The emergence of chaos in productivity distribution dynamics

Abstract

The distribution of productivity levels, and its evolution over time, is a research topic of utmost importance in empirical and theoretical economics. On the theory side, simple analytical models, involving intertemporal optimization, typically characterize agents’ investment decisions about ways to upgrade technology and enhance productivity. The prototypical model endogenously splits the productivity distribution in two: the right-hand side of the distribution is populated by innovators; the left-hand side is occupied by agents who follow a strategy of adoption or imitation. Given the assumptions of the model, the productivity of innovators grows at a constant rate (which directly depends on a constant probability of innovation). The evolution of the productivity of adopters may, in turn, implicate complex dynamics. Because the pace of productivity growth for adopters depends on the shape of the productivity distribution, different distributions might induce distinct growth paths, some of them potentially leading to the emergence of nonlinearities, such as limit cycles and chaos. This study investigates the presence of nonlinearities in technology adoption, for different configurations of the productivity distribution. Under reasonable parameterizations, endogenous fluctuations emerge as a plausible long-term equilibrium.

1 Introduction

The distribution of productivity levels, across countries, industries, firms, or workers, is a much-debated topic of research in economics. On the empirical front, many studies have addressed the issue, most of them engaging in statistical analyses whose primary goal is to identify regularity patterns about the shape and the dynamics of the productivity distribution, in different contexts and locations. Some notable work in this regard includes the contributions of Syverson (2011), Bartelsman et al. (2013), Kahn and Lange (2014), Garicano et al. (2016), Poschke (2018), Van Reenen (2018), and Autor et al. (2020).

Regarding theory, productivity distribution dynamics is typically approached through the analysis of standard optimization models. In these models, although every agent in the economy solves a potentially identical optimal control problem, involving the maximization of consumption utility subject to a constraint on the evolution of productivity, agents are perceived as non-homogeneous entities. The source of heterogeneity is the disparity on the initial endowments of knowledge and skills and, thus, on the initial levels of productivity. Hence, the problem cannot be solved by a representative agent; it must be approached by every individual located along the productivity distribution, with outcomes that will differ given the position occupied by each agent. The control variables in the agents’ problems are the investment levels potentially applied to innovation and/or technology adoption with the purpose of enhancing productivity.

The above-mentioned class of models has been explored essentially by Benhabib et al. (2014, 2020, 2021) and Konig et al. (2016, 2022). The pioneer work, Benhabib et al. (2014), proposes a simple optimal control problem to characterize the choice of the agents between innovation and imitation (or adoption). The decision on whether to invest in the adoption of pre-existent technologies or in original research is attached to the distance to the technological frontier: agents close to the frontier have an advantage in selecting the innovation strategy, while those that are further away from the frontier benefit from a catch-up strategy of imitation (which, for these agents, has a higher probability of success). The corollary of this logical behavior is the endogenous separation of the productivity distribution in two: to the left of some threshold point, agents will find it advantageous to adopt technologies in use by others, while to the right of the mentioned point, innovation is the most rewarding strategy.

Although the contributions that followed Benhabib et al. (2014) have kept the essence of the original model, they have sophisticated the analysis in various directions. One of such directions, and the most relevant in the current context, has been to consider that the reference for imitation is not necessarily the technology frontier (i.e., the right extremity of the productivity distribution), but the entire distribution range. Those who optimally imitate, do not necessarily imitate from the leader but they do it from any individual locating to their right in the distribution (and, thus, holding a higher level of productivity than the imitator itself). Technology adopters are defined, in the context of these models, as agents who randomly establish contact with other agents in the economy, and, in case they meet someone with a higher level of productivity than the one they already hold, then they will absorb such productivity level (i.e., they imitate the corresponding technology).

The particular outcome of the model depends on the assumptions underlying each suggested version. For instance, Benhabib et al. (2021) propose a rich setting, which allows for heterogeneous innovation capabilities and for the possibility of leapfrogging across technological states; despite the sophistication of the framework, it essentially predicts the formation of a balanced growth path such that every agent converges to the technology frontier. In turn, in Konig et al. (2016, 2022), the long-term locus is one of absence of either convergence or divergence: the productivity levels of every agent will potentially grow at the same rate and, thus, the evolution of the productivity distribution occurs under the form of a travelling wave.

Besides the ones already mentioned, other theoretical contributions are relevant in the context of the current study, namely those that put the interplay between innovation and technology diffusion in the forefront of the explanation of productivity growth. Influential literature on the theme includes the works of Luttmer (2012), Perla and Tonetti (2014), Acemoglu et al. (2018, 2022), Akcigit and Kerr (2018), Akcigit et al. (2021), and Perla et al. (2021).

In this study, a prototypical optimization model, inspired in the aforementioned theoretical literature, is set forth. As in such literature, the two fundamental features of the model are the assumption of heterogeneous productivity and the notion that technology adoption occurs through contagion (any agent interacting with another agent holding a better technology is capable of automatically and instantly absorbing the productivity level of the latter). The objective of the study is to investigate the dynamic behavior of productivity, given the intertemporal plan that each agent sets and solves.

The standard result, underlying the model’s intertemporal problem, is such that agents choose a constant over time level of investment in innovation or imitation. Once this investment level is replaced into the dynamic rule that characterizes the pace of growth of productivity, one will end up with a one-dimensional map pertaining to the motion of the productivity ratio (i.e., the ratio between the level of productivity held by the agent and the productivity level at the technology frontier). The evolution of the productivity ratio is contingent on the optimal strategy that is followed. For innovators, the dynamic result is straightforward: because the probability of innovation is constant, the corresponding growth rate of productivity is constant as well. For imitators, complex dynamics are likely to emerge.

Because the growth rate of productivity depends, in the case of technology adopters, on the entire segment of the productivity distribution located to their right, the one-dimensional map that establishes the rule of motion for the productivity ratio will most certainly acquire the form of a nonlinear equation, and its particular shape will change as one admits different configurations for the productivity distribution. The analysis to undertake considers six different types of distributions (uniform, Weibull/exponential, Kumaraswamy, log-logistic, arcsine, and trapezoidal / triangular) and approaches the underlying dynamics for each of them. The emergence of nonlinear dynamics (cycles of different periodicities and chaos) is pervasive across the considered distributions. For some of the distributions, nonlinearities are observed for reasonable parameterizations (i.e., for the assumed benchmark set of parameter values), while for other distributions, one needs to search outside the benchmark parameterization to identify the presence of cyclical or chaotic motion.

Bounded instability, and the potential presence of chaos, identified in the process of technology adoption, is relevant in order to make a fundamental distinction between innovation and imitation. In the context of the proposed model, innovation is a smooth process: when an agent engages in research, there is a probability of success, which generates productivity growth, and there is a probability of failure, that maintains productivity at a given level. Imitation emerges as a potentially tumultuous process: as the imitator randomly interacts with agents in the distribution, and the distribution eventually changes shape, there is the possibility of systematic (and eventually irregular) advances and retreats of the agent’s productivity with respect to the productivity level at the frontier, making it impracticable to predict what the outcome of imitation will be (and, ultimately, what the evolution of the productivity distribution will be).

Since the identification of nonlinearities is an important part of the analysis of the dynamics of the productivity distribution pursued in this paper, this piece of research intends to be, as well, a contribution to the study of endogenous fluctuations in economics. The respective strand of literature gained consistency in the 1980s. The following is a non-comprehensive list of relevant early work on the theme: Stutzer (1980), Benhabib and Day (1981, 1982), Day (1982), Grandmont (1985), Deneckere and Pelikan (1986), and Baumol and Benhabib (1989). In Sect. 2, a brief review of recent work about nonlinearities and chaos in economics is undertaken.

The remainder of the paper is organized as follows. Section 2 presents a short survey on up-to-date research on nonlinear dynamics and economics, and puts, as well, into perspective, how the current study contributes to such literature. Section 3 formalizes the productivity model, which takes the form of an optimal control problem separately solved by each agent occupying a different position in the productivity distribution. In Sect. 4, optimality conditions are derived, and the problem’s underlying dynamics are approached. Section 5 proposes six different shapes for the productivity distribution. The one-dimensional map attached to productivity dynamics is scrutinized in Sect. 6, for each of the six distributions; specific parameterizations conducting to the prevalence of endogenous cycles and chaos are identified and highlighted. In Sect. 7, the productivity equation is transformed into a two-dimensional map with interdependent technologies; the new setting allows to display chaotic attractors for pairs of productivity values. Sect. 8 concludes.

