# How to prove that a concave production imply that the input requirement sets are convex?

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?

• 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 Oct 23 '19 at 19:39

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)