What is the deadweight loss?
Type straight-line demand and supply equations and a per-unit tax, or the efficient quantity, the quantity after a tax or price control, and the price gap. The deadweight loss calculator gives the lost surplus triangle and the tax revenue.
- Deadweight loss
- 37.5
The deadweight loss is 37.5.
- Tax revenue
- 375
- Quantity lost
- 5
- Efficient quantity
- 30
- Quantity with the tax
- 25
- Price buyers pay
- 50
- Price sellers keep
- 35
Deadweight loss: 37.5. The deadweight loss is 37.5.
The results are estimates for information only. They are not financial, tax, or legal advice. Check the numbers with your lender or a qualified professional before you decide. Terms of use
How to calculate
Computes the deadweight loss of a tax or price control from linear demand and supply curves, or from the quantity lost and the price wedge, with the tax revenue.
Example with the default inputs (I have Demand and supply lines, Demand intercept a 100, Demand slope b 2, Supply intercept c 10, Supply slope d 1, Tax per unit 15): The deadweight loss is 37.5.
Method: Lines: Q₀ = (a − c) ÷ (b + d), Q₁ = max(0, (a − c − t) ÷ (b + d)), buyer price = a − bQ₁, seller price = c + dQ₁, DWL = ½ (Q₀ − Q₁)(buyer price − seller price), revenue = t × Q₁. Quantities: DWL = ½ (Q₀ − Q₁) × gap, revenue = gap × Q₁.
- Demand and supply are straight lines, so the lost surplus is a triangle.
- The tax is a fixed amount per unit; who legally pays it does not change the result.
Worked examples
Each example is checked against the calculator on every build.
- I have Demand and supply lines, Demand intercept a 100, Demand slope b 2, Supply intercept c 10, Supply slope d 1, Tax per unit 15 gives Efficient quantity 30, Quantity with the tax 25, Price buyers pay 50, Price sellers keep 35, Deadweight loss 37.5, Tax revenue 375, Quantity lost 5.Source: https://openstax.org/books/principles-economics-3e/pages/3-5-demand-supply-and-efficiency
- I have Quantities and price gap, Efficient quantity 1,000, Quantity with the distortion 800, Price gap per unit 2 gives Deadweight loss 200, Tax revenue 1,600, Quantity lost 200.Source: https://openstax.org/books/principles-economics-3e/pages/3-5-demand-supply-and-efficiency
- I have Demand and supply lines, Demand intercept a 100, Demand slope b 2, Supply intercept c 10, Supply slope d 1, Tax per unit 120 gives Quantity with the tax 0, Deadweight loss 1,350, Tax revenue 0.Source: https://openstax.org/books/principles-economics-3e/pages/3-5-demand-supply-and-efficiency
How it works
OpenStax defines deadweight loss as the loss in social surplus when the market produces an inefficient quantity. With straight-line demand and supply, the lost surplus is a triangle whose base is the quantity lost and whose height is the price gap at the lower quantity.
Demand and supply lines (default). Demand is P = a − b × Q and supply is P = c + d × Q, with b > 0 and d > 0. A tax t is charged per unit.
- Efficient quantity Q₀ = (a − c) ÷ (b + d).
- Quantity with the tax Q₁ = (a − c − t) ÷ (b + d), or 0 if that is below 0 (the tax stops all trade).
- Price buyers pay = a − b × Q₁. Price sellers keep = c + d × Q₁. While Q₁ > 0 they differ by t.
- Deadweight loss = ½ × (Q₀ − Q₁) × (buyer price − seller price).
- Tax revenue = t × Q₁. Quantity lost = Q₀ − Q₁.
Quantities and price gap. Type the efficient quantity Q₀, the quantity traded with the tax or price control Q₁, and the price gap g (the demand price minus the supply price at Q₁; for a tax it is the tax).
- Deadweight loss = ½ × (Q₀ − Q₁) × g.
- Tax revenue = g × Q₁ (meaningful when the gap is a tax). Quantity lost = Q₀ − Q₁.
Rules. Every number is 0 or more, and b and d are at least 0.000001. In lines mode, a must be above c; otherwise there is no answer. In quantities mode, Q₁ cannot be above Q₀.
Rounding. Every value is an exact fraction of the typed decimals, rounded once for display: money-like values to 2 decimals, quantities to 10 significant figures.
Worked examples by hand
A $15 tax (page default). Demand P = 100 − 2Q, supply P = 10 + Q. Q₀ = (100 − 10) ÷ 3 = 30. Q₁ = (100 − 10 − 15) ÷ 3 = 25. Buyers pay 100 − 2 × 25 = 50.00; sellers keep 10 + 25 = 35.00. Deadweight loss = ½ × (30 − 25) × 15 = 37.50. Tax revenue = 15 × 25 = 375.00. Quantity lost 5.
Quantities. Q₀ = 1,000, Q₁ = 800, gap $2. Deadweight loss = ½ × 200 × 2 = 200.00; revenue = 2 × 800 = 1,600.00.
A tax that stops trade. Same lines, t = 120. (90 − 120) ÷ 3 is below 0, so Q₁ = 0. Buyers' price at Q = 0 is 100 and sellers' is 10, a gap of 90. Deadweight loss = ½ × 30 × 90 = 1,350.00, the whole social surplus. Revenue 0.
Other questions people ask
How do I calculate deadweight loss?
With straight-line demand and supply, deadweight loss is a triangle: ½ × (efficient quantity − quantity traded) × (price buyers pay − price sellers keep). If a $2 tax cuts sales from 1,000 to 800 units, the loss is ½ × 200 × 2 = 200.
How do I find deadweight loss from demand and supply equations?
Write demand as P = a − bQ and supply as P = c + dQ. Without a tax, Q₀ = (a − c) ÷ (b + d). With a per-unit tax t, Q₁ = (a − c − t) ÷ (b + d). Deadweight loss = ½ × t × (Q₀ − Q₁), which equals t² ÷ (2(b + d)) while Q₁ is above 0.
What is deadweight loss?
OpenStax defines it as the loss in social surplus that occurs when the economy produces at an inefficient quantity. Trades that buyers and sellers would both have gained from do not happen, and nobody receives that lost value, not even the government.
Does a price ceiling or price floor cause deadweight loss?
Yes, when it binds. Use “Quantities and price gap”: the efficient quantity, the quantity actually traded under the control, and the gap between the demand price and the supply price at that quantity.
Is tax revenue part of the deadweight loss?
No. Tax revenue (tax × quantity traded) moves surplus from buyers and sellers to the government. The deadweight loss is the surplus that no one gets.
Why does a bigger tax cause a much bigger loss?
With straight lines the loss grows with the square of the tax: doubling the tax quadruples the deadweight loss, as long as some trade still happens.