Calculating Monopoly Deadweight Loss

Why Monopoly Deadweight Loss Is a Triangle

A monopolist chooses output to maximize profit, but the output that maximizes the seller’s profit need not maximize the total gains from trade. When a single-price monopolist produces less than the competitive benchmark, some consumers who value an additional unit more than it costs to produce do not receive it. The total surplus from those unrealized trades is deadweight loss.

This tutorial focuses on calculating that loss when inverse demand and marginal cost are linear. The calculation uses the difference between buyers’ marginal willingness to pay and the marginal cost of supplying each unit. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Deadweight loss from monopoly is the total surplus forgone because output is below the efficient quantity. Under the standard model here, inverse demand measures marginal willingness to pay and marginal cost measures the cost of producing each additional unit. For a unit that would be produced competitively but not under monopoly, its contribution to total surplus is demand price minus marginal cost.

Assume a single-price monopolist faces the inverse demand curve $P_D(Q)=a-bQ$, with $a>c$ and $b>0$, and has marginal cost $MC(Q)=c+dQ$, with $d\geq 0$. Assume the firm chooses a positive interior output, there are no external costs or benefits, and the competitive benchmark has firms producing where price equals marginal cost. The formulas below do not cover a case in which the firm shuts down or the relevant quantity is at a boundary.

In this model, monopoly output satisfies marginal revenue equal to marginal cost. With linear inverse demand, total revenue is $TR(Q)=P_D(Q)Q=aQ-bQ^2$, so marginal revenue is $MR(Q)=a-2bQ$. The competitive benchmark is the quantity at which demand price equals marginal cost.

Finding the Monopoly and Competitive Quantities

MR(Q)=a-2bQ,\qquad MC(Q)=c+dQ

Setting $MR=MC$ gives the monopoly quantity. Setting demand equal to marginal cost gives the competitive quantity:

Q_m=\frac{a-c}{2b+d},\qquad Q_c=\frac{a-c}{b+d}

Because $a>c$ and $b>0$, these quantities are positive and $Q_m<Q_c$. The difference $Q_c-Q_m$ is the range of units that would be supplied under the competitive benchmark but are not supplied by the monopolist. The monopoly price is $P_D(Q_m)$; it is not the height of the deadweight-loss triangle.

Worked Example: A Monopoly With Rising Marginal Cost

A hypothetical producer faces inverse demand $P_D(Q)=100-2Q$, where $P$ is dollars per unit and $Q$ is units per period. Its marginal cost is $MC(Q)=20+Q$. Find the monopoly output and price, the competitive quantity, and deadweight loss in dollars per period.

Method. First find the two output quantities, then calculate the area of the surplus lost between them. Because price is dollars per unit and quantity is units per period, the area is dollars per period.

1. Find marginal revenue. Total revenue is $TR(Q)=(100-2Q)Q=100Q-2Q^2$. Therefore $MR(Q)=100-4Q$.

2. Find monopoly output. Set marginal revenue equal to marginal cost:

100-4Q=20+Q \quad\Longrightarrow\quad 80=5Q \quad\Longrightarrow\quad Q_m=16

The monopoly price is $P_D(16)=100-2(16)=68$ dollars per unit. Marginal cost at that quantity is $MC(16)=20+16=36$ dollars per unit.

3. Find competitive output. Set demand price equal to marginal cost:

100-2Q=20+Q \quad\Longrightarrow\quad 80=3Q \quad\Longrightarrow\quad Q_c=\frac{80}{3}

At $Q_c=80/3$, demand price is $100-160/3=140/3$ dollars per unit, and marginal cost is $20+80/3=140/3$ dollars per unit. The equality confirms the competitive benchmark.

4. Calculate deadweight loss. The triangle’s width is $Q_c-Q_m=80/3-16=32/3$ units per period. Its height is the demand–marginal-cost gap at $Q_m$: $68-36=32$ dollars per unit. Thus:

DWL=\frac12\left(\frac{32}{3}\right)(32) =\frac{512}{3} \approx \$170.67\text{ per period}

Check. The demand–marginal-cost gap is $P_D(Q)-MC(Q)=80-3Q$. It is 32 at $Q=16$ and zero at $Q=80/3$, as required for the triangle. The method avoids using the monopoly price as the triangle’s height; the relevant height is demand price minus marginal cost at monopoly output.

