# Mathematical Economics, Minimization

by dracolnyte
Tags: economics, mathematical, minimization
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 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?
 Emeritus Sci Advisor PF Gold P: 16,091 Not in any pleasant fashion. Why would you want to expand it?
 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
 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
Emeritus
Sci Advisor
PF Gold
P: 16,091
 Quote by dracolnyte 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.
 P: 1,104 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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