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Games, expectations, and optimal policy for open economies
Journal of Econumic Dynamics and Control 10 (1986) 45-49. North-Holland
GAMES, EXPECTATIONS, AND O P T I M A L POLICY FOR OPEN ECONOMIES Sean HOLLY London Business School London N W1 4SA. England
1. Introduction In this paper a framework is proposed for calculating the optimal, non-cooperative behaviour of individual country policy-makers. Two main methods are employed. The first, which is a direct method, calculates the 'final form' for a rational expectations model for a given number of players. The second method is an extension of the Penalty Function method of Holly and Zarrop (1983) and involves the generation of the normal final form, but then enforces expectational consistency by an augmenting exterior penalty function at the stage of the optimisation. The solution to a number of problems is provided. First, we have the solution to the single-player game where a policy-maker faces a private sector which forms rational expectations of what the policymaker, is doing. Secondly, we can handle the more difficult problem of how to solve a many-player game, when there are a large number of nonparticipants - the public - who take no direct role in any negotiations which might be going on but who form rational expectations of the outcome. In general, the players can be individual countries or even representatives of trade unions and employers.
2. The single player under rational expectations The linear rational expectations model can be written in reduced form as Yt = A Y t - 1 + B x , + C e , + DyT+l,,,
(1)
where Yt is a vector of g endogenous variables, x t a vector of n policy instruments, e t a vector of m strictly exogenous variables, and Yt+l,t is a vector of rational expectations formed at time t about endogenous variables at time t + 1. The final form of (1) can be obtained if we stack equations up for T time periods in a band matrix form with the matrix D along the upper 0165-1889/86/$3.50©1986, ElsevierSciencePublishers B.V. (North-Holland)
46
S. Holly, Games, expectations and optimal policv
diagonal and A along the lower diagonal, in the form G Y = F X + p. Where G is a g T x g T matrix and F is a n T × nT matrix, p contains the remaining uncontrollable exogenous variables and initial and terminal conditions. Y and X are stacked vectors defined as Y ' = (y~, y~. . . . . y~.),
X ' = (x~, x~,..., x~-),
(2)
so the final form of the rational expectations model can be written as Y = R X + s, where R = G- 1F and s = G- lp. This final form differs from the conventional stacked final form in that R is not lower triangular. The endogenous variables depend not only on current and predetermined policy instruments and exogenous variables but also on future, expected policy instruments and exogenous variables. The policy objective function is the familiar quadratic objective function in stacked form. Where Q is a semidefinite weighting matrix and N is a positive definite weighting matrix, J = 1/2((Y- yd),Q(y_
rd) + ( X -
x d ) ' N ( X - xd)}.
(3)
Given eqs. (2) and (3) the 'optimal' open-loop policy can be written
X * = ( R'QR + N ) - ' { R ' [ Q Y d - Qs] + N X d }.
2.1. A penalty function method For the second method we use the Penalty Function method of Holly and Zarrop (1983). This method has the advantage that it treats the optimal control problem simultaneously with the solution of a rational expectations model. Rewriting the stacked final form as Y= R X + s y e + s, where R is now the conventional lower triangular matrix of dynamic multipliers. S is a lower triangular matrix of the dynamic multipliers between Y and ye, where ye is a stacked vector. An augmented objective function can then be written as J* = 1 / 2 { ( Y - yd)'Q(y_
yd) + ( X - X d ) ' N ( X - X ~)
+ ( y e _ y~),p(y~_ y s ) + ye,Gye),
(4)
where Y~ = V Y + o and V is a null matrix with one's along the upper diagonal, o = (0,0 . . . . . 0), 0 is a set of terminal conditions, and G and P are positive definite matrices. A standard exterior penalty function method is used such
47
S. Holly, Games, expectationsand optimalpolicv
that a sequence of solutions to (4) corresponding to increases in the minimum eigenvalue of P tends towards the convergent saddlepoint consistent expectations path. The optimal X consistent with rational expectations is
X*y~}
=
[[R'QR-P'VR+N] [S'QR-PVR]
x
~
[ R ' Q S + P ( I - V S ) ] -1 [ S ' Q S + P ( I - VS)]
-RQ-R'Q+PV -SQ S Q - P V
P N] P
(5)
.
Lx"J
3. Multi-player games under rational expectations The framework of the previous section can easily be extended to handle a more conventional dynamic game such as that between individual countries. The public would then act as a rational observer of the outcome of the game to which it is not actually a partner. The participants in the game would then have to take account of how the public will react to the outcome of the game. 3.1. A direct method for multi-player games
We can use the previous direct method by stacking up the difference equation with two players as (6)
A o y t - Aljpt_ x - DlYt+ l, t = B1x t q- B2x t + Ce t,
or in stacked notation Y = R1XI + R z X 2 + s.