Why the Lost-Surplus Area Is a Triangle

For every additional unit between $Q_m$ and $Q_c$, demand price is greater than marginal cost. Producing such a unit would add the difference between those two values to total surplus. At $Q_c$, that difference is zero because demand price equals marginal cost. Since both curves are linear, their vertical difference declines linearly from its value at $Q_m$ to zero at $Q_c$.

Identify the unproduced range.
Units from $Q_m$ up to $Q_c$ are not produced by the monopolist but would be produced at the competitive benchmark.

Find the surplus per marginal unit.
At quantity $Q$, the surplus from one more unit is $P_D(Q)-MC(Q)$. At $Q_m$, this gap is positive; at $Q_c$, it is zero.

Use the area of a triangle.
A linear gap that falls from a positive height to zero has a triangular area. Multiply one-half by the output gap and by the height at $Q_m$.

Equivalently, deadweight loss is the area under the gap between demand and marginal cost over the unproduced range. The integral makes the triangle’s height and endpoints explicit:

DWL=\int_{Q_m}^{Q_c}\left[P_D(Q)-MC(Q)\right]\,dQ =\frac12(Q_c-Q_m)\left[P_D(Q_m)-MC(Q_m)\right]

For the general linear curves above, $P_D(Q)-MC(Q)=a-c-(b+d)Q$. At $Q_c$ the gap is zero, while at $Q_m$ it is $(a-c)b/(2b+d)$. The width is $(a-c)b/[(b+d)(2b+d)]$. Substituting these expressions into the triangle formula gives:

DWL= \frac{(a-c)^2b^2}{2(b+d)(2b+d)^2}

This expression is a check on the area calculation, not a substitute for identifying the right quantities and units. If marginal cost is constant, so $d=0$, it simplifies to $DWL=(a-c)^2/(8b)$. If demand and marginal cost do not intersect at a positive quantity, or monopoly output is not an interior solution, the assumptions used for these formulas must be reconsidered.

Worked Example: Checking a Fractional Output Gap

A hypothetical supplier faces $P_D(Q)=84-3Q$ and $MC(Q)=12+Q$, with price in dollars per unit and quantity in units per period. Calculate monopoly output, competitive output, and deadweight loss. Report the final loss to the nearest cent.

Method. Use the marginal-revenue condition for monopoly output and the demand–marginal-cost intersection for competitive output. Keep the fractional quantities exact until the final monetary result.

1. Find monopoly output and price. Total revenue is $TR(Q)=(84-3Q)Q=84Q-3Q^2$, so $MR(Q)=84-6Q$. Equating marginal revenue and marginal cost gives:

84-6Q=12+Q \quad\Longrightarrow\quad 72=7Q \quad\Longrightarrow\quad Q_m=\frac{72}{7}

The monopoly price is $84-3(72/7)=84-216/7=372/7$ dollars per unit. Marginal cost at monopoly output is $12+72/7=156/7$ dollars per unit.

2. Find competitive output. Set demand equal to marginal cost:

84-3Q=12+Q \quad\Longrightarrow\quad 72=4Q \quad\Longrightarrow\quad Q_c=18

At $Q_c=18$, demand price is $84-54=30$ dollars per unit, and marginal cost is $12+18=30$ dollars per unit.

3. Calculate the triangle. The width is $18-72/7=54/7$ units per period. At monopoly output, demand price is $372/7$, marginal cost is $156/7$, and the height is therefore $216/7$ dollars per unit. The loss is:

DWL=\frac12\left(\frac{54}{7}\right)\left(\frac{216}{7}\right) =\frac{5832}{49} \approx \$119.02\text{ per period}

Check. The gap is $P_D(Q)-MC(Q)=72-4Q$. At $Q_m=72/7$, it is $72-288/7=216/7$; at $Q_c=18$, it is zero. The area has the correct positive sign and its units are dollars per period. The fractional values were retained until rounding to two decimal places.

Interpreting the Calculation and Avoiding Traps

Deadweight loss is not the monopolist’s profit, the total amount consumers pay, or the entire reduction in consumer surplus. A higher monopoly price transfers some surplus from buyers to the seller on units that continue to be sold. By contrast, deadweight loss is surplus that neither buyers nor the seller receive: it comes from mutually beneficial units that are not traded. Under the assumptions here, it is the area between demand and marginal cost from $Q_m$ to $Q_c$.

