Elasticity Along a Linear Demand Curve

A straight demand curve looks uniform, so it is tempting to think it bends the same way everywhere. It doesn't, because elasticity isn't slope. Elasticity equals slope times the ratio P/Q at a specific point. The slope (dQ/dP) is locked in for a line, but P/Q keeps changing as you slide along it. Near the top, price is high and quantity is tiny, so P/Q is huge and elasticity is huge: a small percent change in price is a massive percent change in quantity, because quantity started so small. Near the bottom, price is low and quantity is large, so P/Q shrinks toward zero and so does elasticity.

Take P = 20 - Q, so dQ/dP = -1. At Q = 2, P = 18, elasticity = 18/2 = 9, wildly elastic. At Q = 10, P = 10, elasticity = 10/10 = 1, unit elastic, exactly at the midpoint of the line. At Q = 18, P = 2, elasticity = 2/18, about 0.11, deeply inelastic. The pattern is symmetric: elastic above the midpoint, unit elastic at the midpoint, inelastic below it.

This midpoint is exactly where total revenue, P times Q, is maximized. Above it, demand is elastic, so cutting price raises revenue because quantity responds more than proportionally. Below it, demand is inelastic, so cutting price lowers revenue. That link between elasticity and the revenue-maximizing point is the real payoff of this topic, and it sets up later work on how elasticity determines who bears a tax and how much a price floor or ceiling distorts the market.