Endogenous growth in production networks
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Abstract
We investigate the interplay between technological change and macro economic dynamics in an agentbased model of the formation of production networks. On the one hand, production networks form the structure that determines economic dynamics in the short run. On the other hand, their evolution reflects the longterm impacts of competition and innovation on the economy. We account for process innovation via increasing variety in the input mix and hence increasing connectivity in the network. In turn, product innovation induces a direct growth of the firm’s productivity and the potential destruction of links. The interplay between both processes generates complex technological dynamics in which phases of process and product innovation successively dominate. The model reproduces a wealth of stylized facts about industrial dynamics and technological progress, in particular the persistence of heterogeneity among firms and Wright’s law for the growth of productivity within a technological paradigm. We illustrate the potential of the model for the analysis of industrial policy via a preliminary set of policy experiments in which we investigate the impact on innovators’ success of feedin tariffs and of priority market access.
Keywords
Agentbased modeling production networks Endogenous technological changeJEL Classification
C63 D57 D85 L16 L52 O31 O331 Introduction
In contrast with the extremely detailed description of markets and financial interactions that have been developed in the recent literature (see e.g Dawid et al. 2014; Dosi et al. 2015, and references below), the representation of the innovation process has remained relatively stylized in agentbased macroeconomic models. It is usually assumed that technological progress materializes at the micro level through (exponential) growth in the productivity of capital goods over time. Hence the models abstract away from the microeconomics of technological change and fail to inscribe technological processes in the economic state space.
This strongly contrasts with the detailed analysis of the innovation process that has been developed in the evolutionary literature (see Dosi and Nelson 2010, for a recent survey). More importantly, this leaves an important gap open in terms of policy analysis. Indeed, without a detailed representation of innovation and technological processes, agentbased models can hardly be used to analyze policies that involve large impacts on technologies, first and foremost climate change mitigation, which possibly is the most important longterm challenge faced by contemporary economies. Indeed, the direction of technological change and the specific nature of inputs entering the production process are key elements for an assessment of technological change from the point of view of climate policy.
In order to fill part of this gap, we introduce in this paper an agentbased model where technological evolution is modeled in detail through the evolution of production networks. These networks provide a detailed description of the technological and commercial relationships between firms and can easily be mapped to an inputoutput table. On the one hand, they form the structure that determines economic dynamics in the short run. On the other hand, their evolution reflects the longterm impacts of competition and innovation on the economy.
Accordingly, in our model, once the network is given, the dynamics of prices and output follow from the application of simple behavioral rules. Conversely, the evolution of the network reflects the longterm dynamics of the economy driven by competition and innovation processes. Competition materializes via redirections (rewiring) of relationships between firms and hence induces a “horizontal” evolution of the network. Innovation and technological change, which form the core of our model, materialize both via radical (product) innovation and incremental (process) innovation. Radical innovation occurs through the discovery by firms of new technological paradigms that lead to increasingly efficient products. Process innovation materializes, within a technological paradigm, through diversification of the input mix. The interplay between these processes drives the evolution of the network: process innovation through diversification leads to increasing connectivity among firms while radical innovations might render obsolete a very mature technology and hence induce a decrease in connectivity. The inputoutput structure of the model evolves accordingly. We hence approach what is according to Dosi and Nelson (2010) “a quite challenging modeling frontier [that] regards the explicit representation of evolving problemsolving procedures, constrained by paradigmshaped ‘grammars’ and their ensuing dynamics in the more familiar space of input/output coefficients.”
The model is able to reproduce key stylized facts of industrial dynamics with respect to the distribution of firms’ growth rates and size, the persistence of heterogeneity in productivity among firms and the structure of production networks. Also, our representation of process/incremental innovation is consistent with Wright’s law (Wright 1936) and, when combined with radical innovation and imitation á la Nelson and Winter (1982), leads to the emergence of endogenous growth paths and of technologydriven business cycles. These cycles are characterized by the transition between phases of radical/product and incremental/process innovation akin to the one described in the UtterbackAbernathy model (see Utterback 1994). In summary, the model is able to reproduce a rich set of stylized facts and provide bridges between innovation and endogenous growth theories.
Hence, we provide an agentbased framework fit for the analysis of large technological changes in the economy and potentially of innovation policies. In this latter respect, we perform a series of policy experiments in which we investigate the impacts of pricebased measures, akin to feedin tariffs, and quantitybased measures, i.e preferential access to the market, on the survival rate of radical innovators and the growth rate of the economy. Our results emphasize that the impacts of such policy measures heavily depend on the structure of externalities in the innovation process.
The remaining of the paper is organized as follows. In Section 2, we review the related literature. Section 3 gives a detailed description of the model. Section 4 investigates the impact of different innovation processes on the structure of the production network and on macroeconomic dynamics. Section 5, highlights the behavior of the model in a series of policy experiments and Section 6 concludes.
2 Related literature
In most existing agentbased macroeconomic models, the representation of the production process is rather stylized and involves only labor and capital, possibly of heterogeneous kinds. Intermediary consumption or the details of the “recipes” used in production are usually not taken into account. Accordingly, technological progress is embedded in physical capital, the vintages of which grow in productivity over time. This approach is rather generic and followed in particular in Dosi et al. (2010, 2013, 2015), Dawid et al. (2011, 2014) or Ciarli et al. (2010). Mandel et al. (2010) and Wolf et al. (2013) use a different representation of the production process that accounts for intermediary consumption, but productivity growth is driven by cumulative investment and hence totally decoupled from the specifics of the production process.
