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 is very simple, and can be stated as:

Let $y(p)$ be a firm's net supply function in terms of a certain good's price ( $p$ ). Then:

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

for $\pi$ the profit function of the firm in terms of the good's price, assuming that $p>0$ and that derivative exists.

In layman's terms this means: "the change in profits from a change in price is equal to $y^*(p)$ ".

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 $y^*(p)$ is given by the minimum of $\pi (p^*) - p^* y^*(p)$ (cost) at some price, $p^{*}$ , namely where $\frac {\partial \pi (p)}{\partial p} - y = 0$ holds. Thus, $y(p)= \frac {\partial \pi (p)}{\partial p}$ ; QED.

The proof is also a corollary of the envelope theorem.

Application of Hotelling's lemma

This is not an application of Hotelling's lemma, it's the definition of partial derivatives.

let the firm's profit function be:

\pi =py-wx

where:

●

\pi

●

p

y

●

y

●

w

x

●

x

y

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

\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 $y$ , the firm's profits are now:

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

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

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

Empirical evidence for Hotelling's Lemma

A number of criticisms have been made with regards to the use and application of Hotelling's lemma in empirical work, one noteworthy piece on is: Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher by C.Robert Taylor.

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

References

↑ Taylor, C. Robert. “Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher.” Western Journal of Agricultural Economics, vol. 14, no. 2, 1989, pp. 202–203. JSTOR, JSTOR, www.jstor.org/stable/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 0-521-31498-4.
Varian, H. (1992). Microeconomic Analysis (3rd ed.). New York: W. W Norton. pp. 43–45. ISBN 0-393-95735-7.
Taylor, C. Robert. (1989). "Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher". Western Journal of Agricultural Economics. 14 (2): 202–203. JSTOR 40988099.

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[1] Taylor, C. Robert. “Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher.” Western Journal of Agricultural Economics, vol. 14, no. 2, 1989, pp. 202–203. JSTOR, JSTOR, www.jstor.org/stable/40988099.

Hotelling's lemma

Proof For Hotelling's Lemma

Application of Hotelling's lemma

Empirical evidence for Hotelling's Lemma

See also

References