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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.