Finanial Innovation and the Transations Demand for Cash Fernando Alvarez University of Chiago and NBER Franeso Lippi University of Sassari and CEPR September 2007 Abstrat We doument ash management patterns for households that are at odds with the preditions of deterministi inventory models that abstrat from preautionary motives. We extend the Baumol-Tobin ash inventory model to a dynami environment that allows for the possibility of withdrawing ash at random times at a low ost. This modifiation introdues a preautionary motive for holding ash and naturally aptures developments in withdrawal tehnology, suh as the inreasing diffusion of bank branhes and ATM terminals. We haraterize the solution of the model and show that qualitatively it is able to reprodue the empirial patterns. Estimating the strutural parameters we show that the model quantitatively aounts for key features of the data. The estimates are used to quantify the expenditure and interest rate elastiity of money demand, the impat of finanial innovation on money demand, the welfare ost of inflation, the gains of disinflation and the benefit of ATM ownership. JEL Classifiation Numbers: E5 Key Words: money demand, tehnologial progress, inventory models. We thank Alessandro Sehi for his guidane in the onstrution and analysis of the database. We benefited from the omments of Manuel Arellano, V.V. Chari, Bob Luas, Greg Mankiw, Rob Shimer, Pedro Teles and seminar partiipants at the University of Chiago, University of Sassari, Harvard University, FRB of Chiago, FRB of Minneapolis, Bank of Portugal, ECB, Bank of Italy and CEMFI. University of Chiago, 1126 E. 59th St., Chiago, IL 60637. Ente Einaudi, via Due Maelli 73, 00184 Rome, Italy.
1 Introdution There is a large literature arguing that finanial innovation is important for understanding money demand, yet seldom this literature integrates the empirial analysis with an expliit modeling of the finanial innovation. In this paper we develop a dynami inventory model of money demand that expliitly inorporates the effets of finanial innovation on ash management. We estimate the strutural parameters of the model using detailed miro data from Italian households, and use the estimates to revisit several lassi questions on money demand. As standard in the inventory theory we assume that non-negative ash holdings are needed to pay for onsumption purhases. We extend the Baumol-Tobin model to a dynami environment whih allows for the opportunity of withdrawing ash at random times at low or zero ost. Cash withdrawals at any other times involve a fixed ost b. In partiular, the expeted number of suh opportunities per period of time is desribed by a single parameter p. Examples of suh opportunities are finding an ATM that does not harge a fee, or passing by an ATM or bank desk at a time with a low opportunity ost. Another interpretation of p is that it measures the probability that an ATM terminal is working properly or a bank desk is open for business. Finanial innovations suh as the inrease in the number bank branhes and ATM terminals an be modeled by inreases in p and dereases in b. Our model hanges the preditions of the Baumol-Tobin model (BT heneforth) in ways that are onsistent with stylized fats onerning households ash management behavior. The randomness introdued by p gives rise to a preautionary motive for holding ash: when agents have an opportunity to withdraw ash at zero ost they do so even if they have some ash at hand. Thus, the average ash balanes held at the time of a withdrawal relative to the average ash holdings, M/M, is a measure of the strength of the preautionary motive. For larger p the model generates larger values of M/M, ranging between zero and one. Using household data for Italy and the US we doument that M/M is about 0.4, instead of being zero as predited by the BT model. Another property of our model is that the number of withdrawals, n, inreases with p, and the average withdrawal size W dereases, with W/M ranging between zero and two. Using data from Italian households we measure values of W/M smaller than two, the value predited by the BT model. We organize the analysis as follows. In Setion 2 we use a panel data of Italian households to illustrate key ash management patterns, inluding the strength of preautionary motive, to ompare them to the preditions of the BT model, and 1
motivate the analysis that follows. Setions 3, 4 and 5 present the theory. Setion 3 analyzes the effet of finanial diffusion using a version of the BT model where agents have a deterministi number of free withdrawals per period. This model provides a simple illustration of how tehnology affets the level and the shape of the money demand (i.e. its interest and expenditure elastiities). Setion 4 introdues our benhmark stohasti dynami inventory model. In this model agents have random meetings with a finanial intermediary in whih they an withdraw money at no ost, a stohasti version of the model of Setion 3. We solve analytially for the Bellman equation and haraterize its optimal deision rule. We derive the distribution of urreny holdings, the aggregate money demand, the average number of withdrawals, the average size of withdrawals, and the average ash balanes at the time of a withdrawal. We show that a single index of tehnology, b p 2, determines both the shape of the money demand and the strength its preautionary omponent. While tehnologial improvements (higher p and lower b) unambiguously derease the level of money demand, their effet on this index and hene on the shape and the preautionary omponent of money demand is ambiguous. We onlude the setion with the analysis of the welfare impliations of our model and a omparison with the standard analysis as reviewed in Luas (2000). Setion 5 generalizes the model to one where withdrawals upon random meetings involve a small fixed ost f, with 0 < f < b, whih implies a more realisti distribution of withdrawals. Setions 6, 7 and 8 ontain the empirial analysis. In Setion 6 we estimate the model using the panel data for Italian households. The two parameters p and b are overidentified beause we observe four dimensions of household behavior: M, W, M and n. We argue that the model has a satisfatory statistial fit and that the patterns of the estimates are reasonable. For instane, we find that the parameters for the households with an ATM ard indiate their aess to a better tehnology (higher p and lower b). The estimates also indiate that tehnology is better in geographi loations with higher density of ATM terminals and bank branhes. Setion 7 studies the impliations of our findings for the time pattern of tehnology and for the expenditure and interest elastiity of the demand for urreny. The estimated parameters reprodue the sizeable preautionary holdings present in the data, a feature absent in the BT model. Even though our model an generate interest rate elastiities between zero and 1/2, and expenditure elastiities between 1/2 and one, the values implied by our estimates are lose to 1/2 for both, the values of the BT model. We disuss how to reonile our estimates of the interest rate elastiity 2
