Hotelling's lemma

As Hotelling's lemma is known in microeconomics and there, especially in the theory of the firm some characteristics of a profit function . In particular, it implies that the supply function of the goods produced (output goods) and the demand function with regard to the factors used ( input goods ) result directly from the profit function : With optimal production, the partial derivation of the profit function according to the price of goods results in the quantity sold, while the partial derivation Derivation according to the respective factor price is the (negative) factor input. In his assumptions, Hotelling assumed that the prices for the goods produced are determined by the market, while the output volume is determined by the producer.

Mathematically, it is an application of the envelope theorem . The lemma is named after the American statistician and economist Harold Hotelling .

Formal representation

Let be the price of an output good that is produced from input goods. Production is carried out using a given technology by the production function with is represented; this indicates the maximum amount of the output good that can be produced using the factor inputs ( for example, it describes the amount of input factor i used ). Let us also be the vector of the associated factor prices ( for example denotes the price for a unit of input factor i ). ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle f:\mathbb {R} _{+}^{n}\rightarrow \mathbb {R} _{+}}$ ${\displaystyle f=f(x_{1},\ldots ,x_{n})}$ ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}$ ${\displaystyle x_{i}}$ ${\displaystyle \mathbf {w} =(w_{1},\ldots ,w_{n})}$ ${\displaystyle w_{i}}$

It is now the so-called profit function of the enterprise; For given prices of the output goods and the input goods, it shows the maximum profit a company can achieve. ${\displaystyle \pi (p,\mathbf {w} )}$

Hotelling's Lemma (Hotelling 1932): Let f, as usual, be continuous, strictly monotonously increasing, strictly quasi-concave on the and apply . Furthermore, the usual requirements for the profit function are met, i.e. in particular and . In addition, let f be strictly concave on the . Then: ${\displaystyle \mathbb {R} _{+}^{n}}$ ${\displaystyle f(\mathbf {0} )=0}$ ${\displaystyle p\geq 0}$ ${\displaystyle \mathbf {w} \in \mathbb {R} ^{n}\setminus \{\mathbf {0} \}}$ ${\displaystyle \mathbb {R} _{+}^{n}}$

1. ${\displaystyle {\frac {\partial \pi (p,\mathbf {w} )}{\partial p}}=y(p,\mathbf {w} )}$ and
2. ${\displaystyle {\frac {\partial \pi (p,\mathbf {w} )}{\partial w_{i}}}=-x_{i}(p,\mathbf {w} )}$ for all ${\displaystyle i=1,\ldots ,n}$

Derivation

For the sake of simplicity, assume immediately that the secondary condition in the optimization problem for the gain function is fulfilled with equality, that is, that . Of course, one could prove the lemma without this restriction, the result is equivalent in each case (because it would be shown anyway that the entire quantity produced is also offered). ${\displaystyle y=f(\mathbf {x} )}$

Define . The problem with the solution is given . The optimal value function of this is and so . According to the envelope theorem, ${\displaystyle g(\mathbf {x} ;p,\mathbf {w} )=p\cdot f(\mathbf {x} )-\mathbf {w} \cdot \mathbf {x} }$ ${\displaystyle \max _{\mathbf {x} }g(\mathbf {x} ;p,\mathbf {w} )}$ ${\displaystyle \mathbf {x} ^{*}(p,\mathbf {w} )}$ ${\displaystyle v(p,\mathbf {w} )\equiv g\left[\mathbf {x} ^{*}(p,\mathbf {w} );p,\mathbf {w} \right]}$ ${\displaystyle v(p,\mathbf {w} )=\pi (p,\mathbf {w} )}$

${\displaystyle {\frac {\partial \pi (p,\mathbf {w} )}{\partial p}}=\left.{\frac {\partial g(\mathbf {x} ;p,\mathbf {w} )}{\partial p}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(p,\mathbf {w} )}}$

(The prerequisites of the theorem guarantee the necessary differentiability), but this is exactly the same (as can be seen directly from the definition of ) , q. e. d. ${\displaystyle g}$ ${\displaystyle y}$

Analog is also, for everyone , ${\displaystyle i}$

${\displaystyle {\frac {\partial \pi (p,\mathbf {w} )}{\partial w_{i}}}=\left.{\frac {\partial g(\mathbf {x} ;p,\mathbf {w} )}{\partial w_{i}}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(p,\mathbf {w} )}}$,

which in turn corresponds to q. e. d. ${\displaystyle -x_{i}}$

literature

• Harold Hotelling: Edgeworth's taxation paradox and the nature of demand and supply function. In: Political Economy. 40, 1932, pp. 577-616.
• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .

Individual evidence

1. ↑ See Jehle / Reny 2011, p. 148.