<p>I think that $\hat{U}$ from denesp's answer is great, and so is the one by Alecos Papadopoulos. </p>

<p>Another example of a function would be;</p>

<p>$U(x,y) = e^{-(x-x_0)^2-(y-y_0)^2}$ where $x_0,y_0$ is the bliss point. 
This would be how it looks with $x_0=2, y_0=2$:
<a href="https://i.sstatic.net/UUk7x.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UUk7x.jpg" alt="enter image description here"></a></p>

<p>and this with the contour set levels:</p>

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

CC BY-SA 4.0
2018-08-09T15:35:31.123Z
Isn't your $U(x,y)$ a monotonic transformation of my $U(x,y)$?
Giskard
CC BY-SA 4.0
2018-08-09T15:50:02.443Z
Yes I think so. The only reason I put it here is because the shape looks nicer compared to yours. But I think mine satisfies convexity, yes?
erik
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2018-08-09T15:58:34.967Z
They describe the exact same preferences. The indifference curves will look the same, and hence either both or neither will satisfy convexity.
Giskard
CC BY-SA 4.0
2018-08-09T16:00:11.003Z
You are right to point out that the function satisfies convexity, I was wrong there. Your formula was missing a minus sign (so did mine), I edited it.
Giskard
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2018-08-09T17:00:17.493Z
You are correct. I missed copying the minus from my codes. And putting the negative in your U function does take care of it.
erik
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2018-08-09T14:09:14.340Z
2018-08-09T16:04:23.593Z
2018-08-09T16:04:23.593Z
erik
<p>The function
$$u(c) = -ac^2 + bc$$</p>

<p>does the job of initially providing positive utility, then having a maximum, then declininig (negative marginal utility) and eventually turning negative itself. </p>

<p>One should be clear though that these are cardinal utility concepts. I haven't worked out how they would stand up in an ordinal utility framework.</p>

CC BY-SA 4.0
2018-08-09T09:44:24.280Z
2018-08-09T10:34:37.480Z
2018-08-09T10:34:37.480Z
Alecos Papadopoulos
<p>Varian's Intermediate Microeconomics covers a concept called the <a href="https://en.wikipedia.org/wiki/Bliss_point_(economics)" rel="nofollow noreferrer">bliss point</a>.
If the consumed amount of a certain good is under the quantity specified by the bliss point, then the consumer would prefer to consume more of the good, all other things being equal. If the consumed amount is over the quantity specified by the bliss point, then consumer would prefer to consume less of the good, all other things being equal (i.e. it becomes a 'bad').</p>
<p>In case of one good, a utility function that represents this kind of preference is
<span class="math-container">$$
U(x) = -(x - x_b)^2,
$$</span>
where <span class="math-container">$x_b$</span> is the bliss point quantity. For two goods, similar utility functions exist, e.g.
<span class="math-container">$$
U(x,y) = -(x - x_b)^2 - (y - y_b)^2
$$</span>
or
<span class="math-container">$$
\hat{U}(x,y) = -|x - x_b| - |y - y_b|.
$$</span>
Both <span class="math-container">$U$</span> and <span class="math-container">$\hat{U}$</span> violates <a href="https://en.wikipedia.org/wiki/Monotone_preferences" rel="nofollow noreferrer">monotonicity</a>, <span class="math-container">$U$</span> represents strictly <a href="https://en.wikipedia.org/wiki/Convex_preferences" rel="nofollow noreferrer">convex</a> preferences, so averages are still preferred to extremes. <span class="math-container">$\hat{U}$</span> represents weakly convex preferences.</p>

CC BY-SA 4.0
2018-08-09T16:02:57.437Z
Four upvotes and no one pointed out that utility in the bliss point was minimal because of the missing minus sign...
Giskard
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2022-03-26T01:28:11.837Z
$\hat{U}$ does not represents weakly convex preferences. $-\hat{U}$ does.
Amit
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2022-03-26T01:31:04.153Z
$\hat{U}$ also has a missing minus sign.
Amit
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2018-08-09T08:22:47.353Z
2022-03-26T03:29:31.260Z
2022-03-26T03:29:31.260Z
Giskard
<p>In introductory economics courses the concept of marginal utility is illustrated through simple examples like how much benefit one gets from eating another slice of pizza (i.e first slice provides 100 utils, second slice 50 etc). </p>

