To calculate tax incidence, divide the price elasticity of supply by the sum of both elasticities to get the consumer’s share of the tax, and divide the absolute value of the price elasticity of demand by that same sum to get the producer’s share. Multiply each share by the per-unit tax to see the dollar impact on each side. The party that is less price-sensitive ends up carrying more of the tax, regardless of which side the law tells to write the check.
The two formulas are:
- Consumer’s share = Es / (Es + |Ed|)
- Producer’s share = |Ed| / (Es + |Ed|)
Es is the price elasticity of supply. |Ed| is the absolute value of the price elasticity of demand. The two shares always sum to 1.
What You Need Before You Start
Four numbers from the market you’re analyzing, plus the tax itself:
- P1: price before the tax
- Q1: quantity sold before the tax
- P2: price after the tax takes effect
- Q2: quantity sold after the tax takes effect
- The tax amount, in dollars per unit
P1 and Q1 come from sales records or financial statements for the period right before the tax. P2 and Q2 come from comparable data after the market has had time to settle. Your incidence estimate is only as reliable as those four numbers.
For a per-unit excise tax, the tax amount is a fixed dollar figure written into the statute. Federal gasoline excise tax is 18.3 cents per gallon, or 18.4 cents including the Leaking Underground Storage Tank surcharge.1Office of the Law Revision Counsel. 26 USC 4081 – Imposition of Tax Federal cigarette excise tax runs $50.33 per thousand small cigarettes, about $1.01 per pack of 20.2Office of the Law Revision Counsel. 26 USC 5701 – Rate of Tax
Calculate the Elasticities Using the Midpoint Formula
Price elasticity measures how much quantity responds to a price change. The midpoint formula is the standard because it gives the same answer whether you measure from P1 to P2 or P2 to P1.
For the elasticity of demand:
- Find the percentage change in quantity: (Q2 − Q1) divided by the average of Q1 and Q2, which is (Q1 + Q2) / 2.
- Find the percentage change in price the same way: (P2 − P1) divided by (P1 + P2) / 2.
- Divide the percentage change in quantity by the percentage change in price.
That result is Ed. It almost always comes out negative because higher prices reduce quantity demanded. When you plug it into the incidence formulas, drop the sign and use the absolute value.
Run the identical calculation on the supply side using producer cost and quantity data. That gives you Es. Supply elasticity is typically positive because producers offer more at higher prices.
Apply the Incidence Formulas
With Es and |Ed| in hand, the split follows directly. The logic behind the formulas is worth understanding before you crunch numbers: whichever side of the market is more flexible escapes more of the tax, and whichever side is less flexible absorbs more. Consumers who can readily switch to substitutes push the burden onto producers. Producers who can readily cut output or shift capacity push it back onto consumers.
Worked Example
A $2.00 per-unit excise tax is imposed on a product. You’ve calculated Es = 0.3 and Ed = −0.9. So |Ed| = 0.9.
- Consumer’s share: 0.3 / (0.3 + 0.9) = 0.3 / 1.2 = 0.25, or 25%
- Producer’s share: 0.9 / (0.3 + 0.9) = 0.9 / 1.2 = 0.75, or 75%
Consumers have three times the price sensitivity of producers here, so producers absorb three times as much of the tax. Applied to the $2.00 tax:
- Consumers pay $0.50 more per unit (25% of $2.00)
- Producers receive $1.50 less per unit after remitting the tax (75% of $2.00)
If the pre-tax price was $10.00, the buyer now pays $10.50 and the seller retains $8.50 after sending $2.00 to the government. The $2.00 gap between the buyer’s price and the seller’s net price is the tax wedge.
What the Extreme Cases Show
The formulas produce clean results at the boundaries, and those cases are a good sanity check.
When demand is perfectly inelastic (|Ed| = 0), the consumer share becomes Es / (Es + 0) = 1. Buyers bear the entire tax because they buy the same quantity at any price, letting sellers pass the full amount through. Cigarettes approximate this in the real world.
When supply is perfectly elastic (Es approaches infinity), the consumer share also approaches 100%. Producers who can costlessly redirect their output won’t accept lower returns, so the tax lands on buyers.
When supply is perfectly inelastic (Es = 0), producers absorb the whole tax because they can’t reduce quantity. Beachfront hotels locked into fixed seasonal capacity are the classic illustration.
Add Deadweight Loss to the Picture
Incidence tells you who pays the tax. Deadweight loss tells you how much economic value the tax destroys outright. When the tax raises the price buyers face and lowers the return sellers receive, some transactions that would have happened at the old price stop happening. Nobody collects on those vanished trades.
The standard approximation is a triangle:
Deadweight loss = 0.5 × tax amount × (Q1 − Q2)
Using the earlier numbers, if the $2.00 tax pulls quantity from 1,000 units to 950 units, deadweight loss is 0.5 × $2.00 × 50 = $50.00. That $50 doesn’t go to consumers, producers, or the government. It evaporates. More elastic markets produce bigger quantity drops and larger deadweight losses, which is one reason governments prefer taxing goods with inelastic demand.
Where the Standard Calculation Breaks Down
The formulas above assume a competitive market and a per-unit tax, and they treat elasticities as fixed. Three situations complicate that picture.
Ad Valorem Taxes
Specific taxes are a fixed dollar amount per unit. The 18.3-cent gasoline tax stays 18.3 cents whether gas costs $3.00 or $5.00 a gallon.1Office of the Law Revision Counsel. 26 USC 4081 – Imposition of Tax Ad valorem taxes are percentage-based. A 7% sales tax collects $7.00 on a $100 item and $14.00 on a $200 item. The elasticity split still governs who bears the burden, but the dollar amount scales with price, so the supply curve doesn’t shift in parallel the way it does with a per-unit tax. Expect the shift to widen at higher prices.
Short Run vs. Long Run
Elasticities stretch over time. In the short run, a business with a signed lease, hired staff, and purchased inventory can’t easily exit, so supply is relatively inelastic and producers absorb more of a new tax. Consumers locked into habits or contracts may not immediately switch products either. In the long run, both sides find workarounds. Producers close locations, redirect investment, or move into untaxed products. Consumers change habits, find substitutes, or shop across jurisdictions. The burden migrates toward whatever factor stays truly fixed, which is often land or other immovable assets. A soda tax may initially hurt store owners; over years, as businesses and workers relocate, the burden can migrate to commercial landlords whose property can’t leave. If you’re projecting multi-year impacts, assume the burden will drift toward the party with the fewest escape routes.
Market Power
The standard formulas assume many buyers and many sellers. A monopolist or dominant firm doesn’t pass a tax through dollar-for-dollar. Depending on the shape of the demand curve, a firm with pricing power may pass through less than 100% of a per-unit tax, absorbing the rest as reduced profit. In some cases the firm raises prices by more than the tax, a result called over-shifting, when the demand curve is convex enough. For competitive industries the elasticity-based split works well; for a product controlled by one or two firms, treat the straightforward calculation as a starting point, not a final answer.