Hotelling's lemma

Proof for Hotelling's lemma

The lemma is a corollary of the envelope theorem.

Specifically, the maximum profit can be rewritten as ${\displaystyle \pi (p,x^{\ast })=p\cdot f(x^{\ast }(p))-w\cdot x^{\ast }(p)}$ where ${\displaystyle x^{\ast }}$ is the maximizing input corresponding to ${\displaystyle y^{\ast }}$. Due to the optimality, the first order condition gives

By taking the derivative by ${\displaystyle p}$ at ${\displaystyle x^{\ast }}$,

${\displaystyle \frac{d\pi }{dp}=\frac{\mathrm{\partial }\pi }{\mathrm{\partial }x}{|}_{x=x^{\ast }}\frac{\mathrm{\partial }x}{\mathrm{\partial }p}+\frac{\mathrm{\partial }\pi }{\mathrm{\partial }p}=\frac{\mathrm{\partial }\pi }{\mathrm{\partial }p}=f(x^{\ast }(p))=y^{\ast }(p)}$

where the second equality is due to (1). QED

Application of Hotelling's lemma

Consider the following example. Let output ${\displaystyle y}$ have price ${\displaystyle p}$ and inputs ${\displaystyle x_1}$ and ${\displaystyle x_2}$ have prices ${\displaystyle w_1}$ and ${\displaystyle w_2}$. Suppose the production function is ${\displaystyle y=x_1^{1/3}{x}_2^{1/3}}$. The unmaximized profit function is ${\displaystyle \pi (p,w_1,w_2,x_1,x_2)=py-w_1{x}_1-w_2{x}_2}$. From this can be derived the profit-maximizing choices of inputs and the maximized profit function, a function just of the input and output prices, which is

${\displaystyle \pi (p,w_1,w_2)=\frac{1}{27}\frac{p^3}{w_1{w}_2}}$

Hotelling's Lemma says that from the maximized profit function we can find the profit-maximizing choices of output and input by taking partial derivatives:

${\displaystyle \frac{\mathrm{\partial }\pi (p,w_1,w_2)}{\mathrm{\partial }p}=y=\frac{1}{9}\frac{p^2}{w_1{w}_2}}$

${\displaystyle \frac{\mathrm{\partial }\pi (p,w_1,w_2)}{\mathrm{\partial }{w}_1}=-x_1=-\frac{1}{27}\frac{p^3}{w_1^2{w}_2}}$

${\displaystyle \frac{\mathrm{\partial }\pi (p,w_1,w_2)}{\mathrm{\partial }{w}_2}=-x_2=-\frac{1}{27}\frac{p^3}{w_1{w}_2^2}}$

Note that Hotelling's lemma gives the net supplies, which are positive for outputs and negative for inputs, since profit rises with output prices and falls with input prices.

Criticisms and empirical evidence

A number of criticisms have been made with regards to the use and application of Hotelling's lemma in empirical work.

C. Robert Taylor points out that the accuracy of Hotelling's lemma is dependent on the firm maximizing profits, meaning that it is producing profit maximizing output ${\displaystyle y^{\ast }}$ and cost minimizing input ${\displaystyle x^{\ast }}$. If a firm is not producing at these optima, then Hotelling's lemma would not hold.