the price $p$ (usually $x$ falls as $p$ rises). A manager rarely cares only about the direction of that relationship — a much sharper question is how sensitive demand is to a price change, because that sensitivity decides whether raising price raises or lowers total revenue. Price elasticity of demand measures exactly this sensitivity, as a ratio of proportional changes:

e_d = -\dfrac{p}{x}\cdot\dfrac{dx}{dp}

The minus sign is inserted by convention so that $e_d$ comes out positive for an ordinary downward-sloping demand curve (where $dx/dp$ is itself negative).

Note

Reading the Value of $e_d$

$e_d > 1$ : demand is elastic — quantity demanded changes proportionally more than price. A price rise reduces total revenue.
$e_d < 1$ : demand is inelastic — quantity changes proportionally less than price. A price rise increases total revenue.
$e_d = 1$ : demand is of unit elasticity — total revenue is unaffected by a small price change (this is exactly the output at which total revenue $R(x)=xp$ is maximised). …

Definition 6Price Elasticity of Demand (e_d)

A measure of the proportional responsiveness of quantity demanded to a proportional change in price, $e_d = -(p/x)(dx/dp)$ , always taken as a positive number for a normal …

Definition 7Elastic Demand

Demand with $e_d>1$ : quantity responds more than proportionally to a price change; a price increase red …

Definition 8Inelastic Demand

Demand with $e_d<1$ : quantity responds less than proportionally to a price change; a price increase ra …

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