Calculating Revenue Measures From Output Tables

Reading a Revenue Schedule

A revenue schedule reports how much a firm sells at several output levels and the price received at each level. The arithmetic is direct, but interpretation matters: when the price changes with output, total revenue changes because both the quantity sold and the price per unit may have changed. The relevant comparison is between the total revenues at the two output levels, not between the listed prices alone.

Assume each row gives the quantity sold during the same period and the uniform price received for every unit sold at that output level. Ignore taxes, refunds, and other adjustments unless the problem states otherwise. If a schedule instead gives different prices for separate blocks of units, total revenue must be calculated by adding the revenue from each block; multiplying total quantity by one price would not be appropriate.

Definition: At output $Q$, if the uniform price per unit is $P$, total revenue is $TR=PQ$. For positive output, average revenue is $AR=TR/Q$. Between two rows, marginal revenue is the change in total revenue divided by the change in output: $MR=\Delta TR/\Delta Q$. In a discrete table, this is an average revenue change per additional unit over that interval.

Keep the units visible while calculating. If price is dollars per unit and output is units per month, total revenue is dollars per month. Average revenue and interval marginal revenue are dollars per unit. At $Q=0$, total revenue can be zero, but average revenue is undefined because division by zero is not permitted.

From Table Rows to Revenue Measures

For every row, first multiply that row's output by its corresponding price. Then calculate average revenue, if requested, by dividing that row's total revenue by its output. To calculate marginal revenue between rows, subtract the earlier total revenue from the later total revenue and divide by the output increase. Output intervals need not be equal; the denominator must be the actual difference in quantities.

Worked Example: Changing Price Across Unequal Output Intervals

A small equipment supplier's schedule lists output of 2 units at $40 each, 4 units at $36 each, and 7 units at $30 each. Find total revenue at each row, marginal revenue from 2 to 4 and from 4 to 7 units, and average revenue at 7 units.

Method. Treat each listed price as the uniform price for all units sold at that row, calculate total revenue row by row, and then use the actual output change for each marginal-revenue interval.

At $Q=2$, total revenue is $TR=2\times \$40=\$80$. At $Q=4$, it is $TR=4\times \$36=\$144$. At $Q=7$, it is $TR=7\times \$30=\$210$.

From 2 to 4 units, total revenue rises by $\$144-\$80=\$64$, while output rises by $4-2=2$ units. Thus $MR=\$64/2=\$32$ per additional unit over that interval.

From 4 to 7 units, total revenue rises by $\$210-\$144=\$66$, while output rises by $7-4=3$ units. Thus $MR=\$66/3=\$22$ per additional unit over that interval.

At 7 units, average revenue is $AR=\$210/7=\$30$ per unit, equal to the listed price under the uniform-price assumption.

Check. The revenues, $\$80,\ \$144,\ \$210$, match the quantity-price products. The marginal-revenue intervals use output increases of 2 and 3, not the output levels themselves. The common mistake of dividing the first revenue increase by 4 would incorrectly give $\$16$, rather than $\$32$.

This example also shows why marginal revenue is not generally the listed price. In the first interval, the firm sells more units but receives a lower price on all units at the higher output level. Marginal revenue measures the net change in total receipts after both effects are included.

Why the Interval Formula Works

Let the first row have output $Q_1$ and price $P_1$, and the second have output $Q_2$ and price $P_2$. Under the uniform-price assumption, total revenue at each row is the product of its quantity and price. Subtracting those totals and dividing by the output change gives the interval measure:

MR_{1\to2} = \frac{TR_2-TR_1}{Q_2-Q_1} = \frac{P_2Q_2-P_1Q_1}{Q_2-Q_1}, \qquad Q_2>Q_1.

The numerator is the change in the firm's total receipts; the denominator is the number of additional units sold. Therefore the ratio is the average increase in total revenue per additional unit across the interval. If the price is unchanged, $P_1=P_2=P$, then $TR_2-TR_1=P(Q_2-Q_1)$, so interval marginal revenue equals $P$. If price changes, that equality need not hold.

A useful way to see both effects is to add and subtract $P_1Q_2$ in the revenue change:

$$ P_2Q_2-P_1Q_1 = P_1(Q_2-Q_1)+Q_2(P_2-P_1). $$

The first term values the added output at the earlier price. The second term captures the price change applied to all $Q_2$ units at the later row. This is an accounting decomposition, not an assumption that the firm literally sells the added units at the earlier price. It helps explain why a price reduction can offset some or all of the revenue from selling additional output.

Worked Example: Separate the Output and Price Effects

A hypothetical print shop's table gives output of 1, 4, 6, and 10 custom folders at prices of $52, $46, $42, and $35 per folder, respectively. Calculate total revenue at every row, marginal revenue on each interval, and average revenue at 10 folders. Then check the revenue change from 6 to 10 folders using the decomposition above.

Method. Multiply each quantity by its row's uniform price. For each interval, divide the difference in total revenue by the actual output increase.

The total revenues are:

At $Q=1$: $TR=1\times \$52=\$52$.
At $Q=4$: $TR=4\times \$46=\$184$.
At $Q=6$: $TR=6\times \$42=\$252$.
At $Q=10$: $TR=10\times \$35=\$350$.

From 1 to 4 folders, the revenue increase is $\$184-\$52=\$132$ over $4-1=3$ folders, so $MR=\$132/3=\$44$ per folder. From 4 to 6, it is $(\$252-\$184)/(6-4)=\$68/2=\$34$ per folder. From 6 to 10, it is $(\$350-\$252)/(10-6)=\$98/4=\$24.50$ per folder.

