Tax incidence — who really pays a tax — Economics

Before this

This leads to

The government sends the bill to sellers. Who pays it?

A government puts a tax of $5 on every unit of a product. The law says that sellers must pay it: for every unit they sell, they send $5 to the tax office. So the sellers pay the tax. Is that the end of the story?

It is not. Sellers can raise their price, and then buyers pay part of the tax at the till. But if the price rises, some buyers stop buying, and sellers lose sales. So how much of the $5 ends up on buyers, and how much stays with sellers?

The answer surprises many people, including many politicians. Who sends the money to the government does not decide who really pays. The split is decided by how strongly buyers and sellers respond to price. The last lecture ended with exactly this promise: a tax shifts supply, and the split you saw there decides who really pays it. Here is how.

What a tax does to the supply curve

Use the market from the last lecture: $Q_{d} = 100 - 2 P$ and $Q_{s} = 20 + 3 P$ , with equilibrium $P^{*} = 16$ and $Q^{*} = 68$ .

With a tax, there are two prices, not one. Buyers pay one price at the till. Call it $P_{c}$ , the price consumers pay. Sellers hand $5 of it to the government, so they keep $P_{p} = P_{c} - 5$ , the price producers receive. The gap between the two prices is always the tax.

Sellers decide how much to supply by what they keep, not by what buyers pay. Before the tax, keeping $16 per unit brought out 68 units. After the tax, sellers still want to keep $16 to supply 68 units. To keep $16, the price at the till must be $21. This is true at every quantity: each unit now needs a price $5 higher than before. So the whole supply curve moves up by $5.

In the equation, put the price sellers keep into the supply rule:

Q_{s} = 20 + 3 (P_{c} - 5) = 5 + 3 P_{c}

You have seen this supply equation before. In the last lecture, "costs of production rise, so sellers offer 15 fewer units at every price" gave exactly $Q_{s} = 5 + 3 P$ . A $5 tax per unit is a $5 rise in costs per unit. Moving the curve up by $5 is the same as moving it left by $3 \times 5 = 15$ units, because sellers offer 3 units per dollar.

Where the new equilibrium lands

Buyers still follow $Q_{d} = 100 - 2 P_{c}$ . Set the plans equal:

100 - 2 P_{c} 95 P_{c} = 5 + 3 P_{c} = 5 P_{c} = 19

Then $Q = 100 - 38 = 62$ , and sellers keep $P_{p} = 19 - 5 = 14$ . Check with the original supply rule: at a kept price of $14, sellers offer $20 + 3 \times 14 = 62$ . It matches.

A $5 tax on sellers moves supply up by $5 (dashed). Buyers now pay Pc = $19 and sellers keep Pp = $14, so 62 units are sold instead of 68. The tinted rectangle, $5 × 62 = $310, is the tax revenue. The small grey triangle is the deadweight loss.

Now compare with the old price of $16. Buyers pay $3 more than before. Sellers keep $2 less than before. The two parts add up to the tax: $3 + 2 = 5$ . So buyers bear 60% of the tax and sellers bear 40%, even though the law sent the whole bill to sellers.

Tax incidence

The incidence of a tax is how its burden is shared. The buyers' share per unit is $P_{c} - P^{*}$ . The sellers' share per unit is $P^{*} - P_{p}$ . Together they make the whole tax: $(P_{c} - P^{*}) + (P^{*} - P_{p}) = P_{c} - P_{p} = t$ .

The side that responds less to price bears more of the tax.

Why the less responsive side pays more

The last lecture showed how a shift splits between price and quantity. Use the same idea here. At the old price of $16, the tax opens a gap: sellers now offer 15 fewer units, so there is a shortage of 15. The price buyers pay must rise until the gap closes. Each $1 rise closes the gap by 5 units: 2 fewer wanted by buyers and 3 more offered by sellers. So the buyers' price rises by $15 \div 5 = 3$ dollars.

