Vol. 133 (2018) ACTA PHYSICA POLONICA A No. 6

Proc. 13th Econophysics Colloquium (EC) and 9th Symposium of Physics in Economy and Social Sciences (FENS), 2017

The Mechanism of Transformation of Global Business Cycles

into Dynamics of Regional Real Estate Markets

A. Jakimowicza,* and S. Kuleszab

aInstitute of Economics, Polish Academy of Sciences, Palace of Culture and Science,

pl. Delad 1, PL-00-901 Warsaw, PolandbFaculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn,

Sªoneczna 54, PL-10-710 Olsztyn, Poland

The aim of this article is the identication of the occurrence mechanism of sudden quantitative changes inreal-estate market prices, which were observed during the global nancial crisis. Since such phenomena did notoccur to such an intensity during previous crises, it can be assumed that a new economic dynamic type has emergedin real-estate markets. The most promising of the methods of studying such phenomena seems to be the bifurcationmethod and particularly the catastrophe theory. This study analyzes changes in the prices of residential propertybased on cusp catastrophes. Empirical data were t to a stochastic cusp model to visualize the evolutionary pathof real estate market. Two other popular models (linear and logistic) were also estimated to compare results.A comparative analysis proved that the cusp model can best explain structural price instabilities in real-estatemarkets. The results conrmed that the evolution of the real estate market combines two processes: long-termevolution in the area of non-degenerate stability and discontinuous changes in the area of degenerate stability.Structural changes take place in the system only in the area of degenerate stability. The theoretical and practicalresults show that the catastrophe theory may have predictive potential, which could support traditional methodsof predicting changes on real estate markets.

DOI: 10.12693/APhysPolA.133.1351

PACS/topics: housing price dynamics, structural transition, cusp catastrophe, critical point, bifurcation set, equi-librium surface

1. Introduction

The very essence of this paper lies in the diagnosisof dynamic phenomena in the real estate market duringthe global nancial crisis. In mainstream economy dy-namics of economic systems usually have been studiedby focusing on stable equilibrium behavior with the as-sumption that the real estate market can generally exertpositive inuence on the economic stability and growth.In such assumption the dynamics of economic systemsusually have been studied by focusing on stable equi-librium behavior. However, according to Schumpeter,the role of development analyses should be minimized innear-balanced systems, but highly accentuated in transi-tion systems or systems that are far from equilibrium [1].The above approach emphasizes the importance of qual-itative changes in the development process. Qualitativechanges are very dicult to describe in the form of amathematical model. Systems which move from a near-balanced state to a state that is far from equilibrium un-dergo signicant transformations. Dynamical processeslead to evolution. In this case, evolution is the move-ment towards more complex states, and it is accompaniedby structural changes [2]. The list of possible economicsources of complexity is quite extensive [3].This paper introduces an transdisciplinary research

program with the roots in the complexity in eco-

*corresponding author

nomics [4, 5], whereby real estate markets could be seenas complex adaptive systems exhibiting properties spe-cic of open, dynamic systems, which pays the specialattention to the studies of transient phenomena, thatis non-equilibrium states, turbulence, adaptive behavior,self-organization, etc. In this view, the real estate mar-ket is a special type of market with specic rules, and farfrom the denition given by mainstream economics.

The aim of the research presented here is to denea new type of dynamics in property markets whichemerged during the global nancial crisis which featurestwo phases. The rst phase includes a long-term andcontinuous system evolution along the equilibrium path,while the second phase concerns a short-term and discon-tinuous change of the trajectory, consisting of a suddenjump from one equilibrium path to another.

The choice of catastrophe theory as a research methodwas motivated both by practical and theoretical consid-erations. As the review of the economic literature shows,during a global nancial crisis, changes in housing pricesin various countries were synchronized and demonstrateda discontinuous nature. At the same time, sharp uc-tuations in property prices, resembling jumps betweentwo states of equilibrium, were also observed in Poland.Catastrophe theory provides a proper tool to describesuch rapid changes. Therefore, a research hypothesis wasposited that the causes for abrupt changes in propertymarket prices in Poland should be attributed to changesin macroeconomic parameters caused by the global nan-cial crisis, and that cusp catastrophe provides the tran-

(1351)

1352 A. Jakimowicz, S. Kulesza

sition mechanism from the global to the local level. Thechoice of catastrophe theory was also motivated by the-oretical reasons since, according to the classication the-orem, all dynamic changes in low-dimensional space canbe classied into a limited number of patterns which arereferred to as elementary catastrophes [6].We aimed at analyzing the quasi-discontinuous real es-

tate price changes seen during the housing bubble and thefollowing mortgage crisis using the tools of Thom's catas-trophe theory. Although one of the elementary catas-trophes a cusp catastrophe is sometimes used formodelling the housing prices [710], it has never beenused for empirical verication of the transition mecha-nism from the global economy to the local real estatemarkets in Poland. Much empirical evidence has re-cently been gathered to prove that the global nancialcrisis, starting from its source to the results it producedin local property markets, followed a scenario consistentwith a cusp catastrophe. If the same scheme is to re-peat itself in local property markets, the asymmetry andbifurcation factors must be aected by macroeconomicvariables. Thus, a cusp catastrophe transmits macroeco-nomic impulses from the global economy to local propertymarkets. In other words, empirical research based on acusp catastrophe indicates the following direction of thecause-and-eect relation:subprime mortgage crisis [11] → global nancial cri-sis [12, 13]→ global nancial markets [14, 15]→ macroe-conomic parameters→ local property markets in individ-ual countries [1618].Therefore, attaining the assumed objective requires

broader empirical testing of the two latter links of thisscheme for the Polish economy. In our article, we notonly estimate the model based on the cusp catastrophe,but also compare the results to two other models linearand logistic. It appears that the catastrophic model bestmatches reality. The signicance of the method used inour research permits its application in research on hous-ing markets in many countries all over the world.

