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Ann Oper Res (2013) 206:341–366
DOI 10.1007/s10479-013-1379-3
Fiscal and monetary policy interactions: a game theory
approach
Helton Saulo ·Leandro C. Rêgo ·Jose A. Divino
Published online: 23 April 2013
© Springer Science+Business Media New York 2013
Abstract The interaction between fiscal and monetary policy is analyzed by means of a
game theory approach. The coordination between these two policies is essential, since de-
cisions taken by one institution may have disastrous effects on the other one, resulting in
welfare loss for the society. We derived optimal monetary and fiscal policies in context of
three coordination schemes: when each institution independently minimizes its welfare loss
as a Nash equilibrium of a normal form game; when an institution moves first and the other
follows, in a mechanism known as the Stackelberg solution; and, when institutions behave
cooperatively, seeking common goals. In the Brazilian case, a numerical exercise shows that
the smallest welfare loss is obtained under a Stackelberg solution which has the monetary
policy as leader and the fiscal policy as follower. Under the optimal policy, there is evidence
of a strong distaste for inflation by the Brazilian society.
Keywords Fiscal policy ·Monetary policy ·Nash equilibrium ·Stackelberg equilibrium ·
Cooperative solution
1 Introduction
The macroeconomic analysis has experienced wide changes in recent years. The rule-based
policymaking approach has taken the scene in both fiscal and monetary policies all over the
H. Saulo ()
Department of Economics, Federal University of Rio Grande do Sul, 90040-000, Porto Alegre, RS,
Brazil
e-mail: heltonsaulo@gmail.com
L.C. Rêgo
Department of Statistics, Federal University of Pernambuco, 50740-540, Recife, PE, Brazil
e-mail: leandro@de.ufpe.br
J.A. Divino
Department of Economics, Catholic University of Brasilia, 70790-160, Brasilia, DF, Brazil
e-mail: jangelo@ucb.br
342 Ann Oper Res (2013) 206:341–366
world. Woodford (2003), for instance, calls our attention to the adoption of interest rate rules
by the monetary policy aiming at inflation stabilization. The relevance of commitment with
explicit rules is strengthened by the adoption of inflation targeting regimes by many central
banks worldwide, including Central Bank of Brazil, Bank of England, Reserve Bank of
New Zealand, Swedish Riksbank, among others. However, independent movements by the
monetary authority might result in conflicting interests with the fiscal authority. To resolve
this dispute, one might make use of the game theory framework.
Our goal is to analyze the interaction between fiscal and monetary policy, taking into
account different policy regimes, resulting from alternative forms of interactions between
the policy authorities in a game theory environment. We allow for the policymakers to si-
multaneously set their instruments without cooperation as in a normal form game, engage in
a Stackelberg leadership scheme, and simultaneously set their instruments in a cooperative
game to pursue common objectives. Whenever possible, the models are solved analytically.
Otherwise, we provide numerical approximation for the solutions using the Brazilian econ-
omy as a reference.
Backus and Driffill (1985) and Tabellini (1985), for instance, used the theory of repeated
games to demonstrate that, under certain conditions, equilibria with low inflation could also
appear under discretionary policymaking.1The argument is that when the monetary author-
ity fights against inflation by relying on good reputation, it influences private-sector expecta-
tions on future inflation. Thus, even under discretion, the policymakers need to demonstrate
a certain amount of credibility. In this scenario there appeared, for instance, the Taylor rule
(Taylor 1993), establishing a reaction function for the nominal interest rate in response to
variations in inflation and the output gap. Later on, Taylor’s empirical rule was rationalized
by optimizing behavior of individuals and firms in a New Keynesian framework.
Coordination between fiscal and monetary policies has to take care of conflicting inter-
ests, since each policymaker is primarily concerned with his own objectives. In this scenario,
the induced economic spillovers and externalities become very important. This issue is ad-
dressed by Engwerda (1998), Engwerda et al. (1999,2002), who modeled dynamic games
among monetary and fiscal policymakers.
Dixit (2001) built several models of the EMU (European Monetary Union) and the Euro-
pean Central Bank (ECB) in order to analyze the interactions of some countries’ monetary
and fiscal policies. He found that the voting mechanism2achieved moderate and stable in-
flation. In case of a repeated game, the ECM should foresee eventual member drawback and
overcome this perturbation. Dixit (2001) emphasizes the dangerous role played by uncon-
strained national fiscal policies, which can undermine the ECB’s monetary policy commit-
ment.
Van Aarle et al. (2002) implemented a monetary and fiscal policy framework in the EMU
area according to the Engwerda et al. (2002) model. The idea was to study various interac-
tions, spillovers, and externalities involving macroeconomic policies under alternative pol-
icy regimes. Numerical examples were provided for some types of coalitions. It is interesting
to highlight that, in the simulations, full cooperation did not induce a Pareto improvement
for the ECB.
The traditional three-equation Taylor-rule New Keynesian model was extended by Kir-
sanova et al. (2005) to include the fiscal policy and analyze policy coordination. The idea
was to amend the set-up to a five-equation system in order to describe the role of the fiscal
1Macroeconomic policy characterized by absence of commitment.
2This is a mechanism of decision-making in the European Central Bank (ECB).
Ann Oper Res (2013) 206:341–366 343
policy, which might give feed back on the level of debt and helps the monetary authority to
stabilize inflation. Policy interactions were considered in three scenarios: (i) non-cooperative
policies, (ii) partially cooperative policies, and (iii) benevolent policies. The results sug-
gested that, if the authorities are benevolent and cooperative, the monetary authority will
bear all the burden of the stabilization. In addition, the Nash equilibrium will produce large
welfare losses when the monetary authority is benevolent and the fiscal authority discounts
too much the future or aims for an excessive level of output.
Lambertini and Rovelli (2003) also studied monetary and fiscal policy coordination using
a game theory approach. Particularly, they argue that each policy maker prefers to be the
follower in a Stackelberg situation. Moreover, when compared to the Nash equilibrium,
both Stackelberg solutions are preferable. Due to implementation issues, they also claim
that fiscal authorities would naturally behave as leaders in a strategic game with monetary
authorities. Favero (2004), on the other hand, shows that the strategic complementarity or
substitutability between fiscal and monetary policy might depend on the type of shock hitting
the economy. In addition, countercyclical fiscal policy might be welfare-reducing if fiscal
and monetary policy rules are inertial and not coordinated.
Our major contribution states that, in the Brazilian case, the monetary leadership under
the Stackelberg solution yields the smallest welfare loss for the society. The monetary lead-
ership might be associated to the existence of a monetary dominance in the Brazilian econ-
omy during the recent period, as empirically suggested by Tanner and Ramos (2002), Fialho
and Portugal (2005), and Gadelha and Divino (2008). Under the optimal policy, a sensitivity
analysis performed by varying the relative weights placed by themonetary and fiscal author-
ities on their target variables revealed a strong distaste for inflation by the Brazilian society.
In addition, impulse response functions indicated strong reactions of the monetary author-
ity to inflationary pressures. There is also an inflationary effect coming from fiscal shocks,
which contributes to reinforce the key role played by the monetary authority to stabilize the
economy.
The remainder of the paper is organized as follows. The next section details the baseline
macroeconomic model and discusses some elements of game theory. The third section in-
troduces the fiscal and monetary policy games. The fourth section discusses the numerical
approach used to approximate some of the solutions. The numerical results are presented
and analyzed in the fifth section. Finally, the sixth section is dedicated to the concluding
remarks.
2 The baseline model
The New Keynesian framework has been largely used to analyze optimal monetary and fis-
cal policy rules. The system of equations is a linear approximation, in logarithmic form, of
a dynamic stochastic general equilibrium (DSGE) model with sticky prices. The DSGE ap-
proach attempts to explain aggregate economic fluctuations, such as economic growth and
the effects of monetary and fiscal policies, on the basis of macroeconomic models derived
from microeconomic principles. The model is forward-looking and consists of an aggregate
supply equation, also known as the New Keynesian Phillips curve, an aggregate demand
equation, also called the IS curve. Additionally, there is an intertemporal budget constraint,
with which the government should comply, and the optimal monetary and fiscal policy rules.
These two policy rules will be derived later on. The aggregate demand function, represented
by the intertemporal IS curve, results from the first-order conditions of the individual’s op-
timization problem. The IS curve can be modeled taking into account the primary deficit,
344 Ann Oper Res (2013) 206:341–366
as in Nordhaus (1994), the public debt as in Kirsanova et al. (2005) and Bénassy (2007),
or even the level of government expenditures as in Muscatelli et al. (2004). In this paper,
we amend the IS curve proposed by Woodford (2003) in order to capture the effects of the
public debt on aggregate demand. Thus, the set-up considers the following closed economy
IS curve in log-linearized form3
ˆxt=Etˆxt+1−σ(ˆ
it−Etπt+1)+αˆ
bt+ˆrn
t,(1)
where ˆxt=(
Yt−
Yn
t)is the output gap (difference between actual and potential output), ˆ
it
is the nominal interest rate, ˆrn
tis a demand shock, Etrepresents the time texpected value
of the next period inflation rate πt+1and output gap ˆxt+1,btis the real stock of government
debt, σ>0 is the intertemporal elasticity of substitution in private spending, and αmeasures
the sensitivity of the output gap with respect to the debt. Notice that the aggregate demand
relationship depends also on future expected values and not just current ones.
