Open-loop and feed-back equilibrium of a tax competition model Fernando M. M. Ruiz Catholic University of Mons This paper is part of Research Program IAP 6/09  ”Higher Education and Research” of the Belgian Federal Authorities.

Structure Introduction Classical static model of tax competition Dynamic differential game with an open-loop strategy Dynamic differential game with a feed-back strategy Conclusions

Classical conclusions of the tax competition literature: “Independent governments engage in wasteful competition for scarce capital through reductions in tax rates and public expenditure level” (Wilson, 1999).

Results: The tax equilibrium is found at a higher level in a feed-back model than in an open-loop or a static model.

What is an open-loop strategy? If countries use open-loop strategies they design their optimal policies as simple time functions independent of the current state of the system. These time paths are set at the beginning of the game and those actions cannot be changed once the system is running. There is a “precommitment” of the countries not to react to the policies of the competing country during the game, or the governments cannot observe the evolution of taxes in the alternative location (they can just observe the initial tax level).

What is a feed-back strategy? If countries use feed-back strategies they design their optimal policies as decision rules dependent on the state variables of the game. It implies that countries take into account the rivals reactions to their own actions. Countries know the exact state of the system at every point in time and use their control instruments to achieve their goals.

2. A classical static model of tax competition (Wildasin, 1988) The model can be interpreted as a two stage game: In the first stage, 2 countries simultaneously choose their tax rates. In the second stage, the capital owners decide where to invest given taxes.

Second stage Production function Arbitrage condition

First stage Budget constraint Fixed capital stock Equations (1), (2) and (3) implicitly define and

Open economy Private consumption Government’s objective function FOC

Closed economy Total production FOC

Reaction functions Government’s objective function Nash equilibrium

Particular case, open economy Quadratic production function Country 1’s reaction function Nash equilibrium

Particular case, closed economy Taxation level

3. Dynamic differential game Open-loop information structure: depends on the initial state of the system and time. The open-loop strategy relies on a « precommitment » among countries to pick a tax without any regard to the one chosen in the competing country during the game. The countries formulate their tax paths at the moment the system starts to evolve and those taxes cannot be changed once the system is running.

Open-loop Nash equilibrium Government’s objective function Budget constraint

Open-loop Nash equilibrium Budget constraint Equation of motion for the tax rate

Open-loop Nash equilibrium

Open-loop Nash equilibrium

Particular form Open-loop Nash equilibrium Letting Letting

4. Dynamic differential game A feed-back strategy is one which allows a player to choose his actions depending on the current value of the state variables (but cannot recall any of the previous values).

Feed-back Nash equilibrium

Feed-back Nash equilibrium

Feed-back Nash equilibrium Letting Letting

5. Conclusions The tax equilibrium is found at a higher level in a feed-back model than in an open-loop or a static model. When governments observe at every point in time the evolution of the capital tax at home and abroad, and can use the public good provision to control its motion through the budget constraint (feed-back strategy), they will note that if one country increases the public good, the capital tax must go up in the country and capital will move to an alternative location to equalize net returns. However, if that alternative location has time to react (feed-back strategy), it will increase public good provision (and therefore the tax rate) to align the marginal social benefit with the marginal social cost. This induces a capital outflow to the first country and the reaction may begin again.

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