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Journal of Evolutionary Economics

, Volume 29, Issue 1, pp 91–117 | Cite as

Endogenous growth in production networks

  • Stanislao Gualdi
  • Antoine MandelEmail author
Regular Article

Abstract

We investigate the interplay between technological change and macro- economic dynamics in an agent-based 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 long-term 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 feed-in tariffs and of priority market access.

Keywords

Agent-based modeling production networks Endogenous technological change 

JEL Classification

C63 D57 D85 L16 L52 O31 O33 

1 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 agent-based macro-economic 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 micro-economics 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, agent-based 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 long-term 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 agent-based 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 input-output 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 long-term 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 long-term 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 input-output 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 problem-solving procedures, constrained by paradigm-shaped ‘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 technology-driven 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 Utterback-Abernathy 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 agent-based 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 price-based measures, akin to feed-in tariffs, and quantity-based 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 macro-economic 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 agent-based macro-economic 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 input-output space. This allows us to provide the macro-economic 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 micro-founded 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 micro-economic 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 co-exist. 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 pre-existing sectors are complementary and not independent aspects of economic development”. More broadly, we could argue that we operationalize, via a network-based 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 agent-based (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 macro-economic 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 At 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.

Production processes are linked to the structure of the network via production functions, which are assumed to be of the C.E.S type as in the literature on monopolistic competition on the intermediate goods markets (see Ethier 1982; Romer 1990). More precisely, given a production network A, the set of suppliers of firm i is Σi(A) := {jMai,j = 1} and its production possibilities are given by the function:
$$ {f_{i}^{t}}(x_{0},(x_{j})_{j \in \Sigma_{i}(A)}):= x_{0}^{\gamma} \left( \sum\limits_{j \in \Sigma_{i}(A)} \left( {e^{t}_{j}}x_{j}\right)^{\theta}\right)^{(1-\gamma)/\theta} $$
(1)
where γ ∈ (0,1) is the (nominal) share of labor in the input mix, 1/(1 − 𝜃) is the elasticity of substitution, \(x_{0} \in {\mathbb {R}}_{+}\) is the quantity of labor and xj the quantity of input j used in the production process, and ej the productivity of input j.

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 Macro-economic closure

We also attach the evolution of the economy to that of the network. Therefore, we follow Gualdi and Mandel (2015) and consider that the structure of the network determines the dynamics of the economy according to a set of behavioral rules. More precisely, let us denote by At the adjacency structure of the network in period t and denote for every agent i, \({w_{i}^{t}} \in {\mathbb {R}}_{+}\) the wealth it holds, \({q_{i}^{t}} \in {\mathbb {R}}_{+}\) the stock of output it has produced, \({p_{i}^{t}} \in {\mathbb {R}}_{+}\) the price it sets for its output, and \(({\alpha _{j}^{t}})_{j \in \Sigma _{i}(A^{t})} \in {\mathbb {R}}_{+}^{\Sigma _{i}(A^{t})}\) the input shares it chooses. The dynamics of the economy during period t are then completely determined by the structure of the network. More precisely, the following sequence of events take place during a period:
  1. 1.

    Agents receive a nominal demand proportional to the wealths and the input shares of their connections.

     
  2. 2.

    Agents adjust their prices toward their market clearing value (at a rate τp ∈ [0,1]).

     
  3. 3.

    Agents then produce according to the inputs they receive.

     
  4. 4.

    Agents adjust their input shares (at a rate τw ∈ [0,1]) toward their cost-minimizing 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 macro-economic dynamics. We shall consider two main drivers for this evolution: competition and innovation.

