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In 1960 William Alonso completed his dissertation which extended the von Thünen model to urban land uses. His model gives land use, rent, intensity of land use, population and employment as a function of distance to the CBD of the city as a solution of an economic equilibrium for the market for space.
The von Thünen model required considerable modification to apply to residential, commercial and industrial land use. In the von thünen model the bidrent function declined as a result of the increased transportation costs to transport the produce of one unit of land one additional unit of distance.
A preliminary rationalization of a bidrent function for a household came out of the Chicago Transportation Study. There the results indicated that households behaved as though they had a combined rent and transportation budget such that if transportation cost were higher then the amount that they would pay for rent is lower.
A more sophicated formulation assumes that households have preferences given by a set of indifference curves. The bidrent function is the amount that a household could pay for rent at different location (with differing transportation costs) such that the same level of satisfaction is achineved; i.e., the household is on the same indifference curve. This formulation allows for the possibility that different amounts of housing space could be chosen at different locations. Also it allows for the possibility that higher income households end up locating in the suburbs because of the relatively cost of open land space there compared with locations closer the CBD. The bidrent function would not have to be a straight line.
A shift to a higher bidrent function for a household involves the acceptance of a lower indifference curve. This could happen if a household found there was no location where its bidrent function equalled or surpassed the market rent.
Bidrent function theory may be formulated mathematically. Let U(x,h,T) be the utility function of a household where h is the amount of housing space used, T is the amount of leisure time and x is the consumption of other goods and services. The budget faced by the household is that of:
where t is the commuting time, w the wage rate, y_{0} the nonwage income. Given t, r and p the household maximizes utility where
where λ is the Lagrangian multiplier. These first order conditions imply that the household will chose a location where its subjective tradeoff between commuting time and rent per unit space is equal to the ratio of rent per unit space to the wage rate it faces; i.e.,
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