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Mathematical Economics, Minimization 
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#1
Jul2709, 12:01 AM

P: 26

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? 


#2
Jul2709, 12:05 AM

Emeritus
Sci Advisor
PF Gold
P: 16,098

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



#3
Jul2709, 12:10 AM

P: 26

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



#4
Jul2709, 12:17 AM

P: 26

Mathematical Economics, Minimization
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} 


#5
Jul2709, 02:09 AM

Emeritus
Sci Advisor
PF Gold
P: 16,098

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. 


#6
Jul2709, 02:34 AM

P: 1,105

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.



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