Slutsky substitution matrix is negative semidefinite
E746878
The Slutsky substitution matrix is negative semidefinite is a core result in consumer theory stating that, under standard assumptions, the matrix of compensated price effects has non-positive eigenvalues, reflecting the law of demand in a multigood setting.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Slutsky substitution matrix is negative semidefinite canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8630232 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Slutsky substitution matrix is negative semidefinite Context triple: [Hicksian demand, implies, Slutsky substitution matrix is negative semidefinite]
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A.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
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B.
Gale–Nikaidō–Debreu theorem
The Gale–Nikaidō–Debreu theorem is a fundamental result in mathematical economics that provides conditions ensuring the existence (and sometimes uniqueness) of equilibrium in certain nonlinear and general equilibrium models.
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C.
Karush–Kuhn–Tucker conditions
The Karush–Kuhn–Tucker conditions are fundamental optimality criteria in nonlinear programming that generalize Lagrange multipliers to handle inequality constraints.
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D.
Toeplitz matrices
Toeplitz matrices are structured matrices whose entries are constant along each diagonal, playing a central role in operator theory, numerical analysis, and signal processing.
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E.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Slutsky substitution matrix is negative semidefinite Target entity description: The Slutsky substitution matrix is negative semidefinite is a core result in consumer theory stating that, under standard assumptions, the matrix of compensated price effects has non-positive eigenvalues, reflecting the law of demand in a multigood setting.
-
A.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
-
B.
Gale–Nikaidō–Debreu theorem
The Gale–Nikaidō–Debreu theorem is a fundamental result in mathematical economics that provides conditions ensuring the existence (and sometimes uniqueness) of equilibrium in certain nonlinear and general equilibrium models.
-
C.
Karush–Kuhn–Tucker conditions
The Karush–Kuhn–Tucker conditions are fundamental optimality criteria in nonlinear programming that generalize Lagrange multipliers to handle inequality constraints.
-
D.
Toeplitz matrices
Toeplitz matrices are structured matrices whose entries are constant along each diagonal, playing a central role in operator theory, numerical analysis, and signal processing.
-
E.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
result in microeconomic theory
ⓘ
theorem in consumer theory ⓘ |
| appliesTo | finite-dimensional commodity spaces ⓘ |
| assumes |
convex preferences
ⓘ
differentiable utility or demand functions ⓘ locally non-satiated preferences ⓘ standard consumer theory assumptions ⓘ |
| concerns |
Hicksian demand functions
NERFINISHED
ⓘ
Slutsky substitution matrix NERFINISHED ⓘ compensated price effects ⓘ |
| connectedTo | Afriat–Varian revealed preference conditions in differential form ⓘ |
| contrastsWith | positive semidefinite matrices in production theory ⓘ |
| describes | curvature property of the expenditure function ⓘ |
| equivalentTo |
Hicksian demand satisfies the law of compensated demand
NERFINISHED
ⓘ
expenditure function is concave in prices ⓘ |
| field |
consumer theory
ⓘ
microeconomics ⓘ |
| guarantees | consistency of observed Hicksian demand with utility maximization ⓘ |
| historicalOrigin | work of Eugen Slutsky on demand theory ⓘ |
| holdsUnder |
interior solutions to the consumer problem
ⓘ
well-behaved preferences ⓘ |
| implies |
eigenvalues of the Slutsky substitution matrix are non-positive
ⓘ
for any price change vector the associated quadratic form is non-positive ⓘ law of demand in a multigood setting ⓘ own-price compensated effects are non-positive ⓘ substitution terms cannot generate positive quadratic forms ⓘ |
| isPropertyOf | matrix of partial derivatives of Hicksian demand with respect to prices ⓘ |
| isSpecialCaseOf | curvature restrictions in applied demand analysis ⓘ |
| logicalStatus |
necessary condition for rationalizable Hicksian demand
ⓘ
not sufficient alone for full integrability ⓘ |
| mathematicalForm | Slutsky matrix is symmetric and negative semidefinite NERFINISHED ⓘ |
| motivates | imposition of curvature constraints in econometric models of demand ⓘ |
| refersTo | negative semidefiniteness of the Slutsky substitution matrix ⓘ |
| relatedTo |
Hicksian substitution effect
NERFINISHED
ⓘ
Marshallian demand functions NERFINISHED ⓘ Slutsky equation NERFINISHED ⓘ integrability conditions for demand ⓘ |
| requires | twice differentiable expenditure function ⓘ |
| testedIn | empirical demand system estimation ⓘ |
| usedIn |
comparative statics of consumer demand
ⓘ
duality theory in microeconomics ⓘ welfare analysis ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Slutsky substitution matrix is negative semidefinite Description of subject: The Slutsky substitution matrix is negative semidefinite is a core result in consumer theory stating that, under standard assumptions, the matrix of compensated price effects has non-positive eigenvalues, reflecting the law of demand in a multigood setting.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.