Difficulty: Intermediate | Prerequisites: Demand functions and partial derivatives (see companion notes on Demand Functions).
Elasticity is the workhorse concept for translating derivatives into economically meaningful measures. Where a partial derivative tells you the absolute change in quantity per unit change in price, elasticity tells you the percentage change, which makes it comparable across goods with different units and scales. Tax incidence then uses elasticity to determine who really bears the burden of a tax, regardless of who technically writes the cheque. If you are not yet comfortable taking partial derivatives and solving for equilibrium, review those topics first.
Elasticity measures responsiveness in percentage terms. Price elasticity of demand tells you how sensitive buyers are to price changes; cross-price elasticity classifies goods as substitutes or complements; income elasticity classifies goods as normal or inferior. When a tax is imposed, the more inelastic side of the market bears a larger share of the burden, and the tax creates a deadweight loss (a net loss to society).
Price elasticity of demand (ε)
The percentage change in quantity demanded divided by the percentage change in price. Formally: ε = (∂Q/∂p) × (p/Q). A negative number (by the law of demand). When |ε| > 1, demand is elastic; when |ε| < 1, demand is inelastic. Think of it as: how much do buyers flinch when the price moves?
Cross-price elasticity of demand
The percentage change in quantity demanded of good A divided by the percentage change in the price of good B. Formally: ε_AB = (∂Q_A/∂p_B) × (p_B/Q_A). Positive means substitutes; negative means complements.
Income elasticity of demand
The percentage change in quantity demanded divided by the percentage change in income. Formally: ε_Y = (∂Q/∂Y) × (Y/Q). Positive means normal good; negative means inferior good. Above 1 means a luxury good.
Price elasticity of supply
The percentage change in quantity supplied divided by the percentage change in price. Formally: ε_s = (∂Q_s/∂p) × (p/Q_s). Always positive for an upward-sloping supply curve.
Tax incidence
How the burden of a tax is divided between buyers and sellers. It depends on the relative elasticities of demand and supply, not on who is legally required to pay the tax. Think of it as: the law says who writes the cheque, but the market decides who actually pays.
Deadweight loss (DWL)
The net loss in total surplus caused by a tax (or other market distortion). It represents transactions that would have happened in a free market but no longer do because of the tax. Graphically, it is the triangle between the supply and demand curves, from the new (taxed) quantity to the old (free-market) quantity.
Consumer price (p^c) and seller price (p^s)
After a per-unit tax t, the price consumers pay is p^c and the price sellers receive is p^s = p^c - t. The wedge between them is the tax.
The coconut oil demand function (Buschena and Perloff, 1991) provides a clean worked example:
Q^d = 1,200 - 9.5p + 16.2p_p + 0.2Y
where p = price of coconut oil (cents/lb), p_p = price of palm oil (cents/lb), Y = consumer income, and Q is in thousands of metric tons per year.
Given values: p = 45 cents, p_p = 31 cents, Q = 1,275.
Price elasticity of demand
∂Q/∂p = -9.5
ε = (∂Q/∂p) × (p/Q) = -9.5 × (45/1,275) = -9.5 × 0.0353 = -0.335
|ε| < 1, so demand is inelastic.
Interpretation: a 1% increase in the price of coconut oil reduces quantity demanded by about 0.34%. Buyers are not very responsive to price changes, which makes sense for a commodity with limited close substitutes in certain uses.
Cross-price elasticity with respect to palm oil
∂Q/∂p_p = +16.2
ε_cross = (∂Q/∂p_p) × (p_p/Q) = 16.2 × (31/1,275) = 16.2 × 0.0243 = +0.394
The sign is positive, so coconut oil and palm oil are substitutes.
Interpretation: a 1% increase in the price of palm oil raises coconut oil demand by about 0.39%. When palm oil gets more expensive, some buyers switch to coconut oil.
Income elasticity
∂Q/∂Y = +0.2
ε_Y = (∂Q/∂Y) × (Y/Q)
We need Y. We can back it out from the demand function: 1,275 = 1,200 - 9.5(45) + 16.2(31) + 0.2Y, which gives 1,275 = 1,200 - 427.5 + 502.2 + 0.2Y, so 0.2Y = 0.3, and Y = 1.5 (in thousands, so $1,500).
ε_Y = 0.2 × (1.5/1,275) = 0.000235
The sign is positive but very small, so coconut oil is a normal good, though demand is almost completely unresponsive to income changes. This is typical for basic commodities.
