The Demand for Currency versus Debitable Accounts: a Reconsideration

Transcription

1 The Demand for Currency versus Debitabe Accounts: a Reconsideration Bounie D., A. François and N. Houy October 2, 2007 Abstract Payment choice modes based on transaction sizes (TS modes) induce strong predictions about the use of payment instruments. Especiay, a equa-sized transactions shoud be paid with the same payment instrument. Then, for each individua, one shoud observe strict domains of transaction for every payment instruments. Using micro-eve payment data from a representative sampe of the French popuation, we show that TS modes are bad at repicating individua and aggregate payment patterns. First, we show that the predictions of TS modes are not empiricay vaidated on an individua eve. Second, we deveop and test three modes to expain the observed aggregate payment patterns. The first two modes are aggregate versions of TS modes and the third one is an aternative mode based on a payment decision rue depending on cash hoding (CH mode). We find that the third mode gives predictions between 2 and 6 times more precise than the first two modes with notaby ess demanding information on individuas. Key Words: Demand for money, Payment Instruments, Forecasting and Simuation. JEL Cassification: E41, E17, G2. This research benefited from financia support from the Groupement des Cartes Bancaires "CB" and the CNRS under the programme "ATIP Jeunes Chercheurs". We thank Yves Randoux for his support. ESS, Teecom Paris. LARGE, Strasbourg University and ESS, Teecom Paris. THEMA, Cergy-Pontoise University and ESS, Teecom Paris. Corresponding author: nhouy@free.fr.

2 1 Introduction For severa decades, monetary economists have attached a considerabe interest in modeing demand for currency and deposits. 1 Within a wide range of contributions, two distinct ines of anaysis have been proposed. The first one, represented by Santomero (1974, 1979), Santomero and Seater (1996) and their references, have deveoped forma modes in which demand for currency and deposits is directy function of return rates and fixed costs of transaction. Athough important, these modes have never been empiricay tested since they are highy sensitive to discontinuities in payment behavior and since a discontinuous money demand function does not occur in practice (Fokertsma and Hebbink, 1998). The second ine of anaysis promoted by Whitese (1989, 1992) considers that the trade-off between currency and deposits is mainy impacted by the use of payment instruments. More precisey, Whitese s modes show that when the transaction costs for each payment instrument differ and when this difference depends on the transaction size, the demand for currency and deposits may be consideraby affected. A centra impication of these modes is that a equa-sized transactions wi be paid with the same payment instrument and hence, each payment instrument wi be used excusivey on a particuar domain of transaction: cash for transactions with size beow a given threshod and aternative payment instruments (debitabe account) for transactions with size above the same threshod. Hence, these modes are based on transaction sizes and are caed hereafter "TS modes". This second ine of anaysis has been extended in a recent theoretica contribution by Shy and Tarkka (2002) for severa payment instruments. It has aso been tested in empirica studies. Mot and Cramer (1992), Boeschoten (1998), Hayashi and Kee (2003) as we as Bounie and François (2006) have shown that the probabiity to use a given payment instrument was highy infuenced by the 1 Debitabe accounts and deposits are not differentiated in this artice. 1

3 price of the transaction. However, even though the transaction size can be considered as a statisticay significant variabe in expaining the choice of payment instruments, the abiity of TS modes to repicate individua and aggregate payment patterns have, to the best of our knowedge, never been tested. The main question we tacke in this artice is the foowing. How we do TS modes aow to account for the demand for currency and deposits at an individua and at an aggregate eve? In order to answer this question for the individua eve, we use payment data from a representative sampe of the French popuation and we find that 32.3% of the peope who hod cash and an aternative payment instrument do not behave as TS modes predict. Indeed, for those individuas, we do not observe strict domains of transaction for the payment instruments but rather overapping transaction domains. This resut impies that TS modes do not success in repicating the observed payment patterns at an individua eve. In order to answer the question at an aggregate eve, we test three modes for their abiity to repicate the observed aggregated data. The first two modes are TS modes with two different aggregation rues. The first one is naive and considers the whoe popuation as a representative agent satisfying the assumptions of the TS modes. As expected, this mode performs very poory. The second mode considers a representative agent with a probabiistic threshod à a Whitese. The probabiity distribution of the threshod is given by some actua reaistic threshods observed in the data. This atter mode performs much better but is very demanding in terms of information. The third mode we deveop is an aternative mode of payment choice by a representative agent based on a simpe payment decision rue, based on cash hoding ("CH mode"). We find that the CH mode best fits the data of the observed aggregate payments. More precisey, we show that the CH mode gives predictions between 2 and 6 times better than aggregate TS modes with notaby ess requiring information basis on individua payment 2

