Thomas Brenner (2001)
Simulating the Evolution of Localised Industrial Clusters - An Identification of the Basic Mechanisms
Journal of Artificial Societies and Social Simulation
vol. 4, no. 3,
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Received: 13-Oct-00 Accepted: 29-May-01 Published: 30-Jun-01
Table 1: Influence of several local circumstances on the state of firms | |||||||
human capital | spillovers | co-operation | synergies | public opinion | local policies | Venture Capital | |
innovation | a qualified labour force is more innovative | create innovation and increases diffusion | joint R&D leads to more innovation | - | - | - | - |
productivity | qualified labour is more productive | - | joint projects increase productivity | mutually increase productivity | - | support for firms increases their producti-vity | - |
start-up | qualified people are more likely to found a firm | - | - | - | a positive expectation increases the number of start-ups | - | start-ups require venture capital |
(2.1) |
where kq denotes the share of experienced labour force (or human capital) that is available in the region; au (0 < au < 1), the productivity of inexperienced labour divided by the productivity of experienced labour; Cp,qn(t) the co-operative activities; Mqn(t) the mutual profits of firms in the region; Pqn(t) the political support (the dynamics of these aspects will be discussed below); and α-βLn(t) determines whether there are economies of scale. α < 1 is assumed, so that there are never diseconomies of scales for very small sizes of firms. However, if firms increase in size diseconomies might develop (dependent on the value of β). This reflects the fact that the dependence of the production costs on the size of a firm is usually found to be s-shaped.
(2.2) |
v (>0) is a parameter that determines the increase of the innovativeness of firms due to a higher human capital. s (>0) is a parameter that denotes the strength of the influence that spillovers from other firms have on the innovativeness of a firm. Furthermore, spillovers are assumed to be highest if the technological gap between the technology of the firm and the most advanced technology Tmax(t) used by some other firm equals G (> 0). Firms are assumed to profit most from spillovers if they are technologically behind by an amount of G. If the quotient Tmax(t)/Tn(t) is less or more than G the effect of spillovers decreases. gu (>0) determines how much the spillovers decrease for other technological gaps (this aspect of the modelling is taken from Caniëls 1999). In addition, the effect of spillovers decreases with the spatial distance δñn between the firms. η (>0) determines the strength of this effect. The distance between two firms is given by
δñn={[(xqñ - xqn)2 + (yqñ - yqn) 2]} |
(2.3) |
where xq denotes the x-coordinate of region q and yq its y-coordinate (coordinates are taken in the middle of a region). In the simulations the regions are assumed to be located as given in Figure 1.
Figure 1. Location of the 99 regions |
Tn(t+1) = (1+γ) Tn(t) |
(2.4) |
(2.5) |
where Lq(t) denotes the employment of all firms in region q. The dynamics of Kq(t), Vq(t) and Fq(t) are given below. The technology used by a start-up firm equals the average technology used by all existing firms. The initial technology of the first firm is set to 1. The initial number of employment is randomly determined. It ranges between zero and Linit.
(2.6) |
where φ (>0) is a parameter. Spin-offs are often located near to the firm in which the founder has worked before. In this approach it is assumed that the likelihood for the spin-off firm to be located in a certain region decreases exponentially with the distance from the originating firm. The probability for the spin-off firm ñ to be located in region q is given by
(2.7) |
where δqqn is defined analogously to δnñ (see Equation (2.3)) and ςς(>0) is a parameter that characterises the geographical stickiness of spin-offs.
(2.8) |
where D (>0), b (>0) and ρ (>0) are parameters. For a constant wage and interest rate, the production costs are proportional to the inverse productivity. Assuming that firms use markup-pricing, their price is also proportional to the inverse productivity. Thus, the productivity might be used in the demand function as it is done in Equation (2.8). The last term on the right-hand side of Equation (2.8) represents the impact that other firms have on the demand faced by firm n. Yñ(t) denotes the sales of firm ñ at time t. These sales reduce the demand for the products of firm n by ρρ Yñ(t) . Thus, ρ denotes the heterogeneity of the goods. For ρ = 0 the goods are completely different and the demand for the product of one firm does not depend on the behaviour of other firms. If ρ = 1, the goods are identical and an increase of sales by one firm does automatically mean a decrease by the same amount of sales of other firms. All firms are assumed to supply the same market, so that the location of firms does not influence the demand for its products.