with the smaller values typially found in the literature. In Setion 8 we use the estimates to quantify the welfare ost of inflation in partiular the gains from the Italian disinflation in the 1990s and the benefits of ATM ard ownership. In the paper we abstrat from the ash/redit hoie. That is, we abstrat from the hoie of whether to have a redit ard or not, and for those that have a redit ard, whether a partiular purhase is done using ash or redit. Our model studies how to finane a onstant flow of ash expenditures, the value of whih is taken as given both in the theory and in the empirial implementation. Formally, we are assuming separability between ash vs. redit purhases. We are able to study this problem for Italian households beause we have a measure of the onsumption purhases done with ash. We view our paper as an input on the study of ash/redit deisions, an important topi that we plan to address in the future. 2 Cash Holdings Patterns of Italian Households Table 1 presents some statistis on the ash holdings patterns by Italian households based on the Survey of Household Inome and Wealth. 1 For eah year we report ross setion means of statistis where the unit of analysis is the household. We report statistis separately for households with and without ATM ards. All these households have heking aounts that pay interests at rates doumented below. The survey reords the household expenditure paid in ash during the year (we use ash and urreny interhangeably to denote the value of oins and banknotes). The table displays these expenditures as a fration of total onsumption expenditure. The fration paid with ash is smaller for households with an ATM ard, it displays a downward trend for both type of households, though its value remains sizeable as of 2004. These perentages are omparable to those for the US between 1984 and 1995. 2 The table reports the sample mean of the ratio M/, where M is the average urreny held by the household during a year and is the daily expenditure paid 1 This is a periodi survey of the Bank of Italy that ollets information on several soial and eonomi harateristis. The ash management information that we are interested in is only available sine 1993. 2 Humphrey (2004) estimates that the mean share of total expenditures paid with urreny in the US is 36% and 28% in 1984 and 1995, respetively. If expenditures paid with heks are added to those paid with urreny, the resulting statistis is about 85% and 75% in 1984 and 1995, respetively. The measure inluding heks is used by Cooley and Hansen (1991) to ompute the share of ash expenditures for households in the US where, ontrary to the pratie in Italy, heking aounts did not pay an interest. For omparison, the mean share of total expenditures paid with urreny by all Italian households is 65% in 1995. 3
with urreny. We notie that relative to Italian households hold about twie as muh ash than US households between 1984 and 1995. 3 Table 1: Households urreny management Variable 1993 1995 1998 2000 2002 2004 Expenditure share paid w/ urreny a w/o ATM 0.68 0.67 0.63 0.66 0.65 0.63 w. ATM 0.62 0.59 0.56 0.55 0.52 0.47 Curreny b : M/ ( per day) w/o ATM 15 17 19 18 17 18 w. ATM 10 11 13 12 13 14 M per Household, in 2004 euros w/o ATM 430 490 440 440 410 410 w. ATM 370 410 370 340 330 350 Curreny at withdrawals d : M/M w/o ATM 0.41 0.31 0.47 0.46 0.46 na w. ATM 0.42 0.30 0.39 0.45 0.41 na Withdrawal e : W/M w/o ATM 2.3 1.7 1.9 2.0 2.0 1.9 w. ATM 1.5 1.2 1.3 1.4 1.3 1.4 # of withdrawals: n (per year) f w/o ATM 16 17 25 24 23 23 w. ATM 50 51 59 64 58 63 n /(2M) Normalized: ( per year) f w/o ATM 1.2 1.4 2.6 2.0 1.7 2.0 w. ATM 2.4 2.7 3.8 3.8 3.9 4.1 # of observations g 6,938 6,970 6,089 7,005 7,112 7,159 The unit of observation is the household. Entries are sample means omputed using sample weights. Only households with a heking aount and whose head is not self-employed are inluded, whih aounts for about 85% of the sample observations. Notes: - a Ratio of expenditures paid with ash to total expenditures (durables, non-durables and servies). - b Average urreny during the year divided by daily expenditures paid with ash. - The average number of adults per household is 2.3. In 2004 one euro in Italy was equivalent to 1.25 USD in USA, PPP adjusted (Soure: the World Bank ICP tables). - d Average urreny at the time of withdrawal as a ratio to average urreny. - e Average withdrawal during the year as a ratio to average urreny. - f The entries with n = 0 are oded as missing values. - g Number of respondents for whom the urreny and the ash onsumption data are available in eah survey. Data on withdrawals are supplied by a smaller number of respondents. Soure: Bank of Italy - Survey of Household Inome and Wealth. Table 1 reports three statistis whih are useful to assess the empirial performane of deterministi inventory models, suh as the lassi one by Baumol and Tobin. Similar information an be drawn from Figures 3, 4, 5 where eah irle represents the average for households with and without ATM in a given year and 3 Porter and Judson (1996), using urreny and expenditure paid with urreny, estimate that M/ is about 7 days both in 1984 and in 1986, and 10 in 1995. A alulation for Italy following the same methodology yields about 20 and 17 days in 1993 and 1995, respetively. 4
