The distribution of landed property
aa r X i v : . [ phy s i c s . s o c - ph ] J a n The distribution of landed property
Pavel Exner , , Petr ˇSeba , , and Daniel Vaˇsata , Nuclear Physics Institute, Academy of Sciences of the Czech Republic, CZ-25068ˇReˇz near Prague, Czech Republic Doppler Institute for Mathematical Physics and Applied Mathematics, Bˇrehov´a 7,CZ-11519 Prague, Czech Republic University of Hradec Kr´alov´e, V´ıta Nejedl´eho 573 CZ-50002 Hradec Kr´alov´e, CzechRepublic Institute of Physics, Czech Academy of Sciences, Cukrovarnick´a 10, CZ-18000Prague 8, Czech Republic Department of Physics, Faculty of Nuclear Sciences and Physical Engineering,Czech Technical University, Bˇrehov´a 7, CZ-11519 Prague, Czech RepublicE-mail: [email protected], [email protected], [email protected]
Abstract.
The distribution of property is established through various mechanisms.In this paper we study the acreage distribution of land plots owned by natural personsin the Zl´ın Region of the Czech Republic. We show that the data are explained interms of a simple model in which the inheritance and market behavior are combined.PACS numbers: 05.65.+b, 89.65.Gh he distribution of landed property
1. Introduction
Quantitative study of human behavior requires reliable data and tools to analyze them.The latter improved dramatically in recent years when entries in various databasesbecame available in a digital form. From the sociological point this opens way to newhighly interesting empirical findings and theoretical endeavors to understand them.One of the first publicly available records were the telephone books which gave anopportunity to study the distribution of surnames. Interpreting the results in a modelsetting, the recent records found in the telephone book are understood as a result of along process of intermarriages and surnames inheritability, hence the surname statisticsrefers to social and population processes in the past [1]. In particular, it is known thatthe numbers of individuals sharing the same surname follows a scaling rule known as Zipflaw. Let us arrange the surnames in the descending order with respect to the number ofentries found in the telephone book, and let N (1) denote the number of records foundfor the most frequent surname, N (2) the number os records for the second widespreadsurname, etc.; then the Zipf law says that log( N ( k )) ≈ − c log( k ) with some constant c .The mechanism of inheritance from a single parent appears not only in the socialcontext: similarly to surnames it applies to nonrecombining alleles in the genom, henceit is not surprising that there is a close link between the surname distribution and thehuman genome [2]. Furthermore, such a situation is not restricted to surnames butapplies also to groups of people sharing a common native language, to the speciesdiversity in ecological systems and so on – cf. [3]. It demonstrates that culturaltraditions are transmitted from the ancestors to the descendants through a processanalogous to the genetic heredity and display a close parallelism between the culturaland biological evolution – recall that in the evolutionary psychology these mechanismsare studied under the name “memetics” [4, 5].In the field of economics the power law distribution is traditionally named afterthe classical work [6] of Vilfredo Pareto; it is used to describe phenomena such as thestatistics of personal income, or the allocation of wealth in a steady society [7, 8],however, also fluctuations of the stock prices or the land estate display such a behavior[9, 10].Our aim in this paper is to combine these two aspects into a single model capable ofdescribing situations in which both the ancestor–descendant dynamics and the marketbehavior occur. Specifically, we will analyze the distribution of the sizes of land plotsowned by individuals. It is clear that – in contrast to a corporative ownership – anatural person can acquire the land either on the real estate market or to inherit it.On one hand the land is transmitted from an ancestor his/her descendant similarly likesurnames, on the other hand it is a subject to changes by acquiring/selling the landon the real estate market. We are going to demonstrate that such a complex humanbehavior can be reasonably well described by a simple mathematical model.The empirical basis for our investigation are data obtained from the Czech Office forSurveying, Mapping and Cadastre describing the present status of the legal relations to he distribution of landed property