(2.9) |
(2.10) |
where N(t) is the number of firms at time t. This equation looks quite similar to the demand function that is usually applied in oligopoly theory (one might redefine the parameters such that the equation looks more convenient). The only real difference is the dependence on the number of firms N(t) which has to be explicitly considered here because this number changes over time endogenously. In the simulations Equation (2.8) is used to calculate the demand for each firm.
(2.11) |
where κq (>0) is a parameter that denotes the speed of the accumulation of human capital.
(2.12) |
where Iq(t) is the opinion in the population of region q with respect to the industry. The dynamics of Iq(t) are given below.
(2.13) |
(2.14) |
(2.15) |
φ (>0) is a parameter that determines the speed of learning about the advantages of co-operation.
(2.16) |
where µ (>0) represents the overall effect of synergies and χ (>0) determines the local stickiness of synergies.
(2.17) |
where fm (0<fm<1) is a parameter that determines the decay of the memory about founding and close down events. The opinion Fq(t) of the local population with respect to founding a firm influences the number of start-ups and spin-offs as modelled in the Equations (2.5) and (2.6).
(2.18) |
ja,q, je,q and jp,q are parameters of the region q. For small numbers of employees the dependence has a quadratic form. This reflects the aspect that industries are only recognised if they reach a certain level of employment. For larger numbers the opinion levels off, meaning that it is not able to increase above je,q which denotes the maximal effect of this aspect. Furthermore, the public opinion is assumed not to reach this value immediately, but to slowly develop in the direction of this value. The opinion about an industry influences the maximal human capital in a region according to Equation (2.12).
(2.19) |
where Lpol (>0) is a parameter. Pq(t) influences the productivity of all firms of the industry in region q according to Equation (2.1).
(2.20) |
where v (>0) is a parameter and Nq(t) is the number of firms in the considered industry in region q. If the number of firms is very high, the value of Vq(t) converges to one. If, instead, no firm of the considered industry is located in region q, the value of Vq(t) converges to Vinit. The speed with which the availability of venture capital adapts to a new situation is given by v.
Table 2: The ranges for all parameters of the model | ||||||
Parameter | lower bound | upper bound | parameter | lower bound | upper bound | |
Linit | 2 | 30 | g | 0 | 10 | |
α | 1 | 1.3 | η | 0 | 4 | |
β | 0 | 0.01 | cc | 0 | 0.1 | |
D | 10000 | 500000 | φφ | 0.00001 | 0.1 | |
B | 0 | D | pmax,c | 0.00001 | 0.1 | |
ρ | 0.7 | 1 | ca | 0.0003 | 0.02 | |
λ | 0.00001 | 0.003 | ac | 0.00001 | 0.01 | |
εq | 0.00003 | 0.05 | mc | 0.00001 | 0.01 | |
φ | 0.000003 | 10/D | µ | 0.00001 | 0.01 | |
ς | 0 | 4 | χ | 0 | 4 | |
Kinit | 0 | 50 | ja,q | 0.000001 | 0.1 | |
Kmax,q | 50 | 10000 | je,q | 0.001 | 5 | |
κq | 0.001 | 0.03 | jp,q | 50 | 10000 | |
