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Constrained Optimization with equality

  • 15-09-2009 8:22pm
    #1
    machintoshlover
    Closed Accounts Posts: 50 ✭✭


    Hey guys got a fairly simple problem that unfortunetly my simple brain won't work out.

    The question is:

    Consider the following maximization problem:

    max 2X + 3Y
    X,Y

    s.t. PxX + PyY = 40

    (a.) Write down the Lagrangian:

    (b.) Write out the F.O.N.C and solve for the optimal values X* and Y*

    Okay so part (a.) is grand

    L(X,Y) = 2X + 3Y + λ(40 - PxX -PyY)

    In part (b.) the F.O.N.C (First order necessary conditions) are:

    dL(X,Y) = 2 - Px λ
    DX

    dL(X,Y) = 3 - Py λ
    DY

    dL(X,Y) = 40 - PxX - PyY
    dλ

    It is finding the optimal values X* and Y* I am having trouble with:

    If anyone could give me a hint or two to pop on it would be most appreciated!

    Cheers!


Comments

  • #2
    machintoshlover
    Closed Accounts Posts: 50 ✭✭machintoshlover


    Sorry forgot to put what I had down for finding my optimization!

    I work out that:

    Px = 2
    py 3 which I believe is the slope of the budget constraint but my brain stops there - thanks and sorry it is probably very easy but I had a long day!


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