Cash burns: An inventory model with a cash-credit choice Journal of Monetary Economics

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Journal of Monetary Economics

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Cash burns: An inventory model with a cash-credit choice ^R

Fernando Alvarez

^a

, Francesco Lippi

^b,∗

a University of Chicago, NBER, 1126 East 59th Street, Chicago, IL 60637, USA

b EIEF and University of Sassari, Via Sallustiana 62, Rome, Italy

a r t i c l e i n f o

Article history:

Received 18 February 2015 Revised 4 July 2017 Accepted 6 July 2017 Available online 12 July 2017 Keywords:

Inventory models Cash-credit choice Means of payment Money demand Phasing out cash

a b s t r a c t

Adynamiccash-managementmodelisanalysedwhereagentschoosewhethertopaywith cashorcreditatevery pointintime. Inthe modelcreditusagedependsonthecurrent stockofcash,anovelresultthatmatchesrecentmicroevidenceonhouseholds’payment choices.Theoptimalityofsuchadecisionruleisnovelandcannotbeobtainedbymodels wherecash-creditdecisionsaremadeatthe“beginning” ofeachperiod.Wediscusshow tousethemodeltoaccount forcrosscountry-evidenceontheintensityofcreditusage andforseveralstatisticsonthesizeandfrequencyofcashwithdrawals.Themodelisused toassessthehousehold’swelfarecostofphasingoutcash.

© 2017TheAuthors.PublishedbyElsevierB.V.

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1. Introductionandoverview

Adynamicmodelofcash-managementandmeans-of-paymentchoiceisanalysed inwhich optimizinghouseholdsuse bothcash andcredit.Creditis modeled asa paymentinstrumentthat involves acost butrequiresno inventoryathand, suchasadebitoracreditcard,whiletheuseofcashismodeledwithastandardinventorysetup.Acentralfeatureofthe modelisthat ateach momenttheagentcanchoosetopaywitheithercash orcredit.Thisrealisticfeatureisan essential noveltyofourmodel,whichimpliesthatthepreferredpaymentinstrumentdependson thestockofcash holdingsatthe timeofthepurchase:agentsusecashforapurchaseaslongastheyhavecashwiththem,andusecreditotherwise.Agents behaveasif “cashburns” in their hands,a patternthat hasbeen noticed by theempirical literature.¹ Yet, agentsfind it optimalto intermittentlyreplenish their cashholdings,e.g. byasking for“cash-back” afterpaying withcredit,andsoend upusingbothcashandcredit.

Ourmodelisthefirsttohavebothsimultaneoususeofcashandcreditbyhouseholds,aswellascreditusethatdepends on the level of cash holdings. This feature cannot be obtained by models where cash-credit decisions are made at the

“beginning” ofeachperiodsince,byassumption,thosemodelsdonotkeeptrackofthedynamicsofcash balancesandso theuse of creditcannot be conditioned onthe stock ofcash. The model provides a simple analytical mappingbetween thefundamentalparametersandseveralobservablestatisticsonthesizeandfrequencyofcash withdrawals,aswellason

R An earlier version of the paper circulated under the name: “A dynamic cash management and payment choice model”. We benefited from the comments of the participants to “Consumer Finances and Payment Diaries: Theory and Empirics”, Ottawa, Canada, 18–19 October 2012. Philip Barrett and David Argente provided excellent research assistance. Part of the research for this paper was sponsored by the ERC advanced grant 324008.

∗ Corresponding author.

E-mail addresses: falvare@uchicago.edu (F. Alvarez), f.lippi@uniss.it (F. Lippi).

1 See, for example, Stix (2004) , Mooslechner et al. (2006) , Arango et al. (2011) , Arango et al. (2015) and Huynh et al. (2014) . See Appendix A for a brief review of this evidence.

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themeansofpaymentchoicethatdeterminestheshareofpurchasespaidwithcredit.Toillustratetheapplicabilityofthe modelweuseittoassessthehousehold’swelfarecostofphasingoutcash,apolicyendorsedbyRogoff (2016)asameasure againstseveralillegalpractices.

Theanalytic setup consistsoftheBaumol–Tobin cash inventorymodelaugmentedwithtwo features.First,we assume theagentrandomlyreceivessomefreewithdrawalopportunities,asinAlvarezandLippi(2009).Thisassumptiongivesrise toa precautionarymotiveinthedemandforcash,suchasanabove zerocash-balanceatthetime ofcash withdrawals,a patternsystematicallyobservedinhouseholddiary andsurvey evidence.Second,we allowagentstopayfortheir(exoge- nous)constantflowofexpenditureusingeithercashorcredit.Ourchoiceofmodelingexpendituresasaconstantflowhas the interpretationof small-sized purchases.Paying withcash requiresto havepositive cash inventory atthe time ofthe purchase,whichiscostlytoaccumulateandwhichhasanopportunity cost,bothstandard featuresofan inventorymodel.

Payingwithcreditentailsacostpertransaction.²Inprincipleitispossiblefortheagenttopayallofitsconsumptionusing creditbut,aslongastherearesomefreecashwithdrawalopportunities,itwillbeoptimalfortheagenttoalsousecash.

Thepaperhastwomain predictionsforthe optimalchoice ofthemeans ofpayment.First,aslongasthecostofcash withdrawalsislowenoughrelativetothecostofcredit,theagentsneverusecredit.Theytakeadvantageoftheirfreetrips tothebank,possiblysome costlytrips,andusecash only.Second,ifthecostofcashwithdrawalsissufficientlyhigh,then cashiswithdrawneverytimeafreecashwithdrawalopportunityoccurs.Theagentthenusescashuntilsherunsoutofit, andsubsequentlyusescredituntilthenextfreecash withdrawalopportunityoccurs.Intuitively,foranagentwithcash at hand,thecostofobtainingitissunk.Asaresult,usingcashisoptimalsincetheagentpaysonlytheopportunitycost.