4 patterns. Indeed, the CH mode simpy reies on three sorts of information: the distributions of prices and cash withdrawas in the economy and the vaue of a threshod beow which the representative agent goes back to withdraw cash. The CH mode can be considered as a more efficient and ess compex representation of the way a set of individuas pays. The remainder of the paper is structured as foows. In the second section, we present the iterature reated to TS modes and test their performance at the individua eve. We show that our micro-eve data party invaidate TS modes. In the third section, we propose three modes of aggregate payment pattern. Two are derived from TS modes, the ast one, the CH mode, is aternative. We show that the ast mode fits best the observed payment data. In the fourth section, we discuss the possibiity of extending our study for more than two payment instruments. The ast section concudes. 2 The TS modes of individua decisions 2.1 Theory The iterature on transaction size based modes (TS modes) goes back to the transaction demand for money à a Baumo (1952) and Tobin (1956). In these studies, a cost-minimizing consumer has to decide the optima stock of cash to be hed for transaction purposes given the cost of a withdrawa (a fixed fee per withdrawa) and the interest earnings foregone on money hoding. Extending this approach to severa payment instruments, Whitese (1989, 1992) expicity assumes that the consumer s probem during a purchase merey consists in choosing a payment instrument which minimizes the hoding and transaction costs. In Whitese (1989), for instance, an individua is assumed to make purchases of various prices P, with 0 < P <. Transactions are supposed to be uniformy distributed over a continuous unit period. Then, if F(P) represents the vaue of spending on a transactions 3

5 of price P, the expected tota spending is given by Y = F(P)dP. The individua has the possibiity to pay with cash or with an aternative payment instrument. Each transaction carried out with the aternative payment instrument is attended by some costs. Let us note u the fixed cost of using the aternative payment instrument for a transaction of size P and v.p the variabe cost reated to P. The tota cost of a transaction of size P is then C = u + v.p. On the contrary, paying cash is attended ony by a variabe cost w.p for a purchase of size P. This cost is due to interest earnings foregone on money hoding. Then, the aternative payment instrument wi not be used for sma vaue transactions because of u and, if the parameters are we chosen, cash wi be ony used for sma vaue payments, for which the interest earnings foregone wi be minima. 2.2 An empirica verification of the TS modes In this section, we test the performance of the TS modes. In so doing, we use data coected on payment patterns of French representative individuas. First, we present the methodoogy of the survey and the data coected. Second, we perform the test Methodoogy of the survey We administrated a survey from March to May 2005 on a representative sampe of 1,447 French individuas of 18 years and oder. The survey was reaized in two steps. 2 2 We used the quota method of samping; individuas from nine regions (defined by the Nationa Institute of Statistics (INSEE)) and iving in five categories of urban aggomeration were recruited (fewer than 2,000 inhabitants, from 2,000 to 20,000 inhabitants, 20,000 to 100,000 inhabitants, more than 100,000 inhabitants, Paris area). The resut of the crosscomparison of the area and the aggomeration is represented in the sampe in proportion to its demographic weight. The representativeness of the sampe was controed from the foowing quota variabes: "gender" crossed with "work status" (empoyed or not) (in order to avoid an over-representation of "women" within "unempoyed"); "age" in five casses; "socio-professiona category" in eight casses by distinguishing "retired" from "others"; the "type of residence" ("individua house" or "apartment in coective buiding"). 0 4

6 First, we surveyed individuas, during face-to-face interviews, in order to coect information on their payment instrument equipment, their cash management, etc. Second, we asked each respondent to keep a diary in which they reported a information reated to purchases on a daiy basis, for eight days. 3 A purchase is characterized by severa pieces of information such as the amount to be paid (size of transaction) and the type of payment instrument used to sette the purchase (cash, check, payment card, etc.). As a resut, we were abe to cacuate, for each respondent, the smaest and argest transactions paid with each payment instrument. In other words, we can check the existence of transaction domains for each individua. Gobay, we observe that over 1,447 respondents, 1,392 individuas have competed the diary. Overa, we have 16,692 transactions avaiabe containing a the information we need for this study. 4 A description of the distribution of purchases paid cash or with an aternative payment instrument are potted in Figure 1. 5 A few comments may be made. First, the major part of the transactions are sma vaue purchases (beow 20 euros). Second, most of the transactions are paid cash (64%; see Appendix A for more descriptive statistics). Third, as can be seen in Figure 2, and as aready noticed in the Introduction, the amount of a transaction is statisticay important to predict if it wi be paid in cash or using the aternative payment instrument Empirica verification Now, et us try to test the TS modes resuts at an individua eve for peope who have cash and an aternative payment instrument. As in Whitese (1989), these individuas wi use cash or the aternative payment instrument respectivey for sma and arge transactions respectivey. More precisey, for any individua, if P max cash is the maximum vaue of a transaction paid with cash 3 Professiona expenses and bi payments were excuded from diaries. 4 6 individuas have no access to any payment instrument but cash. We did not take them into account. 5 In a the figures we show, the data are summed for price casses of 3 euros thickness. 5

