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Physica A 374 (2007) 349–358

www.elsevier.com/locate/physa

Controlling chaos in an economic model

Liang Chena,�, Guanrong Chenb

aDepartment of Automation, Donghua University, Shanghai 201620, PR ChinabDepartment of Electronic Engineering, City University of HongKong, Kowloon, Hong Kong

Received 16 February 2006; received in revised form 29 April 2006

Available online 4 August 2006

Abstract

A Cournot duopoly, with a bounded inverse demand function and different constant marginal production costs, can be

modeled as a discrete-time dynamical system, which exhibits complex bifurcating and chaotic behaviors. Based on some

essential features of the model, we show how bifurcation and chaos can be controlled via the delayed feedback control

method. We then propose and evaluate an adaptive parameter-tuning algorithm for control. In addition, we discuss

possible economic implications of the chaos control strategies described in the paper.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Chaos; Control; Delayed feedback; Cournot; Oligopoly

1. Introduction

Since the demonstration of Strotz et al. [1] that chaotic dynamics can exist in an economics model,tremendous efforts have been devoted to investigating this kind of complex behaviors in various economicsystems. Significant and transparent theoretical insights have been gained. Recently, it has also been shownthat even oligopolistic markets may become chaotic under certain conditions [2,3].

Oligopoly, with a few firms in the market, is an intermediate structure between the two opposite cases ofmonopoly and perfect competition. Even the duopoly situation in an oligopoly of two producers can be morecomplex than one might imagine since the duopolists have to take into account their actions and reactionswhen decisions are made. Oligopoly theory is one of the oldest branches of mathematical economics datedback to 1838 when its basic model was proposed by Cournot [3]. Research reported in Ref. [4] suggests that aCournot adjustment process of output might be chaotic if the reaction functions are non-monotonic. Thisresult was purely mathematical without substantial economic implication until Puu [5] provided one kind ofeconomic circumstances, i.e., iso-elastic demand with different constant marginal costs for the competitor,under which meaningful unimodal reaction functions were developed. Since then, various modifications havebeen made by numerous economists. Rosser [6] has a good state-of-the-art review of the theoreticaldevelopment of complex oligopoly dynamics.

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2006.07.022

ing author.

ess: lchen_prc@hotmail.com (L. Chen).

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Unstable fluctuations have always been regarded as unfavorable phenomena in traditional economics.Because chaos means unpredictable events in the long time, it is considered to be harmful by decision-makersin the economy. Research on controlling chaos in economic models has already begun and several methodshave been applied to the Cournot model, such as the OGY chaos control method [7,8], the pole placementmethod [9]. These approaches require exact system information before their implementation. That means, tomake an accurate decision, the government or oligolists have to possess enormous amounts of relevanteconomic data, which is impractical or very costly. In contrast, the delayed feedback control (DFC) method[10] can be easily applied without requiring any system information although it has some other drawbacks[11].

In the present paper, the Cournot–Puu model is considered. After describing the nonlinear dynamics of theeconomic model, system state and parameter DFCs of chaos are investigated. Then, an adaptive approach totuning the key system parameter, i.e., the marginal cost according to the observed output oscillation, isproposed to stabilize the output quantity produced by a firm at a desired value. Finally, possible economicimplications corresponding to the proposed control strategies will be discussed.

The paper is organized as follows. In Section 2, the nonlinear economical model and its dynamic propertiesare described. Section 3 is devoted to controlling chaos by using the delayed feedback and the proposedadaptive methods. Theoretical analysis and numerical simulations are included. Conclusions are finallypresented in Section 4.

2. Cournot–Puu model

Alongside several interesting variants of the oligopoly theory emerged from criticism, the Cournot modelwas gradually evolved to be the core. In this paper, we particularly discuss the case of duopoly, that is, twosuppliers facing many consumers on the demand side. Denote the two firms by F 1 and F2, which producequantities q1 and q2, respectively. To determine its production in time period tþ 1, firm F1 has to do twothings. First, it has an expectation on the quantity of the other firm F2 in time period tþ 1. This expectationmight, for example, depend on its own quantity and the quantity of the other firm, both produced in theprevious time period, that is,

qe2ðtþ 1Þ ¼ E2ðq1ðtÞ; q2ðtÞÞ, (1)

where e denotes an expected value. Second, it has to determine the production that can maximize its ownprofit under the above expectation. Let U1ðq1; q

e2Þ be the profit of firm F1. From qU1=qq1 ¼ 0, one obtains the

quantity that leads to the maximal profit in time period tþ 1, as

q1ðtþ 1Þ ¼ V 1;maxU1ðqe

2ðtþ 1ÞÞ.