The objective functions for two players are then j ~ = a / 2 ( ( Y - Y ~ ) Q ~~ (' r - r f ) +
(x~
-
x ~. ') ~ v x ( x ~ - x ~ ) } ,
.12 = 1 / 2 { ( Y - Y ~ ) Qd 2, ( Y - Y ~ ) +
(x2-x~)N d ,2 ( x 2 - x ~ ) )
(7) .
(8~
The open-loop Nash non-cooperative solution for the two-player game with
S. Hally, Games,expectationsand optimalpolicy
48
rational observers is then
X{,} = [[R,1Q1RI + Nx] X~ [[R'2Q2R,]
[R~QlR2] [R'202R2+N21
R'2Q2 -R'2Q2
[R,1Q1S ] ]-t [R~QzS ]
-] s
0
.
(9)
N2 xd I
J 3.2. A penalty function method for multi-player games Alternatively, we can use the penalty function method. Rewriting (6) as
(lO)
Y= R1X1 + R2X 2+ SY ¢ + s, where we augment the problem with the loss function
ys),p(ye_ ys) + 1/2(Ye,Gy~),
je= 1/2(ye_
(11)
where G is a positive-definite matrix. We can then use the same penalty function method approach to compute the non-cooperative Nash solution to this two-player game with rational observers as
[RIQ, R2] [R~Q~R,I
.
[ - (I - VS)'PzR,I
[ w,O_l o
[R'lQtS ]
]-'
[RSO2R2 + N2] [R'2Q=S] [ - ( I - VS)'PVR~] [ ( I - VS)'P(I-
- RiO_l
o
N,
0
R'2Q2 -R'2Q 2
0
0
o
o
[ ( t - vs)e
o
[ - ( t - vs)pv]
VS)+G]
(12) o
Both solutions to the multi-player game assume that the public acts as an observer, taking account of the consequences of the game for the behaviour of the economy. As long as the game is Nash, there are no incentives in the
S. Holly, Games, expectations and optimal policv
49
absence of cooperation for one of the players to depart from the Nash solution as long as we are willing to go along with the assumption that the public is fully aware of what is going on. 4. Conclusions
In this paper we have approached the many problems raised by rational expectations and policy interdependencies from a different perspective to that common in the literature. While analytical tractability is lost, the gains in terms of an ability to carry out policy design with large models is sufficient to justify the use of the techniques described in this paper. The framework we have used enables us to solve the so far intractable problem of how to simultaneously allow for forward looking behaviour on the part of the public, while also allowing governments to formulate economic policies in an interdependent world. References Holly, S. and M. Zarrop, 1983, On optimality and time consistency when expectations are rational, European Economic Review 20, 23-40.
GAMES, EXPECTATIONS, AND O P T I M A L POLICY FOR OPEN ECONOMIES Sean HOLLY London Business School London N W1 4SA. England
1. Introduction In this paper a framework is proposed for calculating the optimal, non-cooperative behaviour of individual country policy-makers. Two main methods are employed. The first, which is a direct method, calculates the 'final form' for a rational expectations model for a given number of players. The second method is an extension of the Penalty Function method of Holly and Zarrop (1983) and involves the generation of the normal final form, but then enforces expectational consistency by an augmenting exterior penalty function at the stage of the optimisation. The solution to a number of problems is provided. First, we have the solution to the single-player game where a policy-maker faces a private sector which forms rational expectations of what the policymaker, is doing. Secondly, we can handle the more difficult problem of how to solve a many-player game, when there are a large number of nonparticipants - the public - who take no direct role in any negotiations which might be going on but who form rational expectations of the outcome. In general, the players can be individual countries or even representatives of trade unions and employers.
2. The single player under rational expectations The linear rational expectations model can be written in reduced form as Yt = A Y t - 1 + B x , + C e , + DyT+l,,,
(1)
where Yt is a vector of g endogenous variables, x t a vector of n policy instruments, e t a vector of m strictly exogenous variables, and Yt+l,t is a vector of rational expectations formed at time t about endogenous variables at time t + 1. The final form of (1) can be obtained if we stack equations up for T time periods in a band matrix form with the matrix D along the upper 0165-1889/86/$3.50©1986, ElsevierSciencePublishers B.V. (North-Holland)
46
S. Holly, Games, expectations and optimal policv
diagonal and A along the lower diagonal, in the form G Y = F X + p. Where G is a g T x g T matrix and F is a n T × nT matrix, p contains the remaining uncontrollable exogenous variables and initial and terminal conditions. Y and X are stacked vectors defined as Y ' = (y~, y~. . . . . y~.),
X ' = (x~, x~,..., x~-),
(2)
so the final form of the rational expectations model can be written as Y = R X + s, where R = G- 1F and s = G- lp. This final form differs from the conventional stacked final form in that R is not lower triangular. The endogenous variables depend not only on current and predetermined policy instruments and exogenous variables but also on future, expected policy instruments and exogenous variables. The policy objective function is the familiar quadratic objective function in stacked form. Where Q is a semidefinite weighting matrix and N is a positive definite weighting matrix, J = 1/2((Y- yd),Q(y_
rd) + ( X -
x d ) ' N ( X - xd)}.
