A Random Utility Model of Demand for Variety under Spatial Differentiation

This paper studies a linear economic geography model under general equilibrium and iceberg transportation costs from a linear random utility point of view. A motivation is given by previous work on spatial product differentiation establishing a connection between demands generated by linear random utility optimization and demands due to CES-utility. Two interpretations of random utility are considered. First, the utility of a representative consumer from one unit of a given variety is assumed to be random e.g. due to a random linear trade cost or unobservable quality. In the second interpretation, as in mainstream random utility theory, the uncertainty is due to the incomplete information of the modeller regarding the utility and/or trade cost parameters. Discrete choice can be assumed within this second linear model. The approach to stochastic utility is based on maximizing linear random utility functions via probabilistic constraints. This allows to deal with both normally and log-normally distributed network link coefficients. The main focus is on the case where the random utility/cost coefficients are independent and identically distributed random variables. Linear random utility in both the first case, where utility/cost is random (or hard to measure), and the second case, where the modeller does not know the utility functions, implies demand for variety. This is different from the outcome under expected linear utility maximization. (Unlike in linear random utility models optimization in this paper is performed before the realization of the random variable, like in models of state dependent utility.) In economic geography the demand for diversity due to linear random utility can explain intra-industry trade between symmetric regions. The outcome of individual stochastic utility optimization coincides with the modeller's predicted solution to an aggregate consumer's income allocation under discrete choice (like in previous work). In addition to the relation to economic geography, the paper is related to recent work on noncooperative network formation under linear utility functions. The main conclusions from the stochastic programming approach to resource allocation in the presence of spatial differentiation can be summarized as follows: (1) A stochastic programming framework for the study of linear random utility models is introduced. The uncertainty in utility can be the consequence of the lack of complete information available either to the modeller or the consumer. In the former case discrete choice can be assumed within this linear model. Unlike in other random linear models, it can be assumed in the stochastic programming framework introduced that any fixed number of varieties (not just one) is demanded. When the utility parameters are i.i.d. normal or lognormal random variables, it is optimal for the representative consumer to demand either all varieties or use all income as money, unless the consumer is restricted to buy at most a single variety. Increasing the product diversity both increases the utility estimated by the modeller and the utility that can be obtained by consumers optimizing stochastic utility. (2) In related work on random utility, the assumption of a joint logconcave distribution of the random utility parameters is a sufficient condition for the existence of a decentralized solution to resource allocation. In this paper it is argued that a logconcave distribution of the random utility/cost coefficients is a sufficient condition for the existence of a Pareto-efficient resource allocation. (3) The linear stochastic programming model in this paper is related to recent work on endogenous network formation under linear utility functions (ref 1). The model in (ref 1) resulting to either an empty network (with no links formed) or a fully connected network (with all possible links) network is extended to firstly allowing for the network link coefficients measuring the decay between the different nodes to be i.i.d. random variables, and secondly, to an endogenous link formation price. In the random utility interpretation, the transactions between a pair of nodes can be interpreted either as communication (ref 1) or more generally as trade in any goods that can be considered substitutes to one another. Ref 1. V. Bala and S. Goyal: "A Noncooperative Model of Network Formation", Econometrica 2000 Ref 2. S. Anderson, A. De Palma and J.-F. Thisse, "Discrete Choice Theory of Product Differentiation"