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An empirical analysis of euro cash payments
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European Economic Review 51 (2007) 1985–1997 www.elsevier.com/locate/eer
An empirical analysis of euro cash payments Philip Hans Fransesa,, Jeanine Kippersb a
Econometric Institute H11-34, Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands b De Nederlandsche Bank, Amsterdam, The Netherlands Received 7 July 2003; accepted 17 January 2007 Available online 26 March 2007
Abstract If the denominational structure of the euro is used in an optimal way, there should be no preferences for certain coins and notes when making cash payments. In Kippers et al. [2003. An empirical study of cash payments. Statistica Neerlandica 57, 484–508] it is documented that the Dutch public did have certain preferences concerning the Dutch guilder in the sense that a few notes and coins were used less often than they should have been. With the advent of the euro, which changed the denominational structure from 1–212–5 (guilder) to 1–2–5 (euro), it is of interest to examine whether there are any preferences for euro coins and notes. In this paper we use a unique dataset for the Netherlands to empirically examine if the euro range is used in an optimal way. We find that there are no preferences for certain euro denominations. r 2007 Elsevier B.V. All rights reserved. JEL classification: E42; E58 Keywords: Euro cash; Truncated Poisson regression model; Payment behavior
1. Introduction and motivation Euro cash consists of seven banknotes with values 500, 200, 100, 50, 20, 10 and 5, and eight coins with values 2, 1, 0.50, 0.20, 0.10, 0.05, 0.02 and 0.01. It is generally accepted that such a 1–2–5 range is the denominational structure that is close to the optimal structure; see also Boeschoten and Fase (1989), Telser (1995) and Wynne (1997).
Corresponding author. Tel.: +31 10 4081273.
E-mail address: [email protected] (P.H. Franses). 0014-2921/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.euroecorev.2007.01.001
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The question is of course if paying individuals behave according to this theoretical prediction, and it is this question that we address in this paper. An optimal range will be reflected in actually observed data by (i) efficient payments and (ii) by the absence of preferences for certain coins or notes. Efficient payments relate to the full transaction, and we follow the definition in Cramer (1983) where efficiency concerns minimizing the number of coins and notes that change hands in the course of a transaction; see also Caianiello et al. (1982), Sumner (1993), Van Hove and Heyndels (1996) and Van Hove (2001). Cash payments basically amount to an individual choice process. An individual has a wallet with banknotes and coins, and he can choose any combination of these notes and coins, provided that it has a combined nominal value of at least the amount to be paid. The choice set is determined by the wallet content and the final choice is a combination of available banknotes and coins. Computational skills are also needed for this choice task. And the choice is constrained by time, as the time between observing the payment amount and the closure of the payment is usually just a matter of seconds. Apparently, paying individuals do not need much time, and do not get much time, to make their choice amongst notes and coins. In this paper, we carry out an empirical analysis of this choice process for the euro currency. Kippers et al. (2003) study a large dataset of cash payments concerning the Dutch guilder. They develop a specific statistical model to summarize the data. The model takes into account (i) the wallet content, as the actual choice process of course depends on it, (ii) the fact that the paying individual makes a choice amongst available denominations and (iii) the impact of the payment amount on the choices the individual can make.1 The conclusion from the analysis of guilder payments in Kippers et al. (2003) is that the 100guilder note was preferred over other notes while in contrast the 50-guilder note was not, across all cases where these notes were equally functional. The current paper is the first empirical study with a statistical analysis of actually observed data for euro payments that simultaneously involves (i) the wallet contents, (ii) the payment amount and (iii) the choice amongst notes and coins. There are many studies on (a) the theoretical aspects of a denominational range and (b) on the choice between various types of money (plastic, electronic), but as of yet there are no empirical studies like ours.2 Our analysis concerns a dataset of euro cash payments that has been collected at two different retail locations in the Netherlands. We decided to collect data on actual payments as it is known from the literature that people may not respond correctly in surveys; see Boeschoten (1992) and Avery et al. (1986) and now we do observe wallet contents, in contrast to Kippers et al. (2003). The outline of our paper is as follows. In Section 2, we describe the characteristics of the unique dataset with actually observed cash payments. We analyze the data of euro cash payments in two steps. First, in Section 3 we see if the payments are efficient or not. Second, in Section 4, we consider the econometric model that has been developed in
1 The data analyzed in Kippers et al. (2003) unfortunately do not incorporate wallet contents, which strongly complicates the statistical analysis. In the present study, we overcome this drawback by actually observing wallet contents. 2 Examples of studies on (a) are Telser (1995), and Wynne (1997) and on (b) are Costa Storti and de Grauwe (2002), Markose and Loke (2001), Humphrey et al. (2001) and Van Hove (2004).
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Kippers et al. (2003). All empirical results are summarized in Section 5 and Section 6 concludes with suggestions for further work. 2. Data In this section, we discuss the properties of the dataset3 we use to empirically examine if there are preferences for certain euro coins or notes. The dataset concerns actually observed cash payments where notes and coins are used (and no plastic money is used). We observe (i) the set of coins and banknotes in wallets prior to the payment, (ii) the amount to be paid, (iii) the notes and coins selected by the individual to make the final payment and (iv) the notes and coins returned as change, if there are any. For example, we observe that someone has a note of 5 euro and two coins of 2 euro in the wallet, that he needs to pay 3.50, and that this person chooses to use the two coins of 2 euro, and that he gets a 0.50 coin as change. Team members were posted at two different retail locations, that is, a supermarket in an urban area and an appliance store (like ‘‘Home Depot’’) in a rural area, on various days in October 2002. They registered the payment amounts and the notes and coins used by individuals paying cash at the cashier. Persons who had just paid cash were to participate in a survey when they left the store. This survey included questions about the contents of their wallets after the cash payment. The answers to these questions allowed us to reconstruct the wallet content prior to the cash payments. This information is critical for our subsequent empirical analysis. The questionnaire also contained other questions. In addition to age and gender, we asked whether the individual had ever worked as a cashier, as in that case that individual should have more experience with paying in cash. Since one might gain more experience by regularly visiting stores, we also asked for the number of times individuals visited stores. Finally, the respondents were given a test to measure computational skills, again with the purpose of testing the influence of these skills on cash payment behavior. The precise nature of this last test is described in the notes to Table 1. After 4 full days of interviewing and recording (each day being a different day in the week during 2 weeks), we obtained information on 272 individuals and their cash payments. Table 1 gives a few summary statistics. Again, the important feature of this dataset is that it concerns actually observed cash payments. Table 2 collects a few characteristics of the relevant data. 3. Descriptive statistics To perform a first analysis of the data, we consider the approach in Cramer (1983). We assume that individuals want to make a payment such that the smallest number of notes and coins is exchanged between payer and receiver, including change. As said, such a payment is called an efficient payment. If we apply the algorithm of Cramer (which is summarized in Appendix A.1) to generate efficient cash payments for the amounts ranging from 0.01 to 100 euro, we find 36,591 3
The dataset to be presented in this section initially contained a very small amount of outliers. We are aware of the possibility that observations that are sometimes called outliers can contain useful information, but in these cases we were quite certain that this would not be the case. Examples of these outliers are an individual who carried 12,000 euro while buying a hammer for about 10 euro, and one individual who used 20 notes of 5 euro.
