1. 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.

2. 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

3. 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).

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

5. 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).

6. 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.

7. 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.

8. Second stage
Production function
Arbitrage condition

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

10. Open economy
Private consumption
Government’s objective function
FOC

11. Closed economy
Total production
FOC

12. Reaction functions
Government’s objective function
Nash equilibrium

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

14. Particular case, closed economy
Taxation level

15. 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.

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

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

18. Open-loop Nash equilibrium

19. Open-loop Nash equilibrium

20. Particular form
Open-loop Nash equilibrium
Letting
Letting

21. 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).

22. Feed-back Nash equilibrium

23. Feed-back Nash equilibrium

24. Feed-back Nash equilibrium
Letting
Letting

25. 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.

26. Thank you