Weanalyticallyderiveseveralpredictionsandcomparethemwithcross-countryevidencegatheredfromhouseholdsur- veysanddiary dataassummarisedine.g.Bagnalletal.(2016).Asmentioned,themodelpredictsthatcreditismorelikely tobeusedwhentheagentisshortofcash,apatternthatisrobustlydocumentedinmicrostudies(seefootnote1).Another relatedphenomenonthatourmodelisuniquelypoisedtoexplain isobtaining“cash-back” attheregisteratthetimeofa credit purchase.Thisisan instance wherethecoexistence isclearandwhere,asinour model,cash isobtainedbecause withdrawingit isparticularlycheap–see ourdiscussionofthe evidenceinKlee(2008)inSection2.Moreover,themodel haspredictionsconcerningtheaverageshareofexpenditurespaidwithcash,thefrequencyofcash withdrawalsandtheir averagesize, theaveragecash holdings(bothunconditionalaswellasatthetime ofcashwithdrawals).Weshow that at lowinflationratesthefivemomentslistedabovearefunctionsof2parametersonly:thenormalizedcostofcredit,namely theratiobetweenthecostofcreditandthecost ofcashwithdrawals, andthefrequencyofthefree withdrawalopportu- nities. Wediscusshowthesetwo parameterscanbe identifiedfromobservationsone.g. thenumberofcash withdrawals andtheaveragecash holdings.Wethen comparewiththedatathemodelpredictionsaboutother momentsnotused for theparameteridentification.Weshowthat,inspiteofitssimplicity,themodelgetstheappropriatemagnitudesforseveral observedmoments.

Finally,weuseourstructuralmodeltoquantifythecostofapolicythatlimitshousehold’scashusage,apolicyendorsed by Rogoff (2016) tofight severalillegal activities which are knownto be cash intensive. Ourobjectiveis to quantifythe welfarecost forhouseholdswho areforcedto movefromtheir optimalcash-credit sharetoone wherethecash shareis zero.Theanalysisshowsthatsuchcostissmall.

1.1. Relatedliterature

Manypapers inthe literature incorporatealternative means ofpayments asinthe seminal workof Lucas andStokey (1987)and Prescott(1987). Howeversuch models donot havean explicit inventorytheoretical modelof money,so they cannot simultaneouslyspeaktoobservationssuch asthefractionofpurchasesmadeincashaswell ascash-management statistics,such thefrequencyandsize ofcash withdrawals.Technically,inthistype ofmodelsthe(exogenouslyimposed) cash-in-advance constraint binds inevery period. As a result, one withdrawal occursevery periodand all cash is spent.

Hence,statisticssuchasfrequencyofwithdrawals,sizeofwithdrawals,cashatthetimeofwithdrawals,areallexogenously determined by the choice of the model’s time-period length. Other models incorporate both cash management and the choice of means ofpayments, whichends up being dictatedby the size ofthe purchases. Examplesof such models are Whitesell(1989)orFreemanandKydland(2000).Yetwhilethesemodelsintroducecash-management,thosechoicesareall

“within” the period,so that agents cannot chooseat every moment whether to use cash or credit.Hence in these models theoptimaluseofcreditcannotdependoncashholdings,contrarytowhatthedatastronglysuggests.³

The closest related models in the literature are Sastry (1970) and Bar-Ilan (1990). Sastry (1970) is one of the earli- est inventory models featuring a sequential cash versus credit choice. In his deterministic Baumol–Tobin model withno

2 The choice of cash vs credit based on the size (i.e. dollar value) of purchases has been addressed in both theoretical and empirical literature, see Arango et al. (2012) and Bouhdaoui and Bounie (2012) . Empirically, smaller transactions are more likely to be paid with cash than with credit, which motivates the assumption of a fixed cost per transaction in the literature. Yet, there are many small transactions paid with either cash or credit. Our assumption of a constant flow of expenditure thus focuses on transactions that are all of the same size and small. It is thus complementary to the explanation in the literature based on size and it is able to address the choice of means of payments for small size transactions. See Alvarez and Lippi (2013) for a complementary analysis of an inventory (cash-only) model that distinguishes between large vs small sized purchases.

3 A close analogy between our sequential formulation and these papers’ simultaneous cash-credit choice is found in the difference between sequential search, as in McCall’s model, versus simultaneous search, as in Stigler’s search model. See Sargent ’s (1987) chapter 2 for a description of the two types of search models.

discountingtheagentisallowedtousecredit,namelyanoverdraft(“negativecash”),sothatwhencashholdingsreachzero theagentmaycontinuetoconsumeandpostponethe paymentofthefixedwithdrawalcost. Bar-Ilan(1990)extendsthis setup to adynamic stochastic inventory model.The main difference withour setup isthat thecost of credit inboth of thesemodelsisassumedtobeproportionaltotheaveragestockofcreditovertheholdingperiod,completelyanalogueto theopportunitycostofcash.Thisimpliesthattheagentusingcreditwillperiodicallydecidetopaytherebalancingcostin ordertokeepthe averagestockofcredit bounded.Notice that underthisassumption itisinfinitelycostlynot topaythe fixedtransanctioncostsince thisimpliesadivergingstockofcredit.Inoursetup,incontrast,thecostofcreditispropor- tionaltotheexpenditureflow,e.g.itisafixedfractionofthepurchasevalue.Usingcreditdoesnotrequireanyfixedcostto

“rebalance” thecash creditstock. Creditpurchasesareimmediatelydebitedontotheagent’scheckingaccount.Inaddition tobeing morerealistic, ourassumption explicitlydistinguishesthe credittechnology fromthe cash technology,andthus makesacredit-onlystrategyofpurchasesfeasiblefortheagent.

1.2.Organizationofthepaper

The structure of the paper is as follows. We illustrate the model’s key idea in Section2 with a simple determinis- ticsteady-state model. Section3 introduces uncertaintyand a proper dynamic treatment of the inventory problemwith payment-choiceandcharacterizestheconditionsunderwhichbothcashandcreditareused.Section4derivesthemodel’s implicationsforthefrequencyandsizeofcashwithdrawalsandtheintensityofcreditusage.Wediscusssomecross-country evidenceto illustratehow themodel canbe calibratedto actual economies.InSection5,we useourstructural modelto quantify the cost of a policy that imposes a zero-cash usage restriction. Section6 extends our model by allowing for a randomcostofcashwithdrawals,afeaturewhichappearsdesirableforempiricalapplications.Section7concludes.

