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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?

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    $\begingroup$ Are you just asking how to maximise that function? What do you know about maximising the value of a function? $\endgroup$ – EnergyNumbers Nov 24 '15 at 9:19
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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?

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  • $\begingroup$ I would appreciate any feedback on improving my answer, if anyone has any. $\endgroup$ – Kitsune Cavalry Dec 25 '15 at 2:28
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    $\begingroup$ I have just seen your comment. The downvote is mine. To be clear: I am not saying the answer is wrong. I cannot remember my particular mindset but I am guessing I strongly disagreed with giving an answer to this question. E.g. recently there has been an uptick of trivial homework questions which are burying some pretty decent questions. I may be out of line here, and I would like you to know I bear no ill will and I am sure you act with the best intentions. I just disagree as to what the results of your method are. Feel free to downvote me somewhere :) $\endgroup$ – Giskard Apr 22 '16 at 22:57
  • $\begingroup$ @denesp Oh denesp, I would not be so petty and downvote you just for this. ;) I'm not here to encourage trivial questions, but I do feel like I can help a little. Also to be fair, I left out the entire second half of this solution (because yeah, it is a homework question of sorts. I guess we do disagree on what's appropriate though. No hard feelings though, thanks for your feedback~ $\endgroup$ – Kitsune Cavalry Apr 23 '16 at 16:52

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