A Monopolist’s Demand Curve and Total Revenue

Why a Monopolist Faces Market Demand

A monopolist is the only seller in a relevant market, but that does not mean it can sell any quantity at any price. Buyers’ willingness to purchase constrains its choices. To sell more, the firm generally must offer a lower price; to charge a higher price, it generally must accept that fewer buyers will purchase. Understanding this constraint is the first step in describing monopoly revenue.

The relevant market identifies the product and the set of buyers being considered. Assume a single seller supplies that market, the product is the same for all buyers, and the seller charges one price per unit. Also hold other influences on buyers’ decisions fixed. Under these assumptions, the quantity the monopolist sells at any given price is the quantity demanded by the entire market at that price.

Definition: A market demand curve shows the total quantity all buyers in a defined market are willing and able to purchase at each possible price, holding other relevant factors fixed. A single-price monopolist serving that market faces this market demand curve: its sales at a price equal the market quantity demanded at that price.

This differs from the demand faced by a price-taking competitive firm. The earlier material on perfect competition distinguishes an individual firm’s demand from market demand: a competitive firm’s own sales are only a portion of the market’s total purchases. A monopolist, by contrast, is the market’s sole seller under the stated assumptions. It cannot sell to the whole market at a price above the market-clearing willingness to pay, nor can it sell more than buyers collectively demand at its chosen price.

If demand is first described for separate groups of buyers, market demand is found by adding their quantities at the same price. This is called horizontal summation. The price is held constant while the quantities purchased by the groups are added. Adding prices across groups would not produce market demand.

Price, Quantity, and Total Revenue

A demand schedule lists the quantity buyers will purchase at each specified price. An inverse demand relation instead gives the price associated with each quantity demanded. Either representation identifies feasible price-quantity combinations; neither allows the seller to select any price and any quantity independently.

Definition: Total revenue is the seller’s total sales receipts from selling $Q$ units at a per-unit price $P$. For a single price applying to every unit, total revenue is $TR=P\times Q$. If price is written as a function of quantity, $P(Q)$, then $TR(Q)=P(Q)Q$.

The multiplication follows directly from the meaning of revenue: each of the $Q$ units brings in $P$ dollars, so adding the receipts from all units gives $P+P+\cdots+P$, with $Q$ terms. Factoring out $P$ gives $P\times Q$. This calculation assumes the same price is charged for every unit and counts sales receipts only; it does not subtract production costs or other expenses.

TR=\underbrace{P+P+\cdots+P}_{Q\text{ units}}=P Q

A downward-sloping demand curve means that, other things held fixed, a greater quantity can be sold only at a lower price. It does not by itself tell us whether total revenue rises or falls as quantity rises. The price falls on every unit sold, while the firm sells more units. The net effect depends on the particular demand schedule or equation, so revenue must be calculated at the relevant price-quantity combinations.

Worked Example: Building Market Demand From Two Buyer Groups

A hypothetical sole seller serves two groups of customers. At a price of $18 per unit, Group A would buy 40 units and Group B would buy 25. At $15, their respective quantities would be 52 and 38. At $12, they would be 65 and 55. Find the market demand schedule and total revenue at each listed price. Assume the seller charges one price to both groups and quantities are measured in units.

Method and assumption. At each price, horizontally sum the two groups’ quantities. Then multiply that market quantity by the common price. The common-price assumption matters: these data do not describe a seller charging each group a different price.

At $18, market quantity is $40+25=65$ units. Total revenue is $18\times65=\$1{,}170$.

At $15, market quantity is $52+38=90$ units. Total revenue is $15\times90=\$1{,}350$.

At $12, market quantity is $65+55=120$ units. Total revenue is $12\times120=\$1{,}440$.

The resulting market schedule is: $18 corresponds to 65 units, $15 to 90 units, and $12 to 120 units. As price falls across these listed points, total revenue rises from $1,170 to $1,350 to $1,440. This conclusion is specific to these data; it is not a rule that every price reduction raises revenue.

Check. The quantities increase as price falls, consistent with the stated downward-sloping demand schedule. The revenue calculations use the combined market quantity at each price, not either group’s quantity alone. A common mistake avoided here is adding the groups’ quantities at different prices rather than at the same price.

Using Inverse Demand to Calculate Revenue

An inverse demand equation makes the price associated with a planned quantity explicit. Substituting the quantity into that equation gives the corresponding price; multiplying that price by quantity gives total revenue. This is a way to compare revenue at different feasible points, not a rule for selecting the firm’s best output. Choosing an output requires additional analysis of costs and is outside the scope of this tutorial.

Worked Example: Comparing Revenue Along a Linear Demand Curve

A hypothetical monopolist faces inverse demand $P(Q)=50-0.5Q$, where $P$ is dollars per unit and $Q$ is units per week. Assume this relation applies for quantities from 0 to 100, the seller charges one price for every unit, and all other demand conditions remain fixed. Calculate price and total revenue at $Q=20$, $Q=50$, and $Q=80$, then identify which of those three quantities gives the greatest revenue.

Method and assumption. Substitute each quantity into inverse demand to obtain its price, then apply $TR=PQ$. Each chosen quantity must be paired with the price specified by demand; quantity and price cannot be chosen independently.

At $Q=20$, the price is $50-(0.5\times20)=50-10=\$40$ per unit. Therefore, $TR=40\times20=\$800$ per week.

