# Quasiconvexity of the indirect utility function

In Mas-Colell, Whinston, and Green's Microeconomics they define the indirect utility function, $v(p,w)$ as

$$v(p,w) := u(x^*)$$

Where $x^* \in x(p,w)$ solves the utility maximization problem.

They state a property of $v(p,w)$ is quasiconvexity, i.e. the set

$$\{(p,w): v(p,w) \leq \bar{v} \}$$

is convex for any $\bar{v}$.

Just the page before they said that convexity of preferences implies that $u(\bullet)$ is quasiconcave, so my question is why when we look at the max of $u$, it's quasiconcave property inflects (can't think of a better word) to quasiconvexity?

The functions and their variables are different, so there is no "inflection" or flipping.

The utility function $u$ which maps from the space of goods $X$ to $\mathbb{R}$ is convex and quasiconcave.

The indirect utility function $v$ which maps from the space of prices to $\mathbb{R}$ is quasiconvex.

Intuitively:
$u$: If you average two consumption bundles your utility is not lower than the average of the utility of the two bundles. Rather than eating just meat one day and just vegetables the other day you prefer to mix these everyday.

$v$: If the price vector is $p$ one day and $p'$ the other day you may be better off than if it was $\frac{p + p'}{2}$ every day. It is easy to check that anything you can buy under the second price regime you can also buy under the first. However there might be consumption bundles that you can only buy under the first price regime.

• Many thanks. Just had the same question today.
– Rumi
Oct 23, 2021 at 2:14