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

<p>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.</p>

CC BY-SA 3.0
2017-11-05T16:05:32.737Z
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?
Aqqqq
CC BY-SA 3.0
2017-11-05T16:09:33.630Z
@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.
Alecos Papadopoulos
CC BY-SA 3.0
2017-11-05T17:15:33.583Z
Thanks. I was unsure as I am new in microeconomics and have seen the term in only one context.
Aqqqq
CC BY-SA 3.0
2017-11-05T15:59:55.287Z
2017-11-05T15:59:55.287Z
Alecos Papadopoulos
<p>According to <a href="https://en.wikipedia.org/wiki/Hotelling%27s_lemma" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Hotelling%27s_lemma</a>, </p>

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

<p>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?</p>

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

<supply-and-demand><production-function><profit-maximization>
Could someone please explain the proof of Hotelling's lemma?
Aqqqq
# Could someone please explain the proof of Hotelling's lemma?

(Asked by: username_0 on 2017-11-05)

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?

**Tags:** supply-and-demand, production-function, profit-maximization


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## 1 answers

### Answer by username_1 on 2017-11-05. Score: 2 | Accepted 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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: @username_0 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.  
> **username_0**: Thanks. I was unsure as I am new in microeconomics and have seen the term in only one context.  

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