Deriving Linear Deadweight Loss

Why Excluded Trades Create a Triangle

When a unit is traded, the buyer’s willingness to pay for it and the resources needed to produce it determine the potential gain from that trade. If willingness to pay exceeds marginal cost, the unit adds to total surplus. If a restriction prevents that unit from being traded, that gain is lost. With linear demand and supply schedules, the lost gains across the excluded units form a triangle.

The key is to measure the gap between marginal willingness to pay and marginal cost for the units that are not traded. The equilibrium price is not that gap: at the efficient quantity, marginal willingness to pay equals marginal cost. The gap is positive at the restricted quantity and falls to zero at the efficient quantity.

Definition: Marginal willingness to pay, written $P_D(Q)$, is the value of the marginal unit to buyers when quantity is $Q$. Marginal cost, written $MC(Q)$, is the cost of producing the marginal unit. For an excluded unit, the potential gain from trade is $P_D(Q)-MC(Q)$, when this difference is positive.

The derivation uses a simple benchmark. Assume units are divisible, the demand curve represents marginal willingness to pay, and the supply curve represents marginal cost. Assume also that units are allocated to the buyers who value them most and produced by the lowest-cost sellers. There are no external costs or benefits, and no other distortion changes which units are traded. Under these assumptions, the efficient quantity $Q^*$ is where marginal willingness to pay equals marginal cost. Consider a restriction that lowers trade to $Q_r<Q^*$, so the excluded units are those between $Q_r$ and $Q^*$.

From Lost Gains to an Integral

For each small additional unit $dQ$ between $Q_r$ and $Q^*$, the gain forgone is the difference between what the buyer would be willing to pay and what it costs to produce that unit. Adding these gains across all excluded units gives the deadweight loss (DWL):

\mathrm{DWL}=\int_{Q_r}^{Q^*}\bigl[P_D(Q)-MC(Q)\bigr]\,dQ.

This is an area under the marginal-gain-from-trade curve $P_D(Q)-MC(Q)$. It is a loss of total surplus, not a payment transferred from one party to another. The integral form applies whenever the marginal curves are sufficiently well-behaved to integrate; linearity is needed for the triangle shortcut, not for the integral itself.

Deriving the Linear Triangle Formula

Suppose inverse demand and marginal cost are linear:

P_D(Q)=A-BQ,\qquad MC(Q)=C+DQ,

where $B>0$, $D>0$, and $A>C$. These conditions give downward-sloping marginal willingness to pay, upward-sloping marginal cost, and a positive efficient quantity. At that quantity the two curves intersect:

A-BQ^*=C+DQ^* \quad\Longrightarrow\quad Q^*=\frac{A-C}{B+D}.

For a restricted quantity $Q_r<Q^*$, the marginal gain from trading the next unit is the vertical distance between the curves at $Q_r$. At $Q^*$, that distance is zero. Because the curves are linear, the distance declines in a straight line over the excluded range. The integral is therefore the area of a triangle.

Find the triangle’s base.
The excluded quantity is $Q^*-Q_r$.

Find its height.
The gap at the restricted quantity is $P_D(Q_r)-MC(Q_r)$. At $Q^*$, the gap is zero.

Calculate the area.
Multiply one-half by the base and the height.

The height can also be expressed using the slopes. Since the gap is $(A-C)-(B+D)Q$, and it equals zero at $Q^*$, its value at $Q_r$ is $(B+D)(Q^*-Q_r)$. Substitution into the triangle area gives the equivalent formula:

\begin{aligned} \mathrm{DWL} &=\frac{1}{2}(Q^*-Q_r)\bigl[P_D(Q_r)-MC(Q_r)\bigr]\\ &=\frac{1}{2}(B+D)(Q^*-Q_r)^2. \end{aligned}

The first expression uses a quantity difference and a marginal-value gap; their product has units of dollars. The second shows that, for given linear slopes, the loss increases with the square of the quantity reduction. This formula is for underproduction relative to $Q^*$. If the schedules are nonlinear, use the integral rather than assuming the loss is a triangle. If the curves represent private rather than social values and costs in a setting with external effects, the intersection and loss must instead be assessed using social marginal benefits and costs.

