Open access peer-reviewed chapter

Optimal Economic Stabilization Policy under Uncertainty

By André A. Keller

Published: October 1st 2009

DOI: 10.5772/8222

Downloaded: 1349

1. Introduction

A macroeconomic model can be analyzed in an economic regulation framework, by using stochastic optimal control techniques [Holbrook, 1972; Chow, 1974; Turnovsky, 1974; Pitchford & Turnovsky, 1977; Hall & Henry, 1988]. This regulator concept is more suitable when uncertainty is involved [Leland, 1974; Bertsekas, 1987]. A macroeconomic model generally consists in difference or differential equations which variables are of three main types: (a) endogenous variables that describe the state of the economy, (b) control variables that are the instruments of economic policy to guide the trajectory towards an equilibrium target, and (c) exogenous variables that describe an uncontrollable environment. Given the sequence of exogenous variables over time, the dynamic optimal stabilization problem consists in finding a sequence of controls, so as to minimize some quadratic objective function [Turnovsky, 1974; Rao, 1987]. The optimal control is one of the possible controllers for a dynamic system, having a linear quadratic regulator and using the Pontryagin’s principle or the dynamic programming method [Preston, 1974; Kamien & Schwartz, 1991; Sørensen & Whitta-Jacobsen, 2005]. A flexible multiplier-accelerator model leads to a linear feedback rule for optimal government expenditures. The resulting linear first order differential equation with time varying coefficients can be integrated in the infinite horizon. It consists in a proportional policy, an exponentially declining weighted integral policy plus other terms depending on the initial conditions [Turnovsky, 1974]. The introduction of stochastic parameters and additional random disturbance leads to the same kind of feedbacks rules [Turnovsky, 1974]. Stochastic disturbances may affect the coefficients (multiplicative disturbances) or the equations (additive residual disturbances), provided that the disturbances are not too great [Poole, 1957; Brainard, 1967; Aström, 1970; Chow, 1972; Turnovsky 1973, 1974, 1977; Bertsekas, 1987]. Nevertheless, this approach encounters difficulties when uncertainties are very high or when the probability calculus is of no help with very imprecise data. The fuzzy logic contributes to a pragmatic solution of such a problem since it operates on fuzzy numbers. In a fuzzy logic, the logical variables take continue values between 0 (false) and 1 (true), while the classical Boolean logic operates on discrete values of either 0 or 1. Fuzzy sets are a natural extension of crisp sets [Klir & Yuan, 1995]. The most common shape of their membership functions is triangular or trapezoidal. A fuzzy controller acts as an artificial decision maker that operates in a closed-loop system in real time [Passino & Yurkovich, 1998]. This contribution is concerned with optimal stabilization policies by using dynamic stochastic systems. To regulate the economy under uncertainty, the assistance of classic stochastic controllers [Aström, 1970; Sage & White, 1977, Kendrick, 2002] and fuzzy controllers [Lee, 1990; Kosko, 1992; Chung & Oh, 1993; Ying, 2000] are considered. The computations are carried out using the packages Mathematica 7.0.1, FuzzyLogic 2 [Kitamoto et al., 1992; Stachowicz & Beall, 2003; Wolfram, 2003], Matlab R2008a & Simulink 7, & Control Systems, & Fuzzy Logic 2 [Lutovac et al., 2001; The MathWorks, 2008]. In this chapter, we shall examine three main points about stabilization problems with macroeconomic models: (a) the stabilization of dynamical systems in a stochastic environment, (b) the PID control of dynamical macroeconomic models with application to the linear multiplier-accelerator Phillips’ model and to the nonlinear Goodwin’s model, (c) the fuzzy control of these two dynamical basic models.

2. Stabilization of dynamical systems under stochastic shocks

2.1 Optimal stabilization of stochastic systems

2.1.1 Standard stabilization problem

The optimal stabilization problem with deterministic coefficients is presented first. This initial form, which does not fit to the application of the control theory, is transformed to a more convenient form. In the control form of the system, the constraints and the objective functions are rewritten. Following Turnovsky, let a system be described by the following matrix equation

