In microeconomics, a consumer's Hicksian demand function (or compensated demand function) represents the quantity of a good demanded when the consumer minimizes expenditure while maintaining a fixed level of utility.
The Hicksian demand function illustrates how a consumer would adjust their demand for a good in response to a price change, assuming their income is adjusted (or compensated) to keep them on the same indifference curve—ensuring their utility remains unchanged.
The Hicksian demand function isolates the effect of relative prices on demand, assuming utility remains constant.
It contrasts with the Marshallian demand function, which accounts for both the substitution effect and the reduction in real income caused by price changes.
Hicksian demand functions are often convenient for mathematical manipulation because they do not require representing income or wealth.
Their derivatives are more fundamentally related by the Slutsky equation.
The two problems are mathematical duals, and hence the Duality Theorem provides a method of proving the relationships described above.
is locally nonsatiated and strictly convex, then by Shephard's lemma it is true that
Note that if there is more than one vector of quantities that minimizes expenditure for the given utility, we have a Hicksian demand correspondence rather than a function.
The substitution effect is the change in quantity demanded due to a price change that alters the slope of the budget constraint but leaves the consumer on the same indifference curve (i.e., at the same level of utility).
The income effect is the change in quantity demanded due to the effect of the price change on the consumer's total buying power.
Since for the Marshallian demand function the consumer's nominal income is held constant, when a price rises his real income falls and he is poorer.
If the good in question is a normal good and its price rises, the income effect from the fall in purchasing power reinforces the substitution effect.
If the good is a Giffen good, the income effect is so strong that the Marshallian quantity demanded rises when the price rises.
The Hicksian demand function isolates the substitution effect by supposing the consumer is compensated with exactly enough extra income after the price rise to purchase some bundle on the same indifference curve.
is continuous and represents a locally nonsatiated preference relation, then the Hicksian demand correspondence