Chapter 3 — Elasticity, Revenue, and Tax Incidence
Chapter 3 — Elasticity, Revenue, and Tax Incidence
Core question
A curve gives a direction. Elasticity asks: how large is the response relative to the change that caused it?
Learning outcomes
You will be able to:
- calculate midpoint elasticity and interpret its sign and magnitude;
- distinguish elasticity from slope;
- connect demand elasticity to total revenue;
- explain why the less elastic side bears more of a tax.
1. Price elasticity of demand
Demand elasticity is normally negative. Report the sign when doing analysis; many introductory exercises classify its absolute value:
| Magnitude | Classification | Meaning |
|---|---|---|
| ` | ε | > 1` |
| ` | ε | = 1` |
| ` | ε | < 1` |
Use the midpoint formula for a discrete change:
2. Worked example: a gym fee
Monthly price rises from £40 to £44; membership falls from 1,000 to 970.
Demand is inelastic over this observed arc.
| Before | After | |
|---|---|---|
| price | £40 | £44 |
| members | 1,000 | 970 |
| revenue | £40,000 | £42,680 |
Revenue rises because price rises proportionally more than quantity falls. Profit need not rise: costs, retention, reputation, and future demand also matter.
3. Elasticity is not a permanent label
Demand is usually more elastic when:
- close substitutes are available;
- the market is narrowly defined;
- the purchase takes a large income share;
- the good can be postponed;
- buyers have time to adjust.
“Petrol demand is inelastic” needs a market, place, group, price range, and time horizon. Commuters may adjust little this week and substantially over five years.
4. Do not infer elasticity from slope alone
Slope uses units; elasticity uses percentages. The same demand curve can be elastic near its high-price end and inelastic near its low-price end.
Changing kilograms to grams changes a numerical slope but not economic responsiveness. Compare elasticity at the relevant point or arc.
5. Total revenue test
For a small price increase:
| Demand | Quantity response | Revenue response |
|---|---|---|
| elastic | proportionally larger fall | falls |
| inelastic | proportionally smaller fall | rises |
| unit elastic | offsetting change | approximately unchanged |
This describes movement along a stable demand curve. If quality, advertising, rivals, or buyer composition also change, the simple test does not isolate the price effect.
6. Other elasticities
| Measure | Formula | Interpretation |
|---|---|---|
| income elasticity | %ΔQ / %Δincome | positive for normal; negative for inferior goods |
| cross-price elasticity | %ΔQ_A / %ΔP_B | positive for substitutes; negative for complements |
| supply elasticity | %ΔQ_s / %ΔP | ability of sellers to expand or contract output |
Supply is often more elastic when spare capacity, inventories, mobile inputs, and time are available. A theatre can add little seating tonight; a manufacturer may add a production line over several years.
7. Tax incidence depends on relative elasticity
A per-unit tax creates a wedge:
where buyers pay P_b and sellers receive P_s.
Return to the coffee market:
With a £2 tax, P_b=P_s+2:
Before tax, price was £4 and quantity 60. Buyers pay £1 more, sellers receive £1 less, and ten trades disappear.
Legal liability does not determine economic incidence. The side with fewer alternatives—the less elastic side—changes behaviour less and bears more of the wedge.
8. Evidence case: sugary-drink taxes
A global systematic review of 86 studies, with 62 in meta-analyses, reported:
- average pass-through to targeted beverage prices: 82%;
- average change in sales: 15% lower;
- estimated sales elasticity: −1.59.
The estimates were highly heterogeneous, and evidence on consumption, health, and subgroup effects was less conclusive. Search coverage ended in June 2020. The lesson is not “every tax reduces consumption by 15%”; it is that price transmission, demand response, substitution, geography, and outcome choice must be measured separately (Andreyeva et al., 2022).
From positive to normative analysis
Positive question: How much do prices and purchases change?
Normative question: Are health gains, revenue use, administrative cost, and distributional effects sufficient to justify the tax?
Elasticity informs both questions but decides neither alone.
9. Deadweight loss and the margin
A tax discourages trades where willingness to pay exceeds production cost but not the tax-inclusive price. The lost gains are deadweight loss in the basic model.
Larger behavioural response usually means:
- larger quantity reduction;
- a smaller tax base than a static calculation predicts;
- larger efficiency cost from the wedge;
- potentially larger correction if the untaxed activity creates an external harm.
Chapter 4 distinguishes a distortion from a correction of an existing market failure.
Practice
Price rises from £5 to £6 and quantity falls from 120 to 90.
- Calculate midpoint demand elasticity.
- Classify demand.
- Compare revenue before and after.
- Predict who bears more of a tax if supply is much more elastic than demand.
- State one reason the estimated elasticity may differ across populations or years.
Answer check
%ΔP = 1/5.5 = 18.18%; %ΔQ = -30/105 = -28.57%; ε ≈ -1.57, so demand is elastic. Revenue falls from £600 to £540. If supply is more elastic, buyers bear more. Transfer requires a comparable market, population, range, and horizon.Quick check
- Elasticity is unit-free and context-specific.
- Revenue and profit are different.
- Statutory and economic tax incidence are different.
- The less elastic side bears more of a tax.
- An empirical average is not a universal structural constant.
Chapter 2 — Demand, Supply, Equilibrium, and Welfare
Market schedules, curve shifts, equilibrium, surplus, price controls, and comparative statics.
Chapter 4 — Firms, Market Power, and Market Failure
Costs, profit, competition, monopoly, oligopoly, digital platforms, externalities, public goods, information, and policy.