<p>This is the utility function @denesp gave in his answer above. It does not have any convexity issues in the range <span class="math-container">$0&lt;x&lt;0.3$</span>, which is the range where the MR will have an U shape. </p>

<p>The function:</p>

<p><a href="https://i.sstatic.net/Lpd9b.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Lpd9b.jpg" alt="enter image description here"></a></p>

CC BY-SA 4.0
2018-10-07T14:20:02.713Z
2018-10-07T14:20:02.713Z
erik
<p>Let <span class="math-container">$v(x) = \int  1 − x + 3x^2 − 3x^3 \text{d}x$</span>.
The quasi linear utility function
<span class="math-container">$$
U(x,y) = v(x) + y
$$</span>
generates the <a href="https://economics.stackexchange.com/a/24834/1601">inverse demand curve given by Ubiquitous</a> if income <span class="math-container">$I$</span> is large enough and <span class="math-container">$p_y = 1$</span>, as
<span class="math-container">\begin{align*}
|MRS(x,y)| &amp; = p_x \\
\\
v'(x) &amp; = p_x \\
\\
1 − x + 3x^2 − 3x^3 &amp; = p_x.
\end{align*}</span>
Hence it will also result in the same <span class="math-container">$MR$</span> for <span class="math-container">$x$</span>.</p>

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

CC BY-SA 4.0
2018-10-06T08:05:30.847Z
@ 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?
erik
CC BY-SA 4.0
2018-10-06T08:10:36.617Z
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.
Giskard
CC BY-SA 4.0
2018-10-06T08:30:39.233Z
@erik I edited my answer.
Giskard
CC BY-SA 4.0
2018-10-07T14:17:14.173Z
@ 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.
erik
CC BY-SA 4.0
2018-10-06T07:50:14.423Z
2018-10-06T08:37:44.397Z
2018-10-06T08:37:44.397Z
Giskard
<p>This is based off an <a href="https://economics.stackexchange.com/a/24834/19094">answer</a> given by @Ubiquitous in here: <a href="https://economics.stackexchange.com/questions/24833/can-marginal-revenue-be-increasing/24834?noredirect=1#comment39250_24834">Can marginal revenue be increasing?</a></p>

<p>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 <span class="math-container">$P(q)=1−q+3q^2−3q^3$</span> (see comments below his answer).</p>

<p>My question is: </p>

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

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

<microeconomics>
Utility function that generates a demand curve which will have an U shaped MR curve
erik
# Utility function that generates a demand curve which will have an U shaped MR curve

(Asked by: username_0 on 2018-10-06)

This is based off an [answer](https://economics.stackexchange.com/a/24834/19094) given by @Ubiquitous in here: [Can marginal revenue be increasing?](https://economics.stackexchange.com/questions/24833/can-marginal-revenue-be-increasing/24834?noredirect=1#comment39250_24834)

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.

**Tags:** microeconomics


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## 2 answers

### Answer by username_1 on 2018-10-06. Score: 3 | Accepted answer

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](https://economics.stackexchange.com/a/24834/1601) 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.

> **Comments:**
> **username_0**: @ 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?  
> **username_1**: 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.  
> **username_1**: @username_0 I edited my answer.  
> **username_0**: @ 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.  

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### Answer by username_0 on 2018-10-07. Score: 0

This is the utility function @denesp gave in his answer above. It does not have any convexity issues in the range $0, which is the range where the MR will have an U shape.

The function:

[![enter image description here](https://i.sstatic.net/Lpd9b.jpg)](https://i.sstatic.net/Lpd9b.jpg)


---

