Question

A firm uses capital and labor to produce output according to the production q = 4VLK (a) Find the marginal product of labor (MPL) and marginal product of capital (MPK). (b) If the wage w=$1/labor-hr. and the rental rate of capital r-$4/machine-hr., what is the least expensive way to produce 16 units of output? (c) What is the minimum cost of producing 16 units? (d) Show that for any level of output, q, the minimum cost of producing q is $q. (e) Find the Total cost function.

Answer 1

A)Q=4√Lk

MPL=∆Q/∆L=2√K/(√L)

MPk=∆Q/∆K=2√L/(√k)

B)cost minimization condition;,

MPk/pk=MpL/pL

2√L/4√K=2√K/√L

L=4K{ cost minimization condition}

16=4√K*4K=8K

K=2

L=2*4=8

C) Total cost=8*1+2*4=8+8=16{ minimum cost of producing 16 units)

D) using cost minimization condition;,

Q=4√k*4k=8K

K=Q/8

L=4*Q/8=Q/2

TC=L*w+K*r=Q/2)*1+Q/8)*4=Q/2+Q/2=Q

So Q units can produce at minimum of Q$.

E)TC=Q{ solved in part d)