2 Chaos in economic theory

Why is the identification of periodic and aperiodic cycles relevant, in the context of dynamic economic models? Barnett et al. (2015) answer this question by claiming that the emergence of such limit cycles provides an endogenous explanation for the apparently stochastic fluctuations often observed in time series pertaining to the evolution of output, asset prices, or any other economic variable. The typology of endogenous fluctuations ranges from period-2 cycles to chaos. Chaos is a specially appealing form of nonlinear outcome, because it is associated with a motion that never exactly repeats itself and that is sensitive to initial conditions. In a chaotic system, it is virtually impossible to anticipate the future: any small initial error or approximation generates an irreparable departure from the actual motion (this is the well-known butterfly effect, remarked, e.g., in Akhmet et al. (2014), and Anufriev et al. (2018); see also Day and Pavlov 2002, regarding the perils of approximations in chaotic systems).

There are various possible definitions of chaos, that apply to systems of different nature and dimension, with the most relevant distinction in the literature being made between topological chaos and ergodic (statistical) chaos (see, e.g., Day and Shafer 1987). Topological chaos might be associated, in low-dimensional dynamic systems, as the ones approached in this study, to the presence of odd-period cycles (Li and Yorke 1975) or, alternatively, with turbulence, a phenomenon that is discernible through the computation of the system’s measure of topological entropy (see Mitra 2001; Deng and Khan 2018; and Deng et al. (2022)). The operational analytical tool used in this study to identify the presence of chaos is the standard Lyapunov characteristic exponent, a measure that quantifies the exponential rate at which nearby orbits drift apart (a positive Lyapunov exponent signifies the presence of chaos, because it indicates that nearby orbits exponentially diverge).

The quest for chaos in economic theory has been nothing short than a bumpy ride. Most dynamic systems one may conceive to explain economic phenomena are linear or have a low degree of associated nonlinearity, thus leading to trivial fixed-point solutions. Limit cycles and chaos only emerge associated with highly nonlinear systems, which are often artificially created by somehow bending what one would recognize as reasonable and sensible economic assumptions. Identifying chaos in models built upon perfectly tenable assumptions is rare, but has been done in some specific circumstances. Gomes (2006) highlighted a few research niches in which chaotic motion intuitively emerges associated with robust economic foundations. The most prominent areas worth mentioning are the following three:

(i) Business cycles analysis in the Kaldor-Hicks-Goodwin tradition. Because of the nonlinear shape of the investment demand function in this class of models, endogenous fluctuations are likely to emerge in relatively simple specifications of the macro economy (see Puu 1986; Dieci et al. 1998; Bischi et al. 2001). Given the intellectual appeal of this simple model, its rich dynamic properties continue to be explored in the literature in the current days (see, e.g., Orlando 2016; Chu et al. 2021; Zhou et al. 2021).

(ii) The assessment of nonlinearities in standard deterministic optimal economic growth models. In this context, limit cycles and chaos can emerge once one goes beyond trivial assumptions, and considers, for instance, non-conventional functional forms for the production function, unreasonably high rates of time preference, or multiple production sectors. Relevant literature on the topic includes Nishimura et al. (1994), Nishimura and Yano (1995), Boldrin et al. (2001), Mattana et al. (2009), and Bella et al. (2017).

(iii) Heterogeneity in expectations and bounded rationality. Brock and Hommes (1997, 1998) initiated an influential research line, whose main underlying assumption is the coexistence of hyper-rational and boundedly rational agents in markets (financial and others). In opposition to the conventional wisdom, these authors have supported the view that fundamentalists do not drive trend-followers out of the market. Instead, they tend to coexist, and their interaction stimulates the generation of nonlinearities and chaotic motion (mainly associated with the evolution of prices). Under this view, markets are inherently chaotic and produce unpredictable outcomes that are hardly compatible with the conventional notion of equilibrium. The Brock-Hommes heterogeneous beliefs theory inspired a relevant strand of literature that has persisted to this day (some recent contributions include Kukacka and Kristoufek 2020; Bao et al. 2021; Hommes 2021; and Vogl 2022).

One example of a novel route to chaos in economics recently explored in the literature is offered by Battaglini (2021), who proposes a model of environmental protection policy. The setting is one in which two political parties alternate in power and select environmental policies strategically, taking into consideration not only the environmental goal, but also the objective of remaining in office. The political process generates a dynamic time inconsistency (between long-run environmental sustainability and short-run political popularity) that deviates the equilibrium solution from a stable point. The time inconsistency, attached to the strategic behavior of agents, is the main source of instability and of the likely formation of limit cycles and chaos.

In the current study, another path, not yet explored in the literature, regarding the emergence of endogenous fluctuations, is undertaken. Nonlinearities are associated with a benchmark utility maximization optimal control problem, in which agents are heterogeneous with regard to their productivity levels. The setting in which these endogenous deterministic cycles germinate is similar to the one proposed by J. Benhabib, M. Konig, and their coauthors, through the series of works that were mentioned in the introduction. As in such works, agents located at a given point in the productivity distribution optimally choose whether to become innovators or, alternatively, adopters of technologies already in use. Nevertheless, unlike what is common in those contributions, the focus of the current study is not concentrated on identifying the formation of ’well-behaved’ balanced growth paths that individual productivities and the whole productivity distribution eventually follow in the long term, given some general configuration of the underlying initial distribution. Instead, by specifying concrete cumulative distribution functions for productivity, it becomes possible to represent the evolution of each agents’ productivity index under an explicit one-dimensional map, which invariably takes the form of a nonlinear difference equation.

The derived nonlinear map uncovers the formation of deterministic cycles (under the form of regular and irregular endogenous fluctuations), which are absent in the standard setup concerning the analysis of productivity growth. When the actions of one agent are contingent on the whole range of states pertaining to every other individual in the economy, complex dynamics are likely to emerge. The analysis pursued in this study illustrates precisely this point, within the scope of a simple optimal control framework of productivity choices, and in a way that so far has been absent from the literature that explores productivity dynamics and optimal choices associated with innovation and imitation.

Bounded instability outcomes do not arise associated with the behavior of every agent, but only with the behavior of technology adopters. The reason, in the context of the model, is that technology adoption requires imitation from those holding higher productivity levels, and these are placed over a distribution that is potentially and most likely nonlinear. Irregular and unpredictable growth patterns of productivity associated with technological adoption, as the ones suggested by the model devised in this study, find support on the empirical literature and they have been documented under a variety of perspectives. At this regard, three illustrative examples can be briefly mentioned:

(i) Lee and Tang (2017) empirically estimate, for a sample of Chinese firms, the impact of choosing imitation as a strategic orientation to serve as a complement or as an alternative to innovation. Their findings point to imitation as a source of dysfunctional competition and technological turbulence, notions that might be associated, within the model, with complex dynamics, namely irregular fluctuations and chaos;

(ii) Im and Shon (2019) attribute the eventual instability of imitation to two countervailing forces that in different periods of time and in different environments may have different strengths, namely the positive externalities that imitation potentially generates and the free-riding issues that also come with the adoption of technologies already in use. If the relative strength of these factors changes fast, then imitation is inherently unstable;

(iii) Liao (2020) measures the search costs associated with imitation. Because the costs of searching for imitation opportunities are influenced by a series of factors (e.g., macroeconomic conditions or the dynamics of the specific industry), one should expect the intensity of imitation to fluctuate over time and, probably, a back-and-forth movement of the productivity of imitators relative to the technology frontier will end up by prevailing.

The sections that follow characterize the model and identify the circumstances in which cycles and chaos eventually emerge.

3 The prototypical productivity model

Conceive an economy populated by a large number of agents, indexed by i. Every agent faces a same planning problem, which consists in maximizing intertemporal utility subject to a constraint on the evolution of productivity. Within this framework, optimal decisions are associated with the choice of how much to invest, over time, in innovation and/or technology adoption. Agents are heterogeneous with regard to their initial productivity levels. Although this is the single source of heterogeneity in the proposed setup, it has relevant implications for the outcome of each individual’s dynamic optimization problem (namely, it will determine whether the agent will be an imitator or an innovator). In this setting, time is discrete.

