competition
David J. Veenstra
August 21, 2013
Title Technological Enclaves and Technological Pluralism Subtitle The effects of imperfect competition
Author David Julian Veenstra
Student Number 10094423
Date 21 August 2013
Program Economics & Business
Specialization Economics
Faculty Faculty of Economics and Business
University University of Amsterdam
Supervisor dr. W.C.M. van Beers
II. Industrial districts . . . 5
III. Technological enclaves. . . 8
IV. Choice of technology . . . 9
V. Ising model and past applications . . . 11
VI. Dynamic model . . . 16
VII. Results . . . 19
VII.a Parametrization . . . 19
VII.b Results and robustness . . . 22
VIII. Conclusion and discussion . . . 27
Appendix A. Derivation transition probability . . . 28
Appendix B. Additional figures and tables. . . 29
Appendix C. Reproducibility . . . 32
Abstract
This paper builds upon the work of David et al. (1998), with the purpose of studying how imperfect competition affects technological enclaves, and to make their model falsifiable. Similar to their work, the Ising model from physics is used to model the industry. The model is analyzed by the use of simulations. The main predictions are that imperfect competition has a higher probability of creating industries with technological enclaves, when holding the industry size constant. However, imperfect competition also has a negative effect on the prob-ability. Because a smaller industry reduces the probability, and as im-perfect competition is associated with lower industry size. The model suggest that the net effect is negative, as the second effect is more pronounced than the second effect.
I. Introduction
The persistent co-existence of strongly dissimilar production technology within an industry, or within a branch or area of a given industry is called technologiaal pluralism. This concept should not be confused with technolog-ical diversity. Technologtechnolog-ical pluralism refers to the situation with dissimilar production technology, while the technological diversity refers to the situation with similar production technology. Even today technological homogeneity, the opposite of technological pluralism, is what is most commonly found in most industries (Barbiroli, 2000).
Traditionally technological pluralism has been seen as anv anomaly. Moreover, it did not fit into standard microeconomic theory (David et al., 1998; Griliches, 1957). Much progress has been made in explaining the oc-currence of technological pluralism. However, technological pluralism is often accompanied by the clustering of firms that use the same production tech-nology (David et al., 1998). An important cause of this empirical regularity is local Marshallian externalities, which are local factor externalities that put downward pressure on the relative price of factors that are extensively used in a given sector or district (Kelly and Hageman, 1999), and externalities that increase knowledge and technological diffusionj.
The simultaneous occurrence of technological pluralism and the clustering of firms with similar production technology have created distinct,industrial districts, whereby in each of these districts there exists aproduction tech-nology that is predominantly used. This type of industrial districts is also
called technological enclaves . Little attention has been given to occurrence of technological enclaves.
David et al. (1998) already have made a significant contribution to fill the gap in the existing literature. In their work they construct a model that is able to account for the existence of technological enclaves, using the Ising model from physics as a basis. However, their model is not without flaws. They have made several simplifying assumptions to keep the model as simple as possible. One of these assumptions is that the market is characterized by symmetric price taking firms. More contemporary, technology intensive industrial district are described more accurately by imperfect competition, Silicon Valley being the prime example. Extending the model in the direction of imperfect competition is an interesting avenue that could increase the heuristic value of the model.
The purpose of this paper is twofold. Firstly, it is to study how imper-fect competition afimper-fects the creating and evolution of technological enclaves. Secondly, it is to extend and modify the model of David et al. (1998) in such a way that it becomes possible to derive testable hypothesis from the model. Similar to their work, a dynamic version of the Ising model will be used to formulate a reduced-form model of an industry. This model will be used to run discrete-time simulations, which will serve to analyze how an indus-try evolves over time, and to analyze the creation of technological enclaves. Two forms of imperfect competition will be considered, namely a form that is comparable to a cournot competition and a form that is comparable to a stackelberg competition. The first is considered because cournot competition is the most standard form of imperfect competition. The second is considered to mimic the situation where a large firm pioneers an industry or a district. In addition, a competition that is similar to perfect competition will be used as well, and will serve as a baseline competition.
This paper is structured as follows. In section II the concept of industrial districts and marshallian externalities will be given a more elaborated de-scription, in addition a few possible reasons will be listed of why economist might have failed to notice the simultaneous occurrence of technological plu-ralism and clustering of technology. A few examples of technological enclaves will be illustrated, in section III. Section IV presents a simple model to gain some intuition of how a firm chooses their production technology, and how
1Technological enclaves are also used to refer to high-tech industrial district. For
they decide to switch to another technology. A brief introduction of the Ising model will be given in section V, moreover some relevant aspects and proper-ties of the Ising model will be highlighted, and some past applications of the model will be presented. Subsequently, in section VI the general equilibrium model will be formulated, as represented by the Ising model. Section VII discusses the specifications of the simulations, and the results are presented and discussed in the subsequent subsection. Lastly, the discussion and a summary of the conclusions will be presented in section VIII.
II. Industrial districts
Marshall (1890) and Weber and Friedrich (1962) go together with the first economists that recognized the benefits of the clustering of firms that performed similar business activities (Paci and Usai, 2000); these clusters are better known as industrial districts. Marshall noted that economics of scale could also exist outside of the firm at the level of the industry, if the firms are clustered in districts. Moreover these districts enjoyed other benefits that spatially diffused industries do not enjoy (Krugman, 1991).
