Mathematical Economics, Minimization

  1. 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?
     
  2. jcsd
  3. Hurkyl

    Hurkyl 16,089
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    Not in any pleasant fashion. Why would you want to expand it?
     
  4. 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
     
  5. 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
     
  6. Hurkyl

    Hurkyl 16,089
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  7. 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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