Elasticity of Demand

Demand for a good depends on its price: usually, the higher the price $p$, the smaller the quantity demanded $x$. The price elasticity of demand measures how responsive demand is to a change in price — a central idea in pricing decisions — and it is defined using the derivative $\dfrac{dx}{dp}$.

Let the demand $x$ be expressed as a function of price $p$, i.e. $x = f(p)$. The elasticity of demand is defined as

The factor $\dfrac{dx}{dp}$ is the rate of change of demand with price; multiplying by $\dfrac{p}{x}$ converts it into a ratio of proportional changes, so $\eta$ compares the percentage change in demand with the percentage change in price. The leading minus sign is included because $\dfrac{dx}{dp}$ is normally negative (demand falls as price rises), and this convention makes $\eta$ come out positive for an ordinary good.

Interpreting the value of $\eta$:

• If $\eta > 1$, demand is elastic — demand changes proportionally more than price. A small price rise causes a large drop in quantity, so total revenue falls when price rises.
• If $\eta < 1$, demand is inelastic — demand changes proportionally less than price. Quantity is relatively insensitive, so total revenue rises when price rises.
• If $\eta = 1$, demand is unit elastic — demand and price change in the same proportion; total revenue is (locally) unchanged by a small price change.

Method.

1. Write demand as $x = f(p)$ and differentiate to get $\dfrac{dx}{dp}$.
2. Substitute the given price $p$ and the corresponding demand $x$ into $\eta = -\dfrac{p}{x}\dfrac{dx}{dp}$. …