provine (the size of the dot is proportional to the number of household observations). There are 103 provines in Italy (the size of a provine is similar to that of a U.S. ounty). The first statisti is the ratio between urreny holdings at the time of a withdrawal (M) and average urreny holdings in eah year (M). While this ratio is zero in deterministi inventory theoretial model, its sample mean in the data is about 0.4. A omparable statisti for US households is about 0.3 in 1984, 1986 and 1995 (see Table 1 in Porter and Judson, 1996). The seond one is the ratio between the withdrawal amount (W ) and average urreny holdings. While this ratio is 2 in the BT model, it is smaller in the data. The sample mean of this ratio for households with an ATM ard is below 1.4, and for those without ATM is slightly below 2. Figure 4 shows that there is substantial variation aross provines and indeed the median aross households (not reported in the table) is about 1.0 for households with and without ATM. 4 The third statisti is the normalized number of withdrawals per year. The normalization is hosen so that in BT this statisti is equal to 1. In partiular, in the BT model the following aounting identity holds, nw =, and sine withdrawals only happen when ash balanes reah zero, then M = W/2. As the table shows the sample mean of this statisti is well above 1, espeially so for households with ATM. The seond statisti, W n, and the third,, are related through the aounting M /(2M) identity = nw. In partiular, if W/M is smaller than 2 and the identity holds then the third statisti must be above 1. Yet we present separate sample means for these statistis beause of the large measurement error in all these variables. This is informative beause W enters in the first statisti but not in the seond and enters in the third but not in the seond. In the estimation setion of the paper we doument and onsider the effet of measurement error systematially, without altering the onlusion about the drawbaks of deterministi inventory theoretial models. For eah year Table 2 reports the mean and standard deviation aross provines for the diffusion of bank branhes and ATM terminals, and for two omponents of the opportunity ost of holding ash: interest rate paid on deposits and the probability of ash being stolen. The diffusion of Bank branhes and ATM terminals varies signifiantly aross provines and is inreasing through time. Differenes in the 4 An alternative soure for the average ATM withdrawal, based on banks reports, an be omputed using Tables 12.1 and 13.1 in the ECB Blue Book (2006). These values are similar, indeed somewhat smaller, than the orresponding values from the household data (see the working paper version for details). 5
Table 2: Finanial innovation and the opportunity ost of ash Variable 1993 1995 1998 2000 2002 2004 Bank branhes a 0.38 0.42 0.47 0.50 0.53 0.55 (0.13) (0.14) (0.16) (0.17) (0.18) (0.18) ATM terminals a 0.31 0.39 0.50 0.57 0.65 0.65 (0.18) (0.19) (0.22) (0.22) (0.23) (0.22) Interest rate on deposits b 6.1 5.4 2.2 1.7 1.1 0.7 (0.4) (0.3) (0.2) (0.2) (0.2) (0.1) Probability of ash being stolen 2.2 1.8 2.1 2.2 2.1 2.2 (2.6) (2.1) (2.4) (2.5) (2.4) (2.6) CPI Inflation 4.6 5.2 2.0 2.6 2.6 2.3 Notes: Mean (standard deviation in parenthesis) aross provines. - a Per thousand residents (Soure: the Supervisory Reports to the Bank of Italy and the Italian Central Credit Register). - b Net nominal interest rates in per ent. Arithmeti average between the selfreported interest on deposit aount (Soure: Survey of Household Inome and Wealth) and the average deposit interest rate reported by banks in the provine (Soure: Central redit register). - We estimate this probability using the time and provine variation from statistis on reported rimes on Purse snathing and pikpoketing. The level is adjusted to take into aount both the fration of unreported rimes as well as the fration of money stolen for different types of rimes using survey data on vitimization rates (Soure: Istat and authors omputations). nominal interest rate aross time are due mainly to the disinflation. The variation nominal interest rates aross provines mostly reflets the segmentation of banking markets. The large differenes in the probability of ash being stolen aross provines reflet variation in rime rates aross rural vs. urban areas, and a higher inidene of suh rimes in the North. Lippi and Sehi (2007) report that the household data display patterns whih are in line with previous empirial studies showing that the demand for urreny dereases with finanial development and that its interest elastiity is below onehalf. 5 From Table 2 we observe that the opportunity ost of ash in 2004 is about 1/3 of the value in 1993 (the orresponding ratio for the nominal interest rate is about 1/9), and that the average of M/ shows an upward trend. Indeed the average of M/ aross households of a given type (with and without ATM ards) is negatively orrelated with the opportunity ost R in the ross setion, in the time series, and the 5 They estimate that the elastiity of ash holdings with respet to the interest rate is about zero for agents who hold an ATM ard and -0.2 for agents without ATM ard. See their Setion 5 for a omparison with the findings of other papers. 6
pool time series-ross setion. Yet the largest estimate for the interest rate elastiity are smaller than 0.25 and in most ases about 0.05 (in absolute values). At the same time, Table 2 shows large inreases in bank branhes and ATM terminals per person. Suh patterns are onsistent with both shifts of the money demand and movements along it. Our model and estimation strategy allows us to quantify eah of them. Another lassi model of money demand is Miller and Orr (1966) who study the optimal inventory poliy for an agent subjet to stohasti ash inflows and outflows. Despite the presene of unertainty, their model, as the one by BT, does not feature a preautionary motive in the sense that M = 0. Unlike in the BT model, they find that the interest rate elastiity is 1/3 and the average withdrawal size W/M is 3/4. In this paper we keep the BT model as a theoretial benhmark beause the Miller and Orr model is more suitable for the problem faed by firms, given the nature of stohasti ash inflows and outflows. Our paper studies urreny demand by households: the theory studies the optimal inventory poliy for an agent that faes deterministi ash outflows (onsumption expenditure) and no ash inflows and the empirial analysis uses the household survey data (exluding entrepreneurs). 