On the aggregate supply curve (Phillips curve), firms face a decision to choose a price
that solves their profit maximization problem. The assumption of price rigidity (Calvo 1983),
according to which a fraction 0 <ϑ<1 of prices remains fixed during each period, allows
the derivation of the following (log-linearized) aggregate supply:
πt=κˆxt+βEtπt+1+νt,(2)
where the current inflation rate (πt)depends on the expected Etinflation rate at t+1, and
the current output gap ˆx. We allow a supply shock νt, as in Woodford (2003), to have a
trade-off between inflation versus output gap stabilization. The parameter κ>0 measures
the sensitivity of inflation with respect to the output gap and β,where0<β<1isthe
intertemporal discount factor.
The debt in Eq. (1) also needs to be modeled. Here, the real stock of debt ˆ
btis treated
as in Kirsanova et al. (2005). Thus, the period treal stock of debt, ˆ
bt, depends on the stock
of debt in the previous period, ˆ
bt−1, flows of interest payments, government spending, and
revenues, such that:
ˆ
bt=1+i∗ˆ
bt−1+bˆ
it+ˆgt−ˆxt+ηt,(3)
where i∗is the equilibrium interest rate, baccounts for the steady state value of the debt, ˆ
it
is the interest rate, ˆgtrepresents the government spending, is the tax rate, ˆxtdenotes the
output gap, and ηtstands for the debt shock.
The monetary policy and fiscal policy variables are interest rate and government spend-
ing, respectively. Through Eq. (1), one can see that the aggregate demand monetary policy
transmission takes place when an increase (decrease) in the interest rate is greater than the
expected increase (decrease) in the inflation rate at t+1. The reduction (rise) in the ag-
gregate demand of the economy lowers (increases) inflation via Eq. (2). On the other hand,
Eq. (3) establishes that an increase (decrease) in the government spending raises (lowers)
the level of debt, which in turn increases (decreases) the level of activity of the economy
through Eq. (1). Everything else constant, the ultimate result is an increase (decrease) in the
inflation rate by Eq. (2). Notice also that a high inflation rate has corrosive effects on the
income coming from public bonds, as argued by Kirsanova et al. (2005).
Equations (1), (3), and (2) define the basic equilibrium conditions of the model. The op-
timization problems of Sect. 3closes the model by deriving optimal rules for both monetary
policy (interest rate rule) and fiscal policy (government spending rule).
3The hat notation is used to denote deviations from the steady state in logarithm form.
Ann Oper Res (2013) 206:341–366 345
2.1 Game theoretic approach
Monetary and fiscal authorities interact with each other in order to minimize their respec-
tive loss functions. We use game theory models to analyze such interactions. The games
analyzed here have two individual players, namely monetary authority (central bank) and
fiscal authority (treasury). Each player has his own instrument, represented by the interest
rate (i) and government spending (g). If players act independently of each other, then we
have a non-cooperative game, while if they coordinate their actions, we have a cooperative
solution.
We consider three different scenarios where the interactions between fiscal and monetary
authorities take place.
2.1.1 Normal form game
When monetary and fiscal policymakers set their instruments simultaneously and non-
cooperatively, we model this situation using a normal form game. In general, a normal form
game is described by the set of players in the game, a set of actions for each player in the
game, and for each player a utility function that assigns a real value to every possible way the
game can be played; the higher the value of the utility function, the better the outcome of the
game for the player. In our case, there are two players: the fiscal and monetary authorities.
The fiscal authority chooses a level of government spending (g) and the monetary authority
chooses the level of interest rate (i). Both players try to minimize their loss function. We
consider the most well-known solution in game theory, the so-called Nash equilibrium of
the game. Intuitively, a pair of interest rate and government spending is a Nash equilibrium
if none of the players can unilaterally deviate from the equilibrium and obtain some gain.
2.1.2 Extensive form game
When the fiscal and monetary authorities move sequentially, we model this situation using an
extensive form game. In general, an extensive form game is described by a game tree where
in each node of the tree a player chooses one of the available actions which are described
by the branches of the tree. We consider only games with perfect information where each
player knows the history of the game each time he moves. A utility value for each player
is associated to each final node of the tree. In economics, if there are only two players and
each one of them moves only once, the player who moves first is known as the Stackelberg
leader and the player who moves last is known as the follower. We use the solution concept
which in game theory is known as subgame perfect equilibrium. Intuitively, in such solution
concept the game is solved from the end of the tree to the beginning. We consider that each
player (fiscal and monetary authorities) acts as the Stackelberg leader and anticipates the
response from the other player.
2.1.3 Cooperative game
When monetary and fiscal policymakers set their instruments simultaneously but in a coop-
erative way in order to pursue a common objective of maximizing social welfare, we model
the situation as a cooperative game.
The cooperation mechanism between the fiscal and monetary authorities occurs indi-
rectly when both authorities associate a positive weight on their instrumental variables. This
mechanism permits a direct adjustment to ongoing actions taken by the other authority. The
authorities face a common optimization problem, i.e. they try to minimize a common loss
function.
346 Ann Oper Res (2013) 206:341–366
3 Fiscal and monetary policy games
In this section, we derive the optimal reaction functions for different regimes of coordina-
tion. The monetary and fiscal authorities minimize their loss functions subject to the equi-
librium conditions of the economy. The authorities solve each optimization problem and
commit themselves to the optimal policy rules, having no incentive to deviate from them.
The rules present relevant properties of time consistency and timelessness. The former char-
acteristic is due to commitment, and the latter hinges on the fact that the monetary and fiscal
authorities need only be committed to determine policies at the later dates by rules that are
optimal from a similar perspective. As stressed by Damjanovic et al. (2005), the timeless
optimal policy is the policy that these authorities would have decided upon for the current
period had such a decision been taking infinitely far in the past.
The technical tools considered throughout this paper follow the general linear-quadratic
policy approach introduced by Giannoni and Woodford (2002a), with applications in Gian-
noni and Woodford (2002b). This approach is widely used in the monetary policy literature.
Particularly, Giannoni and Woodford (2002a) justify the use of this approach since (a) the
policy rule should be consistent with the desired equilibrium, which is a determinate equi-
librium under commitment; (b) the policy rule should be time-invariant and refer only to the
evolution of target variables which represent the authority’s stabilization goals; and (c) the
derived policy rule should continue to be optimal no matter what the statistical properties of
the exogenous disturbances hitting the economy are. Appendix A provides more details on
this technique.
3.1 A normal form game between fiscal and monetary policymakers
The monetary authority, represented by the central bank, tries to minimize a current period
quadratic loss function, with positive weights γπ,γx,andγion deviations of inflation from
the target (zero), output gap, and deviations of the interest rate from the equilibrium rate
(i∗), such that:
LM
t=γππ2
t+γxˆx2
t+γiˆ
it−i∗2,
subject to the equilibrium conditions of the economy.
Thus, the monetary authority’s problem can be written as:
minE01
2
∞
t=0
βtγππ2
t+γxˆx2
t+γiˆ
it−i∗2,
subject to (4)
(1)and(2).
Notice that the equilibrium conditions can be represented by Eqs. (1), (2)and(3). However,
under the current solution, there is no interaction between fiscal and monetary policies,
and the monetary authority takes as given the fiscal variables, which are exogenous to his
choices. Therefore, Eq. (3) is excluded because it defines the dynamics of the debt and the
Ann Oper Res (2013) 206:341–366 347
fiscal side of the economy.4A similar reasoning can be applied to equivalent policy problems
discussed ahead. The Lagrangian for this problem is:5
L=E0⎧
⎪
⎨
⎪
⎩
∞
t=0
βt⎡
⎢
⎣
1
2γππ2
t+1
2γxˆx2
t+1
2γi(ˆ
it−i∗)2
+Λ1,t (ˆxt−ˆxt+1+σ(ˆ
it−πt+1)−αˆ
bt−ˆrn
t)
+Λ2,t (πt−κˆxt−βπt+1−νt)
⎤
⎥
⎦⎫
⎪
⎬
⎪
⎭,(5)
where Λ1,t and Λ2,t are the Lagrange multipliers associated to the constraints in period t.
See Appendix Bfor an explanation on how to derive the results. The first order conditions
yield the following equations:
∂L
∂πt=γππt−β−1σΛ
1,t−1+Λ2,t −Λ2,t −1=0,
∂L
∂ˆxt=γxˆxt+Λ1,t −β−1Λ1,t−1−κΛ2,t =0,
∂L
∂(ˆ
it−i∗)=γiˆ
it−i∗+σΛ
1,t =0.