2. Adeterministicmodelwithmeansofpaymentchoice

Thissection presentsa steady-statedeterministic modelthat highlightsthemainmechanismofthedynamicstochastic modelof Section3. Indeedsome key formulasfrom thissimple modelcoincide with, or are closeto, themore complex decisionrules ofthestochasticmodel.Themaincounterfactualpredictionofthedeterministic modelisthelackofa pre- cautionarymotive,sothatcashbalancesarezeroatthetimeofawithdrawal.⁴

Consideran agentwhoconsumes eperunit oftime andcanpayforthisusingeithercash orcredit.Ifshepayswith creditshe incursa directcost

γ

^{perunit bought. Thequantity}

γ

^{ecan be} interpretedasthe “handling” and“verification andauthorization” costsestimatedin Klee(2008)usinggroceryreceipts data.The ideais that,astheempirical evidence inKlee suggests, usingcredit is moretime consuming than usingcash so that cash is preferredwhen both instruments are available. In particular, Klee uses a dataset of time stamped checkout transactions, from 99stores of a retail chain, containingmorethan6millionscheckouttransactions.Foreachtransactionatacash registerthedatarecordthetime to completion,themeansofpaymentusedforit(cash,credit,debit,check,orother),theamountofthepurchase,theamount paid,whetherthecustomerobtainedcashback(i.e.whethertheyuseitasourceofcash),thenumberofitemsbought,and whethercouponswereused.Table2onpage533ofKlee(2008)estimatesthedifferenceintimethatittakestocomplete it(“toringit”)transactionsthat arepaidwithdifferentinstruments:credit,debit, andchecksrelative tocash, controlling forotherfeatures such thenumberofitem oftransaction.Thesemagnitudes arewhat Kleerefers as“handling cost” and

“authorizationandverificationcosts” —seeTable3onpage535ofKlee(2008).InSection4wediscusshowtoquantifythe creditcost

γ

^{inthelightofempiricalevidence.}

Thetechnologytowithdrawcash(fromaninterestbearingcheckingaccount)isasfollows:atanytimetheagentcanpay afixedcostbandreplenishhercashbalanceswhich,asinthecanonicalBaumol–Tobinmodel,aresubjecttoanopportunity costR whichincludes forgoneinterest ondepositsaswell asthe probabilityof cashtheft. Moreover, theagenthasp≥ 0 withdrawalsperperiodthatcomeforfree.Thelatterassumptionisasimpleparametrizationofthetechnologyforthecash withdrawals,proxyinge.g.forthenumberofATMs(cheapwithdrawals)availabletotheagent.⁵Tounderstandthenatureof theoptimalpoliciesconsideredbelownoticethatinadeterministicsetupanagentwithpositivecashbalanceswillnotpay thefixedcostb towithdrawcash unlesscashbalances arezero.Consider nowthedecisionofwhethertopurchasegoods usingcashorcredit.Foranagentwithpositivecashbalancesm>0itisnotoptimaltopaythecost

γ

^{etousecredit,since}

thecost ofacquiringthe cash issunk at thistime.⁶ Foran agentwithzerocash balances m=0, thereare two possible choices:thefirst oneis topaythe cost

γ

^eandfinanceconsumption usingcredit, waitinguntilthenext free withdrawal opportunitytoreplenishcashbalances.Thesecondchoiceistopaythefixedcostbandwithdrawcash.Wefirstseparately describethesolutionofthesetwocasesandthenanalysethebestchoiceamongthetwo.

4 Readers familiar with continuous time impulse control problems may move directly to Section 3 .

5 See Appendix D for an explicit foundation of how cash theft probability and the nominal interest rate affect R . See Alvarez and Lippi (2009) for estimates of p using Italian household data and for their Appendix C for estimates of cash theft probabilities in Italy and the US.

6 Another possibility is to use credit and deposit the cash to earn a higher interest. With a fixed cost for depositing this is not optimal unless the cash balances are very large, a situation that will not occur along an optimal path.

2.1. Thedeterministiccash-creditmodel

Consideranagentwhofinancesherexpendituresusingcash,andwhopayswithcreditoncecashbalancesaredepleted.

Assumefurtherthatnocostlywithdrawalevertakesplacesothatbisneverpaidandthenumberofwithdrawals,n,equals thenumberofwithdrawalsthatcomeforfree,p.AfteracashwithdrawalofsizeW=m^∗,shespends

τ

^a=m^∗/eunitsoftime payingforconsumptionwithcash,incurringanopportunitycostRm^∗/2,wherem^∗/2istheaveragecashbalanceconditional on m>0 and R is the opportunity cost of cash– which includes the nominalinterest rateas well as the probability of cashtheft.After cashbalanceshitzero,theremainingtime untilafreewithdrawalopportunity,denotedby

τ

^r,isgivenby

τ

^r=1/p− m^∗/e.Notice that

τ

^r+

τ

^a=1/p.Thesteadystatecostineverycycleofduration1/pcanbewrittenas:

τ

^r

γ

^e+

τ

^{a Rm∗}/2.Thecostperunitoftimeisthusp

τ

^r

γ

^e+p

τ

^{a Rm∗}/2.Thustheminimizedcostofthestrategythatusesboth cashandcreditis:

v

r

(

^R,

γ

^,p,e

)

⁼₀ ^min

≤m^∗≤e/pp



(

^{1/p− m∗/e}

) γ

^{e +}

(

^m∗/e

)

^Re

(

^m∗/e

)

2



, (1)

subjecttotheconstraintthat thetime spentusingcreditisnon-negative,i.e.m^∗/e≤ 1/p.Wedenotebysthe“cashshare”, namelytheratiooftheexpenditurepaidwithcashtototalexpenditureperunitoftime,givenby

s=

τ

^a

τ

^a+

τ

^{r =min}



pm^∗ e ,1



. (2)

Denoting theaveragerealbalancesby Mandusingthecash shareswecanwrite M=sm^∗/2.Thecostminimizingpolicy yields ^m_e^∗=^We =min



₁

p,^γ_R



,s=min



1,^γ_R^p



, ^Me =min



1 2 p,₂^p



_γ

R

2



whichimply

v

r

(

^R,

γ

^,p,e

)

⁼

⎧ ⎪

⎨

⎪ ⎩



1−

γ

^p

2R

 γ

^{e ifR≥}

γ

^p



_R

2 p



e ifR<

γ

^{p (3)}

WhenR≥

γ

^{pcreditis“cheap”,sothatbothcashandcreditareused.WhenR}<

γ

^{pcreditisexpensiveanditisnotused.}

2.2. DeterministicBaumol–Tobinmodelwithpfreewithdrawals

LetusconsideramodifiedBaumol–Tobinmodelinwhichtheagentpaysonlyforthewithdrawalsinexcessofthepfree adjustmentsperperiod.Theagentchooses awithdrawalofsizem^∗whencashbalances areexhausted(m=0)^.Thepolicy impliesanaveragecashbalanceM=m^∗/2andanumbern=e/m^∗ ofcashwithdrawals.Theagent’schoiceofm^∗givesthe minimizedcostfunction

v

a

(

^R,b,p,e

)

