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 skilledtype) can only work in sector L. Both types derive utility from income and suffer nodisutility from work.
The government is considering levying a system of sector-specific lump-sumtaxes/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)?

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