Maximization with a binding boundary constraint

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?

• While the question is not off-topic for Economics SE, you may be more likely to get a definitive answer on Mathematics SE. Commented Nov 17, 2021 at 20:52
• I have also asked there but no response. Thanks @Adam Bailey Commented Nov 18, 2021 at 5:27

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.
without taking the constraint into account that would be $$max_x~profit={mx^{2a}-rx~~}$$ FOC $$\frac{d profit}{d x} =2amx^{2a-1} - r = 0$$ $$x = (\frac{r}{2am})^{\frac{1}{2a-1}}$$
• 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. Commented Nov 17, 2021 at 11:44