How to Calculate Deadweight Loss From an Externality
Deadweight loss from an externality is ½ × base × height: base is the gap between market and optimal quantity, height is the external cost at Q optimal.
The Externality DWL formula
Calculator
Enter demand, private cost and social cost to size the welfare triangle the spillover leaves behind.
Demand is P = 40 − 0.5Q, so the intercept is 40.
How far price falls per extra unit. Enter 0.5 for P = 40 − 0.5Q.
MPC = 10 + 0.5Q, which is what the market actually trades on.
How far private cost rises per extra unit.
MSC = MPC plus the external damage, so 10 + 6 = 16.
Normally the same slope as MPC, since the spillover is a constant per unit.
Half of 6 units times $6. Those units cost society more than anyone valued them, and $18 is the size of that mistake.
- Free-market quantity
- 30
- Socially optimal quantity
- 24
- Base of the triangle
- 6
- Height of the triangle
- $6
- Market verdict
- Overproduction
Where marginal private benefit meets marginal private cost, with the spillover ignored.
Where marginal social benefit meets marginal social cost, so the triangle closes to a point here.
The 6 units between the market quantity and the optimal quantity.
The vertical gap between MSC and MSB measured at the market quantity, where it is widest.
How to calculate Externality DWL, step by step
- 1Find the free-market quantity. Set marginal private benefit equal to marginal private cost; this is where the market lands when the spillover is ignored.
- 2Find the socially optimal quantity. Set marginal social benefit equal to marginal social cost. A negative externality puts Q optimal to the left of the market quantity, a positive externality puts it to the right.
- 3Measure the height at the market quantity. At Q market, take the vertical distance between MSC and MSB. With a constant per-unit spillover this distance equals the external cost or benefit per unit.
- 4Compute the triangle. DWL = ½ × base × height, where base is the distance between the two quantities. The triangle always closes to a point at Q optimal, where MSB = MSC.
Worked example: Externality DWL
A steel market has MPC = 10 + 0.5Q and demand (MSB) P = 40 − 0.5Q, and each ton dumps $6 of pollution damage on neighbors, so MSC = 16 + 0.5Q. Free market: 10 + 0.5Q = 40 − 0.5Q gives Q = 30. Social optimum: 16 + 0.5Q = 40 − 0.5Q gives Q = 24. At the market quantity of 30, MSC = 16 + 15 = $31 and MSB = 40 − 15 = $25, so the height is 31 − 25 = $6. The base is 30 − 24 = 6 tons. DWL = ½ × 6 × 6 = $18. Those 6 extra tons cost society more than buyers valued them, and the $18 is the size of that mistake.
Externality DWL questions
Does a positive externality also create deadweight loss?
Yes, the market underproduces, so the triangle sits between the smaller market quantity and the larger optimal quantity. It is measured the same way: ½ × the quantity gap × the external benefit per unit.
Which quantity gives the height of the triangle?
The market quantity, where the gap between marginal social cost and marginal social benefit is widest. At the optimal quantity MSB equals MSC, so the triangle closes to a point there.
Can a tax remove the deadweight loss?
Yes, a per-unit tax equal to the marginal external cost moves output to the optimal quantity and erases the triangle. For a positive externality a per-unit subsidy equal to the external benefit does the same job in reverse.
Get AP Econ exam tips in your inbox
Occasional emails with study tips, new interactive graphs, and exam-season reminders. Free, no spam.
No spam. Unsubscribe anytime. Read our privacy policy.
Keep track of what you have studied
A free EconLearn account adds progress tracking, your quiz history, and achievements. Studying here is free either way, and there is nothing to pay for as a student.
Create a free accountAlready have one? Sign in
Last updated