Competition materializes through the possibility for firms to periodically shift part of their business to more competitive suppliers. More precisely, we consider that at the end of each period, each firm independently receives the opportunity to change one of its suppliers with probability ρchg ∈ [0,1]. If this opportunity materializes for firm i in period t, it selects randomly one of its suppliers \(\overline {j}_{i}\) (in sector \(S_{\overline {j}_{i}}\)) and another random firm j (in the same sector) among those to which it is not already connected. It then shifts its connection from firm \(\overline {j}_{i}\) to firm j if and only if the price (normalized per unit of productivity) of j is less than the one of \(\overline {j}_{i}\). In other words, the adjacency matrix At evolves according to:
$$ \begin{array}{c} a_{i,\overline{j}_{i}}^{t + 1}= \left\{\begin{array}{cc} 1 & \text{if }\frac{p^{t}_{\overline{j}_{i}}}{e_{\overline{j}_{i}}} \leq \frac{{p^{t}_{j}}}{e_{j}} \\0 & \text{otherwise} \end{array}\right.\\ \\ \ a_{i,j}^{t + 1}= 1-a_{i,\overline{j}_{i}}^{t + 1} \end{array} $$
(2)
The actual weight of the new connection is then determined according to an average over other suppliers’ weights.

This competitive process leads to the evolution of the in-degree distribution of the network. In fact, as shown in Gualdi and Mandel (2015), competition leads to the emergence of a scale-free in-degree 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 scale-free 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 (ei,υi) within a paradigm (eii). An incremental innovation for that firm consists in the adoption of a new input mix \(\tilde {\upsilon }_{i}\) within the paradigm (eii) 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.

Changes in the input mix materialize through the addition or the deletion of links. Hence, innovation is embedded within the network and technological progress materializes through the evolution of the network. As for the drivers of technological progress, we consider three possible avenues for productivity growth: process/incremental innovation, product/radical innovation and imitation as in the evolutionary model of Nelson and Winter (1982). More precisely each firm i invests a fixed share 𝜃 of its revenues in R and D and this yields with probability ρinn per period an innovation, i.e every period, a share ρinn of firms are selected uniformly at random to draw an innovation. This innovation can be of three types: incremental, radical or imitative. More precisely, one has:
  • With probability μinc 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 μrad 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, υ0. The maximal number of links for the new paradigm is itself drawn uniformly between υmin and υmax, which are parameters of the model.

  • With probability μim 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 micro-economic implementations of the macro-economic 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 infra-marginal 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 trade-based 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 macro-economic and distributional patterns that emerge from the interplay of these processes.

4 Innovation, growth and the evolution of production networks

Our first objective is to characterize the impact of the different innovation processes under consideration on the macro-economic dynamics and the structural properties of the production network. Therefore, we perform a series of Monte Carlo experiments under varying innovation regimes: incremental only, radical only, incremental and radical in combination. Building on the sensitivity analysis developed in Gualdi and Mandel (2015), we use as default in these experiments the set of parameters given in Table 1, which are representative of the behavior the model in absence of innovation. Unless otherwise specified, we run simulation with fifty different random seeds for each parameter combination in order to average out stochasticity. However, we observe very little variability or macro-economic dynamics with varying seeds : for all the results reported below about output and number of links, the standard error is below 1%. Therefore, we report only the results of a single simulation per set of parameters.
Table 1

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 time-scale at which technological innovation takes place must be somehow separated from the time-scale 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 ρinn 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 ei = 1 and \(\overline {\upsilon }_{i}= 20\).

The results of simulations, illustrated in Fig. 1, show that the qualitative behavior of the model is independent of the choice of parameters. In a setting where innovation occurs only within a fixed technological paradigm, increasing product variety is a transitory process: it lasts until the frontier of the technological paradigm is reached and connectivity saturates. In the absence of saturation, the growth of the number of inputs ought to be linear as the number of inputs (tentatively) added to the production process is constant over time. However, the actual pattern is sub-linear (see left panel of Fig. 1) given that the rate of success of these incremental innovations decreases as more firms approach the frontier of the technological paradigm. Two complementary mechanisms are at play in this process. First, firms that have reached the frontier of the technological paradigm can no longer seize innovation opportunities, such that the number of firms that benefit from incremental innovation decreases over time. Second, as 𝜃 < 1, the marginal returns to incremental innovation are decreasing and hence the probability of success of an incremental innovation decreases with connectivity. The right-panel of Fig. 1 presents the macro-economic counterpart of this connectivity pattern. Output grows at a decreasing rate during a transitory regime and then stabilizes.
Fig. 1

