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The spatial concentration of firms has long been a central issue in economics under both the theoretical and the applied point of view mainly due to the important policy implications. A popular approach to its measurement, which does not suffer from the problem of the arbitrariness of the regional boundaries, makes use of micro data and looks at the firms as if they were dimensionless points distributed in the economic space. However, in practical circumstances the points (firms) observed in the economic space are far from being dimensionless and are conversely characterized by different dimension in terms of the number of employees, the product, the capital, and so on. In the literature, the works that originally introduce such an approach disregard the aspect of the different firm dimension and ignore the fact that a high degree of spatial concentration may result from the case of both many small points clustering in definite portions of space and only few large points clustering together (e.g., few large firms). We refer to this phenomenon as clustering of firms and clustering of economic activities. The present article aims at tackling this problem by adapting the popular K-function to account for the point dimension using the framework of marked point process theory. [ABSTRACT FROM AUTHOR]

A series of recent papers have introduced some explorative methods based on Ripley's K-function (Ripley in J R Stat Soc B 39(2):172-212, ) analyzing the micro-geographical patterns of firms. Often the spatial heterogeneity of an area is handled by referring to a case-control design, in which spatial clusters occur as over-concentrations of firms belonging to a specific industry as opposed to the distribution of firms in the whole economy. Therefore, positive, or negative, spatial dependence between firms occurs when a specific sector of industry is seen to present a more aggregated pattern (or more dispersed) than is common in the economy as a whole. This approach has led to the development of relative measures of spatial concentration which, as a consequence, are not straightforwardly comparable across different economies. In this article, we explore a parametric approach based on the inhomogeneous K-function (Baddeley et al. in Statistica Nederlandica 54(3):329-350, ) that makes it possible to obtain an absolute measure of the industrial agglomeration that is also able to capture spatial heterogeneity. We provide an empirical application of the approach taken with regard to the spatial distribution of high-tech industries in Milan (Italy) in 2001. [ABSTRACT FROM AUTHOR]