SPATIAL PRICE COMPETITION WITH CONSUIMERS ON A PLANE,AT INTERSECTIONS,AND ALONG MAIN ROADWAYS*

ABSTRACT. This paper examines two-dimensional price competition on a plane, with a block metric and a square grid of main roadways. One store is located at each intersection of main roadways. Consumer locations include a uniform distribution over the plane, linear concentrations along main roadways, and point concentrations at intersections. Bertrmd-Nash mill price competition is examined first. The equilibrium price depends on the relative numbers of consumers in the three types of locations (and on travel costs per mile and the spacing between stores). If too many consumers are in each point concentration, then the price equilibrium is undermined by a high-price strategy or by mill-price undercutting. Spatial competition with price discrimination is examined next, and compared to Bertrand-Nash mill price competition.     

Redundancy as an Index of Change in Point Pattern Analysis

Point pattern analysis based on concepts from information theory can go beyond existing techniques. Direct measurement of spatial form is achieved when Thiessen polygons are constructed around the points; in this scheme, each point's proportion of total area may be treated like a probability. Three information-theoretic indices are available for analysis of a distribution of such probabilities. Entropy is density dependent. Redundancy, defined as the difference between observed and maximum entropy, seems to avoid this problem when the number of individuals in a pattern exceeds twenty. Comparisons of prior and posterior redundancy provide an indication of change in overall pattern form. An information gain expectation reflects changes for each individual in a pattern. Here, point-area redundancy parameters are determined for Poisson-generated patterns, using a gamma distribution of polygon areas and a computer-generated set. An application to an urban crime problem illustrates the use of these parameters in the analysis of pattern change.