Let $A_{i,t}\ge 0$ represent the level of productivity of agent i at date t. The income of this agent is a linear function of productivity, i.e., $Y_{i,t}=BA_{i,t}$, with $B>0$ a measure of contextual conditions that allow the agent to generate more or less income for a same level of productivity; the value of this parameter is assumed identical across individuals. Each agent might invest a value $g_{i,t}\ge 0$, per productivity unit, in innovation, and / or a value $s_{i,t}\ge 0$, per productivity unit, in imitation. Variables $g_{i,t}$ and $s_{i,t}$ are the control variables of the intertemporal optimization problem that agent i desires to solve.

The argument of the agent’s utility function is the difference between the individual’s income and the respective investment costs. To simplify the analysis, assume that the utility function is logarithmic (although any other utility function exhibiting constant intertemporal elasticity of substitution would generate similar results): $u_{i,t}=\ln \left[ \left( B-g_{i,t}-s_{i,t}\right) A_{i,t}\right]$. The agent will want to maximize, at $t=0$, the discounted flow of present and future utilities. Specifically, agent i maximizes $U_{0}=\sum _{t=0}^{\infty }\beta ^{t}u_{i,t}$, with $\beta \in (0,1)$ the intertemporal discount factor.

The maximization of $U_{0}$ is constrained by a difference equation that characterizes the time evolution of productivity. Productivity may grow due to innovation or through the adoption of technologies already in use by other agents in the economy. Concerning innovation, the assumption is straightforward: for each unit of investment in research, productivity grows with a probability $p\in (0,1)$. The probability of successful imitation is $q\in (0,1)$; however, imitation also depends on the distance to the technology frontier. The intuition is that agents who potentially imitate will randomly meet with other agents in the economy, and they will only enhance productivity if they interact with someone holding a better technology (or a higher level of productivity, what is the same). Therefore, the closer the individual is to the frontier, the lower will be the respective probability of success in adoption.

To analytically translate the above intuition about the adoption of technology, one needs to take into consideration the cumulative distribution function (CDF) of productivity. The CDF helps in formalizing the idea that the probability of successful adoption is q if the distance to the technology frontier is maximum, and that the probability of successful imitation is zero if the agent locates precisely at the knowledge frontier (at this point the agent has no one from whom to adopt). In between, the probability of successful imitation should fall, from q to zero, as one moves right along the productivity distribution (i.e., as the number of those from whom to potentially imitate with success progressively diminishes).

Under the established arguments, the relevant CDF to consider is $F(x_{i,t})$, with $x_{i,t}=A_{i,t}/Z_{t}$ the ratio between the actual productivity level and the frontier, and $Z_{t}$ the level of productivity at the technology frontier ($Z_{t}$ corresponds to the point at the right extremity of the productivity distribution). Notice that $0\le x_{i,t}\le 1$, and that $F(0)=0$ and $F(1)=1$.

The above reasoning conducts us directly to the difference equation characterizing the motion of the productivity of agent i. This is:

$$\begin{aligned} \frac{A_{i,t+1}-A_{i,t}}{A_{i,t}}=pg_{i,t}+q \left[ 1-F(x_{i,t})\right] s_{i,t}, \qquad A_{i,0}\ \text {given} \end{aligned}$$

(1)

Equation (1) confirms the indication that the investment in innovation generates productivity growth with probability p, while investment in imitation leads to growth at a probability that declines with the proximity to the frontier, given the underlying interaction mechanism.

Although simple in its formulation, the proposed productivity model encloses rich dynamics, which we start exploring in the following section.

4 Optimal solution and baseline dynamics

Because initial endowments of productivity vary across agents, the agents will not solve the optimization problem exactly in the same way. An obvious starting point for the analysis is to inquire how will behave the individuals that locate, from the start, at the technology frontier. Agents in this position will not invest in adoption (they have no one from whom to imitate) and, therefore, they solve the optimization problem characterized in the previous section with an additional constraint, namely $s_{i,t}=0$.

To derive optimality conditions for the dynamic problem under $A_{i,t}=Z_{t}$, begin by writing the respective current-value Hamiltonian function,¹

$$\begin{aligned} H(Z_{t},g_{z,t},v_{z,t})=\ln [(B-g_{z,t})Z_{t}] +\beta v_{z,t+1}pg_{z,t}Z_{t} \end{aligned}$$

(2)

In Eq. (2), $g_{z,t}$ represents the investment that an agent with productivity $Z_{t}$ makes in enhancing productivity further; $v_{z,t}$ is the shadow-price of productivity. The problem must satisfy the transversality condition $\underset{t\rightarrow \infty }{\lim }Z_{t}\beta ^{t}v_{z,t}=0$. First-order optimality conditions are:

$$\begin{aligned}&\frac{\partial H}{\partial g_{z,t}}=0\Rightarrow \frac{1}{B-g_{z,t}}=\beta v_{z,t+1}pZ_{t} \end{aligned}$$

(3)

$$\begin{aligned}&\beta v_{z,t+1}-v_{z,t}=-\frac{\partial H}{\partial Z_{t}} \Rightarrow \left( 1+pg_{z,t}\right) \beta v_{z,t+1}=v_{z,t} -\frac{1}{Z_{t}} \end{aligned}$$

(4)

Combining Eqs. (3) and (4), the following optimal law of motion is derived,²

$$\begin{aligned} B-g_{z,t+1}=\beta \frac{1+pB}{1+pg_{z,t}}\left( B-g_{z,t}\right) \end{aligned}$$

(5)

Expression (5) is a one-dimensional difference equation or scalar map, with investment in innovation as the single endogenous variable. The dynamics associated with this equation is straightforward to scrutinize. First, apply condition $g_{z}^{*}\equiv g_{z,t+1}=g_{z,t}$ to obtain the associated steady state points. Two solutions exist, $g_{1z}^{*}=B$ and $g_{2z}^{*}=\beta B-\frac{1-\beta }{p}$. Note that $g_{2z}^{*}<g_{1z}^{*}$. Figure 1 represents a phase diagram for Eq. (5); the visualization of the diagram is sufficient to notice that $g_{1z}^{*}$ is a stable point, while $g_{2z}^{*}$ corresponds to an unstable equilibrium. However, $g_{1z}^{*}$ is also an unfeasible corner solution, because in this case the utility will diverge to minus infinity. Recall that $g_{z,t}$ is a control variable, and therefore its value can be adjusted by the agent who solves the optimization problem to remain on the feasible steady state point. Hence, the economically meaningful solution is, in fact, $g_{2z}^{*}$.

Fig. 1

Phase diagram (dynamics of the investment in innovation; technology frontier problem)

By replacing $g_{2z}^{*}$ in the productivity equation, (1), one obtains the growth rate of productivity at the frontier, which is a constant value,

$$\begin{aligned} \gamma _{Z}=pg_{2z}^{*}=\beta pB-(1-\beta ) \end{aligned}$$

(6)

Condition $B>\frac{1-\beta }{\beta }\frac{1}{p}$ must hold if one wishes to keep the growth rate of productivity, at the technology frontier, above zero. Because the quotient $\frac{1-\beta }{\beta }$ is the rate of time preference, the growth rate is a nonnegative value if the ratio between the rate of time preference and the probability of successful innovation does not to exceed the level of income per productivity unit.

The following step of the analysis consists in addressing the productivity problem for $g_{i,t}\ge 0$ and $s_{i,t}\ge 0$. Although one is now approaching the general model defined in Sect. 3, without imposing any particular constraint on the possibilities of investment in innovation or imitation, the outcome will not be a general result. On the contrary, the derived solution will be another particular point in the distribution, in this case the point respecting to the level of productivity for which agents are indifferent between investing in innovation or imitation (i.e., the level of productivity for which identical maximum utility levels are obtained regardless of the selected investment strategy).

Let the productivity level at the aforementioned threshold point be $A_{i,t}=\zeta _{t}<Z_{t}$. If, at the specified point, the individual either invests in innovation or imitation with the exact same success, the obvious corollary is the following: for $A_{i,t}<\zeta _{t}$ (i.e., to the left of $\zeta _{t}$ in the distribution), imitation is the only sensible investment strategy to adopt; for $A_{i,t}>\zeta _{t}$ (i.e., to the right of $\zeta _{t}$ in the distribution), the rational strategy is to innovate. Thus, the distribution endogenously splits in two: the right section (to the right of the threshold) is populated by innovators; the left region (to the left of the threshold) is the home of the imitators. This result is the intuitive consequence of the evidence that backwardness favors successful imitation (i.e., it allows for a higher probability of success in adoption).