Evolution of the number of links (left) and of total output (right, log scale) for representative simulations corresponding to a 100% rate of incremental innovation, with varying innovation rates. Other parameters are set as in Table 1

It is clear that linear (or sub-linear) increase in product variety can not lead to exponential growth, even if one abstracts away from the saturation process. In order to provide a quantitative approximation of the transient growth regime, we use the simplifying assumption that each input is used in similar quantity in the production process and has a normalized productivity of 1. The quantity of output obtained by using 1/n units of n distinct varieties of inputs (and of a quantity of labor normalized to unity) is then given by:
$$ f(1,1/n,\cdots,1/n)=(n(1/n)^{\theta})^{(1-\gamma)/\theta}=n^{(1/\theta-1)(1-\gamma)} $$
(3)
Thus, if one discards the saturation process and considers that the number of inputs grows linearly over time, i.e n(t) = kt, productivity (and production) shall grow as
$$ \phi(t)=(kt)^{(1/\theta-1)(1-\gamma)} $$
(4)
consistently with Wright’s law which, in its original form (see Wright 1936) states that production cost decreases (or productivity increases) as a power of the cumulative production (see also Arrow 1962; McNerney et al. 2011). In order to test the validity of this approximation in our framework, we have estimated the time dependency of output in the simulated data (before the establishment of the stationary regime) using a model of the form
$$ y= (at)^{b} $$
(5)
and compared the estimated b exponent with the one predicted by Eq. 4. Table 2 illustrates our results. Equation 5 fits remarkably well the simulated data and the value of the estimated exponent is consistent with the one put forward in Eq. 4, with a downward bias that can be explained by the progressive saturation process.
Table 2

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

R2

0.965

0.965

0.966

Adjusted R2

0.965

0.965

0.966

Residual Std. Error (df = 98999)

0.035

0.054

0.165

p < 0.1; ∗∗p < 0.05; ∗∗∗p < 0.01

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.

Another stylized fact of industrial dynamics that the model closely matches is that the degree distribution of firms’ size, measured via their number of incoming connections, is scale free. See the left panel of Fig. 2 and the related discussion in Gualdi and Mandel (2015).
Fig. 2

Distribution of in-degrees after 2.106 periods for varying innovation rates. Left panel corresponds to incremental innovation only (Section 4.1), right panel to radical innovation only (Section 4.2). Other parameters are set as in Table 1

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 micro-founded 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 counter-factual. 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 macro-economic 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 μrad = 1 while μinc = μim = 0) and the total innovation rate ρinn varies in {10− 2,5.10− 3,10− 3,5.10− 4,10− 4}.

The results of the simulations show that, from the macro-economic perspective, the qualitative behavior of the model is independent of the innovation rate. As illustrated in Fig. 3, radical innovation systematically leads to exponential growth. The growth regime establishes itself rapidly and is remarkably stable. Moreover, in the absence of radical innovation, the average connectivity and the out-degree distribution of the network are also stable. The only source of volatility appears to be the entry and exit process of firms.
Fig. 3

Evolution of the number of links (left) and of total output (right, log scale) for representative simulations corresponding to a 100% rate of radical innovation, with varying innovation rates. Other parameters are set as in Table 1. Note that on the right panel, simulations corresponding to ρinn = 10− 3,5 ⋅ 10− 3 leave the panel box because growth is too rapid

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 in-degree distribution of firms. While the distribution is scale-free 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 ρinn (see right-panel 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 macro-economic 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 υ0 (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 (υ0 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 μinc/μrad 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. μim = 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 ρinn × μrad/μinc.

