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.

  • $\begingroup$ Which edition of MWG are you using? I do not see that statement in Proposition 5.B.2 in my edition. $\endgroup$ – hrrrrrr5602 Dec 21 '20 at 15:41
  • $\begingroup$ @hrrrrrr5602 1995, page 141 $\endgroup$ – B11b Dec 21 '20 at 15:54
  • $\begingroup$ Okay, you mean 5.C.2? $\endgroup$ – hrrrrrr5602 Dec 21 '20 at 16:12
  • $\begingroup$ 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? $\endgroup$ – Bertrand Dec 21 '20 at 16:13
  • $\begingroup$ @hrrrrrr5602 Yes you are right! Sorry for my typo.. $\endgroup$ – B11b Dec 21 '20 at 16:15

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.