Absorptive capacity in a two‐sector neo‐Schumpeterian model: A new role for innovation policy Isabel Almudi University of Zaragoza
Francisco Fatas‐Villafranca University of Zaragoza
Carlos M. Fernandez Universidad Autónoma Madrid
Jason Potts RMIT University Corresponding author (
[email protected])
Francisco J. Vazquez Universidad Autónoma Madrid
Abstract
We propose a new co-evolutionary computational two-sector approach to the design of national innovation policy that recognizes the importance of inter-sectoral absorptive capacity constraints in innovation linkages between sectors in an economy. We show how the innovative capacity of an upstream producer sector can be constrained by the absorptive capacity of the downstream-user sector. This suggests that the low productivity performance of modern innovation policy might in part be understood as a consequence of sectorally unbalanced knowledge evolution, where the problem lies in underinvestment in innovative capabilities in the downstream sector. Our computational two-sector model suggests an important new role for innovation policy to create a balanced, sectorally-targeted approach.
Keywords: Innovation, coevolution, Neo-Schumpeterian models, absorptive capacity.
1
JEL-Code: B52, C63, O31.
1
Introduction
The purpose of this paper is to model the consequences of unbalanced sectoral knowledge. We will develop a model of a two-sector neo-Schumpeterian economy (an ‘upstream’ capital goods or producer sector 1, and a ‘downstream’ consumption goods or user sector 2) in which absorptive capacity limitations in the downstream sector constrain innovation in the upstream sector. Equivalently, innovation in sector one can be constrained by absorbtion capacities of innovation in sector two. Such an economy exhibits what we call unbalanced inter-sectoral knowledge. Our basic argument carries over to a multi-sectoral innovation economy, however the underlying logic is sufficiently demonstrated in two interlinked sectors. Our approach recognizes the importance of inter-sectoral absorptive capacity constraints in innovation linkages between sectors in an economy and in the prospect of overshooting this entails. We combine two distinctly Schumpeterian literatures—the theory of absorptive capacity (Cohen and Levinthal 1990), and the theory of innovation overshooting (Earl and Potts 2013, 2016)—into a multi-sector context using a coevolutionary computational model with micro- and meso-foundations. Our coevolutionary model of (potentially) unbalanced sectoral knowledge suggests a new role and indeed rationale for innovation policy. This new approach may help to explain, among other things, why international trade linkages matter to innovation-led sectoral growth and why advanced technology military procurement can be effective as demand-side innovation policy. Our two-sector neo-Schumpeterian approach to economic change also relates to the multisectoral modelling tradition in economics. Leontief-type input-output (I-O) models that connect sectors of an economy through flows of commodities and payments have long been a workhorse of macroeconomic forecasting, planning and policy (e.g. Chenery 1960). By integrating prices and micro-foundations general equilibrium models (CGE, DSGE) developed in the 1970s have since subsumed the I-O approach. Both types of models, however, are well adapted to primary producing and industrial market-based economies where the main constraints between sectoral flows are factor supply and demand for commodities expressed in income and prices. These models have been optimized to macro industrial planning (I-O models for industry policy) and macro-dynamic forecasting (GE-type models for competition policy, 2
Aghion and Griffith 2005). Less attention however has been afforded to a closely related problem, common in a Schumpeterian economy, namely unbalanced intersectoral knowledge and the innovation constraints this causes. For this reason, the multi-sectoral approach has proven less useful in the context of innovation policy, which has instead tended to rely on an institution-centred innovation systems approach (Nelson 1993, Etzkowitz and Leydesdorff 2000, Foray et al 2009). The standard Arrow-Nelson paradigm of innovation theory tackles questions related to the rate of inventive activity in a better way than those related to its direction (Trajtenberg 2012; Nelson (1959, 1962) and Arrow 1962). Our two-sector model helps us better understand the directin of innovation in a structural context. As Nelson (2012) explains, contemporary innovation economics highlights new features of technical change that were not clear to the paradigm pioneers and that should be at the core of contemporary innovation models. More precisely, a new theory should incorporate the following: 1. Technology develops as an evolutionary process; 2. Extreme inter-field unevenness is an intrinsic characteristic of technical change; 3. Technological progress emerges from the co-evolution of practice and understanding in coupled multisector domains (Potts 2000, Murmann 2003, Dosi and Nelson 2010, Dosi and Grazzi 2010, Metcalfe 2010, Mokyr 2016). We characterize a Schumpeterian knowledge-based economy as one in which the main constraints between sectoral flows turn not on factor supply, income or prices but on knowledge absorptive capacity (Cohen and Levintal 1990, Zahra and George 2002). Knowledge absorptive capacity is an ability to adopt, understand and make use of a technology or capability originating in another sector. We suppose that each sector is engaged in innovation characterised by Schumpeterian competition in which heterogeneous firms develop (and co-evolve from an intersectoral viewpoint) within spaces of performance and price, which are determined by R&D spending and technological learning. In a perfectly balanced economy, each sector can absorb and adopt technologies from every other sector. Sectoral innovation constrains, in a balanced economy, can only be a consequence of variation in resources, prices and income, as in I-O and GE models. Note that the standard absorptive capacity literature focuses on intra-sectoral constraints, not inter-sectoral constraints. In an unbalanced economy, absorptive capacity constraints in a sector—because of prior experiential learning (Dosi et al 2005)—limit 3
its ability to adopt technologies from another sector, which in turn reduces the demand for technologies from that sector. In this way the innovation choices of firms in different sectors interlink. Constrained R&D in a downstream sector can limit innovation in the upstream sector through absorptive capacity constraints. We characterise this as a species of ‘sectoral allocation problem’ in which there is excess innovation spending in one sector and too little absorptive capacity in a connected sector, resulting in wasted innovation spending because the benefits cannot be diffused owing to absorptive capacity constraints. To model this intersectoral dynamic, we develop a new class of computational co-evolutionary multi-sector (agent-based) framework to model these complex inter-linkages. The model is neoSchumpeterian, building on Nelson and Winter (1982), Almudi et al. (2013) and Dosi et al. (2013). However, our model extends the Schumpeterian model to incorporate the prospect of innovating overshooting (Earl and Potts 2013, 2016; Almudi et al. 2017) where the technological capabilities developed in one sector overshoot the absorptive abilities of the other sector. Innovative overshooting is a consequence of unbalanced Schumpeterian competition in between different sectors. We propose that innovation overshooting in a multi-sectoral context of unbalanced Schumpeterian competition offers an explanation for the otherwise puzzling low aggregate productivity measures on public sector R&D and innovation policy that have been widely reported across OECD nations (Jaumotte and Pain 2005). In the simple but still dominant basic-science technology-push model, the practical policy prescription is to target innovation policy at the source of the greatest market failure (i.e. basic science, Bush 1945, Nelson 1959) (Martin and Scott 2000, Trajtenberg 2012). This is equivalent to a single sector innovation target in our multi-sectoral approach. But if, as we argue here, absorbtive capacity between sectors is a more general constraining factor, then the market failure targeting approach may simply lead to overshooting. This will manifest in wasted innovation spending and a dynamic feedback that will drive all sectors to the lowest absorbtive capacity level. Alternatively, our multi-sectoral approach predicts that an unbalanced distribution of innovation policy may produce multi-sectoral innovation blockages, such as a slow-down of productivity growth, frustrated attempts to take off regarding technological change, or even sectoral collapse. The multi-sector computational model not only diagnoses the innovation policy problem but can also be used to redesign innovation policy to target absorptive capacity constraints (similar to the polcy 4
use of the concept of ‘smart specialization’ (Foray et al 2009)). This will improve the productivity of public R&D by redirecting it from a market failure focus on basic science and technology, and to focus on translation layer constraints, particularly in downstream sectors. We thus propose to reinvent sectoral analysis for a dynamic innovation driven economy, and thereby to further advance projects started by Rodrik (2004) and others to reconstruct both industry policy and competition policy as innovation policy. In the final part of our project we suggest the possibility of extending this multi-sectoral model into a (future) multi-country model, in essence seeking to account for the problem of the upstream and downstream sectors being in different countries, and therefore subject to different innovation policies. This introduces a global strategic aspect to innovation policy (Potts 2016). We proceed as follows. Section 2 presents the model (we make continuous references to the timeline of events – written in the pseudocode reported in the Appendix). Section 3 is the computational analysis of the model and results. We explain model details and novelties regarding the role of absorptive capacity in intersectoral co-evolution and innovation. Section 4 concludes with prospects for future research.
