Maximized tax revenue & foreign labor

Question

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 $\eta>0$, $c$ is consumption and $l$ is leisure.

The danish government imposes a tax on the person, $t, 0<t<1$ 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:

$$(3) \ \ \bar{u}>\frac{1}{\eta+1}\left(\frac{\eta}{1+\eta}\frac{\bar{w}}{p}\right)^{\eta+1}$$

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?

Answer 1

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

$$\left(\bar w \frac{\eta}{\eta + 1}\right)^{\eta +1}[1/(\eta + 1)]$$