Difficulty: Intermediate | Prerequisites: Budget constraints, consumer choice, basic demand curves.
Revealed preference and price elasticity are two ways of reasoning about consumer behaviour that come at the problem from different angles. Revealed preference works backwards from what a consumer actually buys to draw conclusions about what they prefer, without ever needing to see their utility function. Elasticity measures how sensitive demand is to price changes, and ties directly to what happens to a firm's revenue when it raises or lowers its price. Both topics are standard intermediate micro material and often appear together on exams.
Revealed preference says: if a consumer chose bundle A when bundle B was affordable, then A is revealed preferred to B. Consistency (WARP) requires they never choose B when A is affordable. Price elasticity of demand measures percentage change in quantity over percentage change in price; when demand is elastic (|E| > 1), a price cut raises total revenue, and when inelastic (|E| < 1), a price cut lowers it. The percentage change in total revenue equals the percentage change in price plus the percentage change in quantity.
Revealed preference
If a consumer chooses bundle A when bundle B is also affordable, then A is directly revealed preferred to A. The idea is that the consumer's choice reveals their preference without the need for a utility function.
Think of it as "actions speak louder than words": what you buy tells us what you like.
Directly revealed preferred
Bundle A is directly revealed preferred to bundle B if A was chosen at prices where B was within the budget.
Weak Axiom of Revealed Preference (WARP)
If A is directly revealed preferred to B, then B must never be directly revealed preferred to A. If WARP is violated, the consumer's behaviour is inconsistent with rational utility maximisation.
In simple terms, WARP says you cannot flip your ranking: if you picked A over B once when both were affordable, you should not later pick B over A when both are affordable again.
Price elasticity of demand
The percentage change in quantity demanded divided by the percentage change in price. Formally: E = (%ΔQ) / (%ΔP).
Think of it as a measure of how jumpy consumers are when prices move.
Elastic demand
Demand with |E| > 1. Quantity responds more than proportionally to a price change. A 1% price increase causes more than a 1% drop in quantity.
Inelastic demand
Demand with |E| < 1. Quantity responds less than proportionally. A 1% price increase causes less than a 1% drop in quantity.
Unit elastic demand
Demand with |E| = 1. The percentage change in quantity exactly offsets the percentage change in price, so total revenue stays constant.
Total revenue (TR)
Price multiplied by quantity: TR = P × Q. How TR changes when price changes depends entirely on the elasticity of demand.
The setup: you observe a consumer's choices under different price and income conditions, and you want to know whether those choices are consistent with some well-behaved utility function.
Suppose you observe two situations:
Situation 1: prices (P1 = 100, P2 = 10), consumer buys bundle B1 = (q1 = 2, q2 = 80)
Situation 2: prices (P1 = 300, P2 = 10), consumer buys bundle B2 = (q1 = 1, q2 = 90)
Step 1: compute the cost of each bundle at each set of prices.
Cost of B1 at Situation 1 prices: 100(2) + 10(80) = 200 + 800 = 1,000
Cost of B2 at Situation 1 prices: 100(1) + 10(90) = 100 + 900 = 1,000... wait, let us recalculate. Actually: cost of B2 at Situation 1 prices = 100(1) + 10(90) = 1,000. But we also check: cost of B1 at Situation 2 prices = 300(2) + 10(80) = 600 + 800 = 1,400. Cost of B2 at Situation 2 prices = 300(1) + 10(90) = 300 + 900 = 1,200.
Step 2: check affordability.
In Situation 1, the consumer chose B1 (cost 1,000). Was B2 affordable? Cost of B2 at Situation 1 prices = 1,000, so yes, B2 was exactly affordable. But the consumer chose B1, so B1 is revealed preferred to B2.
In Situation 2, the consumer chose B2 (cost 1,200). Was B1 affordable? Cost of B1 at Situation 2 prices = 1,400 > 1,200, so no, B1 was not affordable. WARP is satisfied because B1 is not revealed preferred from Situation 2 (it was out of budget).
Conclusion: Bundle 2 is preferred to Bundle 1 would require B1 to be affordable when B2 was chosen, and it was not. The data is consistent with WARP.
The point elasticity formula is:
E = (dQ/dP) × (P/Q)
Or equivalently, for a discrete change:
E = (%ΔQ) / (%ΔP)
From the midterm's example, elasticities were computed at various price points on a demand curve. The key results:
At higher prices (where quantity is low), demand tends to be elastic (|E| > 1).
At lower prices (where quantity is high), demand tends to be inelastic (|E| < 1).
At a price of $8 on the given demand curve, demand was found to be elastic.
This makes intuitive sense: when the price is already high, a further increase drives away a proportionally large share of the remaining buyers.
Total revenue is TR = P × Q. Taking logarithms:
ln TR = ln P + ln Q
Differentiating:
%ΔTR = %ΔP + %ΔQ
This decomposition is the total revenue test:
If demand is elastic (|E| > 1): a price increase causes quantity to fall by more (in percentage terms) than price rises, so TR falls.
If demand is inelastic (|E| < 1): a price increase causes quantity to fall by less than price rises, so TR rises.
If unit elastic (|E| = 1): the effects exactly cancel and TR is unchanged.
This is why firms with inelastic demand (e.g. pharmaceutical companies with patented drugs) can raise prices and earn more revenue, while firms facing elastic demand (e.g. competitive commodity sellers) cannot.