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Table 1 Characteristics of respondents (sample size is 272), where ‘‘Gamma’’ is the appliance store and ‘‘Plusmarkt’’ is a supermarket of fast-moving consumer goods
Average age (years) Percentage of males (%) Average number of shopping trips per week Percentage of respondents with experience as a cashier (%) Percentage of correct answers, series a (%) Percentage of correct answers, series b (%) Percentage of correct answers, series c (%) Percentage of correct answers, series d (%) Average score computing skills
‘‘Gamma’’
‘‘Plusmarkt’’
All observations
47.4 64.0 3.4 22.2
49.9 16.2 3.6 24.3
48.1 50.9 3.5 22.8
90.4 62.1 81.2 46.0 6.2
93.2 47.3 64.9 32.4 4.9
91.2 58.1 76.8 42.3 5.9
The respondents were asked to complete the following four series of numbers:
a. b. c. d.
2-4-6-8-10-___? 5-7-4-6-3-___? 1-2-4-8-16-___? 1-2-4-7-11-___?
We allocate 1 point to series a, 2 to c, 3 to b and 4 to d, to measure complexity. This gives a score in between 0 and 10 for each respondent.
Table 2 Characteristics of payments (272 observations) Amount to be paid Mean Median Maximum Minimum Standard deviation
15.99 10.54 135.59 0.10 18.42
Used notes and coins
Frequency
500 200 100 50 20 10 5 2 1 0.50 0.20 0.10 0.05 0.02 0.01
0 0 5 62 84 97 57 72 62 27 49 34 30 40 31
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Table 3 Efficient payments Efficient payment Wallet content allows for efficient payment Wallet content allows for a restrictive efficient payment Total
80 85 165 (61%)
Inefficient payment 53 54 107 (39%)
Total 133 139 272
Efficiency defined for the case of insufficient wallet content is discussed in Appendix A.2.
efficient payment schemes.4 For 28% of the amounts there is only a single efficient payment scheme. The other amounts can be paid efficiently with more than one payment scheme, with the absolute maximum being 54 different schemes for eight different amounts between 36 and 37 euro, such as 36.67 euro. Furthermore, an average of 5.8 notes and coins is exchanged in efficient payment schemes and no scheme in the given range involves more than eight notes and coins. The principle of least effort, which makes individuals to strive for efficient payments, is appealing but may not hold in practice. First, it requires computational efforts to determine the efficient payment scheme, especially when the scheme becomes more complex. Time pressure can also force individuals to resort to a sub-optimal payment scheme. Finally, an individual’s wallet content is not infinitely large. Cramer’s original algorithm indeed assumes that individuals have access to an unlimited amount of notes and coins. Obviously, in practice this is not the case. Hence, paying individuals have to make a choice amongst the available notes and coins. If the wallet content does not allow for efficiency, an individual has to use the notes and coins as efficiently as possible, which we still call efficiency. To define this type of efficiency, we modify the Cramer algorithm to account for wallet contents, see Appendix A.2. For each transaction in our dataset we generate efficient payment schemes given the wallet restriction and the payment amount. The results for the individuals in our dataset are presented in Table 3. This table shows a number of interesting results. First, 61% of the payments were done efficiently. This constitutes a statistically significant share of the individuals, when compared against the binomial distribution and relying on a 5% significance level. It can, therefore, be concluded that individuals are more inclined to make efficient payments than inefficient payments. Second, all individuals are inclined to make efficient payments, irrespective of whether or not the wallet facilitates the unconstrained efficient payments (80 and 85 out of 133 and 139, respectively). Still, 39% of the individuals in our sample did not pay efficiently. Apart from the reasons mentioned before (computational effort, time pressure) a number of other reasons could exist. Using a logit model, we examine if the characteristics in Table 1 have an effect on the explanatory variable, which is 1 if the payment is efficient and a 0 otherwise. To save space, we summarize the results by stating that none of the covariates has a significant impact on paying efficiently. For example, more computing skills do not lead to more efficiency. 4 For example, the amount 11.30 can be paid efficiently with 10, 1, 0.20 and 0.10, with 10 and 2 with 0.50 and 0.20 as change, and with 10, 1, 0.50, with 0.20 as change. All other payment schemes require five or more notes and coins.
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Inefficiency may also occur if one empties the coins from a wallet, or if one aims at keeping small coins for a subsequent payment. It can also be that there are preferences for certain denominations like in the guilder case (see Kippers et al., 2003). If so, the available range of denominations employed is sub-optimal. In extreme cases of such preferences, it might be worthwhile for monetary institutions to change their denominational structure. We will check any preferences using our model below and we report the outcomes in Section 5. Finally, we take a closer look at the cash payments of the 53 individuals who did not pay efficiently even when their wallet allowed them to do so. Twenty-two individuals paid with only one token, usually a note. Ten individuals paid with too many coins and, considering their wallet contents, were probably aiming to rebalance their wallet content. For the other 21 individuals no logical argument can be formulated. The above counting exercise would become much more involved if we wanted to assess the use of individual denominations relative to others. For example, about 35% of the inefficient payments were paid with one token, and we may now wonder whether this was the obvious denomination given the wallet and the payment amount. Unfortunately, this question cannot be answered with our sample, and also not by means of a simple counting exercise. In the next section, we therefore proceed with a modeling approach to describe the use of denominations. Our intention is to draw model-based conclusions on the use of euro denominations in the Netherlands and to see if there are preferences for some notes or coins such that these are used more than others. 4. Statistical model To examine if there are systematic preferences for denominations in euro cash payments, we consider the statistical model developed and applied in Kippers et al. (2003). The key features of the model are the following. First, it assumes a sequential choice strategy, that is, an individual first checks if the payment amount requires one or more of the highestvalued notes or coins available in the wallet and then he proceeds to adjacent lower-valued notes and coins. A second feature is that it contains a sequence of Poisson regression models as the data are counts. A third feature is that it includes the amount of money that needs to be paid. To illustrate this third feature, assume that the amount to be paid is 15 euros, and that the wallet contains one note of 20, one of 10 and one of 5. If the individual pays with a note of 20, the amount to be left to pay with notes of 10 or 5 is zero. If the individual chooses not to pay with a note of 20, the amount left to pay is 15 euros, which is paid with a note of 10 and a note of 5. Thus, we can compute, for each note and coin in the payment process, which amount still has to be paid after a choice concerning this note or coin has been made. We expect that the larger is the amount left relative to a specific note or coin, the higher is the probability that it will be selected by the individual. In Appendix B, we discuss some of the technical details of the model. If cash payment behavior of individuals reveals no preference for any denomination, then the probability of choosing a denomination for a payment would be equal for all denominations in similar payment situations defined by wallet content and amounts to be paid. Therefore, the question whether individuals are indifferent towards denominations can be answered statistically by testing if the parameter values (intercepts and coefficients on the amount left to pay) are equal across denominations.