Consider Q^d = 30 - 3p and Q^s = 2 + 4p.
Finding equilibrium
Set Q^d = Q^s:
30 - 3p = 2 + 4p
28 = 7p
p* = 4, Q* = 30 - 3(4) = 18
Price elasticity of demand at equilibrium
∂Q^d/∂p = -3
ε_d = -3 × (4/18) = -0.667
|ε_d| = 0.667 < 1, so demand is inelastic at equilibrium.
Price elasticity of supply at equilibrium
∂Q^s/∂p = 4
ε_s = 4 × (4/18) = 0.889
Who bears more tax burden?
The more inelastic side bears the larger share of the tax. Here, demand (|ε| = 0.667) is more inelastic than supply (ε = 0.889). Consumers will bear a larger share of any tax imposed in this market.
The intuition: inelastic demand means buyers cannot easily switch away from the good. They absorb more of the price increase because they have fewer alternatives.
Continuing with Q^d = 30 - 3p and Q^s = 2 + 4p. A $2 per-unit tax is imposed on the seller.
Setting up the tax problem
When the seller pays the tax, the effective price the seller receives is (p - 2), where p is the price the consumer pays. The supply curve, from the seller's perspective, becomes:
Q^s = 2 + 4(p - 2) = 2 + 4p - 8 = -6 + 4p
Equivalently, you can think of the supply curve shifting up by $2.
Finding the new equilibrium
Set Q^d = Q^s:
30 - 3p = -6 + 4p
36 = 7p
p^c = 36/7 ≈ 5.14 (this is what consumers pay)
p^s = p^c - 2 = 36/7 - 2 = 22/7 ≈ 3.14 (this is what sellers receive)
Consumers pay $1.14 more than before the tax (up from $4 to $5.14). Sellers receive $0.86 less than before (down from $4 to $3.14). The $2 tax wedge is split, with consumers bearing the larger share, which is consistent with demand being more inelastic.
New quantity
Q_tax = 30 - 3(36/7) = 30 - 108/7 = 102/7 ≈ 14.57
Quantity has fallen from 18 to about 14.57.
Tax revenue
Tax revenue = tax × quantity = 2 × (102/7) = 204/7 ≈ 29.14
Deadweight loss
DWL is the triangle between the old and new quantities, bounded by the demand and supply curves:
DWL = 0.5 × tax × (Q_old - Q_new)
DWL = 0.5 × 2 × (18 - 102/7) = 0.5 × 2 × (24/7) = 24/7 ≈ 3.43
This $3.43 represents the value of transactions that no longer happen because of the tax. It is a pure loss to society: neither consumers, sellers, nor the government capture it.
Graphing the tax
On your graph, draw the original supply and demand curves intersecting at (Q = 18, p = 4). Then draw the new supply curve (shifted up by $2). Mark p^c = 5.14 on the vertical axis (the price consumers face). Mark p^s = 3.14 (the price sellers receive). The vertical distance between them is the $2 tax. The new quantity is at Q ≈ 14.57. The tax revenue is the rectangle between p^c and p^s, from Q = 0 to Q_tax. The deadweight loss is the triangle to the right of Q_tax, between the original supply and demand curves.
Price elasticity of demand
ε = (∂Q/∂p) × (p/Q)
Cross-price elasticity
ε_AB = (∂Q_A/∂p_B) × (p_B/Q_A)
Income elasticity
ε_Y = (∂Q/∂Y) × (Y/Q)
Price elasticity of supply
ε_s = (∂Q_s/∂p) × (p/Q_s)
Tax wedge
p^c - p^s = t (the per-unit tax)
Tax revenue
TR = t × Q_tax
Deadweight loss (linear case)
DWL = 0.5 × t × (Q_free - Q_tax)
Cross-price elasticity between coconut oil and palm oil matters to agricultural policymakers in Southeast Asia, where both crops are major exports. If a drought hits palm oil production, the cross-price elasticity tells you how much additional demand will flow to coconut oil producers.
Tax incidence analysis is exactly how economists evaluate proposals like carbon taxes, sugary drink taxes, or tobacco excise duties. The key insight, that the legal payer is irrelevant and only relative elasticity determines the actual burden, is one of the most policy-relevant results in microeconomics.
Students often think that if the tax is "paid by the seller," the seller bears the entire burden. This is wrong. Tax incidence is determined by relative elasticities, not by the legal assignment. If the same $2 tax were imposed on the buyer instead, the equilibrium prices p^c and p^s would be identical.
