Consider the problem of minimizing costs rK + wL subject

Consider the problem of minimizing costs rK + wL subject to the production function Y = K αL β, where α, β > 0. Derive the conditional factor demand functions and the cost function; and confirm that Shephard’s lemma holds.

wL rK +min

subject to βαL K Y =

Solving the constraint for L we see that this problem is equivalent to

βαβ//1min -+K wY rK

Taking first order condition we obtain:

0/1=-+-ββαββαK wY r That gives us the conditional demand function for factor 1

βαβαββα++??

????=1

),,(Y r w Y w r K

The other conditional demand function is βαβααβα++-??

????=1

),,(Y r w Y w r L

Plugging back into objective function we obtain cost function:

βαβαββααβααβαββαβα++++-+????

????????????+??????=1),,(Y w r Y w r C

Shephard’s lemma:

K r C =??() ; L w

C =??()

Confir m that Shephard’s lemma is met.

We will use the constant-returns-to-scale assumption that 1=+βα

In this case the cost function reduces to

C(r,w,Y) = y w r αααααα----11)1(

The first order condition is

y w r ααααααα-----111)1(

y w r 1)1(-??

????-ααα

since αβ-=1we get

βαββαβα++??

????r w y 1

= K

For the second conditional demand function (L):

L w

C =??()

C(r,w,Y) = y w r αααααα----11)1(

The first-order function is

y w a r αααααα-----)1()1(1

Thus, we have

y r aw αα-??????-)1(

Since αβ-=1we get

L Y r w =??

????++-βαβααβα1