Shephard’s lemma
E833554
Shephard’s lemma is a result in microeconomics stating that the derivative of a cost (or expenditure) function with respect to input (or price) yields the corresponding conditional factor (or Hicksian demand) demand function.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Shephard’s lemma canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9986293 — 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: Shephard’s lemma Context triple: [Hotelling’s lemma, relatedTo, Shephard’s lemma]
-
A.
Hotelling’s lemma
Hotelling’s lemma is a result in microeconomics that links a firm’s profit function to its supply and factor demand functions via partial derivatives.
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B.
Hicksian demand
Hicksian demand is a concept in microeconomics that describes how a consumer’s demand for goods changes when prices vary while holding utility (satisfaction) constant, often used in welfare and consumer theory.
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C.
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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D.
Marshallian demand
Marshallian demand is the consumer demand function that expresses the quantity of a good chosen as a function of prices and income, derived from utility maximization under a budget constraint.
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E.
Hicks–Kaldor compensation criterion
The Hicks–Kaldor compensation criterion is an economic efficiency test stating that a policy change is desirable if those who gain could in principle compensate those who lose and still be better off, regardless of whether compensation actually occurs.
- 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: Shephard’s lemma Target entity description: Shephard’s lemma is a result in microeconomics stating that the derivative of a cost (or expenditure) function with respect to input (or price) yields the corresponding conditional factor (or Hicksian demand) demand function.
-
A.
Hotelling’s lemma
Hotelling’s lemma is a result in microeconomics that links a firm’s profit function to its supply and factor demand functions via partial derivatives.
-
B.
Hicksian demand
Hicksian demand is a concept in microeconomics that describes how a consumer’s demand for goods changes when prices vary while holding utility (satisfaction) constant, often used in welfare and consumer theory.
-
C.
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.
-
D.
Marshallian demand
Marshallian demand is the consumer demand function that expresses the quantity of a good chosen as a function of prices and income, derived from utility maximization under a budget constraint.
-
E.
Hicks–Kaldor compensation criterion
The Hicks–Kaldor compensation criterion is an economic efficiency test stating that a policy change is desirable if those who gain could in principle compensate those who lose and still be better off, regardless of whether compensation actually occurs.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
microeconomic theorem
ⓘ
result in duality theory ⓘ |
| appliesTo |
cost function
ⓘ
expenditure function ⓘ |
| assumes |
convex technology or convex preferences
ⓘ
interior solutions to the minimization problem ⓘ well-behaved technology or preferences ⓘ |
| characterizes |
relationship between prices and compensated demand
ⓘ
relationship between prices and optimal input choices ⓘ |
| consequenceOf | envelope properties of value functions ⓘ |
| domain |
theory of the consumer
ⓘ
theory of the firm ⓘ |
| field |
consumer theory
ⓘ
microeconomics ⓘ production theory ⓘ |
| holdsUnder |
cost function is the value function of a cost minimization problem
ⓘ
expenditure function is the value function of an expenditure minimization problem ⓘ |
| implies |
cost shares can be obtained from derivatives of the cost function
ⓘ
expenditure shares can be obtained from derivatives of the expenditure function ⓘ |
| mathematicalForm |
∂C(w,y)/∂w_i = x_i(w,y)
ⓘ
∂e(p,u)/∂p_i = h_i(p,u) ⓘ |
| namedAfter |
Ronald Shephard
ⓘ
linked to:
Ron Shephard
|
| originatesIn |
Ronald Shephard’s work on cost and production functions
ⓘ
duality theory of production ⓘ |
| relatedTo |
Hotelling’s lemma
ⓘ
duality between primal and dual optimization problems ⓘ envelope theorem ⓘ |
| relates |
cost function and conditional factor demand
ⓘ
expenditure function and Hicksian demand ⓘ |
| requires |
cost minimization behavior
ⓘ
differentiability of the cost or expenditure function ⓘ expenditure minimization behavior ⓘ |
| statement |
the derivative of the cost function with respect to an input price equals the conditional demand for that input
ⓘ
the derivative of the expenditure function with respect to a good’s price equals the Hicksian demand for that good ⓘ |
| typeOf | comparative statics result ⓘ |
| usedFor |
duality-based consumer analysis
ⓘ
duality-based production analysis ⓘ empirical estimation of demand systems ⓘ recovering Hicksian demand from the expenditure function ⓘ recovering conditional factor demand from the cost function ⓘ |
| usedIn |
derivation of Almost Ideal Demand System
ⓘ
derivation of translog cost functions ⓘ |
| usedToDerive |
compensated price elasticities from expenditure functions
ⓘ
factor demand elasticities from cost functions ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Shephard’s lemma Description of subject: Shephard’s lemma is a result in microeconomics stating that the derivative of a cost (or expenditure) function with respect to input (or price) yields the corresponding conditional factor (or Hicksian demand) demand function.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.