Cash, Paper-based and Electronic Payments: A Theoretical Approach

Transcription

1 Cash, Paper-based and Elecronic Paymens: A Theoreical Approach Francisco Callado y, Jana Hromcová z and Naalia Urero x June 2009 Absrac In his paper we provide a general equilibrium model ha helps explaining paymen choice a a reail level: cash, elecronic and paper-based insrumens. In paricular, i provides heoreical foundaions o reconcile previous empirical evidence on he maer. Paymen srucure of a given counry can be shaped by he cos of each paymen insrumen, he degree of echnology developmen and he ineres rae. Di erences among counries could be based on speci c characerisics relaed o hese variables. Keywords: cash; paymens; human capial; cash-in-advance; JEL classi caion: E42, E4, O42 This research was underaken wih suppor of he Spanish Minisry of Educaion and Science SEJ /ECON, SEJ and SEJ C y Universia de Girona, Deparamen d Economia i Empresa z Corresponding auhor: jhromcova@uao.es. Universia Auònoma de Barcelona, Deparamen d Economia Aplicada. x Universia de Girona, Deparamen d Economia i Empresa

2 Inroducion This paper analyzes he dynamics of he paymen insrumens choice, in paricular he behavior of cash, paper-based and elecronic paymens. The variables ha explain di erences beween counries are he relaive cos of he available paymen insrumens, degree of echnology developmen and moneary policy. The way in which paymens are made in a given counry happens o be a relevan issue a leas in wo respecs. On one side, he cos of a paymen sysem, ha could accoun up o 3% of GDP, Humphrey, Pulley and Vesala (996 and 2000), is clearly a eced by agens choice since he cos of each paymen insrumen di ers. On he oher hand, paymen choice could in uence he funcioning of he nancial sysem and herefore faciliae rade in he real economy. In fac hese decisions could have economic consequences since an imporan par of GDP (around 2/3 in he US) comes from consumer ransacions and hese are compleed wih some mehod of paymen, Schref (2006). The disribuion of paymen insrumens use is usually di eren across counries. Some empirical sudies have a look a he characerisics of his disribuion and also ry o analyze and explain he deerminans of hese di erences. Humphrey e. al. (996) examine he paymen sysems of 4 developed counries and nd ha he use of elecronic means of paymen is clearly increasing in all counries. Besides, hey ry o explain he possible facors behind he di eren srucures across counries. Among hese hey nd as he mos imporan he degree of paymen availabiliy (number of users, erminals, ec) and insiuional and culural di erences (income, new insrumens, ec.). Figure 3 in Hancock and Humphrey (998) provides an evidence of how elecronic means of paymen (credi and debi cards) gain imporance wih respec o checks and cash. An ineresing nding is ha he paymen choice behavior of consumers in he US is a bi di eren han ha of he res of developed counries. Alhough cash use is low, he relevance of elecronic paymens is clearly lower. Humphrey (2004) deepens he analysis of he US by analyzing he subsiuion of cash by cards. In Figure 2 in his sudy we can observe how he relaive imporance of elecronic means of paymens (credi and debi cards) increases over ime. Hankock and Humphrey (998), Figure 3 Humphrey (2004), Figure 2 Humphrey e. al. (200) include price in rying o explain consumer choices using daa from Norway. Alhough producion echniques and consumer needs are very similar across counries, he 2

