The Elasticity Markup Rule

Why Elasticity Determines the Monopoly Markup

A monopolist faces the market demand curve: selling more generally requires accepting a lower price. The key pricing question is therefore not just how much buyers value the product, but how strongly quantity demanded responds when price changes. The elasticity markup rule links that responsiveness to the gap between price and marginal cost.

The rule is a condition for an interior profit maximum, not a complete description of every market outcome. Its standard form assumes one price for all buyers, a differentiable downward-sloping demand curve, and a cost function with a well-defined marginal cost. The familiar conclusion about operating on the elastic portion of demand also uses a nonnegative marginal cost.

Definition: The point price elasticity of demand is the percentage change in quantity demanded associated with a small percentage change in price. At a point with price $P>0$ and quantity $Q>0$, it is $$ \varepsilon=\frac{dQ}{dP}\frac{P}{Q}. $$ For a downward-sloping demand curve, $\varepsilon<0$. Its absolute value, $|\varepsilon|$, is positive. The elasticity markup rule can be written using either the signed elasticity or its absolute value, but the sign must be handled consistently.

Deriving the Elasticity Markup Rule

Let inverse demand give the price $P(Q)$ at which the monopolist can sell quantity $Q$. Total revenue is $R(Q)=P(Q)Q$. Marginal revenue (MR) is the additional revenue from a small increase in output, so the product rule gives $MR=P+Q(dP/dQ)$. This is the marginal-revenue concept developed in the earlier material on deriving marginal revenue from inverse demand.

The demand elasticity is written in terms of $dQ/dP$, while the revenue expression uses $dP/dQ$. At a differentiable point where the demand slope is nonzero, these derivatives are reciprocals: $dP/dQ=1/(dQ/dP)$. Substituting the definition of elasticity converts the slope term into an elasticity term.

\varepsilon=\frac{dQ}{dP}\frac{P}{Q} \quad\Longrightarrow\quad \frac{dP}{dQ}=\frac{P}{\varepsilon Q}

Therefore marginal revenue can be expressed as price multiplied by a factor determined by elasticity:

MR=P+Q\frac{dP}{dQ} =P+\frac{P}{\varepsilon} =P\left(1+\frac{1}{\varepsilon}\right).

For a profit-maximizing monopolist choosing an interior output, the first-order condition is $MR=MC$, where $MC$ is marginal cost. Equating this condition to the expression above and rearranging gives the markup rule.

Theorem: At a differentiable interior profit maximum for a single-price monopolist, $$ \frac{P-MC}{P}=-\frac{1}{\varepsilon} =\frac{1}{|\varepsilon|}. $$ The left-hand side is the price-cost margin as a fraction of price, also called the Lerner index. With positive marginal cost at an interior maximum, $P>MC$, so the fractional markup is positive.

To see the rearrangement, start with $MC=P(1+1/\varepsilon)$. Subtract this expression from $P$, then divide by $P$:

P-MC=P-P\left(1+\frac{1}{\varepsilon}\right) =-\frac{P}{\varepsilon} \quad\Longrightarrow\quad \frac{P-MC}{P}=-\frac{1}{\varepsilon}.

Because demand slopes downward, $\varepsilon$ is negative, which makes $-1/\varepsilon$ positive. Equivalently, replacing the negative signed elasticity by its positive absolute value gives $1/|\varepsilon|$. Do not substitute the absolute value into the signed formula without also changing the sign.

Reading the Rule and Its Elasticity Implication

The markup rule says that a less elastic demand curve at the chosen point permits a larger price-cost margin as a fraction of price. For example, if $|\varepsilon|=4$, the rule gives a margin of one-quarter of price. It does not say that the price is only one-quarter above marginal cost: the denominator is price, not marginal cost. In this case, $MC$ is three-quarters of price.

The rule also explains the elastic-demand result. If demand is inelastic, then $-1<\varepsilon<0$, so $1+1/\varepsilon<0$ and marginal revenue is negative. If marginal cost is nonnegative, negative marginal revenue cannot equal marginal cost at a positive-output interior maximum. At unit elasticity, $\varepsilon=-1$, marginal revenue is zero; that can match marginal cost only when marginal cost is zero. Thus, with positive marginal cost and a differentiable interior maximum, the monopolist operates where $\varepsilon<-1$, the elastic portion of demand.