As seen from (1), q1ðtþ 1Þ generally depends on both q1ðtÞ and q2ðtÞ, that is,

q1ðtþ 1Þ ¼ f 1ðq1ðtÞ; q2ðtÞÞ.

The same is true for firm F2, i.e.,

q2ðtþ 1Þ ¼ f 2ðq1ðtÞ; q2ðtÞÞ.

Here, f ið�Þ ði ¼ 1; 2Þ are called reaction functions.Next, we derive the specific expressions of the above reaction functions under the assumptions of Cournot

and Puu [3].

Cournot assumption. Each firm expects its rival to offer the same quantity for sale in the current period as itdid in the preceding period.

According to this assumption, Eq. (1) becomes qe2ðtþ 1Þ ¼ q2ðtÞ. Consequently, the general reaction

functions are changed to

q1ðtþ 1Þ ¼ f 1ðq2ðtÞÞ; q2ðtþ 1Þ ¼ f 2ðq1ðtÞÞ. (2)

Although Cournot’s assumption may seem rather simplistic, it did produce a more interesting consequencethan its sophisticated revisions. Rand [4] shows that the Cournot model with non-monotonic reaction

ARTICLE IN PRESSL. Chen, G. Chen / Physica A 374 (2007) 349–358 351

functions can give rise to chaotic dynamics, and Puu [5] presents the following economic underpinning whichsupports this kind of reaction functions.

Puu assumption 1. The market demand is assumed to be iso-elastic such that price p is reciprocal to the totaldemand q, i.e., p ¼ 1=q.

Puu assumption 2. Goods are perfect substitutes, such that demand equals supply, i.e., q ¼ q1 þ q2.

Puu assumption 3. The two competitors have constant but different marginal costs, denoted by c1 and c2.

Based on these Puu’s assumptions, the profit of firm F1 becomes

U1ðtþ 1Þ ¼q1ðtþ 1Þ

q1ðtþ 1Þ þ qe2ðtþ 1Þ

� c1q1ðtþ 1Þ. (3)

Maximizing U1 with respect to q1 under the naive Cournot assumption (i.e., qe2ðtþ 1Þ ¼ q2ðtÞ), this yields the

specified unimodal reaction function in (2) as follows:

q1ðtþ 1Þ ¼ f 1ðq2ðtÞÞ ¼

ffiffiffiffiffiffiffiffiffiffiq2ðtÞ

c1

s� q2ðtÞ. (4)

So does firm F 2, that is,

q2ðtþ 1Þ ¼ f 2ðq1ðtÞÞ ¼

ffiffiffiffiffiffiffiffiffiffiq1ðtÞ

c2

s� q1ðtÞ. (5)

This pair of equations (4) and (5) is the central piece of the iterative process in the following study.The dynamic system (4)–(5) has two fixed points: one is the trivial point, ð0; 0Þ, and the other is a non-trivial

point, ðq�1; q�2Þ. Our concern is on the non-trivial point and thus no further consideration is given to the trivial

one. The non-trivial point, called the Cournot equilibrium (or Nash equilibrium), is the intersection of thereaction curves, given by

q�1 ¼c2

ðc1 þ c2Þ2; q�2 ¼

c1

ðc1 þ c2Þ2. (6)

The profits of the duopolists at the Cournot equilibrium can be calculated, as

U1 ¼c22

ðc1 þ c2Þ2; U2 ¼

c21

ðc1 þ c2Þ2. (7)

Stability of the Cournot equilibrium in system (4)–(5) has been studied extensively, e.g., in Ref. [5]. Denote theratio of marginal costs by cr ¼ c2=c1 and, without loss of generality, let c2Xc1. The Cournot equilibrium isstable if and only if

1pcro3þffiffiffi8p

. (8)

Fig. 1. Bifurcation diagram of the firm F2’s production q2 w.r.t. the ratio cr of marginal costs.