2. Real estate market, macroeconomic changes,

and complexity in economics

The real estate market can be dened at dierent lev-els of accuracy, subject to the purpose of the description,the applied criteria, the adopted reference standards andthe required degree of precision. In its most abstract def-inition, real estate market is a complete system of enti-ties (objects) with given attributes remaining under theinuence of mutual interactions that modify the statusof a property (buying, selling, renting). The real es-tate market is the basis for transactions concerning thebuying and selling of land, buildings, houses, etc., henceoperating at the intersection of mutually related areasof law, economic, social and political activities [1921].Real estate market has roots in current microeconomicand macroeconomic reality, but is additionally subjectedto a large variety of external stimuli. Observed inter-actions may dier in their direction and intensity, and

the real estate market in which they can act either as asender or receiver of those stimuli. For example, Tobinand Montz [22] analyzed catastrophic ooding and the re-sponse of the real estate market. According to Fory± [23],the property market together with its economic and so-cial environment form a system of communicating vesselssuch that stagnation in one of the segments and lack ofcommunication between the sectors have a negative im-pact on the remaining elements of the system.

Macroeconomic equilibrium is determined by power-ful relations between the real estate market and the na-tional economy. The above is particularly visible in therelations between the housing sector and the nancialmarket. The situation on international nancial marketsaects property prices in many countries [24]. It couldbe assumed that vast uctuations in real estate pricesobserved in the past decade are not merely transitionaladjustments for long-term global trends. Such eects onthe dynamics of the Asian real estate market were ana-lyzed by Gerlach et al. [25]. Quigley [26] investigated therelations between economic cycles and property prices,whereas Sornette and Woodard [27] examined the originsof the nancial crisis and described the resulting suddendrop in real estate prices.

In the traditional approach, the property market isa factor contributing to increased economic stabilityand growth. It performs those functions by contribut-ing to the gross domestic product, catering to the de-mand for housing, creating new jobs, contributing rev-enue streams from local property taxes, eectively allo-cating land resources and supporting the reinvestment ofcapital through the mortgage system. A strong reverserelationship is also observed: changes in economic pol-icy, in particular monetary policy, evoke a response fromthe real estate market. According to the European Cen-tral Bank [28], changes in interest rates aect propertyprices, and the economic situation in the housing sector(increase in the housing development and modernizationprojects), cost and availability of loans (for householdsand businesses), thus leading to a decrease or increase inconsumer demand.

The authors suggest that the strong uctuations in thereal estate prices observed, in the last decade, in manycountries are not just disruptions of trends or cycles butare an essential part of the market, which destabilizedby changes in the macroeconomic situation seeks anew state of equilibrium through quasi-discrete changesin prices. For example, in Poland, the real estate pricesrose steadily between 1990 and 2003, subject to peri-odic variations. Such a trend and its length and sta-bility pointed to a certain security and stabilization. Itwas not until 2004 (when Poland entered the EuropeanUnion, which, in turn, resulted in easy access to low-costcapital) that the real estate prices started to uctuaterapidly such a situation was especially visible from2006 to 2007 and undermined the belief in real estatemarket stability. The recent real estate market dynamicsand the accompanying changes in the real estate prices

The Mechanism of Transformation of Global Business Cycles. . . 1353

were very intense, multi-phased and, in some periods,could be interpreted as market instability.

The real estate market can be analyzed in terms of a setof its constituent elements (objects property, agents market participants), the relations between those el-ements (e.g. transfer of property, real estate taxes) andspecic attributes of those elements (type of property,location). By sending signals to the environment and re-ceiving stimulatory feedback, the real estate market un-dergoes structural transformation over time, and the tra-jectory of its evolution changes in state space. Real estatemarkets thus fulll the denition of a complex adaptivesystem proposed by Gell-Mann [29]. This means the pos-sibility of studying them using nonlinear dynamics meth-ods and applying catastrophe theory.

If the system's sensitivity threshold to external stimuliis exceeded, it becomes unstable and moves from a nearlystable state to a state that is far from equilibrium. Thesedynamical processes often induce key changes in the sys-tem's trajectory of evolution in state space, leading to thetransformation of one form into another in the process ofdiscontinuous and discrete changes in the system's state.In economic theory, such changes are referred to as dis-crete growth which, according to Jakimowicz [30] implieschanges that put the economy (or its sectors) on a dier-ent path of growth or recession. Cie±lak and Smoluk [31]have argued that discontinuity occurs because an unsta-ble system aims to achieve a new stable state and discon-tinuity somewhat alleviates tension in the system.

Attempts to dene stability and instability were madealready in the 19th century by Maxwell who observed:When the state of things is such that an innitely smallvariation of the present state will alter only by an in-nitely small quantity the state at some future time, thecondition of the system, whether at rest or in motion, issaid to be stable; but when an innitely small variationin the present state may bring about a nite dierencein the state of the system in a nite time, the conditionof the system is said to be unstable [32]. According toArrow [33], the stability of a system implies the existenceof an attribute which guarantees that when the systemleaves equilibrium, a self-regulatory mechanism will beinitiated to bring the system back to the path of equi-librium. Instability is a transitional stage between twostates of systemic equilibrium: the existing state and thenew (target) state.

In this study, an attempt was made to determinewhether phenomena characteristic of structurally unsta-ble markets which, under the inuence of a nitely smallnumber of control parameters, experience discontinuouschanges in their state can be observed in surveys of thereal estate market. The instability of property marketsshould be analyzed considering changes in the princi-ple state variable, namely, the price of property whichchanges far less rapidly than the prices on other mar-kets, such as the stock market.

The fundamental assumption in this paper is that thereal estate market is a dynamic system evolving toward

its equilibrium state, for which intervals of sudden pricechange establish critical points on the evolution path.Critical points mark structural transitions within a sys-tem between a current development scheme with vanish-ing perspectives for further growth and a new one witha number of prospective alternatives. In this case, dueto uctuations of independent variables, the real estatemarket only periodically enters into a state of instability,in the process of searching for a new state of stability.