In contrast, in our setting, growth in productivity is essentially linked to changes in the production process and correlatively in the production network. Hence our approach is closely related to the evolutionary and complex systems literature that have focused on the dynamics of technology. This literature is extensively surveyed in Frenken (2006b) and Dosi and Nelson (2010). In particular Frenken (2006b) identifies three main approaches in the literature: fitness landscape models (e.g Kauffman et al. 2000), percolation models (e.g Silverbeg and Verspagen 2005), and production recipes models (e.g Auerswald et al. 2000). The most relevant contributions from our perspective are those that model the evolution of “production recipes” such as Auerswald et al. (2000), Frenken (2006a) and more recently (McNerney et al. 2011). These contributions are strongly rooted in an engineering/design perspective and, following Kauffman (1993), emphasize the interdependencies among the elements of designs as the key parameter determining the dynamics of technologies. In particular, McNerney et al. (2011) emphasizes that the interplay between complexity of the design, measured via the degree of interdependency, and the increasing difficulty in improving components leads to the emergence of Wright’s law for the rate of technological progress (see Wright 1936; Arrow 1962, in this latter respect). Our approach is slightly more aggregate and distant from engineering considerations as we map the technological process directly in the inputoutput space. This allows us to provide the macroeconomic closure that is missing in the “production recipes” contributions. With respect to Wright’s law, we obtain results similar to those of McNerney et al. (2011).
We go beyond process innovation through the introduction of radical innovation, which leads to the discovery of new technological paradigms and growth in products’ productivity. In our microfounded setting, combining product and process innovation is required to induce exponential growth. This provides an interesting contrast with standard approaches in the endogenous growth literature. Except in degenerate cases (see d’ Autume and Michel 1993), growth models á la Arrow (1962) based on Wright’s law are not conductive to exponential growth. Hence endogenous growth models à la Romer (1990) requires the variety of inputs in the production process to grow exponentially in order to generate sustained growth. This assumption might appear as innocuous when the production process is represented at the aggregate level but leads to major inconsistencies with empirical regularities if implemented in our microeconomic setting. Hence, the emergence of exponential growth requires more radical forms of product innovation of the kind considered in “Schumpeterian” growth models à la Aghion and Howitt (1992). Whereas, this “Schumpeterian” literature considers that the growth process is driven by a succession of monopolies, in our setting different technological “paradigms” at different levels of maturity and diversification coexist. This allows us to preserve competition and heterogeneity in productivity among firms, consistent with empirical observations.
The hybrid nature of our model echoes the considerations about variety put forward in Saviotti et al (1996), in particular Saviotti’s second hypothesis according to which “Variety growth, leading to new sectors, and productivity growth in preexisting sectors are complementary and not independent aspects of economic development”. More broadly, we could argue that we operationalize, via a networkbased approach, Saviotti and Pyka (2008)’s concept of variety, which is broader than that of product variety since it refers to the extent of diversification in the economic system.
Finally, our network perspective on the productive system relates to an expanding stream of literature using both agentbased (Bak et al. 1987; Weisbuch and Battiston 2007; Battiston et al. 2007) and general equilibrium methods (Acemoglu et al. 2012; Carvalho 2014). This literature hasn’t yet approached the issue of growth and technological change but for the notable exception of Carvalho and Voigtländer (2014). These authors put forward a new stylized fact at both the sector and the firm level: producers are more likely to adopt inputs that are already used—directly or indirectly—by their current suppliers. They provide theoretical foundations for this process using the network formation model of Jackson and Rogers (2007), which they adapt by considering that new products/firms entering the economy draw a first part of their inputs at random and a second part from the connections of those drawn in the first phase. Hence their approach is much more precise than ours with respect to the direction of technological change. Yet, our approach is complementary to theirs as it allows the macroeconomic closure of the model and account for the interplay between product and process innovations.
3 The model
3.1 Technological structure
We represent the dynamics of a network consisting of (at most) m firms distributed over S industrial sectors and one aggregate household. We denote the set of firms by M = {1,⋯ ,m}, the household by the index 0 and the set of agents by N = {0,⋯ ,m}. Time is discrete, indexed by \(t \in {\mathbb {N}}\). The network of supply relationships is represented by an adjacency matrix A^{t} such that \(a_{i,j}^{t}= 1\) if and only if j is a supplier of i (and \(a_{i,j}^{t}= 0\) otherwise). The network evolves over time under the influence of competition and innovation.
Technological progress and the evolution of the network will then be closely intertwined. In particular, we shall assume throughout the paper that inputs are substitutable (i.e 𝜃 ∈ [0,1]), and hence productivity will grow with the number of inputs/suppliers combined, that is with the density of the network. As a matter of fact, our results rely on the presence of increasing returns to variety rather than in the choice of a specific functional form for the production function or even the existence, at all, of a production function.
3.2 Macroeconomic closure
 1.
Agents receive a nominal demand proportional to the wealths and the input shares of their connections.
 2.
Agents adjust their prices toward their market clearing value (at a rate τ_{p} ∈ [0,1]).
 3.
Agents then produce according to the inputs they receive.
 4.
Agents adjust their input shares (at a rate τ_{w} ∈ [0,1]) toward their costminimizing value.
A detailed representation of these dynamics is given in the Appendix A.2. One of their salient property is that, if the network is fixed, the economy almost generically converges to the underlying general equilibrium (see Gualdi and Mandel 2015).
3.3 Network Dynamics
Now, our key focus in this paper is the joint evolution of the production network and of the economy, that is, the evolution of the adjacency structure \((A^{t})^{t \in {\mathbb {N}}}\) over time and its impact on macroeconomic dynamics. We shall consider two main drivers for this evolution: competition and innovation.