Look at where the 3 came from. In letters, with demand $Q_{d} = a - b P$ and supply $Q_{s} = c + d P$ , a tax $t$ moves supply left by $d t$ units. The buyers' price rises by $d t \div (b + d)$ . So:

buyers’ share sellers’ share = \frac{d}{b + d} = \frac{b}{b + d}

Here $b = 2$ and $d = 3$ , so buyers bear $\frac{3}{5}$ and sellers $\frac{2}{5}$ . Read the formula carefully, because it is the opposite of what many students expect. The buyers' share depends on $d$ , how strongly sellers respond. If sellers respond strongly (large $d$ ), they simply cut supply when their kept price falls, so the price buyers pay must rise a lot. If buyers respond weakly (small $b$ ), they keep buying at a higher price, so sellers do not need to cut their price much.

That is it. That is the whole idea. The side that can move away from the market more easily leaves more of the tax to the side that cannot.

The same rule can be written with elasticities. At the old equilibrium, price elasticity of demand (PED) is $- 2 \times 16 \div 68 \approx - 0.47$ , and price elasticity of supply (PES) is $3 \times 16 \div 68 \approx 0.71$ . The buyers' share is $PES \div (PES + ∣ PED ∣) = 0.71 \div 1.18 \approx 0.6$ , the same 60%. This version works for any small tax, even when the curves are not straight lines.

Test the rule on the graph below. Change how steep each curve is, keeping the same starting point, and watch who pays the $5.

Graph builder — who pays a $5 tax?

Both curves pass through P* = 16 and Q* = 68. Change b (how many fewer units buyers want for each $1) and d (how many more units sellers offer for each $1). A $5 tax is charged to sellers. Watch the two prices, the shares, the revenue and the loss. Try b = 0, then d = 0.

Buyers' response b (units per $1): 2

Sellers' response d (units per $1): 3

Who pays a $5 tax, for curves through P* = 16 and Q* = 68 with different responses b (buyers) and d (sellers):

b	d	Pc	Pp	Buyers' share	Quantity
2	3	$19	$14	60%	62
0	3	$21	$16	100%	68
2	0	$16	$11	0%	68
8	1	≈ $16.56	≈ $11.56	≈ 11%	≈ 63.6
1	8	≈ $20.44	≈ $15.44	≈ 89%	≈ 63.6

The side with the smaller response bears the larger share. When either side does not respond at all, quantity does not change and there is no deadweight loss.

The two extremes are worth remembering, because the reason is clear. When buyers do not respond at all ( $b = 0$ , a vertical demand curve, as for a medicine with no substitute), buyers pay the whole tax: sellers keep $16 and buyers pay $21. When sellers cannot respond ( $d = 0$ , a vertical supply curve, such as a fixed number of seats or parking spaces), sellers pay it all: buyers still pay $16, and sellers keep only $11.

Does it matter who sends the money?

Make a prediction before you read on.

Predict, then reveal — moving the tax to buyers

The same $5 tax is now charged to buyers instead: they pay the shop's price plus $5 to the government. What happens to the total that buyers pay, compared with the tax on sellers ($19)?

If the $5 tax is charged to buyers instead of sellers, buyers still pay $19 in total and sellers still keep $14. Only the label changes.

Buyers now pay the shop's price plus $5, so demand in terms of the shop's price becomes Qd = 100 − 2(P + 5) = 90 − 2P. With Qs = 20 + 3P: 70 = 5P, so the shop's price is $14, sellers keep $14, buyers pay $14 + $5 = $19, and 62 units are sold.

Economists separate two ideas here. The legal incidence is who must send the money to the government. The economic incidence is who ends up worse off. In a competitive market, the legal incidence does not change the economic incidence. Taxes on wages are a real example. In many countries, the law splits them between employer and worker. But the true split depends on how strongly employers and workers respond to the wage, not on the line in the law.

What the tax costs, and what it collects

The government collects $$5 \times 62 = $310$ per week. This is the tinted rectangle in the figure. Where does it come from? From the lecture on consumer and producer surplus, buyers lose the area between $P^{*}$ and $P_{c}$ , left of the demand curve: $\frac{1}{2} \times (68 + 62) \times 3 = $195$ . Sellers lose the area between $P_{p}$ and $P^{*}$ , left of the supply curve: $\frac{1}{2} \times (68 + 62) \times 2 = $130$ .

