Marginal Revenue and the Price-Cut Effect

For linear demand, MR has twice the slope.

Review Four Quantity and Price Decisions

The worked cases assume a single-price monopolist with a feasible, smooth interior choice and the usual MR–MC crossing from above. Equality is a continuous-output condition, not a requirement for every integer schedule. Check boundaries, total profit and participation when those assumptions do not hold.

Separate the New Sale from the Price Cut

Suppose a seller can sell three units at $10 or four at $9. Revenue rises from $30 to $36, so MR of the fourth unit is $6, below its $9 price. The seller gains $9 from the new unit but loses $1 on each of the previous three. This is why MR lies below demand.

For a single-price seller, lowering price applies to all units sold. The marginal revenue of the fourth unit can be decomposed:

MR = $9 from the new unit – (3 earlier units × $1 lost per unit) = $6.

Compare Sales Plans, Not Past Refunds

The price-cut effect compares two alternative uniform-price sales plans for the same period. It does not assume that a seller refunds earlier completed purchases. It is absent for a price-taking firm because the firm treats its selling price as fixed when varying its own output.

Derive MR for Linear Demand

For inverse linear demand P=a-bQ, with positive a and b and continuous Q, total revenue is TR=aQ-bQ^2, so its derivative is MR=a-2bQ. MR shares the price intercept and has twice the negative slope. Demand reaches zero price at Q = a/b, while derivative MR reaches zero at Q = a/(2b). These slope and intercept rules concern linear demand; do not apply them to arbitrary curves.

Use the Interior Output Rule Carefully

Choose Q_m where MR meets MC under the stated interior assumptions. Then move vertically to demand for P_m. Moving to demand first or setting demand equal to MC usually selects a different quantity. Profit at positive output is (P_m-ATC)Q_m; subtracting total cost from revenue gives the same result.

Distinguish the Efficiency Benchmark

When demand measures marginal benefit and MC measures marginal social cost, with no external effects or other distortions, their intersection is the allocatively efficient quantity. For a positive-cost interior monopoly with the usual curve shapes, MR lies below demand and selects a smaller quantity. This efficiency benchmark does not by itself establish whether the firm can recover fixed cost.

Choose Quantity Before Reading Price

In a continuous-output example, MR equals MC at 30 units and $12. Demand shows consumers will pay $20 for 30. ATC is $15. The monopolist charges $20 and earns (20-15)30=$150. It does not charge the $12 marginal revenue. The equations in the reasoning check below make these values consistent.

Here $12 is the local rate of change in revenue with respect to output, after accounting for the price response; it is not a posted selling price or automatically the exact revenue change for one whole unit. Charging $12 would correspond to a larger quantity on the downward-sloping demand curve. Keep a vertical guide from Q_m to demand.

Keep the Four Values Separate

Classify the Cost Change

If an unavoidable fixed cost rises, ATC rises at each positive output and profit falls, while the best operating output and price remain unchanged when MR and MC do not change. An avoidable operating fee can affect entry or exit. A per-unit tax changes effective MC and generally changes an interior choice; integer output may stay at the same row.

Check Profit and Shutdown

Monopoly power does not guarantee profit. A seller may earn a profit, break even or minimize a loss. If the interior conditions apply, MR = MC identifies the best operating candidate, but compare boundaries and shutdown as well. With fixed cost unavoidable in the short run, operating can beat shutdown when revenue covers variable cost even if total profit is negative.

Distinguish a Finite Step from a Derivative

For a one-unit step from Q − 1 to Q on the linear demand above, the revenue change is a − b(2Q − 1), whereas derivative MR evaluated at Q is a − 2bQ. They differ by b because the finite change covers an interval rather than an infinitesimal adjustment. Label which version a question gives before matching it to a cost change.

Compare Revenue and Profit Peaks

For continuous linear demand with an interior revenue maximum, derivative MR is zero there. With positive MC and the usual crossing, the interior profit maximum occurs at a smaller quantity. For listed integer choices, compare actual totals: ties and rounding can make simple graph shortcuts misleading. Maximizing revenue alone ignores production cost.

Read the Revenue and Output Explanation

OpenStax’s monopoly lesson explains uniform-price sales plans, the revenue effect of a lower price, and the separate quantity and price decisions. The following schedule and calculations are original classroom examples, not copied source exercises.

Test an Integer-Output Schedule

Let demand be P = 18 − Q and total cost be TC = $6 + $10Q. Output is a nonnegative integer, price cannot be negative, and the $6 fixed cost is unavoidable for this period. The MR column reports the exact revenue change from the preceding integer row, not a derivative at the row’s endpoint.

Moving from three to four units adds $11 revenue: $14 from the fourth unit minus a $1 price reduction on each of the other three. The fourth unit adds $1 profit after its $10 marginal cost. The fifth adds only $9 revenue against $10 cost, so it reduces profit. Q = 4 maximizes profit at a $14 price even though its one-unit MR entry is $11, not exactly the $10 MC.

The whole feasible range confirms the choice: profit equals 10 − (Q − 4)² for every integer Q from 0 through 18, so no unlisted larger quantity earns more than $10. Revenue keeps rising after Q = 4, which also shows why a revenue maximum need not be the profit maximum.

Check the Derivative and the $150 Example

In the integer schedule, what is derivative MR at Q = 4 if output were continuous? Separately, can the earlier 30-unit example consistently have price $20, marginal revenue $12, average total cost $15 and profit $150?

Try Three Marginal Revenue Questions

Connect Revenue, Schedules and Output

Practice reading a complete monopoly schedule, compare demand and cost in monopoly output choices, or continue with the CLEP Microeconomics learning hub.