(6)
Isolating and substituting the Lagrange multipliers we obtain the following optimal nom-
inal interest rate rule:6
ˆ
it=−Γ0i∗+Γi,1ˆ
it−1−Γi,2ˆ
it−2+Γπ,0πt+Γx,0ˆxt−Γx,1ˆxt−1,(7)
where the coefficients are Γ0=σκ
β,Γi,1=(σκ
β+1
β+1),Γi,2=1
β,Γπ,0=γπσκ
γi,Γx,0=γxσ
γi,
and Γx,1=γxσ
γi.
The rule (7), which the central bank commits to follow, has contemporaneous and lagged
responses to the output gap. Additionally, it encompasses a history dependence since it de-
pends on past interest rates. The response is inversely related to the size of β. Therefore,
the more importance consumers attach to future variables, the stronger the monetary pol-
icy leverage is. Notice that the steeper the slope of the Phillips curve, measured by κ,the
stronger the interest rate response to inflation deviations from the target. On the other hand,
an increase in the weight placed on interest rate deviations, γi, diminishes the interest rate
reaction to inflation and output gap deviations. The elasticity of intertemporal substitution,
σ, also plays an important role in the monetary authority reaction function. For instance,
a higher value of σimplies stronger responses of the interest rate to deviations in both infla-
tion rate and output gap.
The fiscal side resembles the monetary one, with the difference being that the fiscal au-
thority (treasury) takes into account government spending. So, the period loss function as-
sumes the following form:7
LF
t=ρππ2
t+ρxˆx2
t+ρgˆg2
t,
4Equation (3) might bring on a multiplier effect into the other equations, which is neglected under the current
solution.
5Note that the dating of the expectations operator captures the idea of the policy maker choosing a rule ex-
ante which will be followed in the future. As we have a solution under commitment, the Lagrangian is solved
for expectations at time zero, which characterizes the time when the rule was defined, thereafter followed
without deviations. Thus, we removed the expectations operator on both inflation and output gap at t+1.
6This solution coincides with that proposed by Woodford (2003).
7Kirsanova et al. (2005) and Dixit and Lambertini (2000) use a similar loss function.
348 Ann Oper Res (2013) 206:341–366
where ρπ,ρx,andρgare positive weights placed on deviations of inflation rate, output
gap, and government spending, respectively. The debt does not enter the loss function. The
reason relies on the fact that if the fiscal policy feeds back on debt with a large coefficient,
then it tends to be welfare-reducing, since the economy will exhibit cycles and increase the
volatility of both inflation and output (Kirsanova et al. 2005).
The fiscal authority’s problem is to solve:
minE01
2
∞
t=0
βtρππ2
t+ρxˆx2
t+ρgˆg2
t,
subject to (8)
(1), (2)and(3).
The fiscal authority’s Lagrangian is
L=E0⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
∞
t=0
βt⎡
⎢
⎢
⎢
⎣
1
2ρππ2
t+1
2ρxˆx2
t+1
2ρgˆg2
t
+Λ1,t (ˆxt−ˆxt+1+σ(ˆ
it−πt+1)−αˆ
bt−ˆrn
t)
+Λ2,t (πt−κˆxt−βπt+1−νt)
+Λ3,t (ˆ
bt−(1+i∗)ˆ
bt−1−bˆ
it−ˆgt+ˆxt−ηt)
⎤
⎥
⎥
⎥
⎦⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
.(9)
The associated first order conditions are:
∂L
∂πt=ρππt−β−1σΛ
1,t−1+Λ2,t −Λ2,t −1=0,
∂L
∂ˆxt=ρxˆxt+Λ1,t −β−1Λ1,t−1−κΛ2,t +Λ
3,t =0,
∂L
∂ˆgt=ρgˆgt−Λ3,t =0,
∂L
∂ˆ
bt=−αΛ1,t +Λ3,t −1+i∗βEt(Λ3,t+1)=0.
(10)
Isolating and substituting for the Lagrangian multipliers, we have the optimal nominal
government spending rule:
ˆgt=−Θπ,0πt+Θg,1ˆgt−1−Θg,2ˆgt−2+Θg,+1Etˆgt+1−Θx,0ˆxt+Θx,1ˆxt−1,(11)
where the coefficients are: Θπ,0=ρπακ
ρgB,Θg,1=A
B,Θg,2=1
βB ,Θg,+1=(1+i∗)β
B,Θx,0=
ρxα
ρgB,Θx,1=ρxα
ρgB. Additionally, A=(β−1σκ +1
β+1+(1+i∗)),B=((1+i∗)(σ κ α +
1+β) +α+1).
According to Eq. (11), which the fiscal authority commits to follow, fiscal policy feeds
back on current inflation, current and past output gap, and lagged and expected government
spending. The rule encompasses a forward and backward history dependence since the gov-
ernment spending responds to past and future government spending. Notice that increases
in the weight placed on government spending ρgreduces the reaction to inflation and output
gap deviations.
3.2 Stackelberg leadership
We now address the equilibrium which emerges when the fiscal (monetary) authority moves
first, as a Stackelberg leader, anticipating the response from the monetary (fiscal) authority.
The leader takes into account the follower’s optimal policy, whereas the follower’s optimal
policy remains as a Nash equilibrium solution.
Ann Oper Res (2013) 206:341–366 349
Consider, first, the loss function for the fiscal authority acting as leader. We have the
following problem:
minE01
2
∞
t=0
βtρππ2
t+ρxˆx2
t+ρgˆg2
t,
subject to (12)
(1), (2), (3)and(7).
The corresponding Lagrangian might be written as:
L=E0
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
t=0
βt
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
2ρππ2
t+1
2ρxˆx2
t+1
2ρgˆg2
t
+Λ1,t (ˆxt−ˆxt+1+σ(ˆ
it−πt+1)−αˆ
bt−ˆrn
t)
+Λ2,t (πt−κˆxt−βπt+1−νt)
+Λ3,t (ˆ
bt−(1+i∗)ˆ
bt−1−bˆ
it−ˆgt+ˆxt−ηt)
+Λ4,t (ˆ
it+Γ0i∗−Γi,1ˆ
it−1+Γi,2ˆ
it−2−Γπ,0πt
−Γx,0ˆxt+Γx,1ˆxt−1)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
.(13)
The associated first order conditions are:
∂L
∂πt=ρππt−β−1σΛ
1,t−1+Λ2,t −Λ2,t −1−Γπ,0Λ4,t =0,
∂L
∂ˆxt=ρxˆxt+Λ1,t −β−1Λ1,t−1−κΛ2,t +Λ
3,t −Γπ,0Λ4,t +βΓx,1Et(Λ4,t+1)=0,
∂L
∂ˆgt=ρgˆgt−Λ3,t =0,
∂L
∂ˆ
bt=−αΛ1,t +Λ3,t −1+i∗βEt(Λ3,t+1)=0.
(14)
This optimization problem cannot be solved analytically. Thus, we implement the numer-
ical solution proposed by Juillard and Pelgrin (2007), where a timeless-perspective solution
is derived according to Woodford (2003). The next section will provide further details on
such a problem.
On the other hand, when acting as a leader, the monetary authority aims to minimize:
minE01
2
∞
t=0
βtγππ2
t+γxˆx2
t+γiˆ
it−i∗2,
subject to (15)
(1), (2)and(11).
The corresponding Lagrangian is given by:
L=E0
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∞
t=0
βt⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
2γππ2
t+1
2γxˆx2
t+1
2γi(ˆ
it−i∗)2
+Λ1,t (ˆxt−ˆxt+1+σ(ˆ
it−πt+1)−αˆ
bt−ˆrn
t)
+Λ2,t (πt−κˆxt−βπt+1−νt)
+Λ3,t (ˆgt+Θπ,0πt−Θg,1ˆgt−1+Θg,2ˆgt−2
−Θg,+1Etˆgt+1+Θx,0ˆxt−Θx,1ˆxt−1)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
.(16)
The implied first order conditions are:
350 Ann Oper Res (2013) 206:341–366
∂L
∂πt=γππt−β−1σΛ
1,t−1+Λ2,t −Λ2,t −1+Θπ,0Λ3,t =0,
∂L
∂ˆxt=γxˆxt+Λ1,t −β−1Λ1,t−1−κΛ2,t +Θx,0Λ3,t −βΘx,1Et(Λ3,t+1)=0,
∂L
∂(ˆ
it−i∗)=γiˆ
it−i∗+σΛ
1,t =0.
(17)
Likewise, there is no analytical solution for this problem. The numerical solution, based
on Juillard and Pelgrin (2007), is discussed in the next section.
3.3 Cooperation between policymakers
Here we analyze the outcome which emerges when the fiscal and monetary policymakers
cooperate with each other in pursuing a common objective. This means that the fiscal (mone-
tary) authority takes into account the monetary (fiscal) reaction function. Under cooperation,
both fiscal and monetary authorities face a common optimization problem:
minE01
2
∞
t=0
βtξππ2
t+ξxˆx2
t+ξiˆ
it−i∗2+ξgˆg2
t,
subject to (18)
(1), (2)and(3),
where ξπ=γπ+ρπ,ξx=γx+ρx,ξi=γiand ξg=ρg. That is, the positive weights placed
on the deviations of inflation and output gap are the sum of the weights placed by each
authority on those variables, while the weights on interest rate and government spending
deviations remain unchanged.
