<p>If we try to directly optimize with respect to $t$, given this system (two unknwowns, one equation), we get a nonsensical result.</p>

<p>$$\frac{\partial P}{\partial t}: -s = 0$$
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 $x$.</p>

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

<p>What does this tell you about the optimal value of $t$ if anything?</p>

CC BY-SA 3.0
2015-12-25T02:28:46.987Z
I would appreciate any feedback on improving my answer, if anyone has any.
Kitsune Cavalry
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2016-04-22T22:57:25.577Z
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 :)
Giskard
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2016-04-23T16:52:42.207Z
@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~
Kitsune Cavalry
CC BY-SA 3.0
2015-11-24T20:12:23.150Z
2015-11-24T20:12:23.150Z
Kitsune Cavalry
<p>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?</p>

CC BY-SA 3.0
2015-11-24T09:19:27.467Z
Are you just asking how to maximise that function? What do you know about maximising the value of a function?
410 gone
<microeconomics><principal-agent>
Principal Agent Question Private Benefit
SMK
# Principal Agent Question Private Benefit

(Asked by: username_0 on 2015-11-24)

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?

**Tags:** microeconomics, principal-agent

> **Comments:**
> **username_1**: Are you just asking how to maximise that function? What do you know about maximising the value of a function?  

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## 1 answers

### Answer by username_2 on 2015-11-24. Score: 2

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?

> **Comments:**
> **username_2**: I would appreciate any feedback on improving my answer, if anyone has any.  
> **username_3**: 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 :)  
> **username_2**: @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~  

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