# Example of consumer preferences that switches from being concave to being convex

### Question

Is there an example of consumer preferences over consumption bundles $$(x,y)\in \Bbb R^2$$ that would be concave when $$x$$ is abundant relative to $$y$$ and convex otherwise?

Are there known situations when this happens? I have never heard of that.

### Context

I've been recently thinking a lot about analyzing functions $$f:\Bbb R \to \Bbb R$$ with positive third-order derivative instead of functions with positive second-order derivative (convex functions). Such functions are either entirely convex / concave, or there is a unique point (inflection point) at which concavity switches into convexity. It turns out that such functions have many nice properties, for example they have at most three roots and most one local maximum and one local minimum. Therefore I expect that if a utility function $$u(x,y)$$ had positive/negative third-order derivative along lines in the space of bundles, then it would still be easy to analyze the consumer choice problem as it is in the case of convex preferences.

• What are concave preferences? Convexity of preferences and convexity of functions are very different things. Jan 25 at 8:42
• @MichaelGreinecker I would regard preferences to be concave iff any lower contour set is concave. If the preference is represented by a utility function $u$, then $u$ is quasi-concave. I understand that concavity of $u$ is sufficient condition for concavity of the preferences, but not necessary. Note that function property of having positive third-order derivative can be also generalized to its "quasi" (i.e. ordinal) form. Jan 25 at 9:11
• Quasi-concavity of a representing utility function is equivalent to the convexity (!) of the represented preferences. I don’t know what concave sets are either. Jan 25 at 10:02
• @MichaelGreinecker I remember Varian made a hypothetical example of sardines and ice cream. Each one is delicious however combining them makes people worse off..
– T123
Jan 25 at 11:44
• @T123 There are many reasons why non-convex preferences matter. It is not clear that this specific form is particularly useful. Jan 25 at 12:02

Consider the following form of utility function:

$$U(x,y) = \beta (x-\alpha)^3 + y$$, where $$\alpha,\beta > 0$$ are parameters. It is easy to check this utility function is monotone, in fact, strictly monotone.

For a constant utility level $$\overline{U}$$, we get the indifference curve:

$$\overline{U} = \beta (x-\alpha)^3 + y \implies y = \overline{U} - \beta (x-\alpha)^3$$.

Differentiating $$y$$ w.r.t. $$x$$ in the IC's:

$$y' = - 3 \beta (x-\alpha)^2$$

$$y'' = -6 \beta (x-\alpha)$$

We get that the IC's are convex for $$x < \alpha$$ and concave for $$x > \alpha$$.

Differentiating one more time,

$$y''' = -6 \beta$$.

So the IC's always have a negative third derivative.

Below is the indifference curve for the utility level $$\overline{U} = 8$$, for the functional form above, with parameters $$\alpha = 2, \beta = 1$$.