Maximizing u(x, y) = x^α * y^β subject to Ax + By = m

[Gekkoo]

Homework Statement

I need help to maximize the below function:

Homework Equations

Maximize u(x, y) = x^α * y^β subject to Ax + By = m

Any help is greatly appreciated!

/ Gekkoo

 

[Mute]

Do you know what the method of Lagrange multipliers is? If not, you should look into it, as it's a good way to attack this problem.

http://en.wikipedia.org/wiki/Lagrange_multiplier

 

[HallsofIvy]

I'm a big fan of "Lagrange Multipliers" but if you don't know that method, you could just write y= (m- Ax)/B so that $x^\alpha*y^\beta= x^\alpha*(m-Ax)^\beta/B^\beta$. Now, do you know how to find maxima and minima for that?

 

[Gekkoo]

Thanks for your answers.

1 Solve constraint for y:

y=(m-Ax)/B

2 Plug into objective function:

u=x^α*[(m-Ax)/B]^β

3 Diff w.r.t. x & equate to zero to get critical point:

FOC: x^α*ln(x)*?=0

4 Solve FOC for x:

x=?

5 Plug that into constraint to get value for y:

y = (m-A[?])/B

6 Than I have a candidate solution & need to check SOC of objective function w.r.t x!

But I fail to successfully derive FOC. Can anyone please help me out?