Students confuse the sign of elasticity with its magnitude. A price elasticity of -0.34 does not mean demand is "barely negative." The negative sign is just the law of demand. What matters is the absolute value: 0.34 < 1, so demand is inelastic.
Students sometimes calculate deadweight loss using the tax amount and the original quantity, rather than the change in quantity. DWL is about lost transactions, so you need (Q_old - Q_new), not Q_old.
Students occasionally mix up cross-price elasticity signs. Positive cross-price elasticity means substitutes (price of B goes up, demand for A goes up). Negative means complements. This is the opposite of what some students expect intuitively.
⚠️ Elasticity calculations are almost guaranteed on the exam. You will be given a demand function, values of variables, and asked to compute and interpret an elasticity. Memorise the formula and practise the arithmetic.
⚠️ The "who bears the tax burden" question is a classic exam question. You must be able to compare |ε_d| and ε_s and state which side is more inelastic and therefore bears more of the tax.
⚠️ Expect to compute p^c, p^s, the new quantity, tax revenue, and deadweight loss in a single multi-part problem. These are straightforward but require careful algebra. Show your working.
⚠️ The irrelevance of the legal assignment of the tax (buyer vs seller) is a favourite conceptual question. Be prepared to explain why it does not matter who writes the cheque.
True or false: A price elasticity of demand of -2.5 means demand is inelastic.
False. |ε| = 2.5 > 1, so demand is elastic.
Fill in the blank: If cross-price elasticity between goods A and B is negative, the goods are ______.
Complements.
True or false: If a tax is legally imposed on the buyer, the buyer always bears the full burden of the tax.
False. Tax incidence depends on relative elasticities, not on who is legally required to pay.
Fill in the blank: Deadweight loss measures the value of ______ that no longer occur because of the tax.
Transactions.
True or false: An income elasticity of 0.002 means the good is inferior.
False. The sign is positive, so the good is normal. The small magnitude just means demand barely responds to income.
Q: Given Q^d = 1,200 - 9.5p + 16.2p_p + 0.2Y, with p = 45, p_p = 31, and Q = 1,275, calculate the price elasticity of demand and state whether demand is elastic or inelastic.
A: ε = (-9.5)(45/1,275) = -0.335. Since |ε| < 1, demand is inelastic.
Q: In the market with Q^d = 30 - 3p and Q^s = 2 + 4p, a $2 tax is imposed on the seller. What price do consumers pay, and what price do sellers receive?
A: Set 30 - 3p = -6 + 4p (supply shifts by the tax). p^c = 36/7 ≈ $5.14. p^s = p^c - 2 = 22/7 ≈ $3.14.
Q: In the same market, what is the deadweight loss from the $2 tax?
A: Q_old = 18, Q_new = 102/7 ≈ 14.57. DWL = 0.5 × 2 × (18 - 102/7) = 24/7 ≈ $3.43.
Q: Explain why it does not matter whether the $2 tax is imposed on the buyer or the seller.
A: The equilibrium consumer price and seller price are determined by relative elasticities. If the tax is imposed on the buyer, the demand curve shifts down by $2 instead of the supply curve shifting up. Solving the new equilibrium yields the same p^c and p^s. The market outcome is invariant to the legal assignment of the tax.
Q: The cross-price elasticity of coconut oil with respect to palm oil is +0.394. What does this tell you about the two goods?
A: They are substitutes. A 1% increase in the price of palm oil raises the quantity demanded of coconut oil by about 0.39%.
Elasticity is the bridge between the demand functions covered in the companion notes and the welfare analysis that follows. Once you can compute elasticities, you can evaluate the effects of any policy intervention (taxes, subsidies, price controls) on consumer surplus, producer surplus, and total welfare.
Tax incidence connects to public finance and the design of optimal taxes. The Ramsey rule, which you may encounter later, says that taxes should be levied more heavily on goods with inelastic demand to minimise deadweight loss.
Deadweight loss reappears in monopoly analysis. A monopolist restricts output below the competitive level, creating a DWL triangle that looks very similar to the one from a tax.
Related Terms / Search Tags
price elasticity of demand, cross-price elasticity, income elasticity, elastic, inelastic, unit elastic, normal good, inferior good, luxury good, tax incidence, per-unit tax, ad valorem tax, consumer price, seller price, tax wedge, deadweight loss, welfare loss, tax revenue, Buschena and Perloff, coconut oil demand, palm oil substitute, supply and demand equilibrium, ECON 300A, intermediate microeconomics, University of Washington