Article — Deadweight Loss Calculator
Deadweight loss calculator: the cost of a tax in one triangle
Deadweight loss is the value of trades that fail to happen because a tax, price ceiling, or other intervention pushes price away from the market-clearing level. The standard formula is DWL = ½ × ΔQ × ΔP, where ΔQ is the drop in quantity sold and ΔP is the per-unit price wedge. A $2 tax that reduces sales from 1,000 to 800 units creates deadweight loss of $200. The loss is welfare nobody captures: not buyers, not sellers, not the government.
Arnold Harberger formalised the idea in 1954, drawing the loss as a triangle bounded by the supply curve, the demand curve, and the price wedge. The calculator above accepts pre-tax quantity, post-tax quantity and the price change, and returns the triangle area along with tax revenue and the excess burden ratio.
What is deadweight loss?
Deadweight loss is a welfare concept. In a free market, every buyer who values a good above the price buys it, and every seller whose cost falls below the price sells it. The sum of consumer surplus and producer surplus measures the welfare those trades generate.
Insert a tax and that surplus shrinks. Part transfers to the government as revenue. Part disappears entirely: the trades that would have happened at the original price stop once the tax wedge makes them unprofitable. That vanished surplus is deadweight loss.
How to calculate deadweight loss
DWL = ½ × ΔQ × ΔPΔQ = Q₁ - Q₂Tax revenue = Q₂ × ΔPSuppose a state imposes a $2 excise tax on a product selling 1,000 units at $10. After the tax, sales fall to 800 units. The Harberger triangle has base 200 and height $2. The area is ½ × 200 × 2 = $200 of deadweight loss. The state collects $2 × 800 = $1,600 in revenue. The excess burden ratio is 200 / 1,600 = 12.5%.
The Harberger triangle and its formulation
Harberger published "Monopoly and Resource Allocation" in 1954, and the triangle became the canonical visualisation of welfare loss. Before Harberger, economists discussed deadweight loss in abstract terms.
Harberger's original estimate of monopoly deadweight loss in the United States was about 0.1% of GDP — much smaller than economists had assumed. The number sparked decades of debate. Later studies using broader definitions of monopoly (including markup over marginal cost) push the figure higher, but the small-triangle insight persisted: most market failures are smaller in welfare terms than commentary suggests.
The triangle is geometrically robust. As long as supply and demand are approximately linear over the relevant price range, the formula DWL = ½ × ΔQ × ΔP holds. For larger price changes, an elasticity-based version captures the curvature: DWL = ½ × |e| × Q × (ΔP)² / P. Public finance courses introduce both forms.
Deadweight loss and price elasticity
Elasticity is the lever that determines triangle size. The more responsive quantity is to price, the larger the quantity drop for a given tax, and the larger the deadweight loss. The Ramsey rule, named after Frank Ramsey's 1927 paper, captures this: to minimise total deadweight loss, tax inelastic goods most heavily.
- Insulin (e = -0.05) tiny DWL; quantity barely responds to price
- Salt (e = -0.1) near-zero DWL even at high tax rates
- Gasoline short run (e = -0.25) moderate DWL; long run more elastic
- Tobacco (e = -0.4) modest DWL among addicted users, larger among teens
- Restaurant meals (e = -1.6) large DWL; many substitutes exist
- Luxury watches (e = -2.5) very large DWL per dollar of revenue
- Air travel leisure (e = -2.0) large DWL; booking decisions are price-sensitive
This explains why economists support broad-based consumption taxes rather than narrow excise taxes. A small tax on many goods produces less deadweight loss than a large tax on a few elastic goods.
Deadweight loss from price controls
A binding price ceiling such as rent control creates a shortage. Buyers want more housing than landlords are willing to supply at the capped rent. The trades that would happen at the market rent simply do not occur. Those forgone trades form a deadweight-loss triangle identical in structure to the tax case.
Price floors work the same way in reverse. A minimum wage above the market-clearing wage creates a labour surplus: workers willing to work at the minimum cannot find employment. The unemployed worker-hours are the base of a deadweight-loss triangle. The Congressional Budget Office's 2019 review of a $15 federal minimum wage estimated lost employment of 1.3 million workers, with deadweight loss of roughly $9 billion per year against gains for those who kept their jobs.
Price controls produce more deadweight loss in the long run than the short run. Supply elasticity rises over time as producers exit the market, retool, or build alternatives. Rent control's small initial DWL grows as landlords convert apartments to condos and developers stop building. New York's rent-stabilisation rules, in place since 1969, are the textbook long-run example.
Deadweight loss from monopoly
A monopolist sets price above marginal cost, restricting output to capture more producer surplus. The trades that would occur in a competitive market but do not occur under the monopoly form a deadweight-loss triangle. Modern estimates put US monopoly deadweight loss at 1-3% of GDP, higher than Harberger's original figure because the definition of market power has broadened.
The Federal Trade Commission and Department of Justice use deadweight loss as one criterion in antitrust enforcement, weighing the welfare cost of a merger against expected efficiency gains.
Deadweight loss vs tax revenue
Tax revenue is the transfer from buyers and sellers to the government. Deadweight loss is the welfare that disappears entirely. The ratio between them, the excess burden ratio, lets economists rank tax instruments by efficiency.
A tax with high excess burden may still be the right policy if it addresses an externality or distributes burden fairly. Carbon taxes have high DWL relative to revenue, but the social benefit from reduced emissions can outweigh it. Deadweight loss is one cost on a scorecard, not a final verdict.
The CBO reports excess burden ranges of 15-25% for broad sales taxes, 30-50% for personal income taxes, and 40-60% for corporate income taxes. Lump-sum taxes have an excess burden of zero by construction, which is why economists since Henry George have proposed land-value taxation as the most efficient revenue source.
Common deadweight loss mistakes
Confusing deadweight loss with tax revenue. Revenue is the transfer; DWL is the loss. The two come out of the same surplus pool, but they are not the same thing. A $1 billion tax that funds a project worth $1.5 billion is net positive even if its DWL is $100 million.
Assuming the triangle is linear at all scales. The formula assumes linear supply and demand over the relevant range. For very large taxes (50% or more), the elasticity-based form is more accurate: DWL grows as the square of the price change, not linearly.
Ignoring long-run elasticity. Short-run elasticities understate behavioural response. A gasoline tax may seem to generate little DWL in the first year, but five years out vehicle stock changes and the elasticity doubles or triples.
Treating monopoly DWL as the whole cost. Monopolies impose welfare losses beyond the static triangle: reduced innovation, captured regulators, distortion of complementary markets. Static DWL is a lower bound.
Forgetting general-equilibrium effects. A tax on one good shifts demand and supply in related markets. The deadweight loss measured in the taxed market may understate or overstate the total welfare cost once cross-market effects propagate.