3 A model with deterministi free withdrawals This setion presents a modified version of the BT model to illustrate how tehnologial progress affets the level and interest elastiity of the demand for urreny. Consider an agent who finanes a onsumption flow by making n withdrawals from a deposit aount. We let R be the net nominal interest rate paid on deposits. In a deterministi setting agents ash balanes derease until they hit zero, when a new withdrawal must take plae. Hene the size of eah withdrawal is W = /n and the average ash balane M = W/2. In the BT model agents pay a fixed ost b for eah withdrawal. We modify the latter by assuming that the agent has p free withdrawals, so that if the total number of withdrawals is n then she pays only for the exess of n over p. Setting p = 0 yields the BT ase. Tehnology is thus represented by the parameters b and p. For example, assume that the ost of a withdrawal is proportional to the distane to an ATM or bank branh. In a given period the agent is moving aross loations, for reason unrelated to her ash management, so that p is the number of times that she is in a loation with an ATM or bank branh. At any other time, b is the distane that the agent must travel to withdraw. In this setup an inrease in the 7
density of bank branhes or ATMs inreases p and dereases b. The optimal number of withdrawals solves the minimization problem [ min R n ] 2n + b max(n p, 0). (1) By examining the objetive funtion, it is immediate that the value of n that solves the problem, and its assoiated M/, depends only on β b/ ( R), the ratio of the two osts, and p. The money demand for a tehnology with p 0 is given by M = 1 min (2 ˆb ) 2p R, 1 where ˆb b p 2. (2) To understand the workings of the model, fix b and onsider the effet of inreasing p (so that ˆb inreases). For p = 0 we have the BT setup, so that when R is small the agent deides to eonomize on withdrawals and hoose a large value of M. Now onsider the ase of p > 0. In this ase there is no reason to have less than p withdrawals, sine these are free by assumption. Hene, for all R 2ˆb the agent will hoose the same level money holdings, namely, M = /(2p), sine she is not paying for any withdrawal but is subjet to a positive opportunity ost. Note that the interest elastiity is zero for R 2ˆb. Thus as p (hene ˆb) inreases, then the money demand has a lower level and a lower interest rate elastiity than the money demand from the BT model. Indeed (2) implies that the range of interest rates R for whih the money demand is smaller and has lower interest rate elastiity is inreasing in p. On the other hand, if we fix ˆb and inrease p the only effet is to lower the level of the money demand. The previous disussion makes lear that for fixed p, ˆb ontrols the shape of the money demand, and for fixed ˆb, p ontrols its level. We think of tehnologial improvements as both inreasing p and lowering b: the net effet on ˆb, hene on the slope of the money demand, is in priniple ambiguous. The empirial analysis below allows us to sign and quantify this effet. 4 A model with random free withdrawals This setion presents the main model whih generalizes the example of the previous setion in several dimensions. It takes an expliit aount of the dynami nature of the ash inventory problem, as opposed to minimizing the average steady state ost. It distinguishes between real and nominal variables, as opposed to fi- 8
naning a onstant nominal expenditure, or alternatively assuming zero inflation. Most importantly, we onsider the ase where the agent has a Poisson arrival of free opportunities to withdraw ash at a rate p. Relative to the deterministi model this assumption produes ash management behavior that is loser to the one doumented in Setion 2. The randomness thus introdued gives rise to a preautionary motive, so that some withdrawals our when the agent still has a positive ash balane and the (average) W/M ratio is smaller than two. The model retains the feature (disussed in Setion 3) that the interest rate elastiity is smaller than 1/2 and is dereasing in the parameter p. It also generalizes the sense in whih the shape of the money demand depends on the parameter ˆb = p 2 b/. 4.1 The agent s problem The model we onsider solves the problem of minimizing the ost of finaning a given onstant flow of ash onsumption, denoted by. We assume that agents are subjet to a ash-in-advane onstraint. We use m to denote the non-negative real ash balanes of an agent whih derease due to onsumption and inflation: dm (t) dt = m (t) π (3) for almost all t 0. Agents an withdraw or deposit at any time from an aount that yields real interest r. Transfers from the interest bearing aount to ash balanes are indiated by disontinuities in m: a withdrawal is a jump up on the ash balanes, i.e. m (t + ) m (t ) > 0, and likewise for a deposit. There are two soures or randomness in the environment, desribed by independent Poisson proesses with intensities p 1 and p 2. The first proess desribes the arrivals of free adjustment opportunities (see the Introdution for examples). The seond Poisson proess desribes the arrivals of times where the agent looses (or is stolen) her ash balanes. We assume that a fixed ost b is paid for eah adjustment, unless it happens exatly at the time of a free adjustment opportunity. We an write the problem of the agent as: C (m) = { min E 0 {m(t),τ j } j=0 e r τ j [ I τj b + ( m ( τ + j ) ( ))] } m τ j (4) subjet to (3) and m (t) 0, where τ j denote the stopping times at whih an 9
