Mathematical Economics, Minimization


by dracolnyte
Tags: economics, mathematical, minimization
dracolnyte
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#1
Jul27-09, 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 = [SLp + (1 - S)Kp]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 = [SLp + (1 - S)Kp]1/p expandable?
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Hurkyl
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#2
Jul27-09, 12:05 AM
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Not in any pleasant fashion. Why would you want to expand it?
dracolnyte
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#3
Jul27-09, 12:10 AM
P: 26
because my prof said it would be easier if we let a1 = S1/p and a2 = (1-S)1/p and leave our answers in terms of a1 and a2

dracolnyte
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#4
Jul27-09, 12:17 AM
P: 26

Mathematical Economics, Minimization


im guessing i can make a1p = S and a2p = (1 - S)
then i would get

y = [a1pLp + a2pKp]1/p

y = [(a1L)p + (a2K)p]1/p
Hurkyl
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#5
Jul27-09, 02:09 AM
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Quote Quote by dracolnyte View Post
im guessing i can make a1p = S and a2p = (1 - S)
then i would get

y = [a1pLp + a2pKp]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
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#6
Jul27-09, 02:34 AM
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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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