Yt=A1Yt1+A2Yt2+...+AmYtm+B0Ut+B1Ut1+...+BnUtnE1

The system (1) consists inq1target variables in instantaneous and delayed vectorsYandq2policy instruments in instantaneous and delayed vectorsU. The maximum delays aremandnforYandUrespectively. The squaredq1×q1matricesAare associated to the targets, and theq1×q2matricesBare associated to the instruments. All elements of these matrices are subject to stochastic shocks. Suppose that the objective of the policy maker is to stabilize the system close to the long-run equilibrium, a quadratic objective function will be

t=1(YtY¯)M(YtY¯)+t=1(UtU¯)N(UtU¯)E2

whereMis a strictly positive definite costs matrix associated to the targets andNa positive definite matrix associated to the instruments. According to (1), the two setsY¯andU¯of long-run objectives are required to satisfy

(Ij=1mAj)Y¯=i=0nBiU¯E3

Letting the deviations beYtY¯=ytandUtU¯=ut, the optimal problem is

                           minu(t=1ytMyt+t=1utNut)s.t.yt=A1yt1+A2yt2+...+Amytm+B0ut+B1ut1+...+BnutnE4

2.1.2 State-space form of the system

The constraint (2) is transformed into an equivalent first order system [Preston & Pagan, 1982]

xt=Axt1+BvtE5

wherext=(ytyt1yt2,...,ytm+1utut1,...,utn+1)is theg×1state vector withg=mq1+nq2. The control vector isvt=ut. The block matrixAand the vectorBare defined by

Any stabilization of a linear system requires that the system be dynamically controllable over some time period [Turnovsky, 1977]. The condition for the full controllability of the system states that it is possible to move the system from any state to any other.

Theorem 2.1.2 (Dynamic controllability condition). A necessary and sufficient condition for a system to be dynamically controllable over some time period

Tgis given by the dynamic controllability condition
rank(B|AB|...|Ag1B)=gE6

Proof. In [Turnovsky, 1977], pp. 333-334.

The objective function (3) may be also written as

t=1xt'M*xt+t=1vt'NvtθE7

whereθincludes pasty’s andu’s beforet=1. LettingM˜=M/mandN˜=N/n, the block diagonal matrixM*is defined by

The stabilization problem, (2) is transformed to the control form

minv(t=1xt'M*xt+t=1vt'Nvt)s.t.xt=Axt1+BvtE8

Since the matricesM*andNare strictly positive, the optimal policy exits and is unique.

2.1.3 Backward recursive resolution method

Let a formal stabilization problem be expressed with a discrete-time deterministic system

minxt=1T(yt'Myt+xt'Nxt)MN0s.t.yt=Ayt1+BxtE9

In the quadratic cost function of the problem, thenstate vectoryand themcontrol vectorxare deviations from long-run desired values, the positive semi-definite matricesMn×nandNm×mare costs with having values away from the desired objectives. The constraint of the problem is a first order dynamic system [1] -with matrices of coefficientsAn×nandBn×m. The objective of the policy maker is to stabilize the system close to its long-run equilibrium. To find a sequence of control variables such that the state variablesyt can move from any initialy0to any other stateyT, the dynamically controllable condition is given by a rank of a concatenate matrix equal ton

rank(B|AB|...|An1B)=nE10

The solution is a linear feedback control given by

xt=Rtyt1E11

where we have

Rt=(N+B'StB)1(B'StA)E12
St1=M+Rt'NRt+(A+BRt)'St(A+BRt)E13
ST=ME14

The optimal policy is then determined according a backward recursive procedure from terminal stepTto the initial conditions, such as

stepT:ST=MRT=(N+B'STB)1(B'STA)E15
stepT-1:ST1=M+RT'NRT+(A+BRT)'ST(A+BRT)RT1=(N+B'ST1B)1(B'ST1A)E16
step 1 :       S1=M+R2'NR2+(A+BR2)'S2(A+BR2)R1=(N+B'S1B)1(B'S1A)E17
step 0 : S0=M+R1'NR1+(A+BR1)'S1(A+BR1)R0=(N+B'S0B)1(B'S0A)E18

2.1.4 The stochastic control problem

Uncorrelated multiplicative and additive shocks: The dynamic system is now subject to stochastic disturbances with random coefficients and random additive terms to each equation. The two sets of random deviation variables are supposed to be uncorrelated. [1] -

The problem (3) is transformed to the stochastic formulation ( also [Turnovsky, 1977]).
minxE[ytMyt+xtNxt]s.t.yt=(A+Φt)yt1+(B+Ψt)xt+εtMN0E19

