# 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.

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.

• The RHS of the equation is the indirect utility achieved by consumer when the tax is $t^\star$ the optimal tax - Notice that this is also the utility of the person from Wales were she to move to Denmark. For some reason the person in Wales wants a higher utility before being willing to move. The indirect utility function is decreasing in taxrate so to increase indirect utility for person in Wales you would necessarily have to offer the person a lower taxrate than $t^\star$. – Jesper Hybel Dec 11 '20 at 18:01
• I see! Thanks. How would you draw Person 2's Laffer curve? My suggestion would be that they have the same curve, since they offer the same work, but Person 2's would be cut off somwhere before the optimal tax. – Matt Dec 14 '20 at 10:17
• Person 1's Laffer curve would range from 0 to 1 and have its max at the optimal point. – Matt Dec 14 '20 at 10:24
• That sounds about right (although I think it is strange to talk about the Laffer curve for a single person, but I understand what you mean and agree). Yes, for person two tax revenue just drops to 0 exactly at optimal tax. – Jesper Hybel Dec 14 '20 at 11:26
• I'm also supposed to draw the sum of the two person's Laffer curves, so I just wanted to make sure that I was right about the individual. How would you draw for them both in one graph if one is just on top of the other up until a point? – Matt Dec 14 '20 at 13:49