3 usage of paymen insrumens sill di ers. They nd empirical evidence ha echnology developmen and relaive prices, ogeher wih he relaionship beween cos and echnology, could have a word o say in explaining hese di erences. There is a lack of heoreical lieraure concerning wider choice of paymen insrumens, see Schref (2006) for example. There is a need o develop sound heoreical foundaions for consumer decisions and use i o analyze heir paymen choice behavior, Crowe, Schuh and Savins (2006). Here we propose o ll he gap by focusing on how and why consumers choose which paymen insrumen hey use. We le our consumers use cash, paper-based or elecronic paymens. A consumer facing he decision on he mean of paymen will ake ino accoun he relaive cos of each insrumen. The cos depends on he place and ime of he ransacion. We adap for our purpose he model of Hromcová (2008). We consider ha a moneary policy and a governmen acion (ha favors some of he available means of paymens) may aler agens paymen insrumens choice. We calibrae he model for yearly US daa, We compare he dynamics of he paymen choice in he model and he daa. The model can generae very similar behavior o he one found in he daa. Collecing daa for more counries we will deermine he paricular parameers responsible for di erences in he paymen choice dynamics beween counries. The remainder of he paper is organized as follows. The model and is main properies are saed in secion 2. In secion 3 we describe he balanced growh pah behavior. In secion 4 we discuss he behavior in he ransiion. Final conclusions are summarized in secion 5. Many heoreical models which consider wo paymen insrumens where developed. For example, Schref (992), Gillman (993), Aiagari, Braun and Ecksain (998) show ha he choice beween cash and a paymen via inermediary depends on he mix of he cos of he alernaive mean of paymen and he moneary policy. Ireland (994a) poins ou ha he decrease in he use of cash and is subsiuion by cards is caused by an increase in income. Markose and Loke (2003) show ha his subsiuion is also due o he availabiliy of paymen erminals a he poin of sale. Ireland (994b), Marquis and Re e (994), English (999) and Hromcová (2008) relae he choice of paymen insrumens also o he echnological progress. 3

4 2 Model 2. The Household Problem We follow closely he speci caion of he economy in Hromcová (2008). The economy consiss of a large number of in niely lived households. All households have idenical preferences, producion and rade opporuniies. Households inhabi he following environmen: hey face coninuum of spaially separaed markes, which are indexed by i 2 [0; ]. All households live in marke 0, and he index i indicaes he disance from home. In each marke i a disinc perishable good is produced and sold in every period. Goods are hus indexed by i; which corresponds o he marke of boh producion and rade. The economy has a represenaive household wih preferences given by he uiliy funcion X Z = 0 c (i) di () where c (i) is de ned as he consumpion a period of he good produced in marke i and > 0 is he inverse of he elasiciy of ineremporal subsiuion. The producion and rade is like in Lucas and Sokey (983). Each household is composed of a worker-shopper pair. Prior o any rading governmen xes he gross nominal ineres rae R o be consan in all periods. We will assume ha R. Agens ener he period wih cerain amoun of moneary balances M and he deb B ; carried over from he previous period, and he human capial sock h. A represenaive worker decides o produce on any of he markes i via he ne producion funcion y = wh l (2) where w denoes he marginal produc of human capial in goods producion, and l is he amoun of ime spen working. Human capial accumulaion depends on he ime spen sudying, on he level of human capial and on he depreciaion rae according o h + = ( l ) h + ( ) h (3) where is he e ciency of learning parameer and is he depreciaion rae. Firs, he goods marke opens and consumpion akes place. Worker says a he marke i during he whole period. Shopper visis various markes o acquire consumpion goods carrying all he moneary balances of he household. To acquire consumpion goods agens can use governmen issued money or alernaive means of paymen. When using non-cash paymens, hey can choose from paper-based or elecronic privae securiies. All goods purchased wih governmen issued money will be referred o as cash goods. Goods purchased via paper-based paymens will be referred o as paper-based credi goods and goods purchased via elecronic privae securiies will be referred o as elecronic credi goods. Goods only di er in a way hey are acquired. Nominal moneary balances M can be used o buy goods in some of he markes indexed by i. Cash purchases are subjec o he liquidiy consrain Z 0 [ (i)] c (i)di M p ; (4) 4

