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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), 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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    @GerryMyerson I think I've fixed it... $q$ was a typo2012-10-21

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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.2012-10-21