Average revenue at 10 folders is $AR=\$350/10=\$35$ per folder, which agrees with the listed price. For the last interval, the decomposition gives $42(10-6)+10(35-42)=168-70=\$98$. This equals the direct total-revenue change, $\$350-\$252=\$98$.

Check. The last interval adds four folders, and the price falls by $7 per folder. The $168 output contribution is partly offset by a $70 price effect on the 10 units at the new output level. Dividing the net $98 increase by four gives $24.50 per additional folder. The common mistake of reporting the new price, $35, as marginal revenue ignores that the total receipts increase by only $98.

Using a Schedule Carefully

A table's row labels and units determine what each calculation means. If output rises from 5 to 8 thousand units, the denominator is 3 thousand units, not 3 units and not 8 units. If revenue is reported in thousands of dollars, preserve that scale until the final interpretation. A ratio of thousands of dollars to thousands of units is still dollars per unit.

Average revenue is calculated at one output level; marginal revenue compares two output levels. For a uniform price schedule, average revenue equals the listed price at a positive output because $AR=PQ/Q=P$. This identity does not make average and marginal revenue the same measure. At a changing price, interval marginal revenue depends on both rows' total revenue.

Worked Example: Output Steps and Revenue Units

A hypothetical parts maker reports the following monthly schedule. Output is measured in hundreds of parts, and the listed price is dollars per part.

Output (hundreds of parts)	Price per part
0	$26
3	$24
5	$21
8	$18

Calculate total revenue at each positive output, marginal revenue from 0 to 3, 3 to 5, and 5 to 8 in dollars per part, and average revenue at output 5.

Method. Since output is stated in hundreds, convert each quantity to parts before multiplying by dollars per part. Then take the ratio of each revenue change to the number of additional parts.

At 3 hundred parts, output is 300 parts, so total revenue is $300\times \$24=\$7{,}200$. At 5 hundred parts, it is $500\times \$21=\$10{,}500$. At 8 hundred parts, it is $800\times \$18=\$14{,}400$. At zero output, revenue is zero.

From 0 to 300 parts, $MR=(\$7{,}200-\$0)/(300-0)=\$24$ per part. From 300 to 500 parts, $MR=(\$10{,}500-\$7{,}200)/(500-300)=\$3{,}300/200=\$16.50$ per part. From 500 to 800 parts, $MR=(\$14{,}400-\$10{,}500)/(800-500)=\$3{,}900/300=\$13$ per part.

At 500 parts, average revenue is $\$10{,}500/500=\$21$ per part, matching the row's price. The marginal-revenue answers are already in dollars per part because both numerator and denominator were converted to the corresponding full units.

Check. The revenue figures equal output in parts times price per part. The output increases are 300, 200, and 300 parts. Treating the displayed change from 3 to 5 as a change of only 2 individual parts would produce an incorrect unit calculation.

Common Errors and Their Corrections

Dividing by the later output rather than the output increase. For an interval from $Q_1$ to $Q_2$, use $Q_2-Q_1$. The example's change from 2 to 4 units has an output increase of 2, not 4.
Using the change in price as marginal revenue. A price change is not a revenue change. Calculate both total revenues first, then divide their difference by the output difference.
Using the listed price as marginal revenue when price changes. The listed price is average revenue under the uniform-price assumption. Marginal revenue is computed from the change in total revenue.
Assuming table intervals are equal. A revenue change over 3 additional units must be divided by 3, even if another interval in the same table spans 2 units.
Mixing measurement scales. Convert output consistently, or keep both revenue and quantity in stated scaled units and interpret the ratio correctly.
Applying one price to a block-price schedule. If each block has its own price, calculate each block's receipts and add them; the uniform-price formula $PQ$ requires one price for all units at that row.

Key takeaway: For a uniform-price output table, compute $TR=PQ$ row by row, $AR=TR/Q$ at a positive output, and interval $MR=\Delta TR/\Delta Q$. Always use the actual output difference and compare total revenues rather than prices alone.

Check Your Understanding

For each question, show the quantity-price products before calculating revenue per unit. Prices are uniform at each listed output unless stated otherwise.

A repair-kit seller lists 3 kits at $18 each and 6 kits at $15 each. Find total revenue at both output levels, interval marginal revenue, and average revenue at 6 kits. (Answer: total revenue $54 and $90; marginal revenue $12 per kit; average revenue $15 per kit.)
A ceramic studio lists outputs of 2, 5, and 9 mugs at prices of $50, $44, and $38 per mug. Find total revenue at each level and marginal revenue over both intervals. (Answer: total revenue $100, $220, and $342; marginal revenue $40 per mug from 2 to 5 and $30.50 per mug from 5 to 9.)
A delivery supplier lists 0, 4, and 7 hundred parcels at prices of $32, $29, and $25 per parcel. Calculate total revenue at the positive output levels and marginal revenue from 4 to 7 hundred parcels. (Answer: total revenue $11,600 and $17,500; marginal revenue $19.67 per parcel, rounded to the nearest cent.)
A firm sells 5 units at $24 each and 8 units at $20 each. Calculate total revenue at both levels and the interval marginal revenue. Explain briefly why it is not equal to the new price. (Answer: total revenue $120 and $160; marginal revenue $13.33 per additional unit, rounded to the nearest cent; it is not the new price because it is the change in total revenue divided by three added units.)
A schedule gives 3 units at $28 each. At 6 units, the average revenue is $23 per unit. Find total revenue at each output and marginal revenue from 3 to 6 units. (Answer: total revenue $84 and $138; marginal revenue $18 per additional unit.)

What You'll Learn