Linear Rational Expectations Models: a Frequency-Domain vs. Optimal Control Approach to Policy Design

Copyright © (FAC Dynamic Modelling and Control of I\ational Economies, Budapest, Hungary 1986

LINEAR RATIONAL EXPECTATIONS MODELS: A FREQUENCY -DOMAIN VS. OPTIMAL CONTROL APPROACH TO POLICY DESIGN N. Christodoulakis* and P. Levine** "DI'/I!I rl III I'll 1 of A/ljJlil'd t:CO 11 0111 il'l, C(lIlIiJridgl' ['lIh'l'nil.\', CfllII/llit/gl' UiJ """C"II/I'I' /i)r }:cO)){)lIIic Forl'((Islillg, LO)l{/oll

<)/)1:'. ['/\.

IIl1silll'l.1 Sr/II){)/,

LOIlt/oll, ,\'\rl -IS,4, L'/\

ABSTRACT The paper shows how frequency-domain methods can be used in policy design for linear rational expectations mode ls. A parallel solution with optimal control methods is used to provide guidelines for simple feedback rules. These are then designed in the frequency-domain with the additional requirement of being robust with respect to initial displacements. Employing a linear model with perfect capital mobi l ity and forward looking expectations in the l abour and financial markets, both op timal and sub-optimal simple rules are in turn derived and compared . Keywords. Economic Policy; Rational Expectations; Optimal Contro l; Frequency responses; Robustness.

1.

INTRODUCTION

under rational expectations . Section 3 describes how a parsimonious feedback structure can be derived in the frequency -domain and contains the basic guide lin es for the design to be applied in a rational expectations environment.

This paper is concerned with the design of economic policies for mode l s where expectations are rational in the sense that they are consistent with the model itself. A crucia l standard assumption is that the policy rules are always announced by the Government, and consequentl y the response, or

Section 4 sets out a smal l economic model for an open- economy, where perfect capital mobility is assumed and agents in the capital and labour markets have forward -looking expectations.

Itreaction function lt , of the private sec t or is

derived as the unique saddlepath solut i on that cancels the structural instability of the system. As Currie and Levine ( 198 5) have already argued , this situation calls for policy structu re s which are simple and parsimonious so they can be convinc ingly announced to and monitored by the public.

Assuming that the regulation of real output and price level is the principal policy objective, in Section 5 a welfare loss function is specified and optimal fiscal and monetary policies are derived emp l oying the techniques described in Section 2 . An alternative simple design based on frequency - domain attacks the problem of regulating the same targets. Finally Section 6 summarises the main findings .

Methods based on optimal control, where a welfare loss function is minimised, have so far constituted the main armoury of policy formation exercises . Within this framework the full optimal policy may be obtained as a linear feedback rule which is ind ependen t of the initial displacement (and, in a stochastic environment, the covariance matrix

2.

of the disturbances). If, however, rules are constrained to be parsimoniou s in structure this invariance is lost. A problem of designing rules which are r obust with respect to initial displacements (and in the stochas tic case , covariances) then emerges.

POLICY DESIGN IN THE TIME-DOMAIN

The economic systems considered in this paper have, as general form the following lin ear deterministic differential equation

A

An alternative approach to policy design is based on frequency -domain techniques. Altho ugh there is no explicit specification of a welfare function, the design seeks to achieve a closed-loop system that regulates a given vector of target variables by making sparse use of information about their behaviour.

[:]

+ B w

(2.1)

where z is an (n - m) x 1 vector of predetermined variables , x is an m x 1 vector of nonpredetermined variables which can freely jump in response to

tl

news " and w is an r x 1 vector of

control instruments. The matrices A and B have time-invariant coefficients. All variables are measured as deviations from some long-run trend deterministic equilibrium . Exogenous variables that follow ARIMA processes may be incorporated into (2.1) by a suitable extension of the vector z. The initial conditions of the system at time o are given by z(O).

Additionally, the design may well take into account all the possible dynamic implications of the suggested policies and result in a sys t em which is relatively robust to initial displacements. As such methods have been mainly used so far to design stable closed-loop systems some modifications of the basic tools is needed before applying them to a rational expectations context.