7 Figure 1: Tota number of purchases (soid ine) and number of purchases not paid in cash (dotted ine) in function of the price. and Paternative min is the minimum vaue of a transaction paid with an aternative payment instrument, then we shoud aways observe: P max cash P min aternative. (1) A first anaysis of the payment data shows that some individuas satisfy Equation (1). At the top of Figure 3, we reproduced the payment patterns of such an individua who hods cash and an aternative payment instrument. We can see two strict domains of transaction: the argest transaction paid cash (around 15 euros) is stricty ower than the smaest purchase paid with the aternative payment instrument (around 20 euros). However, surprisingy, we aso find individuas who do not behave as in TS modes. For instance, we reproduced the payment pattern of such a typica individua at the bottom of Figure 3. We can see two overapping domains of transaction. Indeed, we observe that the argest transaction paid cash, around 120 euros, is stricty arger than the smaest purchase paid with 6

8 Figure 2: Share of the purchases paid in cash (beow the soid ine) and with an aternative payment instruments (above the soid ine) in function of the price. the aternative payment instrument (around 7 euros). As a resut, such an individua does not conform to Equation (1) and hence, does not behave as TS modes predict. A more goba anaysis of individua payment patterns shows that TS modes do not account quite we as the way peope pay. In fact, we observe that 32.32% of the peope who get cash and an aternative payment instrument and who carry out at east one transaction do not satisfy Equation (1). In the present section, we have shown that TS modes poory perform at the individua eve. However, it woud be possibe that TS modes remain vaid at an aggregate eve, i.e. when the whoe popuation is considered rather than each individua. The next section shows that TS modes are not good at repicating the data at an aggregate eve either. 7

9 Figure 3: Exampes of use of cash (+) and aternative payment instruments ( ) in function of the price for two individuas. 3 Three modes of aggregate decision In this section, the purpose is to test severa modes at an aggregate eve. In order to do that, we need to specify how individua decisions become aggregate decisions. We propose two ways to do that for TS modes. We aso propose a third mode that considers a representative agent that makes his decision differenty from what TS modes propose. 3.1 The naive TS mode The most natura way to aggregate individua decisions into a popuation decision is to consider a representative agent that behaves ike a singe individua. This aggregation is of course a bit naive but it is a good benchmark to start our study of the use of cash at an aggregate eve. Then, et us consider a representative agent who pays cash if and ony if the price of the good he purchases is smaer than a threshod L and pays with 8

10 the aternative payment instrument if and ony if the price of the good he th purchases is greater than L. Formay, if F L (p) is the theoretica probabiity of paying in cash for a purchase of price p, then F th L (p) = { 1 if p L 0 if p > L. 3.2 The differentiated TS mode Let us now try to improve the naive way to aggregate individua decisions satisfying TS modes predictions by extending the informationa basis. We wi ca this aggregate decision mode, the differentiated TS mode. As we have seen in Section 2.2.2, many individuas do not satisfy the TS modes approach. However, we can try to extract from the individuas behavior some threshods for which they vioate the TS mode impications as rarey as possibe. Let us first give an exampe iustrated in Figure 4. Let us consider an individua i that purchases three goods. The first one, good 1, is 5 euros worth and is paid cash, the second one, good 2, is 10 euros worth and is paid with the aternative payment instrument, the third one, good 3, is 15 euros worth and is paid cash. If we use the TS mode approach as described in Section 2 with a threshod between 0 and 5 euros, 2 data cannot be expained, namey, theoreticay goods 1 and 3 shoud be paid with the aternative payment instrument whereas, by assumption, they are paid cash. If we use a threshod between 5 and 10 euros, 1 data cannot be expained, namey, theoreticay, good 3 shoud be paid with the aternative payment instrument whereas, by assumption, it is paid cash. If we use a threshod between 10 and 15 euros, 2 data cannot be expained, namey, good 2 shoud be paid cash and good 3 shoud be paid with the aternative payment instrument. If we use a threshod above 15 euros, 1 data cannot be expained, namey, good 2 shoud be paid with the aternative payment instrument. Then, the set of prices S i = [5, 10] [15, [ can be interpreted as the vaues for which a threshod woud make TS modes the most acceptabe. 9