adjustment (jump) of m takes plae, and m (0) = m is given. The indiator I τj is zero so the ost is not paid if the adjustment takes plae at a time of a free adjustment opportunity, otherwise is equal to one. The expetation is taken with respet to the two Poisson proesses. The parameters that define this problem are r, π, p 1, p 2, b and. 4.2 Bellman equations and optimal poliy We turn to the haraterization of the Bellman equations and of its assoiated optimal poliy. We will guess, and later verify, that the optimal poliy is desribed by two thresholds for m: 0 < m < m. The threshold m is the value of ash that agents hoose to have after a ontat with a finanial intermediary: we refer to it as the optimal ash replenishment level. The threshold m is a value of ash beyond whih agents will pay the ost b, ontat the intermediary, and make a deposit so as to leave her ash balanes at m. Assuming that the optimal poliy is of this type and that for m (0, m ) the value funtion C is differentiable, it must satisfy: rc (m) = C (m) ( πm) + p 1 min [ ˆm m + C ( ˆm) C (m)] + (5) ˆm 0 + p 2 min [b + ˆm + C ( ˆm) C (m)]. ˆm 0 If the agent hooses not to ontat the intermediary then, as standard, the Bellman equation states that the return on the value funtion rc (m) must equal the flow ost plus the expeted hange per unit of time. The first term of the summation gives the hange in the value funtion per unit of time, onditional on no arrival of either free adjustment or of a loss of ash (theft). This hange is given by the hange in the state m, times the derivative of the value funtion C (m). The seond term gives the expeted hange onditional on the arrival of free adjustment opportunity: an adjustment ˆm m is inurred instantly with its assoiated apital gain C ( ˆm) C (m). Likewise, the third term gives the hange in the value funtion onditional on the money stok m being stolen. In this ase the ost b must be paid and the adjustment equals ˆm, sine m is lost. Regardless of how the agent ends up mathed with a finanial intermediary, upon the math she hooses the optimal level of real balanes, whih we denote by m, whih solves m = arg min ˆm 0 ˆm + C ( ˆm). (6) 10
Note that the optimal replenishment level m is onstant. There are two boundary onditions for this problem. First, if money balanes reah zero (m = 0) the agent must withdraw, otherwise she will violate the non-negativity onstraint in the next instant. Seond, for values of m m we onjeture that the agent hooses to pay b and deposit the extra amount, m m. Combining these boundary onditions with (5) we have: C (m) = b + m + C (m ) if m = 0 C (m) ( + πm) + (p 1 + p 2 ) [m + C (m )] + p 2 b p 1 m r + p 1 + p 2 if m (0, m ) b + m m + C (m ) if m m For the assumed onfiguration to be optimal it must be the ase that the agent prefers not to pay the ost b and adjust money balanes in the relevant range: m + C (m) < b + m + C (m ) all m (0, m ). (8) (7) Summarizing, we say that 0 < m < m, C ( ) solve the Bellman equation for the total ost problem (4) if they satisfy (6)-(7)-(8). We find it onvenient to reformulate this problem so that it is loser to the standard inventory theoretial models. We define a related problem where the agent minimizes the shadow ost V (m) = { min E 0 {m(t),τ j } j=0 e rτ j [ τj+1 τ j I τj b + e rt R m (t + τ j ) dt] } (9) 0 subjet to (3), m (t) 0, where τ j denote the stopping times at whih an adjustment (jump) of m takes plae, and m (0) = m is given. The indiator I τj equals zero if the adjustment takes plae at the time of a free adjustment, otherwise is equal to one. In this formulation R is the opportunity ost of holding ash. In this problem there is only one Poisson proess with intensity p desribing the arrival of a free opportunity to adjust. The parameters of this problem are r, R, π, p, b and. 6 6 The shadow ost formulation is the standard one used in the literature for inventory theoretial models, as in the models of Baumol-Tobin, Miller and Orr (1966), Constantinides (1976), among others. In these papers the problem aims to minimize the steady state ost implied by a stationary inventory poliy. This differs from our formulation, where the agent minimizes the expeted disounted ost in (9). In this regard our analysis follows the one of Constantinides and Rihards (1978). For a related model, Frenkel and Jovanovi (1980) ompare the resulting money demand arising from minimizing the steady state vs. the expeted disounted ost. 11
The derivation of the Bellman equation for an agent unmathed with a finanial intermediary and holding a real value of ash m follows by the same logi used to derive equation (5). The only deision that the agent must make is whether to remain unmathed, or to pay the fixed ost b and be mathed with a finanial intermediary. Denoting by V (m) the derivative of V (m) with respet to m, the Bellman equation satisfies rv (m) = Rm + p min ˆm 0 (V ( ˆm) V (m)) + V (m) ( mπ). (10) Regardless of how the agent ends up mathed with a finanial intermediary, she hooses the optimal adjustment and sets m = m, or V V (m ) = min V ( ˆm). (11) ˆm 0 As in problem (4) we will guess that the optimal poliy is desribed by two threshold values satisfying 0 < m < m. This requires two boundary onditions. At m = 0 the agent must pay the ost b and withdraw, and for m m the agent hooses to pay the ost b and deposit the ash in exess of m. Combining these boundary onditions with (10) we have: V + b if m = 0 Rm + pv V (m) ( + mπ) V (m) = if m (0, m ) (12) r + p V + b if m m To ensure that it is optimal not to pay the ost and ontat the intermediary in the relevant range we require: V (m) < V + b for m (0, m ). (13) Summarizing, we say that 0 < m < m, V, V ( ) solve the Bellman equation for the shadow ost problem (9) if they satisfy (11)- (12)-(13). We are now ready to show that, first, (4) and (9) are equivalent and, seond, the existene and haraterization of the solution. Proposition 1. Assume that the opportunity ost is given by R = r + π + p 2, and that the ontat rate with the finanial intermediary is p = p 1 + p 2. Assume that the 12
