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

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.

• Which edition of MWG are you using? I do not see that statement in Proposition 5.B.2 in my edition. – hrrrrrr5602 Dec 21 '20 at 15:41
• @hrrrrrr5602 1995, page 141 – B11b Dec 21 '20 at 15:54
• Okay, you mean 5.C.2? – hrrrrrr5602 Dec 21 '20 at 16:12
• Can you show us your calculations? You mean the set $\{z \leq 0: f(z) \geq q\}$ ? If there are two elements in this convex set, then any linear combination of these two elements are also in this set. Does it also minimize cost? – Bertrand Dec 21 '20 at 16:13
• @hrrrrrr5602 Yes you are right! Sorry for my typo.. – B11b Dec 21 '20 at 16:15

1 Answer

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.