The constant matricesAn×nandBn×mare the deterministic part of the coefficients. The random components of the coefficients are represented by the matricesΦn×nandΨm×m. Moreover, we have the stochastic assumptions : the elementsϕijtψijtandεitare identically and independently distributed (i.i.d.) over time with zero mean and finite variances and covariances. The elements ofΦtare correlated with those ofΨt, the matricesΦtandΨtare uncorrelated withεt. The solution is a linear feedback control given by

xt=Ryt1E20

[1] -

R=(N+B'SB+E[Ψ'SΨ])1(B'SA+E[Ψ'SΦ])E21
R=(N+B'SB+E[Ψ'SΨ])1(B'SA+E[Ψ'SΦ])p=(N+B'SB+E[Ψ'SΨ])1(B'k+E[Ψ'Sε])E23
andSis a positive semi-definite solution to the matrix equation
S=M+RNR+(A+BR)S(A+BR)+E[(Φ+ΨR)S(Φ+ΨR)]E25

Correlated multiplicative and additive shocks: The assumption of non correlation in the original levels equation, will necessarily imply correlations in the deviations equation. Let the initial system be defined in levels by the first order stochastic equation

Yt=(A+Φt)Yt1+(B+Ψt)Xt+εtE26

and the stationary equation

Y*=AY*+BX*E27

By subtracting these two matrix equations and lettingytYtY*andxtXtX*, we have

yt=(A+Φt)yt1+(B+Ψt)xt+εt'E28

where the additive composite disturbanceε'denotes a correlation between the stochastic component of the coefficients and the additive disturbance. The solution to the stabilization problem takes a similar expression as in the uncorrelated case. We have the solution

xt=Ryt1+pE29

where

andSis positive semi-definite solution to the matrix equation
S=M+R'NR+(A+BR)'S(A+BR)+E[(Φ+ΨR)'S(Φ+ΨR)]E30
andkis solution to the matrix equation
k=(A+BR)'k+E[(Φ+ΨR)'Sε]E31

The optimal policy then consists of a feedback componentRtogether to a fixed componentp. The system will oscillate about the desired targets.

2.2 Stabilization of empirical stochastic systems

2.2.1 Basic stochastic multiplier-accelerator model

Structural model: The discrete time model consists in two equations, one is the final form of output equation issued from a multiplier-accelerator model with additive disturbances, the other is a stabilization rule [Howrey, 1967; Turnovsky, 1977]

Yt+bYt1+cYt2=Gt+εtGt=g1Yt1+g2Yt2+B¯E32

whereYdenotes the total output,Gthe stabilization oriented government expenditures,B¯a time independent term to characterize a full-employment policy [Howrey, 1967] andεrandom disturbances (serially independent with zero mean, constant variance) from decisions only. The policy parameters areg1g2andY¯is a long run equilibrium level [1] -.

Time path of output: Combining the two equations, we obtain a second order linear stochastic difference equation (SDE)

Yt+(bg1)Yt1+(cg2)Yt2=B¯+εtE33

whereB¯is a residual expression. Provided the system is stable [1] -, the solution is given by

Yt=B¯1(bg1)(cg2)+(C1r1t+C2r2t)+j=0t1r1j+1r2j+1r1r2εtjt=1,2E34

whereC1C2are arbitrary constants given the initial conditions andr1r2the roots of the characteristic equation:r1r2=(b±b24c)/2The time path of output is the sum of three terms, expressing a particular solution, a transient response and a random response respectively.

2.2.2 Stabilization of the model

Iso-variance and iso-frequencies loci: Let the problem be simplified to [Howrey, 1967]

Yt+bYt1+cYt2=At+εtE35

Figure 1 shows the iso-variance and the iso-frequencies contours together with the stochastic response to changes in the parametersbandc. Attempts to stabilize the system may increase its variance ratioσy2/σε2. As coefficientbcbeing held constant, the peak is shifted to a higher frequency.

Figure 1.

Iso-variance (a) and iso-frequencies (b) contours

Asymptotic variance of output: Provided the stability conditions are satisfied (the characteristic roots lie within the unit circle in the complex plane), the transient component will tend to zero. The system will fluctuate about the stationary equilibrium rather than converge to it. The asymptotic variance of output is

asyσy2=1+c+g2(1cg2)((1+c+g2)2(b+g1)2)σε2E36

Speed of convergence: The transfer function (TF) of the autoregressive process (4) is given by

T(ω)=(1+beiω+cei2ω)1E37

We then have the asymptotic spectrum

|T(ω)|2=(1+b2+c2+2b(1+c)cosω+2ccos2ω)1E38

The time-dependent spectra are defined by

|T(ωt)|2=j=0t1r1j+1r2j+1r1r2eijωE39

In this application, the parameters take the valuesb=1.1,c=.5,σε2=1as in [Howrey, 1967]. Figure 2 shows how rapid is the convergence of the first ten log-spectra to the asymptotic log- spectrum. [Nerlove et al., 1979].

