Hotelling's lemma

Hotelling's lemma is a result in microeconomics that relates the supply of a good to the profit of the good's producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm.

The lemma can be stated as: The change in profits from a change in price is proportional to the quantity produced.

${\displaystyle {\frac {\partial \pi (p)}{\partial p}}=y(p)}$

For ${\displaystyle \pi }$ the profit function of the firm n terms of the good's price p and ${\displaystyle y}$ the production function in terms of the good's price p, assuming that ${\displaystyle p>0}$ and that derivative exists.

Proof for Hotelling's lemma

The proof of the theorem stems from the fact that for a profit-maximizing firm an under duality, the maximum of the firm's profit at some output ${\displaystyle y^{*}(p)}$ is given by the minimum of ${\displaystyle p^{*}y^{*}(p)-\pi (p^{*})=wx}$ which is the cost at some price, ${\displaystyle p^{*}}$, namely where ${\displaystyle {\frac {\partial \pi (p)}{\partial p}}-y=0}$ holds. Thus, ${\displaystyle y(p)={\frac {\partial \pi (p)}{\partial p}}}$; QED.

The proof is also a corollary of the envelope theorem.

Application of Hotelling's lemma

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.

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 ${\displaystyle y^{*}}$ and cost minimizing input ${\displaystyle x^{*}}$. If a firm is not producing at these optima, then Hotelling's lemma would not hold.[1]

See also

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

References

1. Taylor, C. Robert (1989). "Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher". Western Journal of Agricultural Economics. 14 (2): 200–212. JSTOR 40988099.

• 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.