Why Firms Measure Revenue in Three Ways
A firm needs to know more than the amount of money it receives from sales. It may also ask how much revenue it earns per unit sold and how much its total receipts change when it sells additional units. These are three different questions, answered by total revenue, average revenue, and marginal revenue.
Revenue is the value of output sold during a specified period. It is not the same as profit: revenue records sales receipts, while profit also depends on costs. The distinction matters when a firm compares a sales decision with the resources required to produce and sell the output.
In a single-price sale, the firm charges the same price \(P\) for every unit sold. Then total revenue is \(TR=PQ\), and average revenue is the realized revenue per unit. If the firm changes its price as its sales quantity changes, however, the price received on the additional unit need not equal marginal revenue. The price change may also alter receipts from units that would have been sold anyway.
What Each Measure Represents
| Measure | Question answered | Interpretation |
|---|---|---|
| Total revenue, \(TR\) | How much did sales bring in altogether? | All sales receipts over the stated period |
| Average revenue, \(AR\) | How much revenue was earned per unit sold on average? | Total revenue divided by quantity sold |
| Marginal revenue, \(MR\) | How much did revenue change as sales increased? | Change in total revenue per unit of the quantity change |
These measures use the same quantity and time period. For example, if a firm reports revenue for a month but quantity sold for a quarter, dividing the two does not produce a meaningful average revenue measure. Also, \(AR\) is not defined at \(Q=0\), because division by zero is undefined. A firm with no sales can have zero total revenue, but it cannot calculate revenue per unit sold from that zero-sales observation.
Why Price Equals Average Revenue at a Single Price
Suppose every unit sold receives the same price \(P\), and there are no other receipts included in the revenue measure. Selling \(Q\) units produces \(P\) dollars for each unit, so total revenue is \(PQ\). Dividing by the number of units gives average revenue:
Under this same fixed-price assumption, increasing sales from \(Q\) to \(Q+\Delta Q\) changes total revenue from \(PQ\) to \(P(Q+\Delta Q)\). Subtracting the original revenue gives the change in revenue, and dividing by the quantity change gives marginal revenue:
Thus, when the price stays constant for every unit and the firm can sell additional units at that price, both average revenue and marginal revenue equal price. This conclusion depends on the constant-price assumption. If a firm must lower its price on all units to sell more, marginal revenue can differ from the price charged at the new quantity and can even be negative.
Worked Example: Sales at a Constant Price
A hypothetical equipment supplier sells each replacement filter for \( \$31 \). It sells 25 filters in one week and considers a week in which it sells 26, with the price unchanged. Find total revenue at each quantity, average revenue at 26 filters, and the marginal revenue from the additional filter.
Method. Since the price is fixed at \( \$31 \) for every filter, use \(TR=PQ\). For the change from 25 to 26 filters, use \(MR=\Delta TR/\Delta Q\).
At 25 filters, total revenue is \(TR=31\times25=\$775\).
At 26 filters, total revenue is \(TR=31\times26=\$806\). Average revenue at 26 filters is \(AR=806/26=\$31\) per filter.
The change in total revenue is \(\Delta TR=806-775=\$31\), and the change in quantity is \(\Delta Q=26-25=1\). Therefore, \(MR=31/1=\$31\) per additional filter.
Check. The calculated average and marginal revenue both equal the unchanged price, as the constant-price result predicts. The common mistake this avoids is treating total revenue, \( \$806 \), as if it were revenue per filter.
When a Price Change Affects Earlier Sales
A firm that faces a downward-sloping demand relationship may need to reduce its price to sell more output. In a single-price arrangement, the lower price applies not just to the newly sold unit but also to the units that were already being sold. Total revenue can still rise, but the increase is less than the price of the additional unit would suggest. It can also fall if the price reduction on previous sales is large enough.
For a change from an initial quantity and price to a new quantity and price, calculate total revenue at each outcome first. Then find \(\Delta TR\) and divide by \(\Delta Q\). The new price alone is not enough to determine marginal revenue: the initial price and quantity are also needed.
Worked Example: A Price Reduction Raises Revenue by Less Than the New Price
A hypothetical specialty-food seller initially sells 10 boxes at \( \$50 \) per box. To sell more, it must charge one price to all buyers: \( \$48 \) when it sells 11 boxes, and \( \$45 \) when it sells 12 boxes. Find total and average revenue at each quantity, then find marginal revenue for each one-box increase.
Method. At each sales quantity, multiply that quantity by the price charged to all buyers. Then compare consecutive total-revenue amounts. This accounts for the price reduction on earlier sales.
At 10 boxes, \(TR=10\times50=\$500\), so \(AR=500/10=\$50\) per box.
At 11 boxes, \(TR=11\times48=\$528\), so \(AR=528/11=\$48\) per box. The revenue change from 10 to 11 boxes is \(\Delta TR=528-500=\$28\), with \(\Delta Q=1\). Thus, marginal revenue for this increase is \(MR=28/1=\$28\).
