Path Dependence in Neoclassical Economic Growth Theory A

Vol. 127 (2015) ACTA PHYSICA POLONICA A No. 3-A

Proceedings of the 7th Symposium FENS, Lublin, May 14–17, 2014 Path Dependence in Neoclassical Economic Growth Theory A. Jakimowicz Institute of Economics, Polish Academy of Sciences, pl. Defilad 1, PL–00901 Warsaw, Poland Path dependence is a key feature of complex economic systems. It implies that history matters in the long-term evolution of markets and economies. Path dependence can be viewed as the dynamic version of positive feedback effects. This paper focuses on the nonlinear neoclassical economic growth model with the Cobb–Douglas production function, which accounts for problems related to pollutant emissions. It was found that only selected initial forms have a chance to develop. Present states depend on past states, even though the historical circumstances that had affected the past states may no longer be relevant. The choice among different histories may be a stochastic process. The understanding of economic growth suggested in this study stands in opposition to the neoclassical tradition based on equilibrium states or paths independent of the system’s history. In contemporary economics, the idea of path dependence is most often used in studies on the high-tech industry, where the researchers are focused on such phenomena as innovation processes, monopolization, or the causes of ineffective technical solutions. The analysis of historical conditions is almost entirely carried out with the use of qualitative methods, since the subject of the research is non-formalized. In addition, the theoretical basis for conducting relevant empirical research is still missing. As a result of the development of complexity economics in recent years, numerous dynamic features of complex economic systems can be examined with the application of quantitative methods which, in effect, strengthens the bonds between theory and practice. Rare exceptions include path dependence relations. The aim of this article is to fill this gap and to create a theoretical basis for quantitative research on historical conditions in economics. This is a necessary condition for undertaking empirical research. The theoretical search started with the Keynesian model of the Samuelson–Hicks trade cycle, to demonstrate that conventional economics completely omits the most interesting path-dependence cases. It turns out that only the neoclassical model of economic growth, taking into account two power laws, provides appropriate dynamic characteristics for a full description of path dependence relations. Therefore, appropriate theoretical bases can be provided only by complexity economics. It may seem that, in this work, the dependence on history is restricted to two successive time steps in the case of the Samuelson–Hicks model and a single step in the neoclassical model of economic growth by Day. However, it examines an ordered path dependence, where events are chronologically ordered and the impact of earlier events on the later ones occurs through intermediary events. It should be remembered that events are constantly affected by environmental stimuli that are reflected not only in initial conditions, but also in the values of the parameters for all periods. Thus, it is not a case of short-term memory. DOI: 10.12693/APhysPolA.127.A-86 PACS: 89.65.Gh, 89.75.–k, 05.45.–a, 07.89.+b

1. Introduction traps, the way out from which is only possible through the intervention of a certain external force or a develop- Path dependence is a basic type of market and eco- mental shock. Such phenomena may change the nature of nomic behaviour examined in complexity economics. the system or transform mutual relations between busi- The concept of path dependence puts emphasis on his- ness entities. The occurrence of path dependence indi- torical conditions in the long-run evolution of complex cates the existence in economic systems of various equi- economic systems [1]. Most often, it means either per- librium points, whose reaching may depend on various manent maintenance in the economic consequences of in- fortuitous events. The reaching of such points does not dividual innovations or the absence of a self-correcting have to mean the occurrence of optimal states. Systems mechanism in the development of individual innovations, may sometimes find themselves on undesirable trajecto- through which their outcomes are permanent unless new ries and reach states that endanger their existence. countervailing innovations appear [2]. Therefore, the idea The concept of path dependence finds various applica- of path dependence is based on positive feedback effects. tions in economic sciences. It is used in economic geogra- The research carried out within this field assumes that phy [8], in analyses of decision-making processes [9, 10], the history of institutional economic development is of housing policy [11] and organizational theory [12–14] crucial importance in explaining the evolution of mar- and is useful in examining institutional changes in the kets and economies. While examining the role of posi- development of a welfare state [15] and has signifi- tive feedback effects in economics, it is emphasized that cance in financial markets, where it facilitates the pric- the main sources of economic process ineffectiveness are ing of a certain class of financial contracts, i.e. path- increasing returns and network externalities [3–5]. As a dependent financial instruments [16]. Since path depen- consequence, in path-dependent economies, development dence is a feature of many complex systems, its applica- lock-in by historical events may occur [6, 7]. Therefore, tion is not limited only to economical sciences. Increas- economies may find themselves in certain developmental ingly more studies appear in such disciplines as sociology

