<p>Disclaimer: I'm assuming you are integrating over <span class="math-container">$x$</span>:</p>

<p>I have a problem with your first inequality. Since <span class="math-container">$y(x)$</span> and <span class="math-container">$z(x)$</span> are monotonically increasing (strictly) with respect to <span class="math-container">$x$</span>, there cannot exists an <span class="math-container">$x^*$</span> such that <span class="math-container">$y(x^*)+z(x^*)&gt;y(x)+z(x)$</span> for all <span class="math-container">$x\geq 0$</span>. So the  statement is vacuously true.</p>

<p>The second observation seems correct to me.</p>

<p>In general, you could use the <a href="https://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow noreferrer">Leibniz rule</a> to study the FOC's assuming the solution is interior. Note that your first inequality is not necessary for an interior solution. So the fact that it cannot occur does not mean there cannot be an interior solution.</p>

<p>Note as well that even if <span class="math-container">$y(x)&gt;\bar z$</span> for some values of <span class="math-container">$x$</span>, the solution can still be at <span class="math-container">$x=\infty$</span> (assuming you can choose such a value of <span class="math-container">$x$</span>, of course).</p>

CC BY-SA 4.0
2019-05-23T19:06:21.107Z
2019-05-23T19:27:51.510Z
2019-05-23T19:27:51.510Z
Regio
<p>I would like to know if my understanding about how to find a maximum of the function when some parameters are unknown is correct.</p>

<p>Consider the following maximization problem.</p>

<p><span class="math-container">$\max_{x}V=\int_0^{a(x)}y(x)dF(t)+\int_{a(x)}^{\infty}z(x)dF(t)$</span></p>

<p>where <span class="math-container">$x\geq0$</span>, <span class="math-container">$y'(x)&gt;0$</span>, and <span class="math-container">$z'(x)&gt;0.$</span> If <span class="math-container">$x\to\infty$</span>, <span class="math-container">$z(x)\to \overline{z}$</span>. <span class="math-container">$F$</span> is the cumulative distribution function of <span class="math-container">$t$</span>.</p>

<p>The function <span class="math-container">$a(x)$</span> is non-monotonic in <span class="math-container">$x$</span>:</p>

<p>if <span class="math-container">$x=0$</span>, <span class="math-container">$a(x)=0$</span>,</p>

<p>if <span class="math-container">$0&lt;x&lt;\infty$</span>, <span class="math-container">$a(x)&gt;0$</span>,</p>

<p>and if <span class="math-container">$x\to\infty$</span>, <span class="math-container">$a(x)=0$</span>.</p>

<p>In this maximization problem, if we increase <span class="math-container">$x$</span>, the first term in the right side first increases and then decreases to zero because of the nonmonotonicity of <span class="math-container">$a(x)$</span>.</p>

<p>If <span class="math-container">$y(x^*)+z(x^*)&gt; y(x) +z(x)$</span> for any <span class="math-container">$x$</span>, the objective function <span class="math-container">$V$</span> is maximized at <span class="math-container">$x=x^*$</span>, and the maximum value is <span class="math-container">$\int_0^{a(x^*)}y(x^*)dF(t)+\int_{a(x^*)}^{\infty}z(x^*)dF(t)$</span>.</p>

<p>If <span class="math-container">$y(x)+z(x)&lt;\overline{z}$</span> for any <span class="math-container">$x$</span>, the objective function <span class="math-container">$V$</span> is maximized at <span class="math-container">$x=\infty$</span>, and the maximum value is <span class="math-container">$\int_{0}^{\infty}\overline{z}dF(t)$</span>. The first term in the right side is zero.</p>

<p>I would appreciate it if someone could tell me if my understanding is correct. If it is incorrect, where did I make a mistake?</p>

CC BY-SA 4.0
2019-05-23T18:54:26.843Z
There seems to be a problem with your variable $t$ of integration which does not appear as argument of $y$ and $z$.
Bertrand
CC BY-SA 4.0
2019-05-23T19:00:19.143Z
did you mean to say $dF(x)$?
Regio
<microeconomics><mathematical-economics>
Maximization when parameters are unknown
Tom M.
# Maximization when parameters are unknown

(Asked by: username_0 on 2019-05-23)

I would like to know if my understanding about how to find a maximum of the function when some parameters are unknown is correct.

Consider the following maximization problem.

$\max\_{x}V=\int\_0^{a(x)}y(x)dF(t)+\int\_{a(x)}^{\infty}z(x)dF(t)$

where $x\geq0$, $y'(x)>0$, and $z'(x)>0.$ If $x\to\infty$, $z(x)\to \overline{z}$. $F$ is the cumulative distribution function of $t$.

The function $a(x)$ is non-monotonic in $x$:

if $x=0$, $a(x)=0$,

if $0, $a(x)>0$,

and if $x\to\infty$, $a(x)=0$.

In this maximization problem, if we increase $x$, the first term in the right side first increases and then decreases to zero because of the nonmonotonicity of $a(x)$.

If $y(x^\*)+z(x^\*)> y(x) +z(x)$ for any $x$, the objective function $V$ is maximized at $x=x^\*$, and the maximum value is $\int\_0^{a(x^\*)}y(x^\*)dF(t)+\int\_{a(x^\*)}^{\infty}z(x^\*)dF(t)$.

If $y(x)+z(x)<\overline{z}$ for any $x$, the objective function $V$ is maximized at $x=\infty$, and the maximum value is $\int\_{0}^{\infty}\overline{z}dF(t)$. The first term in the right side is zero.

I would appreciate it if someone could tell me if my understanding is correct. If it is incorrect, where did I make a mistake?

**Tags:** microeconomics, mathematical-economics
> **Comments:**
> **username_1**: There seems to be a problem with your variable $t$ of integration which does not appear as argument of $y$ and $z$.  
> **username_2**: did you mean to say $dF(x)$?  

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## 1 answers

### Answer by username_2 on 2019-05-23. Score: 1 | Accepted answer

Disclaimer: I'm assuming you are integrating over $x$:

I have a problem with your first inequality. Since $y(x)$ and $z(x)$ are monotonically increasing (strictly) with respect to $x$, there cannot exists an $x^\*$ such that $y(x^\*)+z(x^\*)>y(x)+z(x)$ for all $x\geq 0$. So the statement is vacuously true.

The second observation seems correct to me.

In general, you could use the [Leibniz rule](https://en.wikipedia.org/wiki/Leibniz_integral_rule) to study the FOC's assuming the solution is interior. Note that your first inequality is not necessary for an interior solution. So the fact that it cannot occur does not mean there cannot be an interior solution.

Note as well that even if $y(x)>\bar z$ for some values of $x$, the solution can still be at $x=\infty$ (assuming you can choose such a value of $x$, of course).


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