2
The Model
In our computational two-sector neo-Schumpeterian model (Dosi et al. 2013; Metcalfe et al. 2006; Saviotti and Pyka 2013) a population of heterogeneous firms in each sector operates over several dimensions. In Sector 1 (the capital-good sector) different and gradually improved varieties of a capital-good (machines) are produced and sold to Sector 2. In Sector 2 (the consumption-good sector), different varieties of a final consumption good are produced and sold by firms to consumers. Firms in Sector 2 buy different varieties of machines from Sector 1 and produce specific varieties of the consumption good for final consumers. Firms producing and offering machines in Sector 1 compete in price and machine-type (quality) performance. They fix prices according to a modified-pricing rule (Bloch and Metcalfe, 2018; Winter, 1984; Vives, 2001) with the novel feature that the mark-up evolves according to each firm’s estimates of its close competitor’s market power. This is a simple way to incorporate (intra-sectoral) strategic interactions. Each firm then charges an (endogenously-changing) margin on expected unit-cost. 5
The unit cost includes a unit production cost, which is common and constant across firms, and a unit R&D-cost (ex-ante, but realized ex-post). The R&D intensity is a firm-specific behavioral trait, as a lagged proportion of profits. Likewise, we model firm performance in Sector 1 as a relative (and normalized) specific dimension that evolves through innovation. Each firm in Sector 1 produces machines up to the demand point of users from Sector 2. The demand captured by each firm in Sector 1 depends (probabilistically) on, both, the offerings over price and performance dimensions of its machines. Each firm in Sector 2 (the buyers and users of machines) buys at most one machine per period of time. Machines fully depreciate and disappear in one period. At any time period, only profitable firms remain in Sector 1. On the other hand, new firms enter continuously in the sector, although many will fail. Sector 2 consists of a changing number of firms due to entry and exit that produce different varieties of a consumption good and sell these to final consumers. Sector 2 firms use one machine (bought from Sector 1) each to produce their variety of the consumption good, with each quality (variety) of the consumption good dependent on the firm’s production technology (i.e the quality of the corresponding machine). Sector 2 firms have a specific knowledge endowment that evolves with experience, and they observe and assess different parts of the distribution of machines supplied by Sector 1 at any time. They combine price and machine performance from a range of (cogntitively-understandable) options under consideration and choose probabilistically. In this way, demand in Sector 1 (from Sector 2 firms) dynamically evolves. Once firms in Sector 2 buy machines, they set prices and qualities and compete over price and performance to capture final consumers. Only firms with market shares higher than 0.005 in the consumption market remain in the sector. There is also an ongoing process of firm entry in the sector, although many entrants will fail. In Sector 2 firms update their knowledge endowments according to the performance level of their most recent machines. Likewise, each Sector 2 firm has, as a specific behavioral trait, what we call a cognitive radius when scanning the supply of machines: the higher the radius, the wider the scope of innovative search. Thus, firms in Sector 2 have differential absorptive capacity (Cohen and Levintal 1990) as an ability to understand and adopt innovation from Sector 1. Clearly, this absorptive capacity in reality may be constructed over several distinct and partially cumulative mechanisms. The first dimension is the 6
cognitive capacity of each firm and the implications of bounded rationality for organizational learning (Simon 1955, 1991). A second dimension is related to what Kremer (1993) called O-Ring theory, where the absorptive capacity is constrained by the worst performing members of the organizational team. Third, absorptive capacity in Sector 2 may be determined and constrained by the social technologies and institutions needed for training technical people (Nelson and Sampat 2001). Subsections 2.1 and 2.2 formally specify the model assumptions. The pseudocode in the Appendix incorporates all the assumptions in a synthetic way. This pseudocode may be of some help to understand the processes of two-sector coevolution that will emerge from the iterations of the model timeline of events in alternative settings. We also present all parameter and variable notations in the Appendix along with the limiting range of values for relevant simulations in economic terms and the values corresponding to the base-standard setting for the simulations. In running the model, we initialize the process in such a way that, at t=0, the initial number of firms in both sectors is “0”. Then, firms gradually enter (and exit), operate, innovate and coevolve by trading in both sectors.
2.1
The Capital‐good sector (Sector 1)
Prices and performance At any time t, we have a changing set of firms (or companies) in Sector 1:
,
. We denote by
,
, (note subscript i), each individual firm i in Sector 1 (superscript 1) at t. These firms produce different varieties of a capital-good we call machine. We assume boundedly-rational profit-seeking firms that compete in price
,
and machine performance
,
(we normalize performance on the unit interval). We
will assume that firms in Sector 1 set prices using an endogenously-changing mark-up (
,
1) over
expected unit cost. Thus the price set up by firm i at t is: ,
,
(1)
,
A novel component of the model is that we consider a pricing routine (Bloch and Metcalfe, 2018; Winter, 1984) in which the mark up endogenously changes for each firm. More precisely, we assume that each firm i delineates at any time the set of “perceived close rivals” depending on performance distance. The set is determined according to available information from t–1, and this perception is a firm-specific strategic trait. We can define this set of “close/direct perceived rivals” as: 7
Λ
:
,
,
,
,
∈ 0,1 (2)
We assume that we can capture each firm’s perceived sustitability (or perceived direct competition) by adding up the market share of the perceived close-rivals: ∑ this intensity of direct competition ∑
as an element that can make demand more elastic, thus
,
,
. Furthermore, if we now consider
,
,
eroding the perceived market power of the firm, we can state that the mark-up set up by firm i at t, is (Bloch and Metcalfe, 2018): ∑ ,
,
,
∑
,
,
,
,
for new firms, and
,
1
(3) ,
,
otherwise.