<p>I was wondering if there exists a utility function that can allow for the possibility of over consumption (i.e. a function that would produce -50 utils from the 10th slice of pizza ).</p>

<p>I know there would be issues with this since utility functions are only defined up to monotonic transformations, and concavity in our utility functions would violate the "averages preferred to extremes" assumption. </p>

<p>Does such a function exist?</p>

CC BY-SA 4.0
2018-08-08T21:38:46.887Z
@IgnacioValdésZamudio same kind of idea. booze is probably a better example
EconJohn
<microeconomics><utility><consumer-theory>
modelling disutility from over consumption
EconJohn
# modelling disutility from over consumption

(Asked by: username_0 on 2018-08-08)

In introductory economics courses the concept of marginal utility is illustrated through simple examples like how much benefit one gets from eating another slice of pizza (i.e first slice provides 100 utils, second slice 50 etc).

I was wondering if there exists a utility function that can allow for the possibility of over consumption (i.e. a function that would produce -50 utils from the 10th slice of pizza ).

I know there would be issues with this since utility functions are only defined up to monotonic transformations, and concavity in our utility functions would violate the "averages preferred to extremes" assumption.

Does such a function exist?

**Tags:** microeconomics, utility, consumer-theory

> **Comments:**
> **username_0**: @IgnacioValdésZamudio same kind of idea. booze is probably a better example  

---

## 3 answers

### Answer by username_2 on 2018-08-09. Score: 6 | Accepted answer

Varian's Intermediate Microeconomics covers a concept called the [bliss point](https://en.wikipedia.org/wiki/Bliss_point_(economics)).
If the consumed amount of a certain good is under the quantity specified by the bliss point, then the consumer would prefer to consume more of the good, all other things being equal. If the consumed amount is over the quantity specified by the bliss point, then consumer would prefer to consume less of the good, all other things being equal (i.e. it becomes a 'bad').

In case of one good, a utility function that represents this kind of preference is
$$
U(x) = -(x - x\_b)^2,
$$
where $x\_b$ is the bliss point quantity. For two goods, similar utility functions exist, e.g.
$$
U(x,y) = -(x - x\_b)^2 - (y - y\_b)^2
$$
or
$$
\hat{U}(x,y) = -|x - x\_b| - |y - y\_b|.
$$
Both $U$ and $\hat{U}$ violates [monotonicity](https://en.wikipedia.org/wiki/Monotone_preferences), $U$ represents strictly [convex](https://en.wikipedia.org/wiki/Convex_preferences) preferences, so averages are still preferred to extremes. $\hat{U}$ represents weakly convex preferences.

> **Comments:**
> **username_2**: Four upvotes and no one pointed out that utility in the bliss point was minimal because of the missing minus sign...  
> **username_4**: $\hat{U}$ does not represents weakly convex preferences. $-\hat{U}$ does.  
> **username_4**: $\hat{U}$ also has a missing minus sign.  

---

### Answer by username_1 on 2018-08-09. Score: 5

I think that $\hat{U}$ from denesp's answer is great, and so is the one by username_3 username_3.

Another example of a function would be;

$U(x,y) = e^{-(x-x\_0)^2-(y-y\_0)^2}$ where $x\_0,y\_0$ is the bliss point.
This would be how it looks with $x\_0=2, y\_0=2$:
[![enter image description here](https://i.sstatic.net/UUk7x.jpg)](https://i.sstatic.net/UUk7x.jpg)

and this with the contour set levels:

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

> **Comments:**
> **username_2**: Isn't your $U(x,y)$ a monotonic transformation of my $U(x,y)$?  
> **username_1**: Yes I think so. The only reason I put it here is because the shape looks nicer compared to yours. But I think mine satisfies convexity, yes?  
> **username_2**: They describe the exact same preferences. The indifference curves will look the same, and hence either both or neither will satisfy convexity.  
> **username_2**: You are right to point out that the function satisfies convexity, I was wrong there. Your formula was missing a minus sign (so did mine), I edited it.  
> **username_1**: You are correct. I missed copying the minus from my codes. And putting the negative in your U function does take care of it.  

---

### Answer by username_3 on 2018-08-09. Score: 5

The function
$$u(c) = -ac^2 + bc$$

does the job of initially providing positive utility, then having a maximum, then declininig (negative marginal utility) and eventually turning negative itself.

One should be clear though that these are cardinal utility concepts. I haven't worked out how they would stand up in an ordinal utility framework.


---

