December 26, 2006

15:58

WSPC - Proceedings Trim Size: 9in x 6inkichiji-nishibe659
1

POWER LAW DISTRIBUTIONS IN TWO COMMUNITY
CURRENCIES
N. KICHIJI
Department of Engineering, Hokkaido University, Kita 13, Nishi 8, Kita-ku, Sapporo,
Hokkaido 060-8628, Japan,
E-mail: kichiji@topology.coe.hokudai.ac.jp
www.topology.coe.hokudai.ac.jp/
M. NISHIBE
Graduate School of Economics, Hokkaido University, Kita 9, Nishi 7, Kita-ku,
Sapporo, Hokkaido 060-0809, Japan
E-mail: nishibe@econ.hokudai.ac.jp
The purpose of this paper is to highlight certain newly discovered social phenomena that accord with Zipf’s law, in addition to the famous natural andsocial phenomena including word frequencies, earthquake magnitude, city size,
income1 etc. that are already known to follow it. These phenomena have recently been discovered within the transaction amount (payments or receipts)
distributions within two diﬀerent Community Currencies (CC) that had been
initiated as social experiments. One is a local CC circulating in a speciﬁc geographical area, such as a town. The other is a virtual CC used among members
who belong to a certain community of interest (COI) on the Internet. We conducted two empirical studies to estimate the economic vitalization eﬀects they
had on their respective local economies. The results we found were that theamount of transactions (payments and receipts) of the two CCs was distributed
according to a power-law distribution with a unity rank exponent. In addition,
we found diﬀerences between the two CCs with regard to the shapes of theirdistribution over a low-transaction range. The result may originate from thediﬀerence in methods of issuing CCs or in the magnitudes of the minimumvalue unit; however, this result calls for further investigation.
Keywords: Power law; Zipf’s law; Community Currencies; Social Experiment.

1. Introduction
After the full liberalization of capital movements in the early 1980s, freecapital ﬂows have generated new risks of instability. Because of this situ-

December 26, 2006

15:58

WSPC - Proceedings Trim Size: 9in x 6inkichiji-nishibe660
2ation, Community Currencies (CCs)a have spread over the world, particularly in Western countries since the 1980s-the second CC boom after theﬁrst one in the 1930s. However, little research and few studies have been
carried out to quantify how much these CCs vitalize local economies. Since
the two communities of the CC cooperated, we were able to conduct two
empirical studies in which we quantitatively estimated the economic eﬀects
of those two CCs. We discovered that Zipf’s law applies to the distribution of the transaction amounts within those CCs; the results indicate that
economic transactions create Zipf’s law2–4 without considering the type ofCCs, because it is well known that income and wealth distribution withlegal tender (e.g., dollars or yen) follow Zipf’s law. Section 2 deals with twoCCs. Section 3 explains Zipf’s law. Finally, we brieﬂy comment on future
research in this area.
2. Two Community Currencies
We conducted two empirical studies to estimate the economic vitalizationeﬀects of two diﬀerent CCs on local economies. One of these CCs is a local
one that circulates in a speciﬁc geographical area, and the other is a virtual
CC that is used between members who belong to a certain community of interest (COI) on the Internet. The ﬁrst CC is called Tomamae-cho (“town”in Japanese) Chiiki-Tuka (“Community Currency”)b. Tomamae-cho is asmall town in the northern part of Hokkaido in Japan with a population of
about 4300 people. Its main industries are ﬁshing, forestry, and commerce.
Tomamae-cho has recently been faced with problems such as scarcity of job
opportunities and spillover of purchasing power to relatively large neighboring towns (with population of 8000), in addition to long-term depopulation
and aging, as in other rural areas in Japan. In 2004, the Tomamae-choCommerce and Industry Association decided on an experimental introduction of a CC for the purpose of both economic activation and community
revitalization, in collaboration with the Tomamae-cho Municipal Government.
The Tomamae-cho CC, called “P,” has a linear equivalence to “yen,”and the circulation works in two ways: (1) units are issued by the Commercea Community Currencies are also called Local Currencies or Complementary Currencies,
with slightly diﬀerent connotations. For example, Local Currencies often means the CCs
that circulate in conﬁned local areas such as towns or villages. However, we will use only
the term Community Currency to avoid any confusion in terminology.
b This is legally a Gift Certiﬁcate redeemable into yen, so that shoppers may use it again
to others or refund it with money when they receive it from their customers.

