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Essays on the inventory theory of money demand Li, Chen 2007

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ESSAYS ON THE INVENTORY THEORY OF MONEY DEMAND by Chen Li B.E., Ji Lin Institute of Technology, 1998 M.A., University of British Columbia, 2000  A THESIS SUBMITTED IN PARTIAL FUFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in  The Faculty of Graduate Studies (ECONOMICS)  THE UNIVERSITY OF BRITISH COLUMBIA  November, 2007 © Chen Li, 2007  Abstract The goal of this dissertation is to examine the theoretical and empirical implications of the inventory theoretic approach to the demand for money. Chapter 1 reviews the existing inventory theoretic frameworks and empirical money demand literature and provides an overview of this thesis. One of the main conclusions is that the elasticity results from the existing inventory theoretic models are not robust. Chapter 2 develops a partial equilibrium inventory theoretic model, in which a fixed cost is involved per cash transfer. The key feature is that a firm endogenously chooses the frequency of pay periods, which a household takes as given. When the firm must borrow working capital and pay wages by cheque, I show that both the firm and the household choose to transfer cash every payday only. The model keeps the basic result from the classical inventory theoretic approach that both the income and interest elasticity of money demand are 0.5. Chapter 3 extends the partial equilibrium model into a general equilibrium framework and shows that the partial equilibrium elasticity results no longer apply in the general equilibrium. First, the income elasticity is 1 in the general equilibrium. Second, the interest elasticity has two values depending on a threshold interest rate. When interest rates are below this threshold, the model is the Cash-In-Advance model with a constant income velocity of money and zero interest elasticity; otherwise the interest elasticity is close to 0.5 and the velocity fluctuates in response to variations in interest rates. Finally, the general equilibrium elasticity results are robust across alternative specifications of the agent's utility. Chapter 4 calibrates the general equilibrium model to the last 40 years of US data for M1. By constructing a residual measure of money transaction costs from the structural money demand function, I find that a structural break in the transaction costs occurred in 1981 might have been responsible for the instability of long-run money demand. The benefit of this approach is that it can explain this pattern of money demand without appealing to an exogenous structural break in the money demand function.  ii  Table of Contents Abstract ^  ii  Table of Contents.... ^  iii  List of Tables ^  .iv  List of Figures ^  v  Acknowledgements ^  vi  Chapter 1: Literature Review and Thesis Overview 1.1 Introduction ^ 1.2 Literature Review ^  1 ..1  1.2.1 Inventory Theoretic Approach ^  1  1.2.2 Empirical Money Demand.  8  1.3 Thesis Overview ^  .15  Chapter 2: The Frequency of Pay-Periods and Transactions Demand for Money 2.1 Introduction ^  18  2.2 The Firm's Cash Management Problem ^  21  2.3 The Household's Cash Management Problem ^  28  2.4 Money Demand ^  34  2.5 Conclusion  35  Chapter 3: A General Equilibrium Model of Baumol - Tobin Money Demand 3.1 Introduction ^  36  3.2 The Representative Firm's Problem ^  40  3.3 The Representative Household's Problem ^  44  3.4 Market Equilibrium and Money Demand ^  51  3.5 Conclusion  ^60 ii  Chapter 4: A Stable Money Demand Function in a General Equilibrium Inventory Theoretic Model 4.1 Introduction ^  62  4.2 Calibration of Money Demand ^