≡ min_m∗



R m^∗

2 +bmax



_e

m^∗− p,0



. (4)

wherethecostisgivenbythesumoftheopportunitycostofcashholdingsandthecostassociatedwithcashwithdrawals inexcessofp.Theoptimalpolicyforatechnologywithp≥ 0is ^m_e^∗ =¹_p



min



2^b_e^p_R², 1

.Forp>0thereisnoreasontohave lessthanpwithdrawals,sincethesearefree.Hence, forR<2p²b/etheagentwillchooseaconstantlevelmoneyholdings:

m^∗=e/p.Notethattheinterestelasticityofmoneyiszerooverthisrange,whileitisequalto1/2ifR>2p²b/e.Theaverage withdrawalsize Wandtheaverage cashbalances satisfy:W=m^∗, M=2W=2m^∗.Replacingthe optimalm^∗choice in thecostfunctionyields

v

a

(

^R,b,p,e

)

⁼

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

 

2R b e− p^b_e



e ifR≥ 2 p^{2 b}_e andn>p



_R

2 p



e ifR<2 p² b

e andn=p

(5)

wherethetop branchgivesthecost forthecaseinwhichthenumberofwithdrawals exceedsp.Notethatin thisdeter- ministicsetupanagentwithpositivecashbalanceswillnotpaythefixedcostbtowithdrawcashunlesscashbalancesare zero.

2.3. Thefulldeterministicproblem

Wenowanalysetheconditionsunderwhichitisoptimaltousecreditinsteadofwithdrawingfreshcashwhenm=0. Todosowecomparethesteady-statecostofthetwopoliciescomputedabove.Thevaluefunctionfortheproblemisthen

v

(^R,b,

γ

,p,e)=min

{ v

a(^R,b,p,e),

v

r(^R,

γ

,p,e)

}

^{. We define the thresholdfunction b,asthe value of b that equates the}

twominimizedcosts:

v

a(^R,b,p,e)=

v

r(^R,

γ

,p,e)^.Wehavethat b

(

R,

γ

^,p,e

)

=

γ

²

2R e. (6)

whichimpliesthat creditisusedwhen b≥ b providedthat

γ

^p≤ R.⁷ Ifb≥ b and

γ

^p>Rthen creditisnotusedandn=p.

Finally,forb<b credit isnot used andn>psince some costly withdrawalsin excessof thepfree withdrawals are now optimal.

Thenextpropositionsummarisesthebehaviorofthedeterministicmodel.Itconsiderstwocasesdependingonwhether

γ

ࣥ2pb/e,andforeachcaseitanalysesoptimalpolicyasafunctionofR. Proposition1. Letp>0and

γ

>0.ThenW/M=2/sand

If

γ

>2p^b_e,then ifR∈



0, 2p²b e



−

∂

^logM/e

∂

^{logR =0 onlycashused n=p s=1}

ifR∈



2p²b e ,

γ

²

2b/e



−

∂

^logM/e

∂

^{logR =1/2 onlycashused n>p s=1}

ifR∈

 γ

²

2b/e , ∞



−

∂

^logM/e

∂

^{logR =2 cash&creditused n=p s=}

γ

p/R Otherwise,i.e.if

γ

≤ 2p^b_e,then

ifR∈

(

⁰,

γ

^p] −

∂

^logM/e

∂

^{logR =0 onlycashused n=p s=1}

ifR∈

( γ

^p, ∞

)

−

∂

^logM/e

∂

^{logR =2 cash&creditused n=p s=}

γ

^p/R

Thepropositionillustratesthreerobustpropertiesofthemodel.First,themodelhasonlytwoparameters,

γ

^pandp2b/e.

InthemodifiedBaumol–Tobinmodeltheshapeofthemoneydemanddependsonlyonbˆ≡ p²b/e.Foragivenvalueofbˆ, thecash-creditaspectofthemodeldependsonlyon

γ

^{p.Wewillseethatthispropertycontinuestoholdinthestochastic}

modelanalysed below.Second, forcredit tobe used itis necessarythat thecost ofa withdrawalis above thethreshold definedinEq. (6). Ifb>b then the agentuses bothcash and creditto finance her consumption, andcostly withdrawals arenever used.Theconditionforthe optimalityof creditdependsona combinationofthefundamentalparameters R,b and

γ

^pwhich,intuitively,implythatthecostofcash(whichisincreasinginRandb)mustbehighrelativetothecostof credit(whichisincreasingin

γ

^{andp).Third,theinterestrateelasticityofmoneydemandisincreasingintheinterestrate.}

Therearetwocases:thefirstcorrespondstoalargecostofcredit(

γ

^p>2p²b/e),inwhichcasetherearethreequalitatively different behavior depending on the level of interest rates. If interest rates are very low, credit is not used and n=p, resultinginanelasticityofzero.Forintermediate levelofinterestrates,creditisnotused,butn>p,sothelocalbehavior isidenticaltoBaumol–Tobin,producinganinterestrateelasticityof1/2.Forhigherinterestrates,bothcashandcreditare used.Theinterestrateelasticityishigherherebecauseboththecash shareaswellasthesizeofthewithdrawalsreactto interestrates.Ifinsteadthecostofcreditislow (

γ

≤ 2pb/e) thereisnointermediatecase, sincecreditalways dominates theBaumol–Tobintypeofbehavior.

3. Adynamicstochasticmodelwithmeansofpaymentchoice

Inthissectionwe solvea discounted,stochasticdynamicproblemwhichjoinstheoptimalcash managementproblem withthe optimalchoice of means ofpayment. As inthe deterministic problemthe agentfaces a totalconsumption per unit of time e>0 which must be paid with either cash or credit: at each instant the agentcan choose to pay in cash c∈[0,e] andto paytheremaining e− c usingcredit.Ifthe paymentismadeby credit,the agentpaysa flowcost

γ

^per

dollar.⁸ SeeSection2for a descriptionof Klee’s (2008)estimates ofthese costs usingthe time that it takesto conclude differenttransactions.Thequantity

γ

^{ecanbeunderstoodasthetimecostofusingcreditforsmallvalue}transactions,see theprevioussectionforamoredetaileddiscussionofitsnatureandSection4foraquantificationofitsmagnitude.Thestate oftheagent’sproblemissummarised by her realcash balancesm≥ 0.Ifm=0 eithercash mustbe withdrawnorcredit hastobeused.Ifm>0theagentfacesacash/creditchoice.Thelawofmotionofrealbalancesisthendm=−(^c+m

π

)^dt providedthat no adjustment takesplace,where

π

^{isthe constant inflationrate. The agentcan adjusther cash balances}

afterpayingthefixedcostb≥ 0.Additionally,thereisaPoissonprocesswithconstantarrivalratep≥ 0,whichdescribesthe arrivalofafreeadjustmentopportunity.Whensuchanopportunityarisestheagentcanadjusthercashbalancesatnocost.

