Modelling Size Distributions of Rural Land Holdings in Scotland

The paper reports ongoing investigations on the size distribution of rural land holdings in the real world (specifically, in Scotland), and in the output of two simulation programs: a simple numerically-based simulation of the partition and repartition of a fixed-size resource between a fixed set of entities (referred to here as R-SG), and an agent-based model of rural land use and land ownership change, FEARLUS. Introduction: Size Distributions Given any set of entities with associated positive magnitudes (sizes), we can ask how these sizes are distributed. Describing and explaining the statistical properties of size distributions has long been of interest to social scientists, with particular attention having been paid to the distribution of income and wealth (Kleiber and Kotz 2003), and to that of city sizes (Carroll 1982). Recently, this area has been investigated using agent-based modelling (Cederman 2002; Moss 2002), and closely related work has been undertaken on the distribution of different land uses within cities (Batty, Xie et al. 1999). To our knowledge, however, there have been few previous attempts to model the development of size distributions of land holdings, although (Balmann 1997) has modelled processes including the consolidation of farms. We are concerned here with distributions which include many more entities below mean size than above it (as opposed, for example, to the normal distribution, which is symmetrical about the mean); such asymmetical distributions typically occur when the sizes of events such as earthquakes (Bak 1997) and wars (Richardson 1960) are considered, or when members of the set of entities in some sense compete for or ex1 Macaulay Institute, Aberdeen, Scotland 2 Departments of Geography and Environmental Science and Policy, and Center for Social Complexity, George Mason University, Fairfax, Virginia, USA clude each other from resources. Three families of distribution are of particular interest here: 1) Exponential. Here, the probability that a randomly chosen entity has a size X at least equal to a given value X0 (taking X0 greater than or equal to the size of the smallest entity) declines exponentially as X0 increases: ) / ) exp(( ) ( 0 0 b X a X X P − = ≥ , where a and b are constants, with a equal to the smallest entity’s size. 2) Power law. Here, the probability that a randomly chosen entity has a size X at least equal to a given value X0 (greater than or equal to the size of the smallest entity) declines with a (negative) power of X0: b X a X X P 0 0 / ) ( = ≥ , where b > 0. When the logarithm of this probability is plotted against the logarithm of X0, the result is a straight line. 3) Lognormal. As its name suggests, the logarithms of values in a lognormal distribution have a normal (or Gaussian) distribution. Size distributions may be described in several ways. Those used here are: 1) The complementary cumulative density function (CCDF) approach. The CCDF is just the function producing for any value X0 the probability referred to above: that a randomly chosen entity will have a size at least equal to X0. 2) The rank-size approach, in which each entity is assigned a rank, with the largest having rank 1, the next rank 2 and so forth. The ranks can then be plotted against the corresponding sizes. The two approaches give very similar results, but each is sometimes more convenient than the other. Power law size distributions have been reported from a wide range of phenomena involving networks of autonomous but interacting entities, including forest fires, networks of rivers and blood vessels, stock prices, WWW links, scientific publications by authors, wars, personal income and wealth, firm size and market share, city size, and the distribution of land uses within cities (Richardson 1960; Schroeder 1991; Bak 1997; Batty, Xie et al. 1999; Gabaix 1999; Axtell 2001; Barabási 2002; Moss 2002; Reed and McKelvey 2002) – although it is not evident that one general explanation accounts for this. The work reported was begun in the expectation that the sizes of land holdings in Scotland might show this kind of distribution. Rural Land Ownership in Scotland Data on rural land ownership in Scotland is not easy to obtain (Wightman 1996). The sources of real-world data used in the work reported here are Wightman (1996), Wightman (2003) and Wightman (2004). Figure 1 shows the size distribution of the 1411 land holdings of 1,000 acres (approximately 400 hectares) and over, from the most recent information available (the 1970 distribution is not much different). The rank-size approach is employed. At left, a log-log plot is used (logarithms to base 10): a perfect power-law distribution would appear as a straight line; on the right, the logarithm of the rank is plotted against the size: a straight line on this plot would indicate an exponential distribution. The appearance of a downward bending curve at left and an upward bending curve on the right that the true distribution is intermediate between the two; results from regression analysis suggest that a power-law with a slope of around -0.89 gives a rather better fit than any exponential, although a KolmogorovSmirnov test (NIST/SEMATECH 2004) comparing the observed distribution with a constructed power-law distribution shows that the distribution is not a pure power-law with the slope and intercept given by the straight line on the log-log plot (value of the D-statistic ≈.1616, giving a p-value < 10, meaning we can reject the hypothesis that the distribution does follow that specific power-law) 3. Fig. 1. Scottish land holding size distribution. Left: power-law distribution would produce a straight line. Right: exponential distribution would produce a straight line. Further information about the distribution can be gained by considering its “upper tails”: the largest n land holdings, where n ≤1411. Figure 2 shows the R values from regression analysis distributions for the upper tails of lengths between 10 and 1411, assuming a power-law distribution at left, and an exponential on the right (R is a measure of the proportion of the variance that can be explained in terms of a particu3 Note that this procedure can only give us an idea how well a specific power-law distribution fits the data; low p-values do not allow us to reject the hypothesis that the distribution could be produced by some power-law generating process. lar type of distribution). It can be seen that both measures tend to decline as the number of holdings considered (the length of the upper tail) is increased, but that a powerlaw appears to give better results. The same is true for Kolmogorov-Smirnov p-values obtained by comparing the observed distributions with constructed power-law and exponential distributions for the same upper tails, and for the t-ratio of the slope estimate obtained from regression analysis, although in the case of the t-ratio of the intercept estimate, the value for the exponential distribution exceeds that for the powerlaw distribution for upper tails longer than 45. It therefore appears clear that the upper end of the distribution in particular resembles a power-law more closely than an exponential. Fig. 2: Scottish land holding size distribution. Left: R from regression of upper tails of log of holding size rank on log of holding size (assumes power-law distribution). Right: R from regression of upper tails of log of holding size rank on holding size (assumes exponential distribution).