Take the original productivity problem, as characterized in Sect. 3. Consider the corresponding current-value Hamiltonian function:

$$\begin{aligned} H(\zeta _{t},g_{\zeta ,t},s_{\zeta ,t},v_{\zeta ,t}) =\ln [(B-g_{\zeta ,t}-s_{\zeta ,t})\zeta _{t}]+\beta v_{\zeta ,t+1} \left\{ pg_{\zeta ,t}+q \left[ 1-F(x_{\zeta ,t})\right] s_{\zeta ,t}\right\} \zeta _{t} \end{aligned}$$

(7)

In expression (7), the productivity ratio, the investment levels and the co-state variable are indexed by $\zeta$ in order to represent values at the threshold productivity. The following are optimality conditions of the problem (recall that both investment variables are now control variables):

$$\begin{aligned}&\frac{\partial H}{\partial g_{\zeta ,t}}=0\Rightarrow \frac{1}{B-g_{\zeta ,t}-s_{\zeta ,t}} =\beta v_{\zeta ,t+1}p\zeta _{t} \end{aligned}$$

(8)

$$\begin{aligned}&\frac{\partial H}{\partial s_{\zeta ,t}}=0\Rightarrow \frac{1}{B-g_{\zeta ,t}-s_{\zeta ,t}}=\beta v_{\zeta ,t+1}q \left[ 1-F(x_{\zeta ,t})\right] \zeta _{t} \end{aligned}$$

(9)

By combining expressions (8) and (9), one makes the CDF at $\zeta _{t}$ explicit, which in this case is a constant value that indicates the percentage of individuals that optimally choose to be imitators,

$$\begin{aligned} F(x_{\zeta ,t})=1-\frac{p}{q} \end{aligned}$$

(10)

Result (10) imposes a direct constraint on the possibility of coexistence of adopters and innovators. Only if condition $p<q$ is met, it is possible to have a productivity distribution split in two, with imitators falling to the left-hand side of $F(x_{\zeta ,t})$ and innovators falling to the right-hand side of $F(x_{\zeta ,t})$. If the condition is not met, the economy will be entirely populated by innovators. Observe, as well, that the higher is the probability of imitation relative to the probability of innovation, the larger will be the share of imitators. One should also stress that this is a constant value in time; once constant probabilities of success in adoption and research are defined, the optimal choice of the agents is such that the ratio between the number of agents in each category becomes invariant.

As mentioned, at productivity point $A_{i,t}=\zeta _{t}$, it is indifferent for agents to invest in imitation or in innovation. If one replaces (10) into (1), the equation of motion for productivity will be the same, regardless of the chosen investment strategy. In practice, someone located at $A_{i,t}=\zeta _{t}$ solves exactly the same optimal control problem as the agents in the technology frontier, and therefore the outcome in the two locations is precisely the same, namely the investment level $g_{2z}^{*}$ (in this case, in innovation or imitation) and the constant growth rate of productivity in (6) will hold as well.

Because the strategy followed by agents at any $A_{i,t}$ such that $\zeta _{t}<A_{i,t}<Z_{t}$, is the innovation strategy, the same optimality problem is again solved by every agent, and the outcome will be identical for all. Therefore, the dynamics under innovation (including the threshold indifference point) is such that all productivity levels grow at an identical rate, thus preventing the possibility of any process of convergence or divergence across productivity levels. For innovators, the motion of the productivity distribution is just a travelling wave, i.e., the distribution progressively moves right through parallel shifts, given that every point in it grows at rate $\gamma _{Z}$ in (6).

The previous reasoning does not apply for productivity levels below the threshold, $A_{i,t}<\zeta _{t}$. To understand what kind of dynamics is now relevant, one needs to approach once again the optimality problem, now under constraint $g_{i,t}=0$. Proceeding in the same way as before, the current-value Hamiltonian function takes, in this occasion, the form:

$$\begin{aligned} H(A_{i,t},s_{i,t},v_{i,t})=\ln [(B-s_{i,t})A_{i,t}]+\beta v_{i,t+1}q \left[ 1-F(x_{i,t})\right] s_{i,t}A_{i,t} \end{aligned}$$

(11)

Optimality conditions must account for the fact that the productivity variable is present in the expression of the Hamiltonian (11) not only directly but also through the CDF. The optimality conditions are:

$$\begin{aligned} \frac{\partial H}{\partial s_{i,t}}&=0\Rightarrow \frac{1}{B-s_{i,t}} =\beta v_{i,t+1}q\left[ 1-F(x_{i,t})\right] A_{i,t} \end{aligned}$$

(12)

$$\begin{aligned} \beta v_{i,t+1}-v_{i,t}&=-\frac{\partial H}{\partial A_{i,t}} \Rightarrow \left( 1+q\left\{ \left[ 1-F(x_{i,t})\right] -\frac{\partial F(x_{i,t})}{\partial A_{i,t}}A_{i,t}\right\} s_{i,t}\right) \nonumber \\&\quad \beta v_{i,t+1}=v_{i,t}-\frac{1}{A_{i,t}} \end{aligned}$$

(13)

From Eqs. (12) and (13), one derives the following equality,³

$$\begin{aligned}&B-s_{i,t+1}\nonumber \\&\quad =\beta \frac{\left[ 1-F(x_{i,t})\right] \left\{ 1+qB\left[ 1-F(x_{i,t+1})\right] -q\frac{\partial F(x_{i,t+1})}{\partial A_{i,t+1}} A_{i,t+1}s_{i,t+1}\right\} }{\left[ 1-F(x_{i,t+1})\right] \left\{ 1+q\left[ 1-F(x_{i,t})\right] s_{i,t}\right\} } \left( B-s_{i,t}\right) \end{aligned}$$

(14)

As in the case of the technology frontier, two steady state solutions emerge: the corner solution $s_{1i}^{*}=B$, and an unstable steady state, which, in parallel to the innovators case, corresponds to the level of investment in imitation that the agent will actually adopt. This steady state is computed under the assumption that the long-term productivity ratio is constant, i.e., that $A_{i,t}$ grows at the same rate as the technology frontier. This may not be the case if nonlinearities emerge. However, the agent acts as if $x_{i,t}$ were effectively constant in the steady state, choosing the level of investment in adoption accordingly.⁴ Taking this course of action, the agent will select a level of investment in imitation equal to:

$$\begin{aligned} s_{2i}^{*}=\frac{1}{q}\frac{\beta qB\left[ 1-F(x_{i}^{*})\right] -(1-\beta )}{\left[ 1-F(x_{i}^{*})\right] +\beta \frac{\partial F(x_{i}^{*})}{\partial A_{i}^{*}}A_{i}^{*}} \end{aligned}$$

(15)

If the technology adopter chooses, at date t, an investment in imitation given by (15), then the growth rate of productivity, in (1), becomes,

$$\begin{aligned} \frac{A_{i,t+1}-A_{i,t}}{A_{i,t}}=\left[ 1-F(x_{i,t})\right] \frac{\beta qB \left[ 1-F(x_{i,t})\right] -(1-\beta )}{\left[ 1-F(x_{i,t})\right] +\beta \frac{\partial F(x_{i,t})}{\partial A_{i,t}}A_{i,t}} \end{aligned}$$

(16)

The dynamic analysis of the next sections requires modifying Eq. (16). Instead of displaying the dynamics for the productivity level $A_{i,t}$, one writes the equation for the ratio $x_{i,t}$. Given the respective definition, the following equality is equivalent to (16),

$$\begin{aligned} x_{i,t+1}=\left\{ 1+\left[ 1-F(x_{i,t})\right] \frac{\beta qB \left[ 1-F(x_{i,t})\right] -(1-\beta )}{\left[ 1-F(x_{i,t})\right] +\beta \frac{ \partial F(x_{i,t})}{\partial A_{i,t}}A_{i,t}}\right\} \frac{x_{i,t}}{\beta (1+pB)} \end{aligned}$$

(17)

The analysis with specific productivity distributions will reveal that term $\frac{\partial F(x_{i,t})}{\partial A_{i,t}}A_{i,t}$ corresponds, invariably, to an expression where $x_{i,t}$ is the single relevant variable. Therefore, Eq. (17) is a nonlinear one-dimensional map $x_{i,t+1}=f(x_{i,t})$, whose dynamics will be investigated for different shapes of the CDF.

