Difficulty: Introductory to Intermediate | Prerequisites: Budget constraints, basic demand curves, introductory calculus (product rule, total differential).
This set of notes covers three shorter but frequently tested topics from the ECO 300 midterm. Revealed preference lets you infer a consumer's ranking of bundles from their observed choices, without knowing their utility function. Elasticity of demand quantifies how responsive quantity demanded is to price changes. The total revenue proof connects elasticity, price, and quantity using calculus.
These three topics appear as standalone exam questions worth 10 points each. They are conceptually lighter than consumer theory or market equilibrium, but students often lose marks through careless errors or incomplete reasoning.
Revealed preference uses observable choices to rank bundles: if a consumer chose bundle A when B was affordable, A is revealed preferred to B. Price elasticity of demand measures the percentage change in quantity demanded per percentage change in price, and whether it is above or below 1 (in absolute value) tells you whether demand is elastic or inelastic. The total revenue identity shows that the percentage change in revenue equals the percentage change in price plus the percentage change in quantity.
Revealed preference
The principle that a consumer's actual choices reveal their preferences. If a consumer chooses bundle A when bundle B is also affordable, then A is revealed preferred to B.
In simple terms, actions speak louder than utility functions.
Directly revealed preferred
Bundle A is directly revealed preferred to bundle B if A was chosen when B was in the budget set (i.e., the consumer could have afforded B but picked A instead).
Weak Axiom of Revealed Preference (WARP)
If A is revealed preferred to B, then B cannot be revealed preferred to A. A violation of WARP means the consumer's behaviour is inconsistent with rational preference maximisation.
Price elasticity of demand (E_d)
The percentage change in quantity demanded divided by the percentage change in price. Formally: E_d = (dQ/dP) x (P/Q). Elasticity is typically negative (law of demand), but is often discussed in absolute value.
In simple terms, it measures how "stretchy" demand is when you pull on the price.
Elastic demand
Demand where |E_d| > 1. A 1% price increase causes more than a 1% drop in quantity demanded. Total revenue falls when price rises.
Inelastic demand
Demand where |E_d| < 1. A 1% price increase causes less than a 1% drop in quantity demanded. Total revenue rises when price rises.
Unit elastic demand
Demand where |E_d| = 1. A 1% price increase causes exactly a 1% drop in quantity demanded. Total revenue is unchanged.
Total revenue (total expenditure)
P x Q. Revenue to firms equals expenditure by consumers. The relationship between total revenue and price depends on the elasticity of demand.
The exam question: Felix has two situations.
Situation 1: Income = $1,000, p1 = $100, p2 = $10. He chooses bundle A = (q1=2, q2=80).
Cost of A at situation 1 prices: 100(2) + 10(80) = $1,000 (exactly his income).
Situation 2: Income = $1,200, p1 = $300, p2 = $10. He chooses bundle B = (q1=1, q2=90).
Cost of B at situation 2 prices: 300(1) + 10(90) = $1,200 (exactly his income).
The key question: could Felix have afforded bundle B at situation 1 prices?
Cost of B at situation 1 prices: 100(1) + 10(90) = $1,000 = his income in situation 1.
Yes, B was affordable in situation 1 (it cost exactly $1,000). Felix chose A instead, so A is directly revealed preferred to B.
Now check the reverse: could Felix have afforded bundle A at situation 2 prices?
Cost of A at situation 2 prices: 300(2) + 10(80) = $1,400 > $1,200.
No. A was not affordable in situation 2. So B is not revealed preferred to A. There is no contradiction.
Conclusion: A is revealed preferred to B. Felix ranks the original bundle (2, 80) above the new bundle (1, 90). This is consistent with WARP.
Given: Q = 11 - P (a linear demand function).
The point elasticity formula is:
E_d = (dQ/dP) x (P/Q)
Since Q = 11 - P, the derivative dQ/dP = -1. So elasticity simplifies to:
E_d = -1 x (P / Q) = -P / (11 - P)
At P = $8 (Q = 3):
E_d = -8/3 = -2.67
|E_d| = 2.67 > 1, so demand is elastic.
At P = $5 (Q = 6):
E_d = -5/6 = -0.83
|E_d| = 0.83 < 1, so demand is inelastic.
At P = $3 (Q = 8):
E_d = -3/8 = -0.375
|E_d| = 0.375 < 1, so demand is inelastic.
Only at P = $8 is demand elastic. This illustrates a general property of linear demand curves: elasticity varies along the curve. The upper portion (high price, low quantity) is elastic; the lower portion (low price, high quantity) is inelastic. The midpoint (here, P = $5.50) has unit elasticity.
The exam asks you to show that the percentage change in total revenue (TR = P x Q) equals the sum of the percentage change in price and the percentage change in quantity.
Step 1: Take the total differential of TR.
d(PQ) = P·dQ + Q·dP (this is just the product rule from calculus)
Step 2: Divide both sides by PQ.
d(PQ) / (PQ) = (P·dQ) / (PQ) + (Q·dP) / (PQ)
d(PQ) / (PQ) = dQ/Q + dP/P
Step 3: Recognise what each term is.
d(PQ)/(PQ) is the percentage change in total revenue.
dQ/Q is the percentage change in quantity.
dP/P is the percentage change in price.
