How many coins are you carrying in your pocket?

Transcription

1 Physica A 354 (2005) How many coins are you carrying in your pocket? J.C. Nun o a,, C.Grasland c, F.Blasco a, F.Gue rin-pace d, J.Olarrea b, B.Luque b a Dpto. Matemática Aplicada a los Recursos Naturales, E.T.S.I. Montes, Universidad Politécnica de Madrid, Spain b Dpto. Matemática Aplicada y Estadística, E.T.S.I. Aeronáuticos, Universidad Politécnica de Madrid, Spain c Department of Geography, History and Social Sciences, Université Paris, 7 Denis Diderot, France d Institut National d Études Demographiques, France Received 22 November 2004; received in revised form 22 December 2004 Available online 19 April 2005 Abstract A computational model is defined to explain the distribution of coin content of French people s purses as obtained from a sequence of surveys carried out in France during the years 2002 and 2003.The agreement between theoretical and experimental data is noticeable.the model predicts the purse content in both Dollar and Yen monetary areas. r 2005 Elsevier B.V. All rights reserved. Keywords: Computational model; Payments habits; Coins; Monetary system; Optimisation 1. Introduction The Euro Spatial Diffusion Observatory designed a sequence of polls for studying the process of diffusion of euro coins in France over the last 3 years, after the new currency was brought into circulation [1].These surveys have revealed a characteristic distribution of the content of people s purses: an asymmetric curve that peaks at 10 coins, with an average value close to 14, and a long exponential tail Corresponding author. address: juancarlos.nuno@upm.es (J.C. Nun o) /$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi: /j.physa

2 J.C. Nuño et al. / Physica A 354 (2005) (Fig.1).This stationary distribution is a result of the interaction of several factors, the most important being the payment habits of people, the monetary system and the distribution of prices in countries of purchase.although a vast body of literature has been devoted to finding an optimal monetary system [2 5] and to analyzing price variation [6,7], much less attention has been paid to studying the way people behave within a particular monetary context. The lack of interest in which concerns with coins may be due to the fact that the amount of money circulating in coins can be neglected if compared with that in banknotes and has, therefore, little influence in macroeconomics.nevertheless, the eight denominations of euro does not coincide with the ones of the older monetary systems in Euroland.Decisions about how many coins of each denomination should be manufactured were mainly taken in the past in the light of experience accumulated over years.within the new system, this experience is no longer available and Treasury Departments have had serious troubles in accommodating the mint process to the real necessities [8].Needless to say that any deviation from the forecasted coinage represents an appalling business for the National Treasuries.Although originally motivated by the introduction of euro, it is obvious that this problem is extendable to other monetary systems. Fig.1. Distribution of number of coins in French purses is depicted (triangles).as a whole, the number of pockets analyzed is 5098 (from 8813 persons interviewed) containing a total of coins distributed among the 8 denominations.the most probable value is 10 coins, and the average value is around 14 coins.as can be seen, theoretical distribution corresponding to q ¼ 0:63 (circles) fits experimental data extraordinarily well.

3 434 J.C. Nuño et al. / Physica A 354 (2005) The model Taking the latter into account, we have developed a computational model that describes the temporal evolution of a hypothetical purse making cash transactions sequentially.the following simple rules govern the payment behavior: if the value of the purse is not enough to cover the payment, the lowest value banknote is used.if, on the contrary, the value is greater than the price to be paid, then two extreme strategies can be randomly chosen: (a) with probability q, and following the principle of least effort, the payment is made using the lowest number of coins necessary to either equal or exceed, by the minimum quantity, the price; (b) with a probability 1 q, following an emptying process, the highest number of coins is used.in either case, change is always given using the lowest possible number of coins.note that both strategies are opposed: whereas the first one tends to increase the number of coins in the pocket, the second one generally causes a decrease.as an important consequence, the hypothetical purse defined in the model self-regulates: it empties out when full without including any ad hoc strategy. Prices are chosen uniformly at random between 1 and the value of the lowest banknote denomination of the monetary system (e.g. 500 cents ¼ 5 euros in the euro zone; 100 cents ¼ 1 dollar in USA; 1000 yens in Japan).Although real prices could not be distributed uniformly, this survey is justified since the model is dealing with the part of the total price that people usually pay with coins.the simplest Fig.2.Different distributions obtained from the computational model applied to the euro monetary system by tuning the probabilistic parameter q.as q varies from 0 (pure unloading process) to 1 (pure least effort strategy), highly asymmetric bimodal distributions become flatter and peaked around larger numbers of coins.