The controller is concerned about target variables:

Avoiding premature genera li sations , the present paper tries to shed some li ght on the advantages , differences and trade-offs between optima l contro l and frequency - domain techniques(l). This we hope, may eventually lead to a fruitful combination of the two approaches.

q

Cy + Dw

where q is an r x 1 vector , y

(2 . 2)

[:]

and C and

D are time-invariant matrices . The we l fare loss to be minimised at time t = 0 is assumed to be of the form W(z(O)) where

Section 2 summarises the optimal control solution for a l inear system in a deterministic environment

43

N. Christodoulakis and P. Levine

44

From (2.5) and (2.7) we have tha t (2.3)

W(z(O)) = 0.5

[:,]

where p ~ 0 is a discount factor, Q and R have time-invariant coefficients R is symmetric and positive definite, Q is symmetric and non-negative definite, and T denotes transposition. The control position is then to minimise (2.3) subject to (2.1) and (2.2). This has been the subject of a number of recent papers (see Calvo (1978), Driffill (1982), Buiter (1983), Miller and Salmon (1984), Levine and Currie (1984)). The solution, employing Pontryagin's maximum principle, may be outlined as follows. Let us define the Hamil tonian

a

where

[:,]

a

['11 n

4l

n n

14 44

(2.9)

nl3 n

43

n12] N n 42

T T T T T been partitioned conformably with ~z x . P2 P2 ] Then the open-loop form, of (2.8) 1S glven by (2.10)

(2.4) where A(t) is a 1 x n row vector of costate variables. The first order conditions for a m1n1mum are aH/aw=O and A = -aH/ay. Differentiating (2.4), this leads to the following dynamic system under optimal control

A possible difficulty with the feedback form of policy is that it is defined in terms of the nonobservable variable P2' To avoid this we may integrate the lower bIock of (2.9) to obtain

( ) 1o t

P2

t

R + DTQD, Q* = CTQC and

p = e PtA T . The 2n boundary conditions are given by z(O), P2(0) = 0 (partitioning p =~n ) and the transversality condition lim

e- pt pet)

O.

(2 . 6 )

conformally with [zT

(2.7)

Ml~

partitioned conformably with

M2J

[yT pT]T is the matrix of left eigenvectors of n arranged so that the last n rows are associated with the unstable eigenvalues . Using (2.6) and (2.7) the optimal policy may be expressed in feedback form as w

where K

Hence from (2.7) integrating and using W(O)=O we obtain W(z(O)) = -0.5z (O)Nllz(O)

~J

.

-0.5tr(N

11

Z(O)) (2.13)

where Z(O) 3.

POLICY DESIGN IN THE FREQUENCY-DOMAIN

The previous section showe d how an optimal control law of the type (2 . 10) is derived that drives the system towards the desired equilibrium on an optimal trajectory. A feature of (2 . 10) is that the manipulation of policy instruments wet) in general requires information on all the states of the system, therefore making the policy rule complicated from the viewpoint of implement-· ation.(3) The design of parsimonious policy structures can however be effectively dealt with by pursuing a control design based on frequencydomain techniques. Central in the frequency-domain framework is the concept of the transfer-function operator that relates input-output vectors in their Laplace transform q(s) = G(s)w(s) where s is the complex frequence operator and G(s) = C(sI-A)

N

P2 T]T .

(2.12)

T

Assuming that the dynamic matrix in (2.5) has the saddlepoint property (i.e. exactly n eigenvalues with positive real part), the solution to (2.6) takes the form

ruhl

(2.11)

T

dW(z(O)) dz (0)

The

optimal policy itself is given by

M

d

The welfare loss under optimal policy at time t = 0 can be found as follows. By Pontryagin's maximum principle we have

t-

w =

a

a2l z ( T )

Then (2.8) together with P2 given by (2.11) expresses the optimal rule as a feedback on the current value of z plus an integral feedback on a discounted linear combination on past values of z.