funtions C ( ), V ( ) satisfy C (m) = V (m) m + /r + p 2 b/r (14) for all m 0. Then, m, m, C ( ) solve the Bellman equation for the total ost problem (4) if and only if m, m, V, V ( ) solve the Bellman equation for the shadow ost problem (9). Proof. See Appendix A. We briefly omment on the relation between the total and shadow ost problems. Notie that they are desribed by the same number of parameters. They have r, π,, b in ommon, the total ost problem uses p 1 and p 2, while the shadow ost problem uses R and p. That R = r + π + p 2 is quite intuitive: the shadow ost of holding money is given by the real opportunity ost of investing, r, plus the fat that ash holdings loose real value ontinually at a rate π and they are lost entirely with probability p 2 per unit of time. Likewise that p = p 1 + p 2 is lear too: sine the effet of either shok is to fore an adjustment on ash. The relation between C and V in (14) is quite intuitive. First the onstant /r is required, sine even if withdrawals were free (say b = 0) onsumption expenditures must be finaned. Seond, the onstant p 2 b/r is the present value of all the withdrawals ost that is paid after ash is lost. This adjustment is required beause in the shadow ost problem there is no theft. Third, the term m has to be subtrated from V sine this amount has already been extrated from the interest bearing aount. From now on, we use the shadow ost formulation, sine it is loser to the standard inventory deision problem. On the theoretial side, having the effet of theft as part of the opportunity ost allows us to parameterize R as being, at least oneptually, independent of r and π. On the quantitative side we think that, at least for low nominal interest rates, the presene of other opportunity osts may be important. 4.3 Charaterization of the optimal return point m The next proposition gives one non-linear equation whose unique solution determines the ash replenishment value m as a funtion of the model parameters: R, π, r, p, and b. Proposition 2. Assume that r + π + p > 0. The optimal return point m has three arguments: β, r + p, π, where β b. The return point R m is given by the unique 13
positive solution to ( m π + 1 ) 1+ r+p π = m (r + p + π) + 1 + (r + p) (r + p + π) b R. (15) Proof. See Appendix A. Note that, keeping r and π fixed, the solution for m / is a funtion of b/(r), as it is in the steady state money demand of Setion 3. This immediately implies that m is homogenous of degree one in (, b). The next proposition gives a losed form solution for the funtion V ( ), and the salar V in terms of m. Proposition 3. Assume that r + π + p > 0. Let m be the solution of (15). (i) The value for the agents not mathed with a finanial institution, for m (0, m ), is given by the onvex funtion: [ ] [ pv R/ (r + p + π) V (m) = + r + p where A = r+p 2 ( R m + (r + p) b + R r + p + π ) R > 0. r+p+π ] ( ) 2 [ m + A 1 + π m ] r+p π r + p (16) For m = 0 or m m : V (m) = V + b. (ii) The value for the agents mathed with a finanial institution, V, is Proof. See Appendix A. V = R r m. (17) The lose relationship between the value funtion at zero ash and the optimal return point V (0) = (R/r) m + b derived in this proposition will be useful to measure the gains of different finanial arrangements. The next proposition uses the haraterization of the solution for m to ondut some omparative statis. Proposition 4. The optimal return point m has the following properties: (i) m is inreasing in b, m = 0 as b m = 0 and as b. R R R b m (ii) For small, we an approximate by the solution in BT model, or R ( ) m = 2 b b R + o R where o(z)/z 0 as z 0. (iii) Assuming that the Fisher equation holds, in that π = R r, the elastiity of 14
m with respet to p evaluated at zero inflation satisfies 0 p m dm dp π=0 p p + r. (iv) The elastiity of m with respet to R evaluated at zero inflation satisfies 0 R m dm The elastiity is dereasing in p and satisfies: dr π=0 1 2. R m m R π=0 1/2 as ˆb R 0 and R m m R π=0 0 as ˆb R where ˆb (p + r) 2 b/. Proof. See Appendix A. The proposition shows that when b/(r) is small the resulting money demand is well approximated by the one for the BT model. Part (iv) shows that the absolute value of the interest elastiity (when inflation is zero) ranges between zero and 1/2, and that it is dereasing in p. In the limits we use ˆb to write a omparative stati result for the interest elastiity of m with respet to p. Indeed, for r = 0, we have already given an eonomi interpretation to ˆb in Setion 3, to whih we will return in Proposition 8. Sine in Proposition 2 we show that m is a funtion of b/(r), then the elastiity of m with respet to b/ equals the one with respet to R with an opposite sign. 4.4 Number of withdrawals and ash holdings distribution This setion derives the invariant distribution of real ash holdings when the poliy haraterized by the parameters (m, p, ) is followed and the inflation rate is π. Throughout the setion m is treated as a parameter, so that the poliy is to replenish ash holdings up to the return value m, either when a math with a finanial intermediary ours, whih happens at a rate p per unit of time, or when the agent runs out of money (i.e. real balanes hit zero). Our first result is to ompute the expeted number of withdrawals per unit of time, denoted by n. This inludes both the withdrawals that our upon an exogenous ontat with the finanial intermediary and the ones initiated by the agent when her ash balanes reah zero. By the fundamental theorem of Renewal Theory n equals the reiproal of the expeted time between withdrawals, whih after straightforward alulations gives 15
Proposition 5. The expeted number of ash withdrawals per unit of time, n, is ( ) m n, π, p p = 1 ( ) p. (18) 1 + π m π Proof. See Appendix A. As an be seen from expression (18) the ratio n/p 1, sine in addition to the p free withdrawals it inludes the ostly withdrawals that agents do when they exhaust their ash. Note how this formula yields exatly the expression in the BT model when p = π = 0. The next proposition derives the density of the invariant distribution of real ash balanes as a funtion of p, π, and m /. Proposition 6. (i) The density for the real balanes m is: ( p ) [ 1 + π m h (m) = [ 1 + π m ] p π 1 ] p π 1. (19) (ii) Let H (m, m 1) be the CDF of m for a given m. Let m 1 < m 2, then H (m, m 2) H (m, m 1), i.e. H (, m 2) first order stohastially dominates H (, m 1). Proof. See Appendix A. The density of m solves the following ODE (see the proof of Proposition 6) h (m) m (p π) = h (m) (20) (πm + ) for any m (0, m ). There are two fores determining the shape of this density. One is that agents meet a finanial intermediary at a rate p, where they replenish their ash balanes. The other is that inflation eats away the real value of their nominal balanes. Notie that if p = π these two effets anel and the density is onstant. If p < π the density is downward sloping, with more agents at low values of real balanes due to the greater pull of the inflation effet. If p > π, the density is upward sloping due the greater effet of the replenishing of ash balanes. This uses that πm + > 0 in the support of h beause πm + > 0 (see equation 38). We define the average money demand as M = m 0 mh (m) dm. Using the expression for h(m), integration gives M ( ) m, π, p = ( 1 + π m [ ) p π m ( 1+π m ) p+π [ ] 1 + π m p π 1 ] + 1 p+π. (21) 16
