Mathematical Economics, Minimization

[dracolnyte]

Homework Statement

Consider the following general form of a constant elasticity of substitution production function:

y = [SL^p + (1 - S)K^p]^1/p

Assume a firm is trying to minimize the cost of producing any given y. Cost are given by

C = wL + rK

Find the firm's cost minimizing demand function for L. The cost minimizing demand for K is determined simultaneously, so you need both first order conditions. You may assume that nonneggativity constraints on L and K are not binding.

The Attempt at a Solution

Is y = [SL^p + (1 - S)K^p]^1/p expandable?

 

[Hurkyl]

Not in any pleasant fashion. Why would you want to expand it?

 

[dracolnyte]

because my prof said it would be easier if we let a1 = S^1/p and a2 = (1-S)^1/p and leave our answers in terms of a1 and a2

 

[dracolnyte]

im guessing i can make a1^p = S and a2^p = (1 - S)
then i would get

y = [a₁^pL^p + a₂^pK^p]^1/p

y = [(a₁L)^p + (a₂K)^p]^1/p

 

[Hurkyl]

im guessing i can make a1^p = S and a2^p = (1 - S)
then i would get

y = [a₁^pL^p + a₂^pK^p]^1/p

y = [(a₁L)^p + (a₂K)^p]^1/p

This is certainly correct.

FYI, I don't think this change of variable has anything to do with how to go about performing this calculation -- it's just a little optional detail that may (or may not) make it less tedious.

 

[snipez90]

Seems pretty standard. You want to minimize wL + rK over L and K, with y - [SLp + (1 - S)Kp]^1/p = 0 as your constraint. Define the lagrangian and derive the first order conditions by differentiating the lagrangian with respect to L, K, and lambda.