Differentiated or uniform tax

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.