# What is the deadweight loss?

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.

- Page: https://www.acalculator.org/finance/deadweight-loss-calculator
- JSON spec: https://www.acalculator.org/finance/deadweight-loss-calculator.json
- Version: 22fd0b67967b

## Default answer

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.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | I have | Straight-line demand and supply equations with a tax, or the two quantities and the price gap. |
| a | Demand intercept a | Demand is P = a − b × Q: a is the price at which buyers want nothing. |
| b | Demand slope b | How much the demand price falls for each extra unit (b > 0). |
| c | Supply intercept c | Supply is P = c + d × Q: c is the lowest price at which sellers offer anything. |
| d | Supply slope d | How much the supply price rises for each extra unit (d > 0). |
| t | Tax per unit | The per-unit tax that separates the price buyers pay from the price sellers keep. |
| q0 | Efficient quantity | The quantity traded without the tax or price control. |
| q1 | Quantity with the distortion | The quantity traded with the tax or price control. |
| w | Price gap per unit | At the lower quantity: the price buyers would pay minus the price sellers would accept (the tax, for a tax). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| dwl | Deadweight loss | ½ × (efficient quantity − quantity with the distortion) × price gap. |
| revenue | Tax revenue | Price gap × quantity with the distortion (the government’s revenue when the gap is a tax). |
| lost | Quantity lost | Efficient quantity − quantity with the distortion. |
| q0Out | Efficient quantity | Where demand meets supply: (a − c) ÷ (b + d). |
| q1Out | Quantity with the tax | (a − c − t) ÷ (b + d), and 0 if that is below 0. |
| buyerPrice | Price buyers pay | a − b × the quantity with the tax. |
| sellerPrice | Price sellers keep | c + d × the quantity with the tax. |

## 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.

## Assumptions

- 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

1. mode = curves, a = 100, b = 2, c = 10, d = 1, t = 15 gives q0Out = 30, q1Out = 25, buyerPrice = 50, sellerPrice = 35, dwl = 37.5, revenue = 375, lost = 5. Source: https://openstax.org/books/principles-economics-3e/pages/3-5-demand-supply-and-efficiency.
2. mode = quantities, q0 = 1,000, q1 = 800, w = 2 gives dwl = 200, revenue = 1,600, lost = 200. Source: https://openstax.org/books/principles-economics-3e/pages/3-5-demand-supply-and-efficiency.
3. mode = curves, a = 100, b = 2, c = 10, d = 1, t = 120 gives q1Out = 0, dwl = 1,350, revenue = 0. Source: https://openstax.org/books/principles-economics-3e/pages/3-5-demand-supply-and-efficiency.

## FAQ

### 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.

## Sources

- OpenStax. Principles of Economics 3e, §3.5 Demand, Supply, and Efficiency (deadweight loss: the loss in social surplus at an inefficient quantity; price ceiling and price floor examples), retrieved 2026-10-18. https://openstax.org/books/principles-economics-3e/pages/3-5-demand-supply-and-efficiency