(2.5) =

e a22 (t-T)

=

partitioning

say, where U = CTQD, R*

and n has

-1

B +D

(3.1)

is a Dilltrix of ratios of polynomials in s. Using only input-output relations the design problem is concerned with the derivation of dynamic policy rules pes) that will drive the target vector q towards a desired level q*. Denoting measurement errors in q by m we have that instruments vary according to the feedback law:

45

Linear Rational Expec tations Models w(s) = P(s) (q*(s) + m(s) - q(s) )

This results in a new closed-loop transfer function between target vector q, desired levels q* and unanticipated exogenous disturbances ~ impinging upon q, given by q(s) = H(s) (m(s)+q*(s)) +

(I-H(s))~(s)

(3.3)

where H(s) = [I + G(s) p(s))-l G(s)p(s) We observe from (3.3) that by making H(s)+ I one simultaneously achieves the desired target and effectively suppresses the exoBenous disturbances. This however is not feasible in high frequences (s+oo) as it would imply a strong policy reaction to noisy data and have a destabilising effect in the system. Instead, one chooses to satisfy H(s)+ I only in the low freqency region (s+O) which corresponds to precisely achieving the targets in the long-run (t+oo). A convenient assumption in the present exercise is that the model (2.1) is linearised around the long-run equilibrium therefore having q*=O. This enables us to concentrate on efficient regulation from initial conditions. Going back to the optimal controller (2.1 0 ), taking Laplace transforms we have that (3.4)

which has the familiar proportion-plus-integral structure. Guided by this we shall choose p(s) in (3.2) to be of the form P (s)

P

P

In addition we shall derive

+

s-"

simple rules with matrices Pp and PI sparse.

A design criterion for saddlepath solutions

When the policy rules P(s) are meant to be applied in a Rational Expectations context, they have to produce a number of closed-loop unstable poles of the matrix transfer function H(s) equal to the number m of free variables, in order to satisfy the saddlepath condition. Denote the stability matrix of the open-loop and closed-loop system by A and A respectively. We define now the numbers of pole~ that lie in the closed region of the right-half-plane, (RHP) i.e., including eigenvalues at the origin or on the imaginary axis, as follows: mo

number of RHP eigenvalues of A

m k

RHP poles introduced by p(s)

Applying now the Principle of the Argument in the same way as in Macfarlane and Postlethwaithe (1977) we can state the following Lemma of the Generalised Nyquist Criterion assuming that the system has no unstable uncontrollable and/or unobservable modes. Lemma: The number of anti~lockwise encirclements of the critical point (-1 + jO) of the complex plane by the characteristic loci of G(s)p(s) is equal to N(-l) = m o

(3.5)

When the policy rules P(s) are designed to yield a closed-loop system with the unique saddlepath solution, the above Lemma requires: N(-l) = mo + ~ - m

(3.6)

where, we recall, m is the number of non-predetermined variables. Thus when the designed controller is input-output stable (~=O) and the open-loop system . satisf~es mO=m, we must ha~e.no anticlockw1se enc1rclements of the cr1t1cal point. 4.

APPLICATION

The model employed to illustrate the design techniques of the previous two sections is of a small open economy with perfect capital mobility, sluggishly adjusting output and wages. It consists of the following relationships

Y

Wl ("1 (~~+e-~c) - " 2(r- p e)+" 3v - " 4u +" 5Y*-y) (4.1)

This

will give policy rules which are partly decoupled. The input-output stability of the system and the degree of robustness are tested by plotting the open-loop characteristic loci, i.e. the eigenvalues of the loop-gain L(j w) = G(j w) P(j w) for all frequencies w, and checking their position relative to the critical point (-l+jO). An extended recent account of how frequency-domain methods can be used in deriving feedback policies can be found in Vines et al (1983, Part 4) and also in Christodoulakis and Van der Ploeg (1987). What the frequency-domain approach notably lacks is an explicit objective function in the time-domain as that in (2.3). However, seeking good performance in tracking the targets, and stability margins that guarantee a minimum degree of robustness assumes an implicit optimisation that trades-off speed in response with robustness. The design can be performed with the aid of a computer-aided-design package and be based on the optimal approximation of a desired closed-loop matrix H(s) specified in the frequency domain, (See Maciejowski and Macfarlane (1982) and Edmunds (1979)). 3.2.

mc = RHP eigenvalues of Ac

(3.2)

i.e. by using information only about the behaviour of the controlled output vector.