5 where (i) = 0 if a good is purchased on marke i wih cash, or (i) = if a good is purchased on marke i via a non-cash paymen. The nancial inermediary enables paper-based and elecronic paymens a a cos ha is given for each marke i and period. Afer he consumpion akes place, he moneary holdings of agens are augmened by a lump sum ransfer X from he governmen. The amoun X is endogenously deermined in he sysem according o he given nominal ineres rae, so ha he money demand is oally sais ed. As he nex sep he securiies marke opens. During he securiies rading session households choose heir currency holdings M + : They also purchase (or issue) one-period nominally denominaed pure discoun bonds paying B + unis of money a period + while hey cos B + unis of money a R period : Bonds are in zero ne supply. The budge consrain agens are facing can be wrien Z 0 c (i)di + Z 0 (i) (i) ' (i) + [ (i)] e (i) di+ (5) M + p + B + Rp wh l + M p + B p + X p : where (i) = for a non-cash purchase a marke i, (i) = 0 if he good is purchased on marke i via an elecronic paymen and (i) = if he good is bough using a paper-based privae securiy. The real cos of he non-cash ransacion a ime on marke i is ' (i) and e (i) for paper-based and elecronic paymens, respecively. 2.2 Cos of alernaive means of paymens We assume ha he inermediaion cos mus be paid by he buyer, as moivaed in Ireland (994b). To be able o purchase wihou cash, some resources mus be devoed o making he non-cash paymen iself available, checking he ideniy of he buyer and his abiliy o pay. When he shopper is far away from home (marke zero) he communicaion becomes more di cul, and we assume ha he paymen o he inermediary increases wih i. The process of human capial accumulaion gives a poenial for he developmen of new echnologies. I also leads o an increase in income per worker and higher consumpion. Higher purchase means ha more imporance will be given o checking he abiliy of he buyer o pay. The developmen of new echnologies will decrease he processing coss. The real paymen made o he inermediary is characerized by a funcion ha ful lls properies found in some empirical sudies, see Hromcová (2008): he inermediaion cos is lower in richer counries, he cos of inermediaed paymen diminishes over ime, and he cos elasiciy is close o zero (which moivaes he proporional inermediaion cos). Boh paper-based and elecronic paymens ge cheaper as new echnology develop. 2 We specify he inermediaion coss for elecronic and paper-based paymens in he following way e (i) = [(i) + F ] (h ) c (i); e (6) ' (i) = (i) (h ) c (i): ' (7) 2 Check processing and connecions for elecronic ransacions ge beer and cheaper wih an increase in exising echnologies. 5

6 Boh paper-based and elecronic paymens have a ime independen cos (i) ha is increasing wih a disance from home, similarly as in Gillman (993), Ireland (994a) or Hromcová (2003), for example. For he sake of racabiliy we assign i he funcional form used previously in he lieraure, inroduced in Ireland (994a) (i) = i i : (8) The xed cos F is marke and ime independen and i is aribued o a cos of acquiring and mainaining of a machine ha enables elecronic ransacions. This is a relaive cos of paper-based and elecronic paymens and we inerpre i as he governmen insrumens ha can moivae he agens o change he relaive usage of paper-based and elecronic paymens. The ime dependen par of he inermediaion cos, (h ) j cj (i); j = ' or e; embodies properies concerning he proporionaliy o consumpion purchases and he e ec of new echnologies on he cos. I decreases as he level of knowledge increases and i is proporional o consumpion purchases done via inermediary. The funcion =h j re ecs he following: as more human capial is accumulaed, more knowledge is available, beer echnologies can be developed and cheaper inermediaion services can be o ered. The parameer j can be inerpreed as he degree of knowledge di usion ino he paymen sysem. 3 We assume ha 0 < j < : For his inerval of values, any improvemen in he exising echnologies makes i more di cul o lower he cos of inermediaion services. 4 This speci caion means ha for low level of echnology he paper-based paymens on any marke are cheaper han he elecronic ones. Because he degree of knowledge di usion ino he elecronic paymens is higher han he knowledge di usion ino he paper-based ransacions, e > ' : As he economy grows and new echnologies are developed, boh paymens ge cheaper. In he long run he di erences beween he inermediaion cos for employing he elecronic and paper-based paymens disappear. The mean of paymen will be chosen according o is cos. The opporuniy cos of buying wih cash is he nominal ineres rae R ; which does no vary across markes. The opporuniy cos of buying wih paper-based or elecronic securiies is he one corresponding o he inermediaion cos ' (i) and e (i); respecively. We show he behavior of he opporuniy cos of using cash, paper-based or elecronic paymens across markes in Figure. For an economy a a low sage of developmen (low level of human capial) i may happen ha no elecronic goods will be employed, because hey are oo expensive. In Figure a) we can observe ha in all markes wih indexes below paper-based securiies have he lowes cos. In all marke wih indexes above cash will be employed because he opporuniy cos of holding money is lower han he inermediaion cos for eiher non-cash securiy. For a more developed economy, Figure b), paper-based paymens are he 3 For h < higher j increases he inermediaion cos for a given level of human capial. We can undersand ha in our analysis counries capable of performing elecronic ransacions are developed enough so he iniial level of humancapial h > : 4 For example a price of a connecion o he buyers bank accoun (a phone call) is so low ha i is really di cul o lower i even more if beer echnologies are developed, or ha cheaper processing of checks is always more di cul o achieve. 6