First, the clustering of firms with similar characteristics results in a more robust labor market, that is beneficial to both employer and employee (Krug-man, 1991). For the employees it is beneficial if their skills can be employed in many firms in the district. This makes it easier to the employees to find work after being laid off. For the employer, it is beneficial as the labor oppor-tunities attract many workers with employable and specialized skills. Hence, the firms are less likely to suffer from shortage of labor.
Another benefit is that districts make it profitable for local firms to make specialized inputs, or to dedicate the production to the districts (Krugman, 1991). Meaning that the district has inputs that better suits their technology, and, or the cost of inputs might go down. And finally, the geographical clustering of firms has the benefit that it’s easy to share information, and would stimulate the diffusion of technology and knowledge.
From the above it is to be expected that as production increases, the strength of externalities should increase, as an increased production stimu-lates factor input demand.
On the demand-side there are also some other benefits. For districts where the distances to users are essential, there are Hoteling like benefits for being part of the district, as firms gain market share by moving closer to their competitors (Swann and Prevezer, 1996). Moreover, the clustering of
firms into one district reduces the search costs for the users, and facilitates market access (McCormick, 1998; Swann and Prevezer, 1996). This allows some firms to produce their good in quantity, and seems to be an important benefit for poor and low-tech districts (Ceglie and Dini, 1999; McCormick, 1998).
From the empirical work on industrial districts, a few aspects are impor-tant. More high-tech and, or, successful industrial districts have firm-size distributions that are inclined towards large and medium sized firms, while less successful and poor districts exist mostly out of small sized firms (Ceglie and Dini, 1999). Nonetheless, the more high tech and successful industrial district seem to be accompanied by a large set of small firms (Ceglie and Dini, 1999).
Another important aspect that has been observed is that industrial dis-trict often do not form spontaneously (Krugman, 1991). Frequently, it is due to some historical, albeit fairly arbitrary reasons. One interesting example is the tale of the carpet manufacturing industry in Dalton, U.S.A. A young teenage girl made an usual bedspread as a wedding gift, and as a direct conse-quence, a carpet manufacturing industry district was formed in Dalton, and later became the dominant one in the U.S.A. (Krugman, 1991). Historical events can create a platform for the creation of industrial districts. And as a small industrial district is formed, the production increases as there are more firms in the local, which strengthen the externalities, and this could lead to a further growth of the industrial district. A historical element is important in the creation and development of an industrial district, and this historical element will also be found in the dynamic formulation of the model.
One important trend that can be found in the 19th century is the glob-alization of the world economies. A few characteristics of the modern and global economy are: the vertical disintegration of the production, rapid trans-portation, and high speed communication. It might be expected that these things tend to decrease the importance of location. Nonetheless, there are numerous economists who argue the opposite, and some even speak of the re-emergence of regional economies (Martin and Sunley, 2003). The highly specialized industrial districts have strong capacities for innovation and adap-tation, which are essential in the global market (Martin and Sunley, 2003). The discussing whether globalization increases or decreases the importance of location remains incomplete.
Marshall recognized that these industrial districts were potential sources of economic growth and innovation. Nonetheless, the Economists of his era
showed little interest in the concept of industrial districts. During the 1990’s and the end of the 1980’s this changed. The success of various industrial districts in Italy, especially those in the footwear industry (Rabellotti, 1995) and the work of Italian economist on these districts, sparked a real interest amongst economist. The interest in industrial districts was also spurred by the theoretical work of Krugman (1991), and Arthur (1988) (Paci and Usai, 2000). They successfully formalized the external economies of scale, the marshallian externalities, and the clustering of firms. This produced a large and diverse body of literature on industrial districts. The work of David et al. (1998) is part of this literature.
In the past some economists have given an explanation for technical plu-ralism and in some cases they even gave an explanation for the occurrence of technological enclaves. However, they did not succeed to properly account of both concept, as is argued in the work of David et al. (1998). The auteurs list some reasons why this was the case; some of these reasons will be discussed below.
Before the 1950’s the topic of technological enclaves belonged to a small niche within the study of diffusion and technological innovations. In their ex-planatory approach this sub-specialty inherited the approach of sociologists. The explanation focused on the existence of significant inertial lags of human behaviors; diffusion of technology is met by a psychological resistance. And this resistance might form a barrier to the adoption of new technology.
Subsequently, in the 1960’s and 1970’s, the explanation of technical plu-ralism and enclaves was based on the hypothesis that an economy with ratio-nal economic agents can result in a variety of production techniques, if there are heterogeneities in the circumstances of the economic agents. This led to the concept of ‘appropriate technology’, different technology is appropriate under different circumstances. Hence, production technologies that were cre-ated in developed nations might be inappropriate for developing nations, as they have different factor market institutions and different resource endow-ments. But these technologies could still be a source of profit in districts that maintained an environment more similar to that of developed nations, by the support of government subsidies. This would suggest that technological plu-ralism and enclaves is not something that occurs naturally, but instead, is a product of government intervention2. However, this is at odds with what has
2This is also a reason why technological pluralism has been considered as an anomaly
been observed empirically. Technological enclaves have occurred without the influence of the government, and have done so in developed nations (Leunig, 2001; David et al., 1998; Pyke et al., 1990).
In location theory there is one explanation for technological enclaves that is straightforward, and intuitive. The preference of one technology in one district might be dictated by transportation costs or similar regional-specific factors. For example, in the North of the U.S.A, tractors were greatly pre-ferred to the traditional multi-horse teams, as the cold in the north shortened the window when the ground could be ploughed. This kind of explanation has been so intuitive that it seems to have deflected the interest away of the situations where technological enclaves cannot be explained by transporta-tion costs or similar regional-specific factors.