Together, buyers and sellers lose $195 + 130 = $325$ . The government collects only $310. The missing $15 is the deadweight loss (DWL), the small grey triangle. Its base is the fall in quantity, $68 - 62 = 6$ , and its height is the $5 tax:

DWL = \frac{1}{2} \times 6 \times 5 = 15

It comes from the 6 units that are no longer traded. Each was worth more to a buyer than it cost a seller to make, but the $5 tax pushed it out of the market. Notice when the triangle disappears: when either curve is vertical, quantity does not change, and the tax collects revenue with no deadweight loss. This is one reason governments like to tax goods that people keep buying whatever the price.

Stop and think

A $5 tax is charged on a good. A newspaper says: "The tax will raise prices by $5." When is that true?

Show the answer

Only when buyers bear the whole tax. That happens when demand does not respond to price at all. It also happens when supply responds without limit: a horizontal supply curve, as for a product that sellers can make in any amount at a fixed cost. In most markets, the price buyers pay rises by less than the tax, because sellers absorb part of it. In this lecture's market, the price rises by $3, not $5. A rise of the full $5 is a special case, not the rule.

A real case: Philadelphia's tax on sweetened drinks

Worked Example — A city tax and cross-border shopping (United States, 2017)

The policy. From January 2017, the city of Philadelphia taxed sweetened drinks at 1.5 cents per ounce, charged to distributors, not to shoppers.

The prediction. Think about each side of the market inside the city. Supply to one city is very responsive: drink makers and shops can send their drinks to stores just outside the city instead. So $d$ is very large, and the theory says shoppers should bear almost all of the tax. Demand inside the city is also fairly responsive, because shoppers near the edge can cross the border to buy. So the quantity sold in the city should fall a lot.

What happened. A study by Seiler, Tuchman and Yao (Journal of Marketing Research, 2021) found that, on average, 97% of the tax was passed on to shoppers, and prices rose by 34%. Sales of taxed drinks inside the city fell by 46%. More than half of that fall was offset by extra sales in stores just outside the city, so total purchases fell by only about 22%.

The verdict. The law charged distributors, but shoppers paid almost all of it, just as the rule predicts when supply is very responsive. The case also shows a limit of the simple model. Part of what looks like buyers "responding to price" was buyers moving the same purchases to another shop. This mattered for the tax's health goal. People bought fewer taxed drinks, but the fall was much smaller than the city's own sales figures suggested. The lecture on price elasticity of demand described a similar result for Mexico's tax on sugary drinks, which was also passed on to buyers.

Three classic mistakes

Trap 1 — "Whoever is charged, pays"

The law decides who sends the money, not who bears the burden. A tax charged to sellers or to buyers gives the same $P_{c}$ , the same $P_{p}$ and the same quantity. The responses of the two sides decide the split.

Trap 2 — Revenue from the old quantity

Tax revenue is the tax times the quantity after the tax: $5 \times 62 = 310$ , not $5 \times 68 = 340$ . A tax reduces the quantity it is charged on, unless one of the curves is vertical.

Trap 3 — The responsive side pays more

It is the other way round. The side that can respond more easily (by buying elsewhere, or by producing something else) escapes more of the tax. The side that cannot respond bears more of it.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

In a market, $Q_{d} = 200 - 4 P$ and $Q_{s} = 2 P - 40$ , so the equilibrium is $40 and 40 units. A tax of $6 per unit is charged to sellers. Find $P_{c}$ , $P_{p}$ , the new quantity, each side's share, the tax revenue and the deadweight loss.

Hint 1
Sellers keep $P_{c} - 6$ , so $Q_{s} = 2 (P_{c} - 6) - 40 = 2 P_{c} - 52$ .

Hint 2
Solve $200 - 4 P_{c} = 2 P_{c} - 52$ .