This competitive process leads to the evolution of the indegree distribution of the network. In fact, as shown in Gualdi and Mandel (2015), competition leads to the emergence of a scalefree indegree distribution because of two basic facts about the “economy” of suppliers’ switches: the number of incoming business opportunities for a firm is independent of its size (i.e. firms gain link at a constant rate) while the rate at which existing consumers may quit grows linearly with the size of the firm (i.e firms lose links proportionally to their degree). The balance between the flow of incoming and outgoing links lead to the emergence of a scalefree size distribution of incoming links.
Remark 1
The rewiring process described by Eq. 2 also implicitly defines the notion of sector in our setting. A sector is a group of firms the outputs of which are substitutable (from the point of view of the clients). Hence, despite our use of a C.E.S functional form, the substitutability between inputs is actually limited by the sectoral structure of the economy. Furthermore, note that the outputs from the different firms of a sector, though substitutable, are not equivalent, as their productivity/quality might differ.
Innovation materializes through two processes that account respectively for product and process innovation (and, in an extended sense, for radical and incremental innovation). More precisely, two elements characterize a technology in our setting, the productivity \(e \in {\mathbb {R}}_{+}\) of the product produced and the input mix \(\upsilon \in {\mathbb {N}}^{S}\), i.e the number of inputs from the different sectors used in the production process. In turn, a technological paradigm consists in a pair \((e, \Upsilon ) \in {\mathbb {R}}_{+} \times 2^{ {\mathbb {N}}^{S}}\) where e represents the productivity of the product produced and Υ the set of input mixes that can be used to produce the product. In general, we shall assume that Υ is of the form \(\Upsilon := \{\upsilon \in {\mathbb {N}}^{S_{\Upsilon }} \mid S_{\Upsilon }\subset S \wedge \upsilon \leq \overline {\upsilon }\}\) where \(\overline {\upsilon }\) represents the most complex (and hence productive) input mix within the paradigm, that is, the production process can gain in efficiency through diversification up to a maximum amount of diversification, which is technology specific.
Let us consider a firm i that is using a technology (e_{i},υ_{i}) within a paradigm (e_{i},Υ_{i}). An incremental innovation for that firm consists in the adoption of a new input mix \(\tilde {\upsilon }_{i}\) within the paradigm (e_{i},Υ_{i}) such that \(\tilde {\upsilon }_{i} \geq \upsilon _{i}\). A radical innovation consists in the adoption of a new technological paradigm \((\tilde {e}_{i}, \tilde {\Upsilon }_{i})\) such that \(\tilde {e}_{i} \geq e_{i}\) and of an input mix \(\tilde {\upsilon }\) within the new paradigm. Our approach hence builds on Dosi (1982) interpretation of the determinants and direction of technological change.

With probability μ_{inc} an incremental innovation is drawn, in which case a new supplier is drawn at random and added to the input mix (if the current number of suppliers is less than the maximum possible within the paradigm).

With probability μ_{rad} a radical innovation, in which case a new technological paradigm \((e^{t + 1}_{i}, \Upsilon ^{t + 1}_{i})\) is drawn at random. The input mix \(\upsilon ^{t + 1}_{i}\) then is reinitialized by drawing the new number of links according to a binomial distribution the mean of which equals the mean number of links in the initial network, υ_{0}. The maximal number of links for the new paradigm is itself drawn uniformly between υ_{min} and υ_{max}, which are parameters of the model.

With probability μ_{im} the firm imitates one of its peers,^{1} that is, it observes a firm i^{′} at random and adopts its technology \((e_{i^{\prime }}, \upsilon _{i^{\prime }})\) if it is more advanced than its current one in the sense that its more productive \(e_{i^{\prime }} > e_{i}\).

The network is then updated accordingly. That is if the firm has extended its input mix the corresponding number of new suppliers is drawn at random (and the corresponding entries are added to the adjacency matrix). If the firm has adopted a new technological paradigm but with a less elaborate input mix, the corresponding number of links (and the corresponding entries of the adjacency matrix) are selected uniformly at random and deleted.

As a result of this process and of competition, some firms might lose incoming connections, consumers, up to the point where they no longer have any connection in the network. We consider that such a firm goes bankrupt and exits the market. Yet, to sustain competition in the economy, we assume that those exits are compensated by entries of new firms. New firms enter the market with a productivity equal to the average in the economy and with a number of suppliers drawn from a binomial distribution as in Gualdi and Mandel (2015).
These different mechanisms can be seen as microeconomic implementations of the macroeconomic drivers of growth considered in the endogenous growth literature. On the one hand, the incremental/process innovation, which consists in adding inputs to the production process is very similar to the product variety model of endogenous growth à la Romer (1990), as well as to the inframarginal approach to economic growth (see Yang and Borland 1991) or of Adam Smith’s original description of the effects of the division of labor. A number of empirical contributions have also documented the positive impact of increasing input variety on productivity growth: through tradebased measures of variety in Addison (2003), Feenstra et al. (1999), and Funke and Ruhwedel (2001) as well as through direct measures of the variety of inputs used in the production process in the more recent contributions of Amiti and Konings (2007) and Frensch and Wittich (2009).
On the other hand, the product innovation process, which leads to a change of technological paradigm and to a direct increase of productivity, implements a more radical form of innovation. It has strong similarities with Schumpeterian models of endogenous growth in which series of monopolists sequentially push each other out of the market by developing more productive versions of a product (see Aghion and Howitt 1998 and references therein for an extensive description of the “Schumpeterian” approach to endogenous growth as well as Aghion et al. (2013) for a recent review of empirical evidences on the Schumpeterian growth engine).
