Shephard's Lemma
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Shephard's lemma is a result in
microeconomics Microeconomics is a branch of economics that studies the behavior of individuals and Theory of the firm, firms in making decisions regarding the allocation of scarcity, scarce resources and the interactions among these individuals and firms. M ...
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 eco ...
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 pr ...
. The lemma states that if
indifference curves In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
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 expected, required, or given by one party to another in return for goods or services. In some situations, especially when the product is a service rather than a ph ...
p_i is unique. The idea is that a
consumer A consumer is a person or a group who intends to order, or use 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 In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a normative context, utility refers to a goal or objective that we wish ...
given the price of goods in the
market Market is a term used to describe concepts such as: *Market (economics), system in which parties engage in transactions according to supply and demand *Market economy *Marketplace, a physical marketplace or public market *Marketing, the act of sat ...
. The lemma is named after Ronald Shephard, who proved it using the distance formula in his book ''Theory of Cost and Production Functions'' in 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) represents the quantity of a good demanded when the consumer minimizes expenditure while maintaining a fixed level of utility. The Hicksian demand function ...
s for the relevant goods. Similar results had already been derived by
John Hicks Sir John Richard 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 economics ...
(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 use 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 goods, good that consumers are willing and able to purchase at various prices during a given time. In economics "demand" for a commodity is not the same thing as "desire" for it. It refers to both the desi ...
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 * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
\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 eco ...
, 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. 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 * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
\mathbf) and output y. Although Shephard's original proof used the distance formula, modern proofs of Shephard's lemma use the envelope theorem.


Proof for the differentiable case

The proof is stated for the two good cases 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 s ...
.


See also

* Hotelling's lemma *
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". This implies that ...


References


Further reading

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