(A-86) Path Dependence in Neoclassical Economic Growth Theory A-87 or political sciences, where many phenomena cannot be — as it has been demonstrated — is the source of the properly explained without taking into account histor- most interesting path dependence phenomena. The cor- ical events [17, 18]. The methodological trend is also respondence between the coexistence of attractors and becoming of interest through related notions in various repellers in the phase space of dynamical systems and sciences [19–21]. Path-dependent functions, used as field path dependence has not previously been reported in the variables in quantum electrodynamics, are well-known in literature. The article defines a new field of the applica- physics [22]. tion of path dependence in economics, namely the issues of economic growth and development. The Samuelson– Empirical research on historical conditions in eco- Hicks model includes only dependence on history in the nomics must be preceded with formalization of the is- weak version, and it concerns both types of dynamic sue, which means reference to economic theories seek- movements: convergent-to-dynamic equilibrium states, ing to explain path dependence relations observed in the and divergent, when the attractor is a point in infinity. real world. Additionally, the subject of research must be The neoclassical growth model analyses the most inter- quantified to make empirical analyses possible. The for- esting cases in attractors and repellers, while divergent mation of a quantitative basis for examining the path de- trajectories are not examined. pendence phenomenon in economics is not easy at least The Samuelson–Hicks trade cycle model features two for two reasons. First of all, it has been completely omit- time lags, while the neoclassical model of economic ted by classics of this science, and economic history is growth features only one. Nevertheless, it is of no signif- limited to only a selective description of some cases. Sec- icant importance since the article analyses ordered path ondly, the studies concerning the impact of history on dependence. It means that variables are chronologically the processes of creation and development of innovations ordered and the values of variables from previous periods do not have a long tradition in the economics. Therefore, affect the values of the same variables from later periods those issues are analysed mainly with qualitative meth- through the values of intermediate periods. Additionally, ods. In order to determine the theoretical bases necessary the study takes into account the impact of the environ- to make progress in this field, reference is made to two ment, which determines the values of parameters in all well-known models: the Samuelson–Hicks model of busi- periods. Thus, the analysis is not limited to short-term ness cycle fluctuations and the neoclassical model sup- memory cases only. plemented with two power laws (the Cobb–Douglas production function and an element taking into account the emission of pollution). The first model belongs to conven- 2. Definitions of the Path Dependence tional economics and the latter to complexity economics. The Samuelson–Hicks model plays the role of a frame of The literature of the subject provides many definitions reference and was used to demonstrate that strong — of path dependence. From the perspective of this article, and thus the most interesting — path dependence rela- the definitions provided by Page are the most useful, as tions cannot be examined on the basis of conventional they are of high operational importance [23]. A dynamic economics. Therefore, theoretical bases to examine his- process is referred to as path dependent if the outcome torical conditions can be provided only by complexity of any period depends upon the vector of history and can economics. The neoclassical model with the production depend on its order: function of a Cobb–Douglas type and an element related xt+1 = Ot (Ht) , (1) to the reduction of efficiency — as a result of emission where Ht is a vector of historical events, Ot represents of pollutions in production processes — was selected be- the outcome function, while xt+1 is the outcome of pe- cause it is one of the least-explored nonlinear economic riod t + 1. The outcome of each period can also include models. It was developed by Day in the first half of 1980s, additional information on opportunities and events which and its main role consisted in introducing nonlinearity emerged in that period. Thus, a description of the envi- and deterministic chaos to economics. Additionally, aca- ronment in period t is formed, taking into account exoge- demic circles obviously assumed that it would not con- nous factors affecting the outcomes. A history at time t, tribute anything new to the development of economics, denoted by Ht, is a combination of all outcomes xt up which explains its rare presence in economic literature time t together with all other factors also up time t. A dy- in subsequent years. The article proves that this model namic process also has an outcome function Ot, which should be considered a milestone in the development of reflects the impact of the current history on the next economics because of the extremely broad scope of dy- outcome. Since the outcome function can change over namic phenomena that it may generate. It features many time, it is indexed by t. It does not have to be deter- characteristics that are found in much more complex, ministic; it can generate certain probability distribution multi-dimensional systems. At the same time, it is so un- over outcomes. It should be noted that in case of defi- complicated that it can make a convenient starting point nition (1), a change of order x1 and x2 can change the for more advanced quantitative research on path depen- outcome generated by Ot. In a further part of the arti- dence in economics. The article presents various previ- cle, this type of path dependence is referred to as a weak ously unknown dynamic characteristics of this model, version. This means that only condition (1) is satisfied, such as the coexistence of attractors and repellers, which but not condition (2) provided below. A-88 A. Jakimowicz