3. Catastrophe theory

The perception of catastrophe theory in science in-volves a certain paradox. On one hand, it appears tohave been known for long, while on the other, it is notusually attached the importance it deserves. To eluci-date the problem, it was decided to present catastrophetheory in more detail. It has a signicant importancein science, since it reduces all system evolution scenar-ios available in the real world to eleven patterns el-ementary catastrophes if the space dimension is nothigher than ve [34]. In numerous econophysics confer-ences, researchers have presented results which are oftenlow-dimensional cross-sections of forms corresponding tothe elementary catastrophes. In such cases, catastrophetheory permits identifying a more general pattern of dy-namics, and consequently, to gain a deeper insight intothe entirety of the examined phenomenon.In the catastrophe theory, terms crisis, and catastro-

phe are substantially dierent taking into account theirfunctional meaning. According to Thom [35], a crisis pre-cipitates the catastrophe, acts as a forerunner of it andsometimes it might even trigger the catastrophe. In sucha sense, Thom dened the catastrophe as an observeddiscontinuity, whereas the crisis appears to be a hiddenphenomenon [36]. Hence, catastrophe can be understoodin terms of loss of stability of the system that leads toa new, stable state emerging from the state space [37].If so, catastrophe theory aims to determine, how the ob-ject of study (system, process) behaves over time, howit reacts to external stimulation, how it evolves withininstability intervals and how the continuous causes giverise to discontinuous changes.Catastrophe theory has become a subject of increased

research interest in various areas of knowledge. Below theauthors provided only several selected examples. In theeld of biological sciences, this theory was used by Cobband Zacks [38]. Phenomena occurring in urban spaceswere analyzed by Casti and Swain [39], Amson [40], andWilson [41], topics in behavioral sciences were studied bySussmann and Zahler [42], Guastello [43], and Hartelmanet al. [44]; in chemistry it was used by Okni«ski [37] andWales [45] and in physics by Fuhua and Huanmin [46].A number of applications of the catastrophe theory inthe economic sciences were reported by Zeeman [47, 48],Rosser [49, 50], Barunik and Vosvrda [14], Barunik andKukacka [15], Jakimowicz [51], Dou and Ghose [52], andVarian [53].

1354 A. Jakimowicz, S. Kulesza

Cobb and Watson [54] made a signicant contributionto statistical procedure for analyzing catastrophe modelsthrough nonlinear regression while Hartelman [55], andGrasman et al. [56] rebuild Cobb's [57] procedure in theR program language which software was used throughoutthis study [58].

According to catastrophe theory, in four-dimensionalspace-time all structural changes which can take placeproceed according to one of seven patterns, which arecalled elementary catastrophes. The basis of catastro-phe theory is the classication theorem, according towhich all structural changes to any objects can only takeplace according to a strictly-determined and nite num-ber of patterns, named elementary catastrophes. Theproof of this theorem was given by Trotman and Zee-man [59]. There are seven elementary catastrophes infour-dimensional space-time: fold, cusp, swallowtail, but-tery, hyperbolic umbilic, elliptic umbilic, parabolic um-bilic. They are represented with simple polynomials.Relatively simple models are thus obtained, which cangenerate extremely complex dynamics. This is where theusefulness and universality of this method lies.

A catastrophe means a sudden, discontinuous qualita-tive change which moves the analyzed system from oneset of dierential equations to another. This phenomenoncould have a positive or negative transformation direc-tion. The rate of changes in the object's behavior com-pared to the mean change in the past should be takeninto consideration. Catastrophe theory combines manyseemingly contradictory and unconnected methods of de-scribing phenomena into one coherent notional system:evolutionism and revolutionism, continuity and disconti-nuity, stationarity and non-stationarity. The applicationsof catastrophe theory in economics are of great cognitiveimportance, as Rosser [60] points out, they were whatinitiated complexity economics.

Catastrophe theory concerns systems immersed in apotential eld. The term potential (or alternativelypotential function) is a key to this theory, as it entirelydetermines both behavior and stability of the system. Ingeneral, the potential takes the form

V (x, c) = V [(x1, . . . , xn) , (c1, . . . , ck)] , (1)

where (x1, . . . , xn) describe system state variables, while(c1, . . . , ck) describe control parameters. As mentionedpreviously, systems considered in this study always evolvetoward a state of permanent stability being pushed thereby potential forces (force-feedback):

fi = −∂V (x, c)

∂ xi, i = 1, . . . , n. (2)

This balancing process takes place until a stationary stateis reached in which all state variables remain constantover time

∂xi∂ t

= 0, i = 1, . . . , n. (3)

In a stationary state, the rst partial derivatives areall equal to zero and, hence, the potential forcesvanish as well

∂V (x, c)

∂xi= 0, i = 1, . . . , n. (4)

The roots of Eq. (4) are referred to as critical points ofthe system and they form a system equilibrium surfaceM that obeys the following equation:

M =

(x, c) ;

∂ xi∂ t

= −∂V (x, c)

∂ xi; i = 1, . . . , n

. (5)

As mentioned previously, since the system behavior isfully determined by the potential function, the overallcharacteristics of the potential need to be studied. How-ever, one should make a distinction between two extremecases

Ω =

(x, c) ,

∂V

∂xi= 0,

∂2V

∂x2i6= 0, i = 1, . . . , n

, (6)

Σ =

(x, c) ,

∂V

∂xi= 0,

∂2V

∂x2i= 0, i = 1, . . . , n

. (7)

Equation (6) establishes a set Ω containing non-degenerate equilibrium states, while Eq. (7) denes aset Σ consisting of degenerate equilibrium states. Non-empty Σ set is a pre-requisite for the catastrophes,since structural instabilities require degenerate equilib-rium states to occur. On the other hand, potential sys-tems with non-degenerate critical points are always struc-turally stable. As seen, the sum of two sets: Ω and Σ , isthe equilibrium surface M . In turn, projection of Σ ontoa space of control parameters denes bifurcation set Bproviding the structural instability of the system.

