1. The problem statement, all variables and given/known data 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. 3. The attempt at a solution Is y = [SL^{p} + (1 - S)K^{p}]^{1/p} expandable?
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
im guessing i can make a1^{p} = S and a2^{p} = (1 - S) then i would get y = [a_{1}^{p}L^{p} + a_{2}^{p}K^{p}]^{1/p} y = [(a_{1}L)^{p} + (a_{2}K)^{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.
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.