Shephard's lemma
Encyclopedia
Shephard's lemma is a major result in microeconomics
having applications in the theory of the firm
and in consumer
choice.
The lemma
states that if indifference curves of the expenditure or cost function are convex
, then the cost minimizing point of a given good (
) with price
is unique. The idea is that a consumer
will buy a unique ideal amount of each item to minimize the price for obtaining a certain level of utility
given the price of goods in the market
.
The lemma is named after Ronald Shephard
who gave a proof
using the distance formula in his book Theory of Cost and Production Functions (Princeton University Press, 1953).
The equivalent result in the context of consumer theory was first derived by Lionel W. McKenzie
in 1957. It states that the partial derivatives of the expenditure function with respect the prices of goods equal the Hicksian demand functions for the relevant goods. Similar results had already been derived by John Hicks
(1939) and Paul Samuelson
(1947).
theory, Shephard's lemma states that the demand
for a particular good i for a given level of utility u and given prices p, equals the derivative of the expenditure function
with respect to the price of the relevant good:
where hi(p,u) is the Hicksian demand for good
, e(p,u) is the expenditure function, and both functions are in terms of prices (a vector p) and utility 
.
Likewise, in the theory of the firm
, the lemma gives a similar formulation for the conditional factor demand
for each input factor: the derivative of the cost function c(w,y) with respect to the factor price:
where xi(w,y) is the conditional factor demand
for input
, c(w,y) is the cost function, and both functions are in terms of factor prices (a vector w) and output 
.
Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the envelope theorem
.
is the minimand of the constrained optimization problem characterized by the following Lagrangian:
By the envelope theorem
the derivatives of the minimand
with respect to the parameter 
can be computed as such:
where
is the minimizer (i.e. the Hicksian demand function for good 1). This completes the proof.
, which gives a relationship between an indirect utility function
and a corresponding Marshallian demand function
.
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...
having applications in the theory of the firm
Theory of the firm
The theory of the firm consists of a number of economic theories that describe the nature of the firm, company, or corporation, including its existence, behavior, structure, and relationship to the market.-Overview:...
and in consumer
Consumer
Consumer is a broad label for any individuals or households that use goods generated within the economy. The concept of a consumer occurs in different contexts, so that the usage and significance of the term may vary.-Economics and marketing:...
choice.
The lemma
Lemma (mathematics)
In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...
states that if indifference curves of the expenditure or cost function are convex
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
, then the cost minimizing point of a given good (
) with pricePrice
-Definition:In ordinary usage, price is the quantity of payment or compensation given by one party to another in return for goods or services.In modern economies, prices are generally expressed in units of some form of currency...
is unique. The idea is that a consumerConsumer
Consumer is a broad label for any individuals or households that use goods generated within the economy. The concept of a consumer occurs in different contexts, so that the usage and significance of the term may vary.-Economics and marketing:...
will buy a unique ideal amount of each item to minimize the price for obtaining a certain level of utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
given the price of goods in the market
Market
A market is one of many varieties of systems, institutions, procedures, social relations and infrastructures whereby parties engage in exchange. While parties may exchange goods and services by barter, most markets rely on sellers offering their goods or services in exchange for money from buyers...
.
The lemma is named after Ronald Shephard
Ronald Shephard
Ronald William Shephard was Professor of Engineering Science at the University of California, Berkeley.He is best known for two results in economics, now known as Shephard's lemma and the Shephard duality theorem...
who gave a proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
using the distance formula in his book Theory of Cost and Production Functions (Princeton University Press, 1953).
The equivalent result in the context of consumer theory was first derived by Lionel W. McKenzie
Lionel W. McKenzie
Lionel Wilfred McKenzie was the Wilson Professor Emeritus of Economics at the University of Rochester. He was born in Montezuma, Georgia. He completed undergraduate studies at Duke University in 1939 and subsequently moved to Oxford that year as a Rhodes Scholar...
in 1957. It states that the partial derivatives of the expenditure function with respect the prices of goods equal the Hicksian demand functions for the relevant goods. Similar results had already been derived by John Hicks
John Hicks
Sir John Richard Hicks was a British economist and one of the most important and influential economists of the twentieth century. The most familiar of his many contributions in the field of economics were his statement of consumer demand theory in microeconomics, and the IS/LM model , which...
(1939) and Paul Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...
(1947).
Definition
In consumerConsumer
Consumer is a broad label for any individuals or households that use goods generated within the economy. The concept of a consumer occurs in different contexts, so that the usage and significance of the term may vary.-Economics and marketing:...
theory, Shephard's lemma states that the demand
Demand
- Economics :*Demand , the desire to own something and the ability to pay for it*Demand curve, a graphic representation of a demand schedule*Demand deposit, the money in checking accounts...
for a particular good i for a given level of utility u and given prices p, equals the derivative of the expenditure function
Expenditure function
In microeconomics, the expenditure function describes the minimum amount of money an individual needs to achieve some level of utility, given a utility function and prices....
with respect to the price of the relevant good:
where hi(p,u) is the Hicksian demand for good
, e(p,u) is the expenditure function, and both functions are in terms of prices (a vector p) and utility .Likewise, in the theory of the firm
Theory of the firm
The theory of the firm consists of a number of economic theories that describe the nature of the firm, company, or corporation, including its existence, behavior, structure, and relationship to the market.-Overview:...
, the lemma gives a similar formulation for the conditional factor demand
Conditional factor demands
In economics, a conditional factor demand function specifies the cost-minimizing level of an input such as labor or capital, required to produce a given level of output, for given unit input costs of the input factors...
for each input factor: the derivative of the cost function c(w,y) with respect to the factor price:
where xi(w,y) is the conditional factor demand
Conditional factor demands
In economics, a conditional factor demand function specifies the cost-minimizing level of an input such as labor or capital, required to produce a given level of output, for given unit input costs of the input factors...
for input
, c(w,y) is the cost function, and both functions are in terms of factor prices (a vector w) and output .Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the envelope theorem
Envelope theorem
The envelope theorem is a theorem about optimization problems in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity...
.
Proof for the Differentiable Case
The proof is stated for the two-good case for ease of notation. The expenditure function is the minimand of the constrained optimization problem characterized by the following Lagrangian:By the envelope theorem
Envelope theorem
The envelope theorem is a theorem about optimization problems in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity...
the derivatives of the minimand
with respect to the parameter can be computed as such:where
is the minimizer (i.e. the Hicksian demand function for good 1). This completes the proof.Application
Shephard's lemma gives a relationship between expenditure (or cost) functions and Hicksian demand. The lemma can be re-expressed as Roy's identityRoy's identity
Roy's identity is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary demand function to the derivatives of the indirect utility function...
, which gives a relationship between an indirect utility function
Indirect utility function
In economics, a consumer's indirect utility functionv gives the consumer's maximal utility when faced with a price level p and an amount of income w. It represents the consumer's preferences over market conditions....
and a corresponding Marshallian demand function
Marshallian demand function
In microeconomics, a consumer's Marshallian demand function specifies what the consumer would buy in each price and wealth situation, assuming it perfectly solves the utility maximization problem...
.