Deriving The Total-Revenue Test

Why Price and Revenue Can Move in Different Directions

A seller considering a price change needs to account for two effects: the price received per unit changes, and the quantity buyers demand may also change. Total revenue combines both effects. The total-revenue test uses demand elasticity to determine which effect dominates, without treating a price change as if quantity stayed fixed.

The test concerns revenue, not profit. It does not subtract production costs, taxes, or other expenses. Its result is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: If a product sells at price $P$ and quantity demanded is $Q$, total revenue is $R=PQ$. The signed point price elasticity of demand is $\varepsilon_d=(dQ/dP)(P/Q)$, where $dQ/dP$ is the slope of the demand relationship at the price considered. For a downward-sloping demand curve, $\varepsilon_d$ is negative. Its absolute value, $|\varepsilon_d|$, is the elasticity magnitude used to classify demand: elastic if $|\varepsilon_d|>1$, inelastic if $|\varepsilon_d|<1$, and unit-elastic if $|\varepsilon_d|=1$.

Assume quantity demanded is a differentiable function of price, $Q=Q(P)$, over the range being considered. Since revenue is the product $P Q(P)$, the product rule gives its rate of change as price changes.

\frac{dR}{dP} = Q+P\frac{dQ}{dP} = Q\left(1+\frac{P}{Q}\frac{dQ}{dP}\right) = Q(1+\varepsilon_d) = Q(1-|\varepsilon_d|)

The last equality uses downward-sloping demand, so $\varepsilon_d=-|\varepsilon_d|$. Because quantity is positive, the sign of $dR/dP$ is determined by whether the elasticity magnitude is below, equal to, or above one. In plain language, if quantity responds proportionally more than price, its response dominates; if it responds proportionally less, the price effect dominates.

Theorem: For a differentiable demand curve with positive quantity and negative price elasticity, if demand is inelastic at a point, a small price increase raises total revenue and a small price decrease lowers it. If demand is elastic, a small price increase lowers revenue and a small price decrease raises it. If demand is unit-elastic, the instantaneous change in revenue from a small price change is zero.

Write revenue as a product.
Use $R=PQ$, with quantity understood to respond to price.

Connect the change in quantity to elasticity.
By definition, $\varepsilon_d=(dQ/dP)(P/Q)$, so $dQ/dP=(Q/P)\varepsilon_d$.

Check the sign.
Substituting into the product-rule derivative gives $dR/dP=Q(1+\varepsilon_d)$. Compare the elasticity magnitude with one to find whether revenue rises or falls as price rises.

This is a local result: it describes the direction of revenue change for a small price movement at the point where elasticity is evaluated. For a larger movement, elasticity might change along the demand curve. In that case, compare endpoint revenues directly, or establish that the relevant elasticity classification holds throughout the interval.

Worked Applications

Worked Example: A Small Price Increase on Inelastic Demand

A hypothetical market has demand $Q=100-P$, where $P$ is dollars per unit and $Q$ is units per week. At a price of $40, determine the point elasticity, the local revenue effect of a price increase, and the actual revenue change if price rises to $41.

Method. First calculate quantity and point elasticity. Then use the derivative for the local direction and the demand equation for the finite change. This avoids assuming that the local derivative gives the exact revenue change for a one-dollar increase.

At $P=40$, quantity is $Q=100-40=60$. The slope of demand is $dQ/dP=-1$, so $\varepsilon_d=(-1)(40/60)=-2/3$. Thus $|\varepsilon_d|=2/3<1$: demand is inelastic at this point.

Initial revenue is $R_1=40(60)=\$2{,}400$ per week. The derivative is $dR/dP=Q(1-|\varepsilon_d|)=60(1-2/3)=20$. Its positive sign means a small price increase raises revenue at this point.

For the stated finite change, at $P=41$ demand is $Q=100-41=59$. Revenue is $R_2=41(59)=\$2{,}419$. The increase is $R_2-R_1=\$2{,}419-\$2{,}400=\$19$ per week. The actual increase is $19, not $20; the derivative describes the instantaneous rate at the starting price, not the exact one-dollar change.

Check. Quantity fell by one unit while price rose by $1, but the higher price more than offset the lost unit: the new revenue is $19 higher. This agrees with inelastic demand at the starting point.

Worked Example: Elasticity and Revenue at Two Observed Prices

A hypothetical supplier observes that at a price of $10, customers buy 200 units, while at $12 they buy 160 units. Calculate revenue at each price and use midpoint arc elasticity to assess whether the endpoint comparison is consistent with the total-revenue test.

Method. Calculate each endpoint’s revenue, then calculate midpoint arc elasticity. The midpoint calculation summarizes percentage changes using the average price and average quantity; it avoids comparing percentage changes against only one endpoint.

At the first price, revenue is $R_1=10(200)=\$2{,}000$. At the second price, revenue is $R_2=12(160)=\$1{,}920$. Thus price rose by $2 and revenue fell by $80.

For midpoint arc elasticity, the quantity change is $\Delta Q=160-200=-40$, and average quantity is $(200+160)/2=180$. The price change is $\Delta P=12-10=2$, and average price is $(10+12)/2=11$. Therefore,

\varepsilon_{\mathrm{arc}} = \frac{\Delta Q/[(Q_1+Q_2)/2]}{\Delta P/[(P_1+P_2)/2]} = \frac{-40/180}{2/11} = -\frac{2}{9}\cdot\frac{11}{2} = -\frac{11}{9}

The magnitude is $11/9$, or approximately 1.2222, which is greater than one. The arc is elastic by the midpoint measure. The price increase coincided with lower revenue, consistent with the total-revenue test.

