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 16,090
    Staff Emeritus
    Science Advisor
    Gold Member

    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 16,090
    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.
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?
Similar discussions for: Mathematical Economics, Minimization