11 To avoid probems deaing with infinite or 0-vaue threshods, we wi consider the set S i = [P min i i where P min i P max i,pi max ] S i as the set of acceptabe threshods for individua is the vaue of the cheapest good purchased by individua i and is the vaue of the most expensive good purchased by individua i. We define i = max{ S i } and i = min{ S i }. Then, i is the greatest vaue between P min i and P max i for which TS modes make the minimum number of mistakes concerning the purchases of individua i. i is the smaest vaue between P min i and P max i for which TS modes make the minimum number of mistakes concerning the purchases of individua i. We define = { 1,..., 1386 } and = { 1,..., 1386 } the sets of threshods found for the individuas in the popuation we have data for. We show some descriptive statistics about the observed threshods in Tabe 1, page Figure 4: Exampes of TS mode errors for different threshods when using cash (+) or an aternative payment instrument ( ). Instead of a representative agent having a constant threshod L as in the naive aggregation of TS modes, we can now mode a representative agent with a probabiistic threshod. When facing a price p, the representative agent draws by chance a threshod and pays cash if and ony if p. 10

12 The probabiity of drawing = i is given by P( = i ) = n i j n where n i j is the number of transactions made by individua i in the observed data. th Computing this new aggregate decision rue gives the probabiity F (p) for each good worth p euros to be paid cash. Considering instead of in the th previous decision rue gives F. 3.3 The CH mode Let us now describe the CH mode. Unike the TS modes, the CH mode is not ony based on the prices of the goods. More cruciay, it depends on the amount of cash the agent is hoding, hence it is more a cash hoding mode. Let us consider an agent carrying m euros in cash. When facing a price p, the agent pays p euros in cash if and ony if m p. Ese, if m < p, the agent pays p euros using the aternative payment instrument. Then, after each purchase, the agent has m = m p euros (if m p) or m = m euros (if m < p) eft in cash. Then, the agent needs to make a decision concerning his cash hoding eve. Foowing the "bottom inventory" assumption of Boeschoten (1998, p: 43), we assume that the agent withdraws some cash if and ony if his cash hoding is smaer than a threshod, i.e. if m. If the agent decides to withdraw some cash, the amount he withdraws, w, is drawn form the statistica distribution, W(.), of the observed withdrawas in the economy. A compete axiomatization of this mode is given in Bounie et a. (2007). Simpifying a bit, we ist here some crucia assumptions behind this decision rue. 1. The individua is aways interested in buying the good. The surpus the individua gets from the good is superior to the cost of the transaction to buy it. 2. When the individua has the choice between cash and an aternative payment instrument, he prefers to use cash. This behavior can be expained by the cost of cash hoding which mainy depends on the 11

13 interest earnings foregone, the risk of oss and theft as we as the infation cost (Baumo, 1952; Tobin, 1956). 3. The individua goes to an Automated Teer Machine (ATM) when his cash hoding is beow a given threshod. Besides what has been said before, an individua needs to hod a minima amount of cash for severa reasons. First, cash is the ony ega tender in the economy. Second, there is no aternative payment instrument different from cash avaiabe to sette ow vaue transactions since payment cards and checks are not aways accepted beow a threshod by some retaiers. Third, and more generay, aternative payment instruments such as payment cards are not necessariy accepted in the whoe merchant paces. Four, cash can be required to face specific transactions such as those at vending machines. 3.4 Simuations and resuts Method In order to gauge how we the three modes repicate the data, we use numerica simuations. Let us consider a representative agent. At each period of time, the agent is facing a price p. p is drawn by chance from the distribution P(.). Then, the agent makes his decision about the payment instrument according to one of the decision rues described above. If we test the naive TS mode with a threshod L, the agent pays in cash if and ony if p L, ese, he pays with the aternative payment instrument. If we test the differentiated TS mode with threshod distribution,, the agent draws by chance a threshod in as described above and pays in cash if and ony if p is smaer than this threshod. If we test the differentiated TS mode with threshod distribution,, the agent draws by chance a threshod in as described above and pays in cash if and ony if p is smaer than this threshod. If we test the CH mode, 12

