18. The Herbert-Stevens Model

Herbert and Stevens (1960) present an early application of Linear Programming. However, their approach is different from that used in Chaps. 15 and 16. In this chapter, groups of households (users) are assigned to districts of a city on the basis of their ability to pay the most rent (as opposed to least cost). Underlying the Herbert-Stevens model is the idea that each group has a bid rent for each type of real estate. In this chapter, I use the Herbert-Stevens model to help us better understand how a competitive market allocates land (districts) to users (groups). As with any linear program (primal), there is a corresponding dual linear program that solves for the shadow prices on constraints in the primal. In the Herbert-Stevens model, there is one shadow price for each district in the city and one shadow price for each type of user. I here follow Wheaton’s approach of assuming that the bid rent of a household is itself a function of the household’s well being: the higher the bid rent, the less well-off the household (because they have less income left over to buy other things). In equilibrium, the Herbert-Stevens model ensures that every unit of real estate is occupied by the highest bidder, every household get allocated something, and no one can be made better off by altering (reducing) their rent. I then contrast this model with the linear programs models developed in Chaps. 15 through 17 and the Alonso model in Chap. 11.