# Thread: Maximization using Lagrange Multipliers.

1. ## Maximization using Lagrange Multipliers.

Hey guys, is anyone able to help me with this question?

Use Lagrange Multipliers to find expressions for K and L which maximize output given by the production function:

$\displaystyle Q=AK^aL^b$ (A,a,b are positive constants)

subject to a cost constraint

2. Originally Posted by wings1340
Hey guys, is anyone able to help me with this question?

Use Lagrange Multipliers to find expressions for K and L which maximize output given by the production function:

$\displaystyle Q=AK^aL^b$ (A,a,b are positive constants)

subject to a cost constraint
Are we to assume that $\displaystyle P_K$, $\displaystyle P_L$, and M are constant also? If so,
Let [/tex]F(K, L)= AK^aL^b[/tex] and $\displaystyle G(K, L)= P_KK+ P_LL$.

The max and/or min occure when $\displaystyle \nabla F= \nabla G$ for some number $\displaystyle \nabla$. Take the gradients of F and G, put them into that equation, write the equations given by the two components, and find all values K and L that satisfy it. (You do not need to find $\displaystyle \nabla$. I find that dividing one equation by another is a good strategy since it eliminates $\displaystyle \lambda$ and don't for get that $\displaystyle P_KK+ P_LL= M$ is another equation of K and L.)