Please help check answer!?

Question

Suppose a firm is operating in the short run, holding input 2 fixed at x2. Its production function is given by the Cobb-Douglas one: f(x1,x2) =( x1)^a(x2)^b. The firm wants to find out its least cost combination of input for any given output level, y. The firm therefore solves the following optimisation problem:

c(w1, w2, y) = Min {w1x1 +w2x2(bar)} such that f(x1, x2(bar)) = y

the bar means x2 is constant b/c it is SR

What is the short-run cost function, c(w1, w2, y), for the firm?

because it could not be in x2, which made this a lot more complicated, i am not sure if this is correct. please check and help!!! thanks!!!!

since y = x1^a*x2bar^b, y/x1^a = x2bar^b

then (y/x1^a)^1/b = x2bar

to take out x1, i put in a previously solved part for x1, where i found that x1 is ay/w1. please see: http://answers.yahoo.com/question/index;_ylt=AvP2Vyx5csb3HggqC.WSx0Psy6IX;_ylv=3?qid=20080621220737AAv1vUE for verification

so (y/(ay/w1)^a)^1/b = x2bar

and then (y*(ay/w1)^-a)^1/b =

Answer 1

Let:

f(x1;x2)=Q - output

x1=L - labor

x2=C - capital

w - wage (cost of resource x1=L)

r - interest rate (cost of resource x2=C)

MPK - marginal product of capital

MPL - marginal product of labr

TC - total cost

then:

TC=Lw+rC

Q=L^a * C^b

L=(Q/C^b)^1/a

C=(Q/L^a)^1/b

MPK=δQ/δC=(L^a * C^b)' =bL^a *C^(b-1)=

=bL^a *(C^b)/C =bQ/C

MPL=δQ/δL=(L^a * C^b)' =aC^b *L^(a-1)=

=aC^b *(L^a)/L=aQ/L

Equilibrium condition:

MPK/r=MPL/w

bQ/rC=aQ/wL

b/rC=a/wL

C=bwL/ra

L=arC/wb

Now:

Initial modified equation:

L=(Q/C^b)^1/a

C=(Q/L^a)^1/b

Combine with Equilibrium condition:

C=bwL/ra

L=arC/wb

So:

C=C

bwL/ra=(Q/L^a)^1/b

L^(1+a/b)=ra/bw * Q^(1/b)

L=(ra/bw * Q^(1/b))^(1/(1+a/b))

L=L

arC/wb=(Q/C^b)^1/a

C^(1+b/a)=bw/ra * Q^(1/a)

C=(bw/ra * Q^(1/a))^(1/(1+b/a))

So here are your answers:

Quantity of cost-minimizing resources:

x1=(ra/bw * Q^(1/b))^(1/(1+a/b))

x2=(bw/ra * Q^(1/a))^(1/(1+b/a))

♦ Grrr, accidently I have found long-run solution - hold-on a bit and I will modify it to short-run....

Added:

Here it is: since C=2 then Q=(L^a) * (2^b)

so rearrange to L^a=Q/(2^b) and following to:

L=(Q/(2^b))^(1/a)

Answer: Short-run cost-minimizing quantity of x1=(Y/(2^b))^(1/a)

you actually can't minimize here anything - all you can do - modify factor x1 to reach planned/required output Y - because factor x2 is fixed.