Deadweight Loss of a Per-Unit Tax

A tax collects revenue. It also stops some trades happening at all — trades where the buyer valued the good above what it cost to make, but not above that plus the tax. Those trades were worth making to both sides, and they now do not happen. The value that would have been created and is not is the deadweight loss, and the point of this page is that it is separate from, and additional to, the money the tax collects.

The distinction is the whole thing. On every unit still trading, the tax is a TRANSFER: buyers and sellers are worse off, the treasury is better off, and society as a whole is neither. On the units that stopped trading, nobody gains. Buyers lose their surplus on those units, sellers lose theirs, the treasury collects nothing on them because they never happened. That is a pure loss, and it appears in no accounting statement anywhere. Arnold Harberger's work in the early 1960s put numbers on these triangles for the US economy and gave them their other name.

The geometry is a triangle whose height is the tax wedge $t$ — the gap the tax opens between what buyers pay and what sellers receive — and whose base is the fall in quantity. Hence $\tfrac{1}{2}t\,\Delta Q$.

Two properties follow, and both matter more than the formula.

Deadweight loss grows with the SQUARE of the tax rate. This is not obvious from the formula as written, because $\Delta Q$ is hiding a second factor of $t$: a bigger tax opens a bigger wedge AND drives out more trades, so both the height and the base grow together. Double the tax and the loss quadruples. Two consequences: a small tax is nearly free in efficiency terms, and the efficiency case for spreading a given revenue requirement thinly across many goods rather than heavily on a few is arithmetic rather than ideology.

Deadweight loss is driven entirely by elasticity. If neither side can move — perfectly inelastic supply or demand — the quantity does not fall, $\Delta Q$ is zero, and the loss is zero. The tax is then pure transfer. This is the whole efficiency argument for taxing land, and for the observation that taxes on addictive goods raise revenue efficiently whatever else may be said about them. Conversely, taxing something with elastic supply and elastic demand destroys a great deal of trade for very little revenue.

Now the limits, because this triangle gets quoted far past what it can carry. It assumes both curves are straight over the range in question, which for a small tax is a fair local approximation and for a large one is not. It measures ONLY the efficiency cost of the reduced quantity: it says nothing about administration and compliance costs, which for some taxes exceed the deadweight loss outright, and nothing about distribution, since a tax can be efficient and still fall hardest on people least able to pay. It assumes the market was efficient before the tax arrived, which fails wherever an externality or market power is already present — and in that case a tax can REDUCE the total loss rather than create one, which is the entire logic of a carbon price or a congestion charge. And it says nothing whatever about what the revenue is spent on, which is usually the actual question.

Used carefully — comparing two ways of raising the same revenue, say — it is one of the sharpest tools in applied economics. Used as a single number for "the cost of a tax", it is answering a much smaller question than it appears to.