Shephard's lemma

Shephard's Lemma (also lemma Shephard ) states in the household theory that the Hicksian demand function for a commodity the derivative of the output function corresponds to the price of this material. In the theory of the company it says that the conditional factor demand for a production factor corresponds to the derivation of the cost function according to the factor price of this production factor. The two applications are analogous.

The lemma is named after the American economist and statistician Ronald Shephard .

presentation

Household theory

First of all, one starts with a problem of minimizing expenditure

${\displaystyle \min _{(x_{1},\ldots ,x_{n})\in \mathbb {R} _{+}^{n}}\sum _{i=1}^{n}p_{i}x_{i}}$ under the secondary condition ${\displaystyle u(x_{1},\ldots ,x_{n})={\overline {u}}}$

is given, with continuous , differentiable and strictly quasi-concave . The total expenditure for the goods from the shopping cart is minimized, but a certain level of benefit should be preserved. The solution to such an expenditure minimization problem is intended to be a function that indicates which amount of the respective goods should be requested in order to achieve the given level of utility as cost-effectively as possible. It is therefore a function of the price vector and the established level of utility . This is called Hick's demand and it is agreed . ${\displaystyle u(\cdot )}$ ${\displaystyle n}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {x} ^{*}(\mathbf {p} ,{\overline {u}})\equiv \mathbf {x} ^{h}(\mathbf {p} ,{\overline {u}})}$

The so-called optimal value function associated with this is given by the originally minimized function into which the one obtained is now inserted. It is called the expense function : ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle e}$

${\displaystyle e=e(\mathbf {p} ,{\overline {u}})=\sum _{i=1}^{n}p_{i}x_{i}^{h}(\mathbf {p} ,{\overline {u}})}$

It provides the actual expenditure that is to be made in the expenditure minimum for a given benefit level.

Shephard's lemma in household theory: Hick's demand for a good j is given by the partial derivative of the expenditure function according to the price of the good: ${\displaystyle p_{j}}$

${\displaystyle {\frac {\partial e(\mathbf {p} ,{\overline {u}})}{\partial p_{j}}}=x_{j}^{h}(\mathbf {p} ,{\overline {u}})}$

Theory of the company

One starts with a cost minimization problem that goes through

${\displaystyle \min _{(x_{1},\ldots ,x_{n})\in \mathbb {R} _{+}^{n}}\sum _{i=1}^{n}w_{i}x_{i}}$ under the secondary condition ${\displaystyle f(x_{1},\ldots ,x_{n})=y}$

is given, with continuous , differentiable and strictly quasi-concave . The total expenditure for the production factors is minimized, but a certain amount of output should be generated ( is the production function ). The solution to such a cost minimization problem is intended to be a function that indicates which amount of the respective factors should be requested in order to achieve the given production target as cost-effectively as possible. It is therefore a function of the factor price vector and the specified output level . The so given is called conditional factor demand. ${\displaystyle f(\cdot )}$ ${\displaystyle n}$ ${\displaystyle y}$ ${\displaystyle f}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {w} =(w_{1},\ldots ,w_{n})}$ ${\displaystyle y}$ ${\displaystyle \mathbf {x} ^{*}}$

The so-called optimal value function associated with this is given by the originally minimized function into which the one obtained is now inserted. It is called the cost function : ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle c}$

${\displaystyle c=c(\mathbf {w} ,y)=\sum _{i=1}^{n}w_{i}x_{i}^{*}(\mathbf {w} ,y)}$

It provides the actual costs that are incurred in the minimum cost for a given output quantity.

Shephard's lemma in the theory of the enterprise: The conditional factor demand for a production factor j is given by the partial derivation of the cost function according to the price of the production factor: ${\displaystyle w_{j}}$

${\displaystyle {\frac {\partial c(\mathbf {w} ,y)}{\partial w_{j}}}=x_{j}^{*}(\mathbf {w} ,y)}$

Derivation

The lemma is a direct application of the envelope theorem .

See also

• Hotelling's lemma
• Slutsky decomposition
• Envelope theorem

literature

• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Ronald W. Shephard: Cost and Production Functions. Springer, Berlin 1981, ISBN 3-540-11158-1 (reprint of the Princeton edition 1953).

Individual evidence

1. ↑ See Jehle / Reny 2011, pp. 35–39.
2. ↑ See Jehle / Reny 2011, pp. 136-138.