The Lagrangian for this problem is given by:
L=E0⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
∞
t=0
βt⎡
⎢
⎢
⎢
⎢
⎣
1
2ξππ2
t+1
2ξxˆx2
t+1
2ξi(ˆ
it−i∗)2+1
2ξgˆg2
t
+Λ1,t (ˆxt−ˆxt+1+σ(ˆ
it−πt+1)−αˆ
bt−ˆrn
t)
+Λ2,t (πt−κˆxt−βπt+1−νt)
+Λ3,t (ˆ
bt−(1+i∗)ˆ
bt−1−bˆ
it−ˆgt+ˆxt−ηt)
⎤
⎥
⎥
⎥
⎥
⎦
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
,(19)
with the following first order conditions:
∂L
∂πt=ξππt−β−1σΛ
1,t−1+Λ2,t −Λ2,t −1=0,
∂L
∂ˆxt=ξxˆxt+Λ1,t −β−1Λ1,t−1−κΛ2,t +Λ
3,t =0,
∂L
∂(ˆ
it−i∗)=ξiˆ
it−i∗+σΛ
1,t −bΛ3,t =0,
∂L
∂ˆgt=ξgˆgt−Λ3,t =0.
(20)
The resulting optimal nominal interest rate rule is:
ˆ
it=−Γ0i∗+Γi,1ˆ
it−1−Γi,2ˆ
it−2+Γπ,0πt+Γx,0ˆxt
−Γx,1ˆxt−1+Γg,0ˆgt−Γg,1ˆgt−1+Γg,2ˆgt−2,(21)
Ann Oper Res (2013) 206:341–366 351
where the coefficients are: Γ0=σκ
β,Γi,1=(σκ
β+1
β+1),Γi,2=1
β,Γπ,0=ξπσκ
ξi,Γx,0=ξxσ
ξi,
and Γx,1=ξπσ
ξi,Γg,0(bξg
ξi+σξ
g
ξi),Γg,1=(σκbξg
βξi+bξg
βξi+bξg
ξi+σξ
g
ξi),Γg,2=(bξg
βξi).
The resulting optimal government spending rule is given by:
ˆgt=−Θπ,0πt+Θi,0ˆ
it−i∗−Θi,1ˆ
it−1−i∗+Θi,2ˆ
it−2−i∗−Θx,0ˆxt
+Θx,1ˆxt−1+Θg,1ˆgt−1−Θg,2ˆgt−2,(22)
where the coefficients are defined as: Θπ,0=κξπ
C,Θi,0=ξi
σC ,Θi,1=(κξi
βC +ξi
βσC +ξi
σC),
Θi,2=ξi
βσC ,Θx,0=Θx,1=ξx
C,Θg,1=(κbξg
βC +bξg
βσC +bξg
σC +ξ
g
C),Θg,2=bξg
βσC . Finally, C=
(bξg
σ+ξ
g).
The above rules resemble the ones obtained in the normal form game. However, in both
equations, there are cross responses to the other authority policy instrument. That is, the
optimal nominal interest rule responds to current and lagged government spendings, while
the optimal government spending rule reacts to current and lagged interest rates. The coop-
eration occurs via those cross responses. One can also notice that, in the fiscal rule, there
is no response to future government spending, possibly because cooperation eliminated the
forward looking feature of that policy rule.
4 Numerical approach
4.1 Simulation of the Nash equilibrium
The model’s equilibrium is described by ten equations, being five endogenous and five ex-
ogenous processes. The endogenous variables are (ˆxt,π
t,ˆ
bt,ˆ
it,ˆgt)while the exogenous
ones are (ˆrn
t,ν
t,η
t,Ξ
t,O
t). Following the definitions from previous sections, the set of
equations characterizing the equilibrium can be represented as:
IS curve: ˆxt=Etˆxt+1−σ(ˆ
it−Etπt+1)+αˆ
bt+ˆrn
t
Phillips curve: πt=κˆxt+βEtπt+1+νt
Public debt: ˆ
bt=(1+i∗)ˆ
bt−1+bˆ
it+ˆgt−ˆxt+ηt
Monetary rule: ˆ
it=−Γ0i∗+Γi,1ˆ
it−1−Γi,2ˆ
it−2+Γπ,0πt+Γx,0ˆxt−Γx,1ˆxt−1+Ξt
Fiscal rule: ˆgt=−Θπ,0πt+Θg,1ˆgt−1−Θg,2ˆgt−2+Θg,+1Etˆgt+1−Θx,0ˆxt+Θx,1ˆxt−1+Ot
Demand shock: ˆrn
t=χrˆrn
t−1+εr
Supply shock: νt=χννt−1+εν
Debt shock: ηt=χηηt−1+εη
Monetary policy shock: Ξt=χΞΞt−1+εΞ
Fiscal policy shock: Ot=χOOt−1+εO
As usual, the exogenous processes are assumed to follow AR(1)stationary processes.
The AR(1)process reflects, relatively well, the persistence that exists in many macroeco-
nomic time series. Moreover, each εjis independent and identically distributed with zero
mean and variance σ2
j. Based on Brazilian data, we set χj=0.9 in order to capture the high
persistence of those shocks and σ2
j=0.04, ∀j. At this point, we do not allow for nonzero
correlations among the shocks. The DYNARE for MATLAB was used to solve for the ra-
tional expectations model.8
8See Laffargue (1990), Boucekkine (1995), Juillard (1996), Collard and Juillard (2001a,2001b) for details
on how the model can be solved.
352 Ann Oper Res (2013) 206:341–366
4.2 Simulation of the Stackelberg solution
The Stackelberg solution, as stressed earlier, is numerically solved under commitment fol-
lowing Juillard and Pelgrin (2007) and Woodford (2003). The structural equations, which
represent constraints on possible equilibrium outcomes under Stackelberg leadership, are
represented by a system of the form:
Zt+1
Etzt+1=AZt
zt+But+εt+1
0nz×1,(23)
where ztis a nz×1 vector of non-predetermined (forward looking) variables, Ztis a nZ×1
vector of predetermined (backward looking) variables, utis a k×1 vector of policy in-
struments, and εt+1is a nZ×1 vector of zero-mean uncorrelated shocks. The number of
rows of each matrix is, then, n=nz+nZ. The matrices Aand Bare functions of structural
parameters.
The above representation allows us to deal with all structural equations in matrix format.
The intertemporal loss function can be written as:
1
2E1
∞
t=1
βt−1y
tWyt(24)
where, for the sake of simplicity, we can define the vector xt=(Zt,z
t).Also,yt=(x
t,u
t).
The matrix Wis Wxx Wxu
W
xu Wuu (25)
where the matrices Wxx and Wuu are, without loss of generality, assumed to be symmetric.
The Lagrangian that follows from (23)and(24) is given by:
L=E1
∞
t=1
βt−1x
tWxxxt+2x
tWxuut+u
tWuuut+2ρ
t+1(Axt+But+ξt+1−xt+1)(26)
where ξt+1=(εt+1,z
t+1−Ezt+1)and ρt+1is a vector of multipliers. Subsequently, we
should take the first order conditions with respect to ρt+1,xt,andut, such that
⎡
⎢
⎣
In0nxk 0nx n
0nxn 0nx k βA
0kxn 0nxk −B⎤
⎥
⎦⎡
⎢
⎣
xt+1
ut+1
Etρt+1
⎤
⎥
⎦=⎡
⎢
⎣
AB0nxn
−βWxx −βWxu In
W
xu Wuu 0kxn
⎤
⎥
⎦⎡
⎢
⎣
xt
ut
ρt
⎤
⎥
⎦+⎡
⎢
⎣
ξt+1
0xx1
0kx1
⎤
⎥
⎦(27)
where ρ0=0andx0given.
Klein (2000) shows how to solve the above via generalized Schur decomposition. How-
ever, the optimal Ramsey policy, as previously described, is time inconsistent. The time in-
consistency comes from the fact that, in the current period, the policymaker sets the optimal
policy after the private agents’ expectations realization, such that he might have an incentive
to re-optimize in the future. The method proposed by Woodford (2003) to overcome this
issue was numerically implemented by Juillard and Pelgrin (2007). They generalize Wood-
ford’s timeless perspective solution by computing initial values for the Lagrange multipliers.
A timeless perspective hinges on the fact that the equilibrium evolution from time t=t0on-
ward is optimal, subject to the constraint that the economy’s initial evolution be the one
Ann Oper Res (2013) 206:341–366 353
associated with the policy in case (Woodford 2003). That is, the policymaker renounces the
possibility of setting the Lagrange multipliers to zero if he reoptimizes later on (Juillard and
Pelgrin 2007).
Considering now a timeless perspective solution, the Lagrange multipliers can be defined
as
ρt+1=gρ(xt,u
t,ρ
t,ξ
t+1).
