## 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

The system (1) consists in

where

Letting the deviations be

#### 2.1.2 State-space form of the system

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

where

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

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

The objective function (3) may be also written as

where

Since the matrices

#### 2.1.3 Backward recursive resolution method

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

In the quadratic cost function of the problem, the_{t} can move from any initial

The solution is a linear feedback control given by

where we have

The optimal policy is then determined according a backward recursive procedure from terminal step

#### 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 constant matrices

*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

and the stationary equation

By subtracting these two matrix equations and letting

where the additive composite disturbance

where

andThe optimal policy then consists of a feedback component

### 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]

where

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

where

where

#### 2.2.2 Stabilization of the model

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

Figure 1 shows the iso-variance and the iso-frequencies contours together with the stochastic response to changes in the parameters

*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

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

We then have the asymptotic spectrum

The time-dependent spectra are defined by

In this application, the parameters take the values

*Optimal policy:* Policies which minimize the asymptotic variance are such

The output will then ﬂuctuate about

## 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

where

#### 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.

#### 3.1.3. System analysis of the Phillips’ model

The Laplace transform of

Omitting the disturbance

The TF of the system is

Taking a unit investment time-lag with

The constant of the TF is then 4, the zero is at

#### 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 gain

bases the reaction on sum of past errors. The derivative Gain

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

### 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

The aggregate demand

where

The graph of this function is shown in Figure 8.

#### 3.2.2. Block-diagrams of the Goodwin’s model

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

#### 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.

#### 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.

## 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.

#### 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

as in Figure 14. The error is deﬁned by

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

The triangular MF of Figure 15 is defined by

According to the Zadeh operators, we have

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

*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

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" input

#### 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

*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

In this example,

### 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

### 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.

## 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 (with

When

with initial conditions

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

### 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

where

Taking

with the initial conditions

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 for

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

where

Taking the ODE is

with the initial conditions

The graph of the PI-controlled

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

where

Taking

with the initial conditions

The graph of the PID-controlled

## 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.