14 the agent pays in cash if and ony if he hods enough cash and in a second step makes his decision about withdrawing cash as described above. From our data set, we know the distribution of the prices purchased in the economy, P(.). We aso know the distribution of withdrawas, W(.), needed for the CH mode. 6 Let us give a sequence as an exampe for the naive TS mode with L = 25. L = 25 means that the individua pays in cash if and ony if the price is smaer than 25. Assume that, first, the agent faces a buying decision for a good worth 75 euros. Since this good is more expensive than L = 25 euros, it is paid with the aternative payment instrument by the agent. Assume, that, in a second period, the agent faces a buying decision for a good worth 30 euros. Since this good is more expensive than L = 25 euros, it is paid with the aternative payment instrument by the agent. Assume that, in a third period, the agent faces a buying decision for a good worth 21 euros. Since this good is ess expensive than L = 25 euros, it is paid cash by the th agent. If we were stopping our simuation here, we woud have, F 25(21) = 1, F th 25(30) = 0 and F th 25(75) = 0. Obviousy, it is straightforward to check th that with an infinite periods simuation, we have F L (p) = 1 if p L and F L th (p) = 0 if p > L. Let us give the same sequence as an exampe for the differentiated TS mode with. Assume that in the first period, when the price faced is 75 euros, the agent draws by chance a threshod = 80. Then, the agent pays in cash since 75. Assume that in the second period, when the price faced is 30 euros, the agent draws by chance a threshod = 20. The agent pays with the aternative payment instrument since < 20. Assume that in the third period, when the price faced is 21 euros, the agent draws by chance a threshod = 20. The agent pays with the aternative payment instrument since th < 21. If we were stopping our simuation here, we woud have, F (21) = 0, 6 For a statistica description of transactions and withdrawas, see Tabes 2 and 3 in Appendices A and B respectivey. 13

15 F th th (30) = 0 and F (75) = 1. Notice that the representative agent satisfying the differentiated TS mode is not necessariy satisfying the predictions of TS modes for individuas. Indeed, since the threshod can vary, it is possibe that a purchase be paid in cash whereas a cheaper one woud be paid with the aternative payment instrument. Let us give the same sequence as an exampe for the CH mode with = 5. = 5 means that the individua goes to withdraw cash when the amount of cash he hods is beow 5 euros. Moreover, et us assume an agent hoding 100 euros in cash at the beginning of times. 7 When the representative agent is facing the 75 euros good, since the agent can pay with cash, so does he. Then, he has 25 euros remaining in cash. Since the threshod is not reached from above, the individua does not withdraw any cash. When facing the second good, worth 30 euros, since the agent cannot pay with cash, he pays with the aternative payment instrument. He has sti 25 euros in cash and for the same reasons as above, does not withdraw any cash. When facing the third purchase opportunity, worth 21 euros, since the agent can pay in cash, so does he. He has now 4 euros in cash remaining and since his cash stock is beow, he goes to an ATM or to his bank to refi. If he withdraws, say 20 euros, the mode goes on with the agent having 24 euros in cash. Iterating this agorithm an infinite number of times with a representative agent satisfying the naive TS mode, we can compute F th L (p), depending th th on the threshod chosen. We can aso compute F (p) (resp. F (p)), the theoretica frequency with which the representative agent satisfying the differentiated TS mode with threshods distribution (resp. ) pays in cash at each price p. Finay, we can compute F th (p) the theoretica frequency with which the representative agent satisfying the CH mode with threshod pays in cash for each price p. The purpose is now to find a measure to show how we each of these 7 Notice that this information is irreevant for TS modes and it is aso irreevant when the simuation is considering an arbitrariy arge number of periods. 14