Figure 2.

Convergence to the asymptotic log- spectrum

Optimal policy: Policies which minimize the asymptotic variance are such

g1*=bandg2*=c. Then we haveYt=Y¯+εtand
σy2=σε2E40

The output will then fluctuate aboutY¯with varianceσε2.

3. PID control of dynamical macroeconomic models

Stabilization problem are considered with time-continuous multiplier-accelerator models: the linear Phillips fluctuation model and the nonlinear Goodwin’s growth model [1] -.

3.1 The linear Phillips’ model

3.1.1. Structural form of the Phillips’ model

The Phillips’model [Phillips, 1954; Allen, 1955; Phillips, 1957; Turnovsky, 1974; Gandolfo, 1980; Shone, 2002] is described by the continuous-time system

Z(t)=C(t)+I(t)+G(t)(5)C(t)=cY(t)u(t)(6)I˙=β(I(t)vY˙)(7)Y˙=α(Y(t)Z(t))(8)E41

whereI˙andY˙denote the first derivatives w.r.t. time of the continuous-time variablesI(t)andY(t)respectively. All yearly variables are continuous twice-differentiable functions of time and all measured in deviations from the initial equilibrium value. The aggregate demandZconsists in consumptionC, investmentIand autonomous expenditures of governmentGin equation (5). ConsumptionCdepends on incomeYwithout delay and is disturbed by a spontaneous changeuat timet=0in equation (6). The variableu(t)is then defined by the step functionu(t)=0, fort0andu(t)=1fort1. The coefficientcis the marginal propensity to consume. Equation (7) is the linear accelerator of investment, where investment is related to the variation in demand. The coefficientvis the acceleration coefficient andβdenotes the speed of response of investment to changes in production, the time constant of the acceleration lag beingβ1years. Equation (8) describes a continuous gradual production adjustment to demand. The rate of change of productionYat any time is proportional to the difference between demand and production at that time. The coefficientαis the speed of response of production to changes in demand. Simple exponential time lags are then used in this model [1] -.

3.1.2. Block-diagram of the Phillips’ model

The block-diagram of the whole input-output system (without PID tuning) is shown in Figure 3 with simulation results. Figure 4. shows the block-diagram of the linear multiplier-accelerator subsystem. The multiplier-accelerator subsystem shows two distinct feedbacks : the multiplier and the accelerator feedbacks.

Figure 3.

Block-diagram of the system and simulation results

Figure 4.

Block diagram of the linear multiplier-accelerator subsystem

3.1.3. System analysis of the Phillips’ model

The Laplace transform ofX(t)is defined by

X¯(s)L[X(t)]=0estX(t)dtE42
Z¯(s)=C¯(s)+I¯(s)+G¯(s)(9)C¯(s)=cY¯(s)(10)sI¯(s)=βI¯(s)+βvsY¯(s)(11)sY¯(s)=αY¯(s)+αZ¯(s)(12)E43

Omitting the disturbanceu(t), the model (5-8) is transformed to

The TF of the system is

H(s)Y¯(s)G¯(s)=α(s+β)s2+(α(1c)+βαβv)s+αβ(1c)E44

Taking a unit investment time-lag withβ=1together withα=4,c=34andv=35, we have

H(s)=20s+15s22s+5E45

The constant of the TF is then 4, the zero is ats=1and poles are at the complex conjugatess=.2±j. The TF of system is also represented byH(jω). The Bode magnitude and phase, expressed in decibels20log10are plotted with a log-frequency axis. The Bode diagram in Figure 5 shows a low frequency asymptote, a resonant peak and a decreasing high frequency asymptote. The cross-over frequency is 4 (rad/sec). To know how much a frequency will be phase-shifted, the phase (in degrees) is plotted with a log-frequency axis. The phase cross over is near 1 (rad/sec). Whenωvaries, the TF of the system is represented in Figure 5 by the Nyquist diagram on the complex plane.

Ki=0te(τ)dτE46

Figure 5.