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This hypothesis can formally be tested with a likelihood ratio (LR) test. Rejection of the null hypothesis suggests that the parameters significantly differ across denominations. In that case, no denominations are more or less preferred than others and the denominational structure of the currency can be regarded as optimal. In Appendix C, we examine the small sample properties of the relevant LR statistic. These results also suggest a sample size for which the test has reasonable power. It turns out that our sample size of 272 observations is large enough to have reasonable power and hence we can safely rely on the statistical results to be discussed in the next section. 5. Empirical results We estimate the parameters of the cash payment model with the sample data of 272 euro cash payments using maximum likelihood with heteroskedasticity-consistent standard errors. We exclude the observations where individuals have added coins or notes on the request of the cashier, as our model cannot use these for estimation. The final dataset contains 240 observations. Furthermore, given the specification of our model, see again Appendix B, we can only use those observations for which the paying individual has freedom of choice. For example, the case where one needs to pay 20 cents and one only has a 20 cents coin in the wallet, there is no such freedom. These no-free-choice observations do not contribute to the likelihood. Given the hierarchical structure of our model, the number of observations with free choice can decrease as the denominations become lower-valued. In the end, no parameters for the lowest denomination, the 1 euro cent, can be estimated. As the number of observations with free choice quickly becomes quite low for denominations smaller than 1 euro, we focus on the denominations 1, 2, 5, 10, 20, 50 and 100 euros. The largest existing denominations, of 200 and 500 euros, are excluded from analysis lack of data. The estimation results are presented in Table 4. The parameters for the variable measuring ‘‘amount left’’ are all significantly different from zero and they obtain the expected positive sign.5 This means that, when a denomination is needed more, it is also more likely to be used more often. To formally examine if there are differences in preference, we use the LR test. It obtains a value of 5.03, which does not exceed the 5% critical value of 18.31 of the chi-squared distribution with 10 degrees of freedom. We therefore conclude that there are no differences in preferences for one or more denominations. 6. Conclusion In this paper we examined whether the Dutch public has preferences for using certain euro coins and notes. An important motivation is the study of Kippers et al. (2003), who documented that such preferences existed for the Dutch guilder, which had the 1–212–5 based denomination range. Using a unique dataset, a specifically designed model and a 5 It should be mentioned here that the model is highly non-linear in its parameters and that the estimation routine (within Eviews) is susceptible to converging to zero-valued parameters, causing the estimation routine to collapse easily. In view of this and the outcomes of the logit model based on the characteristics of Table 1 discussed earlier, we decided not to include any explanatory variables in the model other than an intercept and the amount left to pay.
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Table 4 Parameter estimates with heteroskedasticity-consistent standard errors (S.E.) for the cash payment model (a sequence of truncated Poisson regression models) Denomination
Intercept
100 50 20 10 5 2 1
Amount left
Effective observations (free choice)
Parameter
S.E.
Parameter
S.E.
0.855 0.519 0.122 0.332 0.139 0.142 0.032
0.829 0.435 0.162 0.161a 0.263 0.262 0.218
2.174a 1.413a 1.019a 1.188a 1.754a 1.633a 1.780a
0.465 0.363 0.187 0.257 0.576 0.390 0.376
10 76 76 66 48 46 39
The effective sample size is 240. a Significant at the 5% level.
statistical test with good properties, we found that for the euro there are no such preferences. We also found that most payments are efficient. The results in our paper indicate that the euro denominational range facilitates efficient payment behavior, more than the guilder range did. Hence, the welfare implications of the introduction of the euro in the Netherlands are positive, both for paying individuals as for the central bank which distributes the euro notes. Acknowledgments We thank Merel van Diepen, Claudia Laheij and Niels Tromp for their help with collecting the data and Marielle Non for her help with programming. We also thank The Editor (von Hagen), three anonymous referees, Kees Koedijk, Herman van Dijk, Mars Cramer and Patrick Groenen for helpful comments. The Gauss and Eviews programs used for the computations are available upon request. Appendix A. Efficient payment schemes A.1. Cramer’s algorithm for infinite-sized wallet contents With A we denote the payment amount and with n(A) the amount of tokens to be used for cash payment. The denomination range is the set {1, 2, y, D} and an element of this set is denoted by d. We denote with n(A, d) the number of tokens of denomination d, so n(A) is Sn(A, d). Finally, v(d) is the value of denomination d. The objective is to minimize n(A) by choosing the ‘‘right’’ tokens out of the set {1, 2, y, D}. More formally, we should solve min
nðAÞ ¼
D P
jnðA; dÞj
d¼1
s:t:
D P d¼1
nðA; dÞvðdÞ ¼ A
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Note that we use the absolute value of n(A, d) to allow for the possibility that notes and coins are returned as change. The algorithm proceeds as follows. We take a range of amounts that are of interest, say 0.01 until 100 euro with intervals of 0.01 euro. The goal of the algorithm is to find for each of these amounts the efficient combinations of tokens: 1. The algorithm starts by taking all amounts that can be paid with only one token. For the euro, the amounts 0.01, 0.02, 0.10, 0.20, 0.50, 1, 2, 5, 10, 20, 50, and 100 are now covered. 2. All amounts that can be paid with two tokens, either two given by the individual as payment or one returned by the retailer as change, are computed. For example, 15 euro can be paid with 10 and 5 euro notes or with 20 euro note with a 5 euro note as change. If we now observe an amount that was already covered with only one token in the previous step, we do not add this pair of two tokens to the list as those tokens are obviously not efficient for this amount. 3. To the pairs that were found efficient in the previous step, we add each token once, both with positive and negative sign, the latter indicating change. For example, to a 10 and 5 euro note we add a 0.20 coin, covering the amounts 15.20 and 14.80 (where in the last case the 0.20 is given as change). For a given pair in Step 2 this gives 2D extra combinations with an additional token each. Adding a token with a positive sign to a combination, which has this same token with a negative sign, would yield a combination with lesser tokens and is therefore ignored. Note also that there is the restriction that the highest-valued token always has a positive sign. With these combinations we cover all resulting amounts, provided they were not already covered by efficient payments with lesser tokens. 4. We repeat step 3 by increasing each time the number of tokens until all amounts between 0.01 and 100 are covered. This algorithm results in efficient combinations (note that there can be more than one!) for each possible amount in the specified range. A.2. Cramer’s algorithm, for restricted wallet contents The original algorithm in A1 assumes that an individual has available all possible notes and coins. This is not realistic, and hence we need to modify the algorithm, which is now based on min
nðAÞ ¼
D P
jnðA; dÞj
d¼1
s:t:
D P
nðA; dÞvðdÞ ¼ A;
d¼1
nðA; dÞpwðdÞ 8d; where we use w(d) to denote the amount of available tokens of denomination d. If a wallet allows for paying according to an efficient scheme, we take this efficient scheme. Otherwise, we need to find the restricted efficient scheme, that is, the combination of tokens that minimizes the numbers of tokens used, given the wallet contents. In this
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case, the following algorithm is executed for a particular amount A and a particular wallet w. Note that this algorithm increases the number of computations substantially. 1. The algorithm starts by computing all amounts that can be covered by a single token. If the wallet content contains the relevant tokens, the combination is stored. 2. All amounts that can be paid with two tokens, either given by the individual or partly returned by the cashier as change, are computed. Again, only if the wallet content allows so, the combination is stored. 3. To the pairs that were found as efficient in the previous step, we add each token once, both with positive and negative sign (indicating change). With these combinations, we fill the list with the resulting amounts, provided they are not already covered by lesser tokens, and provided the combination fits in the wallet. 4. We repeat step 3 for increasing numbers of tokens until the amount A is covered. Note that it is not sufficient to only store the combination which pays the given amount A exactly. We need to store each combination that fits in the individual’s wallet, since we use these as the starting point for the schemes in the next step. As an example, suppose the wallet contains three notes of 5 euro and an individual has to pay 15 euro. The efficient combinations 10 and 5, and 20 and 5 are not available, but the combination 5, 5 and 5 is efficient given this wallet content. Appendix B. The model of Kippers et al. (2003) The model focuses on the probability that an individual i selects a combination of denominations, represented by a D-dimensional discrete random variable Yi, to pay amount Ai, given the wallet content wi. D corresponds to the number of denominations in a denominational range. The realizations of the random variable Yi, are denoted by yi. For modeling purposes a hierarchical structure is imposed. According to this structure the choice of an individual is modeled separately for each denomination, starting from the highest denomination to the lowest. The choice for a certain denomination depends on how previous denominations were used. We consider the probability structure Pr½Y i ¼ yi jwi ¼ Pr½Y D;i ¼ yD;i jwi Pr½Y D1;i ¼ yD1;i jyD;i ; wi . . . Pr½Y 1;i ¼ y1;i jyD;i ; :::; y2;i ; wi .
