Hotelling's lemma

Let ${\displaystyle p}$ denote a variable price, and ${\displaystyle w}$ be a constant cost of each input. Let ${\displaystyle x:{\mathbb {R} ^{+}}\rightarrow X}$ be a mapping from the price to a set of feasible input choices ${\displaystyle X\subset {\mathbb {R} ^{+}}}$. Let ${\displaystyle f:{\mathbb {R} ^{+}}\rightarrow {\mathbb {R} ^{+}}}$ be the production function, and ${\displaystyle y(p)\triangleq f(x(p))}$ be the net supply.

The maximum profit can be written by

${\displaystyle \pi (p)=\max _{x}p\cdot y(p)-w\cdot x(p).}$

Then the lemma states that if the profit ${\displaystyle \pi }$ is differentiable at ${\displaystyle p}$, the maximizing net supply is given by

${\displaystyle y^{*}(p)={\frac {d\pi (p)}{dp}}.}$

The lemma is a corollary of the envelope theorem.

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

${\displaystyle {\frac {\partial \pi }{\partial x}}{\bigg |}_{x=x^{*}}=0.}$

(1)

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

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

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

Consider the following example.[1] 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:

${\displaystyle {\frac {\partial \pi (p,w_{1},w_{2})}{\partial p}}=y={\frac {1}{9}}{\frac {p^{2}}{w_{1}w_{2}}}}$

${\displaystyle {\frac {\partial \pi (p,w_{1},w_{2})}{\partial w_{1}}}=-x_{1}=-{\frac {1}{27}}{\frac {p^{3}}{w_{1}^{2}w_{2}}}}$

${\displaystyle {\frac {\partial \pi (p,w_{1},w_{2})}{\partial w_{2}}}=-x_{2}=-{\frac {1}{27}}{\frac {p^{3}}{w_{1}w_{2}^{2}}}}$

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.

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 ${\displaystyle y^{*}}$ and cost minimizing input ${\displaystyle x^{*}}$. If a firm is not producing at these optima, then Hotelling's lemma would not hold.[2]

• Hotelling's law
• Hotelling's rule
• Supply and demand
• Shephard's lemma

• Hotelling, H. (1932). "Edgeworth's taxation paradox and the nature of demand and supply functions". Journal of Political Economy. 40 (5): 577–616. doi:10.1086/254387. JSTOR 1822600.
• Sakai, Y. (1974). "Substitution and Expansion Effects in Production Theory: The Case of Joint Production". Journal of Economic Theory. 9 (3): 255–274. doi:10.1016/0022-0531(74)90051-9.
• Takayama, A. (1985). Mathematical Economics. New York: Cambridge University Press. pp. 141–144. ISBN 978-0-521-31498-5.
• Varian, H. (1992). Microeconomic Analysis (3rd ed.). New York: W. W Norton. pp. 43–45. ISBN 978-0-393-95735-8.