Optimum Values of X and Y: Lagrange Multiplier Help for Maximizing U=XY

[jack90]

ok this is just an example so you can see where I am having problems with these(it isn't hw)

i need to find the optimum values of X and Y

U= XY

m= Psuby(Y) + Psubx(X)

the first order conditions are

Y +u*Psubx

X+ u*PsubY

m= Psuby(Y) + Psubx(X)


now , where I am having problems is how do i find the optimum values from that. I know you are supposed to plug the FOC in but i can't get the answer my book shows me. could somebody walk me through that part?

 

[LCKurtz]

It's hard to walk you through anything when you haven't stated the problem clearly. You need a function you want to optimize which I assume is U(x,y). Then you need a constraint equation which you haven't stated. Is it P(x,y) = 0? Next you make the auxiliary function:

F(x,y) = U(x,y) + $$\lambda$$P(x,y)
using your formulas for U(x,y) and P(x,y).

I have no idea what your m is and what you call your "first order conditions" aren't even equations; they are just expressions. The equations you need to set up and solve are

F_x = 0
F_y = 0
F_$$\lambda$$ = 0

Three equations in three unknowns.