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0.02

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0.04

q 2

0 20 40 60 80 100 120 140 160 180 200

0

0.02

-0.02

-0.04

t

uq 1

Fig. 2. Time-dependence of the duopolist’s productions q1, q2 and the delayed state feedback control u, with parameters c1 ¼ 1, c2 ¼ 6:25,q1ð0Þ ¼ q2ð0Þ ¼ 0:01 and K ¼ �0:5.

L. Chen, G. Chen / Physica A 374 (2007) 349–358352

Moreover, if 3þffiffiffi8p

oc2=c1p254 , limit cycles and chaos exist. Here, c2=c1p25

4 guarantees positive productionsto both firms. Fig. 1 shows the bifurcation diagram of the firm F2’s production q2 with respect to the ratio cr ofmarginal costs. Fig. 2(b) is the amplified part of the box in Fig. 2(a). It can be seen that after the first cyclethere is a period-doubling cascade to chaos. The general appearance looks like that of the extensively studiedlogistic map.

3. Controlling chaos in the Cournot–Puu model

Over the past decade, there has been tremendous interest in controlling bifurcation and chaos in dynamicalsystems [12]. Although a large number of methods have been proposed, many of them cannot be directlyapplied to the control of oscillations and chaos in economic systems due to the complexity of the economicsand the limitation of the allowable control in applications. The DFC method, however, requires relatively littlesystem information, which is based on a feedback of the difference between the current state and the delayedstate of the system, therefore is easy to apply.

In this section, we apply the DFC to the state (the production of the firm) and the parameter (the marginal cost)of the Cournot–Puu model, respectively. We also develop an adaptive method based on the idea of DFC andproperties of the system. Finally, we will discuss the economic implications of the proposed control strategies.

3.1. State delayed feedback control

Consider the following controlled form of the duopoly model (4)–(5):

q1ðtþ 1Þ ¼

ffiffiffiffiffiffiffiffiffiffiq2ðtÞ

c1

r� q2ðtÞ þ uðtÞ;

q2ðtþ 1Þ ¼

ffiffiffiffiffiffiffiffiffiffiq1ðtÞ

c2

r� q1ðtÞ;

8>>><>>>:

(9)

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with the following DFC law as the control force:

uðtÞ ¼ K ½q1ðtÞ � q1ðt� 1Þ�, (10)

where K is the feedback gain.

Theorem 1. Consider the closed-loop control system described by (9) and (10). The Cournot equilibrium ðq�1; q�2Þ is

locally asymptotically stable if and only if

�1

2�ðcr � 1Þ2

8cr

oKo1�ðcr � 1Þ2

4cr

, (11)

where cr ¼ c2=c1 is the ratio of marginal costs.

Proof. Denote dqiðtÞ ¼ qiðtÞ � q�i ði ¼ 1; 2Þ and e1ðtÞ ¼ q1ðtÞ � q1ðt� 1Þ ¼ dq1ðtÞ � dq1ðt� 1Þ. Linearizing theclosed-loop system (9)–(10) about the fixed point ðq�1; q

�2Þ, we have a third-order linear difference equation as

follows:

dq1ðtþ 1Þ

dq2ðtþ 1Þ

e1ðtþ 1Þ

264

375 ¼

0c2 � c1

2c1K

c1 � c2

2c20 0

�1c2 � c1

2c1K

26666664

37777775

dq1ðtÞ

dq2ðtÞ

e1ðtÞ

264

375. (12)

The stability of (12) is governed by its characteristic equation

l l2 � Klþ K þðcr � 1Þ2

4cr

� �¼ 0. (13)

The Cournot equilibrium is locally asymptotically stable if and only if all solutions of (13) fall inside the unitcircle, i.e., jljo1. This condition is satisfied if and only if (11) holds according to the well-known Jury stabilitytest [13]. &

We have performed some numerical simulations to see how the state delayed feedback methodcontrols the unstable Cournot equilibrium. Parameter values are fixed as c1 ¼ 1 and c2 ¼ 6:25,the initial condition q1ð0Þ ¼ q2ð0Þ ¼ 0:01, and the feedback gain K ¼ �0:5. It is shown in Fig. 2that a chaotic trajectory is stabilized on the Cournot equilibrium and the control force eventuallytends to zero.