7 cheapes way of acquiring consumpion goods in all markes wih indexes below. In all markes wih indexes beween and s elecronic paymens have he lowes cos, and in markes wih indexes above s governmen issued money are he cheapes. (a) (b) Figure : Deerminaion of he marginal markes: when i = ; agens are indi eren beween buying wih cash or paper-based securiies; when i = agens are indi eren beween buying via elecronic or paper-based ransacions; when i = s agens are indi eren beween buying wih cash or via elecronic paymens. (a) Low developmen: only cash and paper-based paymens are used; (b) Higher developmen: all hree paymen insrumens are used. We arbirarily assume ha cash will be used on he marginal markes and s ; and elecronic paymen on he cuo marke : There will exiss a momen in ime for which all hree marginal markes coincide s = =. Thus by checking he relaionship beween s and we can recognize he menioned wo sages of developmen. One in which no elecronic goods will be employed, s < ; and he sage of he coexisence of all hree ypes of considered means of paymens. The basic characerisic of he second sage is ha in he course of ime he elecronic privae securiies will drive ou of he markes cash and paper-based paymens. De ne 8 c 0 (i) when (i) = 0; >< c (i) = c ' (i) when (i) = and (i) = 0; >: c e (i) when (i) = and (i) = : The funcions c 0 (i); c ' (i) and c e (i) characerize he cash, paper-based credi and elecronic credi consumpion per marke i; respecively. Recognizing he wo sages of developmen, we can rewrie he uiliy funcion 2 3 X = 6 4 Z minf ; g 0 c ' (i) di + Z maxf ; s g minf ; g 7 c e (i) di + Z maxf ; s g c 0 (i) 7 di5 ; (9)

8 he budge consrain Z minf ; g [c ' (i) + ' (i)] di + Z maxf ; s g [c e (i) + e (i)] di + Z c 0 (i)di + 0 minf ; g maxf ; s g + M + p + B + Rp wh l + M p + B p + X p (0) and he cash-in-advance consrain Z maxf ; s g c 0 (i)di M p : () 2.3 Equilibrium De niion: Given he se of iniial condiions h ; M ; B and he nominal ineres rae R; he equilibrium consiss of sequences fc 0 (i); c ' (i); c e (i); l ; h + ; M + ; B + ; ; ; s ; X ; p g = such ha (a) a represenaive household is maximizing he discouned uiliy (9) subjec o he budge consrain (0), he cash-in-advance consrain () and he condiion for accumulaion of human capial (3), choosing he sequences fc 0 (i); c ' (i); c e (i); l ; h + ; M + ; B + ; ; ; s g =; (b) markes for goods, money and bonds clear in every period, Z minf ; g c ' (i)di + Z maxf ; s g c e (i)di + Z c 0 (i)di + 0 minf ; g maxf ; s g minf Z ; g + ' (i)di + Z maxf ; s g e (i)di = wh l ; (2) 0 minf ; g M + = M + X ; (3) B + = 0: (4) Le ; ; and be he non-negaive Lagrange mulipliers associaed wih he budge consrain (0), he cash-in-advance consrain (), and he condiion for accumulaion of human capial (3), respecively. The equaions ha characerize he equilibrium are he above menioned marke clearing condiions (2), (3), (4) and he rs order condiions on all ypes of consumpions, labor, 8