Next we analyze how M depends on m and p. The funtion M (, π, p) is inreasing in m, whih follows immediately from part (ii) of Proposition 6: with a higher target replenishment level the agents end up holding more money on average. The next proposition shows that for a fixed m, M is inreasing in p: Proposition 7. The ratio M m is inreasing in p with: M m (π, p) = 1 2 for p = π and M (π, p) 1 as p. m Proof. See Appendix A. It is useful to ompare this result with the orresponding one for the BT ase, whih is obtained when π = p = 0. In this ase agents withdraw m hene M/m = 1/2. The other limit orresponds to the ase where withdrawals happen so often that at all times the average amount of money oinides with the amount just after a withdrawal. The average withdrawal, W, is [ W = m 1 p ] [ p + n n] m (m m) h (m) dm. (22) 0 To understand the expression for W notie that (n p) is the number of withdrawals in a unit of time that our beause the zero balane is reahed, so if we divide it by the total number of withdrawals per unit of time (n) we obtain the fration of withdrawals that our at a zero balane. Eah of these withdrawals is of size m. The omplementary fration gives the withdrawals that our due to a hane meeting with the intermediary. A withdrawal of size m m happens with frequeny h (m). Inspetion of (22) shows that W/ is a funtion of three arguments: m /, π, p. Combining the previous results we an see that for p π, the ratio of withdrawals to average ash holdings is less than two. To see this, using the definition of W we an write W M = m M p n. (23) Sine M/m 1/2, then it follows that W/M 2. Indeed notie that for p large enough this ratio an be smaller than one. We mention this property beause for the Baumol - Tobin model the ratio W/M is exatly two, while in the data of Table 1 for households with an ATM ard the average ratio is below 1.5 and its median value is 1. The intuition for this result in our model is lear: agents take advantage of the free random withdrawals regardless of their ash balanes, hene 17
the withdrawals are distributed on [0, m ], as opposed to be onentrated on m, as in the BT model. We let M be the average amount of money that an agent has at the time of withdrawal. A fration [1 p/n] of the withdrawals happens when m = 0. For the remaining fration, p/n, an agent has money holdings at the time of the withdrawal distributed with density h, so that: M = 0 [ [ 1 n] p + p ] m m h (m) dm. Inspetion of this expression shows that M/ is a funtion of three arguments: m /, π, p. n 0 Simple algebra shows that M = m W or, inserting the definition of M into the expression for M: M = p n M. (24) The ratio M/M is a measure of the preautionary demand for ash: it is zero only when p = 0, it goes to 1 as p and, at least for π = 0, it is inreasing in p. This is beause as p inreases the agent has more opportunities for a free withdrawal, whih diretly inreases M/M (see equations 18 and 24), and from part (iii) in Proposition 4 the indued effet of p on m annot outweigh the diret effet. Other researhers notiing that urreny holdings are positive at the time of withdrawals aount for this feature by adding a onstant M/M to the sawtooth path of a deterministi inventory model, whih implies that the average ash balane is M 1 = M + 0.5 /n or M 2 = M + 0.5 W. See e.g. equations 1 and 2 in Attanasio, Guiso and Jappelli (2002) and Table 1 in Porter and Judson (1996). Instead, when we model the determinants of the preautionary holdings M/M in a random setup, we find that W/2 < M < M + W/2. The leftmost inequality is a onsequene of Proposition 7 and equation (23), the other an be easily derived using the form of the optimal deision rules and the law of motion of ash flows (see the working paper version for details). The disussion above shows that the expressions for the demand for ash proposed in the literature to deal with the preautionary motive are upward biased. Using the data of Table 1 shows that both expressions M 1 and M 2 overestimate the average amount of ash held by Italian households by a large margin. 7 4.5 Comparative statis on M, M, W and welfare We begin with a omparative statis exerise on M, M and W in terms of the primitive parameters b/, p, and R. To do this we ombine the results of Setion 7 The expression for M 1 overestimates the average ash by 20% and 140% for household with and without ATMs, respetively; the one for M 2 by 7% and 40%, respetively. 18
4.3, where we analyzed how the optimal deision rule m / depends on p, b/ and R, with the results of Setion 4.4 where we analyze how M, M, and W hange as a funtion of m / and p. The next proposition defines a one dimensional index ˆb (b/)p 2 that haraterizes the shape of the money demand and the strength of the preautionary motive fousing on π = r = 0. When r 0 our problem is equivalent to minimizing the steady state ost. The hoie of π = r = 0 simplifies the omparison of the analytial results with the ones for the original BT model and with the ones of Setion 3. Proposition 8. Let π = 0 and r 0, the ratios: W/M, M/M and (M/) p are determined by three stritly monotone funtions of ˆb/R that satisfy: Proof. See Appendix A. As ˆb R 0 : W M 2, M M 0, log Mp log ˆb As ˆb R : W M 0, M M 1, log Mp log ˆb R R 1 2. 0. The elastiity of (M/)p with respet to ˆb/R determines the effet of the tehnologial parameters b/ and p on the level of money demand, as well as on the interest rate elastiity of M/ with respet to R sine Diret omputation gives that η(ˆb/r) log(m/)p log(ˆb/r) = log(m/) log R. (25) log(m/) log p = 1 + 2η(ˆb/R) 0 and 0 log(m/) log(b/) = η(ˆb/r). (26) The previous setions showed that p has two opposing effets on M/: for a given m /, the value of M/ inreases with p, but the optimal hoie of m / dereases with p. Proposition 8 and equation (26) show that the net effet is always negative. For low values of ˆb/R, where η 1/2, the elastiity of M/ with respet to p is lose to zero and the one with respet to b/ is lose to 1/2, whih is the BT ase. For large values of ˆb/R, the elastiity of M/ with respet to p goes to 1, and the one with respet to b/ goes to zero. Likewise, equation (26) implies that log M/ log = 1 η and hene that the expenditure elastiity of the money demand ranges between 1/2 (the BT value) and 1 as ˆb/R beomes large. 19