Bode diagram and Nyquist diagram of the transfer function

3.1.4 PID control of the Phillips’ model

The block-diagram of the closed-loop system with PID tuning is shown in Figure 6. The PID controller in Figure 7 invokes three coefficients. The proportional gainKpe(t)determines the reaction to the current error. The integral gain

bases the reaction on sum of past errors. The derivative GainKde˙determines the reaction to the rate of change of error. The PID controller is a weighted sum of the three actions. A largerKpwill induce a faster response and the process will oscillate and be unstable for an excessive gain. A largerKieliminates steady states errors. A largerKddecreases overshoot [Braae & Rutherford, 1978 [1] - A PID controller is also described by the following TF in the continuous s-domain [Cominos & Nunro, 2002]

HC(s)=Kp+Kis+sKdE47

The block-diagram of the PID controller is shown in Figure 7.

Figure 6.

Block diagram of the closed-looped system

Figure 7.

Block diagram of the PID controller

3.2 The nonlinear Goodwin’s model

3.2.1. Structural form of the Goodwin’s model

The extended model of Goodwin [Goodwin, 1951; Allen, 1955; Gabisch & Lorenz, 1989] is a multiplier-accelerator with a nonlinear accelerator. The system is described by the continuous-time system

Z(t)=C(t)+I(t)(13)C(t)=cY(t)u(t)(14)I˙=β(I(t)B(t))(15)B(t)=Φ(vY˙)(16)Y˙=α(Y(t)Z(t))(17)E48

The aggregate demandZin equation (13) is the sum of consumptionCand total investmentI[1] -. The consumption function in equation (14) is not lagged on incomeY. The investment (expenditures and deliveries) is determined in two stages: at the first stage, investmentIin equation (15) depends on the amount of the investment decisionBwith an exponential lag; at the second stage the decision to investBin equation (16) depends non linearly by Φ on the rate of change of the productionY. Equation (17) describes a continuous gradual production adjustment to demand. The rate of change of supplyYis proportional to the difference between demand and production at that time (with speed of responseα). The nonlinear acceleratorΦis defined by

Φ(Y˙)=M(L+MLevY˙+M1)E49

whereMis the scrapping rate of capital equipment andLthe net capacity of the capital-goods trades. It is also subject to the restrictions

B=0ifY˙=0,BLasY˙+,BMas
Y˙E50

The graph of this function is shown in Figure 8.

Figure 8.

Nonlinear accelerator in the Goodwin’s model

3.2.2. Block-diagrams of the Goodwin’s model

The block-diagrams of the nonlinear multiplier-accelerator are described in Figure 9.

Figure 9.

Block-diagram of the nonlinear accelerator

3.2.3 Dynamics of the Goodwin’s model

The simulation results show strong and regular oscillations in Figure 10. The Figure 11 shows how a sinusoidal input is transformed by the nonlinearities. The amplitude is strongly amplified, and the phase is shifted.

Figure 10.

Simulation on the nonlinear accelerator

Figure 11.

Simulation of a sinusoidal input

3.2.4 PID control of the Goodwin’s model

Figure 12 shows the block-diagram of the closed-loop system. It consists of a PID controller and of the subsystem of Figure 9. The simulation results which have the objective to maintain the system at a desired level equal to 2.5. This objective is reached with oscillations within a time-period of three years. Thereafter, the system is completely stabilized.

Figure 12.

Block-diagram and simulation results of the PID controlled Goodwin’s model

4. Fuzzy control of dynamic macroeconomic models

4.1 Elementary fuzzy modeling

4.1.1 Fuzzy logic controller

A fuzzy logic controller (FLC) acts as an artificial decision maker that operates in a closed-loop system in real time [Passino & Yurkovitch, 1998]. Figure 13 shows a simple control problem, keeping a desired value of a single variable. There are two conditions: the error and the derivative of the error. This controller has four components: (a) a fuzzification interface to convert crisp input data into fuzzy values, (b) a static set of "If-Then" control rules which represents the quantification of the expert’s linguistic evaluation of how to achieve a good control, (c) a dynamic inference mechanism to evaluate which control rules are relevant, and (d) the defuzzification interface that converts the fuzzy conclusions into crisp inputs of the process [1] -These are the actions taken by the FLC. The process consists of three main stages: at the input stage 1 the inputs are mapped to appropriate functions, at the processing stage 2 appropriate rules are used and the results are combined, and at the output stage 3 the combined results are converted to a crisp value input for the process.

Figure 13.