(3)
Given eqs. (2) and (3) the 'optimal' open-loop policy can be written
X * = ( R'QR + N ) - ' { R ' [ Q Y d - Qs] + N X d }.
2.1. A penalty function method For the second method we use the Penalty Function method of Holly and Zarrop (1983). This method has the advantage that it treats the optimal control problem simultaneously with the solution of a rational expectations model. Rewriting the stacked final form as Y= R X + s y e + s, where R is now the conventional lower triangular matrix of dynamic multipliers. S is a lower triangular matrix of the dynamic multipliers between Y and ye, where ye is a stacked vector. An augmented objective function can then be written as J* = 1 / 2 { ( Y - yd)'Q(y_
yd) + ( X - X d ) ' N ( X - X ~)
+ ( y e _ y~),p(y~_ y s ) + ye,Gye),
(4)
where Y~ = V Y + o and V is a null matrix with one's along the upper diagonal, o = (0,0 . . . . . 0), 0 is a set of terminal conditions, and G and P are positive definite matrices. A standard exterior penalty function method is used such
47
S. Holly, Games, expectationsand optimalpolicv
that a sequence of solutions to (4) corresponding to increases in the minimum eigenvalue of P tends towards the convergent saddlepoint consistent expectations path. The optimal X consistent with rational expectations is
X*y~}
=
[[R'QR-P'VR+N] [S'QR-PVR]
x
~
[ R ' Q S + P ( I - V S ) ] -1 [ S ' Q S + P ( I - VS)]
-RQ-R'Q+PV -SQ S Q - P V
P N] P
(5)
.
Lx"J
3. Multi-player games under rational expectations The framework of the previous section can easily be extended to handle a more conventional dynamic game such as that between individual countries. The public would then act as a rational observer of the outcome of the game to which it is not actually a partner. The participants in the game would then have to take account of how the public will react to the outcome of the game. 3.1. A direct method for multi-player games
We can use the previous direct method by stacking up the difference equation with two players as (6)
A o y t - Aljpt_ x - DlYt+ l, t = B1x t q- B2x t + Ce t,
or in stacked notation Y = R1XI + R z X 2 + s.
The objective functions for two players are then j ~ = a / 2 ( ( Y - Y ~ ) Q ~~ (' r - r f ) +
(x~
-
x ~. ') ~ v x ( x ~ - x ~ ) } ,
.12 = 1 / 2 { ( Y - Y ~ ) Qd 2, ( Y - Y ~ ) +
(x2-x~)N d ,2 ( x 2 - x ~ ) )
(7) .
(8~
The open-loop Nash non-cooperative solution for the two-player game with
S. Hally, Games,expectationsand optimalpolicy
48
rational observers is then
X{,} = [[R,1Q1RI + Nx] X~ [[R'2Q2R,]
[R~QlR2] [R'202R2+N21
R'2Q2 -R'2Q2
[R,1Q1S ] ]-t [R~QzS ]
-] s
0
.
(9)
N2 xd I
J 3.2. A penalty function method for multi-player games Alternatively, we can use the penalty function method. Rewriting (6) as
(lO)
Y= R1X1 + R2X 2+ SY ¢ + s, where we augment the problem with the loss function
ys),p(ye_ ys) + 1/2(Ye,Gy~),
je= 1/2(ye_
(11)
where G is a positive-definite matrix. We can then use the same penalty function method approach to compute the non-cooperative Nash solution to this two-player game with rational observers as
[RIQ, R2] [R~Q~R,I
.
[ - (I - VS)'PzR,I
[ w,O_l o
[R'lQtS ]
]-'
[RSO2R2 + N2] [R'2Q=S] [ - ( I - VS)'PVR~] [ ( I - VS)'P(I-
- RiO_l
o
N,
0
R'2Q2 -R'2Q 2
0
0
o
o
[ ( t - vs)e
o
[ - ( t - vs)pv]
VS)+G]
(12) o
Both solutions to the multi-player game assume that the public acts as an observer, taking account of the consequences of the game for the behaviour of the economy. As long as the game is Nash, there are no incentives in the
S. Holly, Games, expectations and optimal policv
49
absence of cooperation for one of the players to depart from the Nash solution as long as we are willing to go along with the assumption that the public is fully aware of what is going on. 4. Conclusions
In this paper we have approached the many problems raised by rational expectations and policy interdependencies from a different perspective to that common in the literature. While analytical tractability is lost, the gains in terms of an ability to carry out policy design with large models is sufficient to justify the use of the techniques described in this paper. The framework we have used enables us to solve the so far intractable problem of how to simultaneously allow for forward looking behaviour on the part of the public, while also allowing governments to formulate economic policies in an interdependent world. References Holly, S. and M. Zarrop, 1983, On optimality and time consistency when expectations are rational, European Economic Review 20, 23-40.