In some cases, the notion of strong path dependence where: Y — national income, I — aggregate invest- ind can be useful [23]. A dynamic process is strongly path ments, C — consumption, It — induced investments, aut dependent if for two different vectors of history, the out- It — autonomous investments, c — marginal propen- come functions are different. It can be presented in the sity to consume, s — marginal propensity to save, v — following form: acceleration coefficient, A0 — initial level of autonomous investments, r — rate of growth of autonomous expendi- xt+1 = Ot (Ht) ,Ot (Ht) 6= Ot H¯ t for Ht 6= H¯ t. (2) In other words, two different paths imply various prob- tures. It may seem that path dependence in this model abilities of outcomes. The properties of strong path de- is restricted to two successive time steps present only in pendence result in the path dependence relation. Strong the term describing induced investments. However, as it path dependence may be referred to as order of the path has been previously explained, there are no impediments dependence. Path dependence relations may not be con- to using definition (1) and (2). fused with sensitivity to initial conditions, which is the National income in period t, taking into account essence of the deterministic chaos. Eqs. (3)–(7), can be written in the following form: t In a further part of the article, a dynamic, nonlinear Yt = cYt−1 + v (Yt−1 − Yt−2) + A0 (1 + r) (8) process of economic growth is analysed, taking into ac- or in an equivalent form: count neoclassical postulates. For comparative purposes, Y − (c + v) Y + vY = A (1 + r)t . (9) the analysis of a neoclassical growth model is preceded t t−1 t−2 0 It is a non-homogeneous second order linear difference by presentation of results concerning a search for path equation with constant coefficients. To solve it, a particu- dependence relations in one of the best known dynamic lar integral must be determined, which presents the path models of conventional economics — the Samuelson– of dynamic equilibrium, along with the complementary Hicks trade cycle model. In both cases, conventional and function which determines deviation Y from the trend complex neoclassical cases, the beginning of the history t Y , i.e. Y − Y . of a system is the determination of the initial condition t t t Equilibrium trajectory depends on the course of au- H . Changes in the outcome function O , reflecting the 0 t tonomous investments in time [30]. Usually, three cases evolution of the environment, affect the system through are considered: structural parameters. Since the system is open, exoge- 1. If there are no autonomous expenditures, i.e. nous and stochastic factors affect both the initial condi- Iaut = 0, then the solution of Eq. (9), describing herein tion and the structural parameters of the model. t the system in dynamical equilibrium, takes the following 3. Path dependence in conventional economics form: t — a linear model of the Samuelson–Hicks trade Yt = Y0 (1 + r) . (10) cycle At the same time, the principle of the so-called strong 3.1. Basic equations of the model accelerator is applied: √ 2 Path dependence appears many times in the well- v > 1 + s . (11) known work by J.M. Keynes, The General Theory of In this case, the dynamic equilibrium path (10) is a par- Employment, Interest, and Money, which formed the ba- ticular solution of Eq. (9), which can be proved by sub- sis for development of modern macroeconomics [24, 25]. stituting (10) to (9). Condition (11) means that in such a The conventional Samuelson–Hicks model contains for- situation we are dealing with an even growth of national malization of the key concepts of Keynesian economics in income. the form of a linear dynamic system [26–29]. At the same 2. For an even growth of autonomous investments, fol- time, it should be treated as a milestone in the develop- lowing according to Eq. (7), we also have an dynamic ment of conventional economics, since it combines the equilibrium path described by Eq. (10), while the so- problems of economic fluctuations and dynamic growth. called super multiplier principle must be satisfied: This is possible through assuming that autonomous in- Y0 1 vestments, being the only manifestation of state inter- = c+v v . (12) A0 1 − + 2 vention in the economy, increase according to a certain 1+r (1+r) fixed percentage rate. The basic equations of the model Super multiplier is a term referring to combined effects have the following forms: of a multiplier and the principle of acceleration [30]. 3. In the classical theory of economics, we can also Yt = It + Ct; (3) deal with forced (exogenous) oscillations of autonomous ind aut investments, which are regular and have a constant am- It = It + It ; (4) plitude: aut I = A0 cos (kt − k0) , k = 2π/T, (13) Ct = cYt−1, 0 < c ≤ 1, c = 1 − s; (5) t where: A0 — oscillation amplitude; T — period of the ind autonomous investments; k — oscillation phase. If the It = v (Yt−1 − Yt−2) , v = constans, v > 0; (6) 0 solution of Eq. (9) also contains oscillations, which oc- aut t It = A0 (1 + r) ,A0, r = constans,A0, r ≥ 0; (7) curs for Path Dependence in Neoclassical Economic Growth Theory A-89

relations play a signiﬁcant role in long periods, including √ 2 √ 2 1 − s < v < 1 + s (14) intervals from 30 to 40 years. and the period of oscillation of autonomous investments is longer than the period of internal oscillation of the model, then the particular integral takes the following form:

Yt = Y0 cos (kt − ϑ0) , (15) if the following condition is satisﬁed:

Y0 cos (kt − ϑ0) − (c + v) Y0 cos [k (t − 1) − ϑ0]

+vY0 cos [k (t − 2) − ϑ0] = A0 cos (kt − k0) , (16) where ϑ0 is a phase shift. More information concerning this case can be found in the literature on the subject [30]. A complementary function is determined on the basis of the roots of the following characteristic equation: λ2 − (c + v) λ + v = 0. (17) Its solution allows all possible cases of deviations of national income from the dynamic equilibrium level to be determined. After taking into account relation (17), we obtain a complete description of the dynamics of the system determined by Eqs. (3)–(7).