WeassumethatholdingcashmentailsanopportunitycostRmperunitoftime,whereRcanbeinterpretedasthesumof thenominalinterestrateplusaprobabilitythatcashislostorstolen(seefootnote5).Weassumethattheagentminimises theexpecteddiscounted cost,using theconstant discount rater≥ 0.There arethree substantivedifferences betweenthe modelanalysedinthissectionandthesteady-statedeterministicmodelofSection2.First,wetake intoaccountexplicitly

7 The expression for b comes from equating: vr = γ^{e [1 −}γ^p/ (2 R )] with va = e 

2 R ^b_e − pb.

8 Considering that all expenditures are of the same size, and that the optimal policy is of the bang-bang type, our formulation can be equivalently interpreted as a fixed cost applied to each purchase.

therole ofinflation, ascanbe seen inthelawofmotion.Second, thefree adjustmentopportunities arrivestochastically.

Third,realcostsarediscountedbyanappropriaterater.

Formally we denote by V(m) the minimumexpected discounted cost of supporting a constant flow of expenditure e when thecurrentrealcash athandism≥ 0.Thefunction V, definedinV:R+→R+ mustsolve thefollowing functional equation:

0=min



0min≤c≤eRm+

γ

^[e− c]+pmin

z≥0[V

(

^z

)

− V

(

^m

)

^]− V

(

^m

)(

^c+

π

^m

)

− rV

(

^m

)

, b+min

z≥0V

(

^z

)

− V

(

^m

) 

forall m≥ 0. (7)

Theouter min{· ,·}inthefunctionalEq.(7)chooses betweentwo strategies.The terminthefirstlinerepresentsthecase whereno costlywithdrawals (thatinvolvepayingthefixed costb) occur,although randomfree withdrawalopportunities mayarise,andtheagentchooseswhatfractionofherconsumptiontopayincashversuscredit.Thisisastandardcontin- uous timeBellmanequation,withflowcostRm+

γ

(^e− c)^{andwithexpectedchangesduetoeitherthearrivalofthefree} adjustmentopportunityortothedepletionofcash.Theminimisationwithrespecttocdescribestheagent’schoiceofthe optimalmeans ofpayment.The terminthe second linecorresponds to thestrategy ofexercising control, i.e.payingthe fixedcostbandadjustingcashholdings.Foreachmthevaluefunctionisequaltothevalueofthestrategythatyieldsthe minimumcost.Wheneveranadjustmentismade,eitherpayingthecostborwhenafreeadjustmentopportunityarrives, thepost-adjustmentquantityofcashischosenoptimally.TheoptimalpolicyfortheprobleminEq.(7)consistsofdeciding foreachm≥ 0whetheracostlywithdrawalismadeornotand,ifnoadjustmentismade,whichpaymentinstrument(cash orcredit)touse.Noticethatthisformulationdoesnotimposeanyrestrictionconcerningthetimesofcash withdrawalsor concerningwhendifferentpaymentinstrumentscanbeused.Ateachpointintimetheagentisfreetochoosewhetherto withdrawcashandwhatpaymentinstrumenttouse.Wemaintainthefollowingassumptionthroughoutthissection.

Assumption1. Weletb≥ 0,

γ

≥ 0,

π

≥ 0,p≥ 0,andr+p+

π

>0,e>0,R>0.

Ife=0theproblembecomesuninterestingsincethereisnoexpendituretofinance.Theparametersband

γ

^arecosts,

andp aprobability rate, sothey must be non-negative.The requirementthat r+p+

π

>0 andR>0 are important.For instance,ifr+p+

π

=0thereisnointertemporalincentivestousecash.⁹

3.1. Twocandidatepolicies

ItwillturnoutthattheoptimalpolicyoftheproblemdepictedinEq.(7)isoneoftwotypes,dependingonparameters.

Thefirsttypeisacash-burningpolicy,definedasfollows:

Definition1. Wedefineacashburningpolicyasathresholdm^∗≥ 0forwhich:

1. Creditisonlyusedwhenm=0,andcashisusedforeverym∈(0,m^∗).

2. Cashisonlyadjustedwhenafreeadjustmentopportunityarrives.

3. Immediatelyafteracashadjustment,cashholdingsmtakethevaluem^∗.

Notethatthevalueoffollowingacash-burningpolicywiththresholdm^∗isthefunctionV :[0,m^∗]→R+thatsatisfies:

(

^r+p

)

^V

(

^m

)

=mR+pV

(

^m∗

)

− V

(

^m

)

^[

π

^m+e] forallm∈

(

⁰,m^∗] and (8)

(

^r+p

)

^V

(

⁰

)

=

γ

^e+pV

(

^m∗

)

⁽⁹⁾

ThefirstoneisthestandardBellmanequation,whereweassumeeverythingispaidincash.Thesecondequationsaysthat atm=0agentsusecreditandwaitforafreewithdrawalopportunity.

Thesecondpolicytypeisdefinedasfollows:

Definition2. WedefineaBaumol–Tobinpolicyasathresholdm^∗≥ 0forwhich:

1. Creditisneverused.

2. Cashisadjustedwheneitherafreeadjustmentopportunityarrivesorwhenm=0. 3. Immediatelyafteracashadjustment,cashholdingsmtakethevaluem^∗.

We briefly commentonthe differencesbetweenthe twopolices. There isa sense inwhich cash burnsinthe agent’s hands underbothpolicies,since inboth cases,aslongasitisavailable (m>0), cash isthepreferredmeans ofpayment.

NotethatwhenaBaumol–Tobinpolicyisfollowedtheo.d.e.inEq.(8)holdsintherangeofinaction(0,m^∗].However,under thispolicytheboundaryconditionatm=0isgivenby:

V

(

⁰

)

=b+V

(

^m∗

)

. (10)

9 While here we treat R and r , p , π as independent parameters the value of R and r + p + π can indeed be related –for instance r + π should be the shadow nominal interest rates. We return to this relationship in the next section.

Foracashburningpolicytobeoptimal, i.e.tosolvetheprobleminEq.(7),oneneedstoestablishthatitisoptimalto paywithcash atm∈(0,m^∗] andwithcreditatm=0,wherethe optimalwithdrawalm^∗ must bedetermined. Finally,it hastobe shownthatatm=0itisoptimalto waitfora freeadjustmentopportunity (instead ofpayingb towithdraw).

Likewise,foraBTpolicytobeoptimal,i.e.tosolvetheprobleminEq.(7),oneneedstoestablishthatitisneveroptimalto paywithcreditandthatatm=0itisoptimaltopaybandchoosetheoptimalwithdrawallevelm^∗(seeAppendixBfora formalproofthatallthesepropertiesareverifiedundertheoptimalpolicy).

