Question

Proof : If the set of production possibilities Z is a convex set, the set of necessary inputs is also a convex set.

Answer 1

Product Possibilities (Z) is a Convex set.

Necessary Inputs is also a Convex set

Explanation:

Production Possibilities set Z.

Input requirement set V(z).

Production function f(x)

Z being a Convex set .

(z_{​​​​​​1}​​​​​-x) ε Z

and (z_{​​​​​​1} - x^{​​​​​​¹}) ε Z Equation (1)

Then t (z_{​​​​​​1} - x) + (1-t) (z_{​​​​​​1} - x^{​​​​​​1} ) ε Z

Equation (2) implies

(t_{​​​​​​z}+ (1-t)z_{​​​​​​1}​​​​​ - t_​​​2 - (1-z)z') ε Z

(z_{​​​​​​1} - t_{​​​​​​z}​​​ -(1-t) z')  ε Z

=> tx + (1-t)z' ε v (z) Equation (3)

(1) implies x ε V(z)

x' ε  V(z) Equation (4)

In particular (1) => (2) => (3) => (4)

So (4) => (3) meaning that v(z) is a Convex set.