According to page 7 of this slide, "A convex production set Y implies that the associated input requirement set V(y) is convex".

How can one go about proving it?

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    $\begingroup$ Have you tried applying the definitions (convexity, input requirement set) directly? Could you edit the problems you encounter with that approach into the question? $\endgroup$ – Giskard Oct 23 '19 at 19:39

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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