Difficulty: Intermediate | Prerequisites: Supply and demand basics, linear equations, percentage-change calculations.
Tax incidence is one of the most policy-relevant topics in intermediate micro. It answers a deceptively simple question: when the government slaps a tax on a good, who really pays it? The answer depends not on who physically hands the money to the government, but on the relative elasticities of supply and demand. This set of problems walks through the full chain: find the pre-tax equilibrium, compute elasticities at that point, use those elasticities to split the tax burden, then derive the new price buyers pay and the new quantity traded.
If you are comfortable solving for equilibrium by setting Qs = Qd and can calculate a point elasticity using (dQ/dP) x (P/Q), you are ready for this material.
A per-unit tax drives a wedge between the price buyers pay and the price sellers receive. The side of the market that is less elastic (less responsive to price) bears more of the tax. It does not matter whether the law says buyers or sellers remit the tax: the economic incidence is the same either way.
Equilibrium price and quantity
The price and quantity at which quantity supplied equals quantity demanded. Graphically, it is the intersection of the supply and demand curves. In simple terms, it is the price where the market clears and there is no surplus or shortage.
Price elasticity of demand (Ed)
A measure of how responsive quantity demanded is to a change in price, calculated as (dQd/dP) x (P/Q) at a given point. The result is typically negative (demand slopes down), but we often discuss its absolute value. Think of it as: how much do buyers flinch when the price moves?
Price elasticity of supply (Es)
A measure of how responsive quantity supplied is to a change in price, calculated as (dQs/dP) x (P/Q) at a given point. This is positive for an upward-sloping supply curve. Think of it as: how eagerly do sellers ramp up (or cut back) production when the price shifts?
Tax incidence
The division of the economic burden of a tax between buyers and sellers. Legal incidence (who writes the cheque to the government) can differ from economic incidence (who actually ends up worse off). In simple terms, tax incidence tells you who really pays.
Per-unit (excise) tax
A fixed dollar amount charged per unit of a good sold (e.g. $1 per pack of cigarettes). It creates a vertical wedge of size t between the demand price and the supply price.
Tax wedge
The gap between the price buyers pay (Pb) and the price sellers receive (Ps) after a tax is imposed. For a per-unit tax of t: Pb = Ps + t.
Set quantity supplied equal to quantity demanded: Qs = Qd.
With Qs = -100 + 20P and Qd = 200 - 5P, solve: -100 + 20P = 200 - 5P.
Combine terms: 25P = 300, so P* = 12.
Substitute back: Q* = -100 + 20(12) = 140 (or equivalently, 200 - 5(12) = 140).
Equilibrium: P* = $12, Q* = 140 packs.
The elasticity formula at a point is: E = (dQ/dP) x (P/Q).
For demand: dQd/dP = -5 (the coefficient on P in the demand equation). So Ed = (-5) x (12/140) = -60/140 = -3/7. The absolute value is 3/7, which is less than 1, meaning demand is inelastic at this point.
For supply: dQs/dP = 20 (the coefficient on P in the supply equation). So Es = 20 x (12/140) = 240/140 = 12/7. This is greater than 1, meaning supply is elastic at this point.
Leave answers as fractions: Ed = -3/7, Es = 12/7.
The proportion of a per-unit tax borne by sellers uses the formula: Seller share = |Ed| / (|Ed| + Es).
Seller share = (3/7) / (3/7 + 12/7) = (3/7) / (15/7) = 3/15 = 1/5.
Sellers bear 1/5 (20%) of the tax.
The proportion borne by buyers: Buyer share = Es / (|Ed| + Es).
Buyer share = (12/7) / (15/7) = 12/15 = 4/5.
Buyers bear 4/5 (80%) of the tax.
The less elastic side bears the larger share. Here demand is quite inelastic (smokers do not easily cut back), so buyers shoulder most of the burden.
With a $1 per-pack tax, buyers pay: Pb = P* + (buyer share x t) = 12 + (4/5)(1) = 12 + 4/5 = $12.80.
Sellers receive: Ps = P* - (seller share x t) = 12 - (1/5)(1) = 12 - 1/5 = $11.80.
New quantity sold: plug Pb into the demand equation: Qd = 200 - 5(12.80) = 200 - 64 = 136 packs.
Or plug Ps into supply: Qs = -100 + 20(11.80) = -100 + 236 = 136. Both match.
The tax reduces quantity from 140 to 136, a drop of 4 packs.
A proposal to levy the tax on sellers instead of buyers does not change the economic outcome.
The statutory incidence (who remits the tax) differs, but the economic incidence (who bears the burden in terms of price changes) stays the same.
