Fiscal and monetary policy interactions: a game theory approach

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 decisions 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.

Appendices

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 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 implementation 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.

1.1 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

$$ \phi_{i}i_{t}+\phi_{z}' \bar{z}_{t}+\phi_{Z}'\bar{Z}_{t}+ \phi_{s}'\bar {s}_{t}=\bar{\phi}, $$

(31)

where i _t is the policy instrument, $\bar{z}_{t}$ and $\bar{Z}_{t}$ are the vectors of nonpredetermined and predetermined endogenous variables (e.g., the output gap forecast E _t x _t+k may be an element of $\bar{z}_{t}$), $\bar{s}_{t}$ is a vector of exogenous state variables, and ϕ _i , ϕ _z , ϕ _Z , and ϕ _s , are vectors of coefficients and $\bar{\phi}$ is 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 endogenous 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

$$ E_{t_0}\sum_{t=t_0}^{\infty} \beta^{t-t_0}L_t, $$

(32)

where t ₀ stands for the initial date at which a policy rule is adopted, 0<β<1 denotes the discount factor, and L _t specifies the period loss, that is,

$$ L_t=\frac{1}{2}\bigl(\tau_{t}- \tau^{*}\bigr)'W\bigl(\tau_{t}- \tau^{*}\bigr). $$

(33)

where τ _t is a vector of target variables, τ ^∗ is its corresponding vector of target values, and W is a symmetric, positive-definite matrix. The target variables are assumed to be linear functions

$$ \tau_{t}=Ty_{t}, $$

(34)

where y _t ≡[Z _t z _t i _t ]^⊤, Z _t is a subset of the predetermined variables $\bar{Z}_{t}$, z _t is a subset of the vector of nonpredetermined endogenous variables $\bar{z}_{t}$, and T is a matrix of coefficients. It is assumed that Z _t encompasses all of the predetermined endogenous variables that constrain the possible equilibrium evolution of the variables Z _T and z _T for T≥t. Also, s _t , i.e. the subset of exogenous states, encompasses all of the exogenous states which possess information on the possible future evolution of the variables Z _T and z _T for T≥t.

The endogenous variables z _t and Z _t take the form

$$ \hat{I}\left [ \begin{array}{c} Z_{t+1} \\ E_t{z}_{t+1} \end{array} \right ] = A \left [ \begin{array}{c} Z_{t} \\ z_{t} \end{array} \right ] + B i_{t} + C s_{t}, $$

(35)

where each matrix has n=n _z +n _Z rows, n _z and n _Z denotes the number of nonpredetermined and predetermined endogenous variables, respectively. Note that we may partition the matrices as

$$\hat{I}=\left [ \begin{array}{c@{\quad}c} I & 0 \\ 0 & \tilde{E} \end{array} \right ], \qquad A=\left [ \begin{array}{c@{\quad}c} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array} \right ], \qquad B=\left [ \begin{array}{c} 0 \\ B_2 \end{array} \right ], \qquad C=\left [ \begin{array}{c} 0 \\ C_2 \end{array} \right ], $$

where the upper and lower blocks have n _Z and n _z rows, respectively. The zero restrictions in the upper blocks refer to the fact that the first n _Z equations define the elements of Z _t as elements of z _t−j for some j≥1. It is assumed that A ₂₂ is non-singular in order to let the last n _z equations be solved for z _t as a function of Z _t , s _t , i _t , and E _t z _t+1. In addition, B ₂ is 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 z _t can be expressed as a time-invariant function of a vector of predetermined variables $\bar{Z}_{t}$ and a vector of exogenous variables $\bar{s}_{t}$; that is, a relation of the form $z_{t}=f_{0}+f_{Z}\bar{Z}_{t}+f_{s}\bar {s}_{t}$, applies for all dates t≥t ₀; and (b) the equilibrium evolution of the endogenous variables {y _t } for all dates t≥t ₀ minimizes (32) among the set of all bounded processes, subject to the constraints implied by the economy’s initial state $Z_{t_{0}}$, the requirements for rational expectations equilibrium (i.e., the structural equations (35)), and a set of additional constraints of the form

$$ \tilde{E}z_{t_0}=\tilde{E}[f_{0}+f_{Z} \bar{Z}_{t_0}+f_s\bar{s}_{t_0}], $$

(36)

on the initial behavior of the nonpredetermined endogenous variables.

