Question

Show all work please.

3. Suppose the production function (technology) is given as q = min {2x1,6x2}, where both inputs x1 and x2 are able to vary.

1. What kind of technology is this function?

2. What is the corresponding cost function as a function of w1, w2, and q?

3. Solve for this cost if w1 = 24, w2 = 18, and q = 2.

Answer 1

Answer 3

(a)

A function exhibit constant returns to scale if f(tx₁,tx₂) = t*f(x₁,x₂) for all t > 1.

Here, q = f(x₁,x₂) = min {2x₁,6x₂} => f(tx₁,tx₂) = min {2tx₁,6tx₂} = t*min {2x₁,6x₂} = t*f(x₁,x₂). Hence this function exhibit constant returns to scale.

This producer consider x₁ and x₂ as perfect complements. Thus this production function is a fixed proportion function and exhibit constant returns to scale.

(b)

Production function is given by :

q = f(x₁,x₂) = min {2x₁,6x₂} ----------------------------------(1)

In order to minimize cost for a perfect complement of Leontief production a producer produces at a point where kink of an isoquant will occur.

Kink will occur for this function when 2x₁ = 6x₂

Putting this in (1) we get :

q = min {2x₁,6x₂} = min {2x₁,2x₁} = 2x₁

=> x₁ = q/2 and hence x₂ = q/6

Cost function (C) = w₁x₁ + w₂x₂ = (q/2)w₁ + (q/6)w₂

Hence,Cost function (C) = (q/2)w₁ + (q/6)w₂

(c)

Now  w₁ = 24, w₂ = 18, and q = 2.

=> Cost function (C) = (2/2)*24 + (2/6)*18 = 30

Hence,  if w₁ = 24, w₂ = 18, and q = 2. then Cost = 30.