Further, inserting the above restriction and adopting a timeless perspective policy,
namely, that the choice of the Lagrange multipliers ρ0is governed by the same rule from
time t=t0onwards, we have the Lagrangian for the timeless perspective policy
L=E1
∞
t=1
βt−1x
tWxxxt+2x
tWxuut+u
tWuuut+2ρ
t+1(Axt+But+ξt+1−xt+1)
+β−1ρ
0(y0−y0)(28)
where y0=(x
0,0). The first order conditions remain as before:
⎡
⎢
⎣
In0nxk 0nx n
0nxn 0nx k βA
0kxn 0nxk −B⎤
⎥
⎦⎡
⎢
⎣
xt+1
ut+1
Etρt+1
⎤
⎥
⎦=⎡
⎢
⎣
AB0nxn
−βWxx −βWxu In
W
xu Wuu 0kxn
⎤
⎥
⎦⎡
⎢
⎣
xt
ut
ρt
⎤
⎥
⎦+⎡
⎢
⎣
ξt+1
0xx1
0kx1
⎤
⎥
⎦(29)
where ρ0=0andx0given.
The non-predetermined variables of y0are selected such that (i) the function of the prede-
termined variables exists in the initial period, and (ii) there is a solution for the optimization
problem under the following condition y0=y0,fort>0. The Stackelberg problem is, then,
solved with the help of MATLAB.
4.3 Simulation of the cooperative solution
The cooperative case is similar to the one obtained in the normal form game. The major
differences are in the optimal policy rules (fiscal and monetary). Additional variables enter
those equations modifying thus the optimal responses to other variables.
The equilibrium of the model is described by ten equations, where five are endoge-
nous and the other five are exogenous processes. The endogenous ones are represented by
(ˆxt,π
t,ˆ
bt,ˆ
it,ˆgt)while the exogenous ones are (ˆrn
t,ν
t,η
t,Ξ
t,O
t). Next, we describe each
one of those equations.
IS curve: ˆxt=Etˆxt+1−σ(ˆ
it−Etπt+1)+αˆ
bt+ˆrn
t
Phillips curve: πt=κˆxt+βEtπt+1+νt
Public debt: ˆ
bt=(1+i∗)ˆ
bt−1+bˆ
it+ˆgt−ˆxt+ηt
Monetary rule: ˆ
it=−Γ0i∗+Γi,1ˆ
it−1−Γi,2ˆ
it−2+Γπ,0πt+Γx,0ˆxt−Γx,1ˆxt−1+Γg,0ˆgt−
Γg,1ˆgt−1+Γg,2ˆgt−2+Ξt
Fiscal rule: ˆgt=−Θπ,0πt+Θi,0(ˆ
it−i∗)−Θi,1(ˆ
it−1−i∗)+Θi,2(ˆ
it−2−i∗)−Θx,0ˆxt+
Θx,1ˆxt−1+Θg,1ˆgt−1−Θg,2ˆgt−2+Ot
Demand shock: ˆrn
t=χrˆrn
t−1+εr
Supply shock: νt=χννt−1+εν
Debt shock: ηt=χηηt−1+εη
Monetary policy shock: Ξt=χΞΞt−1+εΞ
Fiscal policy shock: Ot=χOOt−1+εO
354 Ann Oper Res (2013) 206:341–366
Tab le 1 Calibration of the parameters
Parameter Definition Value Reference
σIntertemporal elasticity of
substitution in private
consumption
5.00 Nunes and Portugal (2009)
αSensitivity of output gap to the
debt
0.20 Pires (2008)
κSensitivity of inflation rate to
the output gap
0.50 Gouvea (2007), Walsh (2003)
βSensitivity of agents to the
inflation rate
0.99 Cavallari (2003), Pires (2008)
i∗Natural rate of interest 0.07 Barcelos Neto and Portugal (2009)
bSteady state debt value 0.20 Kirsanova et al. (2005), Nunes and
Portugal (2009)
Tax rate 0.26 Kirsanova et al. (2005), Nunes and
Portugal (2009)
In the cooperative solution, as in the Nash equilibrium one, the exogenous processes are
assumed to follow stationary AR(1)representations, where each εjis independent and iden-
tically distributed with zero mean and constant variance σ2
j. The same calibration described
in Table 1is applied here. The simulation was carried on in Dynare for MATLAB.
5 Numerical results
In order to evaluate the performance of the alternative regime of coordination, we simulate
the models encompassing the Phillips curve, IS curve, government budget constraint, and
optimal monetary and fiscal rules. Additionally, we provide an overview on the social losses
generated by the distinct monetary and fiscal policy arrangements, and compute impulse
response functions. The calibration exercise is meant for the Brazilian economy in the period
after the implementation of the Real Plan.9Following most of the literature, we assume that
each period corresponds to one quarter of a year. The calibrated parameters, along with the
respective sources, are reported in Table 1.
One of the major goals of the simulation is to obtain variances of the variables under
the optimal trajectories, allowing for the computation of the expected social loss associated
to each regime of coordination. As a robustness check, we calculate and plot social losses
generated by alternative monetary and fiscal policy decisions, i.e. by varying the weights
placed on the target variables. We also compute impulse response functions to analyze how
the dynamics of the model behave under shocks of demand, supply, debt, monetary policy,
and fiscal policy. Therefore, the analysis will focus on efficient aspects for macroeconomic
stabilization.
5.1 Social loss analysis
The social loss is defined as the sum of the authorities’ expected individual losses, which
can be easily obtained by computing the unconditional variance.10 Taking, for instance, the
9The Real Plan was edited in June 1994.
10See Woodford (2003) for details.
Ann Oper Res (2013) 206:341–366 355
Tab le 2 Loss values for different coefficients under the Nash solution
σκ
LM=π2
t+0.5ˆx2
t+0.05(ˆ
it−i∗)2
LF=0.5π2
t+ˆx2
t+0.3ˆg2
t
LMLFLS
Variance of
πtˆxtˆ
btˆ
itˆgt
0.50 0.10 12.7452 0.7057 19.1851 27.5664 3.4317 12.9905 4.2008 17.1913
0.50 5.2119 0.2723 7.9440 11.3445 1.4225 5.3084 1.7033 7.0117
0.90 2.7197 0.1369 4.1761 5.9437 0.7486 2.7688 0.8842 3.6500
2.50 0.10 9.6978 2.1892 211.2660 16.3928 11.3685 10.2861 5.6368 15.9229
0.50 2.3218 0.4530 52.7042 4.0432 2.6607 2.4452 1.2729 3.7181
0.90 0.9891 0.1819 22.8264 1.7428 1.1210 1.0389 0.5301 1.5690
5.00 0.10 8.1027 2.9793 414.0482 11.6572 18.6392 8.8767 6.6826 15.5593
0.50 1.5190 0.4980 81.3970 2.2583 3.4490 1.6492 1.1882 2.8374
0.90 0.6059 0.1908 33.0275 0.9111 1.3681 0.6559 0.4654 1.1213
monetary authority period loss function, LM
t=γππ2
t+γxˆx2
t+γi(ˆ
it−i∗)2, it is easy to
calculate the expected loss for the monetary authority, given by:11
LM=γ2
πvar(πt)+γ2
xvar(ˆxt)+γ2
ivarˆ
it−i∗.(30)
Thus, the social loss is given by LS=LM+LF.
The welfare criterion defines a function which depends upon both monetary and fiscal
social losses. We make use of that criterion to analyze the cooperative solution, which occurs
indirectly when both authorities associate a positive weight on their instrumental variables.
The mechanism permits a direct adjustment to ongoing actions taken by the other authority.
Basically, the problem is to maximize a social utility (welfare) or, on the other hand, to
minimize the social loss function LS, which is defined by LS=LM+LF,thatis,thesum
of the authorities’ individual losses.
The results reported in Tables 2,3,4and 5show the variance of each time series and
the losses of each authority for different values of σand κ. The former parameter is the
intertemporal elasticity of substitution in private consumption and the latter one measures
the sensitivity of the inflation rate to the output gap in the Phillips curve. These parameters
came from Eqs. (1)and(2). The reason for choosing these parameters is that they play a
crucial role in both structural equations and policy rules.
According to Table 2, keeping κunchanged, the increases in σtend to reduce the loss for
the monetary but not for the fiscal authority. A high intertemporal elasticity of substitution in
private spending means a preference for future consumption, namely, the agents are willing
to postpone consumption. Under a higher interest rate, aggregate demand experiences a
shrinkage, reducing the output gap and inflation. The monetary policy is more effective,
leading to smaller monetary loss under a higher σ. The fiscal policy also experiences a
similar decrease in loss, but not for all parameter combinations.
Turning now to the parameter κ, when it increases for a given σ, both fiscal and monetary
losses decrease. The idea behind a rise in parameter κis a steeper Phillips curve. Thus,
a higher value of κtends to increase the sensitivity of inflation to the output gap, yielding a
negative effect on the loss. It is interesting to notice that when σ=5.00 and κ=0.90, we
11In order to simplify the notation, we will not distinguish between social loss and expected social loss.