At 12 boxes, \(TR=12\times45=\$540\), so \(AR=540/12=\$45\) per box. The revenue change from 11 to 12 boxes is \(\Delta TR=540-528=\$12\), with \(\Delta Q=1\). Thus, marginal revenue for this increase is \(MR=12/1=\$12\).
Check. Total revenue rises from \( \$500 \) to \( \$528 \) to \( \$540 \), while average revenue falls from \( \$50 \) to \( \$48 \) to \( \$45 \). The marginal revenues, \( \$28 \) and \( \$12 \), are below the respective new prices because the lower price also applies to boxes that were already being sold. The common error is to report \( \$48 \) and \( \$45 \) as marginal revenue simply because those are the new prices.
Marginal Revenue Across a Multi-Unit Change
The discrete formula \(MR=\Delta TR/\Delta Q\) also applies when the quantity change is more than one unit. In that case, it gives the average change in revenue per added unit over the interval. It does not necessarily reveal how much revenue each individual unit contributed if revenue changes unevenly within that interval. State the quantity change explicitly so the units of the calculation remain clear.
Worked Example: Marginal Revenue Is Negative
A hypothetical event supplier sells 40 portable heaters at \( \$12 \) each. It can sell 42 heaters only if it lowers the single price to \( \$11 \) for all buyers. Calculate total revenue at both quantities, average revenue at 42 heaters, and marginal revenue over the two-unit increase.
Method. Calculate total revenue at each price-quantity combination, then divide the change in total revenue by the two-unit change in sales.
At 40 heaters, \(TR=40\times12=\$480\). At 42 heaters, \(TR=42\times11=\$462\). The change in revenue is \(\Delta TR=462-480=-\$18\), and the change in quantity is \(\Delta Q=42-40=2\).
Therefore, marginal revenue over this interval is \(MR=-18/2=-\$9\) per added heater. Average revenue at 42 heaters is \(AR=462/42=\$11\) per heater.
Check. The firm sells two additional heaters, but total revenue falls by \( \$18 \), so the average revenue change per added heater is negative. Average revenue remains positive because the firm still receives \( \$11 \) per heater at the new quantity. The common mistake this avoids is assuming that positive average revenue must imply positive marginal revenue.
Common Mistakes and Practical Checks
- Confusing revenue with profit. Total revenue is sales receipts. It does not subtract production or other costs.
- Using the new price as marginal revenue without checking the pricing rule. This works under a constant-price assumption, but not necessarily when a price change applies to all units.
- Dividing by the wrong quantity change. For a move from 40 to 42 units, divide the change in total revenue by 2, not by 1.
- Mixing units or periods. Keep the currency, output unit, and time interval consistent across the calculation.
- Calling average revenue marginal revenue. \(AR\) describes revenue per unit across all units sold; \(MR\) describes the change in total revenue associated with a change in sales.
Check Your Understanding
For each problem, keep the stated pricing arrangement and quantity interval in view.
- A hypothetical parts shop sells 25 cables at \( \$31 \) each. It sells 26 cables the next week at the same price. Find total revenue at both quantities, average revenue at 26, and marginal revenue for the increase. (Answer: \(TR_{25}=\$775\), \(TR_{26}=\$806\), \(AR_{26}=\$31\) per cable, and \(MR=\$31\) per added cable.)
- A single-price seller receives \( \$70 \) per unit when it sells 8 units and \( \$66 \) per unit when it sells 9. Find total revenue at both quantities and marginal revenue for the increase. (Answer: \(TR_8=\$560\), \(TR_9=\$594\), and \(MR=\$34\) per added unit.)
- A seller receives \( \$9.60 \) per unit for 50 units and must charge \( \$9.25 \) per unit to sell 52 units. Find total revenue at both quantities, average revenue at 52, and marginal revenue over the interval. (Answer: \(TR_{50}=\$480\), \(TR_{52}=\$481\), \(AR_{52}=\$9.25\) per unit, and \(MR=\$0.50\) per added unit.)
- A vendor sells 16 units at \( \$25 \) each, but can sell 18 units only by reducing the single price to \( \$22 \). Calculate total revenue at both outcomes and marginal revenue over the change. (Answer: \(TR_{16}=\$400\), \(TR_{18}=\$396\), and \(MR=-\$2\) per added unit.)
- Two hypothetical firms each increase sales by one unit. Firm A's sales rise from 100 units at \( \$14 \) each to 101 units at the same price. Firm B's sales rise from 100 units at \( \$15 \) each to 101 units at \( \$14.80 \) each, with the new price applying to all units. Find each firm's marginal revenue and identify which has negative marginal revenue. (Answer: Firm A has \(MR=\$14\); Firm B has \(TR\) changing from \( \$1{,}500 \) to \( \$1{,}494.80 \), so \(MR=-\$5.20\); Firm B has negative marginal revenue.)