-~ u-~ y+~ (~*+e-~ )+~ y*-p o 1 2 c c 3

W c w

c

=- S6Wy+2(1- S)6(w -~ ) c c 6(w -~ ) c

c

(4.2)

(4.3) (4.4)

(4.5)

~*c

(4.6) (4.7)

(4.8 )

e = r-r*

(4.9)

where y = real output, r = the nominal rate of interest, v = real net financial wealth of the private sector, u = real autonomous taxation,

p = the ge~eral price level, w = the wage contract, w = the weighted ge8metric average of past wag~ contracts and e = the nominal exchange rate (defined as the price of foreign exchange). All variables are measured in terms of the deviations of their logarithm from equilibrium (which may incorporate trends) except for interest rates which are measured as deviations of proportions. All parameters are positive. A'*' superscript denotes a foreign counterpart to the variable in question. Equation (4.l) is an IS curve with output adjusting sluggishly with mean lag w- l to l

46

N. Christodoulakis and P. Leyinc

competitiveness ~*+e-~, the real interest rate r-pe , real financial wealth v, autonomous taxes u and foreign demand y*. Equation (4.2) determines the change in real wealth from the sum of the government budget constraint and the current account of the balance of payments (but neglects interest payments). Equation (4.3) is a continuous time analogue of the overlapping contract model of Taylor (1980). It may be derived from a model which assumes wage contracts with random durations with the probability that an outstanding contract at time t will last k more periods being e- ok (Calvo (1982)). Thus the expected contract length is 0-1 Equation (4.4) defines the average wage w the weighted geometric average of past wage c5ntracts. Equation (4.5) gives the average wage as a weighted geometric average of past wage contracts. Equation (4.5) gives the price level adjusting to a long-run weighted geometric average of domestic and foreign average wages. Equations (4.6) to (4.8) specify exogenous firstorder autoregressive processes for foreign average wages, the foreign interest rate and foreign output respectively. Finally (4.9) models the exchange rate as asset market determined under conditions of perfect capital mobility and perfect substitutability between domestic and foreign bonds. The model mal be written in the form (2.1) with z = [y y* v w p ~* r*lT as the predetermined variables, x ~ [w ~]T the non-predetermined or "jump" variables and w = [u r]T as the instruments. For the design of policy in the time domain we choose outputs q = [y p]T, no time discounting (p = 0) and Q ~ ~ R=I in (2 03). Then (4.1)

The parameters in the model are given the following values: a = b = 2, ~l = ~2 = as = ~3 = S = ~ = 0.5,

~4

10.0,

a 2 = ~2

0.1,

a

~O

~

3

l = 0.3, = 0 = 1.0, a 4 a

0.4 and

= 1.3

5.

SIMULATIONS AND DISCUSSION

5.1

The Time-Domain

We first report, for each welfare loss function, the eigenvalues, the welfare loss at time t = 0 and the feedback rule for the system under optimal control. The system under control is of order 18 with 9 predetermined variables and 9 nonpredetermined variables. As required for the saddlepath property there are 9 stable roots (9 eigenvalues with negative real part) and 9 unstable roots. These are: ±9.89, 3x(±0.50), ± 0.51 ± 0.87j, ± 0.68, ± 0.20, ± 0.13. Now consider the welfare loss for the two initial displacements y(O) = -1 and W(0)=1. These indicate the relative importance 5f different disturbances to the economic system and in particular the greater impact of demand shocks as opposed to supply shocks. A further use to which the welfare loss under optimal control may be put is as a benchmark against which sub-optimal rules can be assessed. For the system under optimal control the welfare losses turn out to be 0.69 for the output ~hock y(O)= -1 and 0.12 for the 'supply shock' w (0) =1. c To appreciate the need for simple but sub-optimal rules we turn to the optimal policy in feedback form. For the model of section the feedback rule (2.8) becomes

[:]

r2l]

5 e 22 (t-1) 9 21 [y(1)y*(1)v( 1)w- c *(1) ]Td1 9

P22

0

and where K =

~."

