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The formulation of robust policies for rival rational expectations models of the economy
Journal of Economic Dynamics and Control 10 (1986) 93-97. North-Holland
THE FORMULATION OF ROBUST POLICIES FOR RIVAL RATIONAL EXPECTATIONS MODELS OF THE ECONOMY Paul LEVINE London Business School, London NI¥1 4S,4, England
I. Introduction
A problem of utmost importance in macroeconomics is how to design policy when faced with model uncertainty. A number of recent studies have addressed this problem [Becker et al. (1984), Christodoulakis and van de Ploeg (1984), Snower and Wierzbicki (1983)]. But the literature to date has not considered the problems that arise when some or all of the models available assume rational (i.e., model-consistent) expectations (RE). This will soon be of particular importance because the next generation of large econometric models in Britain should see at least four - those at the NIESR, the LBS, City University and Liverpool University - incorporating RE. This paper sets out and demonstrates a general method for designing policies compatible with more than one RE mod.el of the economy. 2. The model The model is of a small open economy with efficient financial markets, sluggishly adjusting output and wages but with forward-looking behaviour in the labour market. It consists of the following eight relationships, five dynamic and three static:
Y= a , ( E - P - P t ) -
a 2 ( R _ pe(t ' t ) ) - a3S,
(1)
Y),
(2)
i f = Ol~+ ( 1 - 0 ) ( E - P + P ' ) ,
(3) (4)
0165-1889/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
J.E.D.C. - D
P. Levine, Policies for rational expectation models
94
W = 8fl f ~ C [ W ¢ ( ~ , t) + ~b(Ye('r, t) - Y*)]e-'(*-t)dr
Jt
+(1 - fl) 14/,
(5)
1~'= 8(1,I,'- W),
(6)
M = P + T1Y- "y2R,
(7)
E=R-R
(8)
r,
where Y = real output, P = the domestic price level, E = the exchange rate (defined as the price of foreign exchange), S = autonomous taxes, W = the wage contract, l~'= the average wage, R = the domestic interest rates, we(% t ) = the expectation of W(~-) f o r m e d a t time t [with ye(~., t) and P(%t) defined similarly], Y* = the natural rate of real output, M = the nominal money stock, all measured in logarithms except interest rates which are proportions. Superscript f denotes a foreign counterpart. By the RE assumption we may put We(% t) = W(~'), ye(~., t) = Y(T) and jbe(~., t) = P('r). Then, in the long-run, P = O W = (1 - O)(E - P + p r ) , y = a l ( E - P + P f ) - a 2 R - a3S, Y = Y*, R = R r and (7). Given that the money stock and S are fixed in the long-run by the authorities, this then gives five relationships which determine the equilibrium values of R, Y, P, W and E. From this point onwards the analysis is conducted in terms of deviations about this long-run equilibrium. Denoting deviations from equilibria by the lower case, differentiating (5) we have. ¢v ~ = 8 f l # y + 28(1 - f l ) ( w - ~v).
(9)
In its stochastic form the model may now be written as
(lO) (dw)*
L(o.°)J where du 1 is a demand disturbance and du 2 a wage bargaining or wage drift disturbance. Substituting out for pe = p from (4) we arrive at a standard stochastic RE model for which the optimal policy may be found by standard methods [e.g., Driffill (1982), Buiter (1983), Miller and Salmon (1983), Levine and Currie (1984)].
95
P. Levine, Poficies for rational expectation models
Suppose now there exists a second RE model in which (1) and (2) are replaced (in deviation form) by dy* = ~ l * [ a l * ( e - p ) - a 2 * ( r - p e ) - a 3 * s - y * ] d t + d u l
*,
(11)
but the remaining relationships remain the same. Suppose that private agents believe the first model with probability Pl and the second with probability 1 - P i . The (9) must be replaced with
#e= _3fl~(ply + (1 - P l ) Y*) + 23(1 - fl)(w - fv).
