<p>Let <span class="math-container">$z_1$</span> and <span class="math-container">$z_2$</span> be <span class="math-container">$\geq 0$</span> and solution to</p>
<p><span class="math-container">$$\min_z \{w^\top z\lvert f(z)\geq q\}$$</span></p>
<p>then clearly <span class="math-container">$f(z_1)\geq q$</span> and <span class="math-container">$f(z_2)\geq q$</span> and since <span class="math-container">$\{z\geq 0\lvert f(z)\geq z \}$</span> is convex it then follows that <span class="math-container">$z_3 := \lambda z_1 + (1-\lambda)z_2$</span> must satisfy the constraint <span class="math-container">$f(z_3)\geq q$</span>.</p>
<p>Since <span class="math-container">$z_1$</span> and <span class="math-container">$z_2$</span> are both minimizers it cannot be the case that <span class="math-container">$w^\top z_1 \not = w^\top z_2$</span> rather it must be the case that <span class="math-container">$w^\top z_1 = w^\top z_2$</span>. This implies that <span class="math-container">$w^\top z_3 = w^\top(\lambda z_1 + (1-\lambda)z_2) = \lambda w^\top z_1 + (1-\lambda) w^\top z_2 = w^\top z_1$</span> hence <span class="math-container">$z_3$</span> is a minimizer.</p>
<p>Since <span class="math-container">$z_3$</span> is minimizer and satisfy constraint it must be in <span class="math-container">$z(w,q)$</span> which therefore must be convex set.</p>
<p>If <span class="math-container">$f$</span> is quasi concave then by definition <span class="math-container">$\{z\geq 0 \lvert f(z) \geq q\}$</span> is convex and therefore using the above argument implies <span class="math-container">$z(w,q)$</span> is convex.</p>

CC BY-SA 4.0
2020-12-22T02:48:19.143Z
2020-12-22T02:48:19.143Z
bomadsen
<p>I have a question on the properties of conditional demand correspondence</p>
<p>Let <span class="math-container">$z(w,q)$</span> be the <strong>conditional factor demand correspondence</strong>, i.e. the solution of the cost minimization problem</p>
<p><span class="math-container">\begin{align}
\min_z \quad&amp; w\cdot z \\ \text{subject to}\quad &amp; f(z)\geq q.
\end{align}</span></p>
<p>In the book of Mas-Colell, Whinston, Green, Proposition 5.C.2 (v) says that</p>
<blockquote>
<p>if the set <span class="math-container">$\{z\ge 0 : f(z)\ge q\} $</span> is convex then <span class="math-container">$z(w,q) $</span> is a convex set.</p>
</blockquote>
<p>Also, in the same property, it states that</p>
<blockquote>
<p>if <span class="math-container">$f$</span> is quasiconcave then <span class="math-container">$z(w,q)$</span> is a convex set for every <span class="math-container">$w&gt;&gt;0$</span>.</p>
</blockquote>
<p>How can I prove these two statements? I would like to understand these two properties. Thanks a lot.</p>

CC BY-SA 4.0
2020-12-21T15:41:05.787Z
Which edition of MWG are you using? I do not see that statement in Proposition 5.B.2 in my edition.
hrrrrrr5602
CC BY-SA 4.0
2020-12-21T15:54:14.880Z
@hrrrrrr5602 1995, page 141
studentp
CC BY-SA 4.0
2020-12-21T16:12:30.230Z
Okay, you mean 5.C.2?
hrrrrrr5602
CC BY-SA 4.0
2020-12-21T16:13:56.430Z
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
CC BY-SA 4.0
2020-12-21T16:15:15.043Z
@hrrrrrr5602 Yes you are right! Sorry for my typo..
studentp
CC BY-SA 4.0
2020-12-21T16:17:18.313Z
MWG states $z \ge 0, f(z) \ge q$, which makes more sense. Also, I do not see the second statement you wrote above in Proposition 5.C.2.
hrrrrrr5602
CC BY-SA 4.0
2020-12-21T22:26:27.713Z
The book is by Mas-Colell, Whinston, and Green, not by someone called Colell.
Michael Greinecker
<microeconomics><self-study><producer-theory><mwg>
Properties on conditional demand correspondence from the textbook of Mas-Colell et al
studentp
# Properties on conditional demand correspondence from the textbook of Mas-Colell et al

(Asked by: username_0 on 2020-12-21)

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.

**Tags:** microeconomics, self-study, producer-theory, mwg

> **Comments:**
> **username_1**: Which edition of MWG are you using? I do not see that statement in Proposition 5.B.2 in my edition.  
> **username_0**: @username_1 1995, page 141  
> **username_1**: Okay, you mean 5.C.2?  
> **username_2**: 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?  
> **username_0**: @username_1 Yes you are right! Sorry for my typo..  
> **username_1**: MWG states $z \ge 0, f(z) \ge q$, which makes more sense. Also, I do not see the second statement you wrote above in Proposition 5.C.2.  
> **username_3**: The book is by Mas-Colell, Whinston, and Green, not by someone called Colell.  

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## 1 answers

### Answer by username_4 on 2020-12-22. Score: 7 | Accepted 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.


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