Revealed Preference, Elasticity of Demand, and Revenue -- ECON 300, Midterm Review
Source: ECON 300 Midterm Examination, Autumn 2016 (University of Washington, Prof. Greg Ellis)
Tags: revealed preference, WARP, SARP, price elasticity of demand, elastic demand, inelastic demand, unit elastic, total revenue, revenue and elasticity, apple growers paradox
Difficulty: Introductory to Intermediate. Prerequisites: understanding of budget constraints, basic algebra, familiarity with the concept of a demand curve.
Big Picture
This document covers three related but distinct topics that all appeared on the same midterm. Revealed preference gives you a way to infer how a consumer ranks bundles without knowing their utility function, just by observing their choices. Elasticity of demand measures how responsive quantity demanded is to price changes, and it is the single most useful concept for connecting theory to real markets. The revenue-elasticity relationship ties these ideas together: whether a firm's revenue rises or falls when output changes depends entirely on whether demand is elastic or inelastic.
Revealed preference lets you rank bundles by checking whether one was affordable when the other was chosen. Price elasticity of demand is the percentage change in quantity divided by the percentage change in price. When demand is inelastic, increasing quantity (and thus lowering price) reduces total revenue, which explains why bumper crops can hurt farmers.
Revealed preference
If a consumer chooses bundle A when bundle B was also affordable, then A is directly revealed preferred to B. The consumer's choice reveals their ranking. In simple terms, if you picked A when you could have picked B, you must like A at least as much.
Weak Axiom of Revealed Preference (WARP)
If A is directly revealed preferred to B, then B cannot be directly revealed preferred to A. A rational consumer's choices should not contradict themselves.
Price elasticity of demand (Ed)
The percentage change in quantity demanded divided by the percentage change in price: Ed = (dQ/dP) x (P/Q). It measures responsiveness. In simple terms, it answers: if the price goes up by 1%, by what percentage does quantity demanded fall?
Elastic demand
Demand is elastic when |Ed| > 1. Quantity responds more than proportionally to price. A 1% price increase causes more than a 1% drop in quantity.
Inelastic demand
Demand is inelastic when |Ed| < 1. Quantity responds less than proportionally to price. A 1% price increase causes less than a 1% drop in quantity.
Unit elastic demand
Demand is unit elastic when |Ed| = 1. The percentage change in quantity exactly matches the percentage change in price. Total revenue does not change.
Total revenue
Price times quantity: TR = P x Q. Whether revenue rises or falls with a price change depends on elasticity.
Felix chooses between clothing (q1) and food (q2).
Situation 1: Income = $1,000. p1 = $100, p2 = $10. He chooses Bundle A: q1 = 2, q2 = 80.
Situation 2: Income = $1,200. p1 = $300, p2 = $10. He chooses Bundle B: q1 = 1, q2 = 90.
Can we determine how Felix ranks Bundle A versus Bundle B?
Step 1: Check whether Bundle B was affordable in Situation 1.
Cost of B at Situation 1 prices: (100)(1) + (10)(90) = 100 + 900 = $1,000. This equals Felix's income in Situation 1, so Bundle B was on the budget line. Felix could have chosen B but chose A instead. Therefore A is directly revealed preferred to B (A ≻ B by revealed preference).
Step 2: Check whether Bundle A was affordable in Situation 2.
Cost of A at Situation 2 prices: (300)(2) + (10)(80) = 600 + 800 = $1,400. Felix's income in Situation 2 is only $1,200. Bundle A was not affordable in Situation 2. So we cannot say B is revealed preferred to A.
Step 3: Conclusion.
Felix ranks A above B. He chose A when B was available, and there is no contradictory evidence (B was chosen in a situation where A was out of reach). WARP is not violated.
The market demand function is Q = 11 - P.
The point elasticity formula is: Ed = (dQ/dP) x (P/Q).
For this linear demand, dQ/dP = -1.
At P = $8, Q = 3:
Ed = (-1)(8/3) = -8/3 ≈ -2.67. |Ed| = 2.67 > 1, so demand is elastic.
At P = $5, Q = 6:
Ed = (-1)(5/6) = -5/6 ≈ -0.83. |Ed| = 0.83 < 1, so demand is inelastic.
At P = $3, Q = 8:
Ed = (-1)(3/8) = -3/8 = -0.375. |Ed| = 0.375 < 1, so demand is inelastic.
Of the three prices, only P = $8 gives elastic demand.
For a linear demand curve, elasticity varies along the curve even though the slope is constant. The upper portion (high P, low Q) is elastic; the lower portion (low P, high Q) is inelastic.
The midpoint of a linear demand curve is unit elastic. For Q = 11 - P, the midpoint is P = $5.50, Q = 5.5. At P = $5 you are just below the midpoint, which is why |Ed| is just below 1.
Washington apple growers produced an unusually large crop, yet their income fell. How?
Apples are a staple food with few close substitutes (at least in the short run). Demand for apples is inelastic. When the bumper crop shifts the supply curve to the right, equilibrium quantity rises and price falls. But because demand is inelastic, the percentage drop in price is larger than the percentage rise in quantity.
Total revenue = P x Q. If price falls by a larger percentage than quantity rises, revenue falls.
If demand is inelastic (|Ed| < 1): a price decrease reduces total revenue. A quantity increase (which forces the price down along the demand curve) reduces revenue.