Let us note that, here, we are incorporating strategic interaction in Sector 1. Regarding performance
,
,
firms improve their machine-varieties through technological innovation. We define this below, along with how firms determine their R&D spending (
).
,
Demand-driven production and costs We assume demand-driven production in Sector 1, so that
,
,
. Likewise, we assume that total costs
include production costs and R&D costs. We consider constant and common unit production costs (c) as firms will differ in their unit R&D efforts. In order to set prices (see (1)), firms use ex-ante expected unit costs. We assume naïve expectations in production, so the expected unit cost is: ,
,
, ,
,
,
0 (4)
Once the structure of demand forms and exchanges between Sectors 1 and 2 have occurred, firms will know the effective production and unit costs. Then, the real profit for firm i at t will be1: ,
,
,
,
;
,
, ,
(5)
Only profitable firms remain in the market (see details in Appendix). We also asume that firms devote a specific proportion of their profits to R&D with a lag, so that: ,
,
, ∈ 0,1
(6)
1
The price will be the one in (1), but the effective unit cost to calculate profits will be the one obtained after exchanges.
8
Every firm in Sector 2 demands, at most, one unit of a specific variety of the capital-good from Sector 1 and uses this unit to produce a consumption good in Sector 2. For simplicity, we assume that every unit of capital (machine) totally depreciates and dissappears at no cost at the end of each period. When selecting a specific capital-good firm at t, firms from Sector 2 assess the prevailing levels of price and performance in Sector 1. In 2.2 we will specify this process of choice (see also Appendix). For the time being, observe that if we define the set of customers for each capital-good firm i at t in Sector 1 as Ω , , we have
,
Ω, .
R&D-based Innovation Let
,
be the flow of new knowledge generated by each firm i in Sector 1 at t. We assume this flow is a
random realization of a (truncated) Pareto distribution, so that
,
~
., with "
" representing the
truncated Pareto distribution, with supporting values L=0 and H=1. We endogenize the typical Paretoparameter (the slope of the density function ) so that
∙
∙
, where (a la Nelson
(1982), Dosi et al. (2013, 2017)) we have: ,
, assimilation of external knowledge from the gap to the frontier;
,
, ,
(7)
, generating knowledge from (normalized) inner R&D;
In expressions (7), we assume that the productivity of R&D reflected in the flow of new knowledge depends on both complementary sources, with a sectoral bias denoted by parameter
,
,
that determines the
relative importance of imitation versus efforts within a specific industry. Notice that the lower the firmspecific value of θ at any time, the higher the probability of obtaining a large flow of new knowledge γ , at that time. Finally, we assume that the relative performance of each Sector 1 firm is updated according to a mechanism in which those firms generating higher than average flows of new knowledge, i.e. ̅
0 , increase their relative performance compared to rivals in Sector 1. Thus: ,
, ,
,
̅ ; ̅ 9
∑
,
,
(8)
,
Firms entry-exit Firms in Sector 1 with profit
,
0 exit the market. Also, at each time step, one new firm enters the
sector. With probability " " (a mutation rate), the new firm’s traits are selected randomly so that with probability " " the new entrant enters into the sector by carrying genuine technological and behavioral novelties. With probability "1
" , the new firm copies one of the incumbents (with probabilities
proportional to market shares and by bearing an implementation cost; details in Appendix).
2.2
The Consumption‐good sector (Sector 2)
Overview At time t, there exist a set of firms/companies in Sector 2,
,
. These firms (each denoted by j)
produce different varieties of a consumption-good (with different prices
,
and quality levels,
,
).
Firms in Sector 2 produce with different techniques or machine-varieties depending on the technological performances of their respective capital-good provider. The technological level (quality) of the machines used by firms in Sector 2 determines the corresponding quality of the supplied variety of consumption good. Firms in Sector 2 with superior machines will produce and supply quality-superior consumption goods. Considering the prevailing distribution of machine-performance levels on the supply-side (from Sector 1 at t), and the distribution of cognitive endowments corresponding to the consumption-good firms in Sector 2
,
,…,
,
), each firm j in Sector 2 decides which capital-good firm from Sector 1 it
will purchase one machine. Assume full-capacity use and total depreciation of machines in one period. Furthermore, for simplicity, the overall production level in Sector 2 is normalized to 1 and fully sold to consumers (the demand-side of Sector 2). As we will see below, the consumption market in Sector 2 is driven by a replicator-dynamics equation (see Dosi et al. 2013).
The process of machine choice by each j-firm in Sector 2 We propose the following process of assessement and choice for each firm j in Sector 2:
10
(1) Firm j delimits the set of (cognitively) feasible options, which will be conditioned by the firm specific cognitive capabilities j (0,1) . This understanding radius is a firm-specific trait. More precisely, the set of feasible providers for firm j is: Ξ ,
:
,
;
,
(2) Firm j chooses a feasible-provider (a cognitively-feasible type of machine) with a probability which is proportional to
(3) The quality of firm j becomes:
1
1
, ,
,
∑
,
,
,
∈ 0,1
,
(4) Each firm in Sector 2 has a cost equal to the price of the machine bought:
,
,
Since this process takes place for all the firms in Sector 2 (see Appendix), we can define now the set of customers for every single firm in Sector 1: Ω ,
.
Likewise, as long as a firm in Sector 2 uses one specific variety of capital-good (with a specific technology level), we asume that this level of performance becomes the firm’s cognitive endowment for the next period, that is:
,
,
.
Market competition in Sector 2 Consider the competition in the consumption-good sector (Sector 2) in which firms in Sector 2 compete in price and quality in the consumption good market. We have already defined how to obtain the quality level of each firm,
,
. Regarding price, we propose that consumption firms also apply a mark-up pricing
routine (Bloch and Metcalfe, 2018), but now we consider a more standard version of the routine. We eschew the role of perceived substitutability in the consumption sector since, normally, consumption goods have less variation than capital goods, and the number of producers in consumption sectors is higher and more competitive than in industrial sectors. Then, we just consider ,
In (9)
,
,
,
,
1 (9)
is the cost of the chosen machine, and the elasticity of demand (>1). To represent the market
process, we define a firm-competitiveness (fitness) level for each firm j that combines quality and price so that: 11
,
,
1
,
1
;
∈ 0,1
.