Check. The endpoint revenue comparison and the arc-elasticity classification agree: revenue fell from $2,000 to $1,920 as price rose. This comparison does not imply that every point along every demand curve has the same elasticity.

Worked Example: Unit-Elastic Demand and Constant Revenue

A hypothetical market follows $Q=1{,}200/P$, with price in dollars per unit and quantity in units per week. Compare revenue at prices of $8 and $10, and verify the point elasticity.

Method. Calculate quantity and revenue at both prices, then derive the elasticity from the demand equation. This checks whether a unit-elastic classification corresponds to unchanged revenue in this specific model.

At $P=8$, quantity is $Q=1{,}200/8=150$, so revenue is $R=8(150)=\$1{,}200$ per week. At $P=10$, quantity is $Q=1{,}200/10=120$, so revenue is $R=10(120)=\$1{,}200$ per week.

Differentiating $Q=1{,}200/P$ gives $dQ/dP=-1{,}200/P^2$. Therefore, $\varepsilon_d=(-1{,}200/P^2)(P/(1{,}200/P))=-1$ at every positive price. Demand is unit-elastic throughout this model, and revenue is constant because $PQ=P(1{,}200/P)=1{,}200$.

Check. Both endpoint calculations give $1,200 per week. The unchanged revenue is not just a local approximation here: the specified demand equation makes revenue constant for every positive price.

Finite Changes and the Midpoint Comparison

For two observed price-and-quantity combinations, the exact revenue change is determined by the products $P_2Q_2$ and $P_1Q_1$. It is not generally enough to look at the percentage change in price alone. The midpoint arc-elasticity comparison has a useful exact relationship to those endpoint revenues when price and quantity move in opposite directions.

Suppose price rises from $P_1$ to $P_2$, while quantity falls from $Q_1$ to $Q_2$, with both prices and quantities positive. The midpoint elasticity magnitude exceeds one exactly when $(Q_1-Q_2)/[(Q_1+Q_2)/2]>(P_2-P_1)/[(P_1+P_2)/2]$. Since the averages are positive, multiplying through and simplifying gives $P_1Q_1>P_2Q_2$. Thus, for this midpoint convention, an elastic arc corresponds exactly to lower revenue at the higher-price endpoint; an inelastic arc corresponds to higher revenue; and a unit-elastic arc corresponds to equal revenue.

This endpoint relationship is distinct from the point-elasticity derivative. Point elasticity gives a local direction; midpoint arc elasticity compares two endpoints. Neither permits treating a demand curve as constant-elasticity unless that assumption is separately specified.

Common Mistake: Comparing Signed Elasticity Directly With One

A frequent error is to compare the negative signed elasticity with positive one without considering its absolute value. For example, if $\varepsilon_d=-1.4$, the incorrect reasoning “$-1.4<1$, so demand is inelastic” misclassifies demand. The correct calculation is $|\varepsilon_d|=1.4>1$, so demand is elastic. Equivalently, retain the signed value in $dR/dP=Q(1+\varepsilon_d)$: here $1+(-1.4)=-0.4<0$, so a small price increase lowers revenue.

Another trap is to say that unit elasticity means revenue is unchanged for every finite price change. The point result says the instantaneous derivative is zero at a unit-elastic point. Revenue remains constant across a range only if the demand relationship supports that conclusion throughout the range, as it does in the constant-elasticity example above. For other demand curves, compare the actual endpoint products.

Key takeaway: With downward-sloping demand, compare the elasticity magnitude with one. Below one, price and revenue move in the same direction locally; above one, they move in opposite directions locally; at one, the local revenue derivative is zero. For a finite change, calculate revenue at the endpoints or use a suitable arc comparison.

Check Your Understanding

For each problem, identify the relevant elasticity or revenue comparison and state the conclusion in context.

A market has demand $Q=90-3P$. At $P=20$, calculate quantity and point elasticity, classify demand, and state the local direction of revenue if price rises. (Answer: $Q=30$, $\varepsilon_d=-2$, elastic; a small price increase lowers revenue.)
A seller observes $P_1=\$15,\ Q_1=80$ and $P_2=\$18,\ Q_2=65$. Calculate both revenues and the midpoint arc-elasticity magnitude, then state whether the result is elastic or inelastic. (Answer: revenues are $1,200 and $1,170; midpoint magnitude is $33/29\approx1.1379$, elastic, and revenue falls as price rises.)
Demand is $Q=1{,}200/P$. Calculate quantities and revenues at $P=\$4$ and $P=\$6$, and verify the point elasticity. (Answer: quantities are 300 and 200; revenue is $1,200 at both prices; $\varepsilon_d=-1$, unit-elastic.)
Between two observations, price falls from $25 to $20 and quantity rises from 40 to 45. Calculate revenue at each endpoint and the midpoint arc-elasticity magnitude. (Answer: revenue rises from $1,000 to $900? Correction: $25(40)=\$1{,}000$ and $20(45)=\$900$, so revenue falls; midpoint magnitude is $9/17\approx0.5294$, inelastic.)
A market has demand $Q=180-2P$. Compare revenue at $P=\$30$ and $P=\$35$, and calculate point elasticity at the lower price. (Answer: revenue rises from $3,600 to $3,850; at $P=30$, $Q=120$ and $\varepsilon_d=-0.5$, so demand is inelastic.)

Deriving The Total-Revenue Test

What You'll Learn

Why Price and Revenue Can Move in Different Directions