When the price of a good rises, its Hicksian demand decreases. Yet for a Giffen good, Marshallian demand *increases* when price rises. How can both statements be true simultaneously?
AGiffen goods violate Shephard's lemma, so Hicksian demand analysis does not apply to them
BHicksian demand isolates only the substitution effect (always negative), while Marshallian demand also includes the income effect, which for a strongly inferior good can be large enough to dominate and reverse the direction
CBoth Hicksian and Marshallian demand slope downward for Giffen goods; the difference is only in the magnitude of the response
DThe Giffen paradox only affects demand measured in nominal terms; in real terms both demands slope downward
Hicksian demand holds utility constant, so it captures only the pure substitution effect: as a good becomes more expensive, the consumer substitutes away from it along the same indifference curve. This substitution effect is always negative (less of the good at higher prices), making Hicksian demand always downward-sloping. Marshallian demand also includes the income effect — rising prices reduce real purchasing power. For a strongly inferior good (a Giffen good), this income effect is positive and large enough to overwhelm the negative substitution effect, producing upward-sloping Marshallian demand. The Slutsky equation formalizes this decomposition.
Question 2 Multiple Choice
A consumer minimizes expenditure subject to achieving utility ū at prices p. Shephard's lemma states that the Hicksian demand for good i equals:
A∂x_i/∂p_i — the change in Marshallian demand when price i changes
B∂e(p, ū)/∂p_i — the partial derivative of the expenditure function with respect to price i
C∂e(p, ū)/∂ū — the marginal cost of raising the utility target
DThe slope of the income-consumption path at prices p
Shephard's lemma is the key technical result connecting Hicksian demand to the expenditure function: h_i(p, ū) = ∂e(p, ū)/∂p_i. It follows from the envelope theorem — at the expenditure-minimizing bundle, a small increase in p_i raises minimum expenditure by exactly the quantity of good i being consumed, because the consumer is already optimizing and adjusts only at the margin. This gives a powerful computational shortcut: derive the expenditure function once, and differentiate with respect to prices to recover all Hicksian demands.
Question 3 True / False
Hicksian demand curves always slope downward, regardless of whether the good is normal, inferior, or Giffen.
TTrue
FFalse
Answer: True
Hicksian demand is derived from expenditure minimization holding utility constant, so it captures only the substitution effect. The substitution effect is always negative (by the concavity of the expenditure function in prices): as a good becomes more expensive relative to alternatives, a utility-maximizing consumer always substitutes away from it along the same indifference curve. There is no income effect to potentially reverse this. This is in contrast to Marshallian demand, which can slope upward for Giffen goods because it includes both substitution and income effects.
Question 4 True / False
Hicksian demand holds income constant while Marshallian demand holds utility constant.
TTrue
FFalse
Answer: False
This is reversed. Marshallian (uncompensated) demand x(p, m) holds *income* m fixed — it is the standard demand function derived from utility maximization subject to a budget constraint. Hicksian (compensated) demand h(p, ū) holds *utility* ū fixed — it is derived from expenditure minimization subject to achieving a given utility level. The term 'compensated' refers to the fact that when prices change, Hicksian demand imagines the consumer's income being adjusted to keep them on the same indifference curve, isolating the pure substitution effect.
Question 5 Short Answer
Explain why Hicksian demand is called 'compensated' demand and what it means for the consumer's income to be 'adjusted' as prices change.
Think about your answer, then reveal below.
Model answer: Hicksian demand is 'compensated' because it imagines compensating the consumer for price changes — adjusting their income just enough to keep them at the same utility level as prices shift. If a price rises and real purchasing power falls, the consumer is hypothetically given extra income to restore their original utility. This compensation removes the income effect from the price response, leaving only the pure substitution effect: the change in the bundle that results from relative prices shifting while the standard of living stays constant. The resulting demand function traces movement along a single indifference curve.
The compensation is hypothetical — it is an analytical device to decompose the total price effect rather than a policy being implemented. In practice, when prices change, income is not literally adjusted. But by imagining this compensation, Hicksian demand isolates the part of demand behavior that is purely about substitution between goods at different relative prices, without the confounding effect of price changes on purchasing power. This isolation is what makes Hicksian demand the right tool for welfare analysis and the Slutsky decomposition.