(12)
The model to be controlled now consists of a composite model of the form
-]
"dy
dy* dp
] I
Y y,
[W]- L J
' = e p-
(dw) e
pe
d , + b [/ bj)*e : i ] dt÷
_(de) e .J
-
we
du~'
[r]dt+
dul 0
0
I
_0
_]
(13)
Note that only pe is involved in the second term, but we present the model in this form to indicate the nature of the general solution. We consider a 'pure' stochastic problem with the system in stochastic equilibrium. The objective function then consists of asymptotic variances which for this model we choose to be asyE(W) =asyE(p~ay2+ (1 -p~)ay*2+bp2+crZ+dgZ),
(14)
where p~ is the government's subjective probability that model 1 is correct. The two RE models for which a robust policy is being sought will have the following properties. Model 1, in which output is described by y following (1) and (2), has output slowly adjusting to its long-run partial equilibrium relationship (i.e., ~ki-1 large) and has a zero interest semi-elasticity (a 2 = 0). This we loosely refer to as the 'Keynesian' model. The second 'Monetarist' model has output rapidly adjusting and a high interest seffli-elasticity (¢i -1 small, a z large). The remaining parameters are common to both models. The parameter values chosen for the simulations were: ~ka= 0.5, ~k~'--10, a 2 = 0, a~ = 0.5, a a = a~ = 0.3, a 3 = a~ =0.4, ~ 2 = 10, 0 = 0.7¢8 = 1, f l = l k =
0.5, a = b = 2, c = d = l, pl=p~=0.5. 3. Simulation results
Four policy regimes for the model in section 2 were computed based on parameter values given above. The four regimes were for states (M, M),
P. Levine, Policiesfor rational expectation models
96
Table 1 Welfare loss for policies and states with welfare function a s y E ( 2 y 2 + 2 p 2 + r 2 + s 2 ). State of orld a
(M, M)
(K, K )
(M, K)
Disturbance dul* du 2
Disturbance du I du 2
Disturbance du* 1 du 2
Disturbance du I du 2
(K, M )
(M, M)
0.078 (22%)
1.281 (50%).
1.780 (30%)
2.626 (43%)
0.078 (22%)
1.415 (48%)
2.000 (0%)
0.713 (60%)
(K, K)
0.100 (0%)
1.400 (44%)
1.586 (37%)
2.335 (497o)
0.100 (0%)
1.603 (41%)
1.848 (8%)
0.595 (67%)
Composite model with p[' =p[~ = 0.5
0.081 (19%)
1.307 (49%)
1.666 (34%)
2.449 (46%)
0.081 (19%)
1.456 (47%)
1.889 (6%)
0.641 (65%)
No control
0.100
2.571
2.528
4.602
0.100
2.728
2.000
1.827
~( A, B) d e n o t e s a world correctly described by model A for given expectations with the private sector using model B.
