The existence of solution for profit maximization problem I'm thinking about the conditions for existence of solution of this profit maximization problem(PMP), i.e.,
$\max_{z \in R_+^{K-1}} pf(z) -wz$,
where $z \geq 0$: input vector, $p&gt;0$: the price of output, $w \gg 0$: an input price vector, and $f:R_+^{K-1} \rightarrow R_+ $: the production function.
Of course, if production set $Y$ is compact, by Weierstrass theorem, we can prove there exists a solution of this PMP. But many cases, $Y$ is closed but not bounded. Then what kind of assumptions on function $f$ are needed to show the existence of a solution, instead of Weierstrass theorem?
A possible approach is to find a compact set $Z$ of inputs and show that the PMP has an optimal solution if and only if the PMP has an optimal solution in $Z$.
If so, we can replace the PMP by the following problem.
$$max_{z \in Z} \,\,p f(z) - w z.$$
If $f$ is continuous and if $Z$ is compact, the existence of a solution follows from the Weierstrass theorem.
An example of a sufficient condition for $Z$ to exist is to assume that $f(0) = 0$ and that there exists an input level $z_0$ such that for all $z &gt; z_0$, $p f(z) - w z &lt; 0$. In words, there is an input level $z_0$ such that having a higher input level will generate negative profits. Then we can set
$$
Z = \{z \in \mathbb{R}^{K-1}_+: z \le z_0\}.
$$
Notice that $Z$ is compact. In order for this to work we need to show that the PMP has an optimal solution if and only if it has an optimal solution in $Z$.
To see that this is true, first notice that $z = 0$ is a feasible solution for the PMP and it is also in $Z$. As such, the optimal solution to the PMP will always generate a profit that is larger or equal to zero, which means that no solution $z$ to the PMP will be outside the set $Z$.