On the continuum approach of spatial and some local public goods or product differentiation models: Some problems

Models with a continuum of consumers and locations such that each consumer can purchase goods in only one location are examined. Such models are common in the literature of the new urban economics. An approximation by economies with a finite number of consumers is shown to fail. In the first two examples, the continuum economy equilibrium exists and is unique, but the finite economy has many equilibria (Example 1) or none at all (Example 2). In Example 3, the continuum economy has no equilibrium, but the finite economy has one. These three examples employ location-dependent utilities. The fourth example illustrates the conceptual difficulties that arise in approximation when transportation cost is introduced, even with location-independent utilities. The results are related to the literatures concerning product differentiation and spatial economies.