The above state DFC can be considered as a ‘private’ or ‘individual’ method to control a chaotic market.The decision-maker of the firm F1 is able to learn and change his policy by observing the differences betweenthe values of the present and the past output. Also note that the control force in (10) only depends on thequantity q1. It implies that firm F 1 alone can stabilize the chaotic market while firm F2 can benefit from itwithout doing anything. In this situation, normally F1 would not like to stabilize the market; otherwise, F 1

will not be able to win the competition over F 2. Due to the symmetry of the model, the roles of F1 and F 2 insystem (9)–(10) can be exchanged.

3.2. Parameter delayed feedback control

The Cournot–Puu model with parameter perturbation is described by

qiðtþ 1Þ ¼ f iðqiðtÞ;ri þ uðtÞÞ; i ¼ 1; 2,

ARTICLE IN PRESSL. Chen, G. Chen / Physica A 374 (2007) 349–358354

where ri is the nominal value of an accessible parameter (in our case, ri ¼ ci), and uðtÞ represents theparameter perturbation given by DFC. Specifically, we consider the following controlled system:

q1ðtþ 1Þ ¼

ffiffiffiffiffiffiffiffiffiffiq2ðtÞ

c1

r� q2ðtÞ;

q2ðtþ 1Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq1ðtÞ

c2 þ uðtÞ

r� q1ðtÞ;

8>>><>>>:

(14)

with the control law

uðtÞ ¼ Kðq2ðtÞ � q2ðt� 1ÞÞ. (15)

Theorem 2. Consider the closed-loop control system described by (14) and (15). The Cournot equilibrium ðq�1; q�2Þ

is locally asymptotically stable if and only if

ðcr � 1Þ2ðcr þ 1Þ

2� 2crðcr þ 1Þo

K

c21oðcr � 1Þ2ðcr þ 1Þ

4þ crðcr þ 1Þ, (16)

where cr ¼ c2=c1 is the ratio of marginal costs.

Proof. Denote dqiðtÞ ¼ qiðtÞ � q�i ði ¼ 1; 2Þ and e2ðtÞ ¼ q2ðtÞ � q2ðt� 1Þ ¼ dq2ðtÞ � dq2ðt� 1Þ. Linearizing theclosed-loop system (14)–(15) about the fixed point ðq�1; q

�2Þ gives a third-order linear difference equation,

dq1ðtþ 1Þ

dq2ðtþ 1Þ

e2ðtþ 1Þ

264

375 ¼

0c2 � c1

2c10

c1 � c2

2c20

�K

2c2ðc1 þ c2Þ

c1 � c2

2c2�1

�K

2c2ðc1 þ c2Þ

266666664

377777775

dq1ðtÞ

dq2ðtÞ

e2ðtÞ

264

375. (17)

The rest of the proof is the same as that of Theorem 1. &

To achieve chaos control by using small perturbations, we can take the control law as

uðtÞ ¼

�; e2ðtÞ4�;

��; e2ðtÞo� �;Ke2ðtÞ otherwise;

8><>: (18)

where the threshold � is a small positive number.Now, we apply this method to control chaos in the Cournot–Puu model. System parameters are chosen as

c1 ¼ 1 and c2 ¼ 6:25, the initial condition q1ð0Þ ¼ q2ð0Þ ¼ 0:01, the feedback gain K ¼ 20, and the threshold� ¼ 0:2. Fig. 3 shows the results of the numerical simulation. One can see that, after a short transient process,the controlled system state stays at the desired Cournot equilibrium.

The above parameter DFC may be thought of as a ‘public’ method in the form of government interventionto stabilize a chaotic market. As defined in (14), the control law directly affects the marginal cost of firm F 2,depending on its behavior. The sign of uðtÞ can be either positive or negative according to the condition thatq2ðtÞ is larger or smaller than q2ðt� 1Þ. The government, therefore, can introduce a production tax (i.e.,uðtÞ40), thus the marginal cost of firm F 2 is increased, aiming at a decrease in firm F 2’s production if it over-produces; and can give a production subsidy (i.e., uðtÞo0), thus the marginal cost of firm F 2 is decreased,aiming at an increase in firm F 2’s production if it under-produces. In this situation, firm F2 can be controlledby the government via a public policy (i.e., tax or subsidy). Due to the symmetry of the model, the roles of F 1

and F2 in system (14)–(15) can be exchanged.