4 J.C. Nuño et al. / Physica A 354 (2005) approximation is to consider this part as the real price modulo the lowest banknote value.as far as we know, there are no results about the behavior of modular distributions and, therefore, in the absence of information the best assumption seems to be uniformity. Initially, the purse is loaded at random with pieces (between 0 and 10) for each denomination.for each q-value, the hypothetical pursue makes 55,000 monetary transactions.to avoid effects of the initial conditions, the first 5000 transactions are not taken into account in the statistical analysis.according to the rules defined above, the purse content evolves with time, reaching a stationary distribution, that depends only on both the particular structure of the monetary system and the parameter q.these distributions, as depicted in Fig.2, are histograms between 0 and the largest content of the purse. 3. Results 3.1. Euro area The theoretical predictions obtained for the euro monetary system, with coins of 1; 2; 5; 10; 20; 50; 100 and 200 cents, are summarized in Fig.2.All distributions are non-symmetric and bimodal, varying from an almost single narrow peak ðq ¼ 0Þ, corresponding to a pure strategy (b), to flatter ones, as q tends to 1, when strategy (a) is predominant.noteably, the distribution corresponding to q ¼ 0:63 (circles) fits experimental data (triangles) extraordinarily well (see Fig.1). Although the simplicity of the model, this agreement is quite remarkable.so, we can stand that our computational model captures the essential properties of cash payments made with coins.the conclusion, from a social economical point of view, is that 63 percent of the times people pay without taking care about the change, and this makes that their purse content is as it is. The two peak shape of the distribution, which is an experimental result, is also an fundamental feature of our model.for each q-value, the purse behavior is a compromize between the two extreme strategies that occurs for q ¼ 0 and q ¼ 1.So, the coin distribution is a weighted mixture of the corresponding extreme distributions which, consequently, leads to bimodality Other monetary zones It would be useful to take into consideration purse content in other monetary zones.as before, it is assumed a uniform distribution of prices and similar payment habits to those in France, i.e., q ¼ 0:63.In dollar areas, with a monetary structure of 1; 5; 10; 25 and 50 cents, the content of citizens purses would be distributed similarly to the euro system, although with a slightly lower average number of about 11 coins. On the other hand, in the Yen area, formed by 7 denominations of 1; 5; 10; 50; 100 and 500 yens, the predicted distribution is broader and the average number of coins

5 436 J.C. Nuño et al. / Physica A 354 (2005) in the purses is around 23.A web-based survey is being designed to check both model predictions. Acknowledgements This work has been partially supported by Grants BFM and SEC E from the Spanish Ministry of Science and Technology. References [1] C.Grasland, F.Gue rin-pace, A.Tostain, Popul.Soc.384 (2002), Data about the distribution of total coins can be found in the web page: [2] A.E. Tschoegl, J. Money Credit Bank. 29 (1997) [3] M.Wynne, Econ.Lett.55 (1997) [4] J.Shallit, Math.Intell.25 (2003) [5] P.Ball, Nature Sci.Update (2003). [6] K.Basu, Econ.Lett.54 (1997) [7] A.Rátfai, Working Papers Series 301, European Central Bank, [8] ECFIN/427/03-EN, European Commission Report, 2003.