User:Editeur24/sandbox2

Let $p$ denote the price of output and $w_{i},i=1,\ldots ,n$ denote the prices of $n$ inputs. Let $x_{i}$ be the amount of input $i$ used in production and $y$ be the output as determined by the production function, so $y=f(x_{1},\ldots x_{n}).$

Profit as a function of the prices is derived by maximizing profit as a function of the prices and the quantity choices:

$\pi (p,w_{1},\ldots ,w_{n})=\max _{x_{1},\ldots ,x_{n}}p\cdot f(x_{1},\ldots ,x_{n})-\sum _{i=1}^{n}w_{i}\cdot x_{i}$

Hotelling's Lemma says that if the profit function is differentiable and positive quantities of all inputs are used at the optimum, the profit-maximizing choices are:

$y^{*}(p,w_{1},\ldots ,w_{n})={\frac {\partial \pi (p,w_{1},\ldots ,w_{n})}{\partial p}}$

$x_{i}^{*}(p,w_{1},\ldots ,w_{n})=-{\frac {\partial \pi (p,w_{1},\ldots ,w_{n})}{\partial w_{i}}}$

Proof of Hotelling's lemma

The lemma uses the same reasoning as the envelope theorem.

The function for maximum profit can be written as

$\pi (p,w_{1},\ldots ,w_{n},x_{1}^{*},\ldots ,x_{n}^{*})=p\cdot f(x_{1}^{*},\ldots ,x_{n}^{*})-\sum _{i=1}^{n}w_{i}\cdot x_{i}^{*},$

where $x_{1}^{*},\ldots ,x_{n}^{*}$ are the maximizing inputs corresponding to the optimal output $y^{*}=f(x_{1}^{*},\ldots ,x_{n}^{*})$. Because the inputs are maximizing profit, the first order conditions hold:

${\frac {\partial \pi }{\partial x_{i}}}{\bigg |}_{x_{i}=x_{i}^{*}}=p{\frac {\partial f}{\partial x_{i}}}{\bigg |}_{x_{i}=x_{i}^{*}}-w_{i}=0.$ (1)

Taking the derivative of profit with respect to $p$ at the optimal values of the inputs yields

${\frac {d\pi }{dp}}=\sum _{j=1}^{n}{\frac {\partial \pi }{\partial x_{j}}}{\bigg |}_{x_{j}=x_{j}^{*}}{\frac {\partial x_{j}}{\partial p}}+{\frac {\partial \pi }{\partial p}}={\frac {\partial \pi }{\partial p}}=f(x_{1}^{*},\ldots ,x_{n}^{*})=y^{*}(p,w_{1},\ldots ,w_{n}),$

where ${\frac {\partial \pi }{\partial x_{j}}}{\bigg |}_{x_{j}=x_{j}^{*}}=0$ for every input $j$ because of (1). Similarly, taking the derivative with respect to input price $w_{i}$ yields

${\frac {d\pi }{dw_{i}}}=\sum _{j=1}^{n}{\frac {\partial \pi }{\partial x_{j}}}{\bigg |}_{x_{j}=x_{j}^{*}}{\frac {\partial x_{j}}{\partial w_{j}}}+{\frac {\partial \pi }{\partial w_{i}}}={\frac {\partial \pi }{\partial w_{i}}}=-x_{i}^{*}$

QED