A model of the learning process with local knowledge externalities illustrated with an integrated graphical framework

In this paper we present a theoretical model of the learning process with knowledge externalities to R&D and other learning inputs within a region, a technological district, an industry or a technological cluster with fast rates of accumulation of new technological knowledge. As there are several definitions of localized technological and learning opportunities (according to the technical space, or to the regional space) and of localized technological knowledge, we can therefore find several possible applications to the generic model. The analysis of the learning firm interacting with a specific region in the production of new technological knowledge is just one of them. The analytical model we develop is amenable to a graphical representation. Thus we provide in the first place a unifying graphical framework, consisting of a four-quadrant picture to analyze the process of knowledge accumulation by learning firms located and operating in a specific region or industry, which simultaneously stresses the nature of the basic learning process and the importance of true knowledge spillovers in the generation of new knowledge. We adopt the following approach to the construction of spillover stocks or pools. First, the magnitude of the state of aggregate knowledge in a region or industry is reconstructed through the historic accumulation of flows of knowledge. Thus, the aggregate level of knowledge can always be updated after every learning loop, or at every moment of discrete time, whose unit of measurement we might assume at the outset of our analysis. Secondly, every firm within a region or industry is treated symmetrically regarding spillover effects and magnitudes. Such statement meaning that the amount of aggregate knowledge borrowed from any available source, either the region or industry under analysis or some other distant region or industry, is regarded as the same by every firm. And finally, we model both the loss of appropriation of benefits from innovation and the distance between different technological bases or regional sources in terms of single parameters, or instantaneous rates of growth, weighting respectively the leakage and the absorption intensities of flows and stocks of knowledge. Several theoretical predictions about the direction and magnitude of the knowledge spillovers can therefore be deducted from parametric changes in the leakage and absorption functions of our model arising from, among other things: - Improvements in information technology and falling communication costs observed in the economic system at general. - Improvements in technological communication systems within specific technological districts. - The establishment of explicit cooperative relations and effective access to the pool of collective knowledge, or instead any improvements of the mutuality and trust conditions, within the group of firms located and operating within a specific region. - The increasing of competitive pressures, or the working of any other mechanism for lowering the appropriation of a firm’s gains from innovation, in an array of industrial sectors. One interesting theoretical result is then derived from our full model. With such purpose in mind, we consider first the existence of a relevant competitive situation where appropriation and communication are both dependent upon the number of receiving and sending firms within the region. Whereas the amount of technological leakage per firm increases with the number of firms effectively operating within the region, ceteris paribus; the extent of absorption per firm also increases with the number of firms effectively communicating within the region, ceteris paribus. Apparently, there is a trade-off between such appropriation conditions and communication conditions. In the long-run, the addition of firms eventually exhausts the net positive effects of taking part in an effective network, and so we can establish an equilibrium number of firms operating in the region.