# Could someone please explain the proof of Hotelling's lemma?

According to https://en.wikipedia.org/wiki/Hotelling%27s_lemma,

the maximum of the firm's profit at some output is given by the minimum of the difference between the profit and the revenue.

However, it does not sound right, as the difference of difference of the profit and the revenue is the negative of the cost of the input. The minimum would correspond to the maximum cost for input incurred. How would the maximum of the fir's profit at this point?

Also what does it mean by "at some output y*(p)"? Is y*(p) fixed here?

## 1 Answer

Well done for spotting this one. It is (was) simply a wrong statement. It is indeed the maximum and not the minimum of the expression as (was in the past) stated in the wikipedia article.

Then the maximum of this expression is the maximum of $-C$ (Cost) which is equivalent to the minimum of $C$, and the equivalence with maximum profit holds under duality.

• Thank you for your answer. Would you mind if you tell me what do you mean by "duality" here? Also what does the article mean by "at some output y*(p)"? Is y*(p) fixed here? Nov 5, 2017 at 16:05
• @Aqqqq If you don't know about duality theory/envelope theorem and how they are applied in microeconomics, you need to study them. Here specifically, duality implies that minimizing costs is equivalent to maximizing profits. Yes, we are looking for the maximum profit with respect to selling price at any given level of output. Nov 5, 2017 at 16:09
• Thanks. I was unsure as I am new in microeconomics and have seen the term in only one context. Nov 5, 2017 at 17:15