For an essay required to complete my high school multivariable calc class I am wanting to write about an approach to an econometric problem (as I probably will study econometrics in uni). I really want to approach the problem using the Lagrange multipliers as I really like and understand well how the method works.
What I intend to do in my investigation is to create a formula in which you just input the constraint and all the constants after which it gives the optimum value for a variable. My best attempt in finding an econometric problem which could be approached using the Lagrange multipliers would be the Cobb-Douglas production function: Output = $A L^\alpha K^\beta$ with the constraint of $\alpha + \beta > 1$ (positive returns to scale). However as $\alpha$ and $\beta$ are constants and $L$ & $K$ are the variable this didn't work out because there was no possibility of $f(x,y)$ & $g(x,y)$.
Therefore I am wondering if any of you can help me with an economic/econometric problem that can be approached using the Lagrange multipliers to create a formula or something similar to maximise the variables!
Thanks in advance