Suppose $x = (x_1, x_2)$ or some longer real vector, $q(x)$ is a concave function, and $~p_q,~ p_1 ~\text{and}~~ p_2~$ are real positive constants.
If $~x^*~$ is a unique solution to the optimization problem:
$$\max_{\text{over}~ x} ~\left\{ p_q * q(x) - (p_1 * x_1 + p_2 * x_2)\right\}~,$$
is it also necessarily a solution to:
$$\max_{\text{over}~ x} ~ \frac{1}{x_1} ~\left\{p_q * q(x) - (p_1 * x_1 + p_2 * x_2)\right\}~~?$$ Alternatively, is it demonstrably not a solution (except perhaps in special cases)? If the latter, can we characterize the solution relative to $~x^*~$ (perhaps in terms of their respective first order conditions)?
Just for a motivating example, this is a modified version of the usual form of the producer's (i.e. the firm's) optimization problem. It is intended to capture the difference between competitive firms run by independent entrepreneurial executives -- that is the standard economic assumption -- and firms run by effectively-controlled agents of the stockholders. My thought is that such agents would wish to maximise profits per unit of capital, rather than profits per se. This feels to me like a minor extension of standard theory, but I keep getting nonsensical answers.
With $q(x) = 1-x_2^2$ and $p_1 = p_2 = 0$ and $p_q = 1$, the maximum of the first problem is finite ($1$) while the second problem is unbounded (simply let $x_1$ tend to $0$ from above, so the answer is no.