Why is the derivative of this SWF non-positive?
This looks like a simple first order condition from constrained optimization. If the maximum is interior, i.e. if $t_i&gt;0$, then the first derivative must be zero. If the maximum is on the boundary, i.e. if $t_i=0$, then the first derivative must be $\leq 0$. If it were $&gt;0$, then we can't have an optimum, because then the value of $U$ could be raised by increasing $t_i$. It can be negative, though, assuming that $t_i$ cannot be negative.