A Polymer Model of the Spatial Patterns of Change in Urban Property Values

Miron Kaufman, Department of Physics,

Sanda Kaufman, Levin College of Urban Affairs

Cleveland State University, Cleveland, OH 44115, USA

Anne Friedrich

ABSTRACT

A polymerization model is proposed for describing changes in time in urban property values. Using Geographic Information Systems mapping of individual properties at two urban locations (Strongsville and Ohio City) in Northeast Ohio, the model is tested for two time periods: 1976 - 1982 and 1982 - 1988. A regularity is found in the relationship between the number and the size of property clusters experiencing change in value. Results of the pilot test suggest the polymer model provides a reliable description of specific aspects of urban property value changes. Policy applications of the model are explored. For example, model predictions could serve as input in decisions regarding the siting of new housing developments in the urban space, or the granting of tax abatements.

Changes in Urban Property Values

How do residential property prices affect each other within the urban space? Popular wisdom advises not to buy the highest-valued property on the block, and not to tolerate neglect on neighbors’ properties, based on the belief that property values are determined not only by their own characteristics but also by those of their neighbors. Is there such a spatial effect of an increase or decrease in the value of a property? Is the effect predictable in extent and location? Answers to these questions can inform intervention initiatives, such as public investment in deteriorating neighborhoods, zoning changes, or environmental remediation of brownfields located in urban areas key to economic development. This project explores the propagation pattern of residential property value changes. We propose a linear polymer model for describing the observed spatial distribution of properties whose value has increased or decreased over a period of time. We discuss the results obtained in a test of this model in two Cleveland area neighborhoods. The last section proposes some directions for further investigation, based on the test results.

Models of Change

We seek to understand how the price of one urban property affects its neighbors’ values. Classical urban models have either not specifically addressed the spatial distribution of such changes, or have addressed it at a scale that does not illuminate the processes taking place at parcel level. A review of the relevant literature is found in Reference 1. Until relatively recently, and certainly at the time when many of these models were proposed, the data storage and computing capabilities necessary for their testing were not available. Geographic Information Systems software has enabled consideration of spatial models using data at parcel level.

What types of models can shed light on the ways in which property price changes propagate through the urban space? The urban network of streets and the property points exert some influence on each other. We propose to represent this influence by a constrained percolation model, whose components are sets of monomers on a lattice, interacting to create linear polymers. A percolation network can be used to describe the travel of some physical property (gas, fluid, electrical current, etc.) through a medium. Percolation theory postulates a lattice, say square, and a quantity moving through the lattice by traveling across the elements of the grid itself. If all the elements of the grid are connected, the quantity will have no difficulty traversing the grid. However, as some elements of the grid are cut, certain paths of travel will no longer be available. As more and more bonds are cut, it will eventually become impossible for the quantity to traverse the grid. Percolation theory predicts at what point (percolation threshold) this will occur.

Constrained percolation models have been applied to physical problems. The polymerization process is an example of a constrained percolation problem. Polymer networks can be seen as analogous to urban networks of streets and roads. This is the basis for our attempt to describe the propagation of property value changes through a residential network using the linear Flory-Huggins polymerization model -- the formation of linear polymers out of monomers located at vertices of an underlying lattice. Neighboring monomers connect to each other with probability k . The size of resulting clusters is governed by the probability h of terminating the polymer growth. This model predicts2 that the most probable distribution of resulting polymer sizes is such that the number of polymers decays exponentially as the polymer size (number of monomers in the same cluster) increases. The following equation links the number of polymers of size n + 1 monomers to the total number of monomers N and to two un-normalized probabilities (or fugacities) for transmission of linkage and termination of this transmission:

Pn= (h /2k )N(k P0)n+1 = (h /2k )Nexp[-(n+1)ln(1/(k P0))] (1)

where: Pn is the number of polymers of n links between monomers; P0 is the number of polymers of size 0, i. e. free monomers; k is the un-normalized probability of transmission of change; h is the un-normalized probability of termination of change. Following the statistical physics terminology, we call k and h transmission and termination fugacities.

Consider the residential properties to be points (vertices) on the underlying network of urban streets. Define a connection between properties to be a value change in the same direction as the change in its neighbor. We propose to match the elements of the Flory-Huggins model to residential property values as follows: (a) the linear aggregation among monomers that form polymers (polymerization) is like the formation of clusters of property value increases (or decreases) around a "seed" (one property whose value has changed.); (b) the transmission probability k associated with formation of a polymer bond among monomers is like the probability that a value increase (or decrease) is transmitted from one property to its neighbor; (c) the termination probability h for polymer growth to stop is like the probability that the value change will stop propagating among neighbors when a cluster has reached a certain size. This model can describe property price changes, whereby the properties (monomers) are said to "polymerize" in a neighborhood if price changes in the same direction -- up or down -- occur in clusters (polymers). Then we would expect the number of clusters of property value changes to decrease exponentially with the size of the clusters. To test this analogy, we fit to our data an equation analogous to Eq. (1):

Pn= A exp[-B(n+1)] (2)

Testing the Model with Real Estate Data

The two sites we selected in the urban space of the Cleveland area, Ohio City and Strongsville, offer some desirable characteristics for testing this model. They are defined, and separated from their surrounding area by roads and topography. They are also socio-economically different from each other and exhibit different trends in property value changes. These differences enabled us to interpret the results, and to shed light on how property value changes "polymerize" in the urban space.

