From Demand Response to Total Revenue
A price change affects revenue through two channels: the amount buyers pay per unit and the number of units they purchase. A price increase raises revenue per unit but may reduce sales; a price decrease lowers revenue per unit but may raise sales. To calculate the net effect, keep both changes in view rather than inferring the revenue outcome from price alone.
The earlier material on demand elasticity establishes how to express quantity responsiveness to price. Here the task is to combine that response with revenue arithmetic. The key practical point is that percentage changes in price and quantity do not generally add to give the exact percentage change in revenue. Their interaction also matters.
Demand elasticity is usually negative: when price rises, quantity demanded falls. Use its signed value when calculating a quantity response. For example, if \(\varepsilon_d=-0.4\) and price rises by 10% under a stated finite-change calculation, the implied quantity change is \((-0.4)(10\%)=-4\%\). Dropping the negative sign would incorrectly predict a quantity increase.
The Exact Revenue Calculation
Let \(p\) and \(q\) be the proportional changes defined above. Then \(P_1=P_0(1+p)\) and \(Q_1=Q_0(1+q)\). Substituting these into revenue gives an exact relationship:
Expanding the product shows why simply adding the two percentage changes is not exact:
The term \(pq\) is the interaction between the changes. When price rises and quantity falls, \(p\) is positive and \(q\) is negative, so this term is negative. When price falls and quantity rises, the signs reverse and the interaction is also negative. For small changes the interaction may be small, which is why adding the percentages can be a useful approximation, but it remains an approximation unless \(pq=0\).
If a finite-change elasticity is calculated using the initial-value percentages, then \(q=\varepsilon_d p\). This gives the exact revenue ratio for those observed changes:
This equation is an accounting identity once the observed \(p\) and \(q\) are known. Using it to predict a future outcome requires an additional assumption: the stated elasticity must describe the quantity response over the particular price change being considered.
Worked Example: Revenue Rises Even Though Quantity Falls
A hypothetical seller charges $12 per unit and sells 500 units in one week. In a later week, the price is $15 and sales are 440 units. Calculate initial and later revenue, the percentage changes, the finite-change demand elasticity using initial values, and the percentage change in revenue.
Method. Calculate revenue directly at each observation, then check the result with the exact percentage-change identity. The data are the two observed price-quantity pairs; no assumption about a demand curve is needed.
Initial revenue. \(R_0=12(500)=\$6{,}000\).
Later revenue. \(R_1=15(440)=\$6{,}600\). The dollar change is \(\$6{,}600-\$6{,}000=\$600\), so the percentage change is \(\$600/\$6{,}000=0.10=10\%\).
Price and quantity changes. The price change is \((15-12)/12=3/12=0.25=25\%\). The quantity change is \((440-500)/500=-60/500=-0.12=-12\%\).
Finite-change elasticity. \(\varepsilon_d=(-0.12)/(0.25)=-0.48\). With these observed changes, price rose by 25% and quantity fell by 12%.
Check using the exact identity. \(p+q+pq=0.25-0.12+(0.25)(-0.12)=0.13-0.03=0.10\), or a 10% revenue increase. Applying this to initial revenue gives \(\$6{,}000(1.10)=\$6{,}600\), matching the direct calculation.
Common mistake avoided. Treating the 25% price rise as a 25% revenue rise ignores the sales decline. Adding \(25\%\) and \(-12\%\) gives 13%, which is also not exact: it omits the \(-3\%\) interaction term.
Using Elasticity to Predict a Finite Revenue Change
In a prediction problem, elasticity may be supplied rather than calculated from two observations. Read carefully how it is intended to apply. If a problem says to use a finite-change elasticity based on initial values, calculate the quantity change as \(q=\varepsilon_d p\), then calculate revenue with \((1+p)(1+q)\). This procedure is exact under that stated response assumption; it is not a universal rule that a point elasticity must produce a quantity change equal to elasticity times any finite price change.
For small changes, the total-revenue test gives a useful local result. Starting from \(R=PQ\), a small change satisfies \(dR=Q\,dP+P\,dQ\). Dividing by \(R=PQ\) gives \(dR/R=dP/P+dQ/Q\). Point price elasticity of demand is \(\varepsilon_d=(dQ/Q)/(dP/P)\), so \(dQ/Q=\varepsilon_d(dP/P)\). Substitution yields:
This derivation explains the local revenue test: near a particular price and quantity, a price rise tends to raise revenue when demand is inelastic, and a price fall tends to raise revenue when demand is elastic. The expression uses small changes at the point where elasticity is measured. For a sizeable price change, do not automatically treat this differential result as an exact percentage change.
Worked Example: A Predicted Quantity Response and Its Revenue Effect
A hypothetical service currently charges $40 per month and has 300 subscribers. For a proposed small-to-moderate price reduction, analysts use a finite-change elasticity of \(-1.4\), applied to the percentage price change relative to the initial price. The price falls by 5%. Calculate the predicted quantity, new revenue, and revenue percentage change under this assumption.
Method. Use the given finite-change elasticity to calculate the quantity response, then multiply the new price by the new quantity. The specified response convention is essential: this is not a claim that a local point elasticity is exact for every finite change.
New price. A 5% decrease from $40 is \(0.05(40)=\$2\), so \(P_1=\$40-\$2=\$38\), and \(p=-0.05\).
Quantity response. \(q=\varepsilon_d p=(-1.4)(-0.05)=0.07\), or a 7% increase. Thus \(Q_1=300(1.07)=321\) subscribers.
Revenue. Initial revenue is \(R_0=40(300)=\$12{,}000\). New revenue is \(R_1=38(321)=\$12{,}198\). The increase is \(\$198\), which as a fraction of initial revenue is \(\$198/\$12{,}000=0.0165=1.65\%\).