(B.1) The choice for a payment with yd,i tokens for denomination d is assumed to be distributed as truncated Poisson, where the choice set is truncated from below by a lower bound lbd,i and from above by an upper bound ubd,i, that is, Y d;i jydþ1;i ; . . . ; yD;i ; wi POIðexpðx0d;i bd ÞÞ I½lbc;i ; ubd;i ,
(B.2)
where bd is a parameter vector and xd,i contains explanatory variables for denomination d. The upper bound represents the maximum number of tokens the individual can use for a payment and it is determined by the wallet content. An individual cannot pay with more tokens than that are available in the wallet. Also, it is assumed that individuals do not pay with a token if it is unnecessary, that is, when it is expected that the same token will be returned as change. The lower bound represents the minimum number of tokens of a denomination the individual has to use for the payment. This minimum is determined by the payment
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amount and the availability of other denominations in the wallet. If the wallet does not contain enough value of lower denominations, then the individual is forced to use a minimum number of tokens of the denomination under consideration. The upper and lower bounds are defined as follows: Ai ubD;i ¼ min ceil ; wD;i , (B.3) vD Ai amount ydþ1;i ; . . . ; yD;i ubd;i ¼ min ceil ; wd;i for d ¼ D 1; . . . ; 1, vd (B.4) Ai amountðw1;i ; . . . ; wD1;i Þ lbD;i ¼ max ceil ;0 , (B.5) vd Ai amountðydþ1;i ; . . . ; yD;i Þ amountðw1;i ; . . . ; wd1;i Þ lbd;i ¼ max ceil ;0 for d ¼ D 1; . . . ; 1 vd
(B.6) with amountðxp ; . . . ; xq Þ ¼
q X
vk xk .
(B.7)
If lbd,i ¼ ubd,i, the individual has no freedom of choice and hence Pr½Y d;i ¼ yd;i ¼ lbd;i ¼ ubd;i jydþ1;i ; . . . ; yD;i ; wi ¼ 1.
(B.8)
k¼p
The main explanatory variable in our model concerns the payment amount that each time when a token is to be chosen is left over to pay. In addition to an intercept, the model therefore requires the explanatory variable, say, ACORR, defined as (a function of) the amount to be paid minus the value of the payments chosen for higher denominations, scaled to its face value, that is, Ai ACORRD;i ¼ ln , vD Ai amountðyD;i ; . . . ; ydþ1 ; iÞ for d ¼ 1; . . . ; D 1. (B.9) ACORRd;i ¼ ln vD Appendix C. Small sample properties of the LR test Starting point for our simulations is the set of efficient payment schemes for amounts 0.01 to 100 euro as generated using the Cramer algorithm. Efficient payments are by definition based only on the face value of denominations and therefore constitute examples of payments with indifference towards denominations. If the parameters in the cash payment model are estimated for a dataset of efficient payments, the LR test of equal parameters should follow the asymptotic chi-squared distribution. The following simulation scheme is executed. 1. For all payments, matching wallets are generated by assuming that the wallet contains the payment, with three tokens of each denomination added.
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2. A sample of size 1.000 is randomly drawn. The cash payment model parameters are estimated with and without the parameter restrictions concerning the 1 euro coin through the 50 euro note. 3. Step 2 is repeated 200 times. The resulting empirical size for the LR test with an asymptotic chi-squared (10) distribution turns out to be 6%, which is rather close to the nominal size of 5%. Given this finding, we can safely rely on the asymptotic chi-squared distribution under the null hypothesis. Next, we examine the power of this test. The starting point for this simulation is again the set of efficient payment schemes for amounts 0.01–100 euro as generated using the Cramer algorithm. The following simulation scheme is executed to examine the power of the LR test for small samples, that is, 1. for a fraction a of these schemes, payments with a single token of denomination d are replaced by payment with two tokens of denominations d+1, provided that denomination d has a face value of two times the denomination d+1; 2. for all payments, matching wallets are generated by assuming that the wallet contains the payment, with three tokens of each denomination added; 3. a sample of size n is randomly drawn. The cash payment model parameters are estimated with and without the restrictions; 4. step 3 is repeated N times. The resulting percentage of rejections measures the empirical power. By the replacement of payments, as executed in step 1 of the simulation scheme, we introduce a preference for certain tokens in the data. We use a replacement of a 20 euro note by two 10 euro notes to simulate a preference for the 10 euro note. The sample size n in step 3 is not simply set to 272, which is the sample size of our euro cash payment dataset. This is due to the fact that for these data there are only 75 effective observations with free choice. Hence, a sample size of n ¼ 75 in the simulation scheme is about comparable to the sample size of our dataset. The results of the simulations for different replacement rates are shown in Table C1. We observe that the empirical power of the test rapidly increases with larger replacement fractions. Also, note that this is already achieved for only 75 effective data points.
Table C1 Empirical power of LR test for varying replacement rates Replacement rate
Rejection frequency No. of replications
25%
50%
75%
0.01 500
0.37 200
0.96 200
Notes: The replacement rate represents a preference and is defined as the percentage of efficient payments in a sample in which payment with one 20 euro note is randomly replaced with two 10 euro notes. The empirical power is defined as the percentage of runs in which the null hypothesis of equal parameters is rejected.
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References Avery, R.B., Elliehausen, G.E., Kennickell, A.B., Spindt, P.A., 1986. The use of cash and transaction accounts by American families. Federal Reserve Bulletin 72, 87–108. Boeschoten, W.C., 1992. Currency Use and Payments Patterns. Kluwer Academic Publishers, Dordrecht. Boeschoten, W.C., Fase, M.M.G., 1989. The way we pay with money. Journal of Business and Economics Statistics 7, 319–326. Caianiello, E., Scarpetta, G., Simoncelli, G., 1982. A systemic study of monetary systems. International Journal of General Systems 8, 81–92. Costa Storti, C., de Grauwe, P., 2002. Electronic money and the optimal size of monetary unions. CEPR Discussion Paper 3391. Cramer, J.S., 1983. Currency by denomination. Economic Letters 12, 299–303. Humphrey, D.B., Kim, M., Vale, B., 2001. Realizing the gains from electronic payments: Costs, pricing, and payment choice. Journal of Money, Credit and Banking 33 (2), 216–234. Kippers, J., van Nierop, J.E.M., Paap, R., Franses, P.H., 2003. An empirical study of cash payments. Statistica Neerlandica 57, 484–508. Markose, S.M., Loke, Y.J., 2001. Network effects on cash-card substitution in transactions and low interest rate regimes. Economic Journal 113, 456–476. Sumner, S., 1993. Privatizing the mint. Journal of Money, Credit and Banking 25, 13–29. Telser, L.G., 1995. Optimal denominations for coins and currency. Economic Letters 49, 425–427. Van Hove, L., 2001. Optimal denominations for coins and bank notes: In defense of the principle of least effort. Journal of Money, Credit and Banking 33, 1015–1021. Van Hove, L., 2004. Electronic purses in Euroland: why do penetration and usage rates differ? SUERF Studies, Nr. 2004/4, The European Money and Finance Forum, Vienna, Austria. Van Hove, L., Heyndels, B., 1996. On the optimal spacing of currency denominations. European Journal of Operations Research 90, 547–552. Wynne, M.A., 1997. More on the optimal denominations for coins and currency. Economics Letters 55, 221–225.
