Question

I've asked the same question before but the answer is wrong so please post right answer this time. Please answer if you know the subject well. Thank you.

Question: Prove that the following function is a cost function and derive a possible production function.

The cost function is :

C(w, r,q) = q(w+ w^1/2r^1/2+ r)

Answer 1

ASSUME THE COST FUNCTION C = C [ W ,R , Q] WHERE W , R ARE TWO INPUTS OF PRODUCTION

SUPPOSE Q = 1 WE CAN USE THE COST FUNCTION TO DERIVE THE ISOQUANT FOR ONE OUTPUT IF YOU CAN DO THIS YOU CAN DO IT FOR ANY OUTPUT LEVEL DEMONSTRATING THAT THE PRODUCTION FUNCTION CAN BE DERVIED FROM THE COST FUNCTION

WE KNOW

C [ W , L , Q ] =Q= F [ K ,L ]

PUTTING THESE TOGETHER E GET

C [ Q, W, R ] Q= F [ K , L ]

WE CAN DO THIS ALBRICALLY

ASSUME C = QW⁵R⁵  AND FINDING IT CORRESPONDING TO Q = F [ K ,L ]

BY SHEPHARD LEMMA

L_C = Q [ R / W ]⁵

K_{C =}Q [ W / R ]⁵

WE CAN SAY

[W / R ] ⁵ =Q / I AND [ W /R ]⁵ = K / Q

THAT IS , Q / I = K / Q

SOLUTION = Q =$\sqrt{}$ [I K ] [ Q ] = - $\sqrt{}$