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

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.