European Economic Review 51 (2007) 1985–1997 www.elsevier.com/locate/eer
An empirical analysis of euro cash payments Philip Hans Fransesa,, Jeanine Kippersb a
Econometric Institute H11-34, Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands b De Nederlandsche Bank, Amsterdam, The Netherlands Received 7 July 2003; accepted 17 January 2007 Available online 26 March 2007
Abstract If the denominational structure of the euro is used in an optimal way, there should be no preferences for certain coins and notes when making cash payments. In Kippers et al. [2003. An empirical study of cash payments. Statistica Neerlandica 57, 484–508] it is documented that the Dutch public did have certain preferences concerning the Dutch guilder in the sense that a few notes and coins were used less often than they should have been. With the advent of the euro, which changed the denominational structure from 1–212–5 (guilder) to 1–2–5 (euro), it is of interest to examine whether there are any preferences for euro coins and notes. In this paper we use a unique dataset for the Netherlands to empirically examine if the euro range is used in an optimal way. We find that there are no preferences for certain euro denominations. r 2007 Elsevier B.V. All rights reserved. JEL classification: E42; E58 Keywords: Euro cash; Truncated Poisson regression model; Payment behavior
1. Introduction and motivation Euro cash consists of seven banknotes with values 500, 200, 100, 50, 20, 10 and 5, and eight coins with values 2, 1, 0.50, 0.20, 0.10, 0.05, 0.02 and 0.01. It is generally accepted that such a 1–2–5 range is the denominational structure that is close to the optimal structure; see also Boeschoten and Fase (1989), Telser (1995) and Wynne (1997).
Corresponding author. Tel.: +31 10 4081273.
E-mail address: [email protected] (P.H. Franses). 0014-2921/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.euroecorev.2007.01.001
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The question is of course if paying individuals behave according to this theoretical prediction, and it is this question that we address in this paper. An optimal range will be reflected in actually observed data by (i) efficient payments and (ii) by the absence of preferences for certain coins or notes. Efficient payments relate to the full transaction, and we follow the definition in Cramer (1983) where efficiency concerns minimizing the number of coins and notes that change hands in the course of a transaction; see also Caianiello et al. (1982), Sumner (1993), Van Hove and Heyndels (1996) and Van Hove (2001). Cash payments basically amount to an individual choice process. An individual has a wallet with banknotes and coins, and he can choose any combination of these notes and coins, provided that it has a combined nominal value of at least the amount to be paid. The choice set is determined by the wallet content and the final choice is a combination of available banknotes and coins. Computational skills are also needed for this choice task. And the choice is constrained by time, as the time between observing the payment amount and the closure of the payment is usually just a matter of seconds. Apparently, paying individuals do not need much time, and do not get much time, to make their choice amongst notes and coins. In this paper, we carry out an empirical analysis of this choice process for the euro currency. Kippers et al. (2003) study a large dataset of cash payments concerning the Dutch guilder. They develop a specific statistical model to summarize the data. The model takes into account (i) the wallet content, as the actual choice process of course depends on it, (ii) the fact that the paying individual makes a choice amongst available denominations and (iii) the impact of the payment amount on the choices the individual can make.1 The conclusion from the analysis of guilder payments in Kippers et al. (2003) is that the 100guilder note was preferred over other notes while in contrast the 50-guilder note was not, across all cases where these notes were equally functional. The current paper is the first empirical study with a statistical analysis of actually observed data for euro payments that simultaneously involves (i) the wallet contents, (ii) the payment amount and (iii) the choice amongst notes and coins. There are many studies on (a) the theoretical aspects of a denominational range and (b) on the choice between various types of money (plastic, electronic), but as of yet there are no empirical studies like ours.2 Our analysis concerns a dataset of euro cash payments that has been collected at two different retail locations in the Netherlands. We decided to collect data on actual payments as it is known from the literature that people may not respond correctly in surveys; see Boeschoten (1992) and Avery et al. (1986) and now we do observe wallet contents, in contrast to Kippers et al. (2003). The outline of our paper is as follows. In Section 2, we describe the characteristics of the unique dataset with actually observed cash payments. We analyze the data of euro cash payments in two steps. First, in Section 3 we see if the payments are efficient or not. Second, in Section 4, we consider the econometric model that has been developed in
1 The data analyzed in Kippers et al. (2003) unfortunately do not incorporate wallet contents, which strongly complicates the statistical analysis. In the present study, we overcome this drawback by actually observing wallet contents. 2 Examples of studies on (a) are Telser (1995), and Wynne (1997) and on (b) are Costa Storti and de Grauwe (2002), Markose and Loke (2001), Humphrey et al. (2001) and Van Hove (2004).
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Kippers et al. (2003). All empirical results are summarized in Section 5 and Section 6 concludes with suggestions for further work. 2. Data In this section, we discuss the properties of the dataset3 we use to empirically examine if there are preferences for certain euro coins or notes. The dataset concerns actually observed cash payments where notes and coins are used (and no plastic money is used). We observe (i) the set of coins and banknotes in wallets prior to the payment, (ii) the amount to be paid, (iii) the notes and coins selected by the individual to make the final payment and (iv) the notes and coins returned as change, if there are any. For example, we observe that someone has a note of 5 euro and two coins of 2 euro in the wallet, that he needs to pay 3.50, and that this person chooses to use the two coins of 2 euro, and that he gets a 0.50 coin as change. Team members were posted at two different retail locations, that is, a supermarket in an urban area and an appliance store (like ‘‘Home Depot’’) in a rural area, on various days in October 2002. They registered the payment amounts and the notes and coins used by individuals paying cash at the cashier. Persons who had just paid cash were to participate in a survey when they left the store. This survey included questions about the contents of their wallets after the cash payment. The answers to these questions allowed us to reconstruct the wallet content prior to the cash payments. This information is critical for our subsequent empirical analysis. The questionnaire also contained other questions. In addition to age and gender, we asked whether the individual had ever worked as a cashier, as in that case that individual should have more experience with paying in cash. Since one might gain more experience by regularly visiting stores, we also asked for the number of times individuals visited stores. Finally, the respondents were given a test to measure computational skills, again with the purpose of testing the influence of these skills on cash payment behavior. The precise nature of this last test is described in the notes to Table 1. After 4 full days of interviewing and recording (each day being a different day in the week during 2 weeks), we obtained information on 272 individuals and their cash payments. Table 1 gives a few summary statistics. Again, the important feature of this dataset is that it concerns actually observed cash payments. Table 2 collects a few characteristics of the relevant data. 3. Descriptive statistics To perform a first analysis of the data, we consider the approach in Cramer (1983). We assume that individuals want to make a payment such that the smallest number of notes and coins is exchanged between payer and receiver, including change. As said, such a payment is called an efficient payment. If we apply the algorithm of Cramer (which is summarized in Appendix A.1) to generate efficient cash payments for the amounts ranging from 0.01 to 100 euro, we find 36,591 3
The dataset to be presented in this section initially contained a very small amount of outliers. We are aware of the possibility that observations that are sometimes called outliers can contain useful information, but in these cases we were quite certain that this would not be the case. Examples of these outliers are an individual who carried 12,000 euro while buying a hammer for about 10 euro, and one individual who used 20 notes of 5 euro.
