# Given $f(x_1,x_2) = \min\{x_1,x_2\}^\alpha$ find the profit-maximizing factor demands, supply function and profit fuction

Given

$$f(x_1,x_2) = \min\{x_1,x_2\}^\alpha$$

I have to find the profit-maximizing demand functions, supply function and profit function but I am not sure how to when the function is given as it is.

I know that I for example have to use Lagrange. Thus

$$L = p \cdot f(x_1,x_2) - w_1 \cdot x_1 - w_2 \cdot x_2 - \lambda \left(p_1x_1 + p_2x_2 - m \right)$$ and then afterwards find the first order conditions. But how I find the derivative in regards to both $$x_1$$ and $$x_2$$ when I have a function such as $$f(x_1,x_2) = \min\{x_1,x_2\}^\alpha$$ ?

Can you help me in the right direction?

• And do you know that all profit maximization problems imply a cost minimization problem as well? If $x_1^*,x_2^*$ are the solution to the profit maximization problem, they are also to solution to the cost minimization problem assuming an output of $y = f(x_1^*,x_2^*)$. So if you can gleam further insights from the cost minimization problem, do so. – Giskard Mar 30 '20 at 13:22
• For cost minimization I know that in optimum we have that $$TRS = \frac{\partial y / \partial x_1}{\partial y / \partial x_2} = \frac{w_1}{w_2}$$ and because we keep a constant output given by $$f(x_1,x_2) = \min \{x_1,x_2\}^\alpha$$ we can solve this equation system. But how do I for example find $$\frac{ \partial y}{ \partial x_1}$$ where $$y = f(x_1,x_2)$$. Thanks in advance. – Mathias Mar 30 '20 at 13:57
• How about using the definition of differentiation? The solution is also quite intuitive, you can get it without differentiation. You can also look around the site, there are similar questions about the $\min$ function. It is also likely that you have covered something like this in class. Anyhow, this seems like the self study part, so I am just going to go. I hope your work will be fruitful. – Giskard Mar 30 '20 at 15:03