# Max of the tax provenue [duplicate]

I have this utility function: $$u(c,l)=c-\frac{\eta}{\eta+1}(24-l)^{\frac{\eta+1}{\eta}}$$ where c is consumption and l is sparetime. Then I also think we must have the condition: $$pc+wl=24w$$ where p is the price and w is the wage. An income tax is then introduced t, 0<t<1 so the the consumer's hourly wage is $$w(1-t)$$.

Then I have to find the highest tax revenue the state can get is?

I think I will use Lagrange. I have proof that I got monotonic convex preferences. Then if I use condition together with the utility function I think I get the Lagrange problem: $$L=max_l \frac{24(w(1-t))}{p}-c-\frac{(w(1-t))l}{p}+(c-\frac{\eta}{\eta+1}(24-l)^{\frac{\eta+1}{\eta}})$$ Then I get the first-order condition: $$\frac{\partial L}{\partial l}=0 <=> -\frac{w\cdot (1-t)}{p}+(24-l)^{\frac{1}{\eta}}=0$$ But I can't see how I can find the highest tax revenue that the state can get. If I isolate t is it probably the tax rate I find, or what? Can someone help me?

• First solve consumer problem to figure out labour supply for a given tax $t$. This implies finding $L^\star = 24 - l^\star$ expressing hours worked. Since you now have $l ^\star$ which is a function of the $t$ you get something like tax revenue is $T = t \bar w (24 - l^\star(t))$. Then diff with respect to $t$ and solve FOC. – Jesper Hybel Dec 10 '20 at 22:14
• See this post for how to solve for $l^\star$. economics.stackexchange.com/questions/41454/… – Jesper Hybel Dec 10 '20 at 22:15