At $Q=50$, the price is $50-(0.5\times50)=50-25=\$25$ per unit. Therefore, $TR=25\times50=\$1{,}250$ per week.

At $Q=80$, the price is $50-(0.5\times80)=50-40=\$10$ per unit. Therefore, $TR=10\times80=\$800$ per week.

Of the three quantities tested, $Q=50$ gives the greatest total revenue: $1,250 per week, compared with $800 at either $Q=20$ or $Q=80$. The result does not claim that 50 units is the revenue-maximizing quantity over every possible quantity; it establishes only the comparison among the three specified alternatives.

Check. Each calculated price is nonnegative and each quantity is within the stated range. Substitution also verifies the demand pairs: at $Q=80$, for example, the equation gives a price of $10, and $10 multiplied by 80 is $800. The common mistake avoided is using the $40 price associated with 20 units to calculate revenue at 80 units.

Why a Downward Slope Does Not Determine Revenue’s Direction

The demand curve’s downward slope describes the relationship between price and quantity. Total revenue combines both values. Increasing quantity puts upward pressure on revenue because more units are sold; lowering price puts downward pressure on revenue because each unit brings in less. To determine the actual total, calculate $P Q$ at each point.

The following example illustrates this directly. It uses a discrete schedule, so the comparison is between the listed sales options; no claim is made about revenue at unlisted prices or quantities.

Worked Example: Tracking Revenue Across Price Reductions

A hypothetical seller has the following weekly demand schedule for a single-price product: at $24, buyers purchase 30 units; at $20, 42 units; at $16, 55 units; and at $12, 70 units. Calculate revenue at every listed price, compare the change in revenue from one option to the next, and state whether revenue rises throughout the schedule.

Method and assumption. For each option, multiply its listed price by its listed quantity. To compare adjacent options, subtract the earlier revenue from the next revenue. The demand schedule is assumed to apply to the same market and time period at each price.

At $24, $TR=24\times30=\$720$ per week.

At $20, $TR=20\times42=\$840$ per week. The change from the $24 option is $840-720=\$120$, an increase.

At $16, $TR=16\times55=\$880$ per week. The change from the $20 option is $880-840=\$40$, also an increase.

At $12, $TR=12\times70=\$840$ per week. The change from the $16 option is $840-880=-\$40$, a decrease of $40.

Thus, revenue rises over the first two price reductions, reaches its highest value among the listed options at $16, and then falls when price drops to $12. The quantity increases at every step, but the revenue pattern is not steadily increasing because the price reduction eventually outweighs the additional sales in the total-revenue calculation.

Check. The revenue sequence is $720, $840, $880, and $840, and the changes are $+\$120$, $+\$40$, and $-\$40$. The common mistake avoided is concluding that revenue must increase just because more units are sold.

Common Mistakes and Why the Distinction Matters

Using the wrong demand curve. Under the single-seller assumptions in this tutorial, the monopolist faces the market demand curve. A competitive firm’s individual demand is not the same thing as total market demand.
Adding quantities at different prices. To construct market demand from groups, add their quantities at one common price, then repeat at another price.
Treating price and quantity as independent choices. A selected quantity must be matched with the price buyers demand at that quantity, or a selected price with the quantity demanded at that price.
Assuming revenue always moves in one direction. A downward-sloping demand curve alone does not determine the direction of total-revenue changes. Calculate $P Q$ at each point.
Confusing revenue with profit. Total revenue is sales receipts. Profit also depends on the firm’s costs, which are not included in $P Q$.

These distinctions matter because a monopolist’s sales and receipts are constrained by buyers’ choices. Demand identifies which price-quantity pairs are feasible; total revenue measures the receipts associated with each pair. Keeping those ideas separate helps prevent a revenue calculation from silently assuming a price or quantity that the market will not support.

Key takeaway: A single-price monopolist serving a defined market faces that market’s demand curve. At each feasible point, total revenue is price multiplied by quantity. Because price and quantity move in opposite directions along downward-sloping demand, compare calculated revenues rather than inferring their direction from the slope alone.

Check Your Understanding

For each problem, use the stated demand information and assume a single price applies to all units unless the problem says otherwise.

At a price of $14, a monopolist sells 36 units. At $11, it sells 50 units. Calculate total revenue at both prices and the change in revenue. (Answer: $504 at $14; $550 at $11; revenue increases by $46.)
At $9, Group A demands 24 units and Group B demands 16 units. At $7, the groups demand 30 and 25 units. Find market quantity and total revenue at each price. (Answer: At $9, 40 units and $360; at $7, 55 units and $385.)
A seller faces inverse demand $P(Q)=36-0.4Q$, for quantities from 0 to 90. Find the price and total revenue at $Q=30$ and $Q=60$, and say which point has greater revenue. (Answer: At 30, price is $24 and revenue is $720; at 60, price is $12 and revenue is $720; the revenues are equal.)
A hypothetical demand schedule gives price-quantity pairs of $30 and 20 units, $25 and 28 units, and $20 and 35 units. Calculate revenue for each option and identify the highest-revenue option among those listed. (Answer: $600, $700, and $700, respectively; the $25 and $20 options tie for the highest listed revenue.)
A candidate calculates revenue at 40 units using an inverse-demand price of $18, then uses that same $18 price to calculate revenue at 70 units. The inverse demand equation is $P(Q)=30-0.3Q$. Correct the calculation and find revenue at both quantities. (Answer: At 40 units, price is $18 and revenue is $720; at 70 units, price is $9 and revenue is $630.)

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