Before advancing to the analysis of the dynamics underlying (17) under a selection of productivity distributions, let us briefly reflect on what can be interpreted as the expectable outcome. Think about the decision of the agents placed at the lowest end of the distribution. These have a probability of success in imitation equal to q and, therefore, their optimal investment level in imitation will be $s_{2i}^{*}=\beta B -\frac{1-\beta }{q}$. The productivity of these agents grows at constant rate $\gamma _{i}=\beta Bq-(1-\beta )$. Because $p<q$, the growth rate of the productivity of imitators in the assumed location is higher than the growth rate of the productivity of innovators. This implies that the productivity level of imitators will converge to the productivity level of innovators, and it will stop converging only when imitators have caught-up with innovators. Although less radical than the case of the individuals locating at the left end of the distribution, a similar process is supposed to be observed for every imitator.

In the above-described scenario, the full dynamics of the productivity distribution is straightforward to characterize: the productivity of adopters would grow faster than the productivity of innovators, and therefore, there would be convergence of all adopters to productivity point $A_{i,t}=\zeta _{t}$. Once this point has been accomplished, every productivity level will evolve at the same rate (6), propelling the productivity distribution to shift right progressively, in an endless process. The dynamic analysis of the following sections unveils the circumstances in which this intuition will or will not hold in practice.

5 Half a dozen distributions

Equation (17) is a discrete-time map involving a single variable, namely the productivity ratio $x_{i,t}$. Hence, given the minimal dimensionality of the model, only a very small set of long-term outcomes for the productivity ratio are expected to emerge. One of the possible outcomes is divergence, i.e., the progressive departure of $x_{i,t}$ from the system’s fixed-point (or from one of its fixed-points) and toward infinity; this implies that the productivity of the imitator grows faster than the productivity of the frontier, what leads to the already described solution of convergence of the adopters’ productivity to the productivity level at the threshold.

If the dynamic outcome is one of stability, $x_{i,t}$ converges to a constant value in the interval (0, 1). This implies that, in the steady state, the productivity of imitators grows the same as the productivity in the frontier and, therefore, no approximation between productivity levels will take place (the value of the ratio expresses the everlasting difference of productivity relative to the frontier). Another possible result, which might emerge for specific parameterizations of the model, is one of periodicity and chaos, which typically follow a bifurcation. Endogenous cycles (periodic or aperiodic) imply that the distance of imitators to the frontier will systematically change as agents continue to follow their optimal strategies.

To get further insights on the long-term dynamics of the productivity ratio, one needs to concretize the shape of the distribution present in Eq. (17). To proceed with the discussion, six different types of distributions are taken into consideration. They are listed below.

i) Uniform distribution. If productivity is uniformly distributed, agents will be evenly spread across every possible level of productivity. Because $x_{i,t}\in [0,1]$, the CDF will write, in this case, simply as $F(x_{i,t})= x_{i,t}$.

ii) Weibull/exponential distribution. The Weibull distribution acquires a wide variety of possible shapes, depending on the values of the parameters that define it. These are an inverse scale parameter $\lambda >0$ and a shape parameter $k>0$. For the particular case $k=1$, the distribution transmutes into an exponential distribution. A hump-shaped form of the Weibull distribution emerges for values of k larger than 1. The Weibull distribution has an infinite support, and, therefore, for our purposes it must be modified to guarantee that its support is confined to the interval [0, 1]. The version of the CDF to consider is:

$$\begin{aligned} F(x_{i,t})=\frac{1-e^{-\left( \lambda x_{i,t}\right) ^{k}}}{1-e^{-\lambda ^{k}}} \end{aligned}$$

(18)

iii) Kumaraswamy distribution. A productivity distribution defined under this category may assume a variety of different shapes (e.g., it can be U-shaped, monotonically decreasing, or hump-shaped) depending on the values of its two defining parameters, $a,b>0$. The support of the distribution is [0, 1], and thus there is no need to modify the original version to apply it to the model under evaluation. The cumulative distribution of productivity is translated into the following functional form,

$$\begin{aligned} F(x_{i,t})=1-\left( 1-x_{i,t}^{a}\right) ^{b} \end{aligned}$$

(19)

iv) Log-logistic distribution. The log-logistic distribution, as the aforementioned distributions, also generates a variety of outcomes, given two associated parameters $a,b>0$. It is, in its original form, an infinite support distribution, and, thus, we normalize it to fit our purpose (i.e., to obey to the boundary values imposed to the productivity ratio). The cumulative distribution is such that:

$$\begin{aligned} F(x_{i,t})=\frac{1+a^{b}}{1+\left( \frac{a}{x_{i,t}}\right) ^{b}} \end{aligned}$$

(20)

v) Arcsine distribution. Under the arcsine distribution, productivity levels concentrate more intensely in the extremities (the probability distribution function is U-shaped). The support of the distribution is the interval [0, 1], what signifies that the commonly used version of the CDF directly applies to our analysis,

$$\begin{aligned} F(x_{i,t})=\frac{2}{\pi }\arcsin \left( \sqrt{x_{i,t}}\right) \end{aligned}$$

(21)

vi) Trapezoidal/triangular distribution. The trapezoidal distribution is a probability distribution with three linear segments, connected by two discontinuity points. The first segment is increasing, the second horizontal, and the third decreasing. If the second segment is missing, the distribution is no longer trapezoidal but, instead, triangular (i.e., the probability distribution function will be tent shaped). The distribution is typically defined in an interval between two positive numbers. Adapting the extremities to zero and 1, the corresponding cumulative distribution can be represented as:

$$\begin{aligned} F(x_{i,t})=\left\{ \begin{array}{l} \frac{1}{1+c-b}\frac{x_{i,t}^{2}}{b},\ \ x_{i,t}<b \\ \frac{1}{1+c-b}\left( 2x_{i,t}-b\right) ,\ \ b\le x_{i,t}<c \\ 1-\frac{1}{1+c-b}\frac{\left( 1-x_{i,t}\right) ^{2}}{1-c}, \ \ x_{i,t}\ge c \end{array}\right. , \ \ 0<b\le c<1 \end{aligned}$$

(22)

Note that if $b=c$, then the trapezoidal distribution transforms into the triangular distribution.

The above functions $F(x_{i,t})$ are a sample of cumulative distributions that can be used to approach the dynamics of technology adoption and the motion of productivity. Although very different from one another, and involving different sets of parameters, the next section conveys the idea that they can all generate nonlinear outcomes. However, the context in which such nonlinearities are formed differ from one distribution to another. For some of the CDFs, it is feasible to arrive to an outcome of periodic cycles and chaos for reasonable values of the parameters that shape the productivity problem (i.e., the probabilities of success of innovation and imitation, the discount factor, and the parameter of the income generating function). For other CDFs, a result different from a fixed-point or the absence of any attainable equilibrium exists only under extreme conditions (i.e., an exaggeratedly high growth rate of productivity under innovation). The main conclusion emerging from the analysis is that the shape of the distribution is crucial in determining the qualitative nature of the evolution of the productivity of the agents that optimally choose to follow the imitation strategy.

The exploration of the dynamics underlying the model will be eminently graphical and it will identify regions of periodic cycles and chaos in the space of parameters, for each of the six assumed CDFs.

6 The quest for chaos

In this section, we investigate the emergence of endogenous fluctuations associated with discrete-time map (17), given each of the CDFs characterized in Sect. 5. To proceed, a benchmark parameterization is taken. Specifically, assume $\beta =0.95$ (what corresponds to a rate of time preference of approximately $\rho =0.053$), $p=0.1$, and $q=0.25$ (for the defined probabilities of success in innovation and imitation, the share of adopters in this economy is $F(x_{\zeta ,t})=0.6$). Let also $B=1$. With these baseline values, the growth rate of the productivity of innovators (including the growth rate at the frontier) is $\gamma =0.045$ (each period, the productivity of innovators increases 4.5 per cent).

Besides the above, one must also specify values for the parameters that shape the various assumed distributions. By letting such parameters take different values within their admissible range, one will uncover a rich set of possible dynamic outcomes, including limit cycles and chaos. As a result, one will be able to conclude that knowledge adoption might follow nonlinear and possibly irregular trajectories, whose essential source is the interaction established among (boundedly) rational agents who seek to maximize intertemporal utility. In the series of examples to explore, the initial value $x_{i,0}=0.25$ is taken (although, in most cases, the basin of attraction is the entire range of possible values of $x_{i,0}$, and, thus, any other $x_{i,0}$ could be equally assumed).

