Math Help - Lagrange Multipliers

1. 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 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.

2. Originally Posted by 413
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).

3. 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?

4. Originally Posted by 413
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.

5. i get it now...thanks.