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Price elasticity and tax incidence
Who really pays a tax? Neither the one who hands it to the revenue service nor half each — and it can be proved.
share borne by the consumer = εs / (εs + εd)Problem / motivation
The state brings in a tax of €3 per pack of cigarettes, collected by the tobacconist. How much of those €3 does the smoker really bear? The answer depends neither on who signs the cheque to the tax office nor on some amicable split: it is entirely determined, and computable.
The key word is ELASTICITY. It measures how sensitive a quantity is to its price: ε = (% change in quantity) / (% change in price). If a 10% rise in the price makes purchases recede by 5%, the elasticity of demand is 0.5 — smokers barely react, and their demand is said to be RIGID or inelastic. Had it made them recede by 20%, it would be 2 and demand would be ELASTIC. Keep the image: elasticity measures the ability to ESCAPE a price rise.
The intuition this course is going to prove then fits in one sentence: whoever can escape avoids the tax, whoever cannot bears it. We shall not be content to assert it — we shall build a market with figures, apply the tax to it, check that the formula comes out right, then go looking for what a percentage split does not say: by how much the quantity sold recedes, what the tax raises, and what it destroys along the way.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
Let us start by measuring. A stylised tobacco market: buyers want Qd = 150 − 5·P thousand packs a day, sellers offer Qs = 20·P − 100 (course “The law of supply and demand” for how to read those relations). The equilibrium is at (€10; 100 thousand), which you find by setting 150 − 5·P = 20·P − 100. The elasticity is then computed AT the equilibrium point: on the demand side, a rise of €1 (i.e. 10% of €10) loses 5 thousand packs (i.e. 5% of 100), so εd = 5%/10% = 0.5. On the supply side, the same 10% rise brings out 20 of them, i.e. 20%, so εs = 20%/10% = 2. We assumed nothing: we READ both elasticities off the two relations. A point of vocabulary, useful for reading other sources: this one is the POINT elasticity, computed at a precise point. When it is computed over a finite movement — two prices, two quantities observed — it is called ARC elasticity, and in general the two coincide only approximately. Good news here: on a straight line, the arc computed from the starting point gives back exactly the point elasticity of that point, and the solving step will check it.
Now comes the tax: €3 per pack, handed to the revenue service by the tobacconist. Two prices now coexist — the one the buyer pays (Pd) and the one the seller keeps (Ps) — separated by exactly the tax: Pd = Ps + 3. That gap is called the TAX WEDGE, and it does all the work. Each of the two prices moves relative to the old equilibrium: the buyer will pay more than €10, the seller will receive less. The question “who pays the tax?” therefore becomes perfectly precise: by how much does Pd rise, and by how much does Ps fall? The two movements, together, must add up to €3.
Here is the proof, which uses nothing but what precedes. (1) On the new market, the quantity bought equals the quantity sold: the change in quantity is the SAME on both sides. (2) Write ΔPd and ΔPs for the changes in the two prices, in EUROS, and P* for the starting price. On the demand side, the relative change in quantity is −εd × (ΔPd/P*); on the supply side, +εs × (ΔPs/P*). Setting them equal, the common P* cancels on both sides: −εd·ΔPd = εs·ΔPs, now in euros. (3) But ΔPs = ΔPd − t, since the tax wedge is t euros. Substituting: −εd·ΔPd = εs·ΔPd − εs·t, i.e. εs·t = ΔPd·(εs + εd). (4) Hence ΔPd = t × εs/(εs + εd): the rise in the price paid is a FRACTION of the tax, and that fraction depends only on the two elasticities. The consumer share is εs/(εs + εd) and the producer share εd/(εs + εd) — the two do add up to 1, since their numerators sum to the denominator.
The numerator of the CONSUMER share is the elasticity of SUPPLY: that comes as a surprise, but it is logical — the more the seller can escape (large εs), the less is left for them, so the more the other side pays. Four readings fix ideas. If εd = 0 (absolutely rigid demand), the consumer share is 100%: they have nowhere to go. If εs = 0 (rigid supply, a building plot for instance), it is 0%: the producer takes the whole hit. If εd = εs, it is exactly 50% — there is the only case in which the poll's “fifty-fifty” is true. And if εd becomes very large, the consumer share tends to 0: a consumer who escapes perfectly is in the same position as a totally rigid producer. (The only case where the formula says nothing: both elasticities nil at once, where you land on 0/0 — a market where nobody reacts to anything has no adjustment mechanism anyway.) Click each term:
Solving / calculation
Let us apply the €3 tax to the market of step 1, without using the formula: we shall recompute the equilibrium by hand, as in the basic course. Then we shall confront the result with the formula — and go looking for the three magnitudes a percentage does not give.
