Part 1 - AL-Tax

Chapter 4Definition 1. The tax policies of the two governments, T A (.) and T B (.), are a Nashequilibrium if and only if:∀∀≠ i , j iT , i(). ⎛⎛ ⎛ I ⎞⎞iargmaxW U 1⎜1Ii1 Ti( Ii1)ν ⎝⎜⎜⎝ω 1⎠⎟⎠⎟ ,U ⎛I T I ⎛I2 i2 i( i2)⎜⎝⎜⎝⎜⎝⎜i2νω2⎞⎞⎞⎠⎟⎠⎟⎠⎟under the budget constraint:p1 . P⎢1 Ti(.), Tj(.) ⎥ Ti( Ii1) p P⎢2 2 Ti(.), Tj(.) ⎥⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ T i( I i2) 0,where for k 1,2:Iikargmax⎡⎛ I I Ti() I ν I⎝⎜⎣⎢ωk⎞⎤⎠⎟.⎦⎥We adopt the Rothschild–Stiglitz–Nash concept of equilibrium. This could becontroversial. Since budget constraint depends on the proportions of both kindsof workers, after migration from one country to the other due to a change in thefiscal policy of one country, the budget constraint is no longer balanced. We keepthis concept of equilibrium for two main reasons. First, there is no consensus onan alternative definition of equilibrium. By choosing another concept of equilibrium,we allow the agents to have a behavior inconsistent with the usual assumption(agents are Nash players) made in the economic literature. Second, as our modelis static, we cannot introduce explicitly public debt. Thus, to keep the model simpleand consistent with previous models, we will make a sharp assumption. Wewill consider that if a government chooses a taxation policy that leads to a publicdeficit, given the policy chosen by the other government, its policy is enforced atthe cost of an infinite welfare loss. Given this last assumption, payoffs are welldefined, and one can apply the Nash concept of equilibrium.We solve this game by using the ‘revelation principle’, i.e. we assume thatthere is no restriction to consider that both governments offer truthful mechanisms.In a more general setting, if competition between two principals is takeninto account there is some loss of generality in restricting the analysis to this setof mechanisms. In our context, the agent cannot deal simultaneously with bothprincipals. An agent works and pays taxes in country A or in country B. In this81