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Math Help - Lagrange Multipliers

  1. #1
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    Lagrange Multipliers

    Paul eats healthy; he spends his entire monthly lunch allowance of $288 on organic arugula salad (X) and seaweed (Y). His utility function is U = 2XY. Both X and Y sells for $6 per unit.
    (a) Calculate Paulís utility-maximizing purchases of X and Y.
    (b) How will his utility change if she who must be obeyed (his wife) reduces his monthly allowance by $2.00?
    (c) As a result of everybody jumping on the organic bandwagon, the price of organic arugula increases to $8.64. By how much must Paulís wife increase his monthly lunch allowance to enable him to maintain his
    initial level of utility as in part (a) above?

    a) i got X=Y=24
    b) Lagrangian multiplier i got is 8, so utility changed by 16
    c) I don't know where to start for this question.
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    Quote Originally Posted by 413 View Post
    Paul eats healthy; he spends his entire monthly lunch allowance of $288 on organic arugula salad (X) and seaweed (Y). His utility function is U = 2XY. Both X and Y sells for $6 per unit.
    (a) Calculate Paul’s utility-maximizing purchases of X and Y.
    (b) How will his utility change if she who must be obeyed (his wife) reduces his monthly allowance by $2.00?
    (c) As a result of everybody jumping on the organic bandwagon, the price of organic arugula increases to $8.64. By how much must Paul’s wife increase his monthly lunch allowance to enable him to maintain his
    initial level of utility as in part (a) above?

    a) i got X=Y=24
    b) Lagrangian multiplier i got is 8, so utility changed by 16
    c) I don't know where to start for this question.
    Your answers to (a) and (b) are OK, although using the multiplier for (b) only gives an approximate answer. The exact answer -15.9444... can be found using the following method to answer (c).

    Start by solving the problem using allowance Z in place of 288 and prices Px and Py in place of 6 and 6. I get the multiplier Z/PxPy and solutions X = Z/2Px, Y = Z/2Py. The maximum utility is U = 2XY = Z^2/2PxPy. To answer (c), solve for Z satisfying Z^2/2x8.64x6 = 288^2/2x6x6, which is the initial maximum utility. The required Z = 288 x sqrt(8.64/6).
    Last edited by JakeD; February 10th 2007 at 10:24 PM.
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    i see your equations are only for this question, what about for other questions? Is there a general rule or equation that i can follow?
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    Quote Originally Posted by 413 View Post
    i see your equations are only for this question, what about for other questions? Is there a general rule or equation that i can follow?
    Did this question come from a calculus class or an economics class?

    There is an economic theory of consumer utility maximization that underlies what I did. For example, the function V(Px,Py,Z) = Z^2/2PxPy giving the maximum utility achievable for given prices and income is called the indirect utility function. So there is a general theory to follow, but the equations must be worked out in each case.
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  5. #5
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    i get it now...thanks.
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