- Before the tax: the equilibrium of step 1
150 − 5·P = 20·P − 100 → 250 = 25·P(P* = €10; Q* = 100 thousand packs) - The tax shifts supply: the seller keeps only P − 3
Qs becomes 20·(P − 3) − 100 = 20·P − 160the supply curve SHIFTS — this is not a slide - The new equilibrium, recomputed by hand
150 − 5·P = 20·P − 160 → 310 = 25·Pprice paid Pd = €12.40; price received Ps = €9.40; Q = 88 - The split, READ off the prices (no formula used)
the buyer pays €2.40 more (12.40 − 10); the seller receives €0.60 less (10 − 9.40)2.40 + 0.60 = €3: 80% for the buyer, 20% for the seller - In passing: the elasticities, recomputed on the observed movement (the promise of step 1)
demand side (−12%)/(+24%); supply side (−12%)/(−6%)0.5 and 2 — exactly the values read at the start: on a straight line, the arc from the initial point gives back the point elasticity - Confronting it with the formula of step 3
εs/(εs + εd) = 2 / (2 + 0.5) = 2 / 2.50.80 → 80%: the formula comes out EXACTLY right - And if the tax were levied on the BUYER? (equivalence, proved)
demand becomes 150 − 5·(P + 3) = 135 − 5·P, and 135 − 5·P = 20·P − 100 → 235 = 25·Pprice received €9.40, price paid €12.40, Q = 88 — STRICTLY the same result - What the percentage does not say (1): quantity and revenue
Q goes from 100 to 88, i.e. −12%; tax revenue = 3 × 88264 thousand euros a day — collected on a shrunken base - What the percentage does not say (2): what each side really loses
consumer surplus 1,000 → 774.4; producer surplus 250 → 193.6losses of 225.6 and 56.4, i.e. 282 in all — split 80/20, exactly like the tax - What the percentage does not say (3): the deadweight loss, by two routes
what is missing from the count: 282 − 264; or the area of the triangle of vanished trades: ½ × €3 × 1218 both ways — value destroyed, which ends up in nobody's pocket
Three lessons the formula alone did not give. First, it is exact, but it was not necessary: recomputing the equilibrium by hand gives the same 80/20, and shows the two prices into the bargain (€12.40 paid, €9.40 received). Second, the legal/economic equivalence is not a slogan: taxing the buyer shifts the demand curve instead of the supply curve, and you land back on the same prices and the same quantity to the cent — the legislator chooses who fills in the form, not who pays. Third, a tax does not merely transfer: out of the 282 of surplus lost by the two sides, the state recovers 264 and 18 evaporate — that is the DEADWEIGHT LOSS, the price of efficiency sacrificed. A remarkable fact: those surplus losses split 80/20 between consumers and producers, exactly like the tax itself. And that is no coincidence peculiar to our figures: on straight lines, each of the two losses is a trapezium of the SAME width — the average quantity (100 + 88)/2 — and of height equal to the movement of the price concerned. Their ratio therefore reduces to the ratio of the two price movements, that is, to the split of the tax. The result holds for any linear market.
consumer share = εs/(εs + εd); Pd = P* + consumer share × t; deadweight loss ≈ ½ × t × ΔQThis simulator does not merely display the split: it recomputes the whole market. Start by moving the tax — the PERCENTAGES do not budge an inch (they depend only on the elasticities) whereas the prices, the quantity, the revenue and the deadweight loss all move. Then make demand more elastic than supply and watch the split tip over. Finally, watch the last line: beyond a certain tax, assumption 4 no longer holds and the model tells you so.
Economic interpretation
Three readings: what this formula is really for, what the deadweight loss implies when choosing a tax, and why time changes everything.
Rigid demand and supple supply: the consumer pays. That is the case for tobacco (few substitutes, addiction), for petrol in the short run, for many medicines. Rigid supply and supple demand: the producer pays. The textbook case is land — the quantity of plots does not respond to price, so a tax on land is borne by the owner, which explains economists' long-standing interest in land taxation. And the intuition to be dismissed for good: “the seller pays the tax since the seller hands it over” is false, and the solving step proved it by redoing the computation both ways. Two honest caveats, all the same. On tobacco precisely, the studies that measure the pass-through of excise duties often find rates close to 100% — that is, a supply even more elastic than in our toy: our 80% is a conservative order of magnitude, calibrated so the arithmetic comes out round. And the equivalence between taxing the buyer and taxing the seller, exact in the model, assumes that the two versions are equally VISIBLE: behavioural research shows that a tax displayed at the till does not produce quite the same behaviour as a tax included in the ticket price.
A tax transfers money, but it also destroys value: the trades that would have taken place without it and no longer do. On our market, 18 out of 282 of lost surplus goes into nobody's pocket. And that loss grows with the elasticities — the more both sides can escape, the more trades vanish — and, for a small tax, it grows as the SQUARE of the tax: doubling the tax roughly quadruples the deadweight loss. Two practical consequences: several small taxes are better than one huge one, and the most inelastic bases (land, tobacco) are the ones that destroy the fewest trades. Note that for tobacco, the fall in quantity is precisely the public-health objective: what the model counts as a loss is here a benefit that was sought.
An elasticity is always relative to a horizon. In the short run, a motorist takes the hit from a rise in fuel: their vehicle, their journey and their job are given, their demand is rigid and they pay most of the tax. Over five years, they can move house, change vehicle, take up cycling: their demand becomes markedly LESS rigid — careful, not outright elastic for all that, long-run estimates for fuel generally stay below 1 — and the burden slides towards producers. The same tax therefore does not have the same incidence depending on the observation window — which is why two serious studies can display different splits without either being wrong.
Limits / critiques
Exercises
On a market, a price rise of 8% makes purchases recede by 6%. What is the price elasticity of demand (in absolute value, two decimals)?
A tax of €4 per unit, |εd| = 1, |εs| = 3. By how many euros does the price paid by the buyer rise?
A tax is brought in on a market where supply is PERFECTLY rigid (εs = 0): the quantity available does not respond to price, as with building plots. Who bears the tax?
On ANOTHER market, a tax of €5 per unit takes the quantity traded from 200 to 170 units. What is the deadweight loss, applying ½ × tax × fall in quantity?
Demand |εd| = 0.75, supply |εs| = 2.25. What share of the tax (in %) does the consumer bear?
In which case, and in which case ONLY, is the tax split exactly fifty-fifty?
True or false: levying the tax on the buyer rather than on the seller changes the real economic split.
True or false: in the long run, the burden of a fuel tax slides further towards producers.