And finally, there is one other possible reason that is not mentioned by David et al. (1998). The broad literature of the 1980’s and 1990’s on indus-trial districts took a fairly aggregate view of the indusindus-trial districts. Papers that give a description of the technology that is being used in a given district are often general or abstract in nature3, obfuscating the possible patterns in
the use of technology that might be present. Often, it is not trivial to verify if technological enclaves exist in a given industry, or not. Moreover, in most literature it is unlikely that these patterns will be seen, unless one is actively looking for it.
III. Technological enclaves
Technological enclaves can be found in both developing and developed nations, in the present and in the past. In the mid-19th century in the U.S.A. there existed two dissimilar methods to farm corn (Griliches, 1957). There was a traditional method of farming corn seed by simply using non-specialized corn seed. The other method was a relatively new method that used hybridized corn seeds, which were specially bred for a specific location. The process of creating a specialized breed was long, but had superior results compared to the traditional method. Nonetheless, the spatial distribution of both methods were segregated, and remained to be so for a long period of time (Griliches, 1957).
3For example see the work of Cooke (2002); Zeller (2001); Rabellotti (1995). Papers
where a more detailed description is given, as in (McCormick, 1998) , or in Pyke et al. (1990), are the exception rather than the norm.
Another example of the past can be found the Lancashire textile-industry during the end of the 18th century and the beginning of the 19th century (Leunig, 2001). Mill-owners had two technologies available that could be used to spin course yarn of cotton: mule spinning, which required skilled labor, and ring spinning which required low skilled labor. The first technique was preferred in the towns of Oldham, Ashton, and Blackburn, while the later technique was preferred in the towns of Bolton and Rockdale. This spatial segregation persisted during the existence of the Lancashire textile-industry (Leunig, 2001).
More contemporary examples, but less specific, can be found in light manufacturing industry in northern Italy in the 1980’s (David et al., 1998; Pyke et al., 1990). Or in the Suami and Kasami districts of metal working in Ghana, where there are clusters using hand tool and clusters that use locally made machines (McCormick, 1998) .
The work of Paci and Usai (2000) shows that there are various high-tech and specialized industrial districts throughout Europe in the industries rang-ing from electronics to the auto manufacturrang-ing industry. Given that these industrial districts are highly specialized, and the existence of marshallian externalities, it is not unreasonable to assume that clustering of technol-ogy also can be found throughout Europe. There is also the suspicion that technological enclaves may be found in the industry of microelectronics and biotechnology (David et al., 1998).
IV. Choice of technology
To gain some understanding of a firm’s choice in technology, a simple model is presented here. For simplicity it is assumed that a firm in an indus-try can choose between two different technologies. All firms have standard convex production functions, and all firms are profit-maximizing firms. The production function of firm k using technology one is denoted by fl,k, and
is a function of input x and input y. The first technology, f1,k, is intensive
in the use of input x, while the other technology is intensive in the input of y. These inputs could be high and low skilled labor, as was the case in the textile industry of Lancashire (Leunig, 2001). To prevent any inherent bias towards a given technology, it is assumed that for a given firm the profitabil-ity of both technologies are similar, and that both technologies are affected similarly by marshallian externalities.
The profit function of firm k, using technology one, is given by:
πl,k = pkfl,k− C(fl,k, vl,k) (1)
Where pk denotes the firms inverse demand function, and C expresses the
cost function. As usual the firms maximize their profit by equating marginal revenue to marginal costs, and the maximized profit is denoted by πm
l,k. A
firm that operates in an industrial district is also affected by the marshal-lian externalities, which influences the profitability in two ways, as discussed in section II. Firstly, the labor market pooling, and the more specialized inputs/input producers tend to lower the costs. Secondly, the increased dif-fusion of knowledge and technology could also decrease the costs by produc-tion technology innovaproduc-tion, but it may also increase the revenue by product innovation. The externalities that affect the costs are denoted by vl,k.
In turn, the cost related externalities are affected by the related input that is intensively used in the immediate local. The increased demand in the inputs, increases the labor market pooling, and, or the profitability of special inputs/input producers. Therefore, v1,k is an increasing function of
the amount of input x that is used in the immediate local, and v2,k is a
function of the amount of input y that is used in the immediate local. For a given firm in industry, the decision to switch to another technology depends on the profit that the firms can achieve with the old technology and with the alternative technology. Given that firms within industrial districts are often rather specialized (Paci and Usai, 2000; Ceglie and Dini, 1999), it is likely that the firms incur some costs to switch to another technology. Considering that firms are assumed to be profit-maximizing, a firm would switch to the alternative technology if the difference in profit is positive, and larger than the switching cost. Hence, for a switching cost, Sk, the decision
to switch technology for firm k is governed by: if πm
1,k− π2,km > Sk switch to production technology 1 (2)
if πm
2,k− π1,km > Sk switch to production technology 2 (3)
The switching costs also create a range where firms do not switch technologies even if πm
a,k < πmb,k
Since the externalities affect the profitability, it also influences the de-cision to switch technology. If production in the immediate starts to use
technology 1 more predominantly, then v1,k is expected to increase, while
v2,k is expected to decrease, as technology 1 is intensive in input x. This
would increase |πm
1 − πm2 | and, hence, increases the likelihood that a firm
switches to the other technology, because both technologies are similar with respect to profitability and by how they are affected by diffusion. The im-portance here is that when one input is starting to be used more relatively to the other input, then the likelihood that firms switch to technology that is intensive in this input, increases. Thus, if a firm is considering switching to another technology it has to take into account what others in the local are doing. Moreover, if the firm switches it increases the probability that others do so as well. Cournot and stackelberg competition have in common that the optimal choice of production depends on what other firms are doing. In this respect, the simple model that is presented here has this aspect in common as well.