According to Woodford (1999), the Lagrangian for the minimization problem can be written as

$$ \mathcal{L}_{t_0}=E_{t_0} \Biggl\{ \sum _{t=t_0}^{\infty} \beta^{t-t_0}\bigl[ L(y_t) + \varphi_{t+1}' \tilde{A} y_t - \beta^{-1} \varphi_{t}' \tilde {I} y_{t} \bigr] \Biggr\}, $$

(37)

where $\tilde{A}\equiv[ A \ B ]$ and $\tilde{I}\equiv[ \hat {I} \ 0 ]$. Note that L(y _t ) denotes the period loss L _t expressed as a quadratic function of y _t and φ _t+1 denotes 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 E _t z _t+1 in these constraints. Set

$$\varphi_{t+1}\equiv \left [ \begin{array}{c} \xi_{t+1} \\ \varXi_{t} \end{array} \right ] $$

and insert the term

$$ \varphi_{t_0}'\tilde{I}y_{t_0}= \xi_{t_0}'Z_{t_0}+\varXi_{t_{0}-1}' \tilde{E}z_{t_0}, $$

(38)

into (37), where $\xi_{t_{0}}'Z_{t_{0}}$ represents the constraints imposed by the given initial values $Z_{t_{0}}$, and $\varXi_{t_{0}-1}'\tilde {E}z_{t_{0}}$ represents the constraints (36). Finally, differentiating the Lagrangian (37) with respect to the endogenous variables y _t , we yield the first-order conditions

$$ \tilde{A}'E_{t}\varphi_{t+1}+T'W \bigl(\tau_{t}-\tau^{*}\bigr)-\beta^{-1}\tilde {I}'\varphi_{t}=0, $$

(39)

for each t≥t ₀. Solving (39) under some assumptions (Giannoni and Woodford 2002a), it is possible to obtain a policy rule of the form expressed in (31).

Appendix B

This appendix explains the solution method used to derive the optimal nominal interest rate rule given by (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:

$$ \mathcal{L}={E}_0\left \{{\displaystyle\sum_{t=0}^{\infty} \beta^{t}} \left [ \begin{array}{@{}l} \frac{1}{2}\gamma_{\pi}\pi_t^2+\frac{1}{2}\gamma_{x}\hat{x}_t^2+\frac {1}{2}\gamma_i(\hat{i}_t-i^{*})^{2} \\\noalign{\vspace{3pt}} \quad {} +\varLambda_{1,t} (\hat{x}_t-{E}_t\hat{x}_{t+1}+\sigma(\hat {i}_t-{E}_t\pi_{t+1})-\alpha\hat{b}_t-\hat{r}_t^n )\\\noalign{\vspace{3pt}} \quad {} +\varLambda_{2,t} (\pi_t-\kappa\hat{x}_t-\beta{E}_t\pi_{t+1}-\nu _t ) \end{array} \right ] \right \}, $$

(40)

where the constraints include Eqs. (1) and (2), and Λ _1,t and Λ _2,t are the Lagrange multipliers.

In order to write the first-order conditions, we need to differentiate this equation with respect to the instrument $(\hat{i}_{t}-i^{*})$ and the state variables π _t and $\hat{x}_{t}$. Before moving forward 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:

(41)

(42)

(43)

Isolating Λ _1,t in (43) and inserting into (42), we obtain

$$ \gamma_x\hat{x}_t-\frac{\gamma_i}{\sigma}\bigl( \hat{i}_t-i^*\bigr)+\frac{\gamma _i}{\beta\sigma}\bigl(\hat{i}_{t-1}-i^* \bigr) - \kappa\varLambda_{2,t}=0, $$

(44)

where $\varLambda_{1,t}=-\frac{\gamma_{i}}{\sigma}(\hat{i}_{t}-i^{*})$ and $\varLambda_{1,t-1}=-\frac{\gamma_{i}}{\sigma}(\hat{i}_{t-1}-i^{*})$. Repeating the procedure for Λ _2,t , we can eliminate all the Lagrange multipliers in (41). Then, isolating $\hat{i}_{t}$ we have

$$ \hat{i}_{t}=-\varGamma_0{i}^{*}+ \varGamma_{i,1}\hat{i}_{t-1}-\varGamma_{i,2}\hat {i}_{t-2}+\varGamma_{\pi,0}\pi_t+ \varGamma_{x,0}\hat{x}_t-\varGamma_{x,1} \hat{x}_{t-1}, $$

(45)

where $\varGamma_{0}=\frac{\sigma\kappa}{\beta}$, $\varGamma_{i,1}= (\frac {\sigma\kappa}{\beta}+\frac{1}{\beta}+1 )$, $\varGamma_{i,2}=\frac {1}{\beta}$, $\varGamma_{\pi,0}=\frac{\gamma_{\pi}\sigma\kappa}{\gamma_{i}}$, $\varGamma _{x,0}=\frac{\gamma_{x}\sigma}{\gamma_{i}}$, and $\varGamma_{x,1}=\frac{\gamma _{x}\sigma}{\gamma_{i}}$.