Full answer
$252 = 6 P_{c}$ , so $P_{c} = 42$ and $P_{p} = 36$ . Quantity: $200 - 168 = 32$ ; check: $2 \times 36 - 40 = 32$ . Buyers bear $42 - 40 = $2$ (one-third); sellers bear $40 - 36 = $4$ (two-thirds). This fits the rule: buyers' share $= d \div (b + d) = 2 \div 6$ . Revenue $= 6 \times 32 = $192$ . Deadweight loss $= \frac{1}{2} \times 8 \times 6 = $24$ .
In the same market, the $6 tax is charged to buyers instead. Show that the outcome is the same.

Hint 1
Buyers pay the shop's price $P$ plus $6, so $Q_{d} = 200 - 4 (P + 6)$ .

Hint 2
Set $176 - 4 P = 2 P - 40$ .

Full answer
$216 = 6 P$ , so the shop's price is $P = 36$ : sellers keep $36. Buyers pay $36 + 6 = $42$ . Quantity: $2 \times 36 - 40 = 32$ . The same $P_{c}$ , $P_{p}$ and quantity as in question 1. Only the legal incidence changed.
A city has exactly 500 parking spaces, and owners rent them all out whatever the price. The city puts a tax of $3 per space per day on owners. Who bears the tax? How much revenue does it raise, and how large is the deadweight loss?

Hint 1
The number of spaces cannot change, so supply is vertical: $d = 0$ .

Hint 2
If the number of spaces stays at 500, can the rent that drivers pay change?

Full answer
Owners bear the whole tax. The quantity stays at 500, so drivers compete for the same 500 spaces and pay the same rent as before. Owners keep $3 less per space. Revenue $= 3 \times 500 = $1500$ per day. There is no deadweight loss, because no space goes unused. This is the formula with $d = 0$ : buyers' share $= 0 \div (b + 0) = 0$ .
At a market's equilibrium, PED is −0.5 and PES is 1.5. A tax of $2 per unit is introduced. Estimate how much of it buyers bear. Then use the Philadelphia result (97% passed on to shoppers) to say how many times more responsive supply was than demand in that city, by this rule.

Hint 1
Buyers' share $= PES \div (PES + ∣ PED ∣)$ .

Hint 2
For Philadelphia, if buyers bear 97%, sellers bear 3%. The ratio of the shares equals $PES \div ∣ PED ∣$ .

Full answer
Buyers' share $= 1.5 \div (1.5 + 0.5) = 0.75$ , so buyers bear about $0.75 \times 2 = $1.50$ and sellers about $0.50. For Philadelphia, buyers' share $\div$ sellers' share $= PES \div ∣ PED ∣$ , so $PES \div ∣ PED ∣ \approx 0.97 \div 0.03 \approx 32$ . By this rule, supply to the city was roughly 30 times more responsive than demand. This is a rough reading, because the rule is for small taxes in a simple competitive market, but it shows how lopsided the two sides were.

The bigger idea

A tax is a wedge between two prices: the price buyers pay and the price sellers keep. Once you see the wedge, the same picture explains other policies. A subsidy is a wedge the other way, with sellers keeping more than buyers pay; the next lectures use exactly this diagram. And when a product harms people outside the market, such as through pollution, a tax can make the market better rather than worse. That idea returns in the lectures on externalities.

The deeper lesson is about laws and outcomes. A law can decide who sends money to the government. It cannot decide who really pays. That is decided by how people respond, and economics is the tool for predicting it.

Key concepts in this lesson

A tax of t per unit charged to sellers moves the supply curve up by t, creating two prices: Pc paid by consumers and Pp kept by producers, with Pc − Pp = t.
The burden is shared: buyers bear Pc − P* per unit and sellers bear P* − Pp, and these add up to the tax.
The side that responds less to price bears more of the tax: with straight lines, buyers' share is d ÷ (b + d), or PES ÷ (PES + |PED|).
Legal incidence (who sends the money) does not decide economic incidence (who is worse off).
Tax revenue is the tax times the new quantity, and the deadweight loss is the surplus lost on units that are no longer traded.