3.2. Path dependence in a weak version The solution of Eq. (17) determines sets of points situated on a plane (v, s), which determine basic types of evolution of national income over time [30, 31]. In the Fig. 1. Typical trajectories of national income: (a) suppressed (or damped) movements for v < 1 and economic system determined by Eqs. (3)–(7), five types (b) exploding movements for v > 1. of dynamics can be distinguished: √ 1. v ≤ (1 − s)2 — asymptotic approach to the dy- A weak version of path dependence does not exclude asymptotic convergence of the outcome functions for two namic equilibrium level Yt, √ 2 different vectors of history. Definition (1) also points 2. (1 − s) < v < 1 — damped oscillation around to another phenomenon, which is not properly empha- Y , t √ sized in conventional economics. The rate and the vari- 3. 1 < v < (1 + s)2 — growing oscillation amplitude, √ 2 ety of economic processes existing within various vectors 4. v ≥ (1 + s) — explosive income growth without of history are determined mainly by the parameter val- oscillation, ues. The importance of initial conditions is not properly 5. v = 1 — regular oscillation with a constant ampli- brought up, as is the case in complexity economics. tude. Omitting case 5, which should be considered as unreal- 4. Path dependence in complexity economics — istic, the remaining cases can be divided into two groups: the neoclassical model of economic growth I ⇒ v < 1 — damped or suppressed movements, modified by Day II ⇒ v > 1 — explosive movements. Figure 1 presents an example course of suppressed and 4.1. Model formulation exploding movements. If v < 1, then we deal with an approach to the dynamic equilibrium path, and if v > 1, The considered dynamic process is described with the then the system attractor is a point in infinity. In both use of one-dimensional irreversible mapping. This type cases, since definition (1) is satisfied, in the model deter- of mapping represents the simplest systems, which are mined by Eqs. (3)–(7) path dependence exists in a weak capable of generating complex dynamics in the form of version. National income in period t + 1, i.e. Yt+1, de- deterministic chaos. Since many phenomena which oc- pends on income in period t: Yt+1 = Ot(Yt). The vector cur in multidimensional systems can also occur in one- of historical events is created by chronologically-ordered dimensional mappings, the latter provide a basic pattern national income volumes in individual periods, starting for dynamic system transformations. One-dimensional ir- from the zero period volume, i.e. Ht = [Y0,Y1, ..., Yt−1]. reversible mappings feature basic chaos scenarios (route Empirical research confirms that cumulated economic to chaos) and major bifurcation phenomena [33]. growth in a specified time period depends on the entire The basis for these reflections is the standard neoclas- vector of history covering this period, and not only on the sical growth model by Solow, formulated in a discrete initial and final values Ht [32]. In particular, historical version with constant marginal propensity to save [34]. A-90 A. Jakimowicz

Let us consider the economy described by the following Day [36, 37] proposes the introduction to the neoclas- set of equations: sical Cobb–Douglas type production function of an ad- φ Yt = Ct + It, (18) ditional, multiplicative element (m − kt) related to efficiency reduction: It = Kt+1, (19) Yt β φ yt = = f (kt) = Bkt (m − kt) , kt ≤ m, (29) St = Yt − Ct = sYt, s > 0, (20) Lt where m, φ are positive parameters. The economic Y = F (K ,L ) , (21) t t t growth and accompanying concentration of capital re- t Lt = (1 + n) L0, n > 0, (22) sult in the emergence of social costs, related mainly to where symbols Y , C, I, K, S, L mean, respectively, degradation of the natural environment. An additional national income, consumption, investment, capital, sav- expression presents the effect of the pollution emission ings, and labour, s is the marginal propensity to save on production per capita. A rational management of re- and n is the natural rate (rate of growth of labour force). sources will help to avoid an increase in pollution, but The symbol F represents a production function, thus production growth will then be reduced. If coefficient φ φ the method in which production factors, Kt and Lt, are is close to zero, then (m − kt) → 1. Parameter m cor- transformed into the national income Yt. Additionally, responds to a certain level of saturation, the reaching of in each period t, the condition of economic equilibrium which (kt = m) will cause a production decrease to zero. St = It is satisfied. From the perspective of environmental protection pro- The neoclassical theory usually applies a linear and grammes, coefficients m and φ can be treated as control homogeneous production function, characterized by parameters. constant returns in relation to scale: Both the neoclassical production function of Cobb– F (λK, λL) = λF (K,L) for λ > 0, (23). Douglas and its modification proposed by Day are examples of power laws. They emerged in economic sci- This means production growth in the same proportions ences for the first time at the end of 19th century as a as labour and capital inputs. The function of this type result of discoveries made by an Italian economist, Vil- is equal to a single variable function, considered in per fredo Pareto, who examined the distributions of income capita terms: K in various societies [38–39]. The dependence he discov- Y = F (K,L) = Lf , (24) ered significantly differed from the Gaussian bell curve, L and additionally demonstrated high stability in space and K K time: the shape of the distribution looked similar for var- where f = F , 1 , L L ious countries and eras. In the opinion of Pareto, the curve of income distribution is an external image of a therefore, y = f(k), where y = Y and k = K . Addition- L L given society and a basic factor of its internal transfor- ally, such a function should be well-behaved [35], which mations. Apparently, the same can be true for the pro- means satisfaction of the following conditions: duction functions of Cobb–Douglas and Day. 0 00 0 f (k) > 0, f (k) < 0, f (k) → ∞, Inclusion of a production function (29) to the growth gdy k → 0, f 0(k) → 0, gdy k → ∞. (25) model (18)–(22) results in a difference equation: A linear and homogeneous production function allows sB β φ kt+1 = kt (m − kt) . (30) the described model to be presented by Eqs. (18)–(22) in 1 + n sB the following form: By introducing a new parameter µ = 1+n , we finally obtain a nonlinear first order difference equation: Kt+1 sF (Kt,Lt) = ⇒ kt+1 (1 + n) = sf (kt) . (26) β φ Lt Lt kt+1 = µ kt (m − kt) = F (k), (31) Detailed information concerning the transformation of which will be subject to computer simulations. Eqs. (18)–(22) into relation (26), with the use of assump- After taking into account power laws in the model (31), tions (23)–(25) can be found in the literature [35, 36]. its dynamic characteristics become completely different With conventional assumptions (23)–(25), the solution from the previously-analysed conventional case. As it of Eq. (26) is an asymptotically stable fixed point: will be demonstrated in the further part of the article, sf (k∗) path dependence is satisfied here both in the weak ver- k∗ = . (27) 1 + n sion (1) and in the strong one (2). This is caused by For example, for the production function of the Cobb– the fact that the production functions of Cobb–Douglas β and Day introduce nonlinearity to the model in the form Douglas type, f(k) = Bkt , all system trajectories are convergent to the path of balanced growth: of a double power law. Consequently, attractors in the 1 form of attracting fixed points and repellers in the forms sB 1−β k∗ = . (28) of repelling fixed points appear in the phase space of 1 + n the model. The co-existence of three fixed points in a Coefficient B may be interpreted as the level of technol- configuration where a point repeller is situated between ogy, while β > 0. two point attractors leads to strong path dependence Path Dependence in Neoclassical Economic Growth Theory A-91