9 capial, nominal balances, nominal bonds and marginal markes indexes, respecively, c 0 (i) = + ; (5) c ' (i) = ; (6) c e (i) = ; (7) w = ; (8) = + wl f ( l + ) + ( )g ; (9) = ; (20) p p + = R + ; (2) p p + c 0 ( ) c e (s ) c ' ( ) c ' ( ) c 0 (s ) c e ( ) = [c ' ( ) + ( )] + ( + ) c 0 ( ); (22) = ( + )c 0 (s ) + [c e (s ) + e (s )] (23) = [c e ( ) + e ( )] + [c ' ( ) + ' ( )] (24) Using (5), (20) and (2), we can rewrie he rs order condiion on cash consumpion as c 0 (i) = R : (25) From he rs order condiions on he marginal markes beween paper-based credi goods and cash goods (22), elecronic credi goods and cash goods (23) and beween paper-based and elecronic credi goods (24) we ge 5 ' h i [(R; h )] = c ' [ (R)] c 0 [R; (R)] + Rc 0 [R; (R)] c ' [ (R)] ; (26) (R) e h i [s (R; h )] = c e [ (R)] c 0 [R; (R)] + Rc 0 [R; (R)] c e [ (R)] ; (27) (R) e [ (R; h )] = ' [ (R; h )] : (28) 5 The iniial level of he Lagrange muliplier on he budge consrain depends on he moneary policy, so we wrie = (R): 9

10 Using he given forms of he cos funcions (6) and (7), we ge for he respecive cuo markes where (R) = (R; h ) = ; (29) (h ) + (R) ' s = s(r; h ) = = (h ) = (R) (h ) F e ; (30) (h ) + (R) e (h ) F e F (3) (h ) e (h ) + F ' 8 ln R for = ; ><! (R) = >: for 6= : As can be seen from he goods marke equilibrium (2), he curren period oupu is spen beween paper-based credi consumpion, elecronic credi consumpion, cash consumpion and paymens o he inermediary for boh ypes of non-cash purchases. The real moneary balances, which equal he amoun of cash consumpion purchased in all markes, are 8 < [ (R; h )] c 0 [R; (R)] for s, or m = m(r; h ; ) = (33) : [ s (R; h )] c 0 [R; (R)] for s > where m = M : (34) p The paper-based credi consumpion in all markes is 8 < (h ) c ' [ (R)] for s, or ' = '(R; h ; ) = (35) : (R; h ) c ' [ (R)] for s > R (32) and elecronic credi consumpion in all markes 8 < 0 for s, or e = e(r; h ; ) = : [s(r; h ) (R; h )] c e [ (R)] for s > : (36) The par of oupu ha goes o he inermediary is he sum of paymens corresponding o paper-based consumpion (solving for he fourh inegral in he goods marke equilibrium (2)), 8 < f (h ) ln [ (h )]g c ' [ (R)] for s, or ' = ' (R; h ; ) = (37) : f (R; h ) ln [ (R; h )]g c ' [ (R)] for s >. 0

11 and he one corresponding o elecronic credi consumpion 8 < 0 for s, or e = : f[(r; h ) s(r; h ) + ln ( ) ln ( s )] + [s ] F e g (h ) c e [ e (R)] for s > : (38) The moneary ransfer from he governmen is given by X = ( + )M where + is he gross growh rae of money supply beween period and +. From (8) and (9) we ge he evoluion of he marginal uiliy of consumpion + = ( + ) : (39) Combining expressions (2), (34) and (39) we can wrie for he growh rae of money supply + = Noice ha he consan in (40) is he gross in aion rae + = p + p = R m + : (40) ( + ) m R ( + ) : (4)