The first distribution to assess is the uniform distribution. The respective CDF has no attached parameters; therefore, its dynamic analysis, for the chosen baseline numerical example, is straightforward. The outcome is one of stability, with $x_{i}^{*}=0.452$ (in the steady state, for the uniform distribution and under the assumed example, the economy converges to a point where the productivity of adopters is 45.2 per cent of the productivity of the agents in the knowledge frontier). Figure 2 plots the corresponding phase diagram. It is evident the process of convergence to the steady state point from any admissible initial state.

Fig. 2

Phase diagram of the productivity map (uniform distribution, benchmark parameterization)

The inspection of the dynamic behavior attached to difference Eq. (17) outside the benchmark example, and under the uniform distribution, allows for the identification of regions, in the parameter space, where periodic and aperiodic endogenous cycles prevail. However, these values would imply unreasonable growth rates of productivity. For instance, if q is high (e.g., $q=0.8$), and B is also high (larger than 12 in the example), limit cycles and chaos set in, through a period-doubling bifurcation (when maintaining fixed the selected values for the remaining parameters).

Specifically, the presence of chaos for some realizations of B can be confirmed by computing the Lyapunov Characteristic Exponent (LCE) associated with the dynamic equation. A positive value for the LCE becomes a common outcome for $B>48$, meaning that above this value of the parameter of the production function, chaos becomes a frequent result. If, e.g., $B=75$, then the corresponding Lyapunov exponent is $LCE=0.367$, what confirms the exponential divergence of nearby orbits and the presence of topological chaos. The setback is that this is not a meaningful economic outcome, since it would correspond to a growth rate of the technology frontier (and of the productivity of any innovator) of 707.5 per cent per period. Nonlinear dynamics pop up under more reasonable circumstances for some of the other assumed distribution functions.

Consider now the Weibull/exponential distribution and confine the analysis to the baseline example. This distribution has two associated parameters, and the values of the parameters will determine the qualitative nature of the dynamics. Table 1 displays the computed LCE for possible combinations of values of the inverse scale parameter and of the shape parameter. The highlighted cells indicate the presence of positive LCEs and, thus, the emergence of chaos for given pairs ($\lambda ,k$). The observation of the numbers in the table allows for the recognition of the presence of chaos for high positive values of k.

Table 1 Lyapunov characteristic exponents (Weibull distribution)

Given the selected values of k and $\lambda$, chaos is identified for $k=95$ and $k=125$, under certain levels of $\lambda$. The cells with no value represent instability outcomes. The cells with negative LCE values may correspond either to a fixed-point or to cycles of a given periodicity. In any case, the relevant result is that chaos is observable for the benchmark set of parameters for specific configurations of the CDF. One should also remark that, for the selected parameter values, chaos is not found for the exponential distribution (no positive LCE was calculated for $k=1$).

To get further insights on productivity dynamics under the Weibull distribution, Fig. 3 displays the system’s bifurcation diagram, taking k as the bifurcation parameter (the inverse scale parameter is set at $\lambda =2.5$). The period-doubling nature of the dynamic process is clearly visible in this case; the first bifurcation occurs around $k=57$, and then the subsequent bifurcations form a path to chaos. Chaos is circumscribed to two intervals of the possible values assumed by k. These two intervals enclose the values of the parameter that were identified in Table 1 as respecting to chaotic outcomes. One also observes that, between the two regions, there is an interval of values of k for which a period-3 cycle is formed, followed by a smaller region characterized by a period-6 cycle. Note, as well, that, for the specified parameterization, bounded instability is confined to a small set of possible realizations of the productivity share (the observation of the graph places the fluctuations of $x_{i,t}$, once the transient phase is overcome, between approximately 0.37 and 0.42).⁵

Fig. 3

Bifurcation diagram (Weibull distribution; $\lambda =2.5$, $0<k<140$)

Figure 4 represents the phase diagram of the one-dimensional map (17), when the CDF is derived from the Weibull distribution. Taking a point in Table 1 for which chaos is identified ($\lambda =2.5,k=95$), one perceives, through the observation of the graphic, how irregular cycles are formed; the nonlinear shape of the map generates a never-ending irregular movement around the steady state equilibrium. Finally, for the same set of parameter values, Fig. 5 unveils the shape of the distribution (the respective probability distribution function). As the figure suggests, chaotic productivity motion requires a productivity distribution where almost all the agents are concentrated at a specific narrow region in the distribution (in this case, around 0.4); the drawn configuration is the direct outcome of taking a very large value for k.

Fig. 4

Phase diagram (Weibull distribution; $\lambda =2.5$, $k=95$)

Fig. 5

Probability distribution function (Weibull distribution, $\lambda =2.5$, $k=95$)

The third scenario to assess is the setting that considers the Kumaraswamy distribution, whose CDF is displayed in Eq. (19). As in the Weibull case, this distribution involves the consideration of two positive parameters. Sticking once again with the baseline example, bounded instability is found, in this case, for relatively high levels of a and b. Table 2 presents the LCEs for selected values of the parameters.

Table 2 Lyapunov characteristic exponents (Kumaraswamy distribution)

The visualization of Table 2 suggests that the relevant bifurcation parameter is a. Changes in the value of b apparently have no decisive role in shaping dynamics, except in what concerns the boundary between chaos and instability (for the value of a that admits chaotic motion, the value of b must be above a given threshold, otherwise the long-term dynamic outcome is one of instability). The nature of the result is further stressed by drawing a bifurcation diagram with a as the bifurcation parameter (Fig. 6).

Fig. 6

Bifurcation diagram (Kumaraswamy distribution; $0<a<125$, $b=100$)

Figure 6 uncovers, as for the Weibull distribution scenario, a period-doubling route to chaos, with chaotic motion concentrating in values of a around 90 to one hundred. The set of values of $x_{i,t}$ for which chaotic motion emerges is a relatively small range, near the upper limit $x_{i,t}=1$. Again, the high values taken by the distribution parameters required to obtain nonlinear dynamic outcomes suggest a distribution shaped similarly to the one represented for the Weibull case (Fig. 5); in this case, the large concentration of agents will occur close to the technology frontier.

The next CDF to account for is the one respecting to the log-logistic distribution. This also encloses two parameters, and nonlinear dynamics can be explored for the range of values these parameters can assume. Again, take the baseline example and search for endogenous fluctuations through the computation of the LCE, for a sample of selected values of a and b (Table 3).

Table 3 Lyapunov characteristic exponents (log-logistic distribution)

The inspection of the LCEs in Table 3 indicates that the dynamics of the map is not significantly affected when changing the value of parameter a. In fact, changes in a are only relevant, for the figures in the table, whenever the value of b is equal or higher than 80. In this region of the parameter space, a relatively low a might lead to the generation of chaotic motion, while relatively high values of this parameter push the dynamics to the instability zone.

The above observations might be corroborated through the presentation of a bifurcation diagram (Fig. 7). Let b be the bifurcation parameter and take $a=10$. The diagram confirms the evidence in Table 3; again, a period-doubling process allows for the generation of chaos for a given interval of values of b. As in the case of the Kumaraswamy distribution, possible realizations of the productivity ratio are near 1, what indicates, as well, that most of the productivity levels in the distribution will locate close to the technology frontier.

Fig. 7

Bifurcation diagram (log-logistic distribution; $a=10$, $0<b<110$)

The arcsine distribution shares with the uniform distribution the evidence that no parameter is associated to the corresponding CDF. This is not the only feature that the two distributions have in common. As for the uniform, the arcsine distribution leads to a fixed-point outcome for the baseline array of parameters. In this case, $x_{i}^{*}=0.485$, a value slightly above the one calculated in the uniform distribution scenario. Also as in the mentioned alternative case, nonlinear dynamics appear associated with large values of q and large values of B (larger than those that could be considered reasonable for per period changes in productivity). To illustrate the outcome, Fig. 8 displays a bifurcation diagram for $B=100$ and for $q\in (p,1)$. The period doubling route to chaos is again identified.