c (and thus to path dependence). The dynamics of sys- near point k2. An arrow marks a continuation of the last tem (31), determined by initial conditions and values of stage of the trajectory presented in Fig. 2a. parameters, may be also more complex. Function (31) has the bell shape with a single maxi- mum, which can be determined by comparing the first derivative to zero: βm k = , (32) max β + φ therefore, m β+φ F (k ) = µ ββφφ. (33) max β + φ The zeroes are as follows: F (0) = 0 and F (m) = 0. To avoid negative values of capital stock per capita, it is required that F (kmax) ≤ m, therefore coefficients have to satisfy the following condition: m β+φ µ ββφφ ≤ m. (34) β + φ In practice, it is not difficult to satisfy condition (34) since, in the real world, the value of parameter µ is significantly lower than one. Otherwise, we would have to deal with such devastation of the natural environment that would involve a highly rapid decrease in the production rate of capital goods. Although this would certainly threaten the existence of the economic system, for- tunately, under normal conditions, such a phenomenon has not yet been observed.

4.2. Weak path dependence

The neoclassical model modified by Day has a certain Fig. 2. Trajectory of the neoclassical growth model for k0 = 0.4, µ = 3.9, β = 2.19, m = 1.45, φ = 8.66, ap- interesting property. Mapping (31) has a fixed point of proaching a fixed point of zero: (a) general view of the zero, which is the most common attracting fixed point. trajectory, (b) enlargement of the origin of the coordi- c After calculating the derivative nate system with a fixed point k2 ≈ 0.024. 0 β−1 φ β φ−1 F (k) = µβk (m − k) − µk φ (m − k) (35) Numerical explorations demonstrated that for other 0 we can see that F (0) = 0 (for β 6= 1). Numerical explo- initial conditions, although the trajectories of the sys- rations have shown that if we assume an initial point tem were different, zero was an attracting fixed point in of k0 = 0.4 and establish that µ = 3.9, m = 1.45, each case. This points to the existence of a weak version 1.3 < φ < 10, 1.2 < β < 2.9, then two point attrac- of path dependency in the economic system (31), since tors may coexist in a phase space of the model and zero x = O (H ) , but for H 6= H¯ , is one of them [40]. The zero attractor can coexist with t+1 t t t t other objects, such as unstable fixed points, or can be ∼ we have Ot (Ht) = Ot H¯ t . (37) the only attractor of the system. This solution only partially corresponds to the neo- Let µ = 3.9, β = 2.19, m = 1.45, φ = 8.66, and classical theory of economic growth, which provides for the initial point remains unchanged. We then have three the existence of distinguished states or equilibrium paths fixed points kc = 0, kc ≈ 0.024 and kc ≈ 0.514; the last 1 2 3 independent of the history of the system. In the exam- two being unstable: ined case, such independence can be referred to only in 0 c 0 c |F (k2)| , |F (k3)| > 1. (36) a global sense. From the economic point of view, such a Figure 2 presents a system trajectory obtained by the solution is surprising, since it means liquidation of the ex- cobweb method. A graph of mapping (31) intersects a isting method of management. On the other hand, in the ◦ c c line inclined at 45 in two points: k2 and k3. The tra- local sense, differences in initial conditions lead to differ- jectory starts at point x0 = 0.4 and after a few dozen ent histories, which implies weak path dependence. Ad- iterations in the 0 < k < 1 range it approaches zero. ditionally, the lengths of the vectors of history ht, which c Figure 2a presents a general course near point k3 and are measured by the number of iterations, were different an arrow marks one of last iterations before reaching for various initial conditions. Global independence of the zero. Figure 2b presents an enlarged fragment of a graph zero state on the history of the system is of little prac- in Fig. 2a and shows the changes occurring in the system tical importance, since economy is an open dynamical A-92 A. Jakimowicz system. Therefore, in a sufficiently long time, environ- µ = 3.9, m = 1.45. Mapping (31) then has three fixed c c c ment impacts may appear which, through the change of points: k1 = 0, k2 ≈ 0.1285 and k3 ≈ 0.4157, while only c parameters, amend the nature of the system and reverse point k2 is unstable. After substituting those values to the negative trends (divergence of the trajectory to zero). Eq. (35), we have: A slightly different possibility is presented in Fig. 3, 0 c 0 c |F (k2)| > 1 and 0 > F (k3) > −1. (38) in which the path of the system for the following values Figure 4 presents two examples of trajectories concern- of parameters is drawn: µ = 3.9, β = 2.87, m = 1.45 ing the described case. The first of them starts at point and φ = 2.9. Since a graph of mapping does not inter- k = 0.1535 and aims towards the fixed point kc ≈ sect the diagonal F (k) = k at all, therefore, the only 01 3 0.4157, while the other, initiated at point k = 0.1 ap- attracting fixed point is zero and the trajectory start- 02 proaches zero. Therefore, we deal in the system with the ing in point k = 0.4 approaches it. From the point of 0 coexistence of two attractors being fixed points. A path view of path dependence relations, there is no significant starting at point k = 0.4 reaches kc, but already for the difference in comparison to the previous case. Defini- 0 3 initial condition k = 0.7 we again have convergence to tion (37) is satisfied. Different vectors of history occur 0 zero. An attracting fixed point of zero may coexist with for different initial conditions, although the global inde- a stable periodic orbit with any period and with a chaotic pendence of an attracting fixed point from the system attractor. history emerges. However, an interesting regularity can be observed. The length of the vectors of history ht di- rectly depend on the initial conditions. The higher the value of the initial capital per capita, the longer the system destruction is.

