Suppose Person 1 is working in Denmark and has the utility function,
$$u(c,l)=c-\frac{\eta}{\eta+1}(24-l)^{\frac{\eta+1}{\eta}}$$
His wage after tax is $w(1-t)$, where tax, t, $0<t<1$.
A second person, Person 2, lives in Wales and has the same utility function. Her utility, $\bar u$, that she has in Wales is greater than the utility Person 1 has in Denmark, where he is taxated with the optimal tax, $t^{\star}=\frac{1}{1+\eta}$.
$$\bar{u}>\frac{1}{\eta+1}\left(\frac{\eta}{1+\eta}\frac{\bar{w}}{p}\right)^{\eta+1}$$
Now, if Person 2 is offered a job in Denmark, she will only move if she is offered a taxation that yields a utility greater than $\bar u$. How would you go about showing that a differentiated tax in Denmark (where Person 1 and 2 is taxated at differented rates) is better than a uniform tax if the government want to maximize tax-revenue?
My intuition says that at a differentiated tax rate, the optimal tax rate for Person 2 is seemingly lower than the tax rate for Person 1 given the inequality. At a uniform tax rate the government would have to lower the income tax rate on Person 1, as increasing it for Person 2 would make her move back to Wales. The only way I can think of showing this is through a drawing - not very mathematical per se.
