Principal Agent Question Private Benefit If an agent can take action which impacts social welfare in a noisy way, the private net benefit of the action is $P(x,t)=x-s(\frac{x^2}{2} + t)$, $s$ is $(0,1)$
How can I determine the privately optimal t?
If we try to directly optimize with respect to $t$, given this system (two unknwowns, one equation), we get a nonsensical result.

$$\frac{\partial P}{\partial t}: -s = 0$$
Which may not be true, and I'm assuming you are taking $s$ as given, not something to choose. You can only optimize with respect to $x$.

$$\frac{\partial P}{\partial x} = 1 - sx = 0 \implies \boxed{x^* = \frac{1}{s}}$$

What does this tell you about the optimal value of $t$ if anything?