Maximized tax revenue & foreign labor I'm dealing with a tricky assignment, and I have no idea of where to begin.
Person 1 lives in Denmark and has a utility function given by,
$$(1) \ \ u(c,l)=c-\frac{\eta}{\eta+1}(24-l)^{\frac{\eta+1}{\eta}}$$
Where $\eta&gt;0$, $c$ is consumption and $l$ is leisure.
The danish government imposes a tax on the person, $t, 0&lt;t&lt;1$ such that his 'post-tax' wage is given by $w=\bar{w}(1-t)$. Tax revenue is given by
$$(2) \ \ T=t\cdot \bar{w}\cdot S(w)$$
where $S(w)$ is the supply of labor and $\bar{w}$ is wage.
First I derived the tax that yields the most tax revenue, $t^*=\frac{1}{(1+\eta)}$.
Here comes the problem.
Suppose that a second person (Person 2) with the same utility function is living in Wales. She is only willing to move to Denmark (for example when a big firm want's to import foreign labor because of qualifications) only when she can get a utility (after tax) that is higher than $\bar{u}$. Suppose that:
$$(3) \ \ \bar{u}&gt;\frac{1}{\eta+1}\left(\frac{\eta}{1+\eta}\frac{\bar{w}}{p}\right)^{\eta+1}$$
I have to show that the tax that is maximizing tax-revenue ($t^*$) is 'too high' for Person 2 to move to Denmark.
I get that I have to show that Person 2's utility is higher when she lives in Wales, than if she lived in Denmark - but how?
I give here the procedure but with price $p=1$ you can do it yourself without that simplification.
Set up Lagrange
$$\mathcal L(c,l,\lambda) = u(c,l) - \lambda (c-\bar w(1-t)(24-l)) $$
clearly
$$\frac{\partial \mathcal L }{\partial c} = \frac{\partial u}{\partial c} - \lambda = 1- \lambda$$
so $\lambda=1$ and constraint is binding. Therefore
$$c^\star = \bar w(1-t)(24-l^\star)$$
Furthermore
$$\frac{\partial L}{\partial l} = (24-l)^{1/\eta} - \lambda \bar w (1-t) = 0$$
since $\lambda = 1$ it follows that
$$l^\star = 24 - (\bar w(1-t))^\eta$$
plugging $l^\star$ and $c^\star$ into utility function, starting with $c^\star$ you get
$$\bar w(1-t)(24 - l^\star) - \frac{\eta}{\eta + 1}(24 - l^\star)^{\frac{\eta + 1}{\eta}}$$
then $l^\star$ to get
$$\bar w(1-t)[\bar w(1-t)]^{\eta} -  \frac{\eta}{\eta + 1}(\bar w(1-t))^{\frac{\eta(\eta + 1)}{\eta}},$$
reduce this to get
$$(\bar w (1-t))^{\eta +1}[1/(\eta + 1)]$$
insert optimal tax and get
$$\left(\bar w \frac{\eta}{\eta + 1}\right)^{\eta +1}[1/(\eta + 1)]$$