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

Let the firm's profit function be:

${\displaystyle \pi =py-wx}$

where:

• ${\displaystyle \pi }$ is profit
• ${\displaystyle p}$ is the price of output ${\displaystyle y}$
• ${\displaystyle y}$ is output
• ${\displaystyle w}$ is input price for input ${\displaystyle x}$
• ${\displaystyle x}$ is the single input needed for producing ${\displaystyle y}$

If a firm produces 10 units of ${\displaystyle y}$ using 5 units of input ${\displaystyle x}$ which cost 1 dollar each and sells each output for 2 dollars. the profit the firm makes is:

${\displaystyle \pi _{1}=(2)(10)-(1)(5)=15}$

If the firm increases the price of the output to 3 dollars and still sells the same amount of ${\displaystyle y}$, the firm's profits are now:

${\displaystyle \pi _{2}=(3)(10)-(1)(5)=25}$

Taking the difference between ${\displaystyle \pi _{1}}$ and ${\displaystyle \pi _{2}}$

${\displaystyle \Delta \pi =\pi _{2}-\pi _{1}=25-15=10}$

The change in profits from a change in price is 10, which is exactly the same as the output produced. thus the statement of ${\displaystyle y(p)={\frac {\partial \pi (p)}{\partial p}}}$ holds.

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.[1]

• 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.