In abstraction, a general model would have many interaction agents that interact with neighboring agents. And as discussed in the previous chapter, history is important to the creation of industrial districts. In the past markov random field theory has been used in Economics to study models that have many interacting agents that interact with a set of neighboring agents, and in some cases to study models that are also history dependent. Similarly to the work of David et al. (1998), the Ising model from the markov random field theory will be used to construct the general equilibrium model.
V. Ising model and past applications
The Ising model is a model that originates from physics. The first work on the model was made by Ernst Ising. In the pursuit of his dissertation, he was given the task of explaining certain empirical facts about ferromagnetic materials (Niss, 2005). Various physicists have worked on the model, and extended it in various directions, and it is now better known as the markov random field theory. In the last century the Ising model has been one the most studied models from physics (Niss, 2005), and the model has been used in broad range of disciplines including economics (Kindermann and Snell, 1980).
Föllmer (1974) pioneered the Ising model in economics. Folmer noted that standard micro-economic theory considered the agents preferences as an a priori given. In reality a person’s preference may have a random element that is influenced by his surroundings. For example the mood of a person
can strongly influence his preference. To study an economy with agents that have random preferences, Folmer used the Ising model. He showed that the economy can almost never be stabilized by a price vector, if there are strong interdependence between preferences. Following Folmer, Allen (1982) used the markov random field theory to model technology diffusion with interdependent demand, and uncertainty in the quality of the innovation, and adoption rate of the adoption rate.
Another interesting application of the markov random field theory can be found in the work of Durlauf (1993). Various dominant theories in macro-economic theory predicted a convergence of per capita output across coun-tries. Nonetheless, this convergences was not (fully) found empirically, even when accounting for various types of microeconomic heterogeneity. Durlauf showed that increasing returns in human capital, and technological comple-mentarities could result in multiple eqilibria, some with high-output, and some with low-output. Moreover, he showed that expansion of the leading industry could push the economy to the high-output equilibrium.
In the simplest representation there are n points on a line, and each point has a spin or orientation, which can be either up or down, denoted by + and − respectively (Kindermann and Snell, 1980). The sample space, as denoted by Ω, exists of all the possible configuration ω of the n points. The spin function σj(ω), gives the spin of point j for configuration ω. Each
configuration is associated with an energy, which is calculated by: U (ω) = −J X pi,j∈P σi(ω)σj(ω) − mH X i∈ω σi(ω)
Here P denotes all the unique pairs of points that are only 1 unit apart. The first summation calculates the energy that is caused by the interactions of the pairs of points, and the summation is taken over all unique pairs. H represents an external magnetic field that might have influence on the system, m and I are a material specific constant.
The sign of J dictates if interaction tends to strengthen pairs that are align the same, or whether the interaction tends to strengthen pairs with opposite spins. The first happens when J is positive, the later happens when J is negative. For the rest of the paper it is assumed that J is positive, as the interaction in technological enclaves tends to reinforce neighboring firms with similar technology. Moreover,
when all spins are aligned the same, because in this case σi(ω)σj(ω) will
always evaluate to +1. The maximum energy occurs when the points in the unique pairs all have opposing spins, hence when the spins of the points on the line alternate in orientation.
Ising assigned a probability to a given state ω proportional to e−
1 kTU (ω)
where T is the temperature and k a universal constant. The Hence, the denominator, or the so called partition function, is given by:
Z =X
ω∈Ω
e− 1 kTU (ω)
Then the probability measure4 defined on sample space Ω is given by:
µ(ω) = e − 1
kTU (ω) Z
Variables such as temperature and the like are only mentioned here to do justice to the Ising model. For this paper only the strength of interaction is of interest, and the strength of interaction may be unique for each unique pair, as the size of the firms, and thus the production, may differ amongst interacting firms. Now assuming that J =, H = 0, and differing strength of interaction, the probability measure can be simplified to:
V (ω) = − X pi,j∈P βi,jσi(ω)σj(ω) (4) µ(ω) = e −V (ω) Z (5)
In this representation the parameter βi,j determines how strong the
in-teractions are between the points, and how strongly the energy is affected by the interactions. For the long-run behavior of the model, parameter βi,j is
of crucial importance. When this parameter is fixed and smaller than some
4A probability measure is a function that assigns a probability to every possible state
critical value, βc, the Ising model converges towards an invariant
probabil-ity distribution from any initial configuration state (Kindermann and Snell, 1980). Hence, the history of the system is of relative little importance, as it always converges to the same probability distribution. This changes when β > βc; a history is instilled into the model. How a system evolves over time
does not only depend on the initial configuration state, it also depends on the previous history of the system. In mathematics this is called nonergod-icity, and in an economic context this dependence on history is called path dependence
A value of β that is larger than βc, is a value that is relatively large,
meaning that the reinforcements of neighbors with similar spins is strong. Given a certain history, a system converges towards one of the extreme states, the two states where all points have the same spin. But it is also possible that the system converges towards a state that is a convex combination of both extreme states (Kindermann and Snell, 1980). In the contest of this paper this means that a system can converge towards a state where all firms adopt the same technology, or that it converges towards a state with technological enclaves.
The story with a varying β is similar. The cases with non-ergodicity occurs for any value of β that is larger than βc. When the minimum value of
βi,j is larger than βc, all βi,j are larger. And by setting the minimum value
larger than βc, the occurrence of non-ergodicity can be forced.