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Table 1 Characteristics of respondents (sample size is 272), where ‘‘Gamma’’ is the appliance store and ‘‘Plusmarkt’’ is a supermarket of fast-moving consumer goods
Average age (years) Percentage of males (%) Average number of shopping trips per week Percentage of respondents with experience as a cashier (%) Percentage of correct answers, series a (%) Percentage of correct answers, series b (%) Percentage of correct answers, series c (%) Percentage of correct answers, series d (%) Average score computing skills
‘‘Gamma’’
‘‘Plusmarkt’’
All observations
47.4 64.0 3.4 22.2
49.9 16.2 3.6 24.3
48.1 50.9 3.5 22.8
90.4 62.1 81.2 46.0 6.2
93.2 47.3 64.9 32.4 4.9
91.2 58.1 76.8 42.3 5.9
The respondents were asked to complete the following four series of numbers:
a. b. c. d.
2-4-6-8-10-___? 5-7-4-6-3-___? 1-2-4-8-16-___? 1-2-4-7-11-___?
We allocate 1 point to series a, 2 to c, 3 to b and 4 to d, to measure complexity. This gives a score in between 0 and 10 for each respondent.
Table 2 Characteristics of payments (272 observations) Amount to be paid Mean Median Maximum Minimum Standard deviation
15.99 10.54 135.59 0.10 18.42
Used notes and coins
Frequency
500 200 100 50 20 10 5 2 1 0.50 0.20 0.10 0.05 0.02 0.01
0 0 5 62 84 97 57 72 62 27 49 34 30 40 31
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Table 3 Efficient payments Efficient payment Wallet content allows for efficient payment Wallet content allows for a restrictive efficient payment Total
80 85 165 (61%)
Inefficient payment 53 54 107 (39%)
Total 133 139 272
Efficiency defined for the case of insufficient wallet content is discussed in Appendix A.2.
efficient payment schemes.4 For 28% of the amounts there is only a single efficient payment scheme. The other amounts can be paid efficiently with more than one payment scheme, with the absolute maximum being 54 different schemes for eight different amounts between 36 and 37 euro, such as 36.67 euro. Furthermore, an average of 5.8 notes and coins is exchanged in efficient payment schemes and no scheme in the given range involves more than eight notes and coins. The principle of least effort, which makes individuals to strive for efficient payments, is appealing but may not hold in practice. First, it requires computational efforts to determine the efficient payment scheme, especially when the scheme becomes more complex. Time pressure can also force individuals to resort to a sub-optimal payment scheme. Finally, an individual’s wallet content is not infinitely large. Cramer’s original algorithm indeed assumes that individuals have access to an unlimited amount of notes and coins. Obviously, in practice this is not the case. Hence, paying individuals have to make a choice amongst the available notes and coins. If the wallet content does not allow for efficiency, an individual has to use the notes and coins as efficiently as possible, which we still call efficiency. To define this type of efficiency, we modify the Cramer algorithm to account for wallet contents, see Appendix A.2. For each transaction in our dataset we generate efficient payment schemes given the wallet restriction and the payment amount. The results for the individuals in our dataset are presented in Table 3. This table shows a number of interesting results. First, 61% of the payments were done efficiently. This constitutes a statistically significant share of the individuals, when compared against the binomial distribution and relying on a 5% significance level. It can, therefore, be concluded that individuals are more inclined to make efficient payments than inefficient payments. Second, all individuals are inclined to make efficient payments, irrespective of whether or not the wallet facilitates the unconstrained efficient payments (80 and 85 out of 133 and 139, respectively). Still, 39% of the individuals in our sample did not pay efficiently. Apart from the reasons mentioned before (computational effort, time pressure) a number of other reasons could exist. Using a logit model, we examine if the characteristics in Table 1 have an effect on the explanatory variable, which is 1 if the payment is efficient and a 0 otherwise. To save space, we summarize the results by stating that none of the covariates has a significant impact on paying efficiently. For example, more computing skills do not lead to more efficiency. 4 For example, the amount 11.30 can be paid efficiently with 10, 1, 0.20 and 0.10, with 10 and 2 with 0.50 and 0.20 as change, and with 10, 1, 0.50, with 0.20 as change. All other payment schemes require five or more notes and coins.
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Inefficiency may also occur if one empties the coins from a wallet, or if one aims at keeping small coins for a subsequent payment. It can also be that there are preferences for certain denominations like in the guilder case (see Kippers et al., 2003). If so, the available range of denominations employed is sub-optimal. In extreme cases of such preferences, it might be worthwhile for monetary institutions to change their denominational structure. We will check any preferences using our model below and we report the outcomes in Section 5. Finally, we take a closer look at the cash payments of the 53 individuals who did not pay efficiently even when their wallet allowed them to do so. Twenty-two individuals paid with only one token, usually a note. Ten individuals paid with too many coins and, considering their wallet contents, were probably aiming to rebalance their wallet content. For the other 21 individuals no logical argument can be formulated. The above counting exercise would become much more involved if we wanted to assess the use of individual denominations relative to others. For example, about 35% of the inefficient payments were paid with one token, and we may now wonder whether this was the obvious denomination given the wallet and the payment amount. Unfortunately, this question cannot be answered with our sample, and also not by means of a simple counting exercise. In the next section, we therefore proceed with a modeling approach to describe the use of denominations. Our intention is to draw model-based conclusions on the use of euro denominations in the Netherlands and to see if there are preferences for some notes or coins such that these are used more than others. 4. Statistical model To examine if there are systematic preferences for denominations in euro cash payments, we consider the statistical model developed and applied in Kippers et al. (2003). The key features of the model are the following. First, it assumes a sequential choice strategy, that is, an individual first checks if the payment amount requires one or more of the highestvalued notes or coins available in the wallet and then he proceeds to adjacent lower-valued notes and coins. A second feature is that it contains a sequence of Poisson regression models as the data are counts. A third feature is that it includes the amount of money that needs to be paid. To illustrate this third feature, assume that the amount to be paid is 15 euros, and that the wallet contains one note of 20, one of 10 and one of 5. If the individual pays with a note of 20, the amount to be left to pay with notes of 10 or 5 is zero. If the individual chooses not to pay with a note of 20, the amount left to pay is 15 euros, which is paid with a note of 10 and a note of 5. Thus, we can compute, for each note and coin in the payment process, which amount still has to be paid after a choice concerning this note or coin has been made. We expect that the larger is the amount left relative to a specific note or coin, the higher is the probability that it will be selected by the individual. In Appendix B, we discuss some of the technical details of the model. If cash payment behavior of individuals reveals no preference for any denomination, then the probability of choosing a denomination for a payment would be equal for all denominations in similar payment situations defined by wallet content and amounts to be paid. Therefore, the question whether individuals are indifferent towards denominations can be answered statistically by testing if the parameter values (intercepts and coefficients on the amount left to pay) are equal across denominations.