It is clear that we are representing both dimensions as related to maximum quality and price in Sector 2 at t. Now, from this fitness indicator, we represent the market process in Sector 2 as follows: ,
,
,
,
̅;
with
̅
∑
,
,
(10)
Firms entry-exit Firms in Sector 2 with a share lower than 0.005 leave the market, while at every time step one new firm enters the sector. With probability " " (a mutation rate), the new firm traits are selected randomly and thus the new entrant carries genuine novel traits. An important element of the model is that when a new firm enters as a genuine mutant in the sector we assume (see Appendix) that a specific feature, the understanding-cognitive radius j (0,1) , is drawn from a Beta distribution with positive parameters
,b . We consider this specific distribution because, as we will see when we undertake a computational analysis of the model, it allows us to represent a wide range of scenarios regarding the absorptive capacity of machine-user firms in Sector 2. Some interesting policy implications follow from this approach. Observe that the expected value and variance of a Beta distribution, given variance Var
0, b 0, are E=
, and
.
To close the model, we consider that with probability "1
" , the entrant firm copies one of the
incumbents (with probabilities proportional to market shares). Likewise, we assume that the initial market share of the new entrant is 0.005, with other market shares being re-calculated accordingly.
3
Computational Analysis of the Model
The model is implemented in JAVA and the statistical analysis is carried out with R-Project. The model dynamics reach limit (stationary) states in approximately 5,000 periods. This is what we have detected as the time-span up to the stationary situations obtained through different methods, including KolmogorovSmirnoff (K-S) test. Because of stochasticity, we run the model 100 times and average the data for each 12
setting of parametric values. From the model presented above, emergent properties regarding bi-sectoral coevolution, innovation and statistical regularities could all be explored (Dosi et al. 2017). However, in this paper we will focus computational analysis on the role played by absorptive capacity in the (machines-user sector) Sector 2, as a key driver of intersectoral coevolution and innovation. More precisely, we want to analyze to what extent the generative stochastic structure Beta (a,b) distribution (in Sector 2) from which machineuser firms with specific understanding radius emerge works to infuence: first, the viability of the twosectors industrial coevolution; and second, the emergent intensity of R&D spending leading to knowledge production in the model. As we will see, our computational results focusing on these two aspects suggest new targets for innovation policy. We will begin by analyzing the absorptive capacity of Sector 2 that depends on certain characteristics of the generative structure Beta (a,b) as a key driver of bi-sectoral coevolution. More precisely, we analyze the influence of the skewness of the Beta (a,b) distribution (measured by the third momentum of the distribution), on the emergent probability of technological overshooting (Earl and Potts 2013, 2016), or what we call probability of collapse in the model. We consider that technological overshooting occurs in the model when the rate of innovation in Sector 1 does not fit with the absorptive capacity of firm-users in Sector 2, thereby collapsing transactions in either sector. We measure the (average) probability of collapse owing to technological overshooting for each setting, as the average number of times in which either Sector 1 or Sector 2 vanish during the 100 initial steps of the average run. This is the probability of collapse that we obtain for each parametric setting. A temporary collapse of intersectoral trading in our model is a consequence of knowledge coordination problems that block the coevolutionary process (Almudi et al. 2018). That is to say, technological overshooting and, its consequence in the model – what we call the intersectoral probability of collapse – are caused by knowledge coordination problems among sectors. When we study the precise origin of these coordination problems in the model in 3.1 below we obtain a statistical relation linking technological overshooting represented by the probability of collapse, and the skewness of the generative distribution Beta (a,b) in Sector 2. We will then show how different intensities in firm-R&D spending in the model emerge depending, again, on the shape of the aforementioned Beta (a,b) generative pattern in Sector 2. This specific mechanism in the model (and which engenders absorptive capacity) arises as a crucial factor which 13
indirectly guides the overall pattern of R&D spending and innovation. We show how the emergent average level of sectoral R&D intensity leading to innovation is clearly related to the generative structure of firmunderstanding radius (in Sector 2) Beta (a,b). The aforementioned results obtained from the computational analysis lead us to propose a new type of innovation policy that is focused on eliminating intersectoral cognitive-coordination problems.
3.1
Relations between absorptive capacity in Sector 2 and the probability of collapse
To what extent does a lack of knowledge absorptive capacity in the user-sector (Sector 2) back-propogate to a collapse of activity in the machine-producing sector (Sector 1)? Of course, these dynamic paths imply some collapse of activity also in Sector 2. When these paths emerge, we say that our two sector economy is experiencing technological overshooting because of knowledge coordination problems. Sector 1 has overshot Sector 2 cuased by the lack of absorbtive capacity in Sector 2, thus collapsing Sector 1. However, our main result is a strong statistical relation explaining the probability of collapse (because of overshooting) in terms of specific skewness patterns of the Beta (a,b) distribution in Sector 2. As we have explained above (see also the pseudocode in Appendix), we always run the model departing from settings in which at time 0 both sectors are empty. Then at any time t, one firm enters each sector (viz. 2.2 and 2.3 above on firm entry-exit and pseudocode). Our computational exploration revealed that although the operative success of firms in both sectors depends on the initial conditions and the parametric values, the probability of collapse may crucially depend on the generative structure Beta, from which new innovative firms in Sector 2 obtain their specific understanding radius ( j (0,1) ). Recall that we define the probability of collapse in a setting as the number of times (over 100 initial periods of the average run) in which either Sector 1 or Sector 2 vanish (they become empty, “0” firms) –at least temporarily, since the Sectors will re-emerge as time passes through new entries. Notice that the Beta (a, b) distribution can be assimilated to different distributions (Uniform, Power Law, Truncated Nornal, Negative Exponential) depending on the (a, b) parameters. To recall this fact, we show in Figure 1 alternative shapes for the Beta-probability density function (PDF) with different values for (a, b). This versatility is the reason we chose the Beta distribution, namely, to play with morphologically different generative structures in Sector 2. 14
Figure 1. Density functions for the Beta (a,b) distribution.
Our first computational analysis consists of trying to relate the probability of collapse in the model to specific shapes (i.e. specific values for the (a,b) parameters) of the probability density function for the Beta (a,b) in Sector 2. Thus, we run the model for different initial conditions and parametric values. Specifically, we depart from what we call in the Appendix the base-setting and we run the model for 79X79=6,241 different parametric combinations (we run the model 100 times for each possibility, since the model is stochastic). Figure 2 shows the probability of collapse related to the specific values of the (a,b) parameters. We can clearly see in the chromograph that the probability of collapse is higher for high values of b, and for low values of the parameter a (notice red-orange-yellow to dark blue colors).
15
Figure 2. Probability of collapse related with the Beta (a,b) parameters.
