Question

The prices of inputs (x1,x2,x3,x4) are (4,1,3,2):

(a) If the production function is given by f(x3,x4) =min⁡{x1+x2,x3+x4} what is the minimum cost of producing one unit of output?

(b) If the production function is given by f(x3,x4)=x1+x2 +min⁡{x3+x4} what is the minimum cost of producing one unit of output?

Answer 1

a) f(x3,x4) =min⁡{x1+x2,x3+x4} = 1 (we need one output)

choose the cheapest among (x1,x2) and (x3,x4). The input quantity combination must also satisfy x1+x2 = x3+x4 =1 (since we want 1 unit of production)

here, we have x1+x2 = x3+x4 = 1

Out of x1 and x2, x2 is cheaper so choose x2. So x1=0, x2=1. Out of x3 and x4, x4 is cheaper so choose x4. So, x3=0 and x4=1

Cost = (p1)(x1)+ (p2)(x2) + (p3)(x3) + (p4)(x4)

= 1 * 1 + 2*1

=3

b) Let A = x1+x2 and B = min⁡{x3+x4}

Hence, f(x3,x4)=A+B

A+B = 1 (required)

In case of perfect substitue, choose among A and B what is cheaper. Find out what is cheaper among A and B.

COST OF A

A = x1+x2

In case of perfect substitue, choose among x1 and x2 what is cheaper. Price of x2 is less so choose x2

Thus, Cost = (p1)(x1) + (p2)(x2) = 4*0 + 1*1 = 1

COST OF B

B = min⁡{x3+x4}

Optimal choice for perfect compliments : x3=x4

We need to produce 1 unit, x3=x4=1

C = (p3)(x3)+ (p4)(x4)

= 3*1 + 2*1

=5

Since out of A and B, A is cheaper

So use A.

Therefore, cost =  1