If the conditions for stationary equilibria are fullled,the system can be studied using elementary (static)catastrophe theory, which best describes qualitative (dis-continuous) changes within the system caused by con-tinuous change in control parameters. Among others, ofparticular interest for models of the dynamics of real es-tate market is a cusp catastrophe, dened using only onestate variable x, and two control variables α and β callednormal and splitting factors or asymmetry and bifurca-tion factors, respectively

V (x, c) ≡ V [x, (α, β)] . (8)

Both control variables are simultaneously canonical vari-ables, which mean that they are built as linear combina-tions of independent variables

α = α0 +

n∑i=1

αixi, (9)

β = β0 +

n∑i=1

βixi. (10)

Several independent variables were joined together toform the asymmetry factor and bifurcation factor andthese canonical variables were then analyzed to obtainthe best possible t.

A potential function in a cusp catastrophe is given inthe form

V : R2 × R1 → R. (11)

The simplest mathematical structure which can fully rep-resent all cusp catastrophe characteristics is a polynomial

The Mechanism of Transformation of Global Business Cycles. . . 1355

structure with the form

V (x, α, β) =1

4x4 +

1

2βx2 + αx. (12)

The 1/4 and 1/2 coecients were selected so as to sim-plify discussion. Thus, Eq. (12) represents the basicmodel used in this paper to study price changes on realestate markets.

Equilibrium surface M (set of critical points) containsroots of the equation

M =

(x, α, β) ;

∂V

∂x= x3 + βx+ α = 0

. (13)

The subset of degenerate equilibrium states Σ is givenby

Σ = M ∩

(x, α, β) ;∂2V

∂x2= 3x2 + β = 0

. (14)

Bifurcation set B consists of control parameters whichfulll the condition

B =

(α, β) ; 4β3 + 27α2 ≤ 0. (15)

On the whole, the evolution path of the system can bethought of as a curve in state space, driven straight ontothe equilibrium surface or in the nearby space.

A cusp is one of the simplest elementary catastrophes.The shape of the equilibrium surface of the cusp catas-trophe shown in Fig. 1 is formed by the set of allextrema of the probability function dening the proba-bility distribution of variable x. The upper and lowersheets in the behavior plane M determine the set of allstate variable distribution maxima and its middle layer the set of minima. This results from the fact that themost probable states are analyzed.

As Fig. 1 shows, when Eq. (12) has three dierentreal roots (D = 4β3 + 27α2 < 0), there are two lo-cally stable conict regions on the catastrophe surfaceM . The system cannot be in both places at the sametime (i.e. simultaneously on the upper and lower sheetsof the equilibrium surface) and this means that a choiceshould somehow be made between the stable regions toprecisely dene the points at which the catastrophe istaking place. One of two heuristic conventions are mostoften adopted to achieve uniqueness: the perfect delayconvention or the Maxwell convention. According to theperfect delay convention, when the system's trajectoryruns on a specic equilibrium surface, the catastrophe(structural change) appears at the moment when one ofthe layers of this surface ends and the next appears. Withthe Maxwell convention, the state of the system is indi-cated by the absolute maximum of the probability distri-bution function.

The probability distribution function Pξ (x), which im-plies the shape of the equilibrium plane characteristic ofthe cusp catastrophe, can be written as follows:

Pξ (x) =

[x4

4+αx2

2+ βx+ g (ξ)

]−1, (16)

where ξ = (α, β) and g is the only function of variable ξsuch that for each ξ the expression in square brackets ispositive and it is true that

Fig. 1. Geometric interpretation of the cusp catastro-phe. Structural changes take place according to the per-fect delay convention.

∞∫−∞

Pξ (x) dx = 1. (17)

Figure 2 shows an example evolution of state variableprobability distribution in time. At the moment t0 wehave a distribution with a single maximum at point x0corresponding to movement along the upper edge of theequilibrium surface, which occurs for β0 (Fig. 1). The ap-pearance of the bifurcation factor (β0) causes the emer-gence of two competing maxima at points x1 and x2,which corresponds to the formation of a fold in the be-havior plane. A catastrophe means a jump of the systemfrom a less to more probable situation, i.e. from point x1to x2 at the instant t1 or from x2 to x1 at the momentt2. The Maxwell convention applies here, due to whichthe system tends toward the absolute maximum of prob-ability distribution. If a perfect delay convention wereadopted, the jump would occur only after the disappear-ance of one of the local maxima.

Fig. 2. State variable probability distributions for thecusp catastrophe.

1356 A. Jakimowicz, S. Kulesza

4. The data source

for real estate market research

The evolution of equilibrium states on the real estatemarket was analyzed based on the example of local hous-ing markets (dwellings) in Poland. The selection of mar-kets was determined by the availability of data, but theproposed methodology can be applied to every local mar-ket operating in line with free-market principles. Thestudy was carried out in the cities of Pozna« and Ol-sztyn, which vary considerably in size, their signicancefor the national economy, income levels, and propertyprices. The key statistical data of the analyzed cities arepresented in Table I.

TABLE I

Statistical data for cities Pozna« and Olsztyn.

