Problem 5: Optimal Taxation with Migration

Problem 5: Optimal Taxation with Migration
Consider an economy with two sectors H (high-tech) and L (low-tech) and two
individuals 1 and 2. Workers in sector H earn an income level of 10 whereas those
working in sector L earn an income level of 6 (labor supply is assumed to be inelastic).
Type 2 (the high skilled guy) can work in both sectors whereas type 1 (the low skilled
type) can only work in sector L. Both types derive utility from income and suffer no
disutility from work.
The government is considering levying a system of sector-specific lump-sum
taxes/transfers on redistributive grounds.
Let TH and TL denote, respectively, the taxes (transfers, if negative) levied on sectors H
and L (the after-tax income in sector H is thus given by 10- TH, whereas that in sector L
is given by 6- TL). The government is seeking to maximize the utility of the least welloff guy, type 1, and is not allowed to run into a fiscal deficit.
a) Formulate the government constrained optimization problem. You should specify
the objective function, the revenue constraint and the incentive constraint associated
with type 2.
b) Solve for the optimal lump-sum taxes/transfers. [Hint: you first need to show which
constraints are binding (satisfied as equality) in the optimal solution for the
government problem].
c) Now suppose that type 2 can migrate to another country where her (after-tax) utility
level would be given by 0<V<10. What would be the modified optimum in the
presence of migration opportunities for type 2 (Hint: incorporate the migration
decision into the incentive constraint associated with type 2)?