Central Path Dynamics and a Model of Competition. II

Growth - the change in number or size - and adaptation - the change in quality or structure - are key attributes of global processes in natural communities, society and economics (see, e.g. Hofbauer and Sigmund, 1988; Freedman, 1991; Young, 1993). In this paper we describe a model with explicit growth-adaptation feedbacks. We treat it in the form of an economic model of competition of two firms (with several departments) on the market. Their size is measured by their capital, and their quality by their productive power (production complexity). It is assumed that the production complexity of a department or firm is a simple function (that is more general than the one considered in Krazhimskii and Stoer, 1999) of its capital. The model works on both the firm level (competition among the departments) and the market level (competition among the firms). The model shows some empirically observable phenomena. Typically, one of the firms will finally cover the market. The winner is not necessarily the firm with the potentially higher maximum productivity. A long-term coexistence of firms may arise in exceptional situations occurring only when the maximum potential productivities (not the actual productivities) are equal. The analysis is also based on the concept of central paths from the interior point optimization theory (see Sonnevend, 1985; and e.g., Ye, 1997).