How high is a monopoly's price above its marginal cost? Combine Monopoly with Calculus with Point Elasticity with Calculus and the answer depends on one number: how elastic demand is at the chosen price.
Key idea
At the profit-maximizing price, $$L = \frac{p - MC}{p} = \frac{1}{|\varepsilon|}.$$ $L$ is the Lerner index, the markup as a share of price. The less elastic demand is, the bigger the markup. A firm with positive marginal cost never chooses a price where demand is inelastic ($|\varepsilon| < 1$).
Why it works
Rewrite marginal revenue with the elasticity: $$MR = p + q\,\frac{dp}{dq} = p\left(1 + \frac{q}{p}\frac{dp}{dq}\right) = p\left(1 + \frac{1}{\varepsilon}\right) = p\left(1 - \frac{1}{|\varepsilon|}\right).$$ Set $MR = MC$: $p - MC = p/|\varepsilon|$, which is the rule. If $|\varepsilon| < 1$, MR is negative: cutting output a little would raise revenue and save cost, so that can't be the best choice. With $MC > 0$ the optimum needs $MR > 0$, so $|\varepsilon| > 1$.
Worked example: Price from the markup rule
At its optimum a firm faces $\varepsilon = -3$, and its marginal cost is \$20. What price does it charge?
- $\dfrac{p - 20}{p} = \dfrac{1}{3}$, so $p - 20 = p/3$.
- $\tfrac{2}{3}p = 20$, so $p = 30$.
Answer: \$30.
Worked example: Elasticity from prices
A profit-maximizing firm charges \$50 with marginal cost \$30. What is the elasticity of demand at that price?
- $L = \dfrac{50 - 30}{50} = 0.4$.
- $|\varepsilon| = 1/0.4 = 2.5$, so $\varepsilon = -2.5$.
Answer: $-2.5$.
Worked example: Checking with a linear demand
For $p = 100 - 2q$ and $MC = 20$, the optimum is $q = 20$, $p = 60$. Verify the rule.
- $\dfrac{dq}{dp} = -\tfrac{1}{2}$, so $\varepsilon = -\tfrac{1}{2}\cdot\dfrac{60}{20} = -1.5$.
- $L = \dfrac{60 - 20}{60} = \dfrac{2}{3} = \dfrac{1}{1.5}$.
Answer: It holds.
For constant-elasticity demand $q = Ap^{-k}$, the rule gives the price directly: $p = MC\cdot\dfrac{k}{k - 1}$. With $k = 2$ the price is twice marginal cost; with $k = 5$ it is 25% above; as $k$ grows the markup shrinks toward zero, the competitive limit.
Common mistake
Thinking a firm with market power picks a price where demand is inelastic, because "buyers can't do without it." If demand were inelastic at its price, a price rise would raise revenue while cutting the quantity it must produce: more revenue, less cost. It would keep raising the price until demand became elastic. Check: compute $|\varepsilon|$ at the claimed optimum. With $MC > 0$ it must exceed 1.
The Lerner index is a common measure of market power, with a caution: in industries with large fixed costs (software, medicines), prices well above marginal cost can go with zero economic profit, because the markup pays for the fixed cost. A high markup alone does not prove high profit.