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

Question

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.

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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.

Answer 1

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 $\alpha = 2, \beta = 1$.