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q 2

0 50 100 150 200 250 300

0

0.5

-0.5

t

u

Fig. 3. Time-dependence of the duopolist’s productions q1, q2 and the delayed parameter feedback control u added to c2, with parameters

c1 ¼ 1, c2 ¼ 6:25, q1ð0Þ ¼ q2ð0Þ ¼ 0:01, K ¼ 20 and � ¼ 0:2.

L. Chen, G. Chen / Physica A 374 (2007) 349–358 355

3.3. Adaptive feedback control

The advantage of the above delayed feedback perturbation scheme is that relatively little priori systeminformation is required to stabilize quantities produced by the duopolists. However, this scheme suffers fromthe limited range of feedback gain K that can stabilize the system. The non-stationary nature of economicsystems implies that an appropriate value of K may drift over time, thereby increasing the likelihood of controlfailure.

As a remedy, we propose a simple approach to controlling chaos in the Cournot–Puu model via adaptivetuning a marginal cost based on the properties of the model. From the bifurcation diagram shown in Fig. 1,one can see that the system stability is heavily dependent on the ratio of marginal costs, cr ¼ c2=c1. If cr islimited in the region ½1; 3þ

ffiffiffi8pÞ, the system is stable (referring to (8)); otherwise, it tends to a period-doubling

bifurcation in cascade to chaos. Once we have detected the system instability, we could decrease the value of cr

by decreasing (or increasing) the value of c2 (or c1) to move the system state back to the stable region. In otherwords, it is possible to use a negative (or positive) perturbation on the parameter c2 (or c1) if necessary, so as toguarantee and maintain the stability of the system. This motivates the following adaptive control law.

Denote e2ðtÞ ¼ q2ðtÞ � q2ðt� 1Þ. If je2ðtÞjo�, then let

c2ðtþ 1Þ ¼ c2ðtÞ � K je2ðtÞj, (19)

otherwise,

c2ðtþ 1Þ ¼ c2ðtÞ, (20)

where K is the control gain and � the small threshold.The basic idea behind the above adaptive algorithm is: if the oscillation persists (i.e., e2ðtÞa0), we know that

cr43þffiffiffi8p

; hence, we should decrease the value of c2 according to the difference between the current and thedelayed production quantities, so as to decrease the value of cr till cro3þ

ffiffiffi8p

. To prevent cr from beingchanged violently into a value less than 1, we choose a small � and K to ensure that c2 is adapted slowly.

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5.5

6

6.5

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c r

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0.05

0.1

0.15

0.2

q 1

0 20 40 60 80 100 120 140 160 180 2000

0.01

0.02

0.03

0.04

q 2

Fig. 4. Control of chaos in the Cournot–Puu model using adaptive algorithm based on delayed feedback control, with parameters c1 ¼ 1,

c2 ¼ 6:25, q1ð0Þ ¼ q2ð0Þ ¼ 0:01, K ¼ 1 and � ¼ 0:1.

L. Chen, G. Chen / Physica A 374 (2007) 349–358356

Now, we apply the above adaptive mechanism to control chaos in the Cournot–Puu model, with parametersc1 ¼ 1 and c2 ¼ 6:25, the initial condition q1ð0Þ ¼ q2ð0Þ ¼ 0:01, the gain K ¼ 1, and the threshold � ¼ 0:1. Itcan be seen from Fig. 4 that quantities of the duopolists’ productions tend to another Cournot equilibriumand cr is changed step by step from the value 6.25 to 5.11.

The above adaptive feedback control scheme may also be thought of as a ‘public’ method. According toeither firm’s behavior, the government can stabilize the chaotic market by introducing a productionsubsidy to firm F 2, i.e., decreasing c2 via (19). Comparing with the public control method mentioned in Section2 above, the adaptive scheme is uni-directional, which can change the original Cournot equilibrium into alower one.

It is natural to raise the question of which firm the government should target. Reconsidering (6) and (7), wecan see that, at the Cournot equilibrium, a firm with lower marginal cost produces more output and makesmore profit than the other firm with higher marginal cost. In the present case (c2Xc1), firm F1 is thought to bestrong because it dominates the market share and earns larger profits. Rewriting the profit formula (7) in termsof cr, as

U1 ¼1

ð1crþ 1Þ2

; U2 ¼1

ð1þ crÞ2, (21)

we can see that both profits are related only to the ratio of marginal costs, where U1 is directly proportional tocr but U2 is inversely proportional to cr. As seen, the chaotic market is stabilized at a higher Cournotequilibrium by the parameter delayed control as compared to the adaptive one. It implies that the profit offirm F1 is higher under the former control policy than that under the latter, while the case for firm F2 is theopposite. Hence, the government may be more favorable to the weaker firm when implementing the adaptivefeedback control policy.