2. A simple real estate trading/inheritance model.
To motivate the model, note first that the Czech cadastral records, which we workwith, do not contain the land plot history, and consequently, the past real estatetransfers cannot be directly extracted from them. There are indirect clues that containinformation on carried transfers: any conveyance of real estate for consideration issubject to the real-estate-transfer tax. On the other hand, the inherited or donatedrealties are liable to the estate and capital transfer tax. Unfortunately, this tax islevied on all assets which makes the information about inherited realties obscured.Nonetheless, the taxing information represents the process of realty conveyance in themetric of the current prices. Recall that the price metric was used, for instance , toanalyze the mechanism leading to the crash of the Japanese land market at the end ofeighties – see [13]. It does not say much, however, about the actual acreage of the plotsand about its historical development. Another factor which makes its use questionableis that the land price is subject to unpredictable fluctuations; to quote an example fromthe same study, the mean price of one square meter of land in Japan increased in theyears 1983 -1993 nearly sevenfold [13]. The dynamics of land prices exhibits a volatilitywhich makes it difficult to describe – we refer to [14] and references therein for moredetails.With these facts in mind, we will focus on the statistical properties of the acreageof the land plots (cadastral units). They represent better objects to study because theplot sizes change little in the course of time and do not yield to market fluctuations.Most often the whole plot is conveyed and its size does not change at all; changes ofthe plot size are rare and are always related to a new surveying. The latter is typically he distribution of landed property
4a complicated and costly procedure, which is one more reason why the acreage is notvulnerable to speculations.We understand the recent records contained in the land registry as a descriptionof a steady state resulting from a long series of land inheritance and land trading.The total acreage is preserved, of course, and can be only redistributed among the newowners. We are going to describe the steady state by an agent-based approach developedoriginally to model the wealth of closed economies – see [15] for a review. For the sakeof simplicity we will use the most elementary version designed initially to describe thesocial stratification – see [8] – it is also known as the “inequality process”.As usual in such situation we use discrete time proceeding in steps typical foran ownership change; one can think of them as of generations. Let S k ( n ) denote theacreage of the cadastral unit k at time n and S k +1 ( n ) be the the acreage of its geometricalneighbor, by that we mean that S k ( n ) and S k +1 ( n ) have a common border. Then themodel cadastral dynamics we propose proceeds to the next generation in the followingway, S k ( n + 1) = λS k ( n ) + a ((1 − λ ) S k ( n ) + (1 − µ ) S k +1 ( n )) (1) S k +1 ( n + 1) = µS k +1 ( n ) + (1 − a ) ((1 − λ ) S k ( n ) + (1 − µ ) S k +1 ( n ))where λ, µ ∈ [0 ,
1] are independent random variables with identical distributions and a is a Bernoulli variable taking values a = 0 ,
1, each with probability 1 /
2. The meaningof the dynamical equations (1) is straightforward. With probability 1 / a = 1in which case the size of the plot k increases, S k ( n + 1) = S k ( n ) + (1 − µ ) S k +1 ( n ), whilethe size S k +1 of the unit k + 1 is reduced, S k +1 ( n + 1) = µS k +1 ( n ). In other words,the cadastral unit k of original acreage S k has incorporated a part of the neighboringland plot k + 1. As a result, we get (after a surveying and introducing into the landregister) two new cadastral units of acreages S k ( n + 1) and S k +1 ( n + 1). With the sameprobability we have a = 0 in which case the plot k shrinks, S k ( n + 1) = λS k ( n ), with apart of it being incorporated into k + 1 giving S k +1 ( n + 1) = S k +1 ( n ) + (1 − λ ) S k ( n ).We are interested in the probability distribution of the steady state to which S k ( n )converges as n → ∞ . A general mathematical result ensures that the limit of such aprocess exist and is unique – see [16]. It depends, of course, on the statistical propertiesof the variables λ and µ and can be numerically evaluated by iterating the mapping(1) – cf. [17]. Moreover, in analogy with [18] one can argue that the convergence isfast coming close to the equilibrium in just a few generations, hence one can expectthe model will be applicable to processes of land plot redistribution provided they runundisturbed for at least a century.The variable λ describes the land conveyance and contains in such a way theinformation about the generation dynamics (we refrain from mentioning the variable µ all the time since it has identical statistical properties). The mechanism is simple:assume that the owner of a cadastral unit has three offsprings. When he or she dies, thechildren become after the appropriate inheritance procedure co-owners of the cadastralunit with the one-third share each; the fact is recorded into the land register. We he distribution of landed property /
3, i.e. we have λ = 1 / p n,m that a fraction n/m with m > n of the cadastral unit is in co-ownership. To take then into accountthe genealogical part od the land conveyance we use this information to define theappropriate distribution of the statistical variable λ : we suppose that it takes the value λ = n/m with probability p n,m whenever m ≥