16 theoretica modes repicate the data. The distance we use is as foows. Let us denote by F obs (p) the observed share of purchases worth p euros paid cash. Let us denote by f th (p), the theoretica share of purchases worth p euros paid cash when one of the modes described above is considered (f th (.) { F L th th th (.), F (.), F (.),F th (.)} depending on the mode we test). The distance D(F obs,f th ) is given by D(F obs,f th ) = p [0, [ n(p) N F obs (p) f th (p). (2) where n(p) is the number of purchases worth p in the data and N = p n(p) is the tota number of purchases observed. Then, this distance sums the differences between F obs and f th for each price p, each difference being weighted by the frequency n(p)/n of purchases of price p. This weight seems quite natura since an error when many transactions are concerned shoud be more crucia than when ony a few transactions are concerned. As an exampe, if one predicts that the individua wi pay cash 95% of the purchases between 3 and 6 euros whereas the actua observed percentage is 96%, the error is more important than predicting that 1% of the purchases between 156 and 159 euros wi be paid cash whereas the actua observed percentage is 0%. Indeed, amost 1/8 of a the purchases are between 3 and 6 euros whereas about 1/3000 of the purchases are between 156 and 159 euros. Some more remarks about the distance used are given in the Concusion Resuts Let us now give the resuts. For the naive aggregation of the TS mode, the observed data show that more than 50% of the purchases are paid cash for prices ess than 18 euros and paid with an aternative payment instrument for prices above 18 euros. Then, the minimum distance D(F obs th, F L ) is obtained for L = 18. More generay, the distance D(F obs th, F L ) in function of the threshod L is shown in Figure 5. We can interpret Figure 5 as foows. For sma vaues of L, the agent often pays with the aternative 15

17 payment instrument. At the imit L 0, the agent never pays cash. The observed probabiity that the agent pays cash is approximatey 64%, 8 hence the difference between the observed data and the theoretica ones is approximatey 64%. On the contrary, for great vaues of L, the agent often pays with cash. At the imit L, the agent never pays with his aternative payment instrument. The observed probabiity that the agent pays with his aternative payment instrument is approximatey 36%, hence the difference between the observed data and the theoretica ones goes to approximatey 36%. 9 In between those vaues, the representative agent s theoretica behavior gets coser to what is observed. The minimum error is reached at the vaue L = 18 where we have D(F obs th, F 18) = 17.1%. In other words, for a price p, the theoretica probabiity that an agent pays cash computed by the naive aggregation of the TS mode mode with L = 18, is on average 17.1% different from the observed probabiity. Figure 6 shows F obs th (p) and F 18(p) for vaues of p between 0 and 160 euros. Not surprisingy, we wi see that the naive aggregation of the TS mode is the worse at repicating the observed aggregate payment. Indeed, as we have shown in Section 2.2.2, the TS mode approach is not necessariy adapted to account for individua payment patterns. It woud have been surprising that a naive aggregation of it perform we. Let us now consider the differentiated aggregation of the TS mode. F th th (p) and F (p), the theoretica frequencies obtained by the differentiated TS mode are shown in Figure 7 together with F obs th (p). Obviousy, F (p) th is above F (p) for each p. Indeed, by definition, i i for a individua i. Then, the threshod above which purchases are made cash in the differentiated TS mode is higher when i is considered rather than when i is. Hence, more purchases are made cash when using than when using. It is now straightforward to compute the distances, D(F obs, 8 See Tabe 2. 9 See Tabe 2. th F ) and 16

18 Figure 5: Distance D(F obs, th F L ) in function of the threshod L. D(F obs th, F ). We find D(F obs th, F ) = 6.4% and D(F obs th, F ) = 10.8%. Hence, the differentiated TS mode is 2 or 3 times better than the naive TS mode at repicating the data. However, we wi see that the CH mode shows better predictive resuts. We now comment on the resuts for the CH mode. The distance D(F obs,f th ) is shown in Figure 8 for different vaues of. We can interpret Figure 8 as foows. According to the CH mode, for sma vaues of, the agent goes rarey to the ATM and then carries itte cash. Then, the representative agent often pays with an aternative payment instrument. At the imit 0, the agent never pays cash. The observed probabiity that the agent pays cash is approximatey 64%, hence the difference between the observed data and the theoretica ones is approximatey 64%. On the contrary, for great vaues of, the agent often withdraws cash. Then, he often pays with cash. At the imit, the agent never pays with any aternative payment instrument. The observed probabiity that the agent pays with his aternative payment 17

19 F th 18 (soid ine) fre- Figure 6: Observed F obs (dotted ine) and theoretica quencies of payment in cash in function of the price. instrument is approximatey 36%, hence the difference between the observed data and the theoretica ones goes to approximatey 36%. In between those vaues, the representative agent s behavior gets coser to what is observed. We can note that the CH mode fits the data with ony an average error of ess than 3.1% for a threshod = 4.1. In other words, for a price p, the theoretica probabiity that a representative agent pays cash computed with the CH mode with = 4.1, is on average ess than 3.1% different from the observed probabiity. This resut suggests that the CH mode has a very good predictive power. Indeed, it performs between 2 and 3 times better than the differentiated TS modes. The theoretica and observed probabiities to pay cash in function of the price of the goods are shown in Figure 9 for the CH mode. We can see that the CH mode behaves particuary we for sma prices (0-60 euros). The behavior for arger prices ( euros) is not as good (the same remark coud be done for the differentiated TS modes). However, since, as can be 18

