What is deadweight loss? The tax example

A city puts a $14 tax on every concert ticket. Most students think: "The money goes from fans and promoters to the city, so nothing is lost, it just changes hands." Most of it does change hands. But a part of the gains from trade vanishes and reaches nobody, not the fans, not the promoters, not the city. That part is the deadweight loss.

The market with no tax

Take the ticket market of Lesson 4. Fans buy fewer tickets the higher the price: at each price, the number they want is (150 − price) × 10. Promoters, the firms that put on the show, offer more the higher the price: (price − 10) × 25. The two meet at $50 and 1,000 tickets.

Lesson 4 measures what each side gains. Consumer surplus (CS) is the most a fan would pay, minus the $50 they do pay, added up over every fan who buys: $50,000. Producer surplus (PS) is the $50 a promoter receives, minus the least they would accept, added up over every ticket sold: $20,000. Total gains: $70,000 a show, and no tax can make that number bigger.

Add the tax: a wedge opens

Now the city taxes promoters $14 a ticket. For a promoter, every ticket now costs $14 more to sell, so the supply line shifts up by $14. The market settles where the gap between what buyers pay and what sellers keep is exactly $14: buyers pay $60, promoters keep $60 − $14 = $46, and 900 tickets sell, down from 1,000. Move the tax with the slider.

Why do fewer tickets sell? At $60 some fans stay home; at $46 some promoters put on one show less. The 100 tickets between 900 and 1,000 were worth more to the fans who wanted them (between $50 and $60) than they cost the promoters who would have sold them (between $46 and $50). Those trades made both sides better off. Now they do not happen.

Where did the $70,000 go?

Fans and promoters lose $9,500 + $3,800 = $13,300. The city gets $12,600. The difference, $700, belongs to nobody: it is the red triangle, ½ × $14 × 100 tickets. The revenue is a transfer: it leaves one pocket and lands in another, and the city can spend it on something useful. The triangle is a loss: it never lands anywhere.

Note the split: of the $14, buyers absorb $10 (the price rose from $50 to $60) and sellers absorb $4 (they keep $46 instead of $50). The side that reacts less to price has fewer ways to escape the tax and carries more of it. Lesson 4 calls this tax incidence.

Double the tax, four times the loss

Halve the tax to $7: 950 tickets sell, 50 fewer than before, and the loss is ½ × $7 × 50 = $175. Double it to $28: 800 tickets sell, 200 fewer, and the loss is ½ × $28 × 200 = $2,800. Twice the tax, four times the loss, because both sides of the triangle double: the tax is its height, the lost tickets are its base. Slide and watch the curve bend upward.

The triangle also depends on how much each side reacts to price. If fans and promoters barely changed their behavior, few trades would be lost and the triangle would be tiny: the tax would be almost a pure transfer.

The trap on tests

• Revenue uses the new quantity. $14 × 900 = $12,600, not $14 × 1,000 = $14,000. The tax itself shrinks the number of tickets that get taxed.
• Revenue is not the deadweight loss. The rectangle is money that changes hands. The triangle is value that disappears: between demand and supply, from the new quantity to the old one.
• The formula. Deadweight loss = ½ × tax × (old quantity − new quantity). The tax is the height, the lost units are the base.
• Who writes the check does not matter. Tax the promoters or tax the fans: the same wedge, the same 900 tickets, the same $700 lost.

The same triangle appears with a price ceiling or a price floor, for the same reason: some trades that helped both sides stop happening. A subsidy, a tax in reverse, also creates one, by pushing quantity above 1,000: the extra tickets cost more to put on than fans value them.

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