356 Ann Oper Res (2013) 206:341–366
Tab le 3 Loss values for different coefficients under the Stackelberg solution: Fiscal leadership
σκ
LM=π2
t+0.5ˆx2
t+0.05(ˆ
it−i∗)2
LF=0.5π2
t+ˆx2
t+0.3ˆg2
t
LMLFLS
Variance of
πtˆxtˆ
btˆ
itˆgt
0.50 0.10 0.0355 0.0077 0.0318 0.0612 0.0034 0.0376 0.0169 0.0545
0.50 0.0219 0.0066 0.0183 0.0365 0.0036 0.0236 0.0124 0.0360
0.90 0.0149 0.0052 0.0120 0.0241 0.0035 0.0163 0.0092 0.0255
2.50 0.10 0.0311 0.0677 0.1868 0.0213 0.0087 0.0481 0.0763 0.1244
0.50 0.0142 0.0311 0.0238 0.0086 0.0034 0.0220 0.0350 0.0570
0.90 0.0080 0.0173 0.0071 0.0047 0.0017 0.0123 0.0195 0.0318
5.00 0.10 0.0296 0.1147 0.1973 0.0095 0.0072 0.0583 0.1227 0.1810
0.50 0.0121 0.0427 0.0148 0.0038 0.0020 0.0228 0.0459 0.0687
0.90 0.0064 0.0217 0.0057 0.0021 0.0010 0.0118 0.0234 0.0352
Tab le 4 Loss values for different coefficients under the Stackelberg solution: Monetary leadership
σκ
LM=π2
t+0.5ˆx2
t+0.05(ˆ
it−i∗)2
LF=0.5π2
t+ˆx2
t+0.3ˆg2
t
LMLFLS
Variance of
πtˆxtˆ
btˆ
itˆgt
0.50 0.10 0.0325 0.0051 1.1690 0.0001 0.0486 0.0338 0.0176 0.0514
0.50 0.0143 0.0119 0.3080 0.0018 0.0337 0.0173 0.0185 0.0358
0.90 0.0073 0.0103 0.1555 0.0050 0.0196 0.0099 0.0139 0.0238
2.50 0.10 0.0325 0.0050 1.1446 0.0013 0.0491 0.0338 0.0175 0.0513
0.50 0.0132 0.0138 0.2759 0.0009 0.0353 0.0167 0.0203 0.0370
0.90 0.0055 0.0132 0.1352 0.0001 0.0212 0.0088 0.0165 0.0253
5.00 0.10 0.0326 0.0050 1.1389 0.0015 0.0490 0.0339 0.0176 0.0515
0.50 0.0133 0.0137 0.2690 0.0018 0.0353 0.0167 0.0202 0.0369
0.90 0.0055 0.0133 0.1305 0.0006 0.0213 0.0088 0.0166 0.0254
obtain the lowest loss (LS=1.1213), meaning that social welfare is maximized under that
parameter combination.
The results reported in Table 3resembles the Nash equilibrium case when we consider
variations in κ. On the other hand, variations in σdo not have clear effects, given that there
are decreases and increases in the monetary loss depending on the value of κ.Thecombi-
nation of σ=0.50 and κ=0.90 provides the lowest loss (LS=0.0255). In addition, the
losses for the fiscal leadership are lower than the losses for the Nash equilibrium, suggesting
that the former is more efficient.
Table 4shows losses similar to what was observed under the fiscal leadership. However,
increases in σhave lower impacts on the fiscal loss. Once again, the pair of values σ=0.50
and κ=0.90 provides the lowest loss (LS=0.0238). Comparing all tables, that value is the
global minimum, which was obtained under a monetary leadership solution.
The coordination scheme presented in Table 5has characteristics similar to the Nash
equilibrium outcome. Thereby, the same analysis can be employed here. The combination of
Ann Oper Res (2013) 206:341–366 357
Tab le 5 Loss values for different coefficients under the cooperative solution
σκ
LM=π2
t+0.5ˆx2
t+0.05(ˆ
it−i∗)2
LF=0.5π2
t+ˆx2
t+0.3ˆg2
t
LMLFLS
Variance of
πtˆxtˆ
btˆ
itˆgt
0.50 0.10 12.1107 0.7355 29.9229 30.4118 4.0704 12.3706 4.1296 16.5002
0.50 4.4734 0.2630 11.2059 11.3251 1.5229 4.5675 1.5184 6.0859
0.90 2.2562 0.1300 5.7000 5.7413 0.7742 2.3030 0.7637 3.0667
2.50 0.10 7.8141 1.6191 364.3679 15.3638 11.8087 8.2573 4.6354 12.8927
0.50 0.1146 0.1094 0.5639 0.1531 0.4110 0.1423 0.1750 0.3173
0.90 1.6526 0.3492 72.2684 3.1152 2.4879 1.7477 0.9862 2.7339
5.00 0.10 5.6878 1.9734 769.0163 9.0333 17.8931 6.2037 5.0057 11.2094
0.50 1.0476 0.3632 112.2076 1.4691 3.0751 1.1421 0.9019 2.0440
0.90 0.4451 0.1524 44.5084 0.6046 1.2813 0.4847 0.3790 0.8637
Fig. 1 Social losses for different weights under the Nash solution
σ=2.50 and κ=0.50 leads to the minimum value for the loss function (LS=0.3173). This
performance, however, is well above the smallest loss obtained under a monetary leadership
in the Stackelberg game.
According to the smallest social loss criterion, the policy regimes might be ordered as
(1) monetary leadership, (2) fiscal leadership, (3) cooperative solution, and (4) Nash equilib-
rium solution. Thus, when the monetary authority moves first as a Stackelberg leader we get
the best scheme of coordination between the authorities. In addition, both Stackelberg so-
lutions are superior to the remaining ones. Finally, comparing the cooperative and the Nash
equilibrium solutions, we can note that the former regime is more efficient in minimizing
the social loss.
5.2 Sensitivity analysis
As a robustness check, we evaluated social losses generated by the three mechanisms of
coordination discussed in the previous section. In each case, it is assumed that the economy
is hit by a supply shock and the weights placed in output gap, inflation, and government
spending vary from 0.10 to 1.50, and in interest rate from 0.05 to 1.00. The resulting losses
are shown in Figs. 1,2,3and 4.
358 Ann Oper Res (2013) 206:341–366
Fig. 2 Social losses for different weights under the fiscal leadership solution
Fig. 3 Social losses for different weights under the monetary leadership solution
Under the Nash equilibrium solution, Fig. 1shows that the monetary loss increases pro-
portionately to the relative weights placed by the central bank on output gap and interest rate.
This is due to the fact that the society dislikes inflation more than the other two variables.
Notice that the distaste for interest rate fluctuations is the smallest, given that its impact on
the social loss is the strongest. Differently, the fiscal loss directly increases with the size
of the relative weights placed by the fiscal authority on government spending and inflation
stabilization. The reason is because the output gap is an important variable for the fiscal
policy. So, the society would prefer that the fiscal authority give more relative importance
to output gap stabilization. Here, changing the relative weight on the government spending
stabilization has the greater impact on the fiscal loss.
Furthermore, Fig. 2reveals that under a fiscal leadership in the Stackelberg solution, the
monetary loss is very sensitive to the relative weight attached by the central bank to output
gap stabilization. On the other hand, the fiscal loss presents a high sensitivity to the weight
on inflation stabilization. Note that Figs. 1and 2strengthen the different consequences on
the loss under the Nash and Stackelberg (fiscal leadership) solutions.
Subsequently, Fig. 3demonstrates that, under a monetary leadership in the Stackelberg
solution, the monetary loss behaves similarly to the case under fiscal leadership when the
central bank changes the relative weights on the output gap and interest rate. The fiscal loss
under a monetary leadership is more sensitive to the relative weight placed on government
spending.
Ann Oper Res (2013) 206:341–366 359
Fig. 4 Social losses for different weights under the cooperative solution
Fig. 5 Impulse responses to a supply shock under the monetary leadership solution
Finally, the losses shown in Fig. 4under the full cooperative solution are quite similar to
the Nash equilibrium case, but to a lower degree. In general, the Nash equilibrium and fully
cooperative solutions display closer responses when the relative weights placed on target
variables are changed.
5.3 Impulse response analysis
We limit our attention to the monetary leadership solution since it is the best scheme of
coordination according to the loss analysis. The impulse responses to alternative exogenous
shocks are presented in Figs. 5,6and 7. The responses of the variables are for one standard
deviation supply, demand or fiscal shock under the monetary leadership solution.
Figure 5shows the effects of a supply shock under monetary leadership. The immediate
effect of the shock is a rise in inflation, which leads the monetary authority to increase the
interest rate. A peak of that policy is reached in the ninth quarter, when the interest rate
360 Ann Oper Res (2013) 206:341–366
Fig. 6 Impulse responses to a demand shock under the monetary leadership solution
Fig. 7 Impulse responses to a fiscal shock under the monetary leadership solution
starts going back to the equilibrium. Due to the strong response of the monetary policy, the
output gap falls and pushes the government expenditure to a lower level. The debt response
is hump-shaped because it follows movements in the interest rate, reaching a peak after six
quarters.