0.22 0.35 -0.23 0.30 -0.01 -0.02 -0.12

.07 0.00 0.01 -0.00 0.00

0.00

0.00

0.06] 0.01 -1.00

and

9 = 2

~'05

0.17 0.22 -0.23

.

0.21 -0.05 -O.ll :-0.17

.

~'BOl

.05 0.01 0.02 -0.04 -0.47 -0.09 -0.18; 0.01 -0.36

[9

21

: 9 22 ]

Clearly the policy-maker will not find it easy to implement policy in this form. The private sector would find it even more difficult to monitor if the authorities did manage to implement the rule. In view of these problems, do our results suggest approximate rules which perform almost as well as the optimal rule but which are simple in design? The values for K and 9 do suggest a rule of the form u = YlY + Y2v + Y3P, r = -Y 4P 22 where P22 -1

to (4.9) can be put in the form of (2 01).

ill = il3

where

(S ol)

-Y P - Y6P 22' This gives r Y (Y 4 Y6 Y - r) or 5 5 6 sluggish adjustment towards r = Bp where -1

8 = Y4Y 6Y5

~

1.3.

A simple proportional price

rule of this form was found by Currie and Levine (1985) to perform extremely well for a model similar to that utilised in this paper. It is possible to us e time-domain techniques to find optimal parameter values for the simple rule described above. This involves the computation of the welfare loss using Lyapunov's equation and the use of a standard numeral minimisation method (Levine and Currie (1984). However, the optimal parameter values will then depend on the initial displacements which is not a feature of the full optimal rule. Fortunately frequen cy domain techniques, to which we now turn, provide a powerful tool for designing a simple rule which is robust in the sense that it performs well irrespective of the initial displacements. 5.2

A Frequency-Domain Design

The structure of the op timal feedback rules suggests that a parsimonious rule where taxes (u) change only in response to changes in real output (y), real wealth (v) and the prices level (p) while the interest rate (r) responds only to changes in the price level. We shall now use the frequencydomain techniques to design a general proportionalintegral type of controller with these features. To keep th e rule rea sonably simple we shall restrict the tax rule further to one that does not depend on v. The open-loop model can easily be shown to have the following eigenvalues {-10, 3x(-0.50), -.31 ± jO.8l, -0.82, 0, 0.20 } therefore making the number of RHP poles m = 2. Given that we do not introduce integra~ control action, we have that ~ = 0 and therefore condition (3.6) requires N(-lj=2+0-2=0 anticlockwise encirclements of th e critical point. (The system has 3 uncontrollable modes with corresponding stable eigenvalues -0.50).

Linear Ratiunal Expectatiuns Mudels Specifying a desired closed loop diagonal matrix as s + 0.40

H(s)

(5.1)

/+1.15s+0.40 and applying frequency-optimisation algorithms available in the Cambridge Linear Analysis and Design Package (CLADP) we finally arrive in the following dynamic structure: 1 [(-1.12S-0.68) P(s)=-s+0.9 0

1

(0.50s+0.20) (-2.48s-l.06)

(5.2)

The policy rules implied by this pes) can be written in the time-domain as:

Jo t

u=-0.9

t

ud T+l.12y+0.68

)~d T -0.50P-0.20

Jord T+2.48P+l.06

t

So

pdT

welfare losses from their reference level as to effectively blur the distinction between optimal and suboptimal rules. Given the appealing neatness of the latter, one may thus opt for them when facing uncertain model specifications. 6.

a.

When there is a clear choice of which targets are to be pursued through a given set of policy instruments, the optimal control solution may provide some strong indication for simpler policy structures. Frequencydomain methods can then be employed to derive them while trying to achieve robustness with respect to initial displacements.

b.