20 Figure 7: Observed F obs (thick dotted ine) and theoretica F th F th (pain ine), (ight dotted ine) frequencies of payment in cash in function of the price. seen in Figure 1, most of the transactions take pace for sma prices, the predictions of the CH modes show a very good fit with the observed data. We sum up our resuts in Tabe 1, page 22. Gobay, we find that the CH mode gives predictions from 2 to 6 times more precise than aggregate TS modes and hence may be considered as more efficient than TS modes. 4 Extension for three payment instruments The previous anaysis suggests that TS modes do not repicate quite we the observed aggregate payment patterns of peope who hod cash and an aternative payment instrument. However peope may hod and use severa aternative payment instruments reated to deposits such as a check and a payment card. Then a question arises: do TS modes fit the payment patterns of peope who hod three payment instruments? 19

21 Figure 8: Distance D(F obs,f th ) in function of the threshod,. In this section, we propose to test the performance of TS modes to repicate observed individua and aggregate payment patterns using our data set for three payment instruments. Hence, we imit our sampe to individuas who hod three payment instruments: cash, check and payment card. 10 Let us first test the performance of the TS modes at an individua eve. For that, we have to identify, as previousy, for each individua, the domains of transaction of the three payment instruments. For peope who hod three payment instruments such as cash, payment card and check, for the same reasons as for two payment instruments, for each individua, we shoud observe: where P max cash P max cash < P min payment card, P max payment card < P min check and P max cash < P min check, (3) max and Ppayment card are respectivey the maximum vaue of a transaction paid with cash and a payment card and P min payment card and P min check are respectivey the minimum vaue of a transaction paid with a payment card 10 83% of the peope in our sampe hod those three payment instruments. 20

22 Figure 9: Observed F obs (dotted ine) and theoretica F th 4.1 (soid ine) frequencies of payment in cash in function of the price. and a check. A first anaysis of the payment data shows that some individuas satisfy Equation (3). At the top of Figure 10, we reproduced the payment patterns of such an individua who hods the three payment instruments. We notice three strict domains of transaction: the argest transaction paid cash is stricty ower than the smaest purchase paid with the payment card and the argest transaction paid with cash and the payment card is stricty ower than the owest transaction paid with check. However, we aso find individuas who do not behave as the TS modes predict. For instance, we reproduced the payment pattern of an individua satisfying none of the inequaities of Equation (3) at the bottom of Figure 10. We notice three overapping domains of transaction. First, we observe that the argest transaction paid cash (around 150 euros) is stricty higher than the smaest purchase paid with the payment card and even higher than the argest purchase paid with the payment card (around 90 euros). Second, we 21

23 Modes % error (in %) threshod CH mode (F th ) Naive TS mode ( F th 18) Differentiated TS mode ( F th ) 6.4 Differentiated TS mode ( F th ) 10.8 min = max = 1, mean()=42.6 min = max = mean() 18.9 Tabe 1: Summing up of the resuts. note that the argest transaction paid with the payment card is higher than the owest transaction paid with check (around 20 euros). Finay, we remark that the argest transaction paid cash is higher than the smaest transaction paid with check. Hence, this individua does not satisfy Equation (3). More generay, when we extend the anaysis to the whoe popuation who hod the three payment instruments and who reaize at east one transaction, we find that 48.55% of them do not satisfy Equation (3). Finay, we coud hardy argue that naive TS modes repicate the data at an aggregate eve with three payment instruments. Indeed, we note in Figure 11 that cash, check and payment card are used for sma vaue transactions as we as for high vaue transactions. Therefore, our data suggest that there are no strict transaction domains for each payment instrument but rather some overapping transaction domains. However, we emphasize two reasons for which our study can hardy be straightforwardy extended for three payment instruments at an aggregate eve. The first reasons is vaid for a the aggregate modes (naive TS, differentiated TS and CH). Our distance to compute how we each theoretica mode fits the data can hardy be adapted for more than two payment instruments. We shoud define a distance that estimates the differences between 22

