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.
For the profit function of the firm n terms of the good's price p and the production function in terms of the good's price p, assuming that 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 is given by the minimum of which is the cost at some price,, namely where holds. Thus, ; QED.
The proof is also a corollary of the envelope theorem.

Application of Hotelling's lemma

Let the firm's profit function be:
where:
If a firm produces 10 units of using 5 units of input which cost 1 dollar each and sells each output for 2 dollars. the profit the firm makes is:
If the firm increases the price of the output to 3 dollars and still sells the same amount of, the firm's profits are now:
Taking the difference between and
The change in profits from a change in price is 10, which is exactly the same as the output produced. thus the statement of 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 and cost minimizing input. If a firm is not producing at these optima, then Hotelling's lemma would not hold.