Hicksian demand function¶

A compensated demand function giving expenditure-minimizing quantities at prices while holding utility fixed.

Version: v3 · 2026-09-19 · History
Domain-specific #: 2001
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Microeconomic Consumer Theory → Economics & Finance
Aliases: Compensated demand function

Core Idea¶

Hicksian demand function is a compensated demand function giving expenditure-minimizing quantities at prices while holding utility fixed.

For prices p and target utility u-bar, Hicksian demand h(p,u-bar) solves the expenditure-minimization problem: choose a consumption bundle achieving at least u-bar at minimum p·x. It is dual to utility maximization, generates the expenditure function, and isolates substitution effects because compensation holds utility rather than nominal income fixed.

Its operative boundary is not supplied by the name alone. Preserve this identity: A compensated demand function giving expenditure-minimizing quantities at prices while holding utility fixed. Validity boundary: Demand must solve the expenditure-minimization problem at fixed utility; Marshallian demand at fixed income is not Hicksian.

How would you explain it like I'm…

Cheapest Way to Stay Happy

Say you want to be exactly as happy as you are with your snacks today. When prices change, you ask: what is the cheapest mix of snacks that still makes me just this happy? Hicksian demand is that cheapest happy-enough mix.

Same Happiness, Least Money

Economists study how people choose what to buy. The Hicksian demand function picks a fixed level of happiness (economists call it 'utility') and asks: at these prices, what is the cheapest mix of goods that reaches it? When prices change, the answer shows how a person would swap one good for another just to stay equally happy. This is different from the usual demand, which keeps the person's money fixed instead of their happiness.

Compensated Demand at Fixed Utility

The Hicksian demand function h(p, ū) gives the quantities of goods a consumer buys when minimizing spending at prices p while still reaching a target utility ū. It is called 'compensated' demand because, as prices change, the consumer is imagined to be compensated so their utility stays fixed. That lets it isolate the substitution effect: how choices shift purely because relative prices changed. Contrast this with Marshallian demand, which holds money income fixed and maximizes utility, mixing substitution with income effects. Plugging Hicksian demand into total cost gives the expenditure function.

The Hicksian (compensated) demand function h(p, ū) solves the expenditure-minimization problem: minimize p·x over bundles x subject to u(x) ≥ ū. It gives cost-minimizing quantities as a function of prices with utility, not income, held fixed. This problem is dual to utility maximization, and its value e(p, ū) = p·h(p, ū) is the expenditure function. Because utility is held constant as prices vary, the responses of h to prices capture pure substitution effects, free of income effects. Marshallian demand, by contrast, maximizes utility at fixed nominal income, so its price responses combine substitution and income effects. The validity boundary is strict: a demand counts as Hicksian only if it solves expenditure minimization at fixed utility.

Scope of Application¶

The abstraction recurs literally within microeconomic analyses of expenditure, compensated price responses, welfare, and dual consumer theory. The following habitats preserve the same recognition machinery; they are not invitations to extend the name metaphorically.

Demand duality. utility maximization and expenditure minimization are related.
Slutsky decomposition. price effects split into substitution and income components.
Cost-of-living indices. expenditure needed for fixed welfare is compared across prices.
Welfare analysis. compensating and equivalent variation use expenditure functions.
Consumer estimation. structural models recover compensated responses under maintained assumptions.

Clarity¶

The function's arguments are prices and utility, not prices and income. The target utility is ordinally represented, so transformations of utility change its numerical label but not the underlying compensated choice when handled consistently.

A practical identification audit begins with the typed roles rather than the title: establish the preference representation, verify the price vector, then test the remaining conditions and exclusions.

Manages Complexity¶

Hicksian demand separates a price change's substitution mechanism from purchasing-power change. The dual problem also converts preference information into an expenditure surface whose derivatives reveal compensated quantities under regularity.

The compression remains accountable because each simplification has a named failure condition. Disagreement can be localized to a missing role, an invalid assumption, an ambiguous measurement, or a neighboring abstraction instead of being hidden inside an unanalyzed label.

Abstract Reasoning¶

R1. State preferences, prices, feasible bundles, and the target utility. R2. Solve cost minimization subject to reaching that utility. R3. Check existence, uniqueness, and interior assumptions. R4. Use Shephard's lemma only where differentiability applies. R5. Compare with Marshallian demand through the correct expenditure or indirect-utility identity.

Knowledge Transfer¶

The function transfers literally across consumer problems satisfying expenditure-minimization structure. Optimization, demand, and price mechanisms are broader parents; a budget-constrained purchase prediction is Marshallian unless utility is explicitly held fixed.

The transfer boundary is explicit: DOMAIN-SPECIFIC PASS / PRIME FAIL: The function recurs across consumers, goods, price vectors, and utility levels in expenditure-minimization analysis. Literal recognition retains the specialist vocabulary and validity conditions of microeconomic consumer theory; outside that setting only broader parent operations transfer.

Relationships to Other Abstractions¶

Current abstraction Hicksian demand function Domain-specific

Parents (3) — outgoing typed relations

Hicksian demand function is a kind of Demand Prime

Demand (prime:demand).
Hicksian demand function is a kind of Optimization Prime

Optimization (prime:optimization).
Hicksian demand function is a kind of Price Mechanism Prime

Price Mechanism (prime:price_mechanism).

Hierarchy paths (4) — routes to 4 parentless roots

Hicksian demand function → is a kind of Demand → presupposes Preference

Show alternative paths (3)

Neighborhood in Abstraction Space¶

Hicksian demand function sits in a moderately populated region (44^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Demand Elasticity & Consumer Response (11 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25