Maximization with a binding boundary constraint

Question

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?

Answer 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 $$\frac{d profit}{d x} =2amx^{2a-1} - r = 0 $$ $$x = (\frac{r}{2am})^{\frac{1}{2a-1}} $$