(M, K), (K, M) and (K, K) where (A, B) denotes the state in which the government uses model A and the private sector model B, the composite model with PiP -Pl-g = 0.5 (both sets of agents give equal weights to both models) and no control (r = s = 0). No control in fact gives a zero root for the model used so that a very slight feedback rule r = 0.0001p was introduced to give the model the required saddlepoint property. For each policy regime and for disturbances dul, du~' and du 2 of unit variances each singly hitting the economy, the welfare loss was computed when the true state of the world was (M, M), (K, K), (M, K) and (K, M). Thus a payoff matrix was constructed and the results are given in table 1. The percentage figures in brackets refer to the welfare gain from optimal control as compared with the no control outcome. The corresponding expected trajectories of the instruments for an initial output wage displacement are given in table 2. Table 2 Instrument trajectories for initial displacement v(0) = v*(0) = 1 with a welfare function a s y E ( 2 y 2 + 2 p 2 42 r 2 + s2). Policy designed for ( M M )
Year 0 1 2 3 4 5
r -
0.82 0.30 0.02 0.01 0.01 0.00
Policy designed for composite model
Policy designed for ( K, K ) s
- 0.65 0.03 0.02 0.01 0.01 0.00
r
-
0.00 0.15 0.17 0.15 0.12 0.09
s
r
s
- 0.72 - 0.127 - 0.06 0.03 0.05 0.05
- 0.34 0.00 - 0.04 - 0.05 - 0.04 - 0.03
- 0.72 - 0.12 - 0.02 0.02 0.03 0.03
P. Let,ine, Policies for rational expectation models
97
4. Conclusion This paper sets out a general method for deriving policies which are robust with respect to several RE models of the economy. A model of a small open economy, with overlapping wage contracts and with two alternative output relationships, provides an illustrative example of the method. Even though the two models have radically different properties and policy prescriptions taken on their own (see the trajectories in table 2), we are able to design a compromise policy that performs reasonably well irrespective of the true state of the world. References Becker, R.F., B. Dwolatzky, E. Karakitsos and B. Rustem, 1984, The simultaneous use of rival models in policy optimization, this issue. Buiter, W.H., 1983, Optimal and time-consistent policies in continuous time rational expectations models, Discussion paper no. A.39 (London School of Economics, London). Christodoulakis, N. and F. van der Ploeg, 1984, Macro-dynamic policy formulation with conflicting views of the economy, Paper presented to the sixth annual conference of the Society of Economic Dynamics and Control, Nice. Driffill, E.J., 1982, Optimal money and exchange rate policies, Greek Economic Review, Dec. Levine, P. and D. Currie, 1984, The design of feedback rules in linear stochastic rational expectations models, Paper presented to the sixth annual conference of the Society of Economic Dynamics and Control, Nice. Miller, M.H. and M.H. Salmon, 1985, Dynamic games and the time-inconsistency of optimal control in open economics, Economic Journal 85, 124-137. Snower, D.J. and A. Wierzbicki, 1983, Imperfect information, simplistic modelling and the robustness of policy rules, Discussion paper no. 131 (Birkbeck College, London).
THE FORMULATION OF ROBUST POLICIES FOR RIVAL RATIONAL EXPECTATIONS MODELS OF THE ECONOMY Paul LEVINE London Business School, London NI¥1 4S,4, England
I. Introduction
A problem of utmost importance in macroeconomics is how to design policy when faced with model uncertainty. A number of recent studies have addressed this problem [Becker et al. (1984), Christodoulakis and van de Ploeg (1984), Snower and Wierzbicki (1983)]. But the literature to date has not considered the problems that arise when some or all of the models available assume rational (i.e., model-consistent) expectations (RE). This will soon be of particular importance because the next generation of large econometric models in Britain should see at least four - those at the NIESR, the LBS, City University and Liverpool University - incorporating RE. This paper sets out and demonstrates a general method for designing policies compatible with more than one RE mod.el of the economy. 2. The model The model is of a small open economy with efficient financial markets, sluggishly adjusting output and wages but with forward-looking behaviour in the labour market. It consists of the following eight relationships, five dynamic and three static:
Y= a , ( E - P - P t ) -
a 2 ( R _ pe(t ' t ) ) - a3S,
(1)
Y),
(2)
i f = Ol~+ ( 1 - 0 ) ( E - P + P ' ) ,
(3) (4)
0165-1889/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
J.E.D.C. - D
P. Levine, Policies for rational expectation models
94
W = 8fl f ~ C [ W ¢ ( ~ , t) + ~b(Ye('r, t) - Y*)]e-'(*-t)dr
Jt
+(1 - fl) 14/,
(5)
1~'= 8(1,I,'- W),
(6)
M = P + T1Y- "y2R,
(7)
E=R-R
(8)
r,
where Y = real output, P = the domestic price level, E = the exchange rate (defined as the price of foreign exchange), S = autonomous taxes, W = the wage contract, l~'= the average wage, R = the domestic interest rates, we(% t ) = the expectation of W(~-) f o r m e d a t time t [with ye(~., t) and P(%t) defined similarly], Y* = the natural rate of real output, M = the nominal money stock, all measured in logarithms except interest rates which are proportions. Superscript f denotes a foreign counterpart. By the RE assumption we may put We(% t) = W(~'), ye(~., t) = Y(T) and jbe(~., t) = P('r). Then, in the long-run, P = O W = (1 - O)(E - P + p r ) , y = a l ( E - P + P f ) - a 2 R - a3S, Y = Y*, R = R r and (7). Given that the money stock and S are fixed in the long-run by the authorities, this then gives five relationships which determine the equilibrium values of R, Y, P, W and E. From this point onwards the analysis is conducted in terms of deviations about this long-run equilibrium. Denoting deviations from equilibria by the lower case, differentiating (5) we have. ¢v ~ = 8 f l # y + 28(1 - f l ) ( w - ~v).