Fig. 8

Bifurcation diagram (arcsine distribution; $B=100$, $0.1<q<0.7$)

Finally, one approaches dynamics under the trapezoidal/triangular distribution. Given admissible values of b and c, and the array of the baseline values of the other parameters in the model, the only feasible outcome is a fixed-point (the value of $x_{i}^{*}$ will depend on b and c, but it will remain bounded in the interval between zero and 1, meaning that the steady state productivity ratio is constant or, identically, that the productivity level of agent i will forever grow at the same rate as the productivity level at the frontier). Once again, complex dynamics eventually arise only for high q and high B. Table 4 displays, for the baseline example and taking different values for parameters b and c, the fixed-point level at which $x_{i,t}$ rests in the long run (the values in the diagonal apply to the triangular distribution).

Table 4 Steady state productivity ratio (trapezoidal / triangular distribution)

The inspection of the productivity dynamics equation under six different forms of the productivity distribution unveiled a rich set of eventual dynamic results, including the possibility of endogenous cycles and bounded instability. These emerged as the direct outcome of the nonlinearities associated with the difference equation once specific distributions are taken into account. One as selected a benchmark set of parameter values to avoid searching for chaos and other nonlinear results in regions of the parameter space with insignificant economic meaning. Although all the distributions allow for the emergence of nonlinearities and chaos, that only occurs, given the set of chosen parameter values, for the Weibull / exponential, the Kumaraswamy, and the log-logistic distributions. For these, one has identified that nonlinear outcomes, and chaos in particular, require setting parameter values of the distribution such that a large portion of the agents concentrate in a relatively small region of the productivity distribution, i.e., a large percentage of agents has very close levels of productivity to one another at the starting date.

The next section will deepen the analysis by assuming complementary technologies and two-dimensional systems, for which one can show strange attractors that are formed in regions of chaotic motion.⁶

7 Interdependent technologies and strange attractors

To briefly explore patterns of nonlinear dynamics and chaos within a more general setting than the one characterized up until now, assume a modified framework, in which two independent sectors make use of two distinct technologies. Agent i has a presence in both sectors, with the corresponding levels of productivity in each of the sectors being denoted by $A_{i,t}\ge 0$ and $\widetilde{A}_{i,t}\ge 0$; productivities of the individual agent may differ across sectors (e.g., if agent i is relatively more productive in the first sector, then $A_{i,t}>$ $\widetilde{A}_{i,t}$). As before, productivity levels can be expressed in the form of ratios to the corresponding frontiers. To variable $x_{i,t}$, as earlier defined, now we add ratio $y_{i,t}\equiv \frac{\widetilde{A}_{i,t}}{\widetilde{Z}_{t}}$, with $\widetilde{Z}_{t}$ the technology frontier for the second type of technology. The agent now solves, independently, two problems, searching for the optimal level of investment in innovation and/or imitation for each technology, in order to attain the trajectory of productivity growth that allows to maximize intertemporal utility. Overall utility, for this agent, is expressed as the sum of the utilities accomplished individually from the presence in each sector. To simplify the analysis, consider identical production functions in the two sectors (in both sectors, output is the product between parameter B and the level of productivity).

If no link exists between sectors, there is no difference between the analysis one can undertake in this scenario and the one pursued thus far. The distinctive feature one now introduces corresponds to a mechanism that promotes the interplay between the two productivity variables. Specifically, we make the probability of successful innovation in one sector to depend on the possibility of interacting with agents holding higher productivity levels in the other sector. The intuition is that copying what others do in a concurrent or complementary sector may offer to the agent ideas to innovate. This will be analytically translated into a probability of successful innovation that is no longer the constant value p; it will be a term $p+\widetilde{p}\left[ 1-F(y_{i,t})\right]$ for the innovation of agents in sector A and a term $p+\widetilde{p}\left[ 1-F(x_{i,t})\right]$ for the innovation of agents in sector $\widetilde{A}$. Positive constant $\widetilde{p}$ is the additional innovation probability that comes from the contact with agents in the alternative sector (and that falls as the possibility of advantageous contact diminishes). Probability $\widetilde{p}$ is bounded by condition $\widetilde{p}<q-p$, as the arguments below justify.

As stated, agents solve two problems, which become interdependent through the probability of innovation. This change in the specification of the innovation probability has impact on the investment in innovation and on the growth rate of the technology frontier. Proceeding as in the original model, the growth rate of productivity for the agent locating at the frontier will be, in each sector:

$$\begin{aligned}&\gamma _{Z}=\beta \left\{ p+\widetilde{p} \left[ 1-F(y_{i,t})\right] \right\} B-(1-\beta ) \end{aligned}$$

(23)

$$\begin{aligned}&\gamma _{\widetilde{Z}}=\beta \left\{ p+\widetilde{p} \left[ 1-F(x_{i,t}) \right] \right\} B-(1-\beta ) \end{aligned}$$

(24)

If one solves the problem for the threshold point of indifference between innovation and imitation, $A_{i,t}=\zeta _{t}$ and $\widetilde{A}_{i,t}= \widetilde{\zeta }_{t}$, the optimality conditions are such that:

$$\begin{aligned} p=q\left[ 1-F(x_{\zeta ,t})\right] -\widetilde{p} \left[ 1-F(y_{\zeta ,t}) \right] =q\left[ 1-F(y_{\zeta ,t})\right] -\widetilde{p}\left[ 1-F(x_{\zeta ,t})\right] \end{aligned}$$

(25)

From equalities (25), one directly obtains the share of agents in each category—imitation and innovation—in each sector, which are identical across sectors and amounts to: $F(x_{\zeta ,t})=F(y_{\zeta ,t})=1- \frac{p}{q-\widetilde{p}}$. Given this result, it follows that for innovators and imitators to both populate the economy, the already presented condition on the probabilities of success on innovation and imitation, $\widetilde{p}<q-p$, must hold.

A two-dimensional map can now be taken into account in order to scrutinize the dynamic behavior of the productivity ratios. Instead of Eq. (17), one should consider, in the current setting, the following two-equation system with endogenous variables $x_{i,t}$ and $y_{i,t}$,

$$\begin{aligned} \left\{ \begin{array}{c} x_{i,t+1}=\left\{ 1+\left[ 1-F(x_{i,t})\right] \frac{\beta qB \left[ 1-F(x_{i,t})\right] -(1-\beta )}{\left[ 1-F(x_{i,t})\right] +\beta \frac{ \partial F(x_{i,t})}{\partial A_{i,t}}A_{i,t}}\right\} \frac{x_{i,t}}{\beta (1+\left\{ p+\widetilde{p} \left[ 1-F(y_{i,t})\right] \right\} B)} \\ y_{i,t+1}=\left\{ 1+\left[ 1-F(y_{i,t})\right] \frac{\beta qB \left[ 1-F(y_{i,t})\right] -(1-\beta )}{\left[ 1-F(y_{i,t})\right] +\beta \frac{\partial F(y_{i,t})}{\partial \widetilde{A}_{i,t}} \widetilde{A}_{i,t}}\right\} \frac{y_{i,t}}{\beta \left( 1+\left\{ p+\widetilde{p}\left[ 1-F(x_{i,t}) \right] \right\} B\right) } \end{array}\right. \end{aligned}$$

(26)

The inspection of the dynamics underlying system (26) demands assuming specific productivity distributions. The analysis, of an eminently graphical nature, will proceed for the three distributions for which chaos was identified under the set of benchmark parameter values, i.e., the Weibull/exponential, the Kumaraswamy, and the log-logistic. This analysis requires taking $x_{i,0}\ne y_{i,0}$, with both initial values positive but lower than 1.

Figures 9, 10, and 11 present bifurcation diagrams for the three distributions, under parameter values for which chaos was identified in the one-dimensional case, now taking $0<\widetilde{p}<0.15$.⁷ The diagrams are drawn for variable $x_{i,t}$, but similar graphs would emerge if the other productivity variable, $y_{i,t}$, had been taken instead. It is visible the formation of regions of chaotic motion, as well as regions of cycles of distinct periodicities.