What we can infer from Figure 2 is that if we take into account that the expected value of the Beta (a,b) distribution is E=
, then Figure 2 is a first indicator suggesting how the Beta distribution may be
crucial in driving the model dynamics. Just looking at the expected value expression, it seems clear that higher values of “a” generate higher levels of average understanding radius in Sector 2, and higher absorptive capacity, leading to lower probabilities of collapse. The opposite effect occurs for parameter “b”. To further explore this intuition, since parameters (a, b) determine the shape of the Beta distribution in each case, we prepared Figure 3 to depict alternative specific shapes of the Beta (a,b) density function, and colored each shape depending on the corresponding (emergent) probability of collapse obtained when running the model. In Figure 3, the probability of collapse is higher (red-orange-yellow) when the Betadensity function (PDF) is convex-shaped and the distribution is closer to the ordinate axis. These shapes correspond to intensely right-tailed (positive skew) distributions. Likewise, when the Beta-density function presents a maximum, it is less likely that collapse emerges (blue and black). In principle, as shown in Figure 3, the probability of collapse is lower the more negative skew (more left-tailed-shape) we have in the Beta-distribution. 16
Figure 3. Beta distributions and probability of collapse
To formalize the graphical results obtained from the computational analysis of the model, we can analyze from an statistical perspective the explicative power of the parameters
, ; and also the skewness of the
Beta distribution. We do so in both cases to explain the probability of collapse from technological overshooting. As we will show, the best fit we find corresponds with a polynomial regression. In Table 1, we show (in columns), first, the polynomial degree in each fit, second, the summand, which appears the polynomial coefficient
, with
,
⋯
. In the third column, we
present the estimated coefficient value (with confidence interval (5% and 95%)) and the p-value (H0: a null coefficient). As we see, the
becomes acceptable (>0.9) for a 3-polynomial degree.
Collapse=P( , ) Degree Coef. 1
Fit-estimates 0.01789939 0.00391692
P-value 0.000576302 0.000092497
17
0.0000000000000002 0.0000000000000002
0.00206380
0.000092498
0.0000000000000002
0.4983 0.00000000000000022
2
0.02059867
0.000748923
0.0000000000000002
0.01215065
0.0002691396
0.0000000000000002
0.00140464
0.00002921586
0.0000000000000002
0.00544907
0.0002691324
0.0000000000000002
0.00080105
0.00002613197
0.0000000000000002
0.00001231
0.00002921566
0.488
0.7918 0.00000000000000022
3
0.021010944
0.000824935
0.0000000000000002
0.022615002
0.0005055976
0.0000000000000002
0.006149604
0.0001197051
0.0000000000000002
0.000465648
0.000009228442
0.0000000000000002
0.009887639
0.0005055976
0.0000000000000002
0.003192800
0.00009537797
0.0000000000000002
0.000008100738
0.0000000000000002
0.000239635
0.000197762 0.000052042 0.000002270
0.000119705
0.000008100326
0.00659 0.0000000000000002
0.000009228393
0.68572
0.9168 0.00000000000000022
Table 1. Polynomial regressions for (a, b). Note that in Table 1 p-values are low and confidence intervals are narrow. Since we are considering two explanatory variables (a,b) both the polynomial degree (three) and the number of coefficients are relatively high, so the parameters have a difficult direct interpretation in the Beta-PDF. It therefore seems obscure to interpret the direct parametric effects on the probability of collapse. Nevertheless, both the 18
results of Table 1 and of Figure 2 indicate something highly informative linking the Beta-shape and the probability of collapse. To sharpen our results, note that Figure 3 shows that the skewness of the Beta distribution could be a good candidate factor to explain the probability of collapse. In order to check and formalize this possibility we analyzed the statistical relationship between the Beta-skew and the probability of collapse. As we show in Table 2 and Figure 4, the best fit seems to be a polynomial regression. Table 2 shows the results for the polynomial regressions with “skew” of the distribution as the unique explanatory variable. Now the interpretation of the coefficients is simpler and more informative. Table 2 reports that a seconddegree polynomial fit notably increases the quality of adjustment
(>0.9), while increasing the
degree adds little to the quality of the adjustment. Therefore, we choose a 2-degree polynomial as the best fit to explain the probaility of collapse in the model in terms of the Beta-skew. Collapse= P(skew) Degree Coef.
1
Fit-estimates
P-value
0.0103665
0.00018184
0.0000000000000002
0.0119287
0.00018898
0.0000000000000002
0.6365 0.00000000000000022
2
0.00664250
0.000086206
0.0000000000000002
0.01192869
0.00008093
0.0000000000000002
0.00402343
0.000039975
0.0000000000000002
0.9334 0.00000000000000022
3
0.00664250
0.000085733
0.0000000000000002
0.01141738
0.000129326
0.0000000000000002
0.00402343
0.0000397563
0.0000000000000002
0.00010179
0.0000201477
0.0000000000000002
0.9341 0.00000000000000022 19
Table 2. Polinomial regressions for skew. Likewise, in Figure 4 we show the fit for a second-order polynomial with Beta distribution skew as the explanatory variable for the probability of collapse. As we already saw in the graphical analysis, Figure 4 shows that a higher skew (a more right-tailed Beta-distribution) produces a higher probability of collapse. This diagram also shows a strong and almost exponential increase in the probability of collapse as we move from skew values “0” upwards. Positive skew (right-tailed) shapes of the Beta-distribution clearly increase the probability of collapse because of what we attribute to be knowledge coordination problems. Whereas for negative skew (left-tailed) distributions the probability of collapse is null.
Figure 4. Second order polinomial fit (Collapse/skew).
