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I know that a profit function is defined by $\pi(p;w)=\max_{x\in\mathbb{R}}pf(x)-w\cdot x$ and $\pi$ is a convex function (i.e. $\pi$ is convex in $(p, w)$. My concern is how we can find the max value of the profit function when the profit function itself is convex.

From my previous knowledge on convex function, I also know that a convex function has a global minimum on its domain (correct me if I was wrong), but how can we find the maximum?

Can anyone please explain this for me? I am so confused. Thanks in advanced!

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You're probably confusing two things: the objective function $pf(x) - wx$ being used in profit maximization problem: $\max pf(x) - wx$, the objective function is concave in $x$ and thus, you can solve the maximization problem. However, a profit function $\pi(p,w)$ being a result of maximization problem identifies the maximum profit given level of prices and wages $\pi^*(p,w)$ (and it's convex in $p$ and $w$ as you said).

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