City RegionArea

[km2]Population

Population

density

[people/km2]

Pozna« Wielkopolska 261.91 551627 2106

OlsztynWarmia

and Mazury88.33 175420 1986

The Registry of Prices and Values maintained by theCity Administration Oce was the source of data relat-ing to 12,000 property transactions (dwellings) in Pozna«and 9,000 property transactions (dwellings) in Olsztyn.Change trends in real estate prices were analyzed for theperiod between January 2001 and October 2011. The dis-tribution of property prices per square meter in Pozna«and Olsztyn is presented in Fig. 3.The graphic representation of the distribution of prop-

erty prices in Pozna« and Olsztyn in Fig. 3 points to ahigh level of proportionality between the directions anddynamics of the trends on the analyzed markets. Regard-less of the initial dierences in price levels in 2001, the ex-amined markets responded nearly identically to changesin the market environment and they underwent similartransformations in time and value. It can be assumedthat local factors were not the cause of sudden price uc-tuations from the middle of 2006 until the middle of 2007in Pozna«, or from 2007 till the beginning of 2008 in Ol-sztyn, but were rather determinants of the rankings ofthese local markets in the national hierarchy.The distribution of property prices in Pozna« and Ol-

sztyn also indicates that the largest increase in propertyprices occurred in both cities in 2007 at the monthly rateof approximately 10%, when the highest dierences be-tween maximum and minimum prices was observed ina given time interval. The dierence reached aroundPLN 6,000 (EUR 1,430) in Pozna« and around PLN4,500 (EUR 1,070) in Olsztyn. This behavior could beindicative of the examined markets' instability in 2007,which results from attempts to reach equilibrium follow-ing changes in its environment and the search for newpaths of evolution in the process of sudden price changes.

Fig. 3. Transactional property prices per square meterin Pozna« and Olsztyn (January 2001 to October 2011).

In 20012006, the increase in property prices in Pozna«and Olsztyn was slow and stable and it was during thisperiod that the smallest dierences between maximumand minimum prices were noted at around PLN 2,000(EUR 480) in Pozna« and PLN 1,500 (EUR 360) in Ol-sztyn. Following a steep increase in prices, the propertymarket gradually stabilized during a gently downwardtrend in 20082011 when a considerable dierence be-tween maximum and minimum prices was reported ataround PLN 4,000 (EUR 950) in Pozna« and PLN 3,500(EUR 830) in Olsztyn. The graphic representation ofthe distribution of property prices in the analyzed citiessupports the observation that a sudden increase in pricesshould be regarded as a critical point in the evolution ofthe studied markets.The applicability of the catastrophe theory to the real

estate market is also justied by the multimodal nature ofempirical data, namely, the presence of at least two statesof stable equilibrium in the system. The distributionof residential property prices in Pozna« and Olsztyn ispresented in Fig. 4.The distribution of the prices of residential property

per square meter, covering 12,000 transactions in Poz-na« and 9,000 transactions in Olsztyn, has a bimodalcharacter. This means that two paths of evolution onthe equilibrium surface can be assigned to a given setof control parameters. In strongly bimodal systems, the

The Mechanism of Transformation of Global Business Cycles. . . 1357

Fig. 4. Histograms of property prices per square meterin Pozna« and Olsztyn (January 2001 to October 2011).

evolutionary path can take the shape of a hysteresis loopwhich moves between the allowed states of equilibrium;alternatively, it can be split into two paths that evolvemore or less independently. In a real system, this wouldimply a rapid increase and a rapid decrease in propertyprices. The prices could be strongly dierentiated to ac-count for the specic attributes of real estate and theprices in one property group would evolve dierently fromother groups (e.g. minor changes in the prices of high-standard property with strong simultaneous uctuationsin the prices of standard property).The statistical interpretation of bimodal distributions

in Fig. 4 could conrm that on the analyzed markets inPozna« and Olsztyn, one trajectory of evolution was re-placed by another in the process of price uctuations. Itimplies that the distribution of empirical data shown inFig. 4 illustrates two completely dierent states of equi-librium which are separated by an unstable state.

5. The cusp catastrophe model

The cusp model was developed according to formula(12) by determining the variable of state (x) and indepen-dent variables in the set of control parameters: the asym-metry coecient (α) and the bifurcation coecient (β).

The property prices, the rate of price changes andchanging trends reect various factors and processes onthe real estate market and in the market environment.Therefore, property prices are robust indicators of thesituation on the property market. Other parameters arealso used to describe the state of the market, includingthe value of property, rental prices, return rates and thehouse price index, but most surveys rely on average trans-action prices. The empirical data shown in Fig. 3 wereused to calculate average transaction prices per squaremeter of residential property in Pozna« and Olsztyn.Transaction prices were applied as state variables (x) infollow-up calculations. These prices were corrected bythe ination rate.The development of the property market, particularly

the housing market, is correlated with economic growth.For this reason, one control variable was built based onthe key measure of national income the gross domesticproduct (GDP). For the property market to develop, in-vestors need ample access to nancing sources. Changesin mortgage rates are a reection of the country's macroe-conomic situation and interest rates set by the centralbank. In this study, control variables were developedin view of the average interest rate of the central na-tional bank (ARN) which comprises the reference rate,the lombard rate and the rediscount rate. Unemploy-ment rate (UR) and the number of new dwellings (NND)were also regarded as important factors. The housingmarket situation is a reection of the nancial situationof households which is determined by their income lev-els. The probability of investments on the real estatemarket increases with income. High unemployment low-ers the demand for housing and increases mortgage de-fault risk. The last independent variable in the set ofcontrol parameters was the number of property transac-tions (NT). Data regarding the analyzed variables wereacquired from the Central Statistical Oce (GDP, URand NND), the National Bank of Poland (ARN) and thedatabase of property prices in Pozna« and Olsztyn (NT).For the control variables GDP and ARN, their real valueswere taken into consideration.