The Ising model is spatially not restricted to one dimension. Various scientists have extended the spatial dimension of the model to use graphs, two dimensional lattices, three dimensional lattices, and more (Kindermann and Snell, 1980). For this paper the two dimensional and finite lattice with a periodic boundary is of interest. A two dimensional lattice could represent the landscape of an industry, and every location could house a firm. The orientation of the spin could denote the technology that a firm is currently using.
Figure 1 demonstrates a 5 by 5 lattice, with the periodic boundary. The neighbors of the point located on the lower left corner are enclosed by a square and are highlighteed with a light grey background. Lattice with these kinds of properties are important, because several problems can be solved analytically, and some important properties have been calculated in the past. Moreover, it prevents boundary effects, and the strength of interactions are in potential the same on every site.
+
-
+
-
+
-
-
+
-
+
+
+
+
-
+
-
+
+
-
+
+
-
-
-
+
Figure 1: Lattice with the periodic boundary
finite lattices that are subject to the periodic boundary principle (David et al., 1998; Kindermann and Snell, 1980). Hence, standard statistical tools can not be used to describe and analyse such lattices. In simulations of stochastic processes, this problem can often be solved by the use of macro-replications. If the parameter of interests does converge towards some constant, then this parameter can be approximated by repeatedly rerunning the same simulation, until the estimate is accurate enough. In this paper macro-replications will be used to give basic description of industries, and to analyse how the probability of technological enclaves differ between the forms of competition.
Similar to the work of David et al. (1998) the Glauber statistic will be used to formulate the dynamic version of the model. Let ¯ωi denote the state
where the spin of site ωi is flipped. Then the transition probability, the
probability that site ωi flips its spin, is given by:
p(ωi) =
µ(¯ωi)
µ(¯ωi) + µ(ω) (6)
This equation can be simplified by using eqs. (4) and (5), and by using the fact that the only different between V (¯ω)i and V (ωi) is given by the energy
created by site ωi and its neighbors . p(ωi) = 1 1 + e−2Vη(ω) (7) Vη(ω) = − X ωj∈ηi βi,jσi(ω)σj(ω) (8)
Where ηi denotes the set of neighbors of site ωi, and the summation runs
over the neighbors. The transition probability, given by (7), will be the basis for the model presented in the following section.
VI. Dynamic model
Suppose an industry with a single product that has n ∗ n firms, repre-sented by a two dimensional lattice that is subject to the periodic boundary principle. The lattice has n ∗ n = N sites, each site can house a firm, and each site has a neighbor in the four cardinal directions, including the sites on the border due to the periodic boundary principle. Each firm has the option between two different technologies, and as in section IV the technologies use up two different input factors and both differ in what input is intensively used.
Moreover, each firm may differ in size. Three different types of competi-tion are considered. Two of these forms will have asymmetric firms, and the size-distribution of the firms is modeled using
s = M
rα (9)
where s is the size, r denotes the rank, one being the largest, and both M and α are some constants. The distribution as described by (9), has often been found empirically (Martin, 2010), and this distribution has been found in high-tech sectors in the U.S.A. (Zhang, 2003). To reduce the number of parameters, (9) will be normalized. As the number of firms in the industry is known, the distribution can be normalized by subtracting the minimum
5Note that the partition function cancels outs. The summation of Z runs over all states
in Ω. Every site can be either + or −, and if there are N possible sites, then a total of
2N states exists. With 100 sites that would result in more that $1030 states. Not having
to calculate Z greatly reduces the computational intensity of the simulations.
size, and by dividing by the difference between the maximum and minimum size. srel = M rα − M Nα M 1 − M Nα srel = N r α − 1 Nα− 1 (10) 0.1 0.5 1 1.5 2.11 0 .2 .4 .6 .8 1 Relative Size 0 25 50 75 100 125 150 175 200 225 Rank
Figure 2: Firm-Size Distribution
Figure 2 displays Srel for different values of α. The most outer, solid
and black line, notes the distribution for an α of 0.1. As α increase the distribution bends inwards, hence, as α increases the number of large firm decreases. A value of 2.11 for α is what is found in high-tech industries in the U.S.A. (Zhang, 2003), and is given by the most inner line. From the graph can also be seen that the relative size quickly goes to zero as the rank increases.
Let the state of the industry be denoted by ω, the technologies be denoted by t ∈ {+, −}, and the size of firm j at site ωj be denoted by sj(ω). A firm’s
decision to switch to another technology depends on what other firms in the local are doing, and is governed by:
p(ωi) =
1
e−2Vηi(ω) (11)
ηi denotes the set of neighbors of site $ωj, andVηi is given by:
Vηi(ω) = −
X
ωj∈ηi
βi,j(si(ω), sj(ω), Si) σi(ω)σj(ω) (12)
Now the spin function σi can be seen as the ‘technology’ function, that
retrieves the technology that is being used at site ωi. The parameter that
determines the strength of the interactions is now a function of the size of both firms. Similarly to section IV, the strength of the externalities depends on the size of production. Given the same industry, it is expected that a firm will produce more if it is large, compared to the situation that the firm were small. Hence, the function βi,j is increasing in the size of both firms.
Standard economic theory predicts that total production will be lower if an industry has imperfect competition, compared to the situation with per-fect competition. Hence, the strength of the externalities should be lower in the situation with imperfect competition. In this model this would result in β’s that in general are lower. The Ising model in this context inherently has one element of imperfect competition; the firms make their decision based on what other firms are doing. Combining this with output restriction, leads to two forms of competition that are similar to cournot and stackelberg compe-tition.
