# The birth model

The first element of the comprehensive spatial microeconometrics approach is constituted by an equation for the birth of new firms. In this methodological framework the observed spatial point pattern of new firms is assumed to be the realization of a point process conditional on the locations of existing firms in that moment.

In order to formalize our model, we rely on the spatial point process methodology (Diggle, 2003) introduced in Chapter 6. Within this framework, a spatial point process is considered as a stochastic mechanism that generates patterns of points on a planar map. The basic characteristic of a spatial point process is the intensity function, introduced in Equation 6.1 and denoted by the symbol Л(х). Thus, by definition, the higher А (л;), the higher the expected concentration of points around л; (see Arbia et al., 2008).

Following the modeling framework originally proposed in a seminal paper by Rathbun and Cressie (1994) and imported by Arbia (2001) into the field of regional economics, the formation process of new firms can be modeled as an inhomogeneous Poisson process (see Diggle, 2003) with intensity function Л(х) driven by the potential interaction effects of the existing firms. The values of Л(х) constitute a realization of a random function parametrically specified by the following model: where a, pi, Рг, рз are parameters to be estimated, d(x) indicates the distance of point x from a conspicuous point (see Section 3.6) and W (x) is a term measuring the sign and the intensity of the interaction between the firm located in point x and the other existing firms and incorporates the idea of non-constant spatial returns. A particular specification for the W(x) function was suggested by Arbia (2001). Furthermore, in Equation 10.1, Z represents a vector of independent variables assumed to be spatially heterogeneous (such as demand or unitary transport costs and regional policy instruments such as local taxation, incentives) that can influence the birth of economic activities in the long-run. Finally Ф{х) is the error term of the model assumed to be spatially stationary, Gaussian and zero mean, but non-zero spatial correlations. Due to the nature of the error term, the estimation of Equation 10.1 presents some problems that will be discussed more thoroughly in Section 10.2.2 where we discuss a numerical application of the model. Notice that Equation 10.1 can be seen as a continuous space version of Krugman’s concentration model that avoids the problems associated with arbitrary geographical partitions (see Krugman, 1991a). An example of the formalization of Equation 10.1 was already presented in Example 8.1.