Properties on conditional demand correspondence from the textbook of Mas-Colell et al

Question

I have a question on the properties of conditional demand correspondence

Let $z(w,q)$ be the conditional factor demand correspondence, i.e. the solution of the cost minimization problem

\begin{align} \min_z \quad& w\cdot z \\ \text{subject to}\quad & f(z)\geq q. \end{align}

In the book of Mas-Colell, Whinston, Green, Proposition 5.C.2 (v) says that

if the set $\{z\ge 0 : f(z)\ge q\} $ is convex then $z(w,q) $ is a convex set.

Also, in the same property, it states that

if $f$ is quasiconcave then $z(w,q)$ is a convex set for every $w>>0$.

How can I prove these two statements? I would like to understand these two properties. Thanks a lot.

Answer 1

Let $z_1$ and $z_2$ be $\geq 0$ and solution to

$$\min_z \{w^\top z\lvert f(z)\geq q\}$$

then clearly $f(z_1)\geq q$ and $f(z_2)\geq q$ and since $\{z\geq 0\lvert f(z)\geq z \}$ is convex it then follows that $z_3 := \lambda z_1 + (1-\lambda)z_2$ must satisfy the constraint $f(z_3)\geq q$.

Since $z_1$ and $z_2$ are both minimizers it cannot be the case that $w^\top z_1 \not = w^\top z_2$ rather it must be the case that $w^\top z_1 = w^\top z_2$. This implies that $w^\top z_3 = w^\top(\lambda z_1 + (1-\lambda)z_2) = \lambda w^\top z_1 + (1-\lambda) w^\top z_2 = w^\top z_1$ hence $z_3$ is a minimizer.

Since $z_3$ is minimizer and satisfy constraint it must be in $z(w,q)$ which therefore must be convex set.

If $f$ is quasi concave then by definition $\{z\geq 0 \lvert f(z) \geq q\}$ is convex and therefore using the above argument implies $z(w,q)$ is convex.