However, the Ising model is not able to model a true cournot and stackel-berg competition. When the market condition change and firms re-evaluate their production schedule, the quantity they produce should change. In the Ising model this should be reflected by changing values of the β’s, but in the Ising model there is no way to explicitly calculate by how much the β’s ought to change. A more sophisticated model, like the Nelson-Wilson model (Nel-son, 1982), would be able to account for this. But that would greatly increase the number of assumption needed, and the number of parameters that have to be chosen. Within the Ising model it is unlikely that the changed market condition would cause a small firm to suddenly produce as if it were a large firm, hence this flaw is unlikely to drive the results. The benefits of a more sophisticated model are likely to be outweighed by the increased parameter
sensitivity. That is why the relatively simple Ising model is preferred. The relatively simplicity of the Ising model results in only five parameters and one function that have to be tuned:
• a base-line value for the β’s, influencing how large the externalities will be in general;
• the βi,j function that determines how the β’s should be influenced by
the size of a firm:
• θ, the parameter that decides the fraction of the largest firms, that gets the first choice in the stackelberg-like competition;
• n, the parameter that describes the number of firms in the district; • α, the parameter that governs the normalized distribution of the size
of the firms.
The relative simplicity of the model makes the comparisons between the different forms of competition and the robustness-checks relatively easy.
VII. Results VII.a Parametrization
All simulations have some parameters in common. To start off it is de-sirable for the simulation to be able to result in a technologically uniform industry, or in an industry with technological enclaves. To achieve this, all simulations will have β’s that have a value that are higher than βc. For the
type of Ising model that is used here, βc = 0.88 (Kindermann and Snell,
1980).
The second parameter that simulation have in common, is the one that governs the size of the industry. This parameter should strike a balance between simulation performance and industry size. With a value of n = 15, the industry exist out of 225 firms. This creates industries of decent size, while not being to computational intensive. Values of n = 12, n = 18 will also be considered.
Another common parameter is the number of interactions between eco-nomic agents that the simulation will iterate over. After multiple runs of the simulation model with a different number of interactions, it turns out that
convergence sets in at 10000 interactions per run. This is enough interac-tions for the industry to settle in a meta-stable configuration, for a 15 by 15 industry. For the smaller 12 by 12 industry 5000 interactions is enough, and for a large 18 by 18, 14000 interactions is enough.
Lastly, the cournot-like and stackelberg-line competition have the βi,j
function in common. This function should be as simple as possible, it should reflect the size of both firms, and it should be easy to control the range of the function. For these reasons, it was chosen to have:
βi,j(ω) = βmin+ c ×
sj(ω) + si(ω)
2
With βmin it is possible to ensure the case of non-ergodicity. The constant c
also profides an absolute roof on the range of βi,j. In addition a small and
large firm should produce a quantity that is similar to the production of two medium sized firms, hence the average is taken of the size of both firms. Given these common parameter, the parametrization of the simulation type with symmetric price taking firms, or perfect-like in short, is relatively easy. The only parameter is the β that all interactions will create. From the previous section it becomes clear that in perfect-like competition the β’s, in general, are higher compared to an industry with imperfect competition. Therefor the value of the β’s should be considerable higher than βc, but should leave
some room for the forms of imperfect competitions to have β’s that are lower in general, but still are larger than 0.89. To fulfill these requirements, βperf ect = 1.2 was chosen.
For the cournot-like competition, the value of c and α have to be parametrized. The parametrization of α is a bit problematic for both forms of imperfect competition. Despite the fact that a single technology switch can ripple throughout the whole industry, which is a result of the periodic boundary principle, the power of a large firm is limited. A large firm can only directly exert its influence on its four neighbors. Choosing a value of α that is too high will result in very few large firms, and given their limit reach, the resulting competition will be strongly similar to perfect-like competition. A value of α = 0.75 is seen as the limit for this model. Looking at fig. 2, a higher value would result in less than 13 firms that are less than eight times as small as the largest firm, and thus less than 13 firms that have at least some influence.
The values of α and c should result in β’s that are lower in general, compared to the perfect-like competition, while the β for interacting pairs
with two large firms should be higher. This achieved by setting c = 3.5, and setting α = 0.5.
Table 1: Cournot Statistics Statistic Mean Median Se. General 1.0588 0.9938 0.0064 25nth 1.4149 1.3978 0.0032
Table 1 shows the average, median and standard deviation of the β’s for 1000 initialized industries. From the table it can be concluded that the β’s are lower in general. The second row of Table 1 gives the descriptive statistics for the 25th largest β. The mean and median for the 24th largest beta should
be comparable, or more likely, higher. Hence, it is safe to assume that the β’s in general are larger for the interactions that involve larger firms, compared to perfect-like competition. Thus, a value of c = 3.5 and α = 0.75 gives the cournot-like competition the envisioned properties.
For the stackelberg-like competition, the value of α, θ, and c have to be parameterized. And as discussed in previous section, the stackelberg-like competition ought to have a higher α, than the cournot-like competition. This creates a narrow range for valid values of α.
Similarly, choosing a value of θ that is too low will result in too few domi-nant firms, and the competition will be strongly similar to that of cournot-like competition with the same value of α, in this case the stackelberg-like com-petition behaves exactly the same as the cournot-like comcom-petition, with the exception of the rare dominant firm. Choosing a value that is too large, on the other hand, would be unrealistic, as the stackelberg-like competition should mirror the industries that were pioneered by few firms.
