<p>It could be good to provide the meaning of the parameters that you are using, this may help in building intuition and for verifying if the problem is well formulated.</p>
<p>You need to check how you defined the profit function
what, m, p, q, x stand for?
is <span class="math-container">$ {mx^{2a}}$</span> total revenue? if <span class="math-container">$x$</span> is output, then what <span class="math-container">$q$</span> is? it is very likely that the constrained is not well formulated.</p>
<p>If you provide the info above, there is a high chance that you will get better solutions.</p>
<p>without taking the constraint into account that would be
<span class="math-container">$$max_x~profit={mx^{2a}-rx~~} $$</span>
FOC
<span class="math-container">$$\frac{d profit}{d x} =2amx^{2a-1} - r = 0 $$</span>
<span class="math-container">$$x = (\frac{r}{2am})^{\frac{1}{2a-1}} $$</span></p>

CC BY-SA 4.0
2021-11-17T11:44:57.037Z
p and q are lower and upper bounds on profit. Profit cannot exceed these intervals. $mx^{2a}$ is revenue and r is per unit cost of x. x is the quantity choice.
Elina Gilbert
CC BY-SA 4.0
2021-11-17T10:36:12.547Z
2021-11-17T11:24:07.110Z
2021-11-17T11:24:07.110Z
Macosso
<p>I have following profit function -</p>
<p><span class="math-container">$$ max_{x} ~ mx^{2a} - rx $$</span>
<span class="math-container">$\text{Subject to,}$</span></p>
<p><span class="math-container">$$ p \geq mx^{2a} - rx \geq q $$</span></p>
<p>Where, <span class="math-container">$ m&gt;r, p&gt;0, q&gt;0$</span> and <span class="math-container">$a&lt; \frac{1}{2}$</span></p>
<p>Since firm always want to choose higher enough profit, upper boundary constraint will be binding.</p>
<p>So, the maximization problem becomes -
<span class="math-container">$$ max_{x} ~ mx^{2a} - rx $$</span>
<span class="math-container">$\text{Subject to,}$</span>
<span class="math-container">$$ mx^{2a} - rx = p $$</span></p>
<p>How can I solve for optimal value of <span class="math-container">$x$</span>?</p>
<p>My working -
<span class="math-container">$$  mx^{2a} - rx = 0 \Leftrightarrow x= \left(\frac{m}{r} \right)^{\frac{1}{1-2a}}$$</span>
Since <span class="math-container">$p&gt;0$</span>
So, <span class="math-container">$x&gt; \left(\frac{m}{r} \right)^{\frac{1}{1-2a}}$</span></p>
<p>My question is - is there a way to solve for exact expression of x?</p>

CC BY-SA 4.0
2021-11-17T20:52:58.640Z
While the question is not off-topic for Economics SE, you may be more likely to get a definitive answer on Mathematics SE.
Adam Bailey
CC BY-SA 4.0
2021-11-18T05:27:13.520Z
I have also asked there but no response. Thanks @Adam Bailey
Elina Gilbert
<optimization><profit-maximization>
Maximization with a binding boundary constraint
Elina Gilbert
# Maximization with a binding boundary constraint

(Asked by: username_0 on 2021-11-17)

I have following profit function -

$$ max\_{x} ~ mx^{2a} - rx $$
$\text{Subject to,}$

$$ p \geq mx^{2a} - rx \geq q $$

Where, $ m>r, p>0, q>0$ and $a< \frac{1}{2}$

Since firm always want to choose higher enough profit, upper boundary constraint will be binding.

So, the maximization problem becomes -
$$ max\_{x} ~ mx^{2a} - rx $$
$\text{Subject to,}$
$$ mx^{2a} - rx = p $$

How can I solve for optimal value of $x$?

My working -
$$ mx^{2a} - rx = 0 \Leftrightarrow x= \left(\frac{m}{r} \right)^{\frac{1}{1-2a}}$$
Since $p>0$
So, $x> \left(\frac{m}{r} \right)^{\frac{1}{1-2a}}$

My question is - is there a way to solve for exact expression of x?

**Tags:** optimization, profit-maximization
> **Comments:**
> **username_1**: While the question is not off-topic for Economics SE, you may be more likely to get a definitive answer on Mathematics SE.  
> **username_0**: I have also asked there but no response. Thanks @Adam Bailey  

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## 1 answers

### Answer by username_2 on 2021-11-17. Score: 1

It could be good to provide the meaning of the parameters that you are using, this may help in building intuition and for verifying if the problem is well formulated.

You need to check how you defined the profit function
what, m, p, q, x stand for?
is $ {mx^{2a}}$ total revenue? if $x$ is output, then what $q$ is? it is very likely that the constrained is not well formulated.

If you provide the info above, there is a high chance that you will get better solutions.

without taking the constraint into account that would be
$$max\_x~profit={mx^{2a}-rx~~} $$
FOC
$$\frac{d profit}{d x} =2amx^{2a-1} - r = 0 $$
$$x = (\frac{r}{2am})^{\frac{1}{2a-1}} $$

> **Comments:**
> **username_0**: p and q are lower and upper bounds on profit. Profit cannot exceed these intervals. $mx^{2a}$ is revenue and r is per unit cost of x. x is the quantity choice.  

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