Check using the exact identity. \(p+q+pq=-0.05+0.07+(-0.05)(0.07)=0.02-0.0035=0.0165\), confirming the 1.65% increase.
Common mistake avoided. The first-order revenue test gives \((1+\varepsilon_d)p=(1-1.4)(-0.05)=0.02\), or a 2% increase. That is a local approximation here, not the exact result under the finite-change response assumption; the interaction reduces the calculated increase to 1.65%.
When Elasticity Is Assumed Constant Across the Interval
A different prediction model assumes that point elasticity stays constant at every price between the starting and ending observations. This assumption implies a particular quantity response, not the simple finite-change rule \(q=\varepsilon_d p\). With constant signed elasticity, the definition in differential form is \(dQ/Q=\varepsilon_d\,dP/P\). Integrating from the initial to the final price gives:
Since \(R=PQ\), revenue then changes according to \(R_1/R_0=(P_1/P_0)^{1+\varepsilon_d}\). This result is exact only under the constant-point-elasticity model over the whole interval. If elasticity is only known at the initial point, the model does not establish the exact quantity response to a large price change.
Worked Example: Revenue Under a Constant-Elasticity Model
A hypothetical product sells for $8 per unit, with 900 units sold. Suppose demand has constant point elasticity \(\varepsilon_d=-2\) over the price interval. The price rises to $12. Calculate the predicted quantity and total revenue at the new price.
Method. Apply the constant-elasticity response \(Q_1/Q_0=(P_1/P_0)^{\varepsilon_d}\), then check revenue directly. The assumption here is constant elasticity throughout the interval.
Price ratio. \(P_1/P_0=12/8=1.5=3/2\).
Quantity ratio. \(Q_1/Q_0=(3/2)^{-2}=(2/3)^2=4/9\). Therefore \(Q_1=900(4/9)=100(4)=400\) units.
Revenue. Initial revenue is \(R_0=8(900)=\$7{,}200\). New revenue is \(R_1=12(400)=\$4{,}800\), a decrease of \(\$2{,}400\). Relative to initial revenue, the change is \(-\$2{,}400/\$7{,}200=-1/3\), or \(-33.33\%\) rounded to two decimal places.
Check with the model's revenue ratio. \(R_1/R_0=(3/2)^{1+(-2)}=(3/2)^{-1}=2/3\). Thus \(R_1=(2/3)(\$7{,}200)=\$4{,}800\), as calculated directly.
Common mistake avoided. Substituting the 50% price increase into \(q=\varepsilon_d p\) would predict a 100% quantity decrease. That formula is not the finite response rule for a constant point elasticity of \(-2\); the specified model instead gives quantity at \(4/9\) of its initial level.
Choosing the Right Calculation
Before calculating, identify whether the problem gives observed endpoints, a finite-change elasticity convention, or a constant point-elasticity model. These are different sources of information and should not be interchanged.
| Information given | Appropriate calculation | Important qualification |
|---|---|---|
| Initial and final prices and quantities | Calculate both revenues directly; use \(p+q+pq\) as a check. | No demand model is needed for the observed revenue change. |
| A finite-change elasticity applied to initial-value percentages | Set \(q=\varepsilon_d p\), then use \(R_1/R_0=(1+p)(1+q)\). | The stated elasticity convention must apply to the finite change being predicted. |
| Constant point elasticity over the interval | Use \(Q_1/Q_0=(P_1/P_0)^{\varepsilon_d}\) and \(R_1/R_0=(P_1/P_0)^{1+\varepsilon_d}\). | This requires elasticity to remain constant throughout the interval. |
A recurring trap is mixing a percentage-change convention with an elasticity estimate defined by a different method. For example, an arc elasticity based on midpoint percentages is not automatically the same as \(q/p\) using initial values. Use the convention specified in the problem; if none is specified and actual endpoint data are available, calculating revenues directly avoids ambiguity.
Total-revenue calculations matter whenever a decision depends on sales receipts rather than price or quantity alone. In a hypothetical insurance setting, for example, a provider considering a change in a monthly service fee would need to consider both the fee and the number of customers choosing the service. The revenue calculation concerns gross receipts only; it does not by itself measure costs, profit, customer welfare, or other effects. The useful habit is to state the response assumption, calculate the new quantity consistently with it, and then multiply the new price and quantity.
Check Your Understanding
For each problem, identify the response convention before calculating revenue.
- A hypothetical seller raises price from $25 to $30 while weekly sales fall from 200 units to 184 units. Find initial and final revenue, the finite-change elasticity using initial-value percentages, and the percentage revenue change. (Answer: $5,000 initially; $5,520 finally; elasticity \(-0.40\); revenue rises 10.4%.)
- A subscription costs $50 and has 400 customers. Price falls by 10%, and a finite-change elasticity of \(-1.3\) is applied to the price change measured from the initial price. Find new customer count and the percentage revenue change. (Answer: 452 customers; revenue rises 1.7%, from $20,000 to $20,340.)
- A product's price changes from $16 to $20, and quantity sold changes from 250 to 225. Calculate initial and final revenue and the observed finite-change elasticity using initial-value percentages. (Answer: $4,000 initially; $4,500 finally; elasticity \(-0.40\); revenue rises 12.5%.)
- A hypothetical product sells for $10 with 360 units sold. Assume constant point elasticity \(-2\) over a price increase to $15. Calculate predicted quantity and percentage revenue change. (Answer: 160 units; revenue falls from $3,600 to $2,400, a decrease of 33.33%.)
- Price rises by 8% and quantity falls by 6%, measured relative to initial values. Initial revenue is $25,000. Calculate final revenue and compare the exact percentage change with the result from adding the two percentages. (Answer: $25,380; exact increase 1.52%, whereas adding the changes gives 2.00%.)