(9)
In its stochastic form the model may now be written as
(lO) (dw)*
L(o.°)J where du 1 is a demand disturbance and du 2 a wage bargaining or wage drift disturbance. Substituting out for pe = p from (4) we arrive at a standard stochastic RE model for which the optimal policy may be found by standard methods [e.g., Driffill (1982), Buiter (1983), Miller and Salmon (1983), Levine and Currie (1984)].
95
P. Levine, Poficies for rational expectation models
Suppose now there exists a second RE model in which (1) and (2) are replaced (in deviation form) by dy* = ~ l * [ a l * ( e - p ) - a 2 * ( r - p e ) - a 3 * s - y * ] d t + d u l
*,
(11)
but the remaining relationships remain the same. Suppose that private agents believe the first model with probability Pl and the second with probability 1 - P i . The (9) must be replaced with
#e= _3fl~(ply + (1 - P l ) Y*) + 23(1 - fl)(w - fv).
(12)
The model to be controlled now consists of a composite model of the form
-]
"dy
dy* dp
] I
Y y,
[W]- L J
' = e p-
(dw) e
pe
d , + b [/ bj)*e : i ] dt÷
_(de) e .J
-
we
du~'
[r]dt+
dul 0
0
I
_0
_]
(13)
Note that only pe is involved in the second term, but we present the model in this form to indicate the nature of the general solution. We consider a 'pure' stochastic problem with the system in stochastic equilibrium. The objective function then consists of asymptotic variances which for this model we choose to be asyE(W) =asyE(p~ay2+ (1 -p~)ay*2+bp2+crZ+dgZ),
(14)
where p~ is the government's subjective probability that model 1 is correct. The two RE models for which a robust policy is being sought will have the following properties. Model 1, in which output is described by y following (1) and (2), has output slowly adjusting to its long-run partial equilibrium relationship (i.e., ~ki-1 large) and has a zero interest semi-elasticity (a 2 = 0). This we loosely refer to as the 'Keynesian' model. The second 'Monetarist' model has output rapidly adjusting and a high interest seffli-elasticity (¢i -1 small, a z large). The remaining parameters are common to both models. The parameter values chosen for the simulations were: ~ka= 0.5, ~k~'--10, a 2 = 0, a~ = 0.5, a a = a~ = 0.3, a 3 = a~ =0.4, ~ 2 = 10, 0 = 0.7¢8 = 1, f l = l k =
0.5, a = b = 2, c = d = l, pl=p~=0.5. 3. Simulation results
Four policy regimes for the model in section 2 were computed based on parameter values given above. The four regimes were for states (M, M),
P. Levine, Policiesfor rational expectation models
96
Table 1 Welfare loss for policies and states with welfare function a s y E ( 2 y 2 + 2 p 2 + r 2 + s 2 ). State of orld a
(M, M)
(K, K )
(M, K)
Disturbance dul* du 2
Disturbance du I du 2
Disturbance du* 1 du 2
Disturbance du I du 2
(K, M )
(M, M)
0.078 (22%)
1.281 (50%).
