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.

Version: v1 · 2026-09-28 · History
Domain-specific #: 11854
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomains: Microeconomics, Consumer Theory → Economics & Finance

Core Idea¶

Roy's identity is treated here as the recurring social_sciences_humanities_arts 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. Specifically, denoting the indirect utility function as v(p,w), the Marshallian demand function for good i can be calculated as.

x_{i}^{m}(p,w)=-\frac{\frac{\partial v}{\partial p_{i}}}{\frac{\partial v}{\partial w}}. where p is the price vector of goods and w is income, and where the superscript {}^m indicates Marshallian demand. The result holds for continuous utility functions representing locally non-satiated and strictly convex preference relations on a convex consumption set, under the additional requirement that the indirect utility function is differentiable in all arguments.

For Roy's identity, the abstraction is narrower than the article's general subject matter: a positive case must preserve 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. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in social_sciences_humanities_arts, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — 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 of u.
Constitutive relation — The indirect utility function v(p_{1},p_{2},w) is the value function of the constrained optimization problem characterized by the following Lagrangian.
Operating condition — 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 the original utility function represents the same preferences.
Recognition evidence — 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.
Admissible variation — 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 to that utility when evaluated at those prices.
Characteristic consequence — Taking the derivative of both sides of this equation with respect to the price of a single good p_i (with the utility level held constant) gives.
Failure boundary — \frac{ \partial v [p, e(p,u)]}{\partial w} \frac{\partial e(p,u)}{\partial p_i} + \frac{\partial v [p, e(p,u)]}{\partial p_i} = 0 .

What It Is Not¶

Not the whole field of social_sciences_humanities_arts. The node requires the specific identity stated by 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.
Not an over-broad reading. The result holds for continuous utility functions representing locally non-satiated and strictly convex preference relations on a convex consumption set, under the additional requirement that the indirect utility function is differentiable in all arguments.
Not an over-broad reading. 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 the original utility function represents the same preferences.
Not an over-broad reading. 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.
Not automatically Hicksian demand function. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Roy's identity applies literally inside social_sciences_humanities_arts wherever the source-defined carrier and relation can be established. Its documented habitats include:

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.
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.
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 of u.
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 to that utility when evaluated at those prices.
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.
Alternative proof using the envelope theorem. By the envelope theorem, the derivatives of the value function v(p_{1},p_{2},w) with respect to the parameters are.

Outside social_sciences_humanities_arts, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

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. The strongest recognition evidence in the frozen account is: 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. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification The result holds for continuous utility functions representing locally non-satiated and strictly convex preference relations on a convex consumption set, under the additional requirement that the indirect utility function is differentiable in all arguments. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Roy's identity compresses multiple social_sciences_humanities_arts 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 p_i (with the utility level held constant) gives. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the social_sciences_humanities_arts 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 the original utility function represents the same preferences.
Demand recognition evidence. 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.
Test variation. Change an implementation or setting while preserving 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 to that utility when evaluated at those prices.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

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 canonical parent is asserted for Roy's identity. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

For expositional ease, consider the two-goods case. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → 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; recognition evidence → 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

Applied / In Practice¶

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. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Derivation of Roy's identity; invariant → 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; boundary → the case exits the class when the result holds for continuous utility functions representing locally non-satiated and strictly convex preference relations on a convex consumption set, under the additional requirement that the indirect utility function is differentiable in all arguments

Structural Tensions¶

T1 — Stable identity versus admissible variation. The result holds for continuous utility functions representing locally non-satiated and strictly convex preference relations on a convex consumption set, under the additional requirement that the indirect utility function is differentiable in all arguments. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. 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 the original utility function represents the same preferences. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. 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. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. 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 of u. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. 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 of u. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Roy's identity literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. The indirect utility function v(p_{1},p_{2},w) is the value function of the constrained optimization problem characterized by the following Lagrangian. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Roy's identity distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Roy's identity is mixed or framed-leaning. Its structural side is the repeatable organization summarized by 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. Its framed side is the social_sciences_humanities_arts vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: 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 the original utility function represents the same preferences. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. 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 stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: 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 of u. The indirect utility function v(p{1},p{2},w) is the value function of the constrained optimization problem characterized by the following Lagrangian. It further constrains recognition and variation through: 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 the original utility function represents the same preferences. 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.

What is domain-bound. social sciences humanities arts supplies the operative entities, technical vocabulary, warrants, and exceptions that make Roy's identity literal. Its documented scope includes the condition that This gives a method of deriving the Marshallian demand function of a good for some consumer from the indirect utility function of that consumer. Another bounded application condition is that 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. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—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 to that utility when evaluated at those prices.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Roy's identity. The reviewed identity 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. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Roy's identity sits in a sparse region of the domain-specific corpus (71^st 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

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish 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?
Hicksian demand function. A compensated demand function giving expenditure-minimizing quantities at prices while holding utility fixed. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Expenditure function. The minimum spending needed at given prices to attain a specified utility level. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Friedman–Savage utility function. A wealth-dependent expected-utility curve with alternating concave and convex regions intended to explain simultaneous insurance purchase and lottery play. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Roy's identity remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside social_sciences_humanities_arts lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Roy%27s_identity (revision 1257711792).
Preserved source candidate: https://archive.org/details/microeconomicana00vari_0/page/106
Preserved source candidate: https://archive.org/details/microeconomicana00vari_0
Preserved source candidate: https://books.google.com/books?id=HO8zAAAAIAAJ&pg=PA45

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.