modelling disutility from over consumption

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?

• I am thinking in an example such as alcohol consumption: first cup could give 100 utils, second one 50, etc. However, 10th one could lead to an ethyl coma, which we can consider as a negative utility, let's say -80 utils. What do you think about it? – Ignacio Valdés Zamudio Aug 8 '18 at 21:34
• @IgnacioValdésZamudio same kind of idea. booze is probably a better example – EconJohn Aug 8 '18 at 21:38

Varian's Intermediate Microeconomics covers a concept called the bliss point. 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, $U$ represents strictly convex preferences, so averages are still preferred to extremes. $\hat{U}$ represents weakly convex preferences.

• Four upvotes and no one pointed out that utility in the bliss point was minimal because of the missing minus sign... – Giskard Aug 9 '18 at 16:02

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.

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

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 $x_0=2, y_0=2$:

and this with the contour set levels:

• Isn't your $U(x,y)$ a monotonic transformation of my $U(x,y)$? – Giskard Aug 9 '18 at 15:35
• 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 Aug 9 '18 at 15:50
• They describe the exact same preferences. The indifference curves will look the same, and hence either both or neither will satisfy convexity. – Giskard Aug 9 '18 at 15:58
• 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 Aug 9 '18 at 16:00
• 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 Aug 9 '18 at 17:00