6. Testing the cusp model

on the real estate markets

Numerical ts presented in this paper were made usingan advanced computation package cusp rst describedin an article by Grasman et al. as an add-on for thestatistical computing language R [56] and recently up-dated by R. Grasman in 2015. The package was down-loaded from the Comprehensive R Archive Network [61].The package implements Cobb's maximum likelihood al-gorithm [38] and oers a number of specialized routinesthat help comprehensive tting of cusp catastrophe mod-els to given data series: from data plotting and ttingvarious models (linear, logistic, catastrophic) to compar-ing t quality measures.The analysis was performed with the use of Hartel-

man's cuspt software [55, 58] which evaluates the qual-

1358 A. Jakimowicz, S. Kulesza

ity of cusp model t to empirical data by comparing theresults for a linear model and a logistic model. The fol-lowing metrics were used to evaluate the t quality: co-ecient of determination R2 (only for linear and the lo-gistic models), pseudo-R2 (only for the cusp model), log-Lik (likelihood ratio test), AIC (Akaike information crite-rion) and BIC (Bayesian information criterion). Unlikethe determination coecient R2 which always assumespositive values, pseudo-R2 can take on both positive andnegative values, and it is calculated according to the fol-lowing formula [57]:

R2 = 1− Var (∆)

Var (y), (18)

where Var(∆) is the variance of dierences between theobserved values and the maximum values of probabilitydistribution which are closest to the observed values andVar(y) is the variance of state variable. The higher thevalue of logLik, the better the model's t to the data,whereas a better t is achieved at lower values of AICand BIC.

In the rst stage of the study, all independent variablespresented in previous section were regarded as signicant.Linear combinations of independent variables determinecanonical control parameters of the analyzed system and,in line with formula (8), they were expressed as α (asym-metry coecient) and β (bifurcation coecient). Theabove assumption was veried by tting the model tothe data, and the signicance of independent variableswas determined in the following order for the asymmetrycoecient (α): GDP, ARN, UR, and for the bifurcationcoecient (β): NT and NND. The nal t of selectedindependent variables to empirical data describing localmarkets in Pozna« and Olsztyn in 20012011 is presentedin Tables II and III.

The tted parameters presented in Tables II and IIIindicate that the behavior of real estate markets is moreeectively described by catastrophe theory and the cuspmodel than other models (linear, logistic). Signicantly,dierent property markets in Pozna« and Olsztyn werealso found to respond similarly to environmental stimuli.

TABLE II

Values of tted parameters and the quality of t of cusp,linear and logistic models to empirical data for Pozna«in 2001-2011.

Model R2 logLik AIC BIC

Linear 0.95 −944 1904 1926

Logistic 0.98 −868 1766 1770

Morphogenetic 0.99 110 −189 −184

Asymmetry

factor (α)

Bifurcation

factor (β)

INTERCEPT 81.900

GDP 0.760

ARN −0.810

UR −1.896

NT −0.023

NND 0.020

TABLE III

Values of tted parameters and the quality of t of cusp,linear and logistic models to empirical data for Olsztynin 2001-2011.

Model R2 logLik AIC BIC

Linear 0.95 −917 1851 1874

Logistic 0.98 −859 1748 1791

Morphogenetic 0.98 78 −125 −79

Asymmetry

factor (α)

Bifurcation

factor (β)

INTERCEPT −63.660

GDP −0.116

ARN −0.136

UR −1.285

NT 0.031

NND 0.068

A comparison of pseudo-R2 values (0.99 for Pozna«and 0.98 for Olsztyn) for the cusp model and R2 val-ues for logistic (0.98) and linear (0.95) models reveals anadvantage of the cusp model. More reliable results areproduced when the remaining parameters are included inthe comparison. The values of tted parameters in lin-ear and logistic models are somewhat dierent. They aresignicantly worse than those noted in the cusp model,which gives further evidence of the cusp model's superi-ority over the remaining models.

The results presented in Tables II and III validate theassumption that sudden uctuations in property pricescan be dened as critical points in a market's evolutionrather than mere adjustments for its long-term trends.During those periods, the trajectory of a market's evolu-tionary path is changed. The previous path of develop-ment cannot be continued and an alternative trajectoryof evolution has to be found. The evolutionary paths ofboth analyzed property markets in the space of controlparameters α and β are represented graphically in Fig. 5.

If we account for minor dierences in the range of coor-dinate axes, both paths appear to be similar. The initial,small variations in property prices place the start pointof each path in the area of structural stability in thelower right quarter of the chart (set Ω of non-degenerateequilibrium states). A high concentration of trajectorypoints in this area corresponds to minor uctuations instate variables (a small dierence between maximum andminimum prices in a given time interval). Evolutionarypaths then move toward the pointed tip where they en-ter the area of structural instability (set Σ of degenerateequilibrium states). This point illustrates the catastro-phe discussed at the beginning of this article, namely, asudden (within a time scale characteristic of the real es-tate market) change in the state of a system which leadsto a rapid increase in property prices. The path of evo-lution enters the area of structural instability relativelydeeply (upper left quarter of the chart) and, as an equallycharacteristic feature, those sudden changes are accom-panied by the path's rapid progression into the system's

The Mechanism of Transformation of Global Business Cycles. . . 1359

Fig. 5. Two-dimensional projection of the evolutionarypath in the phase space of the system in Pozna« andOlsztyn.

phase space. A small cluster of points in the area of thepointed tip and on top of the crease also indicates greatervariation in prices. The only dierence between the stud-ied systems is expressed by the intensity of changes. InOlsztyn, the evolutionary path penetrates the instabilityarea more deeply because values of the asymmetry coef-cient (parameter α) are closer to zero. Toward the endof the analyzed period (October 2011), both paths onceagain approached the edge of the instability area. Thiscould imply that after a period of rapid price changesin 2007 and a stable drop in prices in successive years,the system made a repeated attempt to regain structuralstability.

This discussion indicates that presence of alternativestates of equilibrium and alternative paths of evolutionis not a desirable trait in a system. The above couldresult from inevitable bifurcations in the system which, inthe analyzed case, implies the segmentation of propertyprices (variation that accounts for the specic features ofproperty). Further work is needed to address this issue.