12 3 Balanced Growh Pah To perform our analysis we will assume ha human capial grows a a posiive rae, i.e. h + h > : To characerize he properies of he economy when i reaches he balanced growh pah we look a he behavior of variables as ime goes o in niy. From (29)-(3) we see ha when he human capial accumulaes a a posiive rae and h! he cuo index beween he elecronic and cash goods approaches uniy, s(r; h )! ; and he one beween paper-based paymens and cash goods also (R; h )! ; while he cuo index beween he paper-based and elecronic paymens is approaching zero, (h )! 0: Tha means ha he elecronic paymens are used in more and more markes and he cash and paper-based privae securiies are less and less employed. Le us look a he long run growh raes of all ypes of consumpion and paymens o he inermediary. When aking he limis of he growh raes of hese variables as ime goes o in niy we ge he following: cash consumpion and he paymen o he inermediary for elecronic ransacions in he long run grow a he rae m+ lim! m paper-based credi consumpion = lim! '+ lim! ' = e+ e + = + h h + h h + e ' ; e ; (42) he inermediaion cos for paper-based paymens in he long run does no grow = (43) lim! ' + ' and he elecronic credi consumpion grows a he rae Goods marke equilibrium e+ lim! e = wl = m h implies ha credi consumpion in he long run mus grow like human capial. Tha means ha + + ' h + e h + : ' h + e h (44) h + = : (45) h + Plugging (45) ino (42) and using (39), (3) and (40) we can summarize he resuls in he following proposiion. 2

13 Proposiion On he balanced growh pah cash consumpion and he paymen for elecronic ransacions grow a he rae [ ( + )] ae ; paper-credi consumpion grows a [ ( + )] ae+a' elecronic-credi consumpion and human capial grow a he rae labor is given by and he growh rae of money supply is [ ( + )] ; ( + ) [ ( + )] R ae ( + ) ae : Proof. The labor on he balanced growh pah is obained by equaing he condiion for he human capial accumulaion (3) and he long run growh rae of capial. 4 Transiional Dynamics 4. Numerical Example In order o characerize he dynamics we rewrie he equilibrium equaions. We de ne De ne Then he goods marke equilibrium (2) can be rewrien as ^m = m ; ^' = ' ; ^e = e (46) ^' = ' ; ^e = e : (47) ^G (R; h ) = ^m + ^' + ^e + ^' + ^e : (48) h = ^G (R; h ) : (49) wl Time spen working l can be wrien as a funcion of he growh rae of human capial using he equaion (3). Using (48), (49), (3) and (39) we can characerize he enire equilibrium by a second order di erence equaion in human capial h + h = ^G (R; h + ) ^G (R; h )! h + h + h +2 [ ( + )] : (50) h + + 3

14 4.. Numerical Technique In order o illusrae he behavior of he economy in he ransiion in more deail, we will use a numerical example. Because we know he long run characerisics, we will use backward inducion o solve he di erence equaion in human capial (50). We assume ha for some high enough level of human capial, say h T h ; he economy is on he balanced growh pah, T +2 h T + = lim h+! h = ( + ) : Then we apply he Newon Raphson mehod o nd h knowing h + and h +2 for all : In his way we obain a numerical policy funcion h + = H(h ; R): Given he iniial level of human capial and he policy funcion, he behavior of all oher variables in he economy can be calculaed from he equaions (29)-(38) and (46)-(47) Calibraion We calibrae he model o he yearly US daa, period We will consider ha one period in he model corresponds o one year. We se he long run growh rae of he economy o g = :05, he growh rae of oupu per worker found in he daa. The baseline moneary policy is se o R = :07, as implied by he average value of year reasury bill rae. The depreciaion of human capial will be se o = 0:025 as in Gillman, Kejak, Valeninyi (999). The discoun facor in he uiliy funcion = 0:995: Marginal produc of capial in goods producion w = : Oher parameers are se o mach as closely as possible he average gross in aion rae, daa = :046 and he average gross growh rae of money supply in he daa, daa = :055. We se he e ciency of learning o = 0:05; which implies ha on he balanced growh pah 20% of ime is devoed o working and he res o sudying. The degree of knowledge di usion ino he paper-based and elecronic paymen is se o ' = 0:6 and e = 0:7. These values as se in accordance wih he growh raes of checks and cash found in he daa. The relaive cos beween elecronic and paper-based paymens is se o F = : I leads o he iniial raio of paper-based o elecronic paymens as found in he daa. The inverse of he ineremporal elasiciy of subsiuion is endogenous, = :3: The menioned paramerizaion implies model = :044 and model = :049: 4.2 The dynamics of he paymen insrumen choice We simulae he behavior of he economy using he soluion of he above menioned second order di erence equaion. In Figure 2 we plo he relaive usage of each insrumen in he course of ime. Figure 2a) shows he relaive choice of paymen insrumens over a very long period of ime. The iniial value of human capial in ha case is h 0 = : We begin a a very low sage of developmen during which only wo means of paymens coexis, cash and paper-based ones. Wih he developmen of new echnologies paper-based paymens become relaively cheaper and he fracion of markes where his paymen mehod is chosen increases, meanwhile he fracion of markes where cash is employed decreases. 4