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cr

Pro

fit o

f fir

m F

1

Cournot ProfitAvg. Profit

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0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

cr

Pro

fit o

f fir

m F

2Cournot ProfitAvg. Profit

Fig. 5. Diagram of profits w.r.t. the ratio cr of marginal costs.

L. Chen, G. Chen / Physica A 374 (2007) 349–358 357

3.4. Discussions

The above report is about how to control a chaotic market by using delayed feedback and adaptivemethods. Their corresponding economic implications have also been discussed. One may wonder: is a chaoticmarket always bad? Can a firm be benefited from chaos in the market or from these control strategies? [9] hasdiscussed these kind of questions before. In the following, some details are given in our cases.

Fig. 5 shows profits of firm F1 (left) and firm F2 (right) versus the ratio of marginal costs. The solid line isthe Cournot profit calculated at different Cournot equilibria. The market corresponding to the solid line isalways stable. The dashed line is the average profit calculated from the last 1000 out of 5000 iterations. It canbe seen that below the critical bifurcation value, cr ¼ 3þ

ffiffiffi8p

, the solid line and the dashed line overlap, whichcorresponds to a stable market. But, after that, they both diverge. In the unstable region, the average profit isless than the Cournot profit for firm F1, the stronger firm, but is greater for firm F 2, the weaker one. It mayseem that the stronger firm is disadvantageous in the unstable market while the weaker one is beneficial. Butthis makes sense to the true life where small firms are always trouble-makers who seek to survive and to growin the chaotic market, while on the contrary stronger firms prefer to stabilize the market so as to maintain theiradvantage of benefits.

Regarding the control methods discussed above, as a private control strategy, the state DFC method shouldbe applied by firm F 1 to stabilize the chaotic market so as to keep the better profit. As a public control strategy(the parameter delayed method), if the government uses it to stabilize the market, then it is more favorable tothe stronger firm while scarifying the profit of the weaker one. As to the adaptive method, both theoreticalanalysis and numerical simulation show that when the system is changed from chaotic to stable, cr is changedfrom the value 6.25 to 5.11, or vice versa. It can be observed from the figures that both firms’ profits aredecreased. That means the government’s policy based on the adaptive method seems to be good to stabilize theirregular market. However, no one can benefit from it. Yet, if one adjusts cr more slowly by choosing anappropriate parameter such as the feedback gain K, the system may be stabilized at a higher value of cr, like5.8. In this case, firm F1’s profit is increased, although unfortunately firm F 2’s profit is still decreased, leadingto no ‘win-win’ situation.

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4. Conclusions

Controlling financial markets and economical processes seems to be one of the most important andchallenging tasks facing firm managers, economists, and economical policy makers in the government today.In this paper, we have shown how the delayed feedback method can be applied to control the chaoticbehaviors in a Cournot model describing competition of two firms. One adaptive approach has also beendeveloped to adjust the marginal costs of the firms. This technique is easy to implement and does not requirethe exact knowledge of the underlying economic system.

The control methods based on the delayed feedback principle are applicable to real economical systems. Onthe one hand, the delayed control terms correspond to the behaviors of economical agents, known as rationalexpectations [14], when firms change their policies taking into account the differences between values of thepresent and the past productions. Consequently, there can exist intrinsic market properties that suppress thechaotic behaviors. On the other hand, these delayed terms can be viewed as the actions of the government intrying to stabilize the chaotic market and help weaker firms. Most interestingly, such delayed terms also tendto a well-known property of human decision-making behavior, which can be found in the literature oncognitive psychology under names like ‘status quo bias’ or ‘anchoring and adjustment’. Daniel Kahnemanreceived the 2002 Nobel Prize ‘for having integrated insights from psychological research into economicscience, especially concerning human judgment and decision-making under uncertainty’. The characteristicsand economic implications of delayed feedback control methods implemented in economic dynamic systemsstudied in this paper seem to provide another small piece of case on the combination of psychology andeconomics.

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