Notethat thefeasiblepoliciesconsistentwithEq.(7)aremuchbroaderthanthetwocandidatepoliciesdefinedabove.

Forinstance,onecould consider apolicy inwhichcredit isused forsometime atm=0andacostly withdrawaloccurs afterTperiodsunless afreewithdrawal arrives.Sucha policyisthe optimalone inthemodelsofSastry(1970) andBar- Ilan(1990).Interestingly,asweexplain below,theoptimalpolicyfortheprobleminEq.(7)willeithertaketheformofa modifiedBaumol–Tobinone(wherecreditisnotused)orofacash-burningpolicy.

3.2.Characterizingtheoptimalcash-creditchoice

Thenextpropositioncharacterizestheoptimalcashvscreditchoice.AppendixBprovidestheproofaswellasadetailed analyticcharacterizationofthedecisionrules,includingapproximatesolutionsform^∗inthecaseofzeroinflation.

Proposition2. A cashburningpolicy withm^∗ givenbyEq.(12)is optimalprovidedthatb≥ bwhere thelowerbound forthe fixedcostofadjustmentisgivenby

b= e r+p

 γ

^{− Rm}_e^∗



, (11)

andm^∗/esolves:

0≤ m^∗ e =



1+

(

^r+p+

π )

^γR

_π+^π_r+p

− 1

π

^≤

γ

R. (12)

Insteadifb≤ baBaumol–Tobinpolicyisoptimalandm^∗solves:



1+m^∗

e

π 

1+(r+p)/π

=m^∗

e

(

^r+p+

π )

+1+

(

^r+p

)(

^r+p+

π )

_eR^b. (13) The proposition showsthat there is a threshold b for the fixed cost of adjustment b above which the cash-burning policy isoptimal andbelowwhich the Baumol–Tobin policy isoptimal. When b>b the optimalpolicy consists ofusing both cash credit.A withdrawal ofsize m^∗, as determined by Eq.(12), occurs every time a free withdrawal opportunity arises.Whencasheventuallyhitsm=0,thenitisoptimaltofinanceconsumptionusingcredituntilafreeopportunityfor acash withdrawalarises. Thus,undercash-burningn=p andthefixed costb isneverincurredby theagent. Conversely, whenb<b credit isnot usedand theoptimalpolicy atm=0consistsin payingthe fixedcost b to makea withdrawal ofsize m^∗,asdeterminedbyEq.(13).Notice thattheoptimalthreshold b definedinEq.(11)summarizestheeffectofall fundamentalparameters (

γ

^,R,

π

^{,r,p, e) into one singlefunction that determines the natureof theoptimal policy and,}

inparticular,theoptimalityofusingcredit.Intuitively,Eq.(11)impliesthattheuseofcreditisoptimalwheneverthecost ofusing cash (which isincreasing in Rand b,and decreasingin p) is highrelative to thecost of creditusage (which is increasingin

γ

^).

Finallywenoticethattheoptimalpolicyisofthebang-bangtype,inthesensethatusingcreditstrictlydominates,oris dominatedby,theuseofcostlywithdrawals.ThisresultdiffersfromtheonesofSastry(1970)andBar-Ilan(1990)wherethe useofsome costlycreditaswellassomecostly withdrawalsisoptimal. Asmentionedintheintroduction,thisdifference dependsonthewaythecostofcreditismodeled:weassumethatthecostofcreditisproportionaltotheexpenditureflow:

γ

^{e.Theyassumethecostofcreditis}proportionaltothestockofaccumulatedcredit,theexactanalogueof“negativecash”, whichperiodicallyrequirestheagenttopaythe fixedcost bto rebalancethecredit costwhichwouldotherwisediverge.

Weseeourassumptionasareasonabledescriptionofthecaseofrevolving credit(creditthat isautomaticallydebitedon theagent’scheckingaccountattheendoftheholdingperiod).

Thenextpropositionanalyseshowthethreshold bchangesasafunctionoftheparameters:

Proposition3. Thefunction b≥ 0isboundedabovebye

γ

/(^r+p)^{anditishomogenousofdegreeonein(}

γ

^,R).Moreover

∂

^b

∂γ

^{>0 with γlim→0b=0 and γlim→∞b=∞, (14)}

∂

^b

∂

^{R <0 with Rlim→0b=}

e

γ

(

^r+p

)

^{and Rlim→∞b=0, (15)}

r+limp→0b= eR

π

²

 

1+

γ π

R



log



1+

γ π

R



−

γ π

R



=

γ

²

2Re +

π

^o

 γ

²

R



≥ 0, (16)

πlim→0b= e

γ

r+p



1−log



1+

(

^r+p

)

^γ_R

(

^r+p

)

^γ_R



=

γ

²

2Re +

(

^r+p

)

^o

 γ

²

R



≥ 0, (17)

∂

^b

∂π

^{<0,πlim→∞b=0,}

∂

^b

∂ (

^r+p

)

^{<0, and r+limp→∞b=0. (18)}

Table 1

Selected moments on cash holding patterns: data and theory.

Country: Model fit to US:

Fra Ger Ita US Cash-only Cash-credit Mixed Cash balances ( median ), M / c 8.7 4.9 12.5 3.1 3.1 3.1 3.1

Number of cash withdrawals, n 96 57 55 64 64 64 61

Withdrawal size ( median ), W / M 1.7 2.1 1.3 2.3 1.8 3.7 1.9 Cash at withdrawals ( median ), M / M - 0.3 0.4 0.7 0.1 1.0 0.6 Cash share of expenditures, s = c/e 0.15 0.53 0.52 0.23 1.0 0.4 0.7

The data source is Bagnall et al. (2016) Tables 1 to 4. Entries are sample means (unless otherwise indicated).

The Italian data are from Alvarez and Lippi (2013) for households who posses a ATM card. Cash balances M / c are measured relative to cash expenditures per day computed as c = s ₃₆₅^e . The number of cash withdrawals is per year. The parameters for the mixed model are chosen to the match cash holdings and number of with- drawals. The rate at which b changes is λ= 250 times per year, with a 2/3 probability the cost of withdrawal is 0.2% if daily cash expenditures, otherwise it is 1%.

