cost doubt

[riyaf]

Please can somone explain the folloewing question ?

20.5 (0) The prices of inputs (x1; x2; x3; x4) are (4;1;3;2).
(a) If the production function is given by f(x1; x2) = minfx1; x2g, what
is the minimum cost of producing one unit of output? $5.
(b) If the production function is given by f(x3; x4) = x3+x4, what is the
minimum cost of producing one unit of output? $2.
(c) If the production function is given by f(x1; x2; x3; x4) = minfx1 +
x2; x3 +x4g, what is the minimum cost of producing one unit of output?
$3.
(d) If the production function is given by f(x1; x2) = minfx1; x2g +
minfx3; x4g, what is the minimum cost of producing one unit of output?
$5.
2

[Homer Simpson]

Hello! Please let me know if its done properly.

(a) min(x1,x2) => x1 = x2  and f(x1,x2) = 1 (required output) thus, we get x1 = x2= 1

     C = (p1)(x1)+ (p2)(x2)

     Putting this, C = (4)(1) + (1)(1) = 5

(b) f(x3,x4) = x3+x4 = 1 (required)

     This is a case of perfectly substituted inputs, so i need to choose the cheapest one. x4 is cheaper

     Thus, C = (p3)(x3) + (p4)(x4)
                =  (3)(0) + (2)(1)
                =  2

(c) f(x1,x2,x3,x4) = min(x1+x2, x3+x4) =1 (required)

      here, i have x1+x2 = x3+x4 = 1 - (*)

      Thus, for cost minimization, we choose x1 = 0, x2=1 , x3=0 and x4=1 (satisfies *)

      thus, C = (p1)(x1)+ (p2)(x2) + (p3)(x3) + (p4)(x4)
                  = (1)(1) + (2)(1)
                  = 3

(d) f(x1,x2) = min (x1, x2) + min (x3, x4) = 1 (required) - (*)

      Now, for this to hold, either x1=x2=1 or x3=x4=1

      Thus, C(x1=x2=1) = (p1)(x1)+ (p2)(x2) = (4)(1) + (1)(1) = 5
       and, C(x3=x4=1) = (p3)(x3) + (p4)(x4) = (3)(1) + (2)(1) = 5

      For any other combination satisfying (*), cost shall be higher.

[Akshay Jain]

Its correct.....

[SoniaKapoor]

Akshay ,tuski pls explain me the 3rd part

[Homer Simpson]

for third part, choose the cheapest among x1,x2 and x3,x4. the input quantity combo must also satisfy x1+x2=x3+x4=1

[SoniaKapoor]

Thankuu tsuki

[Anjali]

Great job Tsuki !

[riyaf]

thanks a ton tsuki :)