15 (a) (b) Figure 2: Evoluion of he fracion of cash, paper-based and elecronic paymens as a percenage of all kinds of ransacions for ' = 0:6; e = 0:7 and F = : (a) Behavior of he economy over 000 years, h 0 = ; (b) Behavior of he model economy which corresponds o he US, , h 0 =

16 Elecronic paymens emerge a a cerain level of echnology (in his paricular case displayed in Figure 2, h 343 = 72). Since ha poin agens can choose from hree paymen insrumens. As he cos of elecronic paymens becomes cheaper and cheaper wih he developmen of new echnologies, paper-based insrumens and governmen money are driven away from he economy. However, he process of ransformaion of he paymen sysem is raher slow. In Figure 2b), which is a deail picked from Figure 2a), we can nd similar behavior as observed in he US in he period (Figure 2 in Humphrey (2004), depiced in he inroducion). In he elecronic paymens age, he fracion of markes where cash and paper-based paymens are employed is shrinking, bu i does no mean ha non-elecronic paymens are no growing any more. For e < and e ' < ; cash and paper-based credi consumpions boh grow a posiive raes, as well as he inermediaion paymens for he non-cash ransacions, see Figure 3. We also plo he evoluion of human capial. We can see ha is growh rae does no vary much over ime. Figure 3 helps us o see ha in he long run he growh rae of elecronic credi consumpion converges o he one of human capial, and he growh rae of cash consumpion converges o he growh rae of inermediaion paymens for elecronic ransacions. 6 Figure 3: Evoluion of he cash consumpion, paper-based credi consumpion, elecronic credi consumpion, paymens o inermediary for boh non-cash ransacions and human capial for ' = 0:6, e = 0:7 and F = : In Figure 4 we plo he relaionship beween he raio of cash/all insrumens and elecronic/noncash insrumens. The arrows show he ransformaion ha he paymen sysem undergoes in 25 years period of ime. We compare he resuls delivered by he model wih he US daa. 6 This is a resul found analyicaly for he balanced growh pah. 6

17 Figure 4: Comparison of he ransformaion of he paymen sysem in he model and he US daa, baseline parameers E ecs of moneary policy We can illusrae he e ec of a variaion of he nominal ineres rae, R; in Figure. Imagine ha he opporuniy cos of holding money increases. The cash line in Figure moves up. Agens will wan o economize on heir money holding, subsiuing away from money. In he low sage of developmen he fracion of markes where paper-based paymen insrumens are used increases and where cash is employed decreases, i.e. d > 0. On he higher level of developmen paper-based dr paymens will be used on he same fracion of markes as under low nominal ineres rae, because he cuo marke is no a eced direcly by he moneary policy, see also he equaion (3). This means ha he shif away from cash will go owards elecronic credi consumpion. The behavior of he economy under di eren moneary policies is illusraed in Figure 5. In Figure 5b) we show he dynamics of he relaive usage of each paymen insrumen for wo di eren moneary policies described above. The emergence of elecronic paymens is accompanied by an increase in he growh rae of he economy. As more resources are channeled (unproducively) o he nancial inermediary, agens wan o compensae for he loss by working more (and sudying less). The change in he working ime is relaively higher ha he change in he accumulaion of human capial, and i implies a jump in he growh rae of he economy, see Figure 5c). For a given level of echnology, higher nominal ineres rae implies less cash ransacions. The arrows in Figure 5a) all begin a he same level of human capial. The higher is he opporuniy cos of holding money, he higher is he iniial grade of subsiuing away from cash. Neverheless, he furher ransformaion (over he nex 25 years) is slowing down wih in aion (and higher ineres rae). I is a kind of convergence behavior. The higher is he iniial disance of a counry from is 7