Therefore: %ΔTR = %ΔP + %ΔQ.
This result is exact for infinitesimal changes and a close approximation for small discrete changes. It is the reason elasticity matters for revenue: if |E_d| > 1 (elastic), a price increase causes quantity to fall by a larger percentage than the price rose, so revenue falls. If |E_d| < 1 (inelastic), the price rise dominates and revenue rises.
\text{Revealed preference test: } p_1^A \cdot q_1^B + p_2^A \cdot q_2^B \leq I^A \Rightarrow A \succsim BE_d = \frac{dQ}{dP} \cdot \frac{P}{Q}\text{For } Q = 11 - P: \quad E_d = -\frac{P}{11 - P}d(PQ) = P \cdot dQ + Q \cdot dP\frac{d(PQ)}{PQ} = \frac{dP}{P} + \frac{dQ}{Q}Revealed preference is the foundation of empirical demand estimation: economists observe what people actually buy at given prices and infer preferences from those choices. Elasticity determines pricing strategy in every industry: firms with inelastic demand (pharmaceuticals, petrol) can raise prices without losing much volume, while firms with elastic demand (fast fashion, budget airlines) compete fiercely on price. The total revenue identity is used by analysts to decompose revenue changes into volume effects and price effects, which is standard practice in earnings reports.
Students often confuse "affordable" with "chosen." In revealed preference, the test is whether a bundle was in the budget set (affordable), not whether it was the cheapest option.
A common mistake with elasticity is treating the slope of the demand curve as the elasticity. They are related but different. Elasticity = slope x (P/Q), so it varies along a linear demand curve even though the slope is constant.
Students frequently forget the absolute value when classifying demand as elastic or inelastic. Elasticity is negative (because demand curves slope downward), but we compare |E_d| to 1.
In the total revenue proof, some students use the quotient rule instead of the product rule. TR = P x Q is a product, so the product rule applies.
⚠️ Revealed preference questions require you to check affordability in both directions. Always compute the cost of each bundle at both sets of prices.
⚠️ For elasticity, use the point elasticity formula (dQ/dP) x (P/Q), not the midpoint formula, unless told otherwise. The exam gives you a differentiable demand function, which signals point elasticity.
⚠️ The total revenue proof is a "show that" question. Write every step clearly. Do not skip the division by PQ.
⚠️ Know the link between elasticity and revenue direction. This is the conceptual payoff of the total revenue proof.
True or false: If a consumer chose bundle A when bundle B was affordable, then A is revealed preferred to B.
True.
Fill in the blank: On a linear demand curve, elasticity is highest at the ________ end of the curve.
Upper (high price, low quantity).
True or false: If |E_d| = 0.5, a price increase will reduce total revenue.
False. Demand is inelastic, so total revenue rises with price.
Fill in the blank: The product rule gives d(PQ) = P·dQ + ________.
Q·dP.
True or false: The slope of a linear demand curve and its elasticity are the same thing.
False. Elasticity = slope x (P/Q) and varies along the curve.
Q: Felix chooses (2, 80) at prices (100, 10) with income $1,000. He later chooses (1, 90) at prices (300, 10) with income $1,200. Can you rank the bundles?
A: Yes. Bundle (2, 80) is revealed preferred to (1, 90). At situation 1 prices, (1, 90) costs $1,000, which was affordable, yet Felix chose (2, 80). At situation 2 prices, (2, 80) costs $1,400 > $1,200, so it was not affordable, and there is no reverse revealed preference.
Q: Calculate the elasticity of demand at P = $8 for the demand function Q = 11 - P.
A: E_d = (-1)(8/3) = -8/3 = -2.67. |E_d| > 1, so demand is elastic at this price.
Q: At which of the three prices ($8, $5, $3) is demand elastic?
A: Only at P = $8. At P = $5 (|E_d| = 0.83) and P = $3 (|E_d| = 0.375), demand is inelastic.
Q: Using calculus, show that %ΔTR = %ΔP + %ΔQ.
A: Start with d(PQ) = P·dQ + Q·dP (product rule). Divide both sides by PQ: d(PQ)/(PQ) = dQ/Q + dP/P. The left side is the percentage change in TR; the right side is the sum of the percentage changes in Q and P.
Revealed preference underpins the entire consumer theory framework: if observed choices satisfy WARP (and its stronger cousin SARP), a well-behaved utility function can be constructed to rationalise them. Elasticity connects to tax incidence (more inelastic side bears more tax burden) and to the market equilibrium material in the second set of notes from this exam. The total revenue decomposition reappears in industrial organisation when analysing how firms set prices under different market structures.
Revealed preference, WARP, SARP, directly revealed preferred, budget set, affordable bundle, price elasticity of demand, point elasticity, arc elasticity, midpoint elasticity, elastic demand, inelastic demand, unit elastic, total revenue, total expenditure, product rule, total differential, percentage change decomposition, ECO 300, intermediate microeconomics, demand analysis