Fig. 4. Coexistence of two attractors of mapping (31) being fixed points. The trajectory initiated in point c k01 = 0.1535 is convergent to k3, and the trajectory started in point k02 = 0.1 approaches zero (µ = 3.9, Fig. 3. Trajectory of mapping (31) initiated in point β = 2.9, m = 1.45 and φ = 9.1). k0 = 0.4 approaching the fixed point of zero. No other fixed points (µ = 3.9, β = 2.87, m = 1.45 and φ = 2.9). In the examined case, since definition (2) is satisfied, The two cases examined above point to the need for a there is a strong path-dependence-type relationship. Var- certain modification to the definition of weak path depen- ious vectors of history lead to various outcome functions. dence. This should include the length of the vectors of In the example presented in Fig. 4 we can also see a history, which in the examined model depends on the ini- certain organization of the vectors of history. Initial val- tial conditions. However, we still have two possibilities. ues of the capital stock per capita, which determine in- The first is that the final point can be the same for vari- dividual vectors of history, belong to two disjointed sets. ous vectors of history, which may have different lengths in The first set contains initial conditions whose paths are terms of the number of iterations. The second possibility convergent to zero, while the other set contains initial indicates various target states for various vectors, while conditions whose paths are convergent to an attracting again, the length of individual vectors can be different. fixed point different than zero. In this way, basins of In the second case, we enter a strong path-dependence- attraction of these attractors have been defined. Addi- type relationship, which will be discussed below. tionally, the structure of these sets is complex, as they intermingle. The attractor of zero also emerges for start 4.3. Strong path dependence c points satisfying the condition k0 > k2 ≈ 0.1285. Thus, The situation when the zero attractor coexists with an- we can see that strong path dependence in dynamical other attracting fixed point is slightly different. Let us systems is related to the coexistence of various attrac- examine a point in a four-dimensional space of parame- tors, and their basins of attraction define various vectors ters, with the following coordinates φ = 9.1 and β = 2.9, of history. Path Dependence in Neoclassical Economic Growth Theory A-93

