<p>Intuitively, you'd want the profit function to &quot;peak&quot; at some finite vector <span class="math-container">$\mathbf z^*$</span>. To ensure this, it's sufficient to require that</p>
<ul>
<li><p>the profit function <span class="math-container">$\pi(\mathbf z)=pf(\mathbf z)-\mathbf w\cdot\mathbf z$</span> be concave in <span class="math-container">$\mathbf z$</span>,</p>
</li>
<li><p>the production function <span class="math-container">$f$</span> be increasing and continuously differentiable in <span class="math-container">$\mathbf z$</span>, and</p>
</li>
<li><p>the production function <span class="math-container">$f$</span> satisfy <span class="math-container">$$\lim_{z_i\to\infty}\frac{\partial f(\mathbf z)}{\partial z_i}=0$$</span> for each element <span class="math-container">$z_i$</span> in the vector <span class="math-container">$\mathbf z$</span>.</p>
</li>
</ul>
<p>Given a positive input price vector <span class="math-container">$\mathbf w$</span>, these conditions guarantee a solution to the profit maximization problem.</p>
<p>More generally, you may also want to look at the <a href="https://en.wikipedia.org/wiki/Inada_conditions" rel="nofollow noreferrer">Inada conditions</a>, which are commonly cited in DSGE models.</p>

CC BY-SA 4.0
2020-06-21T21:29:54.983Z
2020-06-21T21:29:54.983Z
Herr K.
<p>A possible approach is to find a compact set <span class="math-container">$Z$</span> of inputs and show that the PMP has an optimal solution if and only if the PMP has an optimal solution in <span class="math-container">$Z$</span>.</p>
<p>If so, we can replace the PMP by the following problem.
<span class="math-container">$$max_{z \in Z} \,\,p f(z) - w z.$$</span>
If <span class="math-container">$f$</span> is continuous and if <span class="math-container">$Z$</span> is compact, the existence of a solution follows from the Weierstrass theorem.</p>
<p>An example of a sufficient condition for <span class="math-container">$Z$</span> to exist is to assume that <span class="math-container">$f(0) = 0$</span> and that there exists an input level <span class="math-container">$z_0$</span> such that for all <span class="math-container">$z &gt; z_0$</span>, <span class="math-container">$p f(z) - w z &lt; 0$</span>. In words, there is an input level <span class="math-container">$z_0$</span> such that having a higher input level will generate negative profits. Then we can set
<span class="math-container">$$
Z = \{z \in \mathbb{R}^{K-1}_+: z \le z_0\}.
$$</span>
Notice that <span class="math-container">$Z$</span> 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 <span class="math-container">$Z$</span>.</p>
<p>To see that this is true, first notice that <span class="math-container">$z = 0$</span> is a feasible solution for the PMP and it is also in <span class="math-container">$Z$</span>. 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 <span class="math-container">$z$</span> to the PMP will be outside the set <span class="math-container">$Z$</span>.</p>

CC BY-SA 4.0
2020-06-22T06:03:12.977Z
2020-06-22T06:03:12.977Z
tdm
<p>I'm thinking about the conditions for existence of solution of this profit maximization problem(PMP), i.e.,</p>
<p><span class="math-container">$\max_{z \in R_+^{K-1}} pf(z) -wz$</span>,</p>
<p>where <span class="math-container">$z \geq 0$</span>: input vector, <span class="math-container">$p&gt;0$</span>: the price of output, <span class="math-container">$w \gg 0$</span>: an input price vector, and <span class="math-container">$f:R_+^{K-1} \rightarrow R_+ $</span>: the production function.</p>
<p>Of course, if production set <span class="math-container">$Y$</span> is compact, by Weierstrass theorem, we can prove there exists a solution of this PMP. But many cases, <span class="math-container">$Y$</span> is closed but not bounded. Then what kind of assumptions on function <span class="math-container">$f$</span> are needed to show the existence of a solution, instead of Weierstrass theorem?</p>

<microeconomics><producer-theory>
The existence of solution for profit maximization problem
Ajisai
# The existence of solution for profit maximization problem

(Asked by: username_0 on 2020-06-21)

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?

**Tags:** microeconomics, producer-theory

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## 2 answers

### Answer by username_2 on 2020-06-22. Score: 4 | Accepted answer

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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### Answer by username_1 on 2020-06-21. Score: 3

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](https://en.wikipedia.org/wiki/Inada_conditions), which are commonly cited in DSGE models.


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