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This hypothesis can formally be tested with a likelihood ratio (LR) test. Rejection of the null hypothesis suggests that the parameters significantly differ across denominations. In that case, no denominations are more or less preferred than others and the denominational structure of the currency can be regarded as optimal. In Appendix C, we examine the small sample properties of the relevant LR statistic. These results also suggest a sample size for which the test has reasonable power. It turns out that our sample size of 272 observations is large enough to have reasonable power and hence we can safely rely on the statistical results to be discussed in the next section. 5. Empirical results We estimate the parameters of the cash payment model with the sample data of 272 euro cash payments using maximum likelihood with heteroskedasticity-consistent standard errors. We exclude the observations where individuals have added coins or notes on the request of the cashier, as our model cannot use these for estimation. The final dataset contains 240 observations. Furthermore, given the specification of our model, see again Appendix B, we can only use those observations for which the paying individual has freedom of choice. For example, the case where one needs to pay 20 cents and one only has a 20 cents coin in the wallet, there is no such freedom. These no-free-choice observations do not contribute to the likelihood. Given the hierarchical structure of our model, the number of observations with free choice can decrease as the denominations become lower-valued. In the end, no parameters for the lowest denomination, the 1 euro cent, can be estimated. As the number of observations with free choice quickly becomes quite low for denominations smaller than 1 euro, we focus on the denominations 1, 2, 5, 10, 20, 50 and 100 euros. The largest existing denominations, of 200 and 500 euros, are excluded from analysis lack of data. The estimation results are presented in Table 4. The parameters for the variable measuring ‘‘amount left’’ are all significantly different from zero and they obtain the expected positive sign.5 This means that, when a denomination is needed more, it is also more likely to be used more often. To formally examine if there are differences in preference, we use the LR test. It obtains a value of 5.03, which does not exceed the 5% critical value of 18.31 of the chi-squared distribution with 10 degrees of freedom. We therefore conclude that there are no differences in preferences for one or more denominations. 6. Conclusion In this paper we examined whether the Dutch public has preferences for using certain euro coins and notes. An important motivation is the study of Kippers et al. (2003), who documented that such preferences existed for the Dutch guilder, which had the 1–212–5 based denomination range. Using a unique dataset, a specifically designed model and a 5 It should be mentioned here that the model is highly non-linear in its parameters and that the estimation routine (within Eviews) is susceptible to converging to zero-valued parameters, causing the estimation routine to collapse easily. In view of this and the outcomes of the logit model based on the characteristics of Table 1 discussed earlier, we decided not to include any explanatory variables in the model other than an intercept and the amount left to pay.
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Table 4 Parameter estimates with heteroskedasticity-consistent standard errors (S.E.) for the cash payment model (a sequence of truncated Poisson regression models) Denomination
Intercept
100 50 20 10 5 2 1
Amount left
Effective observations (free choice)
Parameter
S.E.
Parameter
S.E.
0.855 0.519 0.122 0.332 0.139 0.142 0.032
0.829 0.435 0.162 0.161a 0.263 0.262 0.218
2.174a 1.413a 1.019a 1.188a 1.754a 1.633a 1.780a
0.465 0.363 0.187 0.257 0.576 0.390 0.376
10 76 76 66 48 46 39
The effective sample size is 240. a Significant at the 5% level.
statistical test with good properties, we found that for the euro there are no such preferences. We also found that most payments are efficient. The results in our paper indicate that the euro denominational range facilitates efficient payment behavior, more than the guilder range did. Hence, the welfare implications of the introduction of the euro in the Netherlands are positive, both for paying individuals as for the central bank which distributes the euro notes. Acknowledgments We thank Merel van Diepen, Claudia Laheij and Niels Tromp for their help with collecting the data and Marielle Non for her help with programming. We also thank The Editor (von Hagen), three anonymous referees, Kees Koedijk, Herman van Dijk, Mars Cramer and Patrick Groenen for helpful comments. The Gauss and Eviews programs used for the computations are available upon request. Appendix A. Efficient payment schemes A.1. Cramer’s algorithm for infinite-sized wallet contents With A we denote the payment amount and with n(A) the amount of tokens to be used for cash payment. The denomination range is the set {1, 2, y, D} and an element of this set is denoted by d. We denote with n(A, d) the number of tokens of denomination d, so n(A) is Sn(A, d). Finally, v(d) is the value of denomination d. The objective is to minimize n(A) by choosing the ‘‘right’’ tokens out of the set {1, 2, y, D}. More formally, we should solve min
nðAÞ ¼
D P
jnðA; dÞj
d¼1
s:t:
D P d¼1
nðA; dÞvðdÞ ¼ A
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Note that we use the absolute value of n(A, d) to allow for the possibility that notes and coins are returned as change. The algorithm proceeds as follows. We take a range of amounts that are of interest, say 0.01 until 100 euro with intervals of 0.01 euro. The goal of the algorithm is to find for each of these amounts the efficient combinations of tokens: 1. The algorithm starts by taking all amounts that can be paid with only one token. For the euro, the amounts 0.01, 0.02, 0.10, 0.20, 0.50, 1, 2, 5, 10, 20, 50, and 100 are now covered. 2. All amounts that can be paid with two tokens, either two given by the individual as payment or one returned by the retailer as change, are computed. For example, 15 euro can be paid with 10 and 5 euro notes or with 20 euro note with a 5 euro note as change. If we now observe an amount that was already covered with only one token in the previous step, we do not add this pair of two tokens to the list as those tokens are obviously not efficient for this amount. 3. To the pairs that were found efficient in the previous step, we add each token once, both with positive and negative sign, the latter indicating change. For example, to a 10 and 5 euro note we add a 0.20 coin, covering the amounts 15.20 and 14.80 (where in the last case the 0.20 is given as change). For a given pair in Step 2 this gives 2D extra combinations with an additional token each. Adding a token with a positive sign to a combination, which has this same token with a negative sign, would yield a combination with lesser tokens and is therefore ignored. Note also that there is the restriction that the highest-valued token always has a positive sign. With these combinations we cover all resulting amounts, provided they were not already covered by efficient payments with lesser tokens. 4. We repeat step 3 by increasing each time the number of tokens until all amounts between 0.01 and 100 are covered. This algorithm results in efficient combinations (note that there can be more than one!) for each possible amount in the specified range. A.2. Cramer’s algorithm, for restricted wallet contents The original algorithm in A1 assumes that an individual has available all possible notes and coins. This is not realistic, and hence we need to modify the algorithm, which is now based on min
nðAÞ ¼
D P
jnðA; dÞj
d¼1
s:t:
D P
nðA; dÞvðdÞ ¼ A;
d¼1
nðA; dÞpwðdÞ 8d; where we use w(d) to denote the amount of available tokens of denomination d. If a wallet allows for paying according to an efficient scheme, we take this efficient scheme. Otherwise, we need to find the restricted efficient scheme, that is, the combination of tokens that minimizes the numbers of tokens used, given the wallet contents. In this
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case, the following algorithm is executed for a particular amount A and a particular wallet w. Note that this algorithm increases the number of computations substantially. 1. The algorithm starts by computing all amounts that can be covered by a single token. If the wallet content contains the relevant tokens, the combination is stored. 2. All amounts that can be paid with two tokens, either given by the individual or partly returned by the cashier as change, are computed. Again, only if the wallet content allows so, the combination is stored. 3. To the pairs that were found as efficient in the previous step, we add each token once, both with positive and negative sign (indicating change). With these combinations, we fill the list with the resulting amounts, provided they are not already covered by lesser tokens, and provided the combination fits in the wallet. 4. We repeat step 3 for increasing numbers of tokens until the amount A is covered. Note that it is not sufficient to only store the combination which pays the given amount A exactly. We need to store each combination that fits in the individual’s wallet, since we use these as the starting point for the schemes in the next step. As an example, suppose the wallet contains three notes of 5 euro and an individual has to pay 15 euro. The efficient combinations 10 and 5, and 20 and 5 are not available, but the combination 5, 5 and 5 is efficient given this wallet content. Appendix B. The model of Kippers et al. (2003) The model focuses on the probability that an individual i selects a combination of denominations, represented by a D-dimensional discrete random variable Yi, to pay amount Ai, given the wallet content wi. D corresponds to the number of denominations in a denominational range. The realizations of the random variable Yi, are denoted by yi. For modeling purposes a hierarchical structure is imposed. According to this structure the choice of an individual is modeled separately for each denomination, starting from the highest denomination to the lowest. The choice for a certain denomination depends on how previous denominations were used. We consider the probability structure Pr½Y i ¼ yi jwi ¼ Pr½Y D;i ¼ yD;i jwi Pr½Y D1;i ¼ yD1;i jyD;i ; wi . . . Pr½Y 1;i ¼ y1;i jyD;i ; :::; y2;i ; wi .