Regardless of possible evolution scenarios, the aboveanalysis conrms our assumptions that the evolution ofreal estate markets combines two mutually intertwinedprocesses: long-term evolution, which takes place mostlyin the area of system stability, and sudden changes(short-term) which are dicult to predict, are initiatedin the area of the pointed tip and take place in the areaof instability. A new sudden change can occur as long asthe path is situated in the cusp (fold) area. The path'semergence from the cusp area implies that a new evo-lutionary path was found by the market after a periodof rapid growth and that a stormy phase is sometimesfollowed by a period of smooth growth.

7. Conclusions

The real estate market as an essential part of the eco-nomic system has a signicant impact on each eld of hu-man activity. Our results broadly support the idea thatthe strong uctuations in the real estate prices observedin many countries are not just disruptions of trends orcycles but are an essential part of the market, which destabilized by changes in the macroeconomic situation seeks a new state of equilibrium through quasi-discretechanges in prices. It means that real estate market insta-bility can generate a uctuating, discontinuous change inreal estate prices. The real estate market is a dynamicalsystem in which periods of sudden uctuations in pricescan be dened as critical moments in its evolution. Insuch moments, the ability to develop further inside thecurrent model is exhausted and a search for a new, alter-native path of development commences.The results of the study validate the main hypoth-

esis which postulates the presence of instability inter-vals during the evolution of real estate markets. Unlikesmooth, long-term evolution over the equilibrium surface,the short-term catastrophic inuence of control parame-ters on the system gives rise to discontinuous transition,i.e. sudden changes between dierent equilibrium statesof the system.It seems that the moment of change cannot be pre-

dicted and changes are generally signaled with a certaindelay after data collected in a broader time interval havebeen analyzed. Catastrophe theory can be a useful toolfor developing indicators of a catastrophe. The analyzedsystem's current status and probable development sce-nario, which will take the system in the direction of struc-tural instability or structural stability, can be illustratedwith a graphic representation of the evolutionary path.Hence provide predictive results which allow for moreaccurate economic analyses, results, insurance and longterm nancial gains.The presented research leads to another important con-

clusion that contemporary local real estate markets areincreasingly sensitive to changes in international nan-cial markets. A similar eect of global factors on localeconomic relations is provided by wikinomics, especiallyin the area of public administration [62].

1360 A. Jakimowicz, S. Kulesza

References

[1] J.A. Schumpeter, The Theory of Economic Develop-ment: An Inquiry into Prots, Capital, Credit, Inter-est, and the Business Cycle, Transaction Publishers,New Brunswick 2004.

[2] R. Doma«ski, Evolutionary Spatial Economy,Wydawnictwo Uniwersytetu Ekonomicznego wPoznaniu, Pozna« 2012 (in Polish).

[3] A. Jakimowicz, Int. J. Nonlin. Sci. Num. Simul. 17,1 (2016).

[4] E.D. Beinhocker, The Origin of Wealth: Evolution,Complexity, and the Radical Remaking of Economics,Harvard Business School Press, Boston 2006.

[5] D. Colander, J. Hist. Economic Thought 22, 127(2000).

[6] Y.-C. Lu, Singularity Theory and an Introduction toCatastrophe Theory, Springer-Verlag, New York 1976.

[7] M. Montagnana, F. Prizzon, F. Zorzi, Papers RegionalSci. Assoc. 69, 153 (1989).

[8] X.J. Ge, G. Runeson, A dynamic model of housingmarkets and house prices, Proceedings of the Aus-tralasian Universities Building Educators Association(AUBEA), University of Technology Sydney, 2006,pp. 118. Retrieved April 12, 2017.

[9] X.J. Ge, G. Runeson, A.Y.T. Leung, C.A. Tang,in: Proc. 2006 HKU-NUS Symp. on Real Estate Re-search, Hong Kong University, 2006, p. 229. RetrievedApril 14, 2017.

[10] I. Barens, P. Flaschel, F. Hartmann, A. Röthig, Eu-rop. J. Econom. Economic Policies Intervent. 7, 361(2010).

[11] D. Yang, D. Mu, M. Yao, in: Proc. 2009 Int. Conf. onManagement and Service Science, Wuhan 2009, p. 1.

[12] A. Pleten, in: Market Risk and Financial MarketsModeling, Eds. D. Sornette, S. Ivliev, H. Woodard,Springer-Verlag, Berlin 2012, p. 201.

[13] D. Wesselbaum, Scott. J. Politic. Econ. 64, 376(2017).

[14] J. Barunik, M. Vosvrda, J. Econ. Dyn. Control 33,1824 (2009).

[15] J. Barunik, J. Kukacka, Quantitat. Finance 15, 959(2015).

[16] J. Wang, Ph.D. Thesis, University of Amsterdam,Amsterdam 2015. Retrieved April 24, 2017.

[17] C. Diks, J. Wang, J. Econ. Dyn. Control 69, 68(2016).

[18] M. Beªej, Real Estate Manage. Valuat. 21, 51 (2013).

[19] D. DiPasquale, W.C. Wheaton, Urban Economics andReal Estate Markets, Prentice-Hall, Englewood Clis1996.

[20] N. Carn, J. Rabianski, R. Racster, M. Seldin, RealEstate Market Analysis: Techniques and Applications,Prentice-Hall, Englewood Clis 1988.

[21] P. Hilbers, Q. Lei, L. Zacho, Real Estate MarketDevelopments and Financial Sector Soundness, In-ternational Monetary Fund, IMF Working PaperNo. WP/01/129, September 2001. Retrieved April 25,2017.

[22] G.A. Tobin, B.E. Montz, Soc. Sci. J. 25, 167 (1988).

[23] I. Fory±, Social and Economic Determinants ofthe Housing Market Development in Poland: AQuantication, Wydawnictwo Naukowe UniwersytetuSzczeci«skiego, Szczecin 2011 (in Polish).

[24] D. Venclauskiene, V. Snieska, Econ. Manage. 14,1026 (2009) Retrieved April 20, 2017.