The welfare loss function can still be used in frequency-domain not so much as design criterion any more but as an indication of how far one wishes to trade off simplicity versus optimality.

c.

The two approaches therefore should be considered as complementary ~ather than alternatives in policy design.

(5.3)

The closed-loop system H(s) has the stable poles {-11.16, 3x(-0.50), -.44±j 1.02, -.82, -.57, -O.ll} while the unstable pair is now {0.82, 0.20}. Figure 1 depicts the characteristic loci that ensure the proper number of anti-clockwise encirclements (=0) of the critical point. The fact that they keep well away from that (actually they lie outside the M = 1.4 circle) provides the system with good stability margins against uncertainty. Table 1 shows the key trajectories for the system under optimal control and under the simple controller (5.3) in response to initial unit displacements in real output y and the average wage contract ~. The welfare losses for the simple ru~ corresponding to these displacements are 0.99 and 0.17. These are naturally higher than their optimal counterparts, (0.69 and 0.12 respectively), but only to an extent which may well justify the simplicity in design. 5.3

A test of Robustness

As a way to evaluate the performance of the derived policy sets when they face model uncertainties, we try them under different assumptions about the wage settling regime of equations (4.3) and (4.4). As opposed to the central values of S=0.5 and 0=1.0 we choose two possible alternatives: (a) Sticky wages: B= 0.25, (b) Flexible wages: B= 1,

0= 0.50 0= 2.0

Both the optimal rule (5.1) and the simple rule (5.3) derived on the assumption of central values are applied, and their corresponding welfare losses are reported in Table 2, for the same initial shocks in demand and supply as in Table 1. Welfare losses due to shock in output, do not change markedly from their counterparts in the central case. This is explained by the fact that the suggested parameter changes concern the mechanism of wage adjustment and leave the real economy relatively unaffected. Under a shock in supply, the welfare loss is naturally worsening in a more sticky environment and is improved whenever wages become flexible. Comparing the performance of the two rules, one observes that the optimal policy under the inflexible regime, is barely distinguishable from its simpler alternative. As we now turn to wage flexibility, the optimal rule allows a better dynamic behaviour in the short term and the losses are improved over the simple policy structure. It is worth noting, however, that the presence of model uncertainties may easily so increase the

CONCLUSIONS

The paper has discussed a framework in which frequency-domain methods can be fruitfully used in policy design under rational expectations in order to obtain a controlled economy capable of effectively suppressing exogenous shocks. A parallel exercise with optimal control methods was used both as an indication of how simple policy structures can be specified for the frequency-domain problem, and also as a benchmark of how well one can expect these derived policies to work. Our main findings may be briefly stated as follows:

t

r= -0 09

47

REFERENCES Buiter, W.H. 1983, "Optimal and Time-Consistent Policies in Continuous-time Rational Expectations Models", LSE Discussion Paper No. A39. Christodoulakis, N. and Van der Ploeg, F. 1987. "Macrodynamic Policy Formulation with Conflicting views of the Economy: A synthesis of optimal control and feedback design", Int. Journal of System Science (forthcoming). Currie, D. and Levine, P. 1985, "Simple Macroeconomic Policy Rules in an Open Economy", Economic Journal Supplement. Edmunds, J.M. 1979, "Control Systems Design and analysis using O.L. arrays Int. Journal of Control, Vol. 30, No. 5, pp. 773-802. Levine, P. and Currie, D.A. 1984, "The Design of Feedback Rules in Stochastic Rational Expectations Models", presented to the Nice Conference, QMC PRISM paper No. 20. MacFarlane A.G.J. and Postlewaith 1., "The Generalised Nyquist Stability Criterion and Mu1tivariable Root Loci", Int. Journal of Control, Vol. 25, No. 1, pp. 81-127. Maciejowski, J.M. and MacFarlane A.G.J., 1982. "CLADP": The Cambridge Control Systems Magazine, Vol. 2, No. 4, pp. 3-8. Miller, M. and Salmon, M., "Dynamic Games and the Time inconsistency of Optimal Policy in Open Economies", CEPR Discussion Paper No. 27, London. Tay10r, J.B., (1980). "Aggregate Dynamics and Staggard Contracts", Journal of Political Economy, vol. 88, pp. 1-23. Vines, D., Maciejowski, J. and Meade, J., 1983. "2.4.86 Demand Management Stagflation Vol. 2)", AlIen and Unwin, London. Westaway, P. and Maciejowski, J.M., 1983, "A comparison of frequency-domain and optimal control methods for the design of a macro-economic feedback regulation policy", in T. Baser (ed.) Modelling and Control of National Economics, Pergammon Press, Oxford.