24 Figure 10: Exampes of use of cash (+), payment card ( ) and check ( ) for two individuas. two pairs of ines. Many possibiities exist and further research woud be required in this direction. The second reason for which extending our resuts for more than two payment instruments is not straightforward is vaid for the differentiated TS and the CH modes. Indeed, in order to undertake this generaization, the whoe methodoogy shoud be significanty different from the ones used in the present study. In the case of the differentiated TS mode, the way to find the distribution of the threshods, that are now two-dimensiona, is not obvious. How coud we dea with different pairs of threshods minimizing the number of errors of the TS approach? In the case of the CH mode, the way to consider a third payment instrument is not straightforward either. How can we discriminate between checks and payment cards in a CH mode? Those two questions are eft for further research. 23

25 Figure 11: Share of the transactions paid in cash (beow soid ine), check (between soid and dotted ine) and payment card (above dotted ine) in function of prices. 5 Concusion In this artice, first, we showed that TS modes are not satisfied by the individuas of our data set when they are considered at an individua eve. Second, we provided two aggregate TS modes and a simpe payment rue (CH mode). We showed that the atter better fits the observed aggregate payments patterns than the former. Gobay, we find that the CH mode gives predictions from 2 to 6 times more precise than TS modes at an aggregate eve and hence may be considered as more efficient than them. We now emphasize an important feature of the CH mode that makes it even more appeaing: it is by far ess compex to impement than the TS mode in its differentiated aggregated form since it reies on a very minima information basis. Indeed, the ony free parameter we need to compute is the threshod under which the representative agent goes back to withdraw cash. The same feature is shared by the naive TS mode where the ony 24

26 free parameter is the threshod L above which a purchases are paid with the aternative payment instruments. As we have shown, the former mode performs much better than the atter mode at fitting the observed data. On the contrary, the differentiated TS mode requires the computation of the approximate threshod for each individua. In our study, since we have 1,386 individuas, we need to have a the information about the payment patterns of each of them. Then, the computation requires a 1,386-eements set as an input. Hence, the differentiated TS mode performs ess accuratey than the CH mode and, moreover, shows some obvious informationa and computationa imits. In the same ine, of course, we coud imagine some modes performing better than the CH mode. However, to the best of our understanding, these modes woud require a much broader informationa basis. For instance, a further research woud investigate the performance of another mode that woud mainy consist in the differentiated TS mode with a probabiity distribution among threshods that woud depend on the price faced by the representative agent. The second imitation of our study concerns the distance we used. As aready expained, with this distance, an error of prediction for a given price is weighted by the frequency of the transactions of this price, see Equation (2). Another natura distance woud be worth investigating: D(F obs,f th ) = p n(p) N F obs (p) f th (p). (4) p [0, [ With the atter distance, an error of prediction for a given price is weighted by the frequency of the transactions of this price and by the price itsef. Hence, the "frequency effect" described in the interpretation of Equation (2) sti exists. But there is another effect, the "size effect". According to the atter effect, an error of prediction of 1% for purchases worth 1,000 euros is more important than an error of prediction of 1% for purchases worth 1 euro. Indeed, in the first case, statisticay, the error of prediction is 10 euros arge (1% of 1,000 euros) whereas in the second case, the error of prediction 25

27 is 0.01 euro arge. If the purpose is to estimate the tota amount of cash used in the economy, the distance given in Equation (4) seems more natura. However, the distance shown in Equation (4) overestimates the errors of prediction done for arge transactions in comparison with the distance shown in Equation (2). That is why our data set is not arge enough to aow us to use the distance shown in Equation (4). For instance, in our data set, we have ony one transaction between 20,601 and 20,604 euros. It was paid with an aternative payment instrument, but saying that 0% of the transactions between 20,601 and 20,604 euros is paid in cash is not statisticay significative. This is not important if the distance shown in Equation (2) is used since this transaction represents ony 1/16,193 of the transactions. But if the distance shown in Equation (4) is used, an error of prediction for this very transaction is much more important than an error for any other price. Hence, the resut woud be greaty infuenced by a data that is not statisticay significant. Then, in order to use the distance shown in Equation (4), a arger data set woud be necessary. 26

28 A Descriptive statistics on transactions Aternative Cash Payment Instruments Tota Nb. of transactions 10,419 (64.3%) 5,774 (35.7%) 16,193 (100.0%) Average vaue s.d Max Nb. of transactions for one indiv Nb. of indiv. making at east 1 transaction 1,309 1,165 1,386 Tabe 2: Descriptive statistics on transactions. B Descriptive statistics on withdrawas Cash Nb. of withdrawas 1,785 Average vaue 71.4 s.d Min - Max 10-2,500 Tabe 3: Descriptive statistics on withdrawas. 27

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