Ann Oper Res (2013) 206:341–366 361
The effects of a demand shock under monetary leadership are shown in Fig. 6. On impact,
the shock pushes the output gap upwards, which in turn increases inflation and leads the
monetary authority to raise the interest rate. The peak in the interest rate is reached in the
fourth quarter and convergence to the equilibrium is faster than under a supply shock. The
debt and government spending are less volatile when compared to the supply shock.
A positive fiscal shock, displayed in Fig. 7, increases government spending, debt, and the
output gap. In addition, that shock is inflationary, given that there is a rise in inflation on
impact. The response of the monetary policy is delayed, but the increase in the interest rate
is sufficient to bring the economy back to equilibrium. Also, the debt converges slower than
the government spending due to the effects of high interest rates.
6 Concluding remarks
This paper has applied the game theory approach to a conventional macroeconomic opti-
mization problem to analyze the performance of alternative coordination schemes, repre-
sented by the Nash equilibrium solution, Stackelberg leadership, and the cooperative solu-
tion, in the interaction between fiscal and monetary policies. The comparisons among the
distinct regimes were made in terms of social loss, sensitivity to selected parameters, and
impulse response functions. Whenever possible, analytical solutions were derived for opti-
mal monetary and fiscal rules. In the Stackelberg case, however, due to the complexity of
the solution, only a numerical simulation was obtained.
The numerical approach provided evidence of relative superiority for the monetary lead-
ership in the Stackelberg solution. Thus, when the monetary authority moves first, as a
Stackelberg leader, taking into account the optimal fiscal policy obtained under the Nash
equilibrium solution, one reaches the smallest social loss. This monetary leadership might
be associated to the existence of a monetary dominance in the Brazilian economy during
the recent period. This evidence is supported by empirical findings provided by Tanner and
Ramos (2002), Fialho and Portugal (2005), Gadelha and Divino (2008), among others.
In particular, according to our results, the monetary leadership led to the lowest social
loss. A sensitivity analysis executed by varying the relative weights placed by the monetary
and fiscal authorities on their target variables showed that the Nash equilibrium and cooper-
ative solutions yielded more uniform responses. On the other hand, the monetary leadership
revealed a strong distaste for inflation by the Brazilian society. The impulse response func-
tions, computed for the best coordination scheme, indicated strong reactions of the monetary
authority to inflationary pressures. In addition, there is a clear inflationary effect coming
from fiscal shocks. Under the Stackelberg solution, the time series presented low volatility
and faster convergence to the equilibrium after the alternative exogenous shocks.
For future works, it would be interesting to analyze the performance of coordination
regimes under commitment and discretion, to apply the framework to a bargain problem in
a more complex environment and to extend the model to a block of countries, particularly
in South America, involving a monetary integration with common fiscal targets.
Acknowledgements J.A. Divino and H. Saulo acknowledge CNPq for the financial support. L.C. Rêgo
acknowledges financial support from FACEPE under grants APQ-0150-1.02/06 and APQ-0219-3.08/08, and
from MCT/CNPq under grants 475634/2007-1 and 306358/2010-7.
Appendix A
In this appendix we describe the general linear-quadratic policy approach introduced by
Giannoni and Woodford (2002a) with applications by Giannoni and Woodford (2002a), to
362 Ann Oper Res (2013) 206:341–366
derive an optimal monetary policy rule. Note that this approach can easily be extended to
the fiscal optimization problems discussed in this paper.
Woodford (2003, pp. 23–24) argues that standard dynamic programming methods are
valid only for optimization problems that evolve in response to the current action of the
controller. Hence, they do not apply to problems of monetary stabilization policy since the
central bank’s actions depend on both the sequence of instrument settings in the present
time and the private-sector’s expectations regarding future policies. A direct implementa-
tion of the maximum principle is not indicated, since we have discrete-time problems with
conditional expectations on some variables which affect the solution under commitment.
A.1 General linear-quadratic policy problem
Giannoni and Woodford (2002a) deal with policy problems in which the constraints for the
various state variables can be represented by a system of linear (or log-linear) equations, and
in which a quadratic function of these variables can be used to represent the policymaker’s
objectives. In general, the optimal policy rules considered by the authors take the form
φiit+φ
z¯zt+φ
Z¯
Zt+φ
s¯st=¯
φ, (31)
where itis the policy instrument, ¯ztand ¯
Ztare the vectors of nonpredetermined and prede-
termined endogenous variables (e.g., the output gap forecast Etxt+kmay be an element of
¯zt), ¯stis a vector of exogenous state variables, and φi,φz,φZ,andφs, are vectors of coeffi-
cients and ¯
φis a constant. As pointed out by Buiter (1982), a variable is nonpredetermined
if and only if its current value is a function of current anticipations of future values of en-
dogenous and/or exogenous variables. It is predetermined if its current value depends only
on past values of endogenous and/or exogenous variables.
The discounted quadratic loss function is assumed to have the form
Et0
∞
t=t0
βt−t0Lt,(32)
where t0stands for the initial date at which a policy rule is adopted, 0 <β<1 denotes the
discount factor, and Ltspecifies the period loss, that is,
Lt=1
2τt−τ∗Wτt−τ∗.(33)
where τtis a vector of target variables, τ∗is its corresponding vector of target values, and
Wis a symmetric, positive-definite matrix. The target variables are assumed to be linear
functions
τt=Ty
t,(34)
where yt≡[Ztztit],Ztis a subset of the predetermined variables ¯
Zt,ztis a subset of the
vector of nonpredetermined endogenous variables ¯zt,andTis a matrix of coefficients. It is
assumed that Ztencompasses all of the predetermined endogenous variables that constrain
the possible equilibrium evolution of the variables ZTand zTfor T≥t.Also,st, i.e. the sub-
set of exogenous states, encompasses all of the exogenous states which possess information
on the possible future evolution of the variables ZTand zTfor T≥t.
The endogenous variables ztand Zttake the form
ˆ
IZt+1
Etzt+1=AZt
zt+Bit+Cst,(35)
Ann Oper Res (2013) 206:341–366 363
where each matrix has n=nz+nZrows, nzand nZdenotes the number of nonpredeter-
mined and predetermined endogenous variables, respectively. Note that we may partition
the matrices as
ˆ
I=I0
0˜
E,A=A11 A12
A21 A22 ,B=0
B2,C=0
C2,
where the upper and lower blocks have nZand nzrows, respectively. The zero restrictions
in the upper blocks refer to the fact that the first nZequations define the elements of Ztas
elements of zt−jfor some j≥1. It is assumed that A22 is non-singular in order to let the
last nzequations be solved for ztas a function of Zt,st,it,andEtzt+1. In addition, B2is not
zero in all elements, resulting in an instrument with some effect.
Definition (Giannoni and Woodford 2002a) A policy rule that determines a unique non-
explosive rational expectations equilibrium is optimal from a timeless perspective if the
equilibrium determined by the rule is such that (a) the nonpredetermined endogenous
variables ztcan be expressed as a time-invariant function of a vector of predetermined
variables ¯
Ztand a vector of exogenous variables ¯st; that is, a relation of the form zt=
f0+fZ¯
Zt+fs¯st, applies for all dates t≥t0; and (b) the equilibrium evolution of the en-
dogenous variables {yt}for all dates t≥t0minimizes (32) among the set of all bounded
processes, subject to the constraints implied by the economy’s initial state Zt0, the require-
ments for rational expectations equilibrium (i.e., the structural equations (35)), and a set of
additional constraints of the form
˜
Ezt0=˜
E[f0+fZ¯
Zt0+fs¯st0],(36)
on the initial behavior of the nonpredetermined endogenous variables.
According to Woodford (1999), the Lagrangian for the minimization problem can be
written as
Lt0=Et0∞
t=t0
βt−t0L(yt)+ϕ
t+1˜
Ayt−β−1ϕ
t˜
Iy
t,(37)
where ˜
A≡[AB]and ˜
I≡[ˆ
I0].NotethatL(yt)denotes the period loss Ltexpressed as a
quadratic function of ytand ϕt+1denotes the vector of Lagrange multipliers related to the
constraints (35). Applying the law of iterated expectations, the conditional expectation can
be eliminated from the term Etzt+1in these constraints. Set
ϕt+1≡ξt+1
Ξt
and insert the term
ϕ
t0˜
Iy
t0=ξ
t0Zt0+Ξ
t0−1˜
Ezt0,(38)
into (37), where ξ
t0Zt0represents the constraints imposed by the given initial values Zt0,and
Ξ
t0−1˜
Ezt0represents the constraints (36). Finally, differentiating the Lagrangian (37) with
respect to the endogenous variables yt, we yield the first-order conditions
˜
AEtϕt+1+TWτt−τ∗−β−1˜
Iϕt=0,(39)
for each t≥t0. Solving (39) under some assumptions (Giannoni and Woodford 2002a), it is
possible to obtain a policy rule of the form expressed in (31).
364 Ann Oper Res (2013) 206:341–366
Appendix B
This appendix explains the solution method used to derive the optimal nominal interest rate
rulegivenby(7). Note that a similar procedure can be used to derive the other optimal rules.