Table 2: Stackelberg Statistics Statistic Mean Median Se. General 1.0512 0.9515 0.0085 13nth 1.8481 1.8140 0.0074
In addition the values of the different parameters should result in β’s that are lower in general, compared to the perfect-like competition, while the β’s involving the dominant firms should be considerably higher. This
is accomplished with the values: c = 3.5, α = 0.75, θ = 0.1. Table 2 demonstrates that the β’s are lower in general, but that the largest β’s are considerably higher than those of perfect-like competition, for 1000 initialized industries.
VII.b Results and robustness
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Replications 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Probabilit y Encla ved
Perfect Cournot Stackelberg
Figure 3: Macro-Replication
Figure 3 presents the results of the macro-replication, using the configu-ration that was described in the previous sub-section. As always caution is required, when interpreting the results of a simulation. Nonetheless, some confidence can be gained from the results. The probability that an industry results in an enclaved industry, does converge for the three forms of compe-tition, and seem to be stable after about 3000 replications. In addition, the number of digits that do not vary seem to increase as more replications are performed.
The figure shows an ordering of the p(enclave) probabilities, where the forms of imperfect competition have the highest probability, and the
perfect-line competition has the lowest probability. This ordering does not alter after the probabilities stabilize. This suggests, at least in the model presented here, that technological enclaves occur more often under cournot-like and stackelberg-like competition.
The different parameters for the different forms of competition are highly interconnected. The settings that were used, hit a narrow optimum, and changing one parameter, in most cases, results in the simulations losing their desired properties. For instance, decreasing the value of βperf ect below 1.2,
leaves very little room for the β’s in the forms of imperfect competition to be lower in general, as all β’s have to be larger than 0.88 to ensure the case of non-ergodicity. Increasing it on the other hand, leaves little room for the large, and, or, dominant firms to have β’s that are larger in general. Similarly increasing the βmin, decreasing the α’s, or increasing the values of c for both
forms of imperfect competition, leaves little room for them to have lower β’s in general. As discussed previously, there is an upper limit for the value of α, and the value of θ for the stackelberg-like competition has a narrow band. Changing the value θ does not alter the results in a significant way.
The parameter that can be changed, without the simulation losing their desired properties, is the size of the industry. To see if the results hold if the size of the industry is changed, two additional macro-replications have been performed. The first has small industries that have 144 firms, a decrease of 36%, and the second with large industries that has 324 firms, an increase of 44% percent. The values of the other parameters have been chosen us-ing the same logic that was usus-ing in the previous sub-section. Interestus-ingly enough, the optimum value for the other parameters are the same as the values mentioned in the previous sub-section.
Table 3 gives some descriptive statistics of the distribution of the β’s at the initialization of an industry, by initializing 10000 industries. The table shows that the β’s are lower in general, while the β’s are larger in general for the large firms. Note the general mean is close to 1.2 for both forms of competition. Nonetheless, a more optimal configuration could not be found. Figure 4 shows the results of the two macro-replications with varying industry size. The first thing that is noticeable, is that the smallest industries have the lowest probability, and that the largest industries have the highest probability. The magnitude of change is greater when varying the size of the industry, than when varying the forms of competition. The difference between the smallest and largest industry ranges between 10-25%.
Table 3: Cournot and Stackelberg Statistics for Small and Large Industry Cournot Small
Statistic Mean Median Se. General 1.0988 1.0216 0.0074 25nth 1.4038 1.3918 0.0031
Cournot Large
Statistic Mean Median Se. General 1.0318 0.9761 0.0056 25nth 1.4173 1.4011 0.0030 Stackelberg Small
Statistic Mean Median Se. General 1.1012 0.9765 0.0102 13nth 1.8572 1.8171 0.0076
Stackelberg Large
Statistic Mean Median Se. General 1.0318 0.9761 0.0056 25nth 1.4173 1.4011 0.0030
imperfect competition is small, but the difference is persistent, and again im-perfect competition has a higher probability. The difference between cournot-like and stackelberg-cournot-like competition does disappear. In the larger industries only the probability of the cournot-like competition is persistently higher, than that of perfect-like competition. The difference in probability between stackelberg-like and perfect-like competition, on the other hand, is not per-sistent, and the two probabilities seem to converge to each other.
In summary the probability of technological enclave is persistently higher for cournot-like competition, compared to the situation with perfect-like com-petition. Stackelberg-like competition has a probability that is at least as large as that of perfect competition. And, a stable pattern between the cournot-like and stackelberg-like competition does not seem to exist.
One important difference between the forms of imperfect competition, and perfect competition, is that the former has asymmetric firms, and it has large firms. This implies that imperfect competition has a higher concentra-tion of market power. High market power concentraconcentra-tion has been connected to having a smaller industry size (Martin, 2010; Mueller and Rogers, 1980), and the work of Ceglie and Dini (1999) also seem to indicate that this is also the case for industrial districts. Hence the conclusion should be interpreted conditionally. Holding constant the size of the industry, the model predicts that, on average, an enclaved industry is more common under imperfect com-petition. But imperfect competition may also have let to a smaller industry, which decreases the probability. Whether the net-effect is positive or
nega-tive depends on how much smaller the industry is. Nonetheless, the model is suggestive of the net-effect being negative as the difference in probability is far larger when varying the size of the industry.