We investigate in the following the macroeconomic and distributional patterns that emerge from the interplay of these processes.
4 Innovation, growth and the evolution of production networks
Default parameter values
Parameter  Value 

m  2000 
S  5 
T  500 000 
ρ _{ c h g}  0.05 
τ _{ p}  0.8 
τ _{ w}  0.8 
ρ _{ i n n}  0.001 
𝜃  1/2 
\(\upsilon _{\min }\)  10 
\(\upsilon _{\max }\)  20 
υ _{0}  4 
Also note that, if the speed of innovation becomes too large with respect to the speed of price and quantity adjustment, the economy can’t cope with technological change and the system becomes unstable. In other words, the timescale at which technological innovation takes place must be somehow separated from the timescale of price and quantity adjustment. In our setting, this implies considering relatively low rates of innovation (of the order of 10^{− 3} or 10^{− 4}). In addition, two other mechanisms tend to slow down the diffusion of innovation: not all innovations are successful (because of the disruption it induces in the organization of the firm, innovation might initially make the firm less competitive and possibly lead to failure) and there is a lag between the success of a radical/incremental innovation within a firm and its diffusion through imitation in the economy (as in Fagiolo and Dosi 2003). Indeed each imitation is by itself an innovation for the imitating firm and such an event occurs independently for each firm (and at a relatively low innovation rate). This set of mechanisms implies that, in the simulations presented below, a relatively large number of periods are required for a statistical equilibrium to emerge.
4.1 Incremental innovation
We first focus on the dynamics of the model when the technological paradigm (i.e the maximal number of inputs) is fixed and only incremental innovation occurs, i.e the only source of productivity growth is the diversification of the input mix. In this setting, we perform a series of Monte Carlo simulations focusing on the sensitivity of the model with respect to the rate of innovation and the elasticity of substitution. More precisely, we let ρ_{inn} vary in {10^{− 2},5.10^{− 3},10^{− 3},5.10^{− 4},10^{− 4}} and 𝜃 vary in {3/5,1/2,1/4}. The technological paradigm for each firm is such that e_{i} = 1 and \(\overline {\upsilon }_{i}= 20\).
Wright law exponent for varying elasticity
Dependent variable: \(\log (y)\)  

(𝜃 = 3/5)  (𝜃 = 1/2)  (𝜃 = 1/4)  
\(\log (t)\)  0.207^{∗∗∗}  0.321^{∗∗∗}  1.002^{∗∗∗} 
(0.0001)  (0.0002)  (0.001)  
Constant  − 0.306^{∗∗∗}  − 0.143^{∗∗∗}  0.776^{∗∗∗} 
(0.001)  (0.002)  (0.006)  
Observations  99,001  99,001  99,001 
R^{2}  0.965  0.965  0.966 
Adjusted R^{2}  0.965  0.965  0.966 
Residual Std. Error (df = 98999)  0.035  0.054  0.165 
Hence, the incremental innovation process introduced in the model leads to a behavior consistent with the empirical evidence on the growth of productivity within a technological paradigm, which is summarized by Wright’s law. Empirical estimates of Wright’s law suggest an exponent close to 1/3, corresponding to a value of 𝜃 of 3/5 in our framework.
From a theoretical perspective, the incremental innovation process considered in this Section is closely related to the product variety models à la Romer (1990). However, in our setting, diversification is embedded at the core of the production process whereas in Romer’s type of model the production process is represented in a much more aggregate way and diversification only concerns the production of a final good. From Romer’s aggregate perspective, the assumption that product variety grows exponentially over time, which is required to sustain endogenous growth, does not seem overly problematic. In our microfounded setting, exponential growth of product variety would imply either exponential growth of the network’s density or of the number of firms. Both assumptions clearly are counterfactual. Hence, incremental innovation alone can not sustain endogenous growth except in the corner case where there is infinite complementary between inputs (𝜃 → 0) and the exponent of Wright’s law β = (1/𝜃 − 1)(1 − γ) tends toward infinity (as in d’Autume and Michel 1993).
4.2 Radical innovation
In a second series of experiments, we focus on the effects of radical innovation on industrial and macroeconomic dynamics. Radical innovation yields a direct increase in productivity through product innovation. In this respect, it has similarities with Schumpeterian models of endogenous growth (see Aghion and Howitt 1998, and references therein) in which a series of monopolists sequentially push each other out of the market by developing more productive versions of a product hence putting the economy on an exponential growth path.
In order to characterize the impact of radical/product innovation in our setting, we perform a series of Monte Carlo simulations in which it is the only source of innovation (i.e. we set μ_{rad} = 1 while μ_{inc} = μ_{im} = 0) and the total innovation rate ρ_{inn} varies in {10^{− 2},5.10^{− 3},10^{− 3},5.10^{− 4},10^{− 4}}.
The impact of changes in the innovation rate materialize first by changes in the growth rate of the economy (see Fig. 3). From a more structural perspective, large innovation rates increase volatility in the growth patterns of firms. This materializes in the indegree distribution of firms. While the distribution is scalefree in the presence of incremental innovation only, the presence of radical innovation affects the stability of large firms and shifts the distribution toward exponential tails for large value of the innovation rate ρ_{inn} (see rightpanel of Fig. 2). This feature can be explained by the fact that the strength of competition increases with the speed at which new products, radical innovations, enter the market. Hence, the negative feedback effects on the growth of firms are much more important and lead to the decrease of the tail of the distributions of sizes. This feature of the model might help clarify why conflicting evidences remain about the size distribution of firms (see e.g Cabral and Mata 2003; Axtell 2001). Shifts between different type of distributions might well depend on the growth pattern of the economy.
