<p>Start with definitions:</p>

<ol>
<li>Production (possibilities) set: <span class="math-container">$Y$</span> which you know is convex</li>
<li>Input requirement set: <span class="math-container">$V(y)=\{\mathbf{x}:(y,−\mathbf{x})∈Y\}$</span></li>
</ol>

<p>On page 7. you can see: <span class="math-container">$\mathbf{y}\in Y$</span> and <span class="math-container">$\mathbf{y'} \in Y$</span> which then implies <span class="math-container">$t\mathbf{y}+(1-t)\mathbf{y'} \in Y$</span>. </p>

<p>Hint 1:</p>

<blockquote class="spoiler">
  <p> Remember that: <span class="math-container">$\mathbf{y}=(y,−\mathbf{x})$</span> !</p>
</blockquote>

<p>What does it mean that <span class="math-container">$Y$</span> is a convex set? </p>

<blockquote class="spoiler">
  <p> <span class="math-container">$Y$</span> being a convex set means that if <span class="math-container">$\mathbf{y}=(y,−\mathbf{x}) \in Y$</span> and <span class="math-container">$\mathbf{y'}=(y,−\mathbf{x'}) \in Y$</span> then <span class="math-container">$t(y,−\mathbf{x})+(1-t)(y,−\mathbf{x'}) \in Y$</span> as page 7. would suggest.</p>
</blockquote>

<p>Okay, but what does that actually mean?</p>

<blockquote class="spoiler">
  <p> <span class="math-container">$t(y,−\mathbf{x})+(1-t)(y,−\mathbf{x'}) \in Y \implies (ty+(1-t)y, -t\mathbf{x}-(1-t)\mathbf{x'}) \in Y \implies (y,-t\mathbf{x}-(1-t)\mathbf{x'}) \in Y $</span></p>
</blockquote>

<p>Which is just equivalent of saying that:</p>

<blockquote class="spoiler">
  <p> <span class="math-container">$t\mathbf{x}+(1-t)\mathbf{x'} \in V(y)$</span> because both <span class="math-container">$\mathbf{x}$</span> and <span class="math-container">$\mathbf{x'}$</span> are in <span class="math-container">$V(y)$</span>. (Look: second definition)</p>
</blockquote>

CC BY-SA 4.0
2019-10-23T22:06:24.437Z
2019-10-23T22:06:24.437Z
bajun65537
<p>According to page 7 of <a href="http://folk.uio.no/gasheim/4232f6t2.pdf" rel="nofollow noreferrer">this slide</a>, &quot;A convex production set Y implies that the associated input requirement set V(y) is convex&quot;.</p>
<p>How can one go about proving it?</p>

CC BY-SA 4.0
2019-10-23T19:39:05.427Z
Have you tried applying the definitions (convexity, input requirement set) directly? Could you edit the problems you encounter with that approach into the question?
Giskard
<microeconomics><production-function><input-output>
How to prove that a concave production imply that the input requirement sets are convex?
Aqqqq
# How to prove that a concave production imply that the input requirement sets are convex?

(Asked by: username_0 on 2019-10-23)

According to page 7 of [this slide](http://folk.uio.no/gasheim/4232f6t2.pdf), "A convex production set Y implies that the associated input requirement set V(y) is convex".

How can one go about proving it?

**Tags:** microeconomics, production-function, input-output

> **Comments:**
> **username_1**: Have you tried applying the definitions (convexity, input requirement set) directly? Could you edit the problems you encounter with that approach into the question?  

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## 1 answers

### Answer by username_2 on 2019-10-23. Score: 1 | Accepted answer

Start with definitions:

1. Production (possibilities) set: $Y$ which you know is convex
2. Input requirement set: $V(y)=\{\mathbf{x}:(y,−\mathbf{x})∈Y\}$

On page 7. you can see: $\mathbf{y}\in Y$ and $\mathbf{y'} \in Y$ which then implies $t\mathbf{y}+(1-t)\mathbf{y'} \in Y$.

Hint 1:

> Remember that: $\mathbf{y}=(y,−\mathbf{x})$ !

What does it mean that $Y$ is a convex set?

> $Y$ being a convex set means that if $\mathbf{y}=(y,−\mathbf{x}) \in Y$ and $\mathbf{y'}=(y,−\mathbf{x'}) \in Y$ then $t(y,−\mathbf{x})+(1-t)(y,−\mathbf{x'}) \in Y$ as page 7. would suggest.

Okay, but what does that actually mean?

> $t(y,−\mathbf{x})+(1-t)(y,−\mathbf{x'}) \in Y \implies (ty+(1-t)y, -t\mathbf{x}-(1-t)\mathbf{x'}) \in Y \implies (y,-t\mathbf{x}-(1-t)\mathbf{x'}) \in Y $

Which is just equivalent of saying that:

> $t\mathbf{x}+(1-t)\mathbf{x'} \in V(y)$ because both $\mathbf{x}$ and $\mathbf{x'}$ are in $V(y)$. (Look: second definition)


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