Worked Example: Calculate the Loss by Integrating the Gap

A hypothetical market has inverse demand $P_D(Q)=100-2Q$ dollars per unit and marginal cost $MC(Q)=20+Q$ dollars per unit. A quantity restriction limits trade to $Q_r=20$. Find the deadweight loss. Keep exact values through the calculation and round money to the nearest cent only at the end.

Method. Find the efficient quantity, evaluate the gap at the restriction, and calculate the triangle. Then verify the result by integrating the marginal gain from trade.

Efficient quantity. Set marginal willingness to pay equal to marginal cost:

100-2Q^*=20+Q^* \quad\Longrightarrow\quad 80=3Q^* \quad\Longrightarrow\quad Q^*=\frac{80}{3}.

The restricted quantity is below the efficient quantity: $20<80/3$. The base of the triangle is $80/3-20=20/3$ units. At the restricted quantity, marginal willingness to pay is $100-2(20)=60$ dollars per unit, and marginal cost is $20+20=40$ dollars per unit. The height is $60-40=20$ dollars per unit.

\mathrm{DWL} =\frac{1}{2}\left(\frac{20}{3}\right)(20) =\frac{200}{3} =66.67\text{ dollars, rounded to the nearest cent}.

Check by integration. The marginal gain from trade is $(100-2Q)-(20+Q)=80-3Q$. Its integral from 20 to $80/3$ is:

\begin{aligned} \int_{20}^{80/3}(80-3Q)\,dQ &=\left[80Q-\frac{3}{2}Q^2\right]_{20}^{80/3}\\ &=\left(\frac{6400}{3}-\frac{3200}{3}\right)-(1600-600)\\ &=\frac{3200}{3}-1000 =\frac{200}{3}\text{ dollars}. \end{aligned}

The check agrees with the triangle calculation. Common mistake avoided: using the equilibrium price as the triangle’s height. The equilibrium price is $100-2(80/3)=140/3$ dollars per unit, but the height is the difference between willingness to pay and cost at $Q_r$, which is 20 dollars per unit.

Worked Examples with Linear Schedules

Worked Example: Check the Triangle Against the Full Integral

For a hypothetical service, let $P_D(Q)=150-3Q$ dollars per unit and $MC(Q)=30+Q$ dollars per unit. Trade is restricted to $Q_r=18$. Calculate the deadweight loss, showing both the triangle calculation and an integral check. Round the final dollar amount to the nearest cent.

Method. The intersection gives the efficient quantity; the marginal-value gap at the restricted quantity gives the triangle’s height.

Solve $150-3Q^*=30+Q^*$, so $120=4Q^*$ and $Q^*=30$. The base is $30-18=12$ units. At $Q=18$, willingness to pay is $150-3(18)=96$ dollars per unit, while marginal cost is $30+18=48$ dollars per unit. Thus the height is $96-48=48$ dollars per unit.

\mathrm{DWL} =\frac{1}{2}(12)(48) =288.00\text{ dollars}.

For the check, the marginal gain is $(150-3Q)-(30+Q)=120-4Q$. Integrate over the excluded units:

\begin{aligned} \int_{18}^{30}(120-4Q)\,dQ &=\left[120Q-2Q^2\right]_{18}^{30}\\ &=(3600-1800)-(2160-648)\\ &=1800-1512 =288\text{ dollars}. \end{aligned}

Both methods give a loss of 288.00 dollars. Common mistake avoided: using the demand price at the restricted quantity, 96 dollars per unit, as the triangle’s height. The relevant height is the gap $96-48=48$, not the demand price by itself.

Worked Example: Recover the Quantity Reduction from Deadweight Loss

A hypothetical market has $P_D(Q)=140-2Q$ dollars per unit and $MC(Q)=20+Q$ dollars per unit. Its deadweight loss from a quantity restriction is 150 dollars. Find the restricted quantity, assuming it is below the efficient quantity. Verify the result using the gap at the restricted quantity.

Method. Use the slope form of the linear triangle formula to find the quantity reduction, then subtract it from the efficient quantity.

