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
1. Introduction
2. Classical static model of tax competition
3. Dynamic differential game with an openloop strategy
4. Dynamic differential game with a feedback strategy
5. 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 feedback 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.
Thank you