In all cases, heterogeneity between firms is an emerging property of the model that is in strong contrast with the monopolistic feature of growth models á la Aghion and Howitt (1998) and brings the model much closer to empirical stylized facts about industrial dynamics. In the following, we investigate in more detail how, through the interplay between incremental and radical innovations, the model can also account for complex dynamical patterns of output.
4.3 Mixed innovation regimes and complex dynamics
The experiments performed in the preceding subsections underline the complementary roles of incremental and radical innovations. Incremental innovation accounts for technological progress within a technological paradigm but saturates once the technological frontier is reached. Radical innovation accounts for the development of new products and paradigms and allows endogenous growth to be sustained.
In this section, we investigate the interplay between these processes and in particular the impact on macroeconomic and industrial dynamics of the competition between radical innovators and incumbents. This competition hardly materializes in presence of radical innovation only because all firms, incumbents and innovators, then have the same expected product variety υ_{0} (see Section 3.3) and hence innovators always have a competitive advantage thanks to the increased productivity of their product. When both radical and incremental innovation are present, radical innovators, which initially have a low level of diversification/complexity (υ_{0} in expectation), have to compete with incumbents that have climbed the complexity ladder with an older vintage of the product.
In order to investigate the impact of this competition, we have performed a series of experiments in which the ratio μ_{inc}/μ_{rad} between the rates of incremental and radical innovation varies in {[1,5,10,20,50]}. Other parameters are set as in Table 1 and there is no imitation (i.e. μ_{im} = 0).
The results of the simulations show first that exponential growth is a very robust property of the dynamics: provided the rate of radical innovation is positive, the model eventually settles in an exponential growth regime after a transient period (see Fig. 4). The growth rate is an increasing function of the total innovation rate and a decreasing function of the ratio between incremental and radical innovation, i.e it increases with the rate of radical innovation. Conversely, the length of the transient decreases with the total innovation rate and increases with the ratio between incremental and radical innovation. In fact, a key parameter for the dynamics seems to be the expected frequency of radical innovations, which is proportional to ρ_{inn} × μ_{rad}/μ_{inc}.
In fact, the equilibrium level of connectivity (or equivalently of product variety) is determined by the interplay between radical and product innovation. Indeed, successful incremental innovations increase the number of links while successful radical innovations decrease it. The frequency of success of radical innovations hence determine a rate of decrease of the number of links. The equilibrium level of connectivity corresponds to a level that balance creation and destruction of links. The larger the ratio between incremental and radical innovations, the higher this equilibrium level.
The dynamics of connectivity can then be explained by the presence of decreasing returns to connectivity/product variety (i.e 𝜃 > 0). During the transient period, while connectivity is lower than the equilibrium threshold, productivity gains induced by incremental innovation are large and can not be disrupted by radical innovations. While most firms’ technologies are below this threshold, the increase in product variety is almost unconstrained. In the stable regime, returns to incremental innovation are lower, the model is at an equilibrium where productivity gains (per unit of time) induced by radical innovations are competitive with respect to those induced by incremental innovations. Hence, the increase in connectivity due to incremental innovations is compensated by the decrease triggered by radical innovations.
With a large number of incremental and radical innovations, these mechanisms are only observed indirectly through the stability of the average level of connectivity. In order to characterize them more precisely, we focus on a more stylized version of the model where radical innovation is rare but gets amplified through imitation. Therefore, we run a second series of Monte Carlo simulations in which innovation by imitation is enabled. We let the ratio between incremental and radical innovation, μ_{im}/μ_{rad}, vary in {10,20,50}. We also let the ratio μ_{inc}/μ_{rad} vary in {1,10,20} and set other parameters as in Table 1.
These business cycles correspond to an amplified version of the interactions between the radical and incremental innovation processes analyzed above. The upswings of connectivity cycles correspond to the accumulation of incremental innovations as illustrated in the lower left panel of Fig. 5, which shows the very strong correlation between growth of output and connectivity. The downswings correspond to the occurrence of radical innovations, amplified by imitation, that disrupt the industry. The amplitude and the period of these cycles increase with the μ_{im}/μ_{rad} ratio, that is, as the frequency of radical innovations decreases and the role of imitation increases.
The fluctuations of output are strongly correlated with connectivity. Output grows with connectivity, i.e while technologies get more mature (see the lower left panel of Fig. 5). Accordingly, the growth rate of output increases with the rate of incremental innovation (see the lower right panel of Fig. 5). Radical innovations destroy links and disrupt the production structure. Therefore, they have a negative impact on output in the short term. However, they pave the way for future growth as they allow for a new wave of incremental innovations to occur.
The contrast between Figs. 4 and 5 highlights the crucial role of imitation in the emergence of fluctuations. Indeed, imitation amplifies the synchronization of technological evolution among firms. It leads to the emergence, from the microeconomic behavior of distinct technological phases where product/radical and process/incremental innovation successively dominate. This pattern is reminiscent of empirical observations about the development of technologies described e.g. in the UtterbackAbernathy model (see Utterback 1994).
4.4 Firms’ demographics

As illustrated in the left panel Fig. 7, growth rates of firms are distributed according to a “tentshaped” doubleexponential distribution (see Bottazzi and Secchi 2006). Moreover, the right tail of the distribution thickens with the increasing share of imitative and radical innovation.

