My theoretical model is based upon the one presented in Rosen (1974), which is based on utility maximization. The model assumes competitive equilibrium in a many-dimensional space, representing each of attributes which define a good. In the context of my model, any given house can be represented by the bundle of attributes
where through measure the quantity of the th characteristic present in the good. These attributes include structural characteristics, such as square footage and house age, as well as neighborhood characteristics, such as travel time to work and crime rates.
The Hedonic Price Schedule
In this space, each attribute is valued independently and contributes to the price of the good, . Because is a function of all characteristics, a firm must commit resources to the production of additional goods at an increasing rate. Thus, by holding through constant, and by assuming a perfectly competitive market structure, we can draw the relation between and a given attribute of interest, , as convex. This defines the Hedonic Price Schedule, shown in Figure 1. Holding other attributes fixed, the slope of this function is the implicit price of .
In the context of my model, therefore, the price of a house can be said to be a function of the quantity of tree canopy on and around its property, in addition to its structural and neighborhood characteristics. By constructing house price as an empirical function of those characteristics, the housing price is made to reveal the implicit price of each of the characteristics.
The Consumption Decision
In competitive equilibrium, consumers allocate their income so as to purchase a bundle of goods that maximizes their utility. Because of diminishing returns to consumption intensity, moreover, the utility derived from any given good is considered to increase at a decreasing rate. Consumers’ indifference curves, which represent the utility tradeoffs between an additional quantity of a given good and an alternative quantity of another good, are therefore concave.
In the context of my model, the aforementioned “goods” are really the attributes of housing in question, and all other goods consumed are defined as the numeraire good, . By setting the price of the numeraire good equal to unity (), we measure tradeoffs of with in terms of dollars. We can therefore represent a nonlinear utility function as
Utility therefore depends upon the quantity of each attribute present in the good, increasing at a decreasing rate, as well as on all other goods consumed. Consumers are therefore able to maximize utility tangent to the budget constraint
where represents income, and it is assumed that each consumer only purchases a single house.
We are now able to define a set of willingness-to-pay (WTP) functions, representing consumers’ indifference curves with respect to housing characteristics, given that individual consumers have discreet preferences. By measuring income () relative to the unity-set numeraire good (), we can define a WTP function as
showing consumers’ preferences for housing characteristics given a level of income and utility. Two of these willingness-to-pay curves, $\theta$, are shown tangential to the Hedonic Price Schedule in Figure \ref{fig:HPSwtp}.
At these tangency points, we can measure the ratio of the marginal utility of to the marginal utility of , or . This defines the marginal rate of substitution. If we assume that markets are in equilibrium, we know that these tangencies represent points of maximum utility, where the marginal rate of substitution between the characteristic in question, in this case Tree Canopy, and all other goods is equal to , the change in price given a change in Tree Canopy. Allowing for variable preferences with respect to , we can construct a number of WTP curves tangential to the price condition. These tangencies, more generally, occur where
indicating that the Hedonic Price Schedule lies tangent to a given consumers’ WTP curve (which is an iso-utility curve) at the maximum level of utility. In this model, is proportional to , and therefore equal to the change in WTP for housing divided by the change in characteristic .
Limitations of Model Assumptions
Rosen’s model is based on utility maximization theory, where tangencies between consumers’ budget constraints and indifference curves maximize the utility gained by their income. This model, however, assumes that consumers have sufficient information to differentiate between all possible allocations of their resources—in other words, it assumes perfect information. This is not altogether likely in the residential housing market, where consumers frequently must spend long periods of time investigating possible options or working with real estate agents in order to generate a satisfactory amount of information to make any decision.
Perfect information is just one feature of the market structure of perfect competition, upon whose presence Rosen’s model also relies to guarantee the slope of its curves and to ensure that consumers are able to select the house which allocates their resources so as to maximize utility. Perfect competition, however, is seldom observed in reality, and is certainly not the case in Portland’s housing market. Perfect competition would indicate that the good being sold, housing, is standardized with regard to price, and that there are no transaction costs associated with buying or selling a house. In reality and in realty, houses vary widely with respect to the relationship between quality, characteristics, and price. In addition, real estate firms are frequently employed as middlemen in housing sales, adding to the cost of the transaction. Finally, these same firms exert a substantial amount of control over the housing market, largely through control of the aforementioned imperfect information. In effect, while it is clear that Portland’s housing market does not exhibit perfect competition, the degree to it resembles it will affect the validity of the hedonic model.
Finally, Rosen’s model also assumes that markets are in equilibrium—that is, that supply and demand for housing maximizes economic surplus for both consumers and producers of housing. In the case of housing, a good whose supply takes significant lag time to respond to demand, equilibrium would be disrupted if demand for housing was unable to meet the available supply (such as a sudden decline in the number of prospective homeowners) of if demand exceeded available supply (such a sudden influx of prospective homeowners). In the case of Portland, the latter is more likely to be the case, as Portland is a growing city that has experienced rapid urbanization. Furthermore, Portland has a constraint placed on its housing supply in the form of an Urban Growth Boundary (UGB), which limits the geographic extent of dense development. While it does not present a hard limit to housing growth, it does create a barrier to development, which some studies suggest has increased housing prices in Portland above equilibrium levels.