Calculating Marginal Revenue From a Demand Schedule

From a Demand Schedule to Revenue

A monopolist’s demand schedule connects each quantity customers will buy with the price required to sell that quantity. To study the revenue consequences of moving from one point on the schedule to another, calculate revenue at each point first. Then compare the change in revenue with the change in quantity.

The key distinction is that price is revenue per unit at a particular schedule point, while marginal revenue measures the change in total revenue across two points. When a seller must lower the price to sell more, the lower price applies not only to the additional units but also to units that could have been sold at the earlier, higher price.

Definition: For a single-price seller, total revenue at quantity $Q$ and price $P$ is $TR=P\times Q$. Between two points on a discrete demand schedule, marginal revenue per additional unit is the change in total revenue divided by the change in quantity: $MR=\Delta TR/\Delta Q$, where $\Delta TR=TR_2-TR_1$ and $\Delta Q=Q_2-Q_1>0$. This is an interval measure; it need not be the revenue from any one specific unit.

The schedule is assumed to report the price at which the seller can sell the stated quantity, with that same price charged for every unit sold at that point. The seller has no other source of revenue in the calculation. If the quantity changes by several units between listed points, the interval marginal revenue is the average change in revenue per additional unit over that interval. It does not, by itself, tell us the exact revenue change associated with each individual unit inside the interval.

Calculating Total and Marginal Revenue

For each row, multiply price by quantity to get total revenue. For each move from one row to the next, subtract the earlier total revenue from the later total revenue, then divide by the number of additional units. Keep the units clear: total revenue is measured in dollars, and marginal revenue is dollars per additional unit.

Worked Example: Completing a Discrete Revenue Schedule

A hypothetical seller faces the following demand schedule. Find total revenue at each quantity and marginal revenue for each interval. Then explain why marginal revenue falls as the seller expands sales.

Quantity $Q$	Price $P$	Total revenue $TR=P Q$	Interval marginal revenue
0	$30	$0	—
4	$24	$96	$24 per unit
8	$18	$144	$12 per unit
12	$12	$144	$0 per unit

Step 1: Calculate total revenue. At $Q=0$, revenue is $30\times0=\$0$. At $Q=4$, it is $24\times4=\$96$. At $Q=8$, it is $18\times8=\$144$. At $Q=12$, it is $12\times12=\$144$.

Step 2: Calculate interval marginal revenue. From 0 to 4 units, $\Delta TR=\$96-\$0=\$96$ and $\Delta Q=4$, so $MR=\$96/4=\$24$ per unit. From 4 to 8, $\Delta TR=\$144-\$96=\$48$ and $\Delta Q=4$, so $MR=\$48/4=\$12$ per unit. From 8 to 12, $\Delta TR=\$144-\$144=\$0$ and $\Delta Q=4$, so $MR=\$0/4=\$0$ per unit.

Interpretation and check. In each expansion, the schedule requires a lower price. Revenue increases by $96, then $48, then $0, although four units are added in each interval. The calculation checks: the last interval’s marginal revenue is zero because total revenue is unchanged, not because the units are sold for zero dollars. The common mistake is to equate marginal revenue with the listed price; at $Q=8$, for example, price is $18 but the interval marginal revenue from 4 to 8 is $12 per unit.

Why a Price Cut Reduces Revenue on Earlier Units

The relationship between price and marginal revenue follows directly from the change in total revenue. Suppose the seller moves from an earlier schedule point $(P_1,Q_1)$ to a later point $(P_2,Q_2)$, with $Q_2>Q_1$. The price falls, so $P_2<P_1$. Total revenue changes from $P_1Q_1$ to $P_2Q_2$.

\Delta TR=P_2Q_2-P_1Q_1 =P_2(Q_2-Q_1)+Q_1(P_2-P_1)

The first term, $P_2(Q_2-Q_1)$, is the revenue from the additional units when valued at the new, lower price. The second term, $Q_1(P_2-P_1)$, is the change in revenue on the $Q_1$ units that were already being sold. Because $P_2-P_1<0$, that second term is negative. Dividing the equation by the positive quantity change gives:

MR=P_2+Q_1\frac{P_2-P_1}{Q_2-Q_1}

Since $Q_1\geq0$, $P_2-P_1<0$, and $Q_2-Q_1>0$, the second term is nonpositive. If the earlier quantity is positive and the price cut is strict, the term is negative and interval marginal revenue is below the new price $P_2$. This is a finite-change result for the two schedule points, not a formula for an entire continuous demand curve.

Worked Example: Separating Added Revenue From Lost Revenue

A hypothetical seller can sell 10 units at $24 each or 16 units at $18 each. Calculate the total revenue at both points, the marginal revenue per additional unit, and the two effects of the price cut.

Step 1: Find total revenue at each point. At 10 units, $TR_1=\$24\times10=\$240$. At 16 units, $TR_2=\$18\times16=\$288$. The total change is $\Delta TR=\$288-\$240=\$48$, and the quantity change is $\Delta Q=16-10=6$.

Step 2: Calculate interval marginal revenue. $MR=\Delta TR/\Delta Q=\$48/6=\$8$ per additional unit.

Step 3: Separate the effects. The six additional units bring in $6\times\$18=\$108$ at the new price. But the price cut reduces revenue on the original ten units by $10\times(\$18-\$24)=10\times(-\$6)=-\$60$. Combining the effects gives $\$108-\$60=\$48$, exactly the change found from total revenue. Dividing by six additional units gives $\$48/6=\$8$ per unit.

