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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, $p>0$: 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?

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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 $z > z_0$, $p f(z) - w z < 0$. 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 $Z$.

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Intuitively, you'd want the profit function to "peak" at some finite vector $\mathbf z^*$. To ensure this, it's sufficient to require that

  • the profit function $\pi(\mathbf z)=pf(\mathbf z)-\mathbf w\cdot\mathbf z$ be concave in $\mathbf z$,

  • the production function $f$ be increasing and continuously differentiable in $\mathbf z$, and

  • the production function $f$ satisfy $$\lim_{z_i\to\infty}\frac{\partial f(\mathbf z)}{\partial z_i}=0$$ for each element $z_i$ in the vector $\mathbf z$.

Given a positive input price vector $\mathbf w$, these conditions guarantee a solution to the profit maximization problem.

More generally, you may also want to look at the Inada conditions, which are commonly cited in DSGE models.

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