1.780 (30%)
2.626 (43%)
0.078 (22%)
1.415 (48%)
2.000 (0%)
0.713 (60%)
(K, K)
0.100 (0%)
1.400 (44%)
1.586 (37%)
2.335 (497o)
0.100 (0%)
1.603 (41%)
1.848 (8%)
0.595 (67%)
Composite model with p[' =p[~ = 0.5
0.081 (19%)
1.307 (49%)
1.666 (34%)
2.449 (46%)
0.081 (19%)
1.456 (47%)
1.889 (6%)
0.641 (65%)
No control
0.100
2.571
2.528
4.602
0.100
2.728
2.000
1.827
~( A, B) d e n o t e s a world correctly described by model A for given expectations with the private sector using model B.
(M, K), (K, M) and (K, K) where (A, B) denotes the state in which the government uses model A and the private sector model B, the composite model with PiP -Pl-g = 0.5 (both sets of agents give equal weights to both models) and no control (r = s = 0). No control in fact gives a zero root for the model used so that a very slight feedback rule r = 0.0001p was introduced to give the model the required saddlepoint property. For each policy regime and for disturbances dul, du~' and du 2 of unit variances each singly hitting the economy, the welfare loss was computed when the true state of the world was (M, M), (K, K), (M, K) and (K, M). Thus a payoff matrix was constructed and the results are given in table 1. The percentage figures in brackets refer to the welfare gain from optimal control as compared with the no control outcome. The corresponding expected trajectories of the instruments for an initial output wage displacement are given in table 2. Table 2 Instrument trajectories for initial displacement v(0) = v*(0) = 1 with a welfare function a s y E ( 2 y 2 + 2 p 2 42 r 2 + s2). Policy designed for ( M M )
Year 0 1 2 3 4 5
r -
0.82 0.30 0.02 0.01 0.01 0.00
Policy designed for composite model
Policy designed for ( K, K ) s
- 0.65 0.03 0.02 0.01 0.01 0.00
r
-
0.00 0.15 0.17 0.15 0.12 0.09
s
r
s
- 0.72 - 0.127 - 0.06 0.03 0.05 0.05
- 0.34 0.00 - 0.04 - 0.05 - 0.04 - 0.03
- 0.72 - 0.12 - 0.02 0.02 0.03 0.03
P. Let,ine, Policies for rational expectation models
97
4. Conclusion This paper sets out a general method for deriving policies which are robust with respect to several RE models of the economy. A model of a small open economy, with overlapping wage contracts and with two alternative output relationships, provides an illustrative example of the method. Even though the two models have radically different properties and policy prescriptions taken on their own (see the trajectories in table 2), we are able to design a compromise policy that performs reasonably well irrespective of the true state of the world. References Becker, R.F., B. Dwolatzky, E. Karakitsos and B. Rustem, 1984, The simultaneous use of rival models in policy optimization, this issue. Buiter, W.H., 1983, Optimal and time-consistent policies in continuous time rational expectations models, Discussion paper no. A.39 (London School of Economics, London). Christodoulakis, N. and F. van der Ploeg, 1984, Macro-dynamic policy formulation with conflicting views of the economy, Paper presented to the sixth annual conference of the Society of Economic Dynamics and Control, Nice. Driffill, E.J., 1982, Optimal money and exchange rate policies, Greek Economic Review, Dec. Levine, P. and D. Currie, 1984, The design of feedback rules in linear stochastic rational expectations models, Paper presented to the sixth annual conference of the Society of Economic Dynamics and Control, Nice. Miller, M.H. and M.H. Salmon, 1985, Dynamic games and the time-inconsistency of optimal control in open economics, Economic Journal 85, 124-137. Snower, D.J. and A. Wierzbicki, 1983, Imperfect information, simplistic modelling and the robustness of policy rules, Discussion paper no. 131 (Birkbeck College, London).