Solving Linear-Demand Monopoly Problems Algebraically

From Linear Demand to a Monopoly Solution

A monopoly problem often gives a demand curve and a cost rule, then asks for more than just the quantity produced. A complete solution finds the profit-maximizing quantity, the price buyers pay, total revenue, total cost, and profit. The algebra is systematic: express demand as price in terms of quantity, derive marginal revenue from total revenue, set marginal revenue equal to marginal cost for an interior optimum, and use the demand curve to recover price.

The method here applies to a single seller charging one price per unit. The firm faces the market demand curve, and it cannot choose price and quantity independently: the quantity it sells determines the price buyers are willing to pay. We assume the demand and cost functions are valid over the quantity range considered, and that the firm is choosing an interior output unless a boundary case is explicitly noted. This algebraic approach is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Model: Let inverse demand be $P(Q)=a-bQ$, where $a>0$, $b>0$, and $Q\geq 0$. Total revenue is $TR(Q)=P(Q)Q$. Let total cost be $C(Q)=F+cQ+\frac{k}{2}Q^2$, where $F$ is fixed cost, $c$ is marginal cost at zero output, and $k\geq 0$ describes how marginal cost changes with output. Thus $MC(Q)=c+kQ$. Constant marginal cost is the special case $k=0$.

The model treats the firm as a single-price monopolist: all units sold have the same price, as given by demand. The cost function’s fixed-cost term does not change marginal cost, because it does not change when output changes. For the interior output calculation, we assume the firm is operating and the fixed cost is incurred whether it produces a little more or a little less. Whether the firm should shut down or exit when fixed costs can be avoided is a separate decision.

Deriving Marginal Revenue and the Output Formula

Start with inverse demand. Multiplying price by quantity gives total revenue. Differentiating total revenue with respect to quantity gives marginal revenue. For linear inverse demand, this makes marginal revenue a straight line with the same vertical intercept as demand and twice its slope magnitude.

\begin{aligned} P(Q)&=a-bQ,\\ TR(Q)&=P(Q)Q=(a-bQ)Q=aQ-bQ^2,\\ MR(Q)&=\frac{dTR}{dQ}=a-2bQ. \end{aligned}

Marginal cost follows from differentiating total cost: $MC(Q)=c+kQ$. At an interior profit maximum, the marginal gain from one more unit, $MR$, equals the marginal cost of that unit, $MC$. Equating these expressions gives an equation in one unknown, $Q$.

a-2bQ=c+kQ \quad\Longrightarrow\quad Q^*=\frac{a-c}{2b+k}.

This solution is an interior output when $a>c$, given $b>0$ and $k\geq 0$. If $a\leq c$, the algebraic expression is nonpositive; the unconstrained interior formula is not a feasible positive output. In a problem that restricts output to $Q\geq0$, the best operating output is then at the zero-output boundary. A separate shutdown comparison may still be needed if fixed cost is avoidable.

Why the interior solution is a maximum: Profit is $\pi(Q)=TR(Q)-C(Q)$. Its first derivative is $MR-MC$, and its second derivative under this model is $-2b-k$. Since $b>0$ and $k\geq0$, $-2b-k<0$. Profit is therefore strictly concave in output, so an interior quantity satisfying $MR=MC$ is the unique profit maximum.

The equation $MR=MC$ determines quantity, not price. To find price, substitute the chosen output into the original inverse-demand equation. Then calculate revenue as $P^*Q^*$, total cost from the specified cost function, and profit as revenue minus total cost. Keeping these calculations separate is a useful check on the solution.

A Reliable Calculation Sequence

Use the following sequence to keep the demand, revenue, and cost calculations distinct. In particular, do not substitute marginal revenue for the price charged.

Write inverse demand and total cost.
Record $P(Q)=a-bQ$ and the cost function in terms of $Q$. Identify whether marginal cost is constant or rises with output.

Find marginal revenue and marginal cost.
Form $TR(Q)=P(Q)Q$, differentiate it to obtain $MR$, and differentiate total cost to obtain $MC$.

Solve for output.
For an interior solution, set $MR=MC$ and solve the resulting linear equation. Check that the resulting quantity is positive and lies in the stated demand range.

Calculate the outcome.
Use demand to find price, then calculate total revenue, total cost, and profit. Check that profit equals $TR-C$.