A frequent exam error is to halve the triangle twice. In the preceding example, the correct width is $54/7$ and the correct height is $216/7$. The triangle’s area is:

\frac12\left(\frac{54}{7}\right)\left(\frac{216}{7}\right) =\frac{5832}{49} \approx \$119.02

If the height is incorrectly treated as half of $216/7$ and the triangle formula is then applied as well, the result is:

\frac12\left(\frac{54}{7}\right)\left(\frac{108}{7}\right) =\frac{2916}{49} \approx \$59.51

That calculation is wrong: the gap at monopoly output, $216/7$, is already the triangle’s full height. Its average height is half that amount, but multiplying width by average height is already equivalent to using one-half times width times full height. Do not apply both adjustments.

Use the correct vertical gap. The height is $P_D(Q_m)-MC(Q_m)$, not the monopoly price by itself.
Use the correct horizontal gap. The width is $Q_c-Q_m$, not $Q_m$.
Keep the quantity and price units compatible. Dollars per unit times units per period gives dollars per period.
Check the endpoints. Demand price must equal marginal cost at $Q_c$, so the triangle’s gap is zero there.

Worked Example: A Steeper Marginal-Cost Curve

A hypothetical workshop has inverse demand $P_D(Q)=150-5Q$ and marginal cost $MC(Q)=30+2Q$. Price is measured in dollars per unit and quantity in units per period. Find both benchmark quantities and the deadweight loss, rounded to two decimal places.

Method. Calculate monopoly output using $MR=MC$, then calculate the competitive output using $P_D=MC$. Use the gap at monopoly output as the triangle’s height.

1. Calculate marginal revenue and monopoly output. Total revenue is $TR(Q)=(150-5Q)Q=150Q-5Q^2$, so $MR(Q)=150-10Q$. Then:

150-10Q=30+2Q \quad\Longrightarrow\quad 120=12Q \quad\Longrightarrow\quad Q_m=10

The monopoly price is $P_D(10)=150-50=100$ dollars per unit, while marginal cost at that quantity is $30+20=50$ dollars per unit.

2. Calculate competitive output.

150-5Q=30+2Q \quad\Longrightarrow\quad 120=7Q \quad\Longrightarrow\quad Q_c=\frac{120}{7}

At this quantity, demand price is $150-600/7=450/7$ dollars per unit. Marginal cost is $30+240/7=450/7$ dollars per unit, confirming the intersection. The width is $120/7-10=50/7$ units per period.

3. Calculate and check deadweight loss. At $Q_m=10$, the gap is $100-50=50$ dollars per unit. Therefore:

DWL=\frac12\left(\frac{50}{7}\right)(50) =\frac{1250}{7} \approx \$178.57\text{ per period}

The gap function is $P_D(Q)-MC(Q)=120-7Q$, which equals 50 at $Q=10$ and zero at $Q=120/7$. This check confirms both the height and the width. A common mistake avoided here is using the monopoly price, 100, as the height; the height is the difference between that price and marginal cost, which is 50.

Key takeaway: Under linear demand and marginal cost, monopoly deadweight loss is the area of the triangle between the curves from monopoly output $Q_m$ to competitive output $Q_c$: one-half times $Q_c-Q_m$ times $P_D(Q_m)-MC(Q_m)$. Check both endpoints and keep units consistent.

Check Your Understanding

For each hypothetical market, use inverse demand and marginal cost in dollars per unit, with quantity measured in units per period. Find the monopoly and competitive quantities, then calculate deadweight loss.

A supplier faces $P_D(Q)=90-2Q$ and $MC(Q)=18+Q$. Give the monopoly and competitive quantities and deadweight loss. (Answer: $Q_m=14.4$, $Q_c=24$, $DWL=\$138.24$ per period.)
A producer faces $P_D(Q)=120-4Q$ and $MC(Q)=24+2Q$. Give the two quantities and deadweight loss. (Answer: $Q_m=9.6$, $Q_c=16$, $DWL=\$122.88$ per period.)
A firm faces $P_D(Q)=200-Q$ and $MC(Q)=50+2Q$. Find monopoly output, competitive output, and deadweight loss to the nearest cent. (Answer: $Q_m=37.5$, $Q_c=50$, $DWL=\$234.38$ per period.)
A single-price seller faces $P_D(Q)=96-3Q$ and $MC(Q)=12+Q$. Find both quantities, the monopoly price, and deadweight loss. (Answer: $Q_m=12$, $Q_c=21$, monopoly price $=\$60$ per unit, $DWL=\$162.00$ per period.)
A workshop faces $P_D(Q)=160-4Q$ and $MC(Q)=40+Q$. Retain exact quantities until calculating the triangle, and round the loss to the nearest cent. (Answer: $Q_m=40/3$, $Q_c=24$, $DWL=\$284.44$ per period.)

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