Design of a fuzzy controller

4.1.2 Fuzzyfication and fuzzy rules

Simple control example: Let us consider a simple control example of TISO (Two Inputs Single Output) Mamdani fuzzy controller. The fuzzy controller uses identical input fuzzy sets, namely "Negative", "Zero" and "Positive" MFs. The system output is supposed to follow

x(t)=4+et/5(4cost+36sint)E51

as in Figure 14. The error is defined bye(t)=r(t)x(t)wherer(t)is the reference input, supposed to be constant (a set point) [1] -. Then we havede(t)dte˙=x˙.

Figure 14.

System output, fuzzy rules and phase trajectory

Fuzzification: Membership functions. A membership function (MF) assigns to each element x of the universe of discourse

X, a grade of membershipμ(x)such thatμ:X[0,1]

The triangular MF of Figure 15 is defined byμ(x)=max{min{xabacxcb},0}whereabc. A fuzzy setA˜is then defined as a set of ordered pairsA˜={xμA˜(x)|xX}

According to the Zadeh operators, we have

μ(A˜B˜)=min{μ(A˜)μ(B˜)}μ(A˜B˜)=max{μ(A˜)μ(B˜)}andμ(¬A˜)=1μ(A˜)E52

The overlapping MFs of the two inputs error and change-in-error and the MF of the output control-action show the most common triangular form in Figure 15. The linguistic label of these MFs are Negative", "Zero" and "Positive" over the range[100,100]for the two inputs and over the range[1,1]for the output.

Figure 15.

Membership functions of the two inputs and one output

Fuzzy rules: Fuzzy rules are coming from expert knowledge and consist in "If-Then" statements. An antecedent block is placed between "If" and "Then" and a consequent block is following "Then"[1] -. Let the continuous differentiable variables

e(t)ande˙(t)denote the error and the derivative of error in the simple stabilization problem of Figure 13. The conditional recommendations are of the type
Ifee˙isA×BThenvisCwhere[A×B](xy)=min{A(x)B(y)x[aa]y[bb]}E53

These FAM (Fuzzy Associative Memory)-rules [1] - are those of the Figure 16. These nine rules will cover all the possible situation. According to rule (PL,NL;ZE), the system output is below the set point (positive error) and is increasing at this point. The controller output should then be unchanged. On the contrary, according to rule (NL,NL;NL), the system output is above the set point (negative error) and is increasing at this point. The controller output should then decrease the overshoot. The commonly linguistic states of the TISO model are denoted by the simple linguistic set A={NL,ZE;PL}. The binary input-output FAM-rules are then triples such as (NL,NL;NL): "If" inputeis Negative Large ande˙is Negative Large "Then" control actionvis Negative Large. The antecedent (input) fuzzy sets are implicitly combined with conjunction "And".

Figure 16.

Fuzzy rule base 1: NL-Negative Large, ZE-Zero error, PL-Positive Large

4.1.3 Fuzzy inference and control action

Fuzzy inference: In Figure 17, the system combines logically input crisp values with minimum, since the conjunction "And" is used. Figure 18 produces the output set, combining all the rules of the simple control example, given crisp input values of the pair

(ee˙).

Figure 17.

FAM influence procedure with crisp input measurement

Figure 18.

Output fuzzy set from crisp input measurements

Defuzzyfication: The fuzzy output for all rules are aggregated to a fuzzy set as in Figure 18. Several methods can be used to convert the output fuzzy set into a crisp value for the control-action variable v. The centroid method (or center of gravity (COG) method) is the center of mass of the area under the graph of the MF of the output set in Figure 18. The COG corresponds to the expected value

vc=vμ(v)dvμ(v)dvE54

In this example,vc=.124for the pair of crisp inputs(ee˙)=(55,20)

4.2 Fuzzy control of the Phillips’ model

The closed-loop block-diagram of the Phillips’model is represented in Figure 19 with simulation results. It consists of the FLC block and of the TF of the model. The properties of the FLC controller have been described in Figure 13 (design of the controller), Figure 15 (membership functions), Figure 16 (fuzzy rule base) and Figure 18 (output fuzzy set). Figure 20 shows the efficiency of such a stabilization policy. The range of the fluctuations has been notably reduced with a fuzzy control. Up to six years, the initial range[12,12]goes to

[3,3]E55

Figure 19.

Block diagram of the Phillips model with Fuzzy Control

Figure 20.

Fuzzy stabilization of the Phillips’ model

4.3 Fuzzy control of the Goodwin’s model

Figure 21 shows the block-diagram of the controlled system. It consists of a fuzzy controller and of the subsystem of the Goodwin’s model. The FLC controller is unchanged. The simulation results show an efficient and fast stabilization. The system is stable within five time-periods, and then fluctuates in an explosive way but restricted to an extremely close range.