18 balanced growh pah (he balanced growh pah ransformaion of he paymen sysem), he more rapid is he convergence owards i. (a) (b) (c) (d) Figure 5: Behavior of he model under di eren moneary policies for baseline parameers; R =.07,.7,.27 and.37 correspond o he in aion rae of 4.4%, 4%, 24% and 34% respecively; (a) Subsiuion of cash by paper-based and elecronic paymens; (b) Evoluion of he fracion of cash, paper-based and elecronic paymens as a percenage of all kinds of ransacions; (c) Evoluion of he growh rae of he economy; (d) Evoluion of cash, paper-based and elecronic paymens. In Figure 5d) we can see ha he long run level of cash consumpion is a eced by he opporuniy cos of holding money, as implied also by he equaion (33) and (25). The iniial dependence of paperbased and elecronic credi consumpion on R ends o disappear in he long run, as he fracion of 8

19 markes where elecronic ransacions are employed increases o, lim! = ; and he fracion of markes where paper-based ransacions are employed decreases o 0, lim! s = 0 for all R: E ecs of governmen incenives o a usage of elecronic means of paymens The e ec of a decrease in he relaive cos for elecronic and paper-based paymens, F; can be again deduced from Figure. Moving he elecronic paymens line downward we can see ha ds > 0 dr and d < 0: Elecronic paymens become relaively cheaper and i will be pro able for agens o dr choose his insrumen a lower level of developmen. The subsiuion of cash and paper-based ransacions by elecronic ones will begin sooner. The behavior of he economy for di eren levels of xed cos is illusraed in Figure 6. We can see ha he iniial posiion of a counry and he speed of ransformaion can be a eced in an imporan way by he relaive cos of elecronic and paper-based insrumens, see he arrows in Figure 6a). 7 If he governmen waned o aler he agens paymen insrumen choice, i could subsidize he xed cos for elecronic ransacions. Similarly o he previous case, he emergence of he elecronic ransacions resuls in an increase of he growh rae of he economy. The long run level of cash ransacions will no be a eced. Since he emergence of elecronic insrumens, he level of paper-based ransacions is lower due o lower relaive cos. This di erence persiss in he long run. The long run level of elecronic ransacions converges for di eren xed coss, as all iniial di erences disappear, lim! = and lim! s = 0 for all F. 5 Conclusions This paper ries o improve previous lieraure in several ways. Firs i helps in covering he need for heoreical works by presening a general equilibrium model on paymen choice a a reail level. Second, i also provides heoreical foundaions o reconcile previous empirical evidence on paymen choice among cash, elecronic and paper-based insrumens. Paymen srucure of a given counry can be shaped by he ineres rae, he degree of echnology developmen and he cos of each paymen insrumen. And hird, he model is able o explain why counries wih similar paymen alernaives and consumer needs for paymen insrumens end up wih very di eren usage of hese mechanisms. 7 Again, all arrows begin a he same level of human capial and show he changes in he paymen choice decisions over 25 years period. 9

20 (a) (b) (c) (d) Figure 6: Behavior of he model under di eren xed coss in he inermediaion funcion for elecronic paymens, F =, 0.8, 0.4 and 0.0, for baseline parameers; (a) Subsiuion of cash by paper-based and elecronic paymens; (b) Evoluion of he fracion of cash, paper-based and elecronic paymens as a percenage of all kinds of ransacions; (c) Evoluion of he growh rae of he economy; (d) Evoluion of cash, paper-based and elecronic paymens. 20

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