Question

A) Suppose that your production function is: q = L∙K + K. Find the short-run cost function.

Answer 1

Answer Total cost = r{sqrt(Q*(w/r)} + w{(r/w)[ sqrt(Q*(w/r)] -1}

Q = LK + K

Suppose price of K is r and price of L is w

Then cost function (C) will be

C = rK + wL

Now setting lagrange

Z = rK + wL + λ (Q - LK - K)

Differentiate the above equation with respect to K and equate it to 0

dZ/dK = r - λ (L+1) = 0

r/(L+1) = λ                 -----equation 1

Differentiate the same equation with respect to L and equate it to 0

dZ/dL = w - λK = 0

w/K = λ                   ----------equation 2

Solve equation 1 and 2

w/K = r/(L+1)

L+1 = (r/w)K

L = (r/w)K -1                ------------equation 3

Differentiate the same equation with respect to λ and equate it to 0

dZ/dλ = Q - LK – K = 0

Q = LK + K             ---------------equation 4

Put the value L from equation 3 to equation 4

Q = [(r/w)K -1]K +K

Q = [(r/w)K²

K = sqrt(Q*(w/r))

Note: sqrt = square root

We know that        

L = (r/w)K -1

And K = sqrt(Q*(w/r))

Therefore

L = (r/w)[ sqrt(Q*(w/r)] -1