Simulation study of the extended model

As one can easily see, the extension of model 3.17 cannot be solved analytical.

Even, a conventional stability analysis cannot be conducted due to the inherently high complexity of this model. But it is possible to derive a steady state condition for this model.

Proposition 6 : Given proposition 1 holds than the partial derivatives of ˙Nt, ˙stand C˙_t still exist. Provided ˙N_t = ˙s_t = ˙C_t = 0 holds simultaneously than system 3.34 has

We know, that a steady state must exist. The steady state is reached, if cost conver-gence of both firms has been occurred, thus Γij = 0 is realized. Then we are automat-ically back to model 3.17, from which we know, that a steady state exists. The next section deals with the simulation of the extended version of model 3.17, model 3.34.

3.3.3 Simulation study of the extended model

The simulation setup for model 3.34 is the same as for model 3.17. Thus, it is referred to the same parameter setting as in the simulation of model 3.17. The learning curve parameter have been chosen as follows:

The simulation study has been conducted with Anylogic. To avoid redundancies with respect to simulation results discussion, in the following it is referred mainly on

Parameter Value

χ 0.50

ι 0.50

σ²_ζ 1×10⁻⁶ σ²_ 1×10⁻⁶

Table 3.1: Learning curve parameter setting

simulation meanderings induced by integration of learning aspects. As done before, three simulation scenarios have been performed, for the DRS, CRS and the IRS case.

For the parameter setting of the technological progress it is therefore referred to table 3.2.

With respect to the DRS, CRS, and IRS the inclusion of learning aspects leads to the following observations, based on the simulation study. Again, one can find the impluse responses for the simulation of the extended model in appendix 5, 6 and 7.

The CRS case is depicted in appendix 7 in figures 3.15, 3.16 and 3.17. The IRS case is graphically replicated in appendix 6 with the corresponding figures 3.12, 3.13 and 3.14, whereas the DRS case can be found in appendix 5 in figures 3.8, 3.9, 3.10 and 3.11.

1. Increasing returns to scale

As argued before, the feedback of market selection and the feedback of cost convergence is weightout by inducing a high rate of technological progress with respect to cost reduction. On the other side, the lower θ the more weight is laid to the positive feedback of innovative activity and thus the more cost efficiency firm is crowding out the laggard firm. Now we have to account for a third effect: the positive spillover effect which is driven by learning activities for the laggard firm.

With respect to the simulation results, on the first sight, there is no significant changing with respect to the simulation results based on model 3.17. But if we look more closely, then we observe, that for no feasible parameter constellation of θ the before mentioned monopoly scenario occurs. Soonest this is the case for a small value of technological progress, because the herfindahl takes its largest value for this case. Hence, the higher θ, the lower the herfindahl index HI³³, and thus the more realistic a coexistence of both firms is. For the case of θ = 0.5 a nearly uniformly market segmentation is obtained with a corresponding herfindahl index of HI = 0.50. An interesting results is obtained for a high speed of technological progress, θ = 0.99. In this case without learning effects, the convergence effects outweighs the selection effects, because technological progress is so fast that the

33The herfindahl index HI is defined as HI :=P

is²_i.

lowest cost level is reached until selection effects or diseconomics of scale have time to taken effect in the market. Finally, the less cost efficient firm i = 2 becomes market leader, because the high speed of technological progress leads to a temporal cost leadership of firm i = 2 in a negative sense until cost convergence has reached. Now positive learning effects from which exclusively the laggard firms can benefit, lead to a more turbulent market evolution in the beginning of the simulation study, as one can easily see from the stability index and herfindahl index. But at the end no significant difference can be observed compared to the

”no learning” case. Hence, the inclusion of learning effects in the CRS scenario of model 3.17 results in no significant changes regarding to the simulation results, with the exception that no monopoly scenario occurs. This could be due to the smoothing effects induced by the knowledge spillovers.

2. Constant returns to scale

As for the IRS case, learning effects to not exhibit a significant change on simu-lation results, compared with the case of CRS for model 3.17. But, also for the IRS case mentioned above, no smoothing effects of knowledge spillovers lead to the exclusion of monopoly scenarios.

3. Decreasing returns to scale

For the DRS scenario, we do not observe any significant changes regarding to the DRS case of model 3.17, with two exceptions. First, as mentioned before for the IRS and CRS case respectively, no monopoly scenario occurs for any reasonable value of θ. For any given parameter value of θ, firm i = 2 will remain the market leader at the end. Second, and the more interesting is the special case of θ = 0.02. As argued before, the weak negative feedback causes a slower rate of cost reduction for the leading firm and thus will lead to a surpass by smaller firms. Thus a switching occurs until cost convergence is realized. This mentioned market instability leads to the conclusion, that a final ranking of firms cannot be predicted. If we now integrate learning behaviour we still observe market instability, which is distinct at most with respect to time. But again, the smoothing effects of spillover lead to a more stable structure. No switching phenomena is observed and thus and on contrary as before, the prediction of a final firm ranking seems to be more feasible.

On summary, we can conclude, that integrating learning aspects leads to a more stable market structure at all. Further, this model supports the empirical finding by (Campagni, 1991), which states that inter-firm cooperation based on knowledge sharing can explain the predominance of small firms in the market. This conclusion can be drawn because of the fact that for no simulation scenario, a monopoly market structure

occurs. It is worth to mention, that unreported simulation studies, which are based on a ”constant learning” scenario, reveals that for low speed of technological progress monopolistic market structures occur. Hence, only the inclusion of de facto learning effects leads to an exclusion of monopoly market structures.

3.4 Summary

The early stages of an industry life cycle are characterized by instability and a relatively competitive market environment. This awareness is also labeled in the relevant litera-ture as a stylized fact regarding firm-size dynamics. (Mazzucato, 2000) has shown in a simulation study, based on the replicator dynamics approach, that in fact the before mentioned stylized fact can be replicated by the model assuming decreasing returns to scale which corresponds to the ”Schumpeter-Mark-I”’ hypothesis. The argument of this pretty simple and easy to retrace: firms with a high market share would expire a slower rate of cost reduction potential and thus those firms will be lurched by smaller firms. This process leads to a switching behaviour of market structure, particularly at the beginning of the life cycle of an industry. For other parameter constellations, small firms will be shaked out of the market, especially when increasing returns to scale are assumed.

The lack of the model (Mazzucato, 2000) is that it is not able to cover the fact that a high knowledge transfer intensity, for instance due to cooperations which are based on knowledge transfer, enhances firm innovativeness and hence induces a feedback on market structure. As mentioned by (Campagni, 1991), (Best, 2001), (Porter, 2000) and (Krugman, 1991) in a more spatial context, the exchange of ideas and knowledge lead to a predominance of small firms in the market. For this reason, the question arises how affects learning behaviour the market structure.

For this reason, a model similar to the work of (Mazzucato, 2000) and (Noailly et al., 2003) is setup which is able to replicate the before mentioned stylized fact.

After a simulation has been conducted the model was extended by learning effects and knowledge diffusion. Knowledge diffusion in this context is treated as an endogenous event, driven by psychologically motivated learning endeavours of firms. A further simulation study of the extended model has shown, that for any degree of technological progress small firms still remain in the market, also for the case of IRS, where large firms are in advance. Hence, this model is able to replicate the fact, that small firms are more likely to benefit from knowledge networks and thus from spillovers which define a source of innovativeness, from which large firms cannot profit.

Of course, there are several revenues for further research. For instance, it is planned to embed this rather simple model in a spatial model framework to cover explicit cluster effects of small firms.