Thepropositionshowsthatthecriticalthreshold bisincreasinginthecreditcost

γ

^{anddecreasinginthe}opportunity cost ofusingcash R.In addition,by varying

γ

^{/Rthe threshold b rangesfromzeroto infinity. The}approximationsinthe second to last lineshows that if

γ

^{2/Ris smallthen b coincides with theone of the}deterministic model. Moreover,the threshold bisdecreasingininflation:higherinflationincreasestherangeofparametersforwhichthecashburningpolicy isoptimal.Noticehoweverthatforfinitevaluesof

γ

^{andR, b}>0whichimpliesthatthereexistsasufficientlysmallvalue ofb>0that makesthe useofcreditdominatedby acash-only policy.Alsonoticethat inourmodela credit-onlypolicy, i.e.onewhere cashis notused,doesnot occuraslongasp>0.The lasttwo resultssuggestthat theuseofcash isvery resilient:technicalinnovationsthatreducethecostofcreditarealsolikelytoreducethecostofcashwithdrawals(increase pandorlowerb)sothatcashusageremainsconvenientforagents.

4. Modelpredictionsaboutobservablemoments

This section studies severalstatistics of interest generated by a household who follows the optimal policy described above. Letebe theaverage expenditureperunit oftime, andc theaverage expenditureperunit oftime paidwithcash, sos≡ c/edenotesthecash share,namelythelong runaveragefractionofpurchases(valueweighted)paidwithcash. Let Mbetheaveragecash holdingsofthehousehold, namelythemeanvalueofrealbalances undertheinvariantdistribution impliedby theoptimaldecisionrule.Letntheexpectednumberofwithdrawals perunitoftimeandWtheexpectedsize ofwithdrawals undertheinvariant distribution.Finally,let Mbe theexpectedvalue ofcash atthetime ofawithdrawal.

The leftpanel ofTable1reportssamplemeansforeach ofthesemomentsforselectedOECD economies,withdata taken fromBagnalletal.(2016)andAlvarezandLippi(2013)computedfromhouseholdsurveysanddiaries.¹⁰

Onemaindifference acrosscountryconcernsthefractionofexpenditurespaidwithcashwhichissmallintheUSand France(around20%oftotalexpenditure)andlargerinGermanyandItaly(whereitisaround50%ofthetotalexpenditure), asshowninthelastrowofthetable.Italyalsorecordsthehighestvaluesofcashholdings,equivalenttoabout12daysof cashexpenditures,whiletheUSrecordsthelowestvalue,equivalentto3daysofcashexpenditures(thetableentriesforM areexpressedrelativetothedailycashexpenditure,computedasse/365).

Next,weillustratehowthosestatisticsmapintothefundamentalparametersofthedynamicmodelconsideringahouse- holdwhofollowsthecashburningpolicy,whichisoptimalwhenb>b,aswellasthecasewhenb<bwherecreditisnot used.Westressthatthemainempiricalappealofthecash-burningpolicyistwofold.First,itisconsistentwiththedataas itrationalizestheuseofbothcashandcredit.Second,thepolicyalignswiththeempiricalobservationthathouseholdsare morelikelytousecreditwhentheircashbalancesarerunninglow,afactthatisamplydocumentedinArangoetal.(2011), Kosse andJansen(2012),Huynhetal.(2014) andArangoetal.(2012).Ourmodelalsoprovides a simpleinterpretation of theevidenceinWangandWolman(2016)whichshowsthattheshareoftransactionspaidwithcashdeclinessteadily,from

10Bagnall et al. (2016) analyse data from large-scale payment diary surveys conducted between 2009 and 2012 in Australia, Austria, Canada, France, Germany, the Netherlands and the United States that enable international comparisons. Alvarez and Lippi (2013) focus on Austria and Italy. In Italy we measure e as total household expenditure per day (including durables and services) using SHIW see Alvarez and Lippi (2013) , and in France, Germany and US we measure e as the total expenditure of individuals per day using diary data see Bagnall et al. (2016) .

the1stdayuntilthe15thdayofthemonth,withtheshareofdebitpurchasesbeingan almostexactmirrorimageofthis pattern(see their Section5).WangandWolmaninterpretthispatternasconsistent withhouseholdshavingmorecash at thebeginningofthemonth,duetoapay-dayeffect.Ourmodelimpliessuchabehavior,sinceourcentralresultisthatthe choiceofmeansofpaymentdependsonthelevelofcashbalances.¹¹Thenextpropositionsummarisesthemainresultsand closedformexpressionsfortheobservablesfocusingonthesimplecaseofzeroinflation

π

=0.

Proposition4. Lets≡ c/edenotethecashshare,i.e.theshareofpurchasespaidwithcash.Forthecash-burningandtheBaumol–

Tobinpolicywehave:

(i)Cash-burningpolicy.Letr+p>0,

π

=0,R>0andb≥ b.Then n= p, s=1−



1+

γ (

^r+p

)

R



−_r+^pp

, M e = m^∗

e −s

p, W = e

p s, (19)

thus,usingEq.(12)with

π↓

^{0gives m}e^∗= _r+¹_plog



1+^{γ (r+}_R^p)



≥ 0.Forr

↓

^0theseexpressionsaresimplefunctionsof

γ

^p/R:

s= ^γ

p R

1+^γ_R^p , M e =1

p



log



1+

γ

^p

R



− 1+



1+

γ

^p

R



−1



and (20)

W

M = 1



1+_γ^R_p

log



1+^γ_R^p

− 1 and M/M= 1. (21)

(ii)Baumol–Tobinpolicy.LetR>0,p>0,

π

=r=0andb<b.Thens=1and

n= p

1− e^{−m∗ pe} >p, W M =m^∗

M − p

n ∈

(

⁰,2

)

, (22)

M e =



1

1− e^−epm∗ m^∗

e − 1/p



, and M/M= p

n. (23)

Proposition4givesclosedformexpressionforn,s,M,WandMundereitherofthetwooptimalpoliciesforlow-inflation (

π

→0) andlow-discounting(r→0),both reasonableassumptions fordevelopedeconomies. Themodel isoveridentified since ithas essentially two parameters, pand

γ

^{p/R,that determine 5}observables, aswe discussbelow. Nextwe briefly commentontheeconomicsoftheproposition.

Definethe scalar

θ

≡

γ

^{p/Ras thenormalized costof credit,namely thecost of credit(}

γ

^{) relative to thecost ofcash}

(R/p).Undercash-burning,thecash shareiss= ₁₊^θ_θ whichismonotone increasinginthenormalizedcost ofcredit,with s→1as

θ

→∞andwiths→0as

θ

→0.Theequationshowsthat forthecreditsharetoincrease,duetoe.g.thenewavail- abilityofcheapercreditcards,itisnecessarythatthecostofcreditfallsfasterthanthecostofcash.Thissimpleobservation mayaccountfortheremarkableresilienceofcashusageinseveraldevelopedcountries(asdocumentedbye.g.Bagnalletal.