[25] R. Gerlach, P. Wilson, R. Zurbruegg, J. Int. MoneyFinance 25, 974 (2006).

[26] J.M. Quigley, Int. Real Estate Rev. 2, 1 (1999). Re-trieved April 5, 2017.

[27] D. Sornette, R. Woodard, in: Econophysics Ap-proaches to Large-Scale Business Data and FinancialCrisis, Eds. M. Takayasu, T. Watanabe, H. Takayasu,Springer Science + Business Media, Tokyo 2010,p. 101.

[28] Structural Factors in the European Union HousingMarkets, European Central Bank, Frankfurt am Main2003. Retrieved April 10, 2017.

[29] M. Gell-Mann, The Quark and the Jaguar: Adven-tures in the Simple and the Complex, Holt and Co.,New York 1994.

[30] A. Jakimowicz, From Keynes to Chaos Theory. Evolu-tion of Business Cycle Theories, series: WspóªczesnaEkonomia, Wydawnictwo Naukowe PWN, Warszawa2012 (in Polish).

[31] Fuzzy Sets. Pattern Recognition. Catastrophe The-ory. Selection of Texts, Eds. M. Cie±lak, A. Smoluk,Pa«stwowe Wydawnictwo Naukowe, Warszawa 1988(in Polish).

[32] L. Campbell, W. Garnett, The Life of James ClerkMaxwell: With a Selection from His Correspondenceand Occasional Writings and a Sketch of His Contri-butions to Science, Cambridge University Press, Cam-bridge 2010, Ch. XIV, p. 440.

[33] K.J. Arrow, Collected Papers of Kenneth J. Arrow,Vol. 2, General Equilibrium, The Belknap Press ofHarvard University Press, Cambridge 1983.

[34] R. Gilmore, in: Encyclopedia of Applied Physics,Vol. 3, Calibration and Maintenance of Test and Mea-suring Equipment to Collective Phenomena in Solids,Ed. G.L. Trigg, Wiley-VCH, New York 1992, p. 85.

[35] R. Thom, Paraboles et catastrophes. Entretiens surles mathématiques, la science et la philosophie re-alisés par Giulio Giorello et Simona Morini (Para-bles, Parabolas and Catastrophes: Conversations onMathematics, Science and Philosophy. Interviews andNotes by Giulio Giorello and Simona Morini), Flam-marion, Paris 1983 (in French).

[36] R. Thom, Structural Stability and Morphogenesis: AnOutline of a General Theory of Models, W.A. Ben-jamin, London 1975.

[37] A. Okni«ski, Catastrophe Theory, Polish ScienticPublishers PWN, Warszawa 1992.

[38] L. Cobb, S. Zacks, J. Am. Statist. Assoc. 80, 793(1985).

[39] J. Casti, H. Swain, in: Optimization Techniques Mod-eling and Optimization in the Service of Man, Part 1,Ed. J. Cea, Springer-Verlag, Berlin 1976, p. 388.

[40] J.C. Amson, Environm. Plan. B Urban Analyt. CitySci. 2, 177 (1975).

The Mechanism of Transformation of Global Business Cycles. . . 1361

[41] A.G. Wilson, Catastrophe Theory and Bifurcation:Applications to Urban and Regional Systems, Rout-ledge, New York 2011.

[42] H.J. Sussmann, R.S. Zahler, Behavior. Sci. 23, 383(1978).

[43] S.J. Guastello, Behavior. Sci. 26, 63 (1981).

[44] P.A.I. Hartelman, H.L.J. van der Maas, P.C.M. Mole-naar, Brit. J. Developm. Psychol. 16, 97 (1998).

[45] D.J. Wales, Science 293, 2067 (2001).

[46] L. Fuhua, W. Huanmin, Acta Mech. Sin. 1, 185(1985).

[47] E.C. Zeeman, J. Math. Econ. 1, 39 (1974).

[48] E.C. Zeeman, Sci. Am. 234, 65 (1976).

[49] J.B. Rosser, Jr., From Catastrophe to Chaos: A Gen-eral Theory of Economic Discontinuities, Vol. I,Mathematics, Microeconomics, Macroeconomics, andFinance, 2nd ed., Springer Science + Business Media,New York 2000.

[50] J.B. Rosser, Jr., J. Econ. Dyn. Control 31, 3255(2007).

[51] A. Jakimowicz, Acta Phys. Pol. A 117, 640 (2010).

[52] W. Dou, S. Ghose, J. Busin. Res. 59, 838 (2006).

[53] H.R. Varian, Economic Inquiry 17, 14 (1979).

[54] L. Cobb, B. Watson, Math. Modell. 1, 311 (1980).

[55] P.A.I. Hartelman, Ph.D. Thesis, University of Ams-terdam, Amsterdam 1997.

[56] R.P.P.P. Grasman, H.L.J. van der Maas, E.-J. Wa-genmakers, J. Statist. Software 32, 1 (2009).

[57] L. Cobb, An Introduction to Cusp Surface Analysis,1998. Retrieved May 16, 2017.

[58] E.-J. Wagenmakers, P.C.M. Molenaar, R.P.P.P. Gras-man, P.A.I. Hartelman, H.L.J. van der Maas, PhysicaD 211, 263 (2005).

[59] D.J.A. Trotman, E.C. Zeeman, in: Structural Stabil-ity, the Theory of Catastrophes, and Applications inthe Sciences, Ed. P. Hilton, Springer-Verlag, Berlin1976, p. 263.

[60] J.B. Rosser, Jr., Implications for Teaching Macroeco-nomics of Complex Dynamics, Department of Eco-nomics, James Madison University, Harrisonburg1999. Retrieved May 4, 2017.

[61] Cusp-Catastrophe Model Fitting Using MaximumLikelihood, 2015; Retrieved May 25, 2017.

[62] A. Jakimowicz, D. Rzeczkowski, Acta Phys. Pol. A129, 1011 (2016).