This is a fundamental result in public economics: the economic burden of a tax is determined by relative elasticities, not by legislation.
Equilibrium condition
Set Qs = Qd and solve for P.
Q_S = Q_D \implies -100 + 20P = 200 - 5P \implies P^* = 12,\; Q^* = 140Point elasticity of demand
E_d = \frac{dQ_d}{dP} \times \frac{P}{Q}Point elasticity of supply
E_s = \frac{dQ_s}{dP} \times \frac{P}{Q}Seller's share of the tax
\text{Seller share} = \frac{|E_d|}{|E_d| + E_s}Buyer's share of the tax
\text{Buyer share} = \frac{E_s}{|E_d| + E_s}Post-tax buyer price
P_b = P^* + \left(\frac{E_s}{|E_d| + E_s}\right) \times tPost-tax seller price
P_s = P^* - \left(\frac{|E_d|}{|E_d| + E_s}\right) \times tPost-tax quantity
Plug Pb into the demand equation (or Ps into the supply equation) to get the new quantity.
Students often think that if the law says sellers pay the tax, sellers actually bear the full burden. They do not. The economic burden depends on elasticities, not on the statutory assignment.
Some students forget the absolute value when computing |Ed| for the tax-incidence formula. The demand elasticity is negative, but the formula uses its magnitude.
A common error is computing elasticity as just the slope (dQ/dP) without multiplying by P/Q. The slope and the elasticity are different things. Slope is constant along a linear curve, but elasticity varies at every point.
Students sometimes assume a $1 tax raises the buyer price by exactly $1. That only happens if supply is perfectly elastic. In general, buyers pay less than the full tax, and the rest falls on sellers.
Tax incidence is a perennial exam favourite. Expect to solve for equilibrium, compute elasticities, and split the tax burden in a single multi-part question.
The "does it matter who the tax is levied on?" question (legal vs. economic incidence) appears frequently as a short-answer or true/false item. The answer is always: no, the economic outcome is identical.
You may be asked to leave elasticities as fractions. Do not convert to decimals unless told to.
Know that inelastic demand means buyers bear more of the tax. If supply is relatively more elastic than demand, the burden falls disproportionately on consumers.
True or False: If demand is perfectly inelastic, the entire tax is borne by buyers. (True.)
Fill in the blank: The point elasticity formula is E = (dQ/dP) x ____. (P/Q.)
True or False: A $1 per-unit tax always raises the market price by exactly $1. (False. The price increase to buyers is less than $1 unless supply is perfectly elastic.)
Fill in the blank: When |Ed| < Es, the ____ side of the market bears more of the tax. (demand / buyer.)
True or False: Switching the legal incidence of a tax from buyers to sellers changes who ultimately bears the economic burden. (False.)
Q: Given Qs = -100 + 20P and Qd = 200 - 5P, what are the equilibrium price and quantity?
A: Set Qs = Qd: -100 + 20P = 200 - 5P, so 25P = 300, P* = $12. Then Q* = 200 - 5(12) = 140.
Q: What is the price elasticity of demand at P = 12, Q = 140?
A: Ed = (dQd/dP)(P/Q) = (-5)(12/140) = -60/140 = -3/7.
Q: What is the price elasticity of supply at P = 12, Q = 140?
A: Es = (dQs/dP)(P/Q) = (20)(12/140) = 240/140 = 12/7.
Q: If a $1 per-pack tax is imposed, what proportion is borne by sellers?
A: Seller share = |Ed| / (|Ed| + Es) = (3/7) / (3/7 + 12/7) = 3/15 = 1/5 (20%).
Q: What price do buyers pay after the $1 tax?
A: Pb = 12 + (4/5)(1) = 12.80. Buyers pay $12.80 per pack.
Q: What is the new quantity sold after the tax?
A: Qd = 200 - 5(12.80) = 136 packs.
Q: A politician proposes levying the tax on cigarette companies instead of smokers. Does this change the economic outcome?
A: No. The economic incidence of a tax is determined by the relative elasticities of supply and demand, not by the statutory incidence. The prices buyers pay and sellers receive, and the quantity traded, are all identical regardless of which side the tax is formally imposed on.
Tax incidence connects directly to consumer and producer surplus. The tax creates a deadweight loss (the triangle of lost surplus from the reduced quantity), which you will encounter in welfare analysis. It also ties into government revenue calculations: total tax revenue = t x Q(after tax), split between the buyer and seller portions.
The elasticity concepts here carry into every corner of micro: pricing power (firms facing inelastic demand can pass on costs), international trade (tariff incidence), and labour economics (payroll tax burden between workers and employers).
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