What economic rationale can we infer from these computational results of the model? We have seen that when we depart from strictly convex (right-tailed and very close to the vertical-axis) Beta-distributions – as generative structure of firm-understanding radius in Sector 2 – then the probability mass is mostly concentrated on low radius. Therefore, it makes sense that the low adaptability and low (radius) cognitive20
understanding capacity of user-firms imply a higher emergent probability of collapse. Conversely, when we have peak-shaped and left-tailed Beta-distributions, with the probability mass concentrated on higher values for the entering understanding radius, we obtain lower probabilities of collapse. In terms of the Beta-skewness, the computational results show that negative skew (right-tailed) distributions lead to high probability of collapse. Conversely, positive skew (left-tailed) eliminates collapse. If we consider that strong right-tailedness in the Beta distribution may represent a certain weakness in the absorbtive capacity of new firms in Sector 2 (machine-user sector), any increase in this weakness will rapidly (non-linearly) increase the probability of technological overshooting and therefore the probability of intersectoral collapse and blockage of coevolution. We can connect the shape or skew of the Beta-distribution with the characteristics of the institutional framework that generates new firms in Sector 2 if we recall how we have defined the “firm-specific understanding radius” and its Beta-generative structure in Sector 2 (see section 2 above). This framework involves supporting institutions for the provision of specific skills by training (for instance Universities and technical institutes), but also domain-specific schools for users. Notice that the model suggests the need for increasing knowledge not only where producer-innovation takes place, but also at the level of the consumer good or user-sector. This need to promote absorptive capacity at the Sector 2 level to reduce blockchages at the Sector 1 level (i.e. to mitigate intersectoral knowledge coordination problems) is not usually a key target for innovation policy. Moreover, the role of the Beta-distribution in the model reveals, perhaps, not only a supporting creative role for institutions, but also a new connective role at the intersectoral level by new types of organizations, inter-professional associations, and linking institutions. 3.2 Absorptive capacity, R&D intensity, and innovation In our co-evolutionary model, firm specific R&D to profit ratios are the key behavioral variables to explain innovation efforts and technological change in Sector 1. Notice that the distribution of these firm-specific ratios
∑
at any time, and the average R&D ratio
,
in Sector 1 at t, are dynamic emergent
properties dependent on the overall functioning of the model. Following our exploration of the role of absorptive capacity, the next step is to consider whether we might detect regularities in the computational results of the model connecting R&D intensity (given by 21
∑
,
) in the upstream innovative sector
(Sector 1) and the Beta (a,b) generative distribution in Sector 2. The skewness of the distribution Beta (a,b) in Sector 2 is a good target to explain the emergent limit-stationary value of
∑
,
. Figure 5
shows a surprising result that we denote the slump effect in the model. This effect admits an interesting economic interpretation and suggests a new implication for innovation policy. Specifically Figure 5 illustrates a sigmoidal fit for the limit-stationary data
∑
,
emerging from our runs in Sector 1 as
we change the skew of the (Sector 2) Beta (a,b) distribution. Recall that negative skew means a left-tailed Beta distribution in Sector 2, and positive skew corresponds to a right-tiled Beta distribution in Sector 2. The model reaches stationarity in 5,000 steps.
Figure 5. Sigmoidal fit (Average R&D to profits ratio/skew). We can clearly see in Figure 5 how departing from intense negative skew (highly left-tailed) distributions in Sector 2 (in horizontal axis), we obtain in correspondence high average R&D to profits ratios (in Sector 1) in the limit-stationary situation of the dynamics. This means that high levels of
∑
,
in the
innovative (upstream) Sector 1 emerge not because of subsidies or favorable tax reductions of whatever in Sector 1, but rather because of a left-tailed pattern in Sector 2 – Beta (a,b), the generative structure in the user-sector from which new firms with specific absorptive capabilities (cognitive radius) stochastically are drawn and selected (pro or against) in bi-sectoral coevolution. Thus, we need left-tailed shapes in Beta (a,b) in Sector 2 for high R&D to profits ratios in Sector 1 (upstream) to emerge. Institutional structures 22
dense in probability around relatively high cognitive radius in the downstream sector are driving profitability in the upstream sector. As a policy model this means that to increase the return to R&D in Sector 1, we need to boost absorbtive capacity in Sector 2. ∑
The decreasing relationship explaining lower values for
,
in Sector 1 in terms of
increasing Beta (a,b) skew is clear in Figure 5. The higher the right-tailedness (positive skew) of the Beta generative structure in Sector 2, the lower the R&D intensity and propensity to innovate in Sector 1. Notice that the range of change in the emergent values for
∑
,
is wide: from around 13 percent of net
profits devoted to R&D (on average in Sector 1 as a stationary value) in the best cases, to two percent of profits to R&D on average in Sector 1 (as a stationary emergent property) under less innovative conditions. In Table 3 we observe the numerical results of the sigmoidal fit, depicted in Figure 5, shows a good statistical fit to the data when we use the inverted sigmoidal form (see Table 3). More precisely, the specific functional form that we have estimated for the stationary results of ̅ and skew is: ̅ where
,
,
,
are the coefficients to be estimated. It is a highly nonlinear
regression, so that we paint (in pink in Figure 5; 90% confidence interval) the interval in Figure 5 and we see that the estimation is a good fit. In Table 3 we see the numerical results for the coefficients, intervals and p-values. ̅ = F(Skew) Funtion
Coef.
Sigmoidal
Fit
P‐value
0.0546695 0.00091479
0.0000000000000002
3.0831232 0.1610722 0.0584723
0.11519132 0.03749395 0.00061389
0.0000000000000002 0.00000000000282 0.0000000000000002 0.008813
Table 3. Sigmoidal fit for the result in Figure 5. What is relevant in this result – shown in Figure 5 and Table 3 – is what we call the “slump effect” of R&D intensity. The relationship that emerges from the model linking R&D intensity in the innovative 23
Sector 1 and absorptive capacity in Sector 2 is nonlinear. Moreover, its inverted sigmoidal shape indicates that, as the generative structure in Sector 2 becomes less apt to create aborptive capacity in Sector 2 as Beta (a,b) becomes more righ-tailed, we observe slight initial reductions in ∑
however, we reach a point of skew from which
,
∑
,
. Eventually,
decreases sharply, so that from emergent
values of about 11 percent of profits to R&D, we quickly fall to values of five percent along the vertical axis. This is the slump effect. Absorptive capacity in the user-sector influences the R&D-to-profits emergent ratio in the innovative upstream sector. For policy, we could conclude that small transformations in the Beta (a,b) generative structure in Sector 2 may have a significant effect in increasing R&D intensity and technological change in the upstream sector (Sector 1).
4
Conclusions
We have proposed a computational two-sector model (along the lines of Dosi et al. 2013) in which Sector 1 is an innovative sector producing and selling machines, and Sector 2 is a (user) innovative sector that buys machines from Sector 1, produces consumption goods, and supplies goods to final consumers. The key point of our model is that the cognitive alignment of knowledge creation and absorbtion capabilities among the innovation trajectories of Sector 1 and Sector 2 is not straightforward. In more prosaic terms, it’s not necessarily best to have the smart people and innovative capabilities all working upstream. This will tend to cause overshooting, which will then backpropagate to collapse the upstream sector. This is a problem of balanced innovation economy. We have modeled this process in which firms producing machines in Sector 1 compete in price and machine-type (quality) performance to capture users. They fix prices following a mark-up pricing rule and they spend in R&D-innovation a proportion of profits with a lag. Firm performances in Sector 1 evolve by innovation. On the other side, firms in Sector 2 buy machines, fix prices, and produce varieties of a consumer good. Both sectors have entry/exit mechanisms. An interesting new feature of our model is that the user-firms in Sector 2 have a specific (endogenously-changing) knowledge endowment which allows them to understand (or not), and eventually choose among the set of machine-varieties or formal parts of it offered by Sector 1. Therefore, the dynamics linking both sectors depend on the emergent co-evolution of innovation and absorption activities taking place and developing in Sectors 1 and 2. 24
The computational results of our model highlight the importance of inter-sectoral absorptive capacity constraints in innovation linkages between the two sectors in the economy. Innovation in the upstream sector (Sector one) can be stimulated (but can also be slowed-down or even blocked) depending on the absorptive capacity of the downstream-user sector (Sector two). We found that not only the absorptive capacity of the user-Sector 2 but also the evolution of this sector in interaction with final consumers is crucial for the sustainability of activities in Sector 1 (upstream sector). Our model suggests a new policy framework that may help us to make sense of the widely observed low productivity performance of modern innovation policy as a consequence of sectorally unbalanced knowledge and the presence of frictions in intersectoral co-evolution. In our model, the innovation policy problem lies in aligning innovation and absorptive capacity in a two-sector non-linear stochastic complex framework. New roles for innovation policy are suggested, such as combining supporting and intersectoral connective institutions. In this regard, the slump effect, according to which improving the generative structures in Sector 2 could have sharp positive effects in stimulating R&D and innovation in the upstream sector, highlights a possibility for improving the performance of innovation policy in the near future.