(B.1) The choice for a payment with yd,i tokens for denomination d is assumed to be distributed as truncated Poisson, where the choice set is truncated from below by a lower bound lbd,i and from above by an upper bound ubd,i, that is, Y d;i jydþ1;i ; . . . ; yD;i ; wi POIðexpðx0d;i bd ÞÞ I½lbc;i ; ubd;i ,
(B.2)
where bd is a parameter vector and xd,i contains explanatory variables for denomination d. The upper bound represents the maximum number of tokens the individual can use for a payment and it is determined by the wallet content. An individual cannot pay with more tokens than that are available in the wallet. Also, it is assumed that individuals do not pay with a token if it is unnecessary, that is, when it is expected that the same token will be returned as change. The lower bound represents the minimum number of tokens of a denomination the individual has to use for the payment. This minimum is determined by the payment
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amount and the availability of other denominations in the wallet. If the wallet does not contain enough value of lower denominations, then the individual is forced to use a minimum number of tokens of the denomination under consideration. The upper and lower bounds are defined as follows: Ai ubD;i ¼ min ceil ; wD;i , (B.3) vD Ai amount ydþ1;i ; . . . ; yD;i ubd;i ¼ min ceil ; wd;i for d ¼ D 1; . . . ; 1, vd (B.4) Ai amountðw1;i ; . . . ; wD1;i Þ lbD;i ¼ max ceil ;0 , (B.5) vd Ai amountðydþ1;i ; . . . ; yD;i Þ amountðw1;i ; . . . ; wd1;i Þ lbd;i ¼ max ceil ;0 for d ¼ D 1; . . . ; 1 vd
(B.6) with amountðxp ; . . . ; xq Þ ¼
q X
vk xk .
(B.7)
If lbd,i ¼ ubd,i, the individual has no freedom of choice and hence Pr½Y d;i ¼ yd;i ¼ lbd;i ¼ ubd;i jydþ1;i ; . . . ; yD;i ; wi ¼ 1.
(B.8)
k¼p
The main explanatory variable in our model concerns the payment amount that each time when a token is to be chosen is left over to pay. In addition to an intercept, the model therefore requires the explanatory variable, say, ACORR, defined as (a function of) the amount to be paid minus the value of the payments chosen for higher denominations, scaled to its face value, that is, Ai ACORRD;i ¼ ln , vD Ai amountðyD;i ; . . . ; ydþ1 ; iÞ for d ¼ 1; . . . ; D 1. (B.9) ACORRd;i ¼ ln vD Appendix C. Small sample properties of the LR test Starting point for our simulations is the set of efficient payment schemes for amounts 0.01 to 100 euro as generated using the Cramer algorithm. Efficient payments are by definition based only on the face value of denominations and therefore constitute examples of payments with indifference towards denominations. If the parameters in the cash payment model are estimated for a dataset of efficient payments, the LR test of equal parameters should follow the asymptotic chi-squared distribution. The following simulation scheme is executed. 1. For all payments, matching wallets are generated by assuming that the wallet contains the payment, with three tokens of each denomination added.
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2. A sample of size 1.000 is randomly drawn. The cash payment model parameters are estimated with and without the parameter restrictions concerning the 1 euro coin through the 50 euro note. 3. Step 2 is repeated 200 times. The resulting empirical size for the LR test with an asymptotic chi-squared (10) distribution turns out to be 6%, which is rather close to the nominal size of 5%. Given this finding, we can safely rely on the asymptotic chi-squared distribution under the null hypothesis. Next, we examine the power of this test. The starting point for this simulation is again the set of efficient payment schemes for amounts 0.01–100 euro as generated using the Cramer algorithm. The following simulation scheme is executed to examine the power of the LR test for small samples, that is, 1. for a fraction a of these schemes, payments with a single token of denomination d are replaced by payment with two tokens of denominations d+1, provided that denomination d has a face value of two times the denomination d+1; 2. for all payments, matching wallets are generated by assuming that the wallet contains the payment, with three tokens of each denomination added; 3. a sample of size n is randomly drawn. The cash payment model parameters are estimated with and without the restrictions; 4. step 3 is repeated N times. The resulting percentage of rejections measures the empirical power. By the replacement of payments, as executed in step 1 of the simulation scheme, we introduce a preference for certain tokens in the data. We use a replacement of a 20 euro note by two 10 euro notes to simulate a preference for the 10 euro note. The sample size n in step 3 is not simply set to 272, which is the sample size of our euro cash payment dataset. This is due to the fact that for these data there are only 75 effective observations with free choice. Hence, a sample size of n ¼ 75 in the simulation scheme is about comparable to the sample size of our dataset. The results of the simulations for different replacement rates are shown in Table C1. We observe that the empirical power of the test rapidly increases with larger replacement fractions. Also, note that this is already achieved for only 75 effective data points.
Table C1 Empirical power of LR test for varying replacement rates Replacement rate
Rejection frequency No. of replications
25%
50%
75%
0.01 500
0.37 200
0.96 200
Notes: The replacement rate represents a preference and is defined as the percentage of efficient payments in a sample in which payment with one 20 euro note is randomly replaced with two 10 euro notes. The empirical power is defined as the percentage of runs in which the null hypothesis of equal parameters is rejected.
ARTICLE IN PRESS P.H. Franses, J. Kippers / European Economic Review 51 (2007) 1985–1997
1997
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