There is a negative relation between the variance of growth rate and the size of firms. In the absence of radical innovation, there moreover is a scaling relation, of the form σ(s) = s^{−β} where s is the size of the firm and σ(s) is the variance of growth rates for firms of size s.

As illustrated in the right panel of Fig. 7, the distribution of product’s productivity (the e_{j}s) is heterogeneous ; it exhibits much fatter tails in absence of imitation. Together with the above results about the outdegree distribution of firms, it implies that the model endogenously generate heterogeneity both in terms of process and of product productivity.
5 Policy experiments
The preceding Section puts forward the crucial role of radical innovation in sustaining growth in our model. This strongly echoes the emphasis on innovation and industrial policy in contemporary economies. A prominent example in the current policy debate is the energy industry, where innovations in renewables energy production, which are crucial for climate change mitigation, are seen as potential drivers of “green” economic growth (see e.g Tàbara et al. 2013). A key policy question then is whether growth can be stimulated through measures supporting radical innovations. In the context of energy markets, the main measures put in place were feedin tariffs, which consist in subsidizing the price paid to renewable energy producers, and preferential access to the market for renewable energy producers.^{2}

In the pricesupport scenario, which is akin to feedin tariffs, the prices of radical innovators are subsidized during 500 periods after an innovation occurred. More precisely, if firm i performed a radical innovation less than 500 periods ago, the price paid by its consumers is \((1\tau _{feed}){p_{i}^{t}}\) rather than \({p_{i}^{t}}\). We assume that the difference between buying and selling prices is financed by the government through external deficit.^{3}

In the marketsupport scenario, akin to preferential market access, firms are set to rewire prioritarily to radical innovators when they update their suppliers (see Eq. 2). The length of time after their innovation for which firms are given priority access, T_{pr} is the policy variable.
For each policy scenario, we perform a series of Monte Carlo simulations where we let vary the policy parameter, respectively in {0.1,0.2,0.5} for τ_{feed} and in {100,500,1000} for T_{pr}. Other parameters are set as in Section 4.3 with μ_{im}/μ_{rad} ∈{0,1} and μ_{inc}/μ_{rad} = 1. Simulations are ran for five different seeds for each combination of parameters.
Exit rate of radical innovators(per 10^{3} periods)
Pricesupport scenario  Marketsupport scenario  0  

Parameter  Exit rate  Parameter  Exit rate 
μ_{im} = 0,τ_{feed} = 0.1  3.10^{− 2}  μ_{im} = 0,T_{pr} = 100  3.10^{− 2} 
μ_{im} = 0,τ_{feed} = 0.2  8.10^{− 3}  μ_{im} = 0,T_{pr} = 500  4.10^{− 2} 
μ_{im} = 0,τ_{feed} = 0.5  5.10^{− 4}  μ_{im} = 0,T_{pr} = 1000  5.10^{− 2} 
μ_{im} = 1,τ_{feed} = 0.1  2.10^{− 2}  μ_{im} = 1,T_{pr} = 100  2.10^{− 2} 
μ_{im} = 1,τ_{feed} = 0.2  6.10^{− 3}  μ_{im} = 1,T_{pr} = 500  3.10^{− 2} 
μ_{im} = 1,τ_{feed} = 0.5  3.10^{− 4}  μ_{im} = 1,T_{pr} = 1000  4.10^{− 2} 
It is clearly inappropriate to draw direct policy conclusions from simple experiments performed in such a stylized framework. However, we would argue that our results underline the notion that the impact of policy strongly depends on the kind of externalities in the economy under consideration. The main external effect on innovation in our framework is imitation for which only the most efficient producers matter. If we were to consider stronger complementarities between innovators, the survival of a larger share of innovators might have a much more significant impact on growth.
6 Concluding remarks
We have developed a macroeconomic agentbased model centered on the evolution of production networks. The structure of the network, i.e the structure of the market, constrains firm’s behavior in the shortrun and hence determines shortterm economic dynamics. In turn, competition among firms and technological innovations govern the evolution of the network. Longterm macroeconomic dynamics hence emerge from the microeconomic interactions among firms.
From the theoretical point of view, our main innovation is to provide a detailed microeconomic representation of the production process, accounting for intermediary consumption, within a growing economy. Technological progress is embedded in the structure of the network and we consider two avenues for growth. Process innovation, which materializes through diversification of the input mix and hence increases connectivity in the networks and product innovation, which induces a direct increase of productivity at the expense of a temporary loss of specialization in the production process and hence decreased connectivity. These two processes can, respectively, be interpreted as the decentralization of the two workhorses of endogenous growth theory, product variety model à la Romer (1990) and “Schumpeterian” growth model à la Aghion and Howitt (1992), in a microeconomic setting with boundedly rational agents.
Considering innovation occurs at the microlevel and accounting for the local nature of interactions allow us to reproduce a wealth of stylized facts that the aggregate nature of endogenous growth models discards by construction. Growth is exponential in the aggregate and follows Wright’s law within a technological paradigm. The distribution of productivity among firms is heterogeneous. The distribution of firms’ size exhibit fattails, the thickness of which depends on the aggregate rate of growth.
Additionally, imitation can lead to the synchronization of firms’ innovative behavior and hence to the emergence of growth patterns in which process/incremental and product/radical innovation successively dominate, as in the UtterbackAbernathy model (see Utterback 1994). This cyclicality of the innovation process induces technologically driven business cycles. Process innovation and increasing connectivity coincide with upswings, product innovation and decreasing connectivity with downswings.