In the original BT model W/M = 2, M/M = 0 and log(m/) = 1/2 for log R all b/ and R. These are the values that orrespond to our model as ˆb/R 0. This limit inludes the standard ase where p 0, but it also inludes the ase where b/ is muh smaller than p 2 /R. As ˆb/R grows, our model predits smaller interest rate elastiity than the BT model, and in the limit, as ˆb/R, that the elastiity goes to zero. This result is a smooth version of the one for the model with p deterministi free withdrawal opportunities of Setion 3. In that model the elastiity log(mp/)/ log(ˆb/r) is a step funtion that takes two values, 1/2 for low values of ˆb/R, and zero otherwise. The smoothness is a natural onsequene of the randomness on the free withdrawal opportunities. One key differene is that the deterministi model of Setion 3 has no preautionary motive for money demand, hene W/M =2 and M/M = 0. Instead, as Proposition 8 shows, in the model with random free withdrawal opportunities, the strength of the preautionary motive, as measured by W/M and M/M, is a funtion of ˆb/R. Figure 1 plots W/M, M/M and η as funtions of ˆb/R. This figure ompletely haraterizes the shape of the money demand and the strength of the preautionary motive sine the funtions plotted in it depend only on ˆb/R. The range of the ˆb/R values used in this figure is hosen to span the variation of the estimates presented in Table 5. While this figure is based on results for π = r = 0, the figure obtained using the values of π and r that orrespond to the averages for Italy during 1993-2004 is quantitatively indistinguishable. We onlude this setion with a result on the welfare ost of inflation and the effet tehnologial hange. Let (R, κ) be the vetor of parameters that index the value funtion V (m; R, κ) and the invariant distribution h(m; R, κ), where κ = (π, r, b, p, ). We define the average flow ost of ash purhases borne by households v(r, κ) m rv (m; R, κ)h(m; R, κ)dm. We measure the benefit of 0 lower inflation for households, say as aptured by a lower R and π, or of a better tehnology, say as aptured by a lower b/ or a higher p, by omparing v( ) for the orresponding values of (R, κ). A related onept is l(r, κ), the expeted withdrawal ost borne by households that follow the optimal rule l(r, κ) = [n(m (R, κ), p, π) p] b (27) where n is given in (18) and the expeted number of free withdrawals, p, are subtrated. The value of l(r, κ) measures the resoures wasted trying to eonomize on ash balanes, i.e. the deadweight loss for the soiety orresponding to R. While l 20
2 1.8 1.6 1.4 W/M Figure 1: W/M, M/M, m /M and η = elastiity of (M/)p For π = 0 and r > 0 m*/m η W/M Mlow/M m*/m 1.2 1 0.8 0.6 0.4 Mlow/M η = Elast (M/)p 0.2 0 1 2 3 4 5 6 (b/) p 2 / R is the relevant measure of the ost for the soiety, we find useful to define v separately to measure the onsumers benefit of using ATM ards. The next proposition haraterizes l(r, κ) and v(r, κ) as r 0. This limit is useful for omparison with the BT model and it also turns out to be an exellent approximation for the values of r that we use in our estimation. Proposition 9. Let r 0: (i) v(r, κ) = R m (R, κ); (ii) v(r, κ) = R 0 M( R, κ)d R, and (iii) l(r, κ) = v(r, κ) R M(R, κ). Proof. See Appendix A. This proposition allows us to estimate the effet of inflation or tehnology on agents welfare using data on W and M, sine W + M = m. In the BT model l = RM = Rb/2 sine m = W = 2M. In our model m /M = W/M + M/M < 2, as an be seen in Figure 1, thus using RM as an estimate of R(m M) produes an overestimate of the ost of inflation l. For instane, for ˆb/R = 1.8, the BT welfare ost measure overestimates the ost of inflation by about 60%, sine m /M = 1.6. Clearly the loss for soiety is smaller than the ost for households; using (i)-(iii) and Figure 1 the two an be easily ompared. As ˆb/R ranges from zero to, the 21
ratio of the osts l/v dereases from 1/2, the BT value, to zero. Not surprisingly (ii)-(iii) implies that the loss for soiety oinides with the onsumer surplus that an be gained by reduing R to zero, i.e. l(r) = R 0 M( R)d R RM(R). This extends the result of Luas (2000), derived from a money-in-the-utility-funtion model, to an expliit inventory-theoreti model. Measuring the welfare ost of inflation using the onsumer surplus requires the estimation of the money demand for different interest rates, while the approah using (i) and (iii) an be done using information on M, W and M. Setion 8 presents an appliation of these results and a omparison with the ones by Luas (2000). 8 5 A model with ostly random withdrawals The dynami model disussed above has the unrealisti feature that agents withdraw every time a math with a finanial intermediary ours, so that many of the withdrawals have a very small size. This setion extends the model to the ase where the withdrawals done upon the random ontats are subjet to a fixed ost f, assuming 0 < f < b. The model produes a more realisti distribution of withdrawals, by limiting the minimum withdrawal size. We skip the formulation of the total ost problem, that is exatly parallel to the one for the ase of f = 0. With notation analogous to the one used above, the Bellman equation when the agent is not mathed with an intermediary is rv (m) = Rm + p min {V + f V (m), 0} + V (m) ( mπ) (28) where V min ˆm V ( ˆm) and min {V + f V (m), 0} takes into aount that it may not be optimal to withdraw for all ontats. Indeed, whether the agent hooses to do so depend on the level of ash balanes. It an be shown that V ( ) is stritly dereasing at m = 0 with a unique value of m suh that V (m ) = 0. Then there will be two thresholds, m and m, that satisfy V +f = V (m) = V ( m). Thus solving the Bellman equation is equivalent to finding 5 numbers m, m, m, m, V and a funtion V ( ) suh that: V = V (m ), 8 In (ii)-(iii) we measure welfare and onsumer surplus with respet to variations in R, keeping π fixed. The effet on M and v of hanges in π for a onstant R are quantitatively small. 22