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How do I optimise this type of problem? I have read something about applying the chain rule to find the first order condition but can't imagine how to express it or solve it.

Can somebody give me a hint?

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As you requested, I will just give you a hint and let you proceed from there.

The agent's choice variable is the number of working hours $w$ that he provides in a day. If he works $w$ hours,

  • he earns $w$ euros and therefore consumes a quantity $Y=w$ of goods (there is no savings in your problem, so all income is spent on consumption).
  • he has $L=24-w$ hours of leisure.

Thus, how can you write the agent's total utility from working $w$ hours?

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  • $\begingroup$ I'm going to need more than that I think :) $\endgroup$ – itsjpv Mar 23 '17 at 10:41
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    $\begingroup$ @itsjpv You need to show some effort first, we are not a homework-doing machine. Please read our policy on homework questions and our general FAQ. $\endgroup$ – Oliv Mar 23 '17 at 10:57

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