Revealed preference test: Compute cost of each bundle at each price vector. If bundle A costs no more than the budget when bundle B was chosen, then B is revealed preferred to A (since the consumer chose B when A was affordable).
WARP check: If A is revealed preferred to B, then B must not be revealed preferred to A.
Point elasticity: E = (dQ/dP) × (P/Q)
Arc elasticity (midpoint method): E = ((Q2 - Q1) / ((Q2 + Q1)/2)) / ((P2 - P1) / ((P2 + P1)/2))
Total revenue decomposition: %ΔTR = %ΔP + %ΔQ
Total revenue test:
|E| > 1 (elastic): price up → TR down; price down → TR up
|E| < 1 (inelastic): price up → TR up; price down → TR down
|E| = 1 (unit elastic): TR unchanged
Revealed preference is the theoretical foundation for empirical demand analysis: economists observe what people buy at various prices and use that data to infer preferences, rather than asking people to fill in utility surveys. The elasticity-revenue link drives real pricing strategy: airlines price business-class tickets high (inelastic business travellers) and discount leisure tickets (elastic holiday travellers), precisely because the two groups sit at different points on the elasticity spectrum.
Students often confuse "affordable" with "chosen." A bundle being affordable at given prices does not mean it was preferred; it only becomes relevant if it was affordable but not chosen, because then the chosen bundle is revealed preferred to it.
A common error is treating elasticity as always negative and forgetting the absolute value when classifying. Demand curves slope down, so dQ/dP is negative, but when we say "elastic" we mean |E| > 1.
Students sometimes think a good with elastic demand at one price must be elastic everywhere. Elasticity changes along a linear demand curve: elastic at high prices, unit elastic at the midpoint, inelastic at low prices.
Confusing the total revenue test direction is a frequent exam mistake. Remember: elastic demand means quantity dominates, so a price cut raises revenue.
⚠️ Revealed preference questions require careful bookkeeping. Always compute the cost of every bundle at every price vector before drawing conclusions. A misplaced number breaks the whole argument.
⚠️ Know WARP cold. If the question asks "is the consumer's behaviour consistent with utility maximisation?" it is asking you to check WARP.
⚠️ Be ready to classify demand as elastic, inelastic, or unit elastic at a specific price point, and to explain what that means for total revenue.
⚠️ The log-differentiation derivation (%ΔTR = %ΔP + %ΔQ) is a common short-answer or proof question.
True or false: If a consumer chose bundle A when bundle B was more expensive, then A is revealed preferred to B.
False. For revealed preference, B must have been affordable (within the budget), not just more expensive. "More expensive" and "not affordable" are different things.
Fill in the blank: When |E| > 1, demand is ____.
Elastic.
True or false: If a firm faces inelastic demand, cutting its price will raise its total revenue.
False. With inelastic demand, cutting the price reduces revenue because the quantity increase does not offset the lower price.
True or false: Elasticity is constant along a linear demand curve.
False. Elasticity varies: it is higher (more elastic) at higher prices and lower (more inelastic) at lower prices.
Q: A consumer buys (3, 10) at prices (5, 2) and (4, 8) at prices (4, 3). Is the behaviour consistent with WARP?
A: Cost of (3, 10) at prices (5, 2) = 15 + 20 = 35. Cost of (4, 8) at prices (5, 2) = 20 + 16 = 36 > 35 (not affordable in Situation 1, so no revealed preference from Situation 1). Cost of (3, 10) at prices (4, 3) = 12 + 30 = 42. Cost of (4, 8) at prices (4, 3) = 16 + 24 = 40 (chosen). Since (3, 10) costs 42 > 40, it is not affordable in Situation 2 either. Neither bundle is revealed preferred to the other from these two observations, so WARP is trivially satisfied.
Q: Demand is Q = 100 - 2P. Compute the elasticity at P = 20.
A: Q = 100 - 2(20) = 60. dQ/dP = -2. E = (-2)(20/60) = -2/3. |E| = 2/3 < 1, so demand is inelastic at P = 20.
Q: At P = 40 on the same demand curve, is demand elastic or inelastic?
A: Q = 100 - 2(40) = 20. E = (-2)(40/20) = -4. |E| = 4 > 1, so demand is elastic at P = 40.
Q: If a firm raises its price by 5% and quantity demanded falls by 8%, what happens to total revenue?
A: %ΔTR = %ΔP + %ΔQ = 5% + (-8%) = -3%. Revenue falls by approximately 3%. Since quantity fell proportionally more than price rose, demand is elastic in this range.
Revealed preference connects to utility theory and consumer choice: it provides the empirical foundation, showing that consistent observed behaviour is enough to infer a utility function exists (via the representation theorem).
Elasticity connects to tax incidence: the more inelastic side of the market bears a larger share of a tax, which is the same elasticity concept applied to a different question.
Both topics feed into welfare economics: revealed preference underpins demand estimation, and elasticity determines how large the deadweight loss from a given tax or monopoly markup will be.
Revealed preference, WARP, Weak Axiom of Revealed Preference, SARP, Strong Axiom, directly revealed preferred, budget set, affordable bundle, price elasticity of demand, elastic, inelastic, unit elastic, total revenue test, percentage change decomposition, midpoint elasticity, arc elasticity, point elasticity, log-differentiation of revenue, demand sensitivity, revenue maximisation