Worked Applications

Worked Example: Constant Marginal Cost

A hypothetical sole supplier faces inverse demand $P(Q)=120-2Q$. Its total cost is $C(Q)=100+20Q$, so marginal cost is constant at 20 dollars per unit. Find the interior profit-maximizing quantity and price, then calculate revenue, total cost, and profit. Quantities are units and money is in dollars.

Method. Derive marginal revenue from total revenue, set it equal to marginal cost, and then evaluate price and the accounting totals at the resulting quantity. This avoids the common mistake of treating marginal revenue as the price.

1. Find total revenue and marginal revenue. Total revenue is $TR(Q)=(120-2Q)Q=120Q-2Q^2$. Therefore, $MR(Q)=120-4Q$.

2. Solve for quantity. Set $MR=MC$: $120-4Q=20$. Thus $100=4Q$, so $Q^*=25$ units.

3. Find price from demand. $P^*=120-2(25)=120-50=70$ dollars per unit.

4. Calculate revenue, cost, and profit. $TR^*=70(25)=1{,}750$ dollars. Total cost is $C(25)=100+20(25)=100+500=600$ dollars. Therefore, $\pi^*=1{,}750-600=1{,}150$ dollars.

Check. The calculated price and quantity satisfy demand, and $MR(25)=120-4(25)=20=MC$. Also, $TR^*-C(25)=1{,}750-600=1{,}150$. The solution is internally consistent.

Worked Example: Increasing Linear Marginal Cost

A hypothetical monopolist faces inverse demand $P(Q)=90-Q$. Its total cost is $C(Q)=80+10Q+Q^2$. Find the profit-maximizing quantity and calculate price, revenue, total cost, and profit.

Method. Differentiate both total revenue and total cost. This example tests whether the quantity-dependent part of marginal cost is included when solving $MR=MC$.

1. Find marginal revenue. $TR(Q)=(90-Q)Q=90Q-Q^2$, so $MR(Q)=90-2Q$.

2. Find marginal cost. Differentiating $C(Q)=80+10Q+Q^2$ gives $MC(Q)=10+2Q$. The fixed cost of 80 does not enter marginal cost.

3. Solve for quantity. Set $90-2Q=10+2Q$. Then $80=4Q$, so $Q^*=20$ units.

4. Find price and revenue. $P^*=90-20=70$ dollars per unit. Revenue is $TR^*=70(20)=1{,}400$ dollars.

5. Find total cost and profit. $C(20)=80+10(20)+20^2=80+200+400=680$ dollars. Thus, $\pi^*=1{,}400-680=720$ dollars.

Check. At $Q=20$, $MR=90-2(20)=50$ and $MC=10+2(20)=50$. Profit is $720$ dollars by the revenue-minus-cost calculation.

Worked Example: Convert Direct Demand Before Solving

A hypothetical supplier is given market demand as $Q=100-2P$, rather than as inverse demand. Its total cost is $C(Q)=120+5Q+\frac{1}{4}Q^2$. Find the monopoly outcome and profit.

Method. Rearrange demand to express price as a function of quantity before deriving total revenue. The trap is to use the direct-demand equation as though it were inverse demand.

1. Convert demand to inverse demand. From $Q=100-2P$, we get $2P=100-Q$, hence $P=50-\frac{1}{2}Q$.

2. Derive marginal revenue and marginal cost. $TR(Q)=(50-\frac{1}{2}Q)Q=50Q-\frac{1}{2}Q^2$, so $MR(Q)=50-Q$. Differentiating total cost gives $MC(Q)=5+\frac{1}{2}Q$.

3. Solve for quantity. Set $50-Q=5+\frac{1}{2}Q$. Then $45=\frac{3}{2}Q$, so $Q^*=30$ units.

4. Find price and revenue. $P^*=50-\frac{1}{2}(30)=50-15=35$ dollars per unit. Thus $TR^*=35(30)=1{,}050$ dollars.

5. Find total cost and profit. $C(30)=120+5(30)+\frac{1}{4}(30^2)=120+150+225=495$ dollars. Profit is $\pi^*=1{,}050-495=555$ dollars.

Check. Substituting $P=35$ into the original demand gives $Q=100-2(35)=30$, matching the calculated output. At that quantity, $MR=50-30=20$ and $MC=5+\frac{1}{2}(30)=20$.

