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Shephard's lemma is a major result in microeconomics having applications in the
theory of the firm The theory of the firm consists of a number of economic theories that explain and predict the nature of the firm, company, or corporation, including its existence, behaviour, structure, and relationship to the market. Firms are key drivers in ec ...
and in
consumer choice The theory of consumer choice is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption as measured by their pre ...
. The
lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a ...
states that if indifference curves of the expenditure or cost function are
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
, then the cost minimizing point of a given good (i) with
price A price is the (usually not negative) quantity of payment or compensation given by one party to another in return for goods or services. In some situations, the price of production has a different name. If the product is a "good" in t ...
p_i is unique. The idea is that a
consumer A consumer is a person or a group who intends to order, or uses purchased goods, products, or services primarily for personal, social, family, household and similar needs, who is not directly related to entrepreneurial or business activities. ...
will buy a unique ideal amount of each item to minimize the price for obtaining a certain level of
utility As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
given the price of goods in the market. The lemma is named after Ronald Shephard who gave a
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...
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 to the prices of goods equal the
Hicksian demand function In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is his quantity demanded as part of the solution to minimizing his expenditure on all goods while delivering a fixed level of utility. Essenti ...
s for the relevant goods. Similar results had already been derived by
John Hicks Sir John Richards Hicks (8 April 1904 – 20 May 1989) was a British economist. He is considered one of the most important and influential economists of the twentieth century. The most familiar of his many contributions in the field of economic ...
(1939) and
Paul Samuelson Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he "h ...
(1947).


Definition

In
consumer A consumer is a person or a group who intends to order, or uses purchased goods, products, or services primarily for personal, social, family, household and similar needs, who is not directly related to entrepreneurial or business activities. ...
theory, Shephard's lemma states that the
demand In economics, demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given time. The relationship between price and quantity demand is also called the demand curve. Demand for a specific item ...
for a particular good i for a given level of utility u and given prices \mathbf, equals the derivative of the expenditure function with respect to the price of the relevant good: :h_i(\mathbf, u) = \frac where h_i(\mathbf, u) is the Hicksian demand for good i, e (\mathbf, u) is the expenditure function, and both functions are in terms of prices (a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
\mathbf) and utility u. Likewise, in the
theory of the firm The theory of the firm consists of a number of economic theories that explain and predict the nature of the firm, company, or corporation, including its existence, behaviour, structure, and relationship to the market. Firms are key drivers in ec ...
, the lemma gives a similar formulation for the conditional factor demand for each input factor: the derivative of the cost function c (\mathbf, y) with respect to the factor price: :x_i(\mathbf, y) = \frac where x_i(\mathbf, y) is the conditional factor demand for input i, c (\mathbf, y) is the cost function, and both functions are in terms of factor prices (a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
\mathbf) and output y. Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the envelope theorem.


Proof for the differentiable case

The proof is stated for the two-good case for ease of notation. The expenditure function e(p_,p_,u) is the value function of the constrained optimization problem characterized by the following Lagrangian: :\mathcal=p_x_ + p_x_ + \lambda(u-U(x_,x_)) By the envelope theorem the derivatives of the value function e(p_,p_,u) with respect to the parameter p_ are: :\frac=\frac=x_^ where x_^ 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 identity, which gives a relationship between an indirect utility function and a corresponding
Marshallian demand function In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) is the quantity they demand of a particular good as a function of its price, their income, and the prices of other goods, a more technical exposition of the ...
.


See also

*
Hotelling's lemma Hotelling's lemma is a result in microeconomics that relates the supply of a good to the maximum profit of the producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm. Specifically, it states: ''The rate of a ...
*
Convex preferences In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly ...


References


Further reading

* {{DEFAULTSORT:Shephard's Lemma Consumer theory Economics theorems