The monetary authority minimizes the constrained loss function given by:
L=E0⎧
⎪
⎨
⎪
⎩
∞
t=0
βt⎡
⎢
⎣
1
2γππ2
t+1
2γxˆx2
t+1
2γi(ˆ
it−i∗)2
+Λ1,t (ˆxt−Etˆxt+1+σ(ˆ
it−Etπt+1)−αˆ
bt−ˆrn
t)
+Λ2,t (πt−κˆxt−βEtπt+1−νt)
⎤
⎥
⎦⎫
⎪
⎬
⎪
⎭,(40)
where the constraints include Eqs. (1)and(2), and Λ1,t and Λ2,t are the Lagrange multipli-
ers.
In order to write the first-order conditions, we need to differentiate this equation with
respect to the instrument (ˆ
it−i∗)and the state variables πtand ˆxt.Beforemovingforward
we need to consider how to deal with the expectation terms within the constraint. Since this
is a policy under commitment, the dating of the expectations operator captures the idea of
the policymaker choosing an ex-ante rule which will be followed in the future. Hence, the
expectations operator on inflation and the output gap at t+1 are removed. For example, if
the inflation rate which the policymaker sets influences both actual and expected inflation,
then he may directly optimize over the two. The first-order conditions are:
∂L
∂πt=βtγππt−βt−1σΛ
1,t−1+βtΛ2,t −βt−1Λ2,t −1(β) =0,(41)
∂L
∂ˆxt=βtγxˆxt+βtΛ1,t −βt−1Λ1,t −1−βtκΛ2,t =0,(42)
∂L
∂(ˆ
it−i∗)=βtγiˆ
it−i∗+βtσΛ
1,t =0.(43)
Isolating Λ1,t in (43) and inserting into (42), we obtain
γxˆxt−γi
σˆ
it−i∗+γi
βσ ˆ
it−1−i∗−κΛ2,t =0,(44)
where Λ1,t =−γi
σ(ˆ
it−i∗)and Λ1,t−1=−γi
σ(ˆ
it−1−i∗). Repeating the procedure for Λ2,t ,
we can eliminate all the Lagrange multipliers in (41). Then, isolating ˆ
itwe have
ˆ
it=−Γ0i∗+Γi,1ˆ
it−1−Γi,2ˆ
it−2+Γπ,0πt+Γx,0ˆxt−Γx,1ˆxt−1,(45)
where Γ0=σκ
β,Γi,1=(σκ
β+1
β+1),Γi,2=1
β,Γπ,0=γπσκ
γi,Γx,0=γxσ
γi,andΓx,1=γxσ
γi.
References
Backus, D., & Driffill, D. (1985). Rational expectations and policy credibility following a change in regime.
Review of Economic Studies,52, 211–221.
Barcelos Neto, P. C. F., & Portugal, M. S. (2009). The natural rate of interest in Brazil between 1999 and 2005.
Revista Brasileira de Economia,63(2), 103–118.
Boucekkine, R. (1995). An alternative methodology for solving nonlinear forward-looking variables. Journal
of Economic Dynamics & Control,19, 711–734.
Buiter, W. H. (1982). Predetermined and non-predetermined variables in rational expectations models. Eco-
nomics Letters,10, 49–54.
Bénassy, J. P. (2007). Money, interest and policy: dynamic general equilibrium in a non-Ricardian world.
Cambridge: MIT Press.
Ann Oper Res (2013) 206:341–366 365
Calvo, G. A. (1983). Staggered prices in a utility maximizing framework. Journal of Monetary Economics,
12, 383–398.
Cavallari, M. C. L. (2003). A coordenação das políticas fiscal e monetária ótimas. In Proceedings of the 31st
Brazilian economics meeting.
Collard, F., & Juillard, M. (2001a). Accuracy of stochastic perturbation methods: the case of asset pricing
models. Journal of Economic Dynamics & Control,25, 979–999.
Collard, F., & Juillard, M. (2001b). A higher-order Taylor expansion approach to simulation of stochastic
forward-looking models with and application to a non-linear Phillips curve. Computational Economics,
17, 127–139.
Damjanovic, T., Damjanovic, V., & Nolan, C. (2005). Optimal monetary policy rules from a timeless per-
spective.Centre for Dynamic Macroeconomic analysis working papers series.
Dixit, A. (2001). Games of monetary and fiscal interactions in the EMU. European Economic Review,45,
589–613.
Dixit, A., & Lambertini, L. (2000). Fiscal discretion destroys monetary commitment. Available at SSRN
http://ssrn.com/abstract=232654.
Engwerda, J. C. (1998). On the open-loop Nash equilibrium in LQ-games. Journal of Economic Dynamics &
Control,22, 1487–1506.
Engwerda, J. C., van Aarle, B., & Plasmans, J. (1999). The (in)finite horizon open-loop Nash LQ-game: an
application to the EMU. Annals of Operations Research,88, 251–273.
Engwerda, J. C., van Aarle, B., & Plasmans, J. (2002). Cooperative and non-cooperative fiscal stabilization
policies in the EMU. Journal of Economic Dynamics & Control,26, 451–481.
Favero, C. A. (2004). Comments on fiscal and monetary policy interactions: empirical evidence on optimal
policy using a structural new-Keynesian model. Journal of Macroeconomics,26, 281–285.
Fialho, M. L., & Portugal, M. S. (2005). Monetary and fiscal policy interactions in Brazil: an application to
the fiscal theory of the price level. Estudos Econômicos,35(4), 657–685.
Gadelha, S. R. B., & Divino, J. A. (2008). Dominância fiscal ou dominância monetária no Brasil? uma análise
de causalidade. Brazilian Journal of Applied Economics,12(4), 659–675.
Giannoni, M. P., & Woodford, M. (2002a). NBER working paper series: Vol. 9419.Optimal interest-rate
rules: I. General theory.
Giannoni, M. P., & Woodford, M. (2002b). NBER working paper series: Vol. 9420.Optimal interest-rate
rules: II. Applications.
Gouvea, S. (2007). Central Bank of Brazil working papers: Vol. 143.Price rigidity in Brazil: evidence from
CPI micro data. Available at http://ideas.repec.org/p/bcb/wpaper/143.html.
Juillard, M. (1996). CEPREMAP working papers (Couverture Orange): Vol. 9602.Dynare: a program for
the resolution and simulation of dynamic models with forward variables through the use of relaxation
algorithm. Available at http://ideas.repec.org/p/cpm/cepmap/9602.html.
Juillard, M., & Pelgrin, F. (2007). Computing optimal policy in a timeless-perspective: an application to a
small-open economy. Working papers, 07–32, Bank of Canada.
Kirsanova, T., Stehn, S. J., & Vines, D. (2005). The interactions between fiscal policy and monetary policy.
Oxford Review of Economic Policy,21(4), 532–564.
Klein, P. (2000). Using the generalized Schur form to solve a multivariate linear rational expectations model.
Journal of Economic Dynamics & Control,24(10), 1405–1423.
Laffargue, J. (1990). Résolution d’un modèle macroéconomique avec anticipations rationnelles. Annales
d’Économie et de Statistique,17, 97–119.
Lambertini, L., & Rovelli, R. (2003). Monetary and fiscal policy coordination and macroeconomic stabiliza-
tion. A theoretical analysis. Available at SSRN http://dx.doi.org/10.2139/ssrn.380322.
Muscatelli, V., Tirelli, P., & Trecroci, C. (2004). Fiscal and monetary policy interactions: empirical evidence
and optimal policy using a structural new-Keynesian model. Journal of Macroeconomics,26, 257–280.
Nordhaus, W. (1994). Policy games: coordination and independence in monetary and fiscal polices. Brookings
Papers on Economic Activity,2, 139–216.
Nunes, A. F. N., & Portugal, M. S. (2009). Active and passive fiscal and monetary policies: an analysis for
Brazil after the inflation targeting regime. In Proceedings of the 37th Brazilian economics meeting.
Pires, M. C. C. (2008). Interação entre política monetária e fiscal no Brasil em modelos robustos a pequenas
amostras. Ph.D. Dissertation, Department of Economics, Brasília, University of Brasília.
Tabellini, G. (1985). Rational expectations and policy credibility following a change in regime. Giornali degli
Economisti e Annali di Economia,44, 389–425.
Tanner, E., & Ramos, A. M. (2002). Fiscal sustainability and monetary versus fiscal dominance: evidence
from Brazil 1991–2000. IMF Working Paper, 02/5.
Taylor, J. (1993). Discretion versus policy rules in practice. Carnegie-Rochester Conference Series on Public
Policy,39, 195–214.
366 Ann Oper Res (2013) 206:341–366
Van Aarle, B., Engwerda, J., & Plasmans, J. (2002). Monetary and fiscal policy interactions in the EMU:
a dynamic game approach. Annals of Operations Research,109, 229–264.
Walsh, C. E. (2003). Monetary theory and policy. Cambridge: MIT Press.
Woodford, M. (1999). NBER Working paper series: Vol. 7261.Optimal monetary policy inertia.
Woodford, M. (2003). Interest and prices: foundations of a theory of monetary policy. Princeton: Princeton
University Press.