The large number of stylized facts the model is able to reproduce and the richness of the dynamical patterns observed in simulations suggest that the model could be a useful testbed for the analysis of industrial and innovation policies, in particular in the context of the energy transition. On this view, we perform a first series of policy experiments in which we investigate the impact of feedin tariffs and of priority access to the market on the survival rate of innovators and growth. Our results underline the fact that the impact of policy crucially depends on the nature of externalities among innovators. If imitation dominates, only the most efficient firms matter and these can survive without public support.
Yet, an important avenue for future research is to account for other form of external effects in the innovation process, for the role of institutions and for the broader socioeconomic landscape in which innovation is developed (see e.g Saxenian 1996, in these respects). Another important aspect that requires further investigation is the role of the demand in the development of innovations. In this respect, it might be worth investigating the emergence of demand among heterogeneous households that might not necessarily be characterized by preferences for goods but rather by Lancasterian preferences for characteristics (Lancaster 1966).
Footnotes
 1.
Here and in the following, we always consider implicitly that the vector (μ_{inc},μ_{im},μ_{rad}) is normalized so that μ_{rad} + μ_{inc} + μ_{im} = 1
 2.
A cautionary note in this respect is that, as argued by Lamperti et al. (2015), marketbased policies may not be sufficient to prevent environmental disasters while Commandand Control policies are fully effective.
 3.
In fact, everything goes as if the economy were receiving an external subsidy.
Notes
Funding
This study was funded by the European Commission through the FP7 project IMPRESSIONS (603416) and the H2020 project Dolfins (640772), and through the Agence Nationale de la Recherche via Labex Louis Bachelier (ANR 11LABX0019) and Labex OSE (ANR10LABX9301).
Compliance with Ethical Standards
Conflict of interests
the authors declare that they have no conflict of interest.
References
 Acemoglu D, Carvalho VM, Ozdaglar A, TahbazSalehi A (2012) The network origins of aggregate fluctuations. Econometrica 80(5):1977–2016. http://ideas.repec.org/a/ecm/emetrp/v80y2012i5p19772016.html CrossRefGoogle Scholar
 Addison DM (2003) Productivity growth and product variety: gains from imitation and education world bank policy research paper 3023. Technical report, The World BankGoogle Scholar
 Aghion P, Howitt P (1992) A model of growth through creative destruction. Econometrica 60(2):323–351CrossRefGoogle Scholar
 Aghion P, Howitt P (1998) Endogenous growth theory. MIT Press, CambridgeGoogle Scholar
 Aghion P, Akcigit U, Howitt P (2013) What do we learn from schumpeterian growth theory? Technical report, National Bureau of Economic ResearchGoogle Scholar
 Amiti M, Konings J (2007) Trade liberalization, intermediate inputs, and productivity: evidence from indonesia. Am Econ Rev 97(5):1611–1638CrossRefGoogle Scholar
 Arrow KJ (1962) The economic implications of learning by doing. Rev Econ Stud 29(3):155–173CrossRefGoogle Scholar
 Auerswald P, Kauffman S, Lobo J, Shell K (2000) The production recipes approach to modeling technological innovation: an application to learning by doing. J Econ Dyn Control 24(3):389–450CrossRefGoogle Scholar
 Axtell RL (2001) US firm sizes are zipf distributed. Science 93:1818–1820CrossRefGoogle Scholar
 Bak P, Tang C, Wiesenfeld K (1987) Selforganized criticality: an explanation of the 1/f noise. Phys Rev Lett 59(4):381CrossRefGoogle Scholar
 Battiston S, Gatti DD, Gallegati M, Greenwald B, Stiglitz JE (2007) Credit chains and bankruptcy propagation in production networks. J Econ Dyn Control 31 (6):2061–2084. http://ideas.repec.org/a/eee/dyncon/v31y2007i6p20612084.html CrossRefGoogle Scholar
 Bottazzi G, Secchi A (2006) Explaining the distribution of firm growth rates. RAND J Econ 37(2):235–256CrossRefGoogle Scholar
 Cabral LMB, Mata J (2003) On the evolution of the firm size distribution: facts and theory. Am Econ Rev 98(1):1075–1090CrossRefGoogle Scholar
 Carvalho VM (2014) From micro to macro via production networks. J Econ Perspect 28(4):23–47CrossRefGoogle Scholar
 Carvalho VM, Voigtländer N (2014) Input diffusion and the evolution of production networks. Technical report, National Bureau of Economic ResearchGoogle Scholar
 Ciarli T, Lorentz A, Savona M, Valente M (2010) The effect of consumption and production structure on growth and distribution. A micro to macro model. Metroeconomica 61(1):180–218CrossRefGoogle Scholar
 Coad A (2009) The growth of firms: a survey of theories and empirical evidence. Edward Elgar Publishing, CheltenhamCrossRefGoogle Scholar
 d’Autume A, Michel P (1993) Endogenous growth in arrow’s learning by doing model. Eur Econ Rev 37(6):1175–1184CrossRefGoogle Scholar
 Dawid H, Gemkow S, Harting P, van der Hoog S, Neugart M (2011) The eurace@ unibi model: an agentbased macroeconomic model for economic policy analysis. Technical Report, Working Paper. Universität BielefeldGoogle Scholar
