Roy's identity¶
Roy's identity (named after French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm.
Core Idea¶
Roy's identity is treated here as the recurring socialscienceshumanitiesarts identity summarized by this source-grounded definition: Roy's identity (named after French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. Roy's identity (named after French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary (Marshallian) demand function to the derivatives of the indirect utility function.
Scope of Application¶
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Application. This gives a method of deriving the Marshallian demand function of a good for some consumer from the indirect utility function of that consumer.
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Derivation of Roy's identity. Roy's identity reformulates Shephard's lemma in order to get a Marshallian demand function for an individual and a good ( i ) from some indirect utility function.
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Derivation of Roy's identity. The first step is to consider the trivial identity obtained by substituting the expenditure function for wealth or income w in the indirect utility function v (p, w) , at a utility.
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Derivation of Roy's identity. This says that the indirect utility function evaluated in such a way that minimizes the cost for achieving a certain utility given a set of prices (a vector p ) is equal.
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Alternative proof using the envelope theorem. The indirect utility function v(p{1},p{2},w) is the value function of the constrained optimization problem characterized by the following Lagrangian.
Clarity¶
A clear use of Roy's identity names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is Roy's identity (named after French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm.
Manages Complexity¶
Roy's identity compresses multiple socialscienceshumanitiesarts details into a stable diagnostic relation. The source shows both the central mechanism—the indirect utility function v(p{1},p{2},w) is the value function of the constrained optimization problem characterized by the following Lagrangian.—and the practical consequence—taking the derivative of both sides of this equation with respect to the price of a single good pi (with the utility level held.
Abstract Reasoning¶
- Type the carrier. Identify the socialscienceshumanitiesarts entities to which the claim applies.
- State the relation. Use the source-grounded identity: Roy's identity (named after French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm.
- Check operation and conditions. The additional step of dividing by the wealth derivative of the indirect utility function in Roy's identity is necessary since the indirect utility function, unlike the expenditure function, has an ordinal interpretation: any strictly increasing transformation of.
Knowledge Transfer¶
Within the home domain. Knowledge about Roy's identity transfers literally when a new case preserves the same carrier type, relation, and recognition test. This gives a method of deriving the Marshallian demand function of a good for some consumer from the indirect utility function of that consumer. Roy's identity reformulates Shephard's lemma in order to get a Marshallian demand function for an individual and a good ( i ) from some indirect utility function. Beyond the home domain. No.
Neighborhood in Abstraction Space¶
Roy's identity sits in a sparse region of the domain-specific corpus (71st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Microeconomic Theory & Welfare Criteria (13 abstractions)
Nearest neighbors
- Nash welfare rule — 0.84
- Elasticity of intertemporal substitution — 0.84
- Abstract economy — 0.83
- Valuation (geometry) — 0.83
- Filling radius — 0.83
Computed from structural-signature embeddings · 2026-10-08