The neoclassical model of economic growth supple- In light of the above results, economic interpretation of mented with two power laws, the production function of the neoclassical model seems to be a crucial issue. I do the Cobb–Douglas type and a multiplicative element re- not suppose that the convergence of the trajectory to flecting the impact of pollution on economic growth, con- zero would reduce its applicability, since reaching this tains all important elements of neoclassical economics. state requires time and, consequently, the performance Therefore, it can be considered a milestone in the history of a certain number of iterations. Such a path can be of economic thought. It was formulated at the beginning considered equivalent to impermanent, temporary pro- of 1980s, but has not yet been subject to thorough dy- duction methods. Also, not everything depends on the namic research. Perhaps the academic community has value of parameters. The system has certain restrictions decided that it has already satisfied its basic task, which imposed on initial conditions: only specific initial forms was the introduction of complexity to economics. Models have an opportunity to develop, and not all of them. such as this one have certainly made researchers aware The situation presented in Fig. 4, where a repelling fixed of the role of nonlinearity and deterministic chaos in eco- point occurs between two stable fixed points, may prove nomics. Unfortunately, they were later a bit forgotten. the existence of a growth barrier difficult to be over- This article makes an attempt to fill the existing gap and come for less developed countries (starting from a lower c presents an in-depth analysis of system (31), taking into level). On the other hand, if the limit k2 ≈ 0.1285 is account various initial values and parametric sensitivity crossed, then the economic system will reveal flexibility issues. This led to discovery of previously unknown dy- and an ability to survive, even if stable cycles or chaotic namic features of the model, such as the co-existence of behaviour emerge. However, high initial values are also attractors and repellers and the occurrence of path de- not desirable, as such gigantomania leads to the decline pendence in a weak and strong version. of the system (the basins of attraction of both attractors intermingle). Coexistence of attractors can also in- 5. Conclusions dicate the frailty of contemporary economies, since their survival depends on an appropriate combination of pa- Traditional economics, based on linear conventional rameter values. A zero attractor represents the potential models, reveal a path-dependence relation in a weak threat of a decline of the present method of management version. This is related to the existence of only two based on uncontrolled use of resources and leading to types of attractors: either a dynamic state of equilib- environmental pollution. Analogically, divergent trajec- rium, or a point in infinity, to which system trajectories tories can be treated as forbidden states of the system. are asymptotically convergent, regardless of the initial The ranges determined by parameters and initial con- conditions. In conventional economics, the rate and va- ditions in which the system can exist and develop are riety of events within various vectors of history mainly not too broad. To summarize, it seems that this simple depend on the parameter values. Complexity economics, nonlinear model is highly useful for describing reality. taking into account nonlinear phenomena, broadens the In so far as the phenomenon of path dependence is one catalogue of types of behaviour demonstrated by mar- of the basic features of complex dynamic systems which kets and economies. A strong path dependence already in the hard sciences are subject to formalization and anal- exists in simple one-dimensional nonlinear systems, as yses in a quantitative way, the issue of historical condi- shown by a modified version of the Solow model of eco- tions in economics and social sciences is examined almost nomic growth. Path dependence, regardless of whether entirely with the use of qualitative methods. Path de- it is considered in a weak or strong version, should not pendence ideas are used mainly for describing advanced be identified with deterministic chaos. The neoclassical technology markets where the causes of market failures model of economic growth also confirms the existence and the emergence of monopolies offering non-optimal so- of a weak version this relation, while the lengths of the lutions are analysed. One of the best known examples is vectors of history do not depend only upon the value of the QWERTY layout keyboard, which is less efficient for parameters — as in the case of conventional models — typing in comparison to other, better but lesser-known but upon initial conditions. solutions. For these reasons, the study of path depen- The results of computer simulations of dynamic dence in economics is still in the non-formalized phase. changes in the neoclassical model are slightly surprising. This article is one of the first theoretical attempts to es- According to Eq. (31), a decrease of φ to zero guarantees tablish necessary foundations for empirical analyses of φ path dependence phenomena in economics. It also shows that a multiplicative element (m − kt) will approach unity and the production function (29) will be similar to another inventive application of path dependence, con- β sisting in an analysis of risks resulting from environmen- the traditional Cobb–Douglas function f(k) = Bkt [41]. tal pollution. In addition, a known solution in the form of a sustain- able growth path can be expected. However, a decrease Acknowledgments in parameter φ causes quite unexpected changes in system behaviour (31). If β = 2.15 and φ = 10, then the I would like to thank the anonymous reviewer for in- trajectory is convergent to zero, since it is the only at- depth and detailed comments concerning the original ver- tracting fixed point. sion of the text. A-94 A. Jakimowicz

