# Thread: help with lagrange in economics

1. ## help with lagrange in economics

I have a profit function = -100 + 80a - 0.1a^2 + 100b - 0.2b^2

the question states relationg to the output of two products if total production (a + b = 500) find the solution that will maximize profit. solve the lagrange multiplier also.

so i have reworked the formula to show = -100 + 80a - 0.1a^2 + 100b - 0.2b^2 - λ(500 - a - b)
fa = 80 -0.2a - λ
fb = 100 -0.4b - λ
fλ = -500 + a + b

i know i have to set fa, fb and fλ to 0 to solve the maximization point so im just not sure how to go from here using substitution to solve each p.derivative.

if someone could help would be great

2. Hello

You have the function:

$\displaystyle f(a,b,\lambda)=-100+80a-0.1a^2+100b-0.2b^2-\lambda(500-a-b)$

then, consider the equations:

$\displaystyle 0=\dfrac{\partial f}{\partial a}=80-0.2a+\lambda$

$\displaystyle 0=\dfrac{\partial f}{\partial a}=100-0.4b+\lambda$

$\displaystyle 0=\dfrac{\partial f}{\partial a}=a+b-500$

From the first and second relation:

$\displaystyle a=400+5\lambda$
$\displaystyle b=250+\dfrac{5\lambda}{2}$

And substiting in the last one:

$\displaystyle 400+5\lambda+250+\dfrac{5\lambda}{2}=500$

Now, I suppose that you can conitnue...

Best regards.