Does the input requirement set V (y) = {(x1, x2, x3) | x1 + min {x2,...

Does the input requirement set

V (y) = {(x1, x2, x3) | x1 + min {x2, x3} ≥ 3y, xi ≥ 0 ∀ i = 1, 2, 3}

corresponds to a regular (closed and non-empty) input requirement set?
Does the technology satisfies free disposal? Is the technology convex?

Solutions

Expert Solution

The graphical representation is done in the photo uploaded.The second diagram shows the isoquant of x₂ and x₃. Min(x₂, x₃) gives us L shaped isoquants as they are in fixed proportions. The first diagram shows us the relationship with x₁ and x_3, given the 3Y constraint.

It is clearly understandable from the diagram that it is a closed and non empty set. It is also a convex set because if we join any two points within the set, it will lie inside the set only which is at par with the definition of convex set.

The property of free disposal is actually equivalent to the monotonicity of the production function. It is because free disposal is actually a situation where a firm can get rid off excess factors of production. As a result firm's output cannot decrease because they are not being penalised due to excess inputs. Thus the property states that having more and more inputs will result in at least as much as previous output, making it equivalent to the monotonicity property.

Q= x₁ + min(x₂,x₃) is the production function.

Either x₂ is minimum or x₃ is.

When x₂< x₃, Q= x₁ + x₂.

When x₁ increases or x₂ increases, Q increases.

Similary when x₃<x₂ , Q= x₁ + x₃.

when x₁ or x₃ increases, Q increases.

Thus this technology is monotonic i.e. it satsifies free disposal.

Rahul Sunny answered 1 year ago

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