Interpretation and check. The interval marginal revenue of $8 is below the new price of $18 because $60 of revenue is lost on the original units. The common mistake is to count only the $108 received from new units and report $108/6=\$18$ as marginal revenue. That omits the loss on earlier sales; the correct net change is $48.

Different Quantity Steps and the Average-Revenue Trap

Rows in a schedule do not have to be equally spaced. Always use the actual quantity difference for the interval being analyzed. Also distinguish marginal revenue from average revenue: at a positive quantity, average revenue is total revenue divided by quantity. Under the single-price assumption, $TR/Q=P$. That identity does not make price the marginal revenue from moving between schedule points.

Worked Example: Unequal Quantity Intervals

A hypothetical producer’s demand schedule includes the points shown below. Calculate total revenue and marginal revenue for each successive interval, and evaluate the claim that marginal revenue at seven units is $34 because the price there is $34.

Quantity $Q$	Price $P$	Total revenue
0	$50	$0
3	$42	$126
7	$34	$238
10	$25	$250

Step 1: Calculate total revenue. At 0 units, $TR=\$50\times0=\$0$. At 3 units, $TR=\$42\times3=\$126$. At 7 units, $TR=\$34\times7=\$238$. At 10 units, $TR=\$25\times10=\$250$.

Step 2: Calculate marginal revenue for each interval. From 0 to 3, the revenue change is $\$126-\$0=\$126$, over 3 units, so $MR=\$42$ per unit. From 3 to 7, the change is $\$238-\$126=\$112$, over $7-3=4$ units, so $MR=\$112/4=\$28$ per unit. From 7 to 10, the change is $\$250-\$238=\$12$, over $10-7=3$ units, so $MR=\$12/3=\$4$ per unit.

Step 3: Assess the claim. The $34 figure is the price at seven units. It is also average revenue at that point, since $\$238/7=\$34$. But the interval marginal revenue from 3 to 7 units is $28 per unit, not $34. Marginal revenue uses the change in total revenue divided by the change in quantity; average revenue uses total revenue divided by the quantity at one point.

Check. Total revenue rises by $112 over the four-unit interval, and $4\times\$28=\$112$. The final three-unit interval adds only $12 in total, and $3\times\$4=\$12$. The common mistake is dividing total revenue at seven units by seven and calling the result marginal revenue. That calculation gives price (average revenue), not the revenue change across an interval.

A Reliable Schedule Method

For a discrete schedule, a short, consistent sequence prevents most calculation errors:

Record the schedule points.
For each point, identify its price and quantity. Confirm which point is earlier and which is later in the comparison.

Calculate total revenue at each point.
Multiply that point’s price by its quantity. At zero quantity, total revenue is zero.

Find both changes.
Subtract earlier total revenue from later total revenue, and earlier quantity from later quantity. For an expansion, the quantity change should be positive.

Divide and interpret.
Divide the revenue change by the quantity change. State the result as dollars per additional unit over the interval, and remember that a price reduction also changes revenue on units already sold.

Marginal revenue can be positive, zero, or negative. It is positive when total revenue rises as quantity expands, zero when total revenue is unchanged, and negative when total revenue falls. Those descriptions are about total revenue across the chosen interval; they do not by themselves state whether producing the added units is profitable, because that would also require cost information.

Common Mistakes to Avoid

Using price as marginal revenue. Price is revenue per unit at a schedule point. Marginal revenue is the change in total revenue divided by the change in quantity.
Ignoring the units already sold. When the price falls, revenue on earlier units falls too. Calculate the net change in total revenue rather than counting only sales of additional units.
Dividing by the wrong quantity change. If quantity moves from 3 to 7, the change is 4, not 7. Use the interval width.
Confusing average and marginal revenue. Total revenue divided by quantity gives average revenue; for a single-price seller, that equals price. It is not interval marginal revenue.
Describing an interval result as an exact unit-by-unit result. A discrete schedule supports an average marginal revenue over the listed interval. It may not identify revenue from each individual unit in between.

Key takeaway: Calculate $TR=P Q$ at each schedule point, then calculate $MR=\Delta TR/\Delta Q$ between points. When expansion requires a price cut, the additional sales create revenue while the lower price reduces revenue on previously sold units; marginal revenue reflects both effects.

Check Your Understanding

For each schedule, calculate total revenue at the relevant quantities and interval marginal revenue. Treat price as applying to every unit sold at that schedule point.

A seller’s schedule lists $Q=0,3,6$ at prices $20, $17, and $14. Find total revenue at 3 and 6 units and marginal revenue on each interval. (Answer: total revenue is $51 at 3 units and $84 at 6 units; interval marginal revenues are $17 and $11 per unit.)
A seller can sell 2 units at $45 or 5 units at $33. Calculate total revenue at each point and marginal revenue from 2 to 5 units. (Answer: total revenue is $90 and $165; marginal revenue is $25 per additional unit.)
At one schedule point, quantity is 4 and price is $26; at the next, quantity is 9 and price is $20. Find both total revenues and the interval marginal revenue. (Answer: total revenues are $104 and $180; marginal revenue is $15.20 per additional unit.)
A schedule lists $Q=0,5,9$ at prices $32, $27, and $21. Calculate the total revenues and marginal revenues for both intervals. (Answer: total revenues are $0, $135, and $189; interval marginal revenues are $27 and $13.50 per unit.)
A seller moves from 4 units at $25 to 8 units at $19. Calculate the change in total revenue and marginal revenue. Then separate the added-unit revenue at the new price from the revenue lost on the first four units. (Answer: total revenue rises from $100 to $152, a change of $52; marginal revenue is $13 per additional unit; added-unit revenue is $76 and the loss on the first four units is $24.)

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