2. We can leave out the case m = 1 whenthe land plot is inherited as a whole, S k ( n + 1) = S k ( n ), noting that the identity of theowner is not important.The inheritance does not tell us the whole story, of course, since a part of the realestate is traded on the market. This concerns, in contrast to the above, the plots with n = m = 1 having a sole proprietorship with λ = 1. Such plots – being fully ownedby a single person – are subject to free trading. For simplicity we will assume thatthe trading mechanism is completely random in full analogy with the closed economymodels [15]). In other words, we suppose that on the subset of plots with n = m = 1the quantity λ is a uniformly distributed random variable.To summarize this part, we have formulated the model of exchange of cadastral unitsgoverned by the dynamical map (1). For units that are in a co-ownership of differentpersons the variable λ equals to the corresponding share and enters the equations withthe probability matching the relative appearance of the given share among all thecadastral records. On the other hand, for cadastral units have a sole proprietorshipthe variable λ is random and uniformly distributed. In such a way if a piece of realestate is co-owned only the appropriater share can be traded, while those with a singleproprietorship can be on the other hand traded without limit in a fully random way.The corresponding dynamics can be now investigated numerically. One finds thatthe process converges fast, as we argued above, and the resulting equilibrium distributionis not sensitive to the initial state of the cadastre. In the next section that will comparethe steady result of this linear model with the data and show that it leads to an amazingagreement with the acreage distribution extracted from the Czech Land Registry. he distribution of landed property
3. The results
To begin with we note that if the random part of the dynamics (1) is omitted theland possession becomes fractalized. As a mathematical result about iterative mapswith Bernoulli variables it was demonstrated not so long ago – see, for instance, [19].The essence of the effect, however, was a fact well known to our ancestors and it wasbehind various juridical acts attempting to prevent a land structure disintegration.One can mention the so-called birthright edict – a part of the legal reform of theHabsburg Emperor Joseph II dated 1787 – which did not allow to split the land below40 “measures”, which is roughly 7.6 ha or 19 acres. This law was valid until the year1869 when the rural land market – representing the random part of the dynamics (1)– became vivid enough and the edict became obsolete. The imperial act allowing thefree divisibility and exchangeability of land (in Moravia, from which our data are taken,it was valid already one year earlier, in 1868) caused an unprecedented increase of theland exchange. For example, according to [20] in average 10% of estates changed theproprietary relations yearly during the period 1869–87.What we find in the present cadastral data is thus a result of a process that startedin 1868, some five generations ago. At the beginning it was marked by an boostedland exchange and the dynamics became steady only about a generation later. Whilethese fact are documented, one may speculate about other factors having impact of thedynamics, in particular, changes in the population structure. It is obvious that thatthe natality and the number of offsprings living up to the adulthood were considerablyhigher at the beginning of the process, and as a result, the distribution of the landco-ownership was different from the present data. Unfortunately, the digitalized Czechland registry does not contain such a history record – all we see is a snapshot of therecent situation. In the light of these comments indicating how simplified our modeldynamics (1)) is, it is surprising and worth of attention how well do its result agree withthe observed acreage probability distribution.One more comment is needed before the results can be presented. We are dealingwith all the plots contained in digitalized cadastral map with the exception of the buildup areas . The reason for this methodical choice is simple: build up areas are placeswhere building stand. In the cadastral map, however, we see only the ground plane,i.e. the projection of this building which does not tell us what the latter is like: it canbe a simple one floor house as well as a high-rise construction. An information on thebuildings is not a part of topographical data collection being contained in a separateregistry, and some parts of the building (flats) can be traded leaving the ground planunchanged. This is why we exclude the build up areas from the considerations and willdiscuss them in a separate paper.For the comparison we have used the cadastral records of the Zl´ın Region, aterritorial unit of the Czech republic of the area 3964 km and population of 590thousand. We selected all the plots owned by individuals, excluding those owned bycompanies. It is clear that the plots owned in a co-ownership, or in the so-called he distribution of landed property −4 −3 The plot acreage p r ob a b ili t y d e n s i t y the cadastral datamodel results Figure 1.
The probability density of the a given plot acreage found in the cadastralrecords (black crosses) is compared with the distribution obtained by iterating the map(1) (blue points). undivided co-ownership of spouses are recorded repeatedly times in the data, hencethe duplications were removed to get a meaningful size distribution. As indicate aboveall the build up areas were deleted from the sample. This left us with the collectionof 1200121 plots which we could use to work out the size statistics and to evaluate theacreage probability density.The same cadastral data were used to evaluate the probabilities p n,m of theparticular shares λ = n/m describing the plot co-ownership as described in the previoussection. The map (1) was then iterated starting from the uniform distribution, S k (1) = 1for all k . The result was subsequently rescaled to the mean which equals the mean plotsize found in the cadastre and the two probability densities were compared; the resultis plotted on the Figure 1.We see that the two match perfectly; it is worth to stress that the model containsno free parameters . On the other hand, it is obvious that the data do not displaythe Pareto behavior. This fact is easily explained; it is enough to realize that the datarepresent the acreage probability of the plots and not the the probability of the individualbelongings. The point is that one person can own more plots and their acreage (countingof course with the related co-ownership fraction) has to be summed. If we do this thePareto behavior is restored as can be seen from the appropriate distribution plot on theFigure 2. he distribution of landed property −5 −4 −3 The acreage ot total landed property p r ob a b ili t y d e n s i t y Figure 2.
The probability density describing the acreage of the total landed propertyowned by individuals in the Zl´ın Region. The Pareto behavior is clearly visible
Acknowledgments
The research was supported by the Czech Ministry of Education, Youth and Sportswithin the project LC06002 and the Grant Agency of the Czech Republic No.202/08/H072. We are indebted to Helena ˇSandov´a and Petr Souˇcek from the
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