Figure 4: Macro-replications 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Replications 0.28 0.29 0.30 0.31 0.32 0.33 0.34 Probabilit y Encla ved
Perfect Cournot Stackelberg
(a) Small Industry
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Replications 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 Probabilit y Encla ved
Perfect Cournot Stackelberg
VIII. Conclusion and discussion
This paper sets out to study how imperfect competition affects the cre-ation of technological enclaves, and to extend and modify the model of David et al. (1998) in such a way that it becomes possible to derive testable hy-pothesis from the model. Similar to their work, the Ising model is used to model an industry.
In their model they only use a form of competition that is similar to perfect competition. In this paper two other forms of competition are intro-duced. The consideration of what other firms are doing is inherent in the dynamic model, and together with the introduction of large firms, and dom-inant firms that are the first to act, result in forms of competition that are similar to cournot and stackelberg competition. The introduction of these forms of competition creates different scenarios, and thus creates the possi-bilities to make comparisons, which may yield testable hypotheses.
The central limit theorem does not apply to the kind of Ising model that was used here. Macro-replications have been used to overcome this problem, and to compare the three forms of competition. The predictions are condi-tional. Holding constant the size of the industry, the cournot-like competition should have a higher probability of creating enclaved industries, compared to the case of perfect-like competition. And stackelberg-competition should have a probability that is at least as high as the case of perfect-like com-petition. However the smaller the industry, the lower is the probability of an enclaved industry. Imperfect competition has a higher concentration of market power. A negative correlation between market power and industry size have been often found. Thus there are two effects, one that increases the probability, and one that the reduces the probability. The model presented here suggests that the industry-size effect dominates, as the industry size seems to be of greater effect.
The predictions are a bit ambiguous, but nonetheless they can still be falsified. Finding out that smaller sized industries with perfect-like compe-tition have, in general, a higher probability of creating enclaved industries, compared to larger sized industries with imperfect competition, would be a clear rejection of the model. Similarly, finding out that perfect competition has a higher probability, in general when controlling for industry size, com-pared to imperfect competition, would also be a rejection of the model. And other similar scenarios exist.
firms can only directly exert its influence on its four neighbors, this limit the influence that a large firm has. As a result the size distribution of the firms cannot take on values of α that are observed empirically, as this would reduce the forms of imperfect competition to the form of perfect-like competition. Secondly, it should be remembered that this was mostly a theoretical exercise. The question whether the model is accurate or not remains to be answered in a future study.
A. Derivation transition probability
The equation (7) can be derived by pluggin in (5) in (6), and using (4). p(ωi) = e−V (¯ωi) e−V (¯ωi)+ e−V (ωi) p(ωi) = 1 1 + eV (¯ωi)−V ω
The difference between V (ω) and V ( ¯ωi)is only determined by the unique
pairs that contain ωi. Hence, the transition probability can be simplified
further to: p(ωi) = 1 1 + e − X ωj∈ηi βi,j[σi(¯ω)σj(ω) − σi(ω)σj(ω)]
Where ηi denotes the set of neighbors of site ωi. Moreover, since every site
has four neighbors the set of products of σi(ω)σj(ω), for ωj ∈ ηi, can be
denoted by a set of four + or ˘. Inversing a spin is done by multiplying the spin with −1, hence σi(¯ωi) = −σi(ω), and the set of the products of
σi(¯ω)σj(ω), for ωj ∈ ηi, is simply the negative of S or S0 = {−s|s ∈ S}.
Therefor Vηi(ω) = − X ωj∈ηi βi,jσi(ω)σj(ω) − X ωj∈ηi βi,jσi(¯ω)σj(ω) = −Vηi(ω)
p(ωi) =
1 1 + e−2Vηi(ωi)
B. Additional figures and tables
Table 4 gives the descriptive statistics for 1000 initialized 15 by 15 indus-tries. The same settings where used as in section VII, but the α of cournot-like competition was decreased to 0.25, for stackelberg-cournot-like competition the α was reduced to 0.5. The table shows that the general mean is close to 1.2, hence it can’t be really said that β’s are lower in general for the forms of imperfect competition.
Table 4: Macro-Replication with decreased α Simulation Statistic Mean Median Se. Cournot General 1.1827 1.1080 0.0081 Cournot 25nth 1.6770 1.6670 0.0032
Stackelberg General 1.1479 1.0303 0.0099 Stackelberg 13nth 2.0751 2.0101 0.0061
In Figure 5 the same settings where used as in section VII, but with varying θ. Varying the θ does not change the results in a significant way.
Similarly, increasing the value of βperf ect above 1.2 does not changes the
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Replications 0.0 0.1 0.2 0.3 0.4 0.5 Probabilit y Encla ved θ = 0.01 θ = 0.05 θ = 0.1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Replications 0.0 0.2 0.4 0.6 0.8 1.0 Probabilit y Encla ved β = 0.9 β = 1.2 β = 1.5 β = 1.8
C. Reproducibility
The simulations of this paper were implemented in the language python, because python allows for quick development and debugging (Bilina and Law-ford, 2012), and because it has a mature and robust statistical library.
To make this paper as reproducible as possible, the Emacs org-mode was used to produce this paper. Org-mode allows one to seemingly integrate the evaluation of code with LATEX. In the pdf version of this paper, the full org
and python source code is provided in a zip file . If any datafile is missing upon the exportation of the org file to pdf, then the missing datafiles will be recreated by redoing the necessary simulations. The zip that is provided does not contain any datafile, hence it will redo every simulation upon exportation. Note that performing all the simulations takes time. Depending on the speed of the computer, this may take up a day or a few days to complete.
The scipy stack is required to compile the org file. It is also required that the python shell command points toward to python version 3+.
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