Figure 21.

Block-diagram and simulation results of the fuzzy controlled Goodwin’s model

5. Conclusion

Compared to a PID control, the simulation results of a linear and nonlinear multiplier-accelerator model show a more efficient stabilization of the economy within an acceptable time-period of few years in a fuzzy environment. Do the economic policies have the ability to stabilize the economy ? Sørensen and Whitta-Jacobsen [Sørensen & Whitta-Jacobsen, 2005] identify three major limits: the credibility of the policy authorities’ commitments by rational private agents, the imperfect information about the state of the economy, and the time lags occurring in the decision making process. The effects of these limits are studied using an aggregate supply–aggregate demand (AS-AD) model and a Taylor’s rule.

6. Appendix A: Analytical regulation of the Phillips’ model

6.1 Unregulated model dynamics

The unregulated model (withG=0andu=1) is governed by a linear second order ordinary differential equation (ODE) inY, deduced from the system (5-8). We have

Y¨+(α(1c)+βαβv)Y˙+αβ(1c)Y(t)=αβE56

Whent0with the initial conditionsY(0)=0,Y˙(0)=α. Taking the following values for the parameters:c=3/4,v=3/5,α=4(T=α1=3months) andβ=1(time constant of the lag 1 year), the ODE is

5Y¨2Y˙+5Y(t)=20,t0,E57

with initial conditionsY(0)=0,Y˙(0)=4The solution of the unregulated model is

Y(t)=4+2et/5(2cos265t6sin265t)t0E58
or
Y(t)=4+6.32et/5cos(.98t+.68)t0E59

The phase diagram in Figure 22 shows an unstable equilibrium for which stabilization policies are justified.

Figure 22.

Phase diagram of the Phillips’ model

6.2 Stabilization policies

The stabilization of the model proposed by [Phillips, 1954] consists in three additive policies: the proportional P-stabilization policy, the proportional + integral PI-stabilization policy, the proportional + integral + derivative PID-stabilization policy. Modifications are introduced by adding terms to the consumption equation (6).

P-stabilization policy: For a P-stabilization, the consumption equation is

C(t)=cY(t)u(t)λD+λKpY(t)E60

whereKpdenotes the proportional correction factor andλthe speed of response of policy demand to changes in potential policy demand [1] -λ=2(a correction lag with time constant of 6 months). The dynamic equation of the model is a linear third order ODE inY

Y(3)+(α(1c)+β+λαβv)Y¨+(βλ+α(1c)(β+λ)+αλKpαβλv)Y˙+αβλ(1c+Kp)Y(t)=αβλu(t)E61

Takingc=3/4,v=3/5,α=4,β=1,λ=2,Kp=2,u=1the ODE is

5Y(3)+8Y¨+81Y˙+90Y(t)=40,t0E62

with the initial conditionsY(0)=0,Y˙(0)=4,Y¨(0)=5.6. The solution fort0is

Y(t)=.44.03e1.15t1.1e.23tsin(3.96t+44)E63

The graph of the P-controlled is plotted in Figure 23(b). The system is stable according to the Routh-Hurwitz stability conditions [1] -Moreover, the stability conditions forKpareKp.25and

Kp.35E64

Y(t)=.07e1.43t.13e.69t+1.08e.26tsin(4.03t+.19)E65

PI-stabilization policy: For a PI-stabilization policy, the consumption equation is

C(t)=cY(t)u(t)λD+λ{KpY(t)+KiY(t)dt}E66
Y(4)+(α(1c)+β+λαβv)Y(3)+(βλ+α(1c)(β+λ)+αλKpαβλv)Y¨+(αβλ(1c)+αβλKp+αλKi)Y˙+αβλKiY(t)=0E67

whereKidenotes the integral correction factor. The dynamic equation of the model is a linear fourth order ODE inYis deduced

c=3/4,v=3/5,α=4,β=1,λ=2,Kp=Ki=2,u=1E68

Taking the ODE is

5Y(4)+8Y(3)+81Y¨+170Y˙+80Y(t)=0,t0E69

with the initial conditionsY(0)=0,Y˙(0)=4,Y¨(0)=5.6,Y(3)(0)=96. The solution fort0is

Y(4)+6Y(3)+20.6Y¨+34Y˙+16Y(t)=0,t0E70

The graph of the PI-controlledY(t)is plotted in Figure A.2(c). The system is unstable since the Routh-Hurwitz stability conditions are not all satisfied [1] -GivenKp=2, the stability conditions onKiare