December 26, 2006

15:58

WSPC - Proceedings Trim Size: 9in x 6inkichiji-nishibe661
3and Industry Association and the Municipal Government of Tomamae-cho
when participants buy CC with yenc , (2) participants get paid with CC for
the service or goods they supply to others.
We have participated in the CC experiment and investigated its circulation ﬂow in a unique way. Before the experimental circulation in 2004,
our investigation team requested the issuing agents to devise an entry spaceon the back of each CC note for its recipients to record transaction dates,
their names and addresses, and the purposes of use, for a maximum of ﬁve
recipients. We also asked the participants to agree to cooperate in our investigation. After the circulation experiment of the Tomamae-cho CC, wecollected all the currency notes that had returned to the issuing agents inexchange for yen, and we then created the adjacent matrix from the acquired data to analyze the circulation ﬂow of the notes of the CC by using
a social network theory.
The virtual CC is called LETS-Qd . Q is the “monetary” unit of LETSQ; although it is not exchangeable with yen, it is equivalent to yen. TheCC system was initiated in Japan in November 2001 and is managed by anadministrative committee called Q-hive, comprising individuals such as professors, teachers, farmers, students, artists, and musicians. One of the authors, Makoto Nishibe, is the initiator of LETS-Q and an ex-representativeof Q-hive. Coﬀee shops and bakeries are also members of LETS-Q apartfrom individual members, and are scattered all over the world. Some ofthem are in Tokyo and Osaka, and others in Hokkaido and even in New
York. The other author, Nozomi Kichiji, is also a member of LETS-Q.
The characteristic feature of LETS-Q lies in the realization of the same
type of multilateral settlement of accounts in the Internet as in the caseof usual LETS systemse . Using this technique, participants are relieved ofthe constraints of geographically localized communities such that they caninvolve in LETS-Q irrespective of their location. Therefore, we call this typeof online LETS “open communities” or “glocal” currency. Since LETS-Q
was implemented on the Internet, and all related data are stored in thec 500 ‘Yen’ can buy 500 ‘P’ note and 10 ‘P’ stamps as its 2 percent premium.

d LETS stands for a Local Exchange Trading System. This system embodies the mostpopular type of Community Currency; it was initiated in 1983 by Michael Linton in
Comox Valley, Vancouver Island, Canada, and is now utilised in more than 2000 districts
worldwide.
e All participants in LETS start from zero in their account and cumulatively record
plus and minus transactions within their accounts when they oﬀer and receive goods orservices from other participants. It is a characteristic feature of the LETS system thatthe summation of all accounts remains constantly at zero.

March 2, 2007

15:37

WSPC - Proceedings Trim Size: 9in x 6inkichiji-nishibe6462

LETS-Q server, we have been able, with the help of Q-hive, to access theLETS-Q data; these data contain only the ID numbers of participants, thetransaction amount, and the time stamps of settlement on the web. It does
not include personal data such as name, address, and age. Using this data,
we can analyze economic phenomena at the meso level of a small-sizedgroup.

3. Zipf ’s Law
Zipf’s law, named after the Harvard linguistic professor George Kingsley
Zipf, is the observation that frequency of occurrence of a certain event(P ),as a function of the rank (i) when the rank is determined by the above
frequency of occurrence, is a power-law function Pi ∼ i1a with the exponent
close to unity (1). It is widely known that Zipf’s law applies to city size,income, word frequency, and earthquake magnitude. However, the mechanism of Zipf’s law has not been clariﬁed enough; accordingly, it is desirable
to ﬁnd more events to which Zipf’s law applies. In this paper, we havepresented four such events. We examined the distribution of payments andreceipts of Tomamae-cho CC and LETS-Q over a certain period. The results are illustrated in Fig. 1 and Fig. 2. All ﬁgures show the rank-size
relationships of the transaction amounts, the x axis is the rank, and y axis
is the size. These relationships are approximately linear on a log-log plot; inother words, the ranked slopes in both the cases are nearly -1, which makes
them Zipf. Thus, we discovered Zipf’s law distributions in the transaction
amounts within the two diﬀerent CCs (local and virtual). These results indicate that transaction amounts follow Zipf’s law irrespective of the typeof CCs.
However, the slope of the line of best ﬁt of the Tomamae-cho CC (-1.13)
is steeper than that of LETS-Q (-1.00). This indicates that the LETSQ is slightly more level and even than the Tomamae-cho CC in terms oftransaction amounts. Moreover, the ways the two graphs ﬁt to linear lines,
as is clear from the comparison between Fig. 1 and Fig. 2, are quite diﬀerent.
The distribution of transaction amounts of the Tomamae-cho CC almost
perfectly ﬁts a straight line, but that of the LETS-Q deviates from it after
rank 100. It can be seen that the overall Gini coeﬃcientsf of the Tomamae-f The Gini coeﬃcient is a measure of inequality of a distribution, deﬁned as the ratio of

area between the Lorenz curve of the distribution and the curve of the uniform distribution, to the area under the uniform distribution.