During the transient period, the dynamics of output and of connectivity, illustrated in Fig. 4, are very similar to those observed in presence of incremental innovation only (see Section 4.1): the growth of output is driven by the increasing product variety in the production process (or equivalently by the increasing connectivity in the network). The growth pattern is also consistent with Wright’s law. The transition to the stable regime occurs smoothly when the growth rate has reached an “equilibrium” value, which depends on the frequency of radical innovations. The end of the transient regime is also marked by the stabilization of connectivity. This “equilibrium” level of connectivity does not in general saturate the constraint of the technological paradigms (maximal number of suppliers) and decreases with the frequency of radical innovation (see Fig. 4). It is independent of the elasticity of substitution and of the maximal number of links.
Fig. 4

Evolution of the number of links (left) and of total output (right, log scale) for representative simulations with varying ratios between the rates of incremental and radical innovation (upper panel) and varying level of innovation rates (lower panel). Other parameters are set as in Table 1

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, μim/μrad, vary in {10,20,50}. We also let the ratio μinc/μrad vary in {1,10,20} and set other parameters as in Table 1.

Our analysis mainly focuses on the properties of the stable regime, which is still characterized by exponential growth in the long-run. Yet, a key feature that emerges is the presence of technologically driven-business cycles, which materialize via fluctuations of the output and of the connectivity in the network, i.e of the average product variety of the technologies used (see Fig. 5, in particular upper-left and lower-right panels).
Fig. 5

Evolution of the number of links (upper left) and of total output (upper and lower right) for representative simulations with varying imitation and incremental innovation rates. The lower left panel displays the joint evolution over 100000 periods of the growth rate of links and output for a representative simulation. Other parameters are set 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 μim/μrad 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 micro-economic 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 Utterback-Abernathy model (see Utterback 1994).

The structural evolution of the network is also aligned with key empirical facts about the demographics of firms. The distribution of firms’ sizes, measured through their in-degree distribution in Fig. 6, is characterized by fatter tails than normal, though the scale-free character is absent given that we consider here relatively large rate of innovation that prevent the formation of very large firms. The out-degree distribution of firms in the right panel of Fig. 6 measures the variability of productivity (through variety in the input mix) in the population of firms. This distribution underlines the persistence of heterogeneity among firms in terms of productivity and the fact that this heterogeneity increases with the rate of incremental innovation that allows a deeper exploration of the technological paradigm.
Fig. 6

Distribution of in-degrees (left, log-linear) and out-degrees (right) after 2.106 periods for varying imitation and incremental innovation rates. Other parameters are set as in Table 1

4.4 Firms’ demographics

More broadly, the model is able to replicate a wealth of stylized facts about industrial dynamics (see e.g Coad 2009):
  • As illustrated in the left panel Fig. 7, growth rates of firms are distributed according to a “tent-shaped” double-exponential 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) = sβ 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 ejs) is heterogeneous ; it exhibits much fatter tails in absence of imitation. Together with the above results about the out-degree distribution of firms, it implies that the model endogenously generate heterogeneity both in terms of process and of product productivity.

Fig. 7

Distribution of growth rates (left) and of products’ productivity (right) for varying rates of innovation

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 feed-in tariffs, which consist in subsidizing the price paid to renewable energy producers, and preferential access to the market for renewable energy producers.2

Accordingly, we investigate in the model the impact of price-based and market access-based measures on the survival rate of radical innovators and on the growth rate of the economy. In a first series of experiments, we focus on the aggregate impact of these measures and do not differentiate between sectors. More precisely, we consider the following scenarios.
  • In the price-support scenario, which is akin to feed-in 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 market-support 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, Tpr 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 τfeed and in {100,500,1000} for Tpr. Other parameters are set as in Section 4.3 with μim/μrad ∈{0,1} and μinc/μrad = 1. Simulations are ran for five different seeds for each combination of parameters.

As illustrated in Fig. 8 and Table 3, respectively, only the price-support policy has an impact on survival probabilities and hence on output. This suggests that price-support measures have, in our framework, a stronger impact on the competitive position of firms. Moreover, the market-support policy only shifts demand within the economy, while the price-support policy provides, at the aggregate level, a subsidy to the economy by financing externally the price reduction for radical innovators. This subsidy can yield a demand-push and indirectly trigger multiplier effects.
Table 3

Exit rate of radical innovators(per 103 periods)

Price-support scenario

 