We identified residential property price changes within a bounded urban area (lattice of streets). We counted the clusters of same-direction change of various sizes. We fitted Eq. (2) to the data to determine the values of A and B in equation (2).= h / 2k = ln(1/x) We computed the fugacities k and h by using the values of N, P0, A and B, and the following relations obtained by comparing Eq. (2) to Eq. (1):

A = (h /2k )N and B = ln[1/(k P0)]. (3)

We prepared maps of residential property value changes at the two sites (reference 3). The data are from the Cuyahoga County Auditor's files of assessed values for residential property. The years used in this project were 1976, 1982, and 1988. We selected them because they are the years in which records of property value changes are based on an actual re-appraisal of all properties, while in other years, values were assessed by computer projections.

The first test area is "Ohio City," an older Cleveland neighborhood with mostly low property values for years. It provides a good test situation, because numerous scattered sites have seen substantial changes in value over the time period covered by the data. The second area, Strongsville, is a suburban community south-western of Cleveland. It has gone from being a sparsely developed, relatively "rural" area to a far more "urban" place in the last two decades. We selected it because a considerable portion of its residential development has occurred during the test time. Changes to 4,331 properties were tracked between 1976 and 1982, and changes to 4,368 properties were mapped for 1982 to 1988. We compared the property values in constant dollars across the years 1976-1982 and 1982-1988 for each parcel. New variables were created by combining existing ones. The property address was created by combining the fields for numerical and street addresses. The property value was created by adding the fields for building and land value. Once the base data were identified, each pair of years (1976-1982 and 1982-1988) was run through a comparison process. Parcels which showed an increase in value were separated from parcels which showed no real increase in values, or a real decrease in value, yielding eight maps. We defined Files for properties which increased in value were labeled UP. Files for properties which did not increase were labeled DO (for down). Names of files for Ohio City contain an OC, and names of files for Strongsville contain a ST. Filenames for the 1976 to 1982 time period end with 7682 and filenames for the 1982 to 1988 time period end with 8288.clusters as adjoining parcels -- side by side, across the street from one another, or back to back. We recorded the number of clusters of each size.

To test the polymerization model, we graphed cluster size against the number of clusters of each size (see Figure1) and fitted curves to each of the eight situations. The resulting cluster size distribution, far from random, has an exponential distribution as predicted by the model. We estimate the parameters k and h for each of the eight cases. Strongsville had the largest transmission fugacity k , and the largest individual cluster suggesting it is approaching "percolation" or the transmission of value change through the entire system, which then becomes one large cluster.

While differences in k between the eight cases do not follow any pattern, differences in h are interesting. The values for h are substantially larger in Ohio City than they are in Strongsville. In fact (Figure 2) the k and h pairs for the eight cases appear to fall into two groups, reflecting some key differences between the two locations yet to be explored. While we know the two locations to be very different along some dimensions, it is unclear from this project which of these, if any, affect the k and h values. We can only speculate on reasons for the observed results. For example, the larger probability of termination h observed in Ohio City may be due to the street geometry; similarly, the larger k values observed in Strongsville may reflect the intense sales activity in that area. However, only the analysis of further locations and model refinement can really support such hypotheses.

Conclusions

We have proposed an analogy between properties located on a network of urban streets and monomers located on a lattice, in terms of the effect of elements on their close neighbors. Neighboring monomers form links to become linear polymers. Neighboring urban properties form clusters of increasing or decreasing value in time. A test of this analogy in two time periods was performed at two sites, Strongsville and Ohio City, to derive the direction of value change. We found that: (a) Property value changes (up or down) at the two sites in the two time periods occurred in a non-random manner. Specifically, clusters of various sizes emerged such that the number of clusters of a given size decreases exponentially with cluster size. (b) The proposed constrained percolation process appears to predict the observed exponential distribution of cluster numbers by cluster size. Two parameters, the transmission fugacity k and the termination fugacity h characterize completely this distribution.

The results are rather surprising, not least because the analogy between linear polymer formation and residential property value changes is limited by a key factor: the latter term is driven by human behavior, which in most if not all respects is far less predictable than the forces governing the physical world. Another interesting feature of the test results is the excellent fit obtained with a rather parsimonious model.

The relationship between number and size of clusters of property price change in Strongsville and Ohio City can be used in a variety of decisions. However, more confirmatory analyses are necessary to ensure the relationship prevails at any location, and to understand the factors underlying this relationship. Such factors include: (a) geometry effects of street layouts and street hierarchy (wide-narrow), as well as the effect of strong physical features that might limit transmission (e.g., large highways, steep slopes or a body of water); (b) differences in the nature of the processes of transmitting value changes in specific areas; (c) factors affecting the transmission and termination fugacities (e.g., the level of economic activity in a region, the mix of land uses, etc.); (d) dynamic effects (e.g., the time it takes for property value changes to percolate through a whole neighborhood).

After the observations from this test have been confirmed and better understood, the model can be used in several ways. The linear polymer model could help understand the way in which special events (targeted investments in deteriorating neighborhoods or brownfields in developing or stagnant areas) affect property values in space and time. The model could assist public or private sector decision-makers in exploring the effects of various intervention decisions and in optimizing their investments spatially and temporally for the greatest effect on urban property values. This tool could be calibrated for a specific urban area and would enable decision makers to ask "what if’ questions regarding proposed policies. Advocates of specific policies could use quantitative reasoning based on this tool to argue their positions. Researchers and policy makers could investigate the effects on property values of forecasted population size and distribution over the metropolitan areas.

References

1. A. Friederich, S. Kaufman, M. Kaufman, Fractals, 2, 469-471 (1994).

2. M. Kaufman, Physical Review B 39, 6898-6906 (1989).

FIGURE 1: Cluster Size Distribution for Strongsville Increases, 1976-1982
(A. Friederich, M. Sc. in Urban Studies Thesis, Cleveland State University, 1994).

FIGURE 2: Termination and Transmission Fugacities: Diamonds for Ohio City and Crosses for Strongsville.