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This is based off an answer given by @Ubiquitous in here: Can marginal revenue be increasing?

The solution he proposed involved a MR curve that sloped down, then up and then down. His equation for the demand curve (for example) to generate such a MR is $P(q)=1−q+3q^2−3q^3$ (see comments below his answer).

My question is:

Is there an Utility function that can generate a demand curve which gives, say an U-shape looking MR?

Note: I am not looking for an utility function that replicates Ubiquitous's MR.

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Let $v(x) = \int 1 − x + 3x^2 − 3x^3 \text{d}x$. The quasi linear utility function $$ U(x,y) = v(x) + y $$ generates the inverse demand curve given by Ubiquitous if income $I$ is large enough and $p_y = 1$, as \begin{align*} |MRS(x,y)| & = p_x \\ \\ v'(x) & = p_x \\ \\ 1 − x + 3x^2 − 3x^3 & = p_x. \end{align*} Hence it will also result in the same $MR$ for $x$.

P.s.: I have not checked if $U(x,y)$ is convex, so second order conditions may need to be examined here. Note that $U(x,y)$ need not be convex everywhere, only above the $MR$'s U shape. Everywhere else it can be altered as we see fit.

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  • $\begingroup$ @ Denesp: Thank you for your answer. Based on my calculations for the interval (0,8), if $Q = \frac{32}{16-p}$ then $P = 16 - \frac{32}{Q}$. This means that revenue is $PQ = 16Q-32$, implying MR = 16, a constant. So MR is not U shaped here, yes? $\endgroup$ – erik Oct 6 '18 at 8:05
  • $\begingroup$ You are indeed correct. My guess is that going from constant do increasing would not be very difficult, by tweaking the utility function a bit, but I forgot that you want to have a U shape, not an upside down U shape. $\endgroup$ – Giskard Oct 6 '18 at 8:10
  • $\begingroup$ @erik I edited my answer. $\endgroup$ – Giskard Oct 6 '18 at 8:30
  • $\begingroup$ @ denesp: Your solution is very nice. I feel like I should have thought of it. And your function does not have any convexity issues. I will post a picture. Thank you. $\endgroup$ – erik Oct 7 '18 at 14:17
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This is the utility function @denesp gave in his answer above. It does not have any convexity issues in the range $0<x<0.3$, which is the range where the MR will have an U shape.

The function:

enter image description here

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