Market-support scenario

0

Parameter

Exit rate

Parameter

Exit rate

μim = 0,τfeed = 0.1

3.10− 2

μim = 0,Tpr = 100

3.10− 2

μim = 0,τfeed = 0.2

8.10− 3

μim = 0,Tpr = 500

4.10− 2

μim = 0,τfeed = 0.5

5.10− 4

μim = 0,Tpr = 1000

5.10− 2

μim = 1,τfeed = 0.1

2.10− 2

μim = 1,Tpr = 100

2.10− 2

μim = 1,τfeed = 0.2

6.10− 3

μim = 1,Tpr = 500

3.10− 2

μim = 1,τfeed = 0.5

3.10− 4

μim = 1,Tpr = 1000

4.10− 2

The impact on output in the price-support scenario materializes, both with and without imitation, by the increase of the output per productivity unit ratio, where the latter is measured by the average product productivity (see Fig. 8). This implies that firms have, on average, a more efficient/diversified production process, i.e that they have seized more incremental innovations. This feature is clearly consistent with the decreased rate of exit for radical innovators: if firms leave longer, they have more opportunities to seize incremental innovations.
Fig. 8

Evolution of outptut (normalized by average product productivity J) for the price-support (left) and the market-support (right) scenarios. In both cases, one has ρinn = 10− 3, μincr = μimi = μrad and other parameters are set as in Table 1

The impact on aggregate output is more ambiguous. Figure 9 suggests that the price-support policy has a negative impact in the absence of imitation and a neutral (or slightly positive) impact in the presence of imitation. This presumably is the counterpart of the increased lifespan of radical innovators. If incumbent innovators are more efficient, it is harder for a new radical innovation to succeed and hence the growth rate of productivity is reduced. This effect is partly offset by imitation, which allows the diffusion of increased productivity directly among incumbents.
Fig. 9

Evolution of outptut (log scale) for the price-support scenario with (left) and without (right) imitation

In the context of climate change, policy also aims at directing technological change toward specific sectors (e.g. green vs fossil energy use). We investigate this more specific type of policies in a second series of experiments where we consider that the price-support policy is implemented only in sector 1. We consider varying subsidy rates, from 10 to 50%, and repeat each experiment in 30 different Monte-Carlo simulations (all the other parameters are set as in the previous policy experiments and imitation is enabled ). We do not find substantial aggregate variability in the simulation results (the standard errors are lees than one percent) and therefore report only the results of a representative run in Fig. 10. The experiments clearly show that feed-in policy can direct technological change toward an increase use of the output in sector 1 in the production process. Hence, as illustrated in the left panel of Fig. 10, the share of sector 1 in total output increases with the feed-in rate. Moreover, the policy appears to have a slightly positive impact on total output (see right panel of Fig. 10). This last result is at odds with the one obtained in the case where all the sectors were equally subsidized and suggests that directed technological change might be more beneficial to the economy as it limits the disruption induced by radical innovations.
Fig. 10

Share of sector 1 in total output (left) and total output (right) for varying level of subsidization in sector 1

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 macro-economic agent-based 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 short-run and hence determines short-term economic dynamics. In turn, competition among firms and technological innovations govern the evolution of the network. Long-term macro-economic dynamics hence emerge from the micro-economic interactions among firms.

From the theoretical point of view, our main innovation is to provide a detailed micro-economic 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 micro-economic setting with boundedly rational agents.

Considering innovation occurs at the micro-level 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 fat-tails, 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 Utterback-Abernathy 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 feed-in 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 socio-economic 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. 1.

    Here and in the following, we always consider implicitly that the vector (μinc,μim,μrad) is normalized so that μrad + μinc + μim = 1

  2. 2.

    A cautionary note in this respect is that, as argued by Lamperti et al. (2015), market-based policies may not be sufficient to prevent environmental disasters while Command-and- Control policies are fully effective.

  3. 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 11-LABX-0019) and Labex OSE (ANR-10-LABX-93-01).

Compliance with Ethical Standards

Conflict of interests

the authors declare that they have no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CentraleSupélecChâtenay-MalabryFrance
  2. 2.Paris School of EconomicsUniversité Paris I Panthéon-SorbonneParisFrance

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