I have the following problem $$\underset{x_s, x_r}{Max} \space x_r(1-q)(m-1)+x_sq(m-1)\\subject \space to:x_r(1-q)+x_sq\leq\alpha$$.
Where $x_s,x_r\in[0,1]$, $q\in(0,1)$ and $m>1. $ I find the following solution: $x_r^*=\frac{\alpha-x_s^*q}{1-q}$ and $x_s^*=\frac{\alpha-x_r^*(1-q)}{q}.$ The constraint, naturally, binds. Is this a correct characterization? Thank you.