Deadweight Loss of Taxes

A tax reduces the quantity traded from Q* to Qt (the post-tax quantity). The government collects revenue = tax $\times$ Qt, but this is a transfer, not a loss. The deadweight loss is the value of the trades between Qt and Q* that no longer happen. These were trades where buyers valued the good more than it cost to produce, but the tax made them uneconomical.

Interactive — Tax wedge: drag the tax to see DWL grow

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Same market as the worked example (D: P = 50 − Q, S: P = 10 + Q). Drag the tax slider and watch the wedge open up.

Now the math. The DWL is the area of the red triangle. Its base is the lost quantity $Q^{*} - Q_{t}$ and its height is the tax $t$ :

Deadweight loss of a per-unit tax t: the triangle between the old and new quantities, with height equal to the tax

Trap · Warning

DWL grows with the square of the tax

When the tax doubles, the lost quantity $Q^{*} - Q_{t}$ also roughly doubles. The triangle is twice as tall AND twice as wide — so its area quadruples. In symbols, $D W L \propto t^{2}$ . This is why economists prefer broad-based taxes with low rates over narrow taxes with high rates.

Tax revenue = $t \times Q_{t}$ is the gray rectangle. As the tax rises, Qt falls. Revenue rises at first (higher rate on still-decent volume), but eventually the tax kills so much activity that revenue falls. That’s the idea behind the Laffer Curve — there’s a tax rate that maximises revenue, and going beyond it is counterproductive.

Worked Example

Same market: Demand P = 50 - Q, Supply P = 10 + Q, tax = 10 dollars. Q* = 20, Qt = 15. Find DWL and tax revenue.

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Why does DWL grow with the square of the tax?

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DWL of taxation:

Tax revenue is a transfer, not a loss. The DWL represents:

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