References [18] P. Pierson, Am. Polit. Sci. Rev. 94, 251 (2000). [19] L. Dobusch J. Kapeller, Schmalenbach Business Re- [ ] W.B. Arthur, Complexity Economics: A Different 1 view 65, 288 (2013). Framework for Economic Thought, Santa Fe Institute Working Paper: 2013-04-012 (2013). [20] J.-P. Vergne, R. Durand, J. Manage. Stud. 47, 736 (2010). [2] S.N. Durlauf, What Should Policymakers Know About Economic Complexity?, Santa Fe Institute [21] J. Sydow, A. Windeler, G. Müller-Seitz, K. Lange, Working Paper: 97-10-080 (1997). Bus. Res. 5, 155 (2012). [3] W.B. Arthur, Increasing Returns and Path Depen- [22] R.I. Khrapko, Path-Dependent Functions, Theor. dence in the Economy, University of Michigan Press, Math. Phys. 65, 1196 (1985). Ann Arbor 1994. [23] S.E. Page, Q. J. Political Science 1, 87 (2006). [4] C. Antonelli, Path Dependence, Localized Technologi- [24] J.M. Keynes, The General Theory of Employment, cal Change and the Quest for Dynamic Efficiency, in: Interest, and Money, 5th ed., MacMillan, London New Frontiers in the Economics of Innovation and 1946. New Technology: Essays in Honour of Paul A. David, [25] A. Herscovici, Nature of Capital, Long-Run Expec- Eds. C. Antonelli, D. Foray, B.H. Hall, W.E. Stein- tations and Path Dependence in the General The- mueller, Edward Elgar Publishing, Cheltenham 2006, ory: Some Epistemological Observations, The Third p. 51. Nordic Post-Keynesian Conference, Aalborg Univer- [5] W. Kwaśnicki, Skanalizowane ścieżki rozwoju prze- sity, Aalborg 2014. mysłu, XX Szkoła Symulacji Systemów Gospodar- [26] P.A. Samuelson, A Synthesis of the Principle of Ac- czych, Polanica Zdrój 2003. celeration and the Multiplier, J. Polit. Econ. 47, [6] P.A. David, Path Dependence, its Critics and the 786 (1939). Quest for ‘Historical Economics’, in: Evolution [27] P.A. Samuelson, Rev. Econ. Stat. 21, 75 (1939). and Path Dependence in Economic Ideas: Past and Present, Eds. P. Garrouste, S. Ioannides, Edward El- [28] A. Jakimowicz, Analiza porównawcza teorii cyklu gar Publishing, Cheltenham 2001, p. 15. koniunkturalnego w ujęciu J.M. Keynesa i P.A. Samuelsona [A Comparative Analysis of J.M. Keynes’ [ ] S.J. Liebowitz, S.E. Margolis (Eds.), Path Depen- 7 and P.A. Samuelson’s Business Cycle Theories], dence and Lock-In, Edward Elgar Publishing, Chel- Wydawnictwo Adam Marszałek, Toruń 1991. tenham 2014. [8] R. Martin, P. Sunley, The Place of Path Depen- [29] J.R. Hicks, A Contribution to the Theory of the Trade dence in an Evolutionary Perspective on the Eco- Cycle, Clarendon Press, Oxford 1951. nomic Landscape, in: The Handbook of Evolutionary [30] A. Jakimowicz, Od Keynesa do teorii chaosu. Economic Geography, Eds. R. Boschma, R. Martin, Ewolucja teorii wahań koniunkturalnych [From Edward Elgar Publishing, Cheltenham 2010, p. 62. Keynes to Chaos Theory. Evolution of Business [9] J. Koch, M. Eisend, A. Petermann, Bus. Res. 2, 67 Cycle Theories], Wydawnictwo Naukowe PWN, (2009). Seria: “Współczesna Ekonomia”, Warszawa 2012. [10] C. List, Am. Polit. Sci. Rev. 98, 495 (2004). [31] R.G.D. Allen, Ekonomia matematyczna, PWN, Warszawa 1961. [11] P. Malpass, Housing, Theory and Society 28, 305 [32] J. Bellaïche, J. Math. Econ. 46, 163 (2010). (2011). [33] E. Ott, Chaos in Dynamical Systems, 2nd ed., Cam- [12] C. Castaldi, G. Dosi, E. Paraskevopoulou, Path De- bridge University Press, Cambridge 2002. pendence in Technologies and Organizations: A Con- cise Guide, Working Paper 11.04, Eindhoven Centre [34] R.M. Solow, Q. J. Econ. 70, 65 (1956). for Innovation Studies, School of Innovation Sciences, [35] R.G.D. Allen, Teoria makroekonomiczna. Ujęcie Eindhoven University of Technology, Eindhoven 2011. matematyczne, PWN, Warszawa 1975. [13] K.J. Meier, R.D. Wrinkle, S. Robinson, J.L. Poli- [36] R.H. Day, Am. Econ. Rev. 72, 406 (1982). nard, Path Dependence and Organizational Perfor- [37] R.H. Day, The Emergence of Chaos from Classical mance: Representative Bureaucracy, Prior Decisions, Economic Growth, Q. J. Econ. 98, 201 (1983). and Academic Outcomes, 59th National Meeting of [38] V. Pareto, Cours d’économie politique, vol. 1–2, in: the Midwest Political Science Association, Chicago Oeuvres complètes, Eds. G.-H. Bousquet, G. Busino, 2001. vol. 1, Librairie Droz, Genève 1964. [14] I. Greener, Theorising Path Dependence: How does History Come to Matter in Organisations, and what [39] V. Pareto, Manual of Political Economy, Macmillan, Can We Do about It?, Working Paper 3, Department London 1971. of Management Studies, University of York, York [40] A. Jakimowicz, Podstawy interwencjonizmu państ- 2004. wowego. Historiozofia ekonomii [Foundations of [15] B. Ebbinghaus, Can Path Dependence Explain Insti- State Interventionism: Philosophy of Economics His- tutional Change? Two Approaches Applied to Welfare tory], Seria: Współczesna Ekonomia, Wydawnictwo State Reform, Discussion Paper 05/2, Max Planck In- Naukowe PWN, Warszawa 2012. stitute for the Study of Societies, Cologne 2005. [41] G. Abraham-Frois, Dynamique économique, Dalloz, [16] R. Hochreiter, G.Ch. Pflug, Computational Eco- Paris 1995. nomics 28, 291 (2006). [17] J. Mahoney, Theory and Society 29, 507 (2000).