Fig. 9

Bifurcation diagram (two-sector model, Weibull distribution; $\lambda =2.5$, $k=95$, $0<\tilde{p}<0.15$)

Fig. 10

Bifurcation diagram (two-sector model, Kumaraswamy distribution; $a=100$, $b=100$, $0<\tilde{p}<0.15$)

Fig. 11

Bifurcation diagram (two-sector model, log-logistic distribution; $a=10$, $b=100$, $0<\tilde{p}<0.15$)

To complete the visual analysis, three other pictures are drawn. These are strange attractors found for values of $\widetilde{p}$ such that chaotic motion holds, in the case of each distribution. One confirms that technology adoption generates, in fact, complex outcomes, that become more evident as one introduces into the analysis two interdependent technologies. The strange attractors are displayed in Figs. 12, 13, and 14.

Fig. 12

Strange attractor (two-sector model, Weibull distribution; $\lambda =2.5$, $k=95$, $\tilde{p}=0.095$)

Fig. 13

Strange attractor (two-sector model, Kumaraswamy distribution; $a=100$, $b=100$, $\tilde{p}=0.047$)

Fig. 14

Strange attractor (two-sector model, log-logistic distribution; $a=10$, $b=100$, $\tilde{p}=0.067$)

8 Conclusion

Inspired by the existing literature on the topic, this study has formulated a simple optimal control problem that every agent in the economy must solve in order to maximize intertemporal utility. The single source of heterogeneity in the proposed environment is the productivity level that each agent holds at the initial date in which the planning problem is approached. The choice of the agents consists in selecting an investment strategy. They can invest in innovation or, alternatively, the investment can be made in technology adoption.

The above-mentioned choice is endogenous, and it is determined by the specific conditions underlying the model. Namely, the probability of successful innovation is assumed exogenous and constant, while the productivity of success in imitation is determined by a process of social interaction, where upgrades in productivity are the direct consequence of meeting and interacting with someone in the productivity distribution holding a higher productivity level (or a better technological capacity, what is the same in the context of the model).

Approaching the simple optimization model for different initial levels of productivity yields a series of interesting results. First, one derives a constant productivity growth rate for the individuals locating at the technology frontier (i.e., the agents holding the best technology available in the economy, and therefore the agents who necessarily need to innovate if they wish to expand their productivity). Second, there is a threshold point in the distribution of productivity, which is endogenously determined, and that separates imitation from innovation. Imitators fall to the left of the threshold point, while innovators will be placed to the right. This is a point of indifference between investment strategies: investing in innovation or imitation yields, in this particular state, a same intertemporal utility level.

Third, all innovators, i.e., all agents with productivity levels between the indifference point and the knowledge frontier, will see their productivity grow at the same rate, i.e., the rate of growth at the frontier. Fourth, there is no straightforward solution for the problem of the imitators. Because productivity growth of technology adopters depends on the productivity levels of agents locating to their right in the distribution, results will be contingent on the shape of the distribution. Thus, there is no guarantee of a stability outcome.

To address the dynamics of productivity for technology adopters, six specific distributions were taken: uniform, Weibull/exponential, Kumaraswamy, log-logistic, arcsine, and trapezoidal/triangular. The adoption of these distributions uncovered, in every case, a nonlinear equation of motion for the productivity ratio. The numerical study of this one-dimensional map has led, invariably, to a variety of stability results for different parameter values shaping the distribution: fixed-point stability holds in some circumstances, but periodic cycles and chaotic motion are not uncommon results, even for combinations of parameter values that are reasonable and empirically defendable.

The emergence of chaos is particularly relevant in this context and from the perspective of economic intuition. It signifies that adapting technologies already in use by other agents in the economy may conduct to an unpredictable and boundedly unstable outcome that is strongly determined by the profile of agents holding better technologies and how they spread over a conceivable productivity distribution. To have an exact or approximated notion of the implications of the investment in imitation, one needs to know the true shape of the distribution of productivity and how this evolves over time. As suggested in the penultimate section of the paper, the issue becomes more complex if one considers additional dimensions, associated, e.g., with concurrent and intertwined technologies.

Appendix - Derivation of Eqs. (5) and (14)

In this appendix, a detailed derivation of dynamic equations (5) and (14) is undertaken, given the optimality conditions associated to each of the optimal control problems subject to analysis (i.e., the planning problem of innovators and the planning problem of imitators).

To obtain the expression of difference Eq. (5), begin by multiplying both sides of Eq. (4) by the variable representing the knowledge frontier,

$$\begin{aligned} \left( 1+pg_{z,t}\right) \beta v_{z,t+1}Z_{t}=v_{z,t}Z_{t}-1 \end{aligned}$$

(27)

Next, recall that, by definition (i.e., because agents at the frontier are necessarily innovators), the following equality holds: $Z_{t}=\left( 1+pg_{z,t-1}\right) Z_{t-1}$. Hence, Eq. (27) is equivalent to:

$$\begin{aligned} 1+\left( 1+pg_{z,t}\right) \beta v_{z,t+1}Z_{t} =\left( 1+pg_{z,t-1}\right) v_{z,t}Z_{t-1} \end{aligned}$$

(28)

In the following step, terms $v_{z,t+1}Z_{t}$ and $v_{z,t}Z_{t-1}$ can be suppressed from the equality, by taking into consideration condition (3),

$$\begin{aligned} 1+\frac{1+pg_{z,t}}{p\left( B-g_{z,t}\right) } =\frac{1+pg_{z,t-1}}{\beta p\left( B-g_{z,t-1}\right) } \end{aligned}$$

(29)

Rearranging expression (29), an equivalent equation is obtained,

$$\begin{aligned} \frac{1+pB}{B-g_{z,t}}=\frac{1+pg_{z,t-1}}{\beta \left( B-g_{z,t-1}\right) } \end{aligned}$$

(30)

Rearranging further,

$$\begin{aligned} \beta \frac{1+pB}{1+pg_{z,t-1}}\left( B-g_{z,t-1}\right) =B-g_{z,t} \end{aligned}$$

(31)

Advancing one period in time, (31) becomes the same expression as the one displayed in the main text, (5).

The steps required to obtain (14) are similar to those employed in the previous case. First, multiply both members of optimality condition (13) by $A_{i,t}$, to obtain:

$$\begin{aligned} \left( 1+q\left\{ \left[ 1-F(x_{i,t})\right] -\frac{\partial F(x_{i,t})}{\partial A_{i,t}} A_{i,t}\right\} s_{i,t}\right) \beta v_{i,t+1} A_{i,t}=v_{i,t}A_{i,t}-1 \end{aligned}$$

(32)

Second, recall that, for imitators, the following rule for the motion of productivity holds (this is the version of difference Eq. (1) that is appropriate in this case): $A_{i,t}=\left\{ 1+q\left[ 1-F(x_{i,t-1}) \right] s_{i,t-1}\right\} A_{i,t-1}$. Therefore, (32) is equivalent to:

$$\begin{aligned}&1+\left( 1+q\left\{ \left[ 1-F(x_{i,t})\right] -\frac{\partial F(x_{i,t})}{ \partial A_{i,t}}A_{i,t}\right\} s_{i,t}\right) \beta v_{i,t+1}A_{i,t}\nonumber \\&\quad =\left\{ 1+q \left[ 1-F(x_{i,t-1})\right] s_{i,t-1}\right\} v_{i,t}A_{i,t-1} \end{aligned}$$

(33)

The product between the productivity variable and the respective shadow-price is then overwritten from the expression, taking into consideration condition (12),

$$\begin{aligned} 1+\frac{1+q\left\{ \left[ 1-F(x_{i,t})\right] -\frac{\partial F(x_{i,t})}{\partial A_{i,t}}A_{i,t}\right\} s_{i,t}}{q\left( B-s_{i,t}\right) \left[ 1-F(x_{i,t})\right] } =\frac{1+q\left[ 1-F(x_{i,t-1})\right] s_{i,t-1}}{\beta q \left( B-s_{i,t-1}\right) \left[ 1-F(x_{i,t-1})\right] } \end{aligned}$$

(34)

Equality (34) is equivalent to the following expression,

$$\begin{aligned} \beta \frac{1+qB\left[ 1-F(x_{i,t})\right] -q\frac{\partial F(x_{i,t})}{\partial A_{i,t}}A_{i,t}s_{i,t}}{\left[ 1-F(x_{i,t})\right] \left\{ 1+q \left[ 1-F(x_{i,t-1})\right] s_{i,t-1}\right\} } \left( B-s_{i,t-1}\right) \left[ 1-F(x_{i,t-1})\right] =B-s_{i,t} \end{aligned}$$

(35)

Equation (35) is the same as Eq. (14) in two consecutive time periods.