March 2, 2007

15:37

WSPC - Proceedings Trim Size: 9in x 6inkichiji-nishibe6563

cho CC are smaller than those of the LETS-Qg . However, when calculating
the Gini coeﬃcients for LETS-Q for the range of rank 1 to 100, we obtainresults smaller than those for Tomamae-cho CCh . These results indicatethat the Tomamae-cho CC is more equal than the LETS-Q on the whole
but LETS-Q is more equal than the Tomamae-cho CC under rank 100;
this twist is caused by the aforementioned deviation of the distribution oftransaction amounts with LETS-Q from the line of best ﬁt over a lowtransaction range.
The subject of concern is the reason for such a deviation in the case ofLETS-Q. There are two possible explanations for this. The ﬁrst is due to thediﬀering rules under which the CCs are issued. The Tomamae-cho CC can
be bought and redeemed with yen, although the same is neither permitted
nor possible for LETS-Q. This creates the diﬀerence of constraints on theboundary conditions of the two CCs. The other reason is more technical;
in the Tomamae-cho CC, the minimum value unit is 500 P, because only
500 P notes are issued, but for LETS-Q it is just 1Q, because transaction
amounts with arbitrary numbers can be recorded in the participants’ accounts. This seems to make the lower-right tail of the LETS-Q longer thanthat of the Tomamae-cho CC. It would appear that the two types of ﬁtof the distributions to Zipf’s law reﬂect the diﬀerences between the innermechanisms and the dynamics of these two CC systems.
4. Summary
It is widely recognized that Zipf’s law applies to income and wealth distribution of individuals or corporations with legal tenders. In this paper, it
has been shown that Zipf’s law applies to diﬀerent kinds of socio-economic
phenomena, such as transactions in CCs. The results suggest that a genericeconomic transaction can create Zipf’s law, regardless of the type of currency. Moreover, we also refer to the diﬀerence with regard to how well thedistributions ﬁt Zipf’s law, signifying the diﬀerence of inner mechanisms
and dynamics of the two CC systems. We observe that the distributionsof transaction amounts of both CCs evolve over time, but further study isrequired on the relation between the evolution of distributions and Zipf’slaw.
g The

overall Gini coeﬃcients for receipts/payments with Tomamae-cho CC are0.757641/0.719131, and those for LETS-Q are 0.772328/0.735642, respectively.
h The Gini coeﬃcients for receipts/payments with LETS-Q for the range of ranks 1-100
are 0.62885/0.558051.

December 26, 2006

15:58

WSPC - Proceedings Trim Size: 9in x 6inkichiji-nishibe664
65

RANKED SLOPE=−1.17−1.21

10

4

10

Amount of Payment
Amount of Receipt

610
Amount of Transaction

Amount of Transaction

Amount of Payment
Amount of Receipt

RANKED SLOPE=−1.00−1.11

410231010011010210

Rank of Amount of Transaction

Fig. 1. Individual Rank by its transaction amount in Tomamae-cho CC.
Rank of amount of transactions andthe amount of transactions are plottedin the log-log scale. The plotted data
are for Nov.2004-Feb.2005. The total
amount of transactions is 1,385,500 P.
The ranked slopes are from -1.17 to 1.21.

01011010

2

Rank of Amount of Transaction

Fig. 2. Individual Rank by its transaction amount in LETS-Q. The rankof amount of transactions and theamount of transactions are plotted inthe log-log scale. The plotted data
are for Dec.2001-Oct.2002. The total
amount of transactions is 10,220,305Q.
The ranked slopes are from -1.00 to 1.11.

Acknowledgments
We like to express our appreciation to LETS-Q, Q-hive, Tomamae-cho Commerce and Industry Association and Tomamae-cho Municipal Government
for their support and cooperation in our research; and to Takayoshi Kusago,
Ippei Hozumi, Kenichi Kurita, Kenichi Yamamoto, Masayuki Yoshida, andSatoshi Yoshii for their collaborations in Tomamae-cho CC experiments.
We also thank Satoshi Tanda and Victor M. Yakovenko for their adviceand comments. This work was supported by Hokkaido University’s 21stcentury COE program “Topological Science and Technology.”
References
1. V. Pareto, Le Cours d’Économie Politique (Macmillan, London, 1897).
2. G. Zipf, Human Behavior and the Principles of Least Eﬀort (Addison-Wesley,Cambridge, 1949).
3. L. Adamic and B. Huberman, Glottometrics 3, 143 (2002).
4. L. A. Adamic, Zipf, power-laws and pareto - a ranking tutorial (2002), http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html.