25
5
References
Aghion P. and Griffith, R. (2005). Competition and Growth. MIT Press. Cambridge, MA. Almudi, I., Fatas-Villafranca, F. and Izquierdo, L. R. (2013). Industry dynamics, technological regimes and the role of demand. Journal of Evolutionary Economics, 21 (2), 345-375. Almudi, I., Fatas-Villafranca, F., Potts, J and Thomas, S. (2018). Absorptive capacity of demand in sports innovation. Economics of Innovation and New Technology, 27 (3), 1-15. Arrow, K.J. (1962). The economic implications of learning by doing. Review of Economic Studies, 29 (1), 155-173. Bloch, H. and Metcalfe, J. S. (2018). Innovation, creative destruction and price theory. Industrial and Coporate Change, 27(1), 1-13. Bush, V. (1945). Science the endless frontier. US Government Printing Office. Washington. Cohen, W.M. and Levinthal, D.A. (1990). Absorptive capacity: A new perspective on learning and innovation. Administrative Science Quarterly, 35 (1), 128-152. Chenery, H (1960). Patterns of industrial growth. American Economic Review, 50, 624–54 Dosi, G. and Grazzi, M. (2010). On the nature of technologies. Cambridge Journal of Economics, 34 (1), 173-184. Dosi, G. and Nelson, R.R. (2010). Technical change and industrial dynamics as an evolutionary processes. In Hall, B.H. and Rosenberg, N. (ed), Handbook of the Economics of Innovation. Elsevier, North-Holland, Amsterdam. Dosi, G. Marengo L. and Fagiolo (2005). Learning in evolutionary environments. In Dopfer K. (ed). The Evolutionary Foundations of Economics. Cambridge University Press. Cambridge. Dosi, G. Fagiolo, G. Napoletano, M. and Roventini, A. (2013). Income distribution, credit and fiscal policies in an agent-based Keynesian model. Journal of Economic Dynamics and Control, 37 (8), 15981625. Dosi, G. Pereira, M. and Virgillito, M.E. (2017). The footprint of evolutionary processses of learning and selection upon the statistical properties of industrial dynamics. Industrial and Corporate Change, 26 (2), 187-210. Earl, P. and Potts, J. (2013). The creative instability hypothesis. Journal of Cultural Economics, 37 (2), 153-173. Earl, P. and Potts, J. (2016). The management of creative vision and the economics of creative cycles. 26
Management and Decision Economics, 37 (7), 474-484. Etzkowitz, S. and Leydesdorff, L. (2000) The dynamics of innovation: from National Systems and “Mode 2” to Triple Helix of university–industry–government relations. Research Policy, 29(2), 109-23. Foray D., David P. and Hall B. (2009). Smart specialization: The concept. In Knowledge for Growth. European Comission. Brussels. Jaumotte, F. and Pain, N. (2005). From ideas to development. OECD Economics Dep. Working Paper. Doi: 10.1787/18151973. Kremer, M (1993). O-Ring theory of economic development. Quarterly Journal of Economics, 108 (3), 551-575. Martin, S. and Scott, J. (2000). The nature of innovation market failure and the design of public support for private innovation. Research Policy, 29(4), 437-447. Metcalfe, J.S. (2010). Technology and economic theory. Cambridge Journal of Economics, 34(1), 153171. Metcalfe, J.S., Foster, J. and Ramlogan, R. (2006). Adaptive Economic Growth. Cambridge Journal of Economics, 30, 7-32. Mokyr, J. (2016). A Culture of Growth: Origins of the Modern Economy. Princeton University Press: Princeton. Murmann, P. (2003) Knowledge and Competitive Advantage. Cambridge university Press: Cambridge. Nelson, R.R. (1959). The simple economics of basic scientific research. Journal of Political Economy, 77, 297-306. Nelson, R.R. (ed) (1962). The Rate and Direction of Inventive Activity. NBER. Princeton. Nelson, R.R. (1982). The Role of Knowledge in R&D Efficiency.Quarterly Journal of Economics, 97(3), 453-470. Nelson, R.R. (ed) (1993). National Innovation Systems. Oxford University Press. Oxford. Nelson, R.R. (2012). Some features of research by economists foreshadowed by The Rate and Direction of Inventive Activity. pp. 35-42 in Lerner and Stern (eds) (2012). The Rate and Direction of Inventive Activity Revisited. University of Chicago Press. Chicago. Nelson, R.R. and Winter, S.G. (1982). An Evolutionary Theory of Economic Change. Harvard University Press. Cambridge, MA. 27
Nelson, R. and Sampat, B. (2001). Making sense of institutions as a factor shaping economic performance. Journal of Economic Behavior and Organization, 44 (1), 31-54. Potts, J. (2000). The New Evolutionary Microeconomics. Edward Elgar. Cheltenham. Potts, J. (2016). ‘Innovation policy in a global economy’ Journal of Entrepreneurship and Public Policy 5(3): 308-24. Thomas S., Potts J. (2015) ‘Industry sustainability under technological evolution: a case study of the overshooting hypothesis in sports’ Procedia Engineering, 112, 562-7. Rodrik, D. (2004). Industrial policy for the 21st Century. Harvard University Working Papers Series. RWP04-047. Saviotti, P.P. and Pyka, A. (2013). The coevolution of innovation, demand and growth. Economics of Innovation and New Technology, 22 (5), 461-482. Simon, H. (1955) ‘A behavioural model of rational choice’ Quarterly Journal of Economics, 69 (1), 99118. Simon, H (1957). Models of Man. Wiley, New York. Simon, H. (1991) Bounded Rationality and Organizational Learning. Organization Science 2(1), 125– 134. Trajtenberg, M. (2012). Can the Nelson-Arrow paradigm still be the beacon of innovation policy? pp. 679684. In J. Lerner and S. Stern (eds) (2012). The Rate and Direction of Inventive Activity Revisited. University of Chicago Press. Chicago. Vives, X. (2001). Oligopoly Pricing. The MIT Press. Cambridge MA. Winter, S.G. (1984) Schumpeterian competition in alternative technological regimes. Journal of Economic Behavior and Organization, 5(3), 287–320. Zahra, S.A. and George, G. (2002). Absorptive capacity. Academy of Management Review, 27 (2), 185203.
