Calculating tax incidence
Whoever hands a tax to the state does not necessarily bear it. The burden splits according to the elasticities: the side of the market that can avoid it less bears more.
Formulas and rules
- Tax t paid by sellers: Q_s = c + d·(p − t)
- Consumer price p_c from the new equilibrium, producer price p_p = p_c − t
- Consumers’ share = (p_c − p*) / t = d / (b + d)
- Tax revenue = t × new quantity
- Deadweight loss = ½ × t × (q* − new quantity)
Your turn
Three problems on this topic. Type numbers like 12.5 or 1,250 and elasticities with their sign.
Task
In a market: demand Q_d = 277 − 5p, supply Q_s = 97 + 4p (p in €). The government levies a tax of €18.00 per unit, paid by the sellers. Find the price consumers pay, the price producers keep, the new quantity, the tax revenue and the share of the tax borne by consumers.
Price paid by consumers in €
Price kept by producers in €
New quantity
Tax revenue in €
Consumers’ share of the tax in %
Numbers like 12.5 or 1,250; elasticities with their sign (−0.75); percentages as a number (12.5). Round to 2 decimal places.
Worked example
In a market: demand Q_d = 209 − 5p, supply Q_s = 2 + 4p (p in €). The government levies a tax of €18.00 per unit, paid by the sellers. Find the price consumers pay, the price producers keep, the new quantity, the tax revenue and the share of the tax borne by consumers.
Worked solution
- 1.Without the tax: p* = 23, q* = 94
- 2.Sellers keep only p − 18: Q_s = 2 + 4 × (p − 18) = 4p − 70
- 3.Set equal to demand: 279 = 9p → consumer price 31
- 4.Producer price 31 − 18 = 13; quantity 209 − 5 × 31 = 54
- 5.Tax revenue 18 × 54 = €972.00
- 6.Consumer share = (31 − 23) / 18 = 44.44%. The less elastic side bears more.
Typical mistake: Assuming whoever hands the tax to the state also bears it. The burden splits according to the elasticities, whichever side pays the state.
In a market: demand Q_d = 318 − 5p, supply Q_s = 96 + p (p in €). A tax of €18.00 per unit is introduced. Work out the new quantity, the tax revenue and the deadweight loss.
Worked solution
- 1.Without the tax: q* = 133, p* = 37
- 2.With the tax: 318 − 5p = 96 + 1(p − 18) → consumer price 40, quantity 118
- 3.Tax revenue 18 × 118 = €2,124.00
- 4.Deadweight loss = ½ × tax × fall in quantity = ½ × 18 × 15 = €135.00
- 5.Consumers bear 1 / (5 + 1) of the tax → less than half
Typical mistake: Counting the tax revenue as a loss. Revenue goes to the state; only the triangle of units no longer traded is lost.
Common questions
Does it matter whether buyers or sellers pay the tax to the state?
Not for prices, quantity or who bears the burden. It only changes which price is measured before and which after tax.
Why is there a deadweight loss?
The tax crowds out trades that were worthwhile for both sides. Those gains disappear and do not show up in the tax revenue either.