Price Elasticity of Demand measured elasticity between two points with the midpoint method. With a demand formula and the Constant, Power and Sum Rules, we can measure it at a single point, and we see why it changes along a straight line. In this course elasticity keeps its sign: negative for a demand curve.
Key idea
$$\varepsilon = \frac{dQ}{dP}\cdot\frac{P}{Q}.$$ For linear demand $Q = a - bP$, $\varepsilon = \dfrac{-bP}{a - bP}$, which runs from 0 to $-\infty$ along the line and equals $-1$ at the midpoint. For constant-elasticity demand $Q = AP^{-k}$, $\varepsilon = -k$ at every price.
Why it works
Elasticity compares percent changes: $\dfrac{\Delta Q/Q}{\Delta P/P} = \dfrac{\Delta Q}{\Delta P}\cdot\dfrac{P}{Q}$. As the price step shrinks, $\Delta Q/\Delta P$ becomes the derivative and the midpoint becomes the point itself. Another way to write it: $\varepsilon = \dfrac{d\ln Q}{d\ln P}$, the slope on a log–log graph. For $Q = AP^{-k}$, $\ln Q = \ln A - k\ln P$ is a straight line of slope $-k$, which is why that curve has the same elasticity everywhere. That is also why economists often fit log–log regressions: the coefficient on $\ln P$ is an elasticity (see Logs, Dummies and Interactions in Econometrics).
Worked example: Linear demand
$Q = 100 - 4P$. Find $\varepsilon$ at $P = 15$.
- $\dfrac{dQ}{dP} = -4$, and $Q = 100 - 60 = 40$.
- $\varepsilon = -4\cdot\dfrac{15}{40} = -1.5$.
- Elastic: a 1% price rise cuts quantity by about 1.5%, so revenue falls.
Answer: $\varepsilon = -1.5$.
Worked example: Inverse demand
$P = 50 - 2Q$. Find $\varepsilon$ at $Q = 10$.
- At $Q = 10$, $P = 30$.
- The formula needs $dQ/dP$. Here $dP/dQ = -2$, so $dQ/dP = -\tfrac{1}{2}$.
- $\varepsilon = -\tfrac{1}{2}\cdot\dfrac{30}{10} = -1.5$.
Answer: $\varepsilon = -1.5$.
Worked example: Constant elasticity
$Q = 500P^{-2}$. Find $\varepsilon$ at $P = 5$ and at $P = 20$.
- $\dfrac{dQ}{dP} = -1000P^{-3}$.
- $\varepsilon = -1000P^{-3}\cdot\dfrac{P}{500P^{-2}} = -2$.
- The same at both prices: a 1% price rise always cuts quantity by about 2%.
Answer: $-2$ at every price.
Elasticity and revenue are linked by one line of calculus. Revenue is $R = PQ$, so $$\frac{dR}{dP} = Q + P\frac{dQ}{dP} = Q\,(1 + \varepsilon).$$ When demand is elastic ($\varepsilon < -1$) a price rise lowers revenue; when it is inelastic ($-1 < \varepsilon < 0$) a price rise raises revenue; at $\varepsilon = -1$ revenue is at its peak. This is the total revenue test, now with a proof.
Common mistake
Using the slope alone as the elasticity, or using $dP/dQ$ when demand is written as $P$ in terms of $Q$. The slope $-4$ of $Q = 100 - 4P$ is the same everywhere, but the elasticity is $-0.25$ at $P = 5$ and $-3$ at $P = 18.75$. And for $P = 50 - 2Q$ the slope you read off, $-2$, is $dP/dQ$; the formula needs its reciprocal. Check: elasticity has no units. If your answer would change when you measure price in cents instead of dollars, you used a slope, not an elasticity.