Population and Land Use

The neoclassical economic model of population and land use originates from von Thünen's theory of agricultural land use, wherein the “highest and best use” of land depends on the cost of transporting the various crops it can yield to a central marketplace. Because different candidates (crops) have different transport costs—and offsetting market values—those with relatively high (low) transport costs end up being produced close to (far from) the marketplace. So, for example, if tomatoes cost more to transport than wheat, tomato farmers will outbid wheat farmers for locations adjacent to the marketplace.

In the 1960s, William Alonso, Edwin Mills, and Richard Muth (jointly credited but working independently) generalized the Thünen theory of agricultural land use to a theory of modern human settlement patterns. The framework describes a locational rent gradient that falls away from its peak around a business district located at the center of a circular region with a dense transportation network that is situated on an otherwise flat, featureless plain. At market equilibrium, all households, which are assumed to be identical, attain exactly the same level of “utility” (satisfaction received from consuming a good or service), so the rent gradient reflects the trade-off between location and the cost of travel to and from the central business district. Meanwhile, a corresponding density gradient emerges as a result of households occupying greater amounts of space toward the urban fringe. To maintain a fixed level of utility, households consume more land through substitution with other goods as rent declines. The density gradient, and, with it, development, ends altogether once locational rent reaches zero, and the highest and best use of land is for agriculture or some other natural resource–oriented activity.

Somewhat more formally, the theory underpinning the models of population and land use assumes that households have a common utility function, say U(z,s), which contains a composite good, or numeraire, z, and urban space, or land, s. An individual household's budgetary constraint is determined by its income, y, less the cost of travel, k, between the central business district and its location at radial distance, d:

where k(d) is a continuous function that increases with d and r(d) is the locational rent per unit of space at d. Within this framework, households maximize their utility by choosing some combination of the numeraire and land, subject to their particular—spatially explicit—budgetary constraint. The outcome of this choice is a household's bid-rent, ρ(d,u), which is the maximum price it is willing to pay for land at distance d while maintaining a fixed level of utility, u. Simply put, bid-rent is the most that a household's members are willing to pay per unit of space to secure the right to occupy their location of choice, given that they derive happiness from both the land and other forms of consumption. In addition to land prices, bid-rent yields a household's optimal amount of land consumption, or lot size, ς(d,u), which is what ultimately gives the built environment its character (Figure 1).

Figure 1 illustrates how the bidding process translates into a physical pattern of settlement. It displays the marginal rate of substitution, described [Page 2237]by an indifference curve (the arc) for a fixed level of utility, u, between the numeraire, z, and land, s, plus the budget constraints (the dashed lines) and corresponding consumption bundles (the dotted lines) associated with two households located at distances d1 and d2 from the central business district, where d1 < d2. Because the cost of travel to and from the central business district, k(d), is lower at d1 than it is at d2, the net income of the household located at d1 is greater than the net income of the household at d2, or y–k(d1) > y–k(d2). The two budget constraints, which must be tangent to the indifference curve for their respective households to each achieve utility level, u, show that (a) the bid-rent, which is equivalent to the slope of the budget constraint, for the household located at d1 is greater than the bid-rent for the household located at d2, or ρ(d1,u) > ρ(d2,u), and (b) the optimal lot size for the household located at d1 is less than the optimal lot size for the household located at d2, or ς(d1,u) > ς(d2,u).

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