This conclusion is conditional. A zero-output choice, a binding capacity limit, a discontinuity or kink in demand, or other constraints can prevent the interior condition from applying. Negative marginal cost can also change the elastic-portion conclusion. The rule is not a claim that every point on a demand curve is a profit maximum.

Find the point elasticity.
Use the demand slope at the price and quantity in question: $\varepsilon=(dQ/dP)(P/Q)$. Keep the negative sign for downward-sloping demand.

Apply the correct form.
Use $MR=P(1+1/\varepsilon)$, or use the markup form $(P-MC)/P=1/|\varepsilon|$ when the interior profit-maximizing condition applies.

Check the economic conditions.
Confirm that the choice is an interior point, demand is differentiable, and the stated cost and market assumptions fit the calculation.

Applying the Rule

Worked Example: Checking a Candidate Price and Output

A hypothetical monopolist faces inverse demand $P=120-2Q$ and has total cost $C(Q)=40Q$. At $Q=20$, calculate price, point elasticity, and marginal revenue; then check whether this output satisfies the interior profit-maximizing condition. Name the method and a mistake it avoids.

Method. Use inverse demand to find the price, calculate elasticity from the demand slope, and then apply the marginal-revenue condition. This method avoids using the market-demand slope as though it were elasticity; elasticity also depends on the price and quantity at the point.

At $Q=20$, price is $P=120-2(20)=80$. Writing demand as $Q=60-0.5P$ shows that $dQ/dP=-0.5$. Therefore $\varepsilon=(-0.5)(80/20)=-2$, and $|\varepsilon|=2$.

Marginal cost is $MC=40$, since $C(Q)=40Q$. Marginal revenue from the elasticity expression is $MR=80(1+1/(-2))=80(0.5)=40$. Thus $MR=MC$, as required for an interior candidate.

The markup condition gives $(80-40)/80=40/80=0.5$, while $1/|\varepsilon|=1/2=0.5$. The two sides agree. To check that this candidate is a maximum in this example, profit is $\pi(Q)=(120-2Q)Q-40Q=80Q-2Q^2$. Its derivative is $80-4Q$, which is zero at $Q=20$, and its second derivative is $-4<0$. The candidate is a profit maximum.

Worked Example: Using Constant-Elasticity Demand

A hypothetical firm has demand $Q=2000P^{-2}$ and total cost $C(Q)=10Q$. At price $P=20$, calculate quantity, elasticity, marginal revenue, and profit. Verify that the point satisfies the interior condition and is a maximum.

Method. Use the demand function to calculate the point quantity and elasticity, then check the first- and second-order profit conditions. This avoids treating a stated elasticity as though it were automatically the elasticity at every price for any demand curve.

At $P=20$, quantity is $Q=2000/(20^2)=2000/400=5$. Differentiating the demand function gives $dQ/dP=-4000P^{-3}$. At $P=20$, this is $-4000/8000=-0.5$. Hence $\varepsilon=(-0.5)(20/5)=-2$.

The elasticity expression gives $MR=20(1+1/(-2))=10$. Since $C(Q)=10Q$, marginal cost is $MC=10$, so $MR=MC$. Revenue is $PQ=20(5)=100$, cost is $10(5)=50$, and profit is $100-50=50$.

For a maximum check, write inverse demand as $P=\sqrt{2000/Q}$. Then revenue is $R(Q)=Q\sqrt{2000/Q}=\sqrt{2000Q}$, so profit is $\pi(Q)=\sqrt{2000Q}-10Q$. Its first derivative is $\tfrac{1}{2}\sqrt{2000/Q}-10$, which at $Q=5$ is $10-10=0$. Its second derivative is negative for $Q>0$, so this stationary point is a maximum.

