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In economic models we assume that firms which face a decreasing demand curve also face a decreasing marginal revenue curve. I just realized that I wasn't sure that this is true, so I tried to prove it, however I'm finding this difficult.

Let $p(x)$ be an inverse demand curve such that $p'(x)<0$ (so, if you're unfamiliar with demand curves and things, $p(x)$ is the price that would elicit consumers to buy $x$ units of a good). Then total revenue is given by $xp(x)$, and so the marginal revenue, which is just the derivative of total revenue, would be $\frac{d}{dx}(xp(x))=p(x)+xp'(x)$. I would like, from the assumption that $p'(x)<0$, to show that $\frac{d}{dx}[p(x)+xp'(x)]<0$... but I suspect from looking at it that it's not even necessarily true. Taking the derivative of that gives that it's true as long as $2p'(x)<xp''(x)$, but that's not very meaningful to me. Can you see from that what sorts of functions might violate that condition, or if it's always true?

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The question in the title doesn't match the question in the body. Please edit to bring the two into agreement. Also, what's $q$? – Gerry Myerson Oct 21 '12 at 23:35
@GerryMyerson I think I've fixed it... $q$ was a typo – crf Oct 21 '12 at 23:40
up vote 0 down vote accepted

Let $p(x)=e^{-x}$. Then $p'(x)=-e^{-x}\lt0$ for all $x$. $xp(x)=xe^{-x}$. $(xp(x))'=p(x)+xp'(x)=-xe^{-x}+e^{-x}=(1-x)e^{-x}$. $(p(x)+xp'(x))'=(x-2)e^{-x}$. This is positive for $x\gt2$.

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Huh, yeah, I figured that that must exist. So every time an economics professor says that marginal revenue is decreasing because the demand curve is downward sloping, they're missing part of the story. Thank you. – crf Oct 21 '12 at 23:51

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