48

N. Christodoulakis and P. Lcyille

a) Initial Displacement y(O) = -1;

NOTES

We lfare Loss 0.68 (controller (i); Welfare Loss = 0.99 (controller (ii»; Time (ye a rs)

o

(ii) -1.00 -0.30 0.28 0.32 0.10 -0.07

-1.00 -0.34

1 2

4

0.11 0.21 0.13

5

0.02

3

(ii) 0.00 -0.05 -0.03 0.01 0.02 0.02

(i)

0.00 -0.04 -0.01 0.02 0.03 0.02

3

4 5

b)

(ii)

-0.44 0.18 0.28 0.13 -0.01 -0.07

1 2

(ii) 0.00 0.14 0.25 0.21 0.14 0.09

(i) 0.00 -0.01 -0.01 0.14 0.16 0.14

r

u

(i)

o

w c

p

y

(i)

(1 ) A comparison between frequency-domain and

-1.10 -0.20 0.35 0.29 0.03 -0.11

(i)

(ii) -0.03

-0.07 -0.01 -0.01 -0.01 -0.01 -0.00

-0.09

-0.02 0.04 0.05 0.03

Initial Displacement w(O)=l Welfare Loss 0.12 (Controller (i»; Welfare Loss = 0.17 (Controller (ii»;

Time (years)

o 1 2 3

4 5

y

0.00 -0.16 -0.15 -0.08 -0.03 -0.01

-0.19

-0.19 -0.08 0.00 0.01

(i) 0.00 0.04 0.00 0.01 -0.01 -0.00

o

-0.23

1

-0.19

2

-0.08 -0.01 0.00 -0.01

3 I,

5

Table 1.

c

(ii)

(i)

(ii)

0.00 0.07 0.06 0.05 0.05 0.05

1.00 0.84

1.00

0.71

0.63 0.56 0.50

0.75 0.67 0.62 0.57 0.51

r

u

(i)

w

p

(ii) 0.00

(i)

(ii)

(i)

(ii) 0.07 0.13 0.08 0.06

-0.003 -0.00 -0.21 -0.05 -0.17 0.07 0.07 -0.05 0.07 0.01 0.01 0.06

0.06

0.06

Time Responses to Initial Displacements

~ Output (y) and the average wage (wcl.

(i) = Optimal Controller:

Optimal Rule

Simple Rule

y(O)= -1 W,(O)= 1

0.68 0.12

0.99 0.17

y(O)= -1 we(O)= 1

0.71 0.66

1.02 0.665

y(O)= -1 we(O)= 1

0.63 0.05

0.93 0.10

Assumptions

Displacement

Central

~ ~

Flexible

~

(ii) = Simple Controller

Table 2. Welfare losses under different model assumptions. 2,8

~ IrtA (T

,'\ f'".-., :'.c.~i....

··2 Y - - - - - -

~.

The Nyquist diagram.

---n

optimal control methods has been presented by Westaway and Maciejowski (1983) but there was no consideration of rational expectations and the associated problems of saddle-path cond i t ions. (2) Although we only consider the deterministic control problem in this paper it is quite straightforward to generalise our results to the stochastic case (see Levine and Currie (1984) . (3) There also are cases where the use of information on states may be completely incomprehensible. This arises when the linear model is obtained as a reduced order linear approximation of a large econometric model. The states of the reduced system are then compressed combinations of the original economic variables and bear no me aningful interpretation.