First, set demand equal to marginal cost: $140-2Q^*=20+Q^*$, so $120=3Q^*$ and $Q^*=40$. Here the demand slope magnitude is $B=2$, and the marginal-cost slope is $D=1$. Let the reduction be $x=Q^*-Q_r$. The slope form gives:

150=\frac{1}{2}(2+1)x^2 \quad\Longrightarrow\quad 150=\frac{3}{2}x^2 \quad\Longrightarrow\quad x^2=100 \quad\Longrightarrow\quad x=10.

The reduction must be positive, so $x=10$ units and $Q_r=40-10=30$. At $Q=30$, marginal willingness to pay is $140-2(30)=80$ dollars per unit, and marginal cost is $20+30=50$ dollars per unit. The height is $80-50=30$ dollars per unit. The triangle check is $\frac{1}{2}(10)(30)=150$ dollars.

The result is consistent with the stated loss. Common mistake avoided: taking the square root before accounting for the one-half and the combined slope. The quantity reduction is $\sqrt{2\mathrm{DWL}/(B+D)}$, not simply the square root of the loss.

Interpretation and Common Traps

The triangle formula is a compact way to add the gains from every excluded unit. Its base measures how many units are missing; its height measures the gain from the first marginal unit just beyond the restriction. The gain shrinks across the excluded range and reaches zero at the efficient quantity. This logic is why the area uses one-half rather than multiplying the full base by the initial gap.

Using the wrong height. Use $P_D(Q_r)-MC(Q_r)$, not either curve’s price or value alone.
Using the wrong endpoint. The base is $Q^*-Q_r$, not the restricted quantity itself.
Confusing loss with a transfer. The integral counts gains from units that are not traded; it does not count a payment that merely changes hands.
Applying the triangle to nonlinear curves. For nonlinear curves the gap need not decline in a straight line. Integrate the gap unless the relevant area is demonstrably triangular.
Using private curves when social values differ. The calculation requires curves that represent the marginal benefits and costs relevant to total surplus under the stated model.

Deriving the formula makes its conditions and interpretation visible, which is useful for problems aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. A reliable calculation states the efficient and restricted quantities, identifies both curves’ values at the restriction, and checks that the resulting area is positive when $Q_r<Q^*$.

Key takeaway: With linear marginal willingness to pay and marginal cost, a restriction from $Q^*$ to $Q_r$ creates a triangular loss: one-half times the excluded quantity $Q^*-Q_r$ times the gap $P_D(Q_r)-MC(Q_r)$. This is equivalent to one-half times the sum of the curve slopes times the square of the quantity reduction.

Check Your Understanding

For each problem, assume the stated linear demand curve represents marginal willingness to pay and the stated cost curve represents marginal cost. Unless a problem says otherwise, calculate the loss from a restriction below the efficient quantity. Round final dollar amounts to the nearest cent where needed.

A hypothetical market has $P_D(Q)=84-2Q$ and $MC(Q)=12+Q$, with trade restricted to 20 units. Find the efficient quantity, the gap at the restriction, and deadweight loss. (Answer: $Q^*=24$; gap $=12$ dollars per unit; DWL $=\$24.00$.)
Let $P_D(Q)=110-Q$ and $MC(Q)=20+2Q$. If the restriction reduces trade by 6 units from the efficient quantity, calculate deadweight loss using the triangle formula. (Answer: $\$54.00$.)
Let $P_D(Q)=160-3Q$ and $MC(Q)=20+Q$. A restriction creates a deadweight loss of 200 dollars. Find the efficient and restricted quantities. (Answer: $Q^*=35$, $Q_r=25$.)
For $P_D(Q)=96-Q$ and $MC(Q)=16+Q$, trade is restricted to 28 units. Find the triangle’s base and height, then calculate the loss. (Answer: base $=12$ units, height $=24$ dollars per unit, DWL $=\$144.00$.)
For $P_D(Q)=180-2Q$ and $MC(Q)=30+Q$, a restriction causes a loss of 216 dollars. Recover the quantity reduction and the restricted quantity. (Answer: reduction $=12$ units and $Q_r=38$.)

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