Common Errors and Interpretation

The algebra is short, but several distinctions matter. Demand gives the price associated with a quantity; marginal revenue gives the change in revenue from selling one more unit. For a downward-sloping linear demand curve, these are different at positive output. Likewise, total cost and marginal cost serve different purposes: total cost is needed to calculate profit, while marginal cost is used to find the interior quantity.

Common error	Incorrect approach	Correction
Using demand as marginal revenue	For $P=120-2Q$, set $120-2Q=MC$.	First calculate $TR=120Q-2Q^2$, so $MR=120-4Q$.
Reporting marginal revenue as price	After solving $MR=MC$, call that common value the selling price.	Substitute the chosen quantity into inverse demand to find $P$.
Dropping the cost curvature	For $C(Q)=F+cQ+\frac{k}{2}Q^2$, use $MC=c$.	Differentiate the full cost function: $MC=c+kQ$.
Using marginal cost to calculate total cost	Multiply the marginal cost at the chosen output by quantity, then add fixed cost.	Evaluate the given total-cost function. With changing marginal cost, $MC(Q)Q$ is generally not variable cost.

For example, in the increasing-cost case above, marginal cost at the chosen quantity is 50 dollars per unit. Multiplying $50$ by $20$ and adding fixed cost would give $1{,}080$ dollars, not the correct total cost of $680$ dollars. The error is that marginal cost at the final unit is not the average variable cost of all units. The specified total-cost function gives the correct total directly.

Key takeaway: With inverse demand $P=a-bQ$ and marginal cost $MC=c+kQ$, an interior monopoly quantity is $Q^*=(a-c)/(2b+k)$, when this quantity is positive and within the model’s valid range. Then find price from demand, revenue from $PQ$, cost from the total-cost function, and profit from $TR-C$.

Check Your Understanding

For each problem, show the marginal-revenue and marginal-cost equation before calculating the outcome. Unless stated otherwise, assume the firm is operating and report dollar amounts to the nearest cent.

A hypothetical monopolist faces $P(Q)=84-3Q$ and has total cost $C(Q)=60+12Q$. Find its interior quantity, price, total revenue, total cost, and profit. (Answer: $Q^*=12$, $P^*=\$48$, $TR^*=\$576$, $C=\$204$, $\pi^*=\$372$.)
Market demand is $Q=120-2P$, and total cost is $C(Q)=50+12Q+\frac{1}{2}Q^2$. Convert demand to inverse form and find quantity, price, revenue, total cost, and profit. (Answer: $P=60-\frac{1}{2}Q$, $Q^*=24$, $P^*=\$48$, $TR^*=\$1{,}152$, $C=\$626$, $\pi^*=\$526$.)
A hypothetical seller faces $P(Q)=150-2Q$, with total cost $C(Q)=200+30Q+\frac{1}{2}Q^2$. Calculate the interior monopoly outcome and verify $MR=MC$ at the chosen quantity. (Answer: $Q^*=24$, $P^*=\$102$, $TR^*=\$2{,}448$, $C=\$1{,}208$, $\pi^*=\$1{,}240$; $MR=MC=\$54$.)
For inverse demand $P(Q)=40-Q$, marginal cost is $MC(Q)=50+2Q$, and output must be nonnegative. Does the interior equation give a feasible positive quantity? State the best operating quantity under the nonnegative-output constraint, and explain the role of the boundary. (Answer: The equation gives $Q^*=-2.5$, which is infeasible; the best constrained operating quantity is $Q=0$.)

Common error	Incorrect approach	Correction
Using demand as marginal revenue	For \(P=120-2Q\), set \(120-2Q=MC\).	First calculate \(TR=120Q-2Q^2\), so \(MR=120-4Q\).
Reporting marginal revenue as price	After solving \(MR=MC\), call that common value the selling price.	Substitute the chosen quantity into inverse demand to find \(P\).
Dropping the cost curvature	For \(C(Q)=F+cQ+\frac{k}{2}Q^2\), use \(MC=c\).	Differentiate the full cost function: \(MC=c+kQ\).
Using marginal cost to calculate total cost	Multiply the marginal cost at the chosen output by quantity, then add fixed cost.	Evaluate the given total-cost function. With changing marginal cost, \(MC(Q)Q\) is generally not variable cost.

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