If demand is elastic (|Ed| > 1): a price decrease increases total revenue.
If demand is unit elastic (|Ed| = 1): total revenue is unchanged.
On a supply-and-demand diagram, the supply curve shifts right. The new equilibrium has a lower price and a higher quantity. The revenue rectangle (P x Q) gets wider but shorter. When demand is inelastic, it gets shorter faster than it gets wider, so the area shrinks. The lost revenue rectangle on the price side is taller than the gained revenue rectangle on the quantity side.
Revealed preference test: if p1A·q1B + p2A·q2B ≤ IA and bundle A was chosen, then A is revealed preferred to B.
Point elasticity of demand: Ed = (dQ/dP) x (P/Q)
For Q = 11 - P: dQ/dP = -1, so Ed = -P/(11 - P)
Total revenue: TR = P x Q
Revenue change rule: if |Ed| < 1, price and revenue move in the same direction. If |Ed| > 1, price and revenue move in opposite directions.
The apple growers paradox is not unique to apples. Agricultural commodities in general tend to have inelastic demand, which is why good harvests often depress farm incomes. This is the economic logic behind government crop subsidies and supply management programmes: by restricting output, producers can keep prices high enough to increase total revenue.
Revealed preference is the empirical backbone of demand estimation. Economists observe what consumers actually buy at different prices and incomes, and from those choices they infer preferences without ever needing to ask about utility directly. This is how the Bureau of Labour Statistics constructs consumer price indices.
Students often confuse the slope of a demand curve with elasticity. A linear demand curve has a constant slope but varying elasticity. Elasticity depends on both the slope and the ratio P/Q, which changes as you move along the curve.
Students sometimes think that if a consumer chose B under new conditions, B must be preferred. That conclusion only holds if the other bundle was also affordable. If A was out of reach, the choice of B tells you nothing about the ranking.
Students frequently forget to take absolute values when classifying elasticity. Ed is negative (price and quantity move in opposite directions), but we classify based on |Ed|.
Students often think a bigger harvest always means more income. The revenue-elasticity relationship shows this is only true when demand is elastic. For inelastic goods, more output means less revenue.
⚠️ Revealed preference questions require you to check affordability in both directions. Always compute the cost of each bundle under each set of prices. Do not skip a direction.
⚠️ Use the point elasticity formula, not the midpoint formula, unless the question says otherwise. ECON 300 typically uses Ed = (dQ/dP)(P/Q).
⚠️ For the revenue-elasticity essay question, a graph is required. Draw supply-and-demand, show the rightward shift, label both equilibria, and indicate the revenue rectangles.
⚠️ Clearly state whether demand is elastic, inelastic, or unit elastic at each requested price. One-word classification plus the number is what earns marks.
True or False: If bundle A was chosen when bundle B was affordable, then A is revealed preferred to B.
Fill in the blank: For Q = 11 - P, the elasticity at P = $8 is ______.
True or False: A linear demand curve has constant elasticity.
True or False: When demand is inelastic, a price decrease increases total revenue.
Fill in the blank: The midpoint of the demand curve Q = 11 - P occurs at P = ______.
Answers: 1. True. 2. -8/3 (or about -2.67; elastic). 3. False. 4. False (revenue decreases). 5. $5.50.
Q: Felix chooses Bundle A (q1=2, q2=80) at prices p1=$100, p2=$10, income=$1,000. He later chooses Bundle B (q1=1, q2=90) at prices p1=$300, p2=$10, income=$1,200. Using revealed preference, how does he rank the bundles?
A: Cost of B at Situation 1 prices = 100(1) + 10(90) = $1,000 = his income. B was affordable when A was chosen, so A is revealed preferred to B. Cost of A at Situation 2 prices = 300(2) + 10(80) = $1,400 > $1,200. A was not affordable when B was chosen. No contradiction. Felix prefers A to B.
Q: For Q = 11 - P, calculate the elasticity of demand at P = $5.
A: At P = 5, Q = 6. Ed = (-1)(5/6) = -5/6 ≈ -0.83. |Ed| < 1, so demand is inelastic.
Q: At which price is demand elastic: P = $8, P = $5, or P = $3?
A: Only at P = $8, where |Ed| = 8/3 ≈ 2.67 > 1.
Q: Washington apple growers had a bumper crop but their income fell. Explain using elasticity.
A: Demand for apples is inelastic. The bumper crop shifted supply right, lowering price and raising quantity. With inelastic demand, the percentage fall in price exceeds the percentage rise in quantity, so total revenue (P x Q) falls. The farmers receive a lower price on every unit, and the extra units sold do not compensate.
Q: On a linear demand curve, where is demand unit elastic?
A: At the midpoint. For Q = 11 - P, this is at P = $5.50, Q = 5.5.
Revealed preference connects to consumer theory (Doc 1): it provides the empirical test for the utility-maximisation model. If observed choices violate WARP, the rational consumer model does not fit the data. Elasticity connects directly to tax incidence (Doc 2): the side of the market with less elastic behaviour bears more of the tax burden. The revenue-elasticity link is also the foundation for understanding monopoly pricing: a profit-maximising monopolist never operates on the inelastic portion of the demand curve, because they could raise price and increase revenue simultaneously.
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