Worked Example: Inferring Price From Elasticity and Cost

A hypothetical service provider faces constant-elasticity demand $Q=46656P^{-3}$ and has total cost $C(Q)=24Q$. Find the interior profit-maximizing price and quantity, and calculate the price-cost margin. State why the constant-elasticity assumption matters.

Method. Use the markup condition with the demand elasticity, then substitute the resulting price into demand. This avoids confusing a percentage markup on price with a markup on marginal cost. The demand function matters because it establishes that elasticity is $-3$ at each price.

For this demand, $dQ/dP=-3(46656)P^{-4}$, so $\varepsilon=(dQ/dP)(P/Q)=-3$. Marginal cost is $24$. From the signed elasticity markup rule, $(P-24)/P=-1/(-3)=1/3$. Therefore $1-24/P=1/3$, so $24/P=2/3$, and $P=36$.

Quantity at this price is $Q=46656/(36^3)=46656/46656=1$. Marginal revenue is $MR=36(1+1/(-3))=36(2/3)=24=MC$. Revenue is $36(1)=36$, cost is $24(1)=24$, and profit is $12$.

The price-cost margin is $(36-24)/36=12/36=1/3$, matching $1/|\varepsilon|=1/3$. Since the elasticity is constant, the marginal-revenue condition gives $MR=2P/3$; as output rises, price falls and marginal revenue falls. The stationary choice is therefore a maximum for this demand and linear cost.

Common Mistakes and Practical Limits

A frequent exam trap is dropping the negative sign in the signed elasticity formula. Suppose $\varepsilon=-2$. The correct calculation is $MR=P(1+1/(-2))=P/2$. The incorrect calculation $P(1+1/2)=1.5P$ treats the signed elasticity as positive and gives the wrong result. If using the positive absolute elasticity instead, the equivalent expression is $MR=P(1-1/|\varepsilon|)$.

Another error is to apply the markup formula to a point that has not been shown to be profit-maximizing. The equality $(P-MC)/P=1/|\varepsilon|$ follows from the interior condition $MR=MC$. It is not an identity that must hold at every price and quantity. A candidate point can be tested with the relationship, but a full maximum claim may require checking the profit function, boundaries, and any constraints.

Finally, read the margin carefully. The Lerner index divides the difference between price and marginal cost by price. If the index is $1/4$, then marginal cost is $3/4$ of price; it does not mean price is $1/4$ above marginal cost. Rearranging the formula before substituting numbers is a useful way to prevent that denominator mistake.

Key takeaway: At an interior single-price monopoly optimum, the fractional price-cost margin equals the reciprocal of the absolute point elasticity of demand. With positive marginal cost, this condition requires elastic demand at the chosen point. Always preserve the signed elasticity in $MR=P(1+1/\varepsilon)$, or use the absolute-value form consistently.

Check Your Understanding

For each problem, state any condition needed to interpret the result as a profit-maximizing choice.

At a point, price is $30$, quantity is $60$, and $dQ/dP=-1.5$. Calculate elasticity and marginal revenue. If marginal cost is nonnegative, can this be an interior profit maximum? (Answer: $\varepsilon=-0.75$, $MR=-10$; no, not with nonnegative marginal cost.)
At an interior monopoly optimum, demand elasticity is $-4$ and price is $50$. Find marginal cost and the price-cost margin as a fraction of price. (Answer: $MC=37.50$; margin $=0.25$, or 25%.)
A firm has marginal cost $18$ at an interior optimum, and demand elasticity there is $-1.5$. Find price and the dollar markup $P-MC$. (Answer: $P=54$; markup $=36$.)
A hypothetical firm faces demand $Q=1000P^{-2}$ and constant marginal cost $8$. Find the interior optimum price and quantity, assuming the interior condition gives a maximum. (Answer: $P=16$, $Q=1000/256=3.90625$.)
At a current point, $P=40$, $Q=10$, and $dQ/dP=-0.5$. Marginal cost is $15$. Calculate elasticity and marginal revenue, then say whether a small increase in output raises or lowers profit at this point. (Answer: $\varepsilon=-2$, $MR=20$; a small output increase raises profit because $MR>MC$.)

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