Rejoinder to casetti’s “critical notes on an alleged operationalization of the concept of spatial utility”

Abstract

In a paper completed some three years ago, "On Place Utility and the Normative Allocation of Intra-Urban Migrants" (Brown, Horton, and Wittick, 1970), a linear programming model of intra-urban migration was presented. This model was designed to operationalize concepts put forth in an earlier conceptual model (Brown and Moore, 1970); these are principally ones relating to the role of aspirations and search in the residential site selection process. The interpretation of the dual of this model proved to be interesting in that it linked household aspiration and search behaviors to place utility, a concept that has received much attention in the intraurban migration literature but has proved difficult to operationalize (Brown and Longbrake, 1970). Casetti objects to this work on two accounts, the more noteworthy of which concerns the interpretation of the model's dual variables as place utilities. This needs to be examined more closely since in our view the model is a useful and informative one, and its elaboration constitutes the major thrust of our paper. The primal portion of the model allocates migrant households to va.cancies so as to minimize the "search effort for a place i household to identify (or find) a vacancy at place j" (p. 178) (.aij*), while maximizing "the level of return or satisfaction which is likely to accrue to a place i household were it to relocate to a vacancy in place j" (p. 178), both for the total migrant population. In operationalization the aspiration factor is measured by an index (dij) in which "lower numbers indicate combinations of origin and destination zone types which are more satisfactory in terms of the aspiration sets of migrant households" (p. 180), so that the model actually minimizes the total cost (c+j), aij*dij. "Thus migrant households will be allocated among O-D zones such as to minimize their search effort and their dissatisfaction with respect to a new residence site" (p. 180). Clearly, there is some confusion in the Brown-Horton-Wittick article over the definition of dij. Casetti interprets it as a measure of satisfaction ("the smaller the dij the more the aspirations of a migrant from zone i match the characteristics of zone j"), rather than as an aspiration factor. We feel similarly, except that because of the way in which dij varies, the model is made most clear by interpreting it as measuring the "dissatisfaction with respect to a new residence site". This will be used below in our response to Casetti's comments on the model's dual variables. A useful rationale for our interpretation of the dual is presented in Stevens's (1961) treatment of location rent in a linear programming context. Paraphrasing his argument, let pij be the gain or the increase in place utility in migrating from i to j and