28
6
Appendix: List of symbols and pseudocode.
We use subscript for any firm in Sector 1; for firms in Sector 2 ; we use
otherwise (generic index).
Notation regarding Sector 1 ,
: firm/company i from Sector 1 at time . : set of firms or companies operating in Sector 1 at .
,
0.5 1.5 0.01 0.5 0.75 0.05
~ ~
,
∈ 0,1 0 , , ,
0 0
1 0 , ∈ 0,1 0 ∈ 0,1 0 , 0 , ,
,
0,1 0,1
Parameters in Sector 1 and base-setting Price/performance sensitivity of demand Common parameter in pricing routine Unit production cost Relative importance imitation vs inner R&D. Entry cost for new imitative entrants Probability of entering doing innovation (identical in both sectors)
Firm-specific parameters in Sector 1 Share of profits devoted to R&D Radius delimiting perceived direct competitors
Firm-specific variables Sector 1 Technological level in relative terms Expected sales (in real terms = number of expected customers) R&D spending Expected total unit cost Unit profit mark-up on costs Price Firm knowledge Sales (in real units = number of customers) Market Share Total unit cost (ex post) Total firm profit
Notation regarding Sector 2 ,
: firm/company j in Sector 2 at .
29
: Set of firms in Sector 2 at .
,
Parameters in Sector 2 and base-setting Price/performance sensitivity of demand Probability of entry doing innovation (equal in both sectors) Common parameter in pricing routine Parameter beta-distribution Parameter beta-distribution
0.5 0.05 1.06 1 1
Firm-specific parameter Sector 2 Cognitive absorptive capacity (as a radius)
~Beta ,b
Firm-specific variables in Sector 2 Knowledge to manage machines
∈ 0,1
, ,
0
Cost of the machine
,
0
Machine quality
,
0
Price of the variety of consumption good
,
∈ 0,1
Consumption good firm-specific fitness (tradeoff quality/price)
,
∈ 0,1
Market Share
0
,
Firm profit
Aggregates
Number of firms in each sector:
Industrial concentration index (Herfindhal) in each sector:
and
. and
Parametric conditions when departing from the standard (base) setting
0
1;
1;
0;
0
1; 0;
0
0
1; 1; 30
.
1;
0;
0;
Pseudocode (Algorithm) 1. Initialize: ∅).
1.1. Sector 1 (machines production). Initially empty sector- no firms (
1.2. Sector 2 (consumption goods production). Initially empty sector- no firms (
∅).
1.3. END; 2. For any : 2.1. Call Entry_Sector1(); 2.2. Call Entry_Sector2(); 2.3. Call Operate _Sector1(); 2.4. Call Operate_Sector2(); 2.5. Call Apply_Replicator_Sector1(); 2.6. Call Apply_Replicator_Sector2(); 2.7. Call Exit_Sector1(); 2.8. Call Exit_Sector2(); 2.9. END; 3. END;
Define Entry_Sector1(): ∪
1. Entry Sector 1: One new firm enters in Sector 1 ( 2. With probability , or if the sector is empty, then ~
0,1 ;
~
0,1 ;
,
~
0,
,
if
1 otherwise;
exists, or
31
,
,
);
random initialization of traits:
3. If the new entrant copies, then: it copies firm share , , so that:
, with probability proportional to market
;
; ,
,
;
4. Normalize ∑
,
1. The new entrant affects sectoral technology levels;
5. END;
Define Entry_Sector2(): ∪
1. Entry Sector 2: A new firm j enters Sector 2 ( 2. Recalculate market shares:
0.005, ∑
,
1
,
,
,
); 0.995;
3. With probability , or if the sector is empty, random initialization of traits:, ~Beta ,b ; ,
0,
~
,
if
exists, or
1 otherwise;
4. If the new entrant copies, then: It copies firm in Sector 2, with a probability which is proportional to its market share , , and: ; ,
,
;
5. END;
Define Operate_Sector1(): 1. For each firm in Sector 1: 1.1. R&D Investment: 1.1.1. If it is a new imitative entrant: 1.1.2. otherwise:
,
,
,
;
1.2. Expected unit cost: 32
,
;
1.2.1. If it is a new imitative entrant:
,
,
,
1.2.2. If it is a new entrant but it does not imitate: 1.2.3. otherwise:
,
,
,
,
,
,
;
:
,
;
,
;
,
,
,
1.2.4. END; 1.3. Perception and delimitation of direct rivals ( ∑
1.4. Set Cournot mark-up:
∑
1.5. Pricing:
,
,
,
,
, ,
,
otherwise;
,
,
,
,
):
,
,
,
;
,
for new firms and
;
1.6. New knowledge at t: , ~ ., with " ." representing a (truncated) Pareto distribution, with supporting values 0, 1 and parameter (slope of density function). The higher the value of , the more probable it is that we get small knowledge improvements (small , ). We have in our model as being determined by: ∙
;
∙ , ,
, ,
, that is, assimilation of knowledge from the gap to the frontier;
, that is, knowledge obtained from inner R&D;
2. END;
Define Operate_Sector2(): 1. For each firm : 1.1. Re-scaling 0,
,
to be comparable with the values
, whereas the values
,
,
vary in 0,
, since values
,
range within
, since we have an additional firm
in the current period: ,
,
∙
;
1.2. Selection of understandable machines:
33
,
:
,
,
;
1.3. Buy a machine from with probability proportional to: (generation of demand for Sector 1 firms) 1
,
,
,
1
∑
;
,
,
;
, ,
,
;
,
,
;
1.4. Pricing:
,
;
,
,
1.5. Firm competitivenes (fitness) in the consumption goods market: ,
,
1
,
1
;
2. END;
Define Apply_ Replicator_Sector1(): 1. For each firm in Sector 1, update performance by:
,
, ,
,
̅, ̅
2. END;
Define Apply_Replicator_Sector2(): 1. For each in Sector 2, calculate its market share from the replicator equation: ,
, ,
,
̅,
̅
∑
,
,
;
2. END;
Define Exit_Sector1(): 1. For each firm in Sector 1: 1.1. Calculate ex post unit cost:
,
, ,
;
34
∑
,
,
;
1.2. Calculate profit:
,
,
1.3. Firm exists the market when
,
, ,
; 0;
∑ , 1. Note that firm exist, alters the relative values of 1.4. Normalize: ∑ , technological levels in the sector, both, in the current period, and in the next one; 1.5. Communicate to Sector 2 the re-scaling in the previous step; 2. END;
Define_Exit Sector2(): 1. Each firm
in Sector 2 exists the market when:
2. Normalize: ∑
,
1;
3. END;
35
,
0.005;