 Dawid H, Harting P, Neugart M (2014) Economic convergence: policy implications from a heterogeneous agent model. J Econ Dyn Control 44:54–80CrossRefGoogle Scholar
 Dosi G (1982) Technological paradigms and technological trajectories: a suggested interpretation of the determinants and directions of technical change. Res Policy 11 (3):147–162CrossRefGoogle Scholar
 Dosi G, Nelson RR (2010) Technical change and industrial dynamics as evolutionary processes. Handbook of the Economics of Innovation 1:51–127CrossRefGoogle Scholar
 Dosi G, Fagiolo G, Roventini A (2010) Schumpeter meeting keynes: a policyfriendly model of endogenous growth and business cycles. J Econ Dyn Control 34(9):1748–1767. http://ideas.repec.org/a/eee/dyncon/v34y2010i9p17481767.html CrossRefGoogle Scholar
 Dosi G, Fagiolo G, Napoletano M, Roventini A (2013) Income distribution, credit and fiscal policies in an agentbased Keynesian model. J Econ Dyn Control 37 (8):1598–1625CrossRefGoogle Scholar
 Dosi G, Fagiolo G, Napoletano M, Roventini A, Treibich T (2015) Fiscal and monetary policies in complex evolving economies. J Econ Dyn Control 52:166–189CrossRefGoogle Scholar
 Ethier WJ (1982) National and international returns to scale in the modern theory of international trade. Am Econ Rev 72(3):389–405Google Scholar
 Fagiolo G, Dosi G (2003) Exploitation, exploration and innovation in a model of endogenous growth with locally interacting agents. Struct Chang Econ Dyn 14 (3):237–273CrossRefGoogle Scholar
 Feenstra RC, Madani D, Yang T H, Liang C Y (1999) Testing endogenous growth in South Korea and Taiwan. J Dev Econ 60(2):317–341CrossRefGoogle Scholar
 Frenken K (2006a) A fitness landscape approach to technological complexity, modularity, and vertical disintegration. Struct Chang Econ Dyn 17(3):288–305Google Scholar
 Frenken K (2006b) Technological innovation and complexity theory. Econ Innov New Technol 15(2):137–155Google Scholar
 Frensch R, Wittich VG (2009) Product variety and technical change. J Dev Econ 88(2):242–257CrossRefGoogle Scholar
 Funke M, Ruhwedel R (2001) Product variety and economic growth: empirical evidence for the oecd countries. IMF Staff Pap 48(2):225–242Google Scholar
 Gualdi S, Mandel A (2015) On the emergence of scalefree production networks. J Econ Dyn Control 73:61–77CrossRefGoogle Scholar
 Jackson MO, Rogers BW (2007) Meeting strangers and friends of friends: how random are social networks?. Am Econ Rev 97(3):890–915CrossRefGoogle Scholar
 Kauffman S (1993) The origins of order: self organization and selection in evolution. Oxford University Press, USAGoogle Scholar
 Kauffman S, Lobo J, Macready WG (2000) Optimal search on a technology landscape. J Econ Behav Organ 43(2):141–166CrossRefGoogle Scholar
 Lamperti F, Napoletano M, Roventini A (2015) Preventing environmental disasters: marketbased vs. commandandcontrol policies. LEM working papers, 34Google Scholar
 Lancaster KJ (1966) A new approach to consumer theory. J Polit Econ 74 (2):132–157CrossRefGoogle Scholar
 Mandel A, Jaeger C, Fürst S, Lass W, Lincke D, Meissner F, PabloMarti F, Wolf S (2010) Agentbased dynamics in disaggregated growth models. Documents de Travail du Centre d’Economie de la Sorbonne 10077, Université PanthéonSorbonne (Paris 1) Centre d’Economie de la Sorbonne. http://ideas.repec.org/p/mse/cesdoc/10077.html
 McNerney J, Farmer JD, Redner S, Trancik JE (2011) Role of design complexity in technology improvement. Proc Natl Acad Sci 108(22):9008–9013CrossRefGoogle Scholar
 Nelson RR, Winter SG (1982) An evolutionary theory of economic change. Harvard University Press, HarvardGoogle Scholar
 Romer PM (1990) Endogenous technological change. J Polit Econ 98(5 pt 2):71–102CrossRefGoogle Scholar
 Saviotti P, Pyka A (2008) Product variety, competition and economic growth. J Evol Econ 3(18):323– 347CrossRefGoogle Scholar
 Saviotti P P, et al (1996) Technological evolution, variety and the economy. BooksGoogle Scholar
 Saxenian AL (1996) Regional advantage. Harvard University Press, CambridgeGoogle Scholar
 Silverberg G, Verspagen B (2005) A percolation model of innovation in complex technology spaces. J Econ Dyn Control 29(1):225–244CrossRefGoogle Scholar
 Tàbara J D, Mangalagiu D, Kupers R, Jaeger CC, Mandel A, Paroussos L (2013) Transformative targets in sustainability policy making. J Environ Plan Manag 56(8):1180–1191CrossRefGoogle Scholar
 Utterback JM (1994) Mastering the dynamics of innovation: how companies can seize opportunities in the face of technological change. Harvard Business School Press, Boston. ISBN 0875843425Google Scholar
 Weisbuch G, Battiston S (2007) From production networks to geographical economics. J Econ Behav Organ 64(3):448–469CrossRefGoogle Scholar
 Wolf S, Fuerst S, Mandel A, Lass W, Lincke D, PabloMarti F, Jaeger C (2013) A multiagent model of several economic regions. Environ Model Softw 44:25–43. https://doi.org/10.1016/j.envsoft.2012.12.012. ISSN 13648152. http://www.sciencedirect.com/science/article/pii/S1364815213000029 CrossRefGoogle Scholar
 Wright TP (1936) Factors affecting the cost of airplanes. Journal of the Aeronautical Sciences 3(4):122–128CrossRefGoogle Scholar
 Yang X, Borland J (1991) A microeconomic mechanism for economic growth. J Polit Econ 99(3):460–482CrossRefGoogle Scholar