Source: ECON 300 E, University of Washington
Difficulty: Intermediate | Prerequisites: Budget constraints, indifference curves, basic understanding of utility maximisation and perfect substitutes.
Revealed preference and compensated demand are two ways of getting at consumer preferences from different starting points. Revealed preference works backwards from observed choices: if someone could have bought bundle A but chose bundle B instead, they must prefer B. Compensated (Hicksian) demand approaches from the cost-minimisation side, asking what is the cheapest way to reach a given utility level. Both topics test whether you understand the logic of consumer theory beyond just plugging into formulas.
Revealed preference lets you rank bundles without knowing the utility function, just by checking which bundles were affordable when each was chosen. Compensated demand for perfect substitutes is a corner-solution problem: the consumer buys only the cheaper good (per unit of utility), and the expenditure function has different expressions depending on the price ratio.
Revealed preference
The idea that a consumer's choices reveal their preferences. If bundle A was chosen when bundle B was also affordable, then A is "directly revealed preferred" to B. In simple terms, your shopping receipt tells me what you like best.
Directly revealed preferred
Bundle A is directly revealed preferred to bundle B if A was chosen when B was affordable (i.e., the cost of B at the prices when A was chosen is less than or equal to the consumer's income).
Weak Axiom of Revealed Preference (WARP)
If bundle A is directly revealed preferred to B, then B cannot be directly revealed preferred to A. This is a consistency requirement: a rational consumer should not contradict themselves.
Compensated (Hicksian) demand
The quantity of each good that minimises expenditure for a given utility level. Think of it as: "what would you buy if we took away just enough money to keep you at the same happiness, then let you re-optimise?"
Perfect substitutes
Goods with a constant marginal rate of substitution. The indifference curves are straight lines. The consumer always buys only the cheaper good (per unit of utility), leading to corner solutions. Think of it as: two goods that do the same job, so you just pick whichever is cheaper.
Corner solution
An optimum where the consumer buys zero of one good and spends everything on the other. This occurs with perfect substitutes when the price ratio differs from the MRS.
Marginal rate of substitution (MRS)
The rate at which the consumer is willing to trade one good for another while staying on the same indifference curve. For U = q1 + 2q2, the MRS is 1/2 (one unit of q1 is worth half a unit of q2 in utility terms).
Remi chooses between clothing (q1) and food (q2).
Situation 1: I = $1000, p1 = $100, p2 = $10. She chooses q1 = 2, q2 = 80.
Situation 2: I = $1200, p1 = $150, p2 = $10. She chooses q1 = 1, q2 = 105.
To rank the two bundles, check whether each was affordable under the other's budget.
Is Bundle 2 (1, 105) affordable under Budget 1? Cost = $100(1) + $10(105) = $1,150 > $1,000. No, it was not affordable.
Is Bundle 1 (2, 80) affordable under Budget 2? Cost = $150(2) + $10(80) = $1,100 < $1,200. Yes, it was affordable.
Under Budget 2, Remi could have bought Bundle 1 but chose Bundle 2 instead. By revealed preference, she prefers Bundle 2 to Bundle 1.
We cannot say the reverse (that Bundle 1 is preferred to Bundle 2), because Bundle 2 was not affordable under Budget 1. There is no contradiction, and WARP is not violated.
If you chose B when A was affordable, you must prefer B to A. If A was never affordable when B was chosen, you learn nothing from that direction.
Sylvan's utility: U(q1, q2) = q1 + 2q2. This is perfect substitutes with MRS = 1/2.
Minimise expenditure E = p1·q1 + p2·q2 subject to U = q1 + 2q2.
The key step is comparing p1/p2 to the MRS (1/2).
Case 1: p1/p2 < 1/2 (q1 is relatively cheap). Buy only q1. Set q1 = U, q2 = 0. Expenditure: E = p1·U.
Case 2: p1/p2 > 1/2 (q2 is relatively cheap per unit of utility). Buy only q2. Set q2 = U/2, q1 = 0. Expenditure: E = p2·U/2.
Case 3: p1/p2 = 1/2 (indifferent). Any combination on the utility constraint works. E = p1·q1 + p2·q2 for any (q1, q2) satisfying q1 + 2q2 = U.
With perfect substitutes, the indifference curves are straight lines. The budget line is also straight. Unless they have exactly the same slope (Case 3), the cheapest point is always at a corner.
Plug the Hicksian demands into E = p1·q1 + p2·q2:
Case 1 (p1/p2 < 1/2): E = p1·U.
Case 2 (p1/p2 > 1/2): E = p2·U/2.
Case 3 (p1/p2 = 1/2): E = p1·q1* + p2·q2* for any bundle on U = q1 + 2q2. All such bundles cost the same because the prices and MRS are aligned.
The expenditure function for perfect substitutes is piecewise: you always use the cheapest "route" to utility. When q1 delivers utility more cheaply per pound spent, you buy only q1. When q2 does, you buy only q2. When they are tied, any mix works.