Ki[0,.8987]E71

PID-stabilization policy: For a PID-stabilization policy, the consumption equation is

C(t)=cY(t)u(t)λD+λ{KpY(t)+KiY(t)dt+KdDY(t)}E72

whereKddenotes the derivative correction factor. The dynamic equation of the model is a linear fourth order ODE inY

Y(4)+(α(1c)+β+λ+αλKdαβv)Y(3)+(βλ+(1c+λKdλv)αβ+(1c+Kp)αλ)Y¨+(αβλ(1c+Kp)+αλKi)Y˙+αβλKiY(t)=0E73

Takingc=3/4,v=3/5,α=4,β=1,λ=2,Kp=Ki=2,Kd=.55,u=1, the fourth order ODE nYis

with the initial conditionsY(0)=0,Y˙(0)=4,Y¨(0)=12,Y(3)(0)=2.4. The solution (fort0) is

Y(t)=.07e2.16t.12e.74t+1.40e1.55tcos(2.76t+1.54)E74

Figure 23.

Stabilization policies over a 3-6 years period: (a) no stabilization policy, (b) P-stabilization policy, (c) PI-stabilization policy, (d) PID-stabilization policy

The graph of the PID-controlledY(t)is plotted in Figure A.2(d). The system is stable since the Routh-Hurwitz stability conditions are all satisfied [1] -GivenKp=Ki=2, the stability conditions onKdareKd3.92andKd.07. The curve Figure A.2(a) without stabilization policy shows the response of the activityYto the unit initial decrease of demand. The acceleration coefficient (v=.8) generates explosive fluctuations [1] -.The proportional tuning corrects the level of production but not the oscillations. The oscillations grow worse by the integral tuning. The combined PI-stabilization [1] - renders the system unstable. The additional derivative stabilization is then introduced and the combined PID-policy stabilizes the system.

Notes

  • Any higher order system has an equivalent augmented first-order system, as shown in 2.1.2 . Let a second-order system be the matrix equation
  • The deviations are about some desired and constant objectives such that and .
  • The stabilization rule may be considered of the proportional-derivative type [Turnovsky, 1977] rewriting as
  • A necessary and sufficient condition of a linear system is that the characteristic roots lie within the unit circle in the complex plane. In this case, the autoregressive coefficients will satisfy the set of inequalities The region to the right of the parabola in Figure 1 corresponds to values of coefficients and which yield complex characteristic roots.
  • The use of closed-loop theory in economics is due to Tustin [Tustin, 1953].
  • The differential form of the delay is the production lag where the operator is the differentiation w.r.t. time. The distribution form is Given the weighting function , the response function is for the path of following a unit step-change in
  • The Ziegler-Nichols method is a formal PID tuning method: the and gains are first set to zero. The gain is then increased until to a critical gain at which the output of the loop starts to oscillate. Let denote by the oscillation period, the gains are set to
  • The autonomous constant component is ignored since is measured from a stationary level.
  • The commonly used centroid method will take the center of mass. It favors the rule with the output of greatest area. The height method takes the value of the biggest contributor.
  • Scaling factors may be used to modify easily the universe of discourse of inputs. We then have the scaled inputs and .
  • See [Braee & Rutherford, 1978] for fuzzy relations in a FLC and their influences to select more appropriate operations
  • Choosing an appropriate dimension of the rule is discussed by [Chopra et al., 2005]. Rules bases of dimension 9 (for 3MFs), 25 (5MFs), 49 (7 MFs), 81 (9 MFs) and 121 (11 MFs) are compared.
  • The time constant of the correction lag is years.. In the numerical applications, we will retain
  • Let be the polynomial equation with real coefficientsThe Routh-Hurwitz theorem states that necessary and sufficient conditions to have negative real part are given by the conditions that all the leading principal minors of a matrix must be positive. In this case, the matrix isWe have all the positive leading principal minors: and .
  • The leading principal minors are: .
  • The leading principal minors are :
  • Damped oscillations are obtained when the acceleration coefficient lies in the closed interval .
  • The integral correction is rarely used alone.

© 2009 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited and derivative works building on this content are distributed under the same license.

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André A. Keller (October 1st 2009). Optimal Economic Stabilization Policy under Uncertainty, Advanced Technologies, Kankesu Jayanthakumaran, IntechOpen, DOI: 10.5772/8222. Available from:

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