(2016)).Iftechnicalprogressinpaymentinstrumentsreducesthecostofcreditaswell asthecostofcash,such that

θ

^is

constant,thencashusageisunaffected.

ThemoneydemandM/eunderthecash-burningpolicyisafunctionoftwoparameters:

θ

≡

γ

^{p/Randp.12Theleftpanel}

ofFig.1plotstheaveragemoneyholdingsMrelativetodailytotalexpenditurese/365forp=50and p=100.Recall that palsoequalsthe numberofcash withdrawalsninthe cash-creditmodel.The rightpanelof thefigureplots theaverage withdrawalsizerelativetotheaveragemoneyholding,W/M,underacash-burningpolicy.TheratioW/Misdecreasingin

θ

^,

rangingfromW/M→0as

θ

→∞,toW/M→∞as

θ

→0,asshowninFig.1.Forcomparisons,whenb<bandthemodified Baumol-Tobinpolicyisoptimal,wehavethatW/M≤ 2.

Thepropositionallowsustointerpretthevariationinoutcomes,suchasthecross-countrydatareportedintheleftpanel ofTable1.Underacash-creditpolicy differentfrequencyofcashwithdrawals willthenbetriggered bydifferentefficiency ofthecashwithdrawaltechnology(n=p).ForinstancethelargenumberofcashwithdrawalsinFranceandtheUSrequires highvalues ofp,which onecan interpretasa proxy forthe densityofe.g.ATM machines.¹³ Likewise, differentsharesof cash expenditures(s) willbe ascribed to differentvaluesof thenormalized costof credit,

θ

≡

γ

^{p/R,whichcombines the}

costofcredit(atechnologicalparameter) withthenominalinterestrate. Forinstance,thedatasuggestsa relativelymore efficientcreditcost(smaller

θ

^{)inFrancethaninItaly.}

TherightpanelofTable1presentsacalibrationofthemodelfortheUnitedStates,bothfortheBaumol-Tobinpolicyas well asforthe cash-burningpolicy.The two parametersofthe model,

θ

^{andp,arechosen to matchthestatistics inthe}

11 The deterministic model of Section 2 , where deterministic cash inflows are triggered by p cash withdrawals at equally spaced intervals, is the simplest framework to see how this effect works.

12 For the cash credit model we normalize the cash holdings by the total expenditure e . Notice the difference from the normalization used in traditional cash inventory model where the scale variable is the total cash expenditure c = s e .

13 In Alvarez and Lippi (2009) we found a strong correlation between estimates of p and measures of the geographic density of bank branches and ATMs in Italy (see Table 4).

Fig. 1. Moments under cash-credit policy Note: Cash balances are measured relative to total expenditures per day, e ^d = e/ 365 .

first2 rowsofthedatatable,namelythe cashholdings (normalized bycash expenditures,seefootnote 12),M/c,andthe numberofcashwithdrawals,n.ThestatisticsintheremainingrowsoftheTablecanthusbeseenasatestofthemodel.The differentmodelshavemixedsuccessdependingonwhatstatisticsonefocusesone.Thecash-onlymodeldoesafairlygood jobatcapturingthewithdrawalsize,whichis2.3inthedataanditis1.8inthecash-onlymodel.Howeverthecash-only modelpredictsaverylowvalueofcashatthetimeofwithdrawals(smallprecautionarymotive)equalto0.1(versus0.7in thedata),andbydefinitionitismuteabouttheshareofcash expenditures(i.e.s=1inTable1forthecash-onlymodel).

Thecash-creditmodelpredictsashareofcashexpendituresof0.4(versus0.2inthedata),andahighprecautionarymotive (1versus0.7inthedata).Howeverthemodelpredictionconcerningthesizeofwithdrawalsisonthehighside.Considering itsparsimonioussetupwefindthisresultsofinterest.InSection6wediscussanextensionthatimprovesthemodelability tofitthedata.

4.1. Elasticitiesofobservableswithrespectto

θ

We conclude witha discussion of the modelpredictions concerning the interest rateelasticity of some key statistics whichhavebeenconsideredintheempiricalliterature,suchastheinterestelasticityofmoneydemandwhichisessential to measure thewelfare costs ofinflation, see e.g. Lucas (2000). Theelasticity ofswith respectto

θ

≡

γ

^{p/R is 1}/(¹+

θ

), whichimpliesthattheinterestelasticityofsisgivenby _∂^∂_log^log_θ^s =1− s,oroneminusthecashshare,sothattheelasticityis decreasingin

θ

^{,andsmallerthanone.Weusetheexpressionforthecashsharetoillustratethetensionbetweentheuseof}

cashvs.creditandtheroleofp.Ifb<bonlycashisused,i.e.s=1.Ifb>bbothcashandcreditareused.Yet,ifptendsto zero,then bisfinite,sotherearevaluesforwhichb>bandonlycreditisused,i.e.s=0.Thus,wefindthattheinteresting caseistheonewhereb>bandp>0,sothat0<s<1.

TheelasticityofMp/eis:0≤^∂log_∂log^Mp_θ^/e=

 θ 1+θ

2

log(1+θ )−θ/(1+θ )≤ 2and isdecreasing in

θ

^.Thusmoney demandis decreasingin theopportunitycostR,withaninterestrateelasticityincreasinginRsatisfying:

0≤ −

∂

^logM/e

∂

^{logR =}

∂

^logMp/e

∂

^log

θ

^{≤ 2 (24)}

Forcomparison,themodelwithcashpurchasesonly(c=e)summarizedinpart(ii)ofProposition4,hasan(absolutevalue) oftheinterestrateelasticityofmoneytocashconsumption,M/c,thatisincreasinginthelevelR,butboundedaboveby1/2.

Thisdifferencereflectstheelasticityofthecashshares,whichisbetween0and1,andtheelasticityofthemoneydemand relativetocashconsumptionM/c,alsobetween0and1:

0≤ −

∂

^logM/c

∂

^{logR =}

γ

^p/R

1+

γ

^p/R

1

log

(

¹⁺

γ

^p/R

)

^{≤ 1. (25)}

Thehigherinterestelasticityofmoneydemandproducedbythemodelwithacashcreditmarginillustratestheimportance ofjointlymodelingthecash-inventoryproblemandthecash-creditchoice.Bydoingsotheagentchoosesboththeextensive margin (how muchcash vscreditto use)aswell astheintensive margin (howmanycash withdrawals tomake). Failure toaccountfortheinteractionbetweenthesetwomargins,asdoneforinstanceinAlvarezandLippi(2009)wherethecash expenditureistakenasexogenous,leadstounderestimatingtheinterestrateelasticityofmoneydemand.