ξ | 0.00004 | 0.002 | f + | 0 | 0.01 | |
au | 0 | 1 | f - | 0 | 0.02 | |
γγ | 0.000001 | 0.1 | fm | 0.001 | 0.1 | |
M0 | 0.0003 | 0.03 | Lpol | 50 | 10000 | |
ML | 0.0000015 | 0.00003 | π | 0.00001 | 0.1 | |
S | 0.0000015 | 0.00003 | Vinit | 0.01 | 1 | |
G | 1 | 100 | v | 0.000001 | 0.01 | |
(3.1) |
Table 3: Influences of the parameters on the variables that characterise the result of the simulation runs. The first value of each entry in the table denotes the number of runs in which significant (significance level: 0,01) positive impact is found. The second value denotes the number of runs in which a significant (significance level: 0,01) negative impact is found. | ||||||
parameter | runs | N | g | Ci | ||
Linit | 20 | 8/3 | 8/3 | 5/5 | 1/4,7 | 6/4 |
α | 20 | 2/10 | 6/6 | 7/4 | 3/0,7 | 5/5 |
β | 20 | 10/- | 2/4 | 1/7 | 0,7/2,7 | 1/5 |
D | 20 | 8/- | 11/- | 1/7 | 3/2 | 1/- |
b | 20 | -/1 | 2/1 | -/1 | 0,3/0,3 | -/1 |
ρ | 20 | -/4 | 1/6 | 3/- | -/2,3 | 2/1 |
λ | 20 | 2/10 | 13/1 | 15/- | 3,3/1 | 10/- |
εq | 20 | 18/- | 13/- | -/17 | 2/7,3 | 1/12 |
Φ | 20 | 13/- | 8/5 | 2/5 | 0,7/5,3 | 7/1 |
ζ | 20 | 2/5 | 1/4 | 12/- | 3,7/- | 11/- |
Kinit,q | 20 | 6/1 | 3/2 | -/7 | 0,3/3,7 | -/7 |
Kmaxq | 20 | 4/5 | 5/3 | 10/1 | 5,3/0,3 | 10/- |
κq | 20 | 3/5 | 5/6 | 8/2 | 5/1,7 | 8/2 |
ξ | 20 | 2/4 | 3/4 | 3/2 | 0,3/1,3 | 1/2 |
au | 20 | 4/2 | 9/2 | 3/5 | 2,3/2,7 | 4/5 |
γ | 20 | 1/3 | -/6 | 8/- | 1,3/2,3 | 9/- |
m0 | 20 | -/2 | -/3 | -/- | -/1 | -/- |
mL | 20 | 2/3 | -/3 | 4/- | 1,3/- | 4/1 |
s | 20 | -/1 | -/- | 1/- | 0,7/- | 1/- |
g | 20 | -/- | -/- | -/- | 0,3/0,3 | -/- |
G | 20 | -/1 | -/1 | 1/- | 0,3/- | -/- |
η | 20 | -/- | -/- | -/- | -/- | -/- |
Table 4: Influences of the parameters on the variables that characterise the results of the simulation runs. The first value of each entry in the table denotes the number of runs in which a significant (significance level: 0.01) positive impact is found. The second value denotes the number of runs in which a significant (significance level: 0.01) negative impact is found. | ||||||
parameter | runs | N | g | Ci | ||
cc | 20 | -/3 | -/7 | 9/- | 2/0,3 | 8/- |
φ | 20 | -/- | -/- | -/- | -/- | -/1 |
pmax,c | 20 | 1/- | 1/- | -/1 | -/0,3 | -/- |
ca | 20 | 2/- | 1/2 | -/2 | 1,3/0,3 | -/2 |
ac | 20 | 3/1 | 3/3 | 4/- | 0,7/1,3 | 4/- |
mc | 20 | -/1 | -/- | -/- | -/- | 1/- |
µ | 20 | 1/1 | 2/1 | 6/- | 1/0,7 | 6/- |
χ | 20 | 2/- | 3/2 | 2/1 | 1/0,3 | 2/1 |
ja,q | 20 | -/1 | -/1 | 1/- | 1/- | -/- |
je,q | 20 | -/1 | -/1 | 1/1 | 0,3/- | 1/1 |
jp,q | 20 | -/- | -/- | -/- | 0,7/- | -/- |
f+ | 20 | -/- | -/- | -/- | -/- | -/- |
f- | 20 | -/- | -/- | -/- | -/0,3 | -/1 |
fm | 20 | 1/- | -/1 | -/1 | 1/- | -/- |
Lpol | 20 | -/- | -/1 | 1/- | -/- | -/- |
π | 20 | 2/1 | 1/1 | 3/- | 0,3/- | 2/- |
Vinit | 20 | 14/- | 8/- | -/12 | 1/6 | 1/11 |
v | 20 | 10/- | 9/- | 1/5 | -/1,7 | 5/2 |
2 Of course, this statement in principle only holds if an infinite number of random sets is chosen. However, the more sets of parameters are chosen, the more reliable is the obtained result.
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