Revealed preference test: Bundle A is directly revealed preferred to B if p_A · B ≤ I_A (B was affordable when A was chosen).
Perfect substitutes utility: U(q1, q2) = q1 + 2q2, MRS = 1/2.
Hicksian demand (perfect substitutes, U = q1 + 2q2):
p1/p2 < 1/2: q1* = U, q2* = 0.
p1/p2 > 1/2: q1* = 0, q2* = U/2.
p1/p2 = 1/2: any (q1*, q2*) on q1 + 2q2 = U.
Expenditure function:
p1/p2 < 1/2: E = p1·U.
p1/p2 > 1/2: E = p2·U/2.
p1/p2 = 1/2: E = p1·q1* + p2·q2* (same for all feasible bundles).
Revealed preference is the empirical backbone of demand estimation. Economists observe what people buy at different prices and incomes, then test whether the data are consistent with rational choice. The perfect-substitutes model applies whenever two goods are functionally interchangeable: generic vs. brand-name paracetamol, two streaming services with the same catalogue, or two bus routes covering the same journey. Consumers pick whichever is cheaper, and demand flips entirely when relative prices cross the threshold.
Students often test affordability in the wrong direction. The question is whether the other bundle was affordable under the budget when the chosen bundle was picked. Check both directions separately.
With perfect substitutes, students sometimes set up a Lagrangian and look for an interior solution. There is no interior solution (except in the knife-edge case where p1/p2 = MRS). Compare the price ratio to the MRS and go straight to corner solutions.
Students sometimes confuse "revealed preferred" with "preferred." Revealed preference only tells you about ranking when a bundle was affordable and not chosen. If a bundle was not affordable, you cannot draw any conclusion from the fact that it was not chosen.
For the expenditure function with perfect substitutes, students sometimes write a single expression. It is piecewise, with different formulas for different price ratios.
⚠️ Revealed preference questions test logical reasoning, not calculation. Show each affordability check step by step: state the cost of the bundle, compare it to income, and draw the conclusion.
⚠️ For perfect substitutes, the exam will test whether you can identify the three cases (p1/p2 less than, greater than, or equal to MRS) and write the correct demand and expenditure function for each.
⚠️ Do not confuse Marshallian demand (maximise utility given income) with Hicksian demand (minimise expenditure given utility). Know which problem you are solving.
⚠️ Revealed preference arguments require you to state clearly which bundle was chosen, which was affordable, and what that implies. Vague reasoning loses marks.
True or false: If bundle A was chosen when bundle B was not affordable, we can conclude A is preferred to B.
Fill in the blank: For perfect substitutes with U = q1 + 2q2, the MRS is ______.
True or false: The compensated demand for perfect substitutes always involves buying both goods.
Fill in the blank: The expenditure function for perfect substitutes is ______ (one word describing its mathematical form).
True or false: If both bundles are affordable under both budgets, you can always rank them using revealed preference.
Q: Remi has I = $1000, p1 = $100, p2 = $10 and chooses (2, 80). Later, I = $1200, p1 = $150, p2 = $10 and she chooses (1, 105). Which bundle does she prefer?
A: She prefers (1, 105). Under the new budget, (2, 80) costs $1,100, which is affordable (income is $1,200), but she chose (1, 105) instead. So (1, 105) is revealed preferred. The reverse test fails: (1, 105) costs $1,150 under the original budget, which exceeds $1,000, so it was not affordable.
Q: For U = q1 + 2q2, what is the Hicksian demand when p1/p2 < 1/2?
A: q1* = U, q2* = 0. Good 1 delivers utility more cheaply, so the consumer buys only q1.
Q: What is the expenditure function when p1/p2 > 1/2?
A: E = p2 · U/2. The consumer buys only q2, and needs U/2 units of it (since each unit of q2 contributes 2 to utility).
Q: If p1 = 3 and p2 = 8, which good does the consumer buy to minimise expenditure?
A: p1/p2 = 3/8 = 0.375 < 0.5. So the consumer buys only q1. Expenditure is E = 3U.
Q: Could you observe a WARP violation in Remi's data? Why or why not?
A: No. A WARP violation would require each bundle to be affordable when the other was chosen. Here, (1, 105) was not affordable under Budget 1 ($1,150 > $1,000), so the data are consistent with rational choice.
Revealed preference connects to the theory of the consumer: if a consumer truly maximises a well-behaved utility function, their choices will always satisfy WARP. Compensated demand connects to the expenditure function and Shephard's lemma (the derivative of the expenditure function with respect to a price gives the Hicksian demand for that good). The perfect-substitutes case is the polar opposite of perfect complements, where the consumer always buys goods in fixed proportions.
Revealed preference, directly revealed preferred, WARP, weak axiom of revealed preference, SARP, affordability test, budget constraint, compensated demand, Hicksian demand, expenditure minimisation, perfect substitutes, corner solution, interior solution, marginal rate of substitution, MRS, piecewise function, Shephard's lemma, ECON 300, intermediate microeconomics.