# How to prove that profit function is convex in price (with smaller price)?

According to this site, if output price increases from $$p*$$ to $$p'$$ and factor prices remain constant, then a new production bundle chosen must yield at least the same amount of profits as the old bundle, as the old profit would yield at least as much profit in higher output price. The reasoning behind which is understandable.

However, how does it also apply for the case when output price decreases from $$p*$$ to e.g. $$p"$$? How does one be sure that the maximum profit at $$p"$$ would definitely be higher than the one at $$p*$$ (instead of lower)?

Also what does it mean by "reverse convexity" in "A reverse convexity will be obtained if we plot $$\pi (p, w)$$ against a particular factor price, $$w_i$$." in the aforementioned proof on the same site? I can not even find a proper definition of the term on internet.

I don't think it is a case. On a page you linked we have a proof that the profit function is non-decreasing in $$p$$. If the output price decreases, $$p \geq p'$$ and factor prices remain constant, $$w_i \leq w'_i$$ for all inputs, then the profit would be less or equal to the previous one.

Then, in connection to your previous question: How to prove that a concave production imply that the input requirement sets are convex? you can let $$\mathbf{y}=(y,-\mathbf{x})$$ be maximizing profit at $$(p,\mathbf{w})$$ so that your profit function becomes $$\pi(p,\mathbf{w})=py-\mathbf{wx}$$. On page 12. of the Lectures in Microeconomic Theory Fall 2009, Part 3 you have linked, you can see a proof of that (probably the same as on the page you linked just now but not sure as the page does not open properly to me (cannot really see the mathematical equations).

As for your second question - "reverse convexity" is a little bit strange expression, to show what the author meant I will use a picture. As you can see it is still convex in input prices $$w$$. However, in comparison to plot with output prices $$p$$, it is decreasing. Algebraically, you can see it that way:

1. Profit function is convex in both prices: $$\frac{\partial^2\pi}{\partial p^2}>0 \;\;\; \textrm{and} \;\;\; \frac{\partial^2\pi}{\partial w^2}>0$$
2. Profit function is increasing in output prices $$p$$: $$\frac{\partial\pi}{\partial p}>0$$
3. Profit function is decreasing in input prices $$w$$: $$\frac{\partial\pi}{\partial w}<0$$
• Thank you for your answer. Regarding Lectures in Microeconomic Theory Fall 2009, Part 3, it was not the site I linked. I went to here: eml.berkeley.edu/~webfac/dellavigna/e101a_f09/lecture03.pdf, which does not seem to correspond with what you said about the page. Would you mind if you post the link? – Aqqqq Oct 31 '19 at 10:10
• Yes of course! I meant Lectures in Microeconomic Theory Fall 2009, Part 3 that you can find here: folk.uio.no/gasheim/4230f9t3.pdf . It just looked "almost" the same as Lectures in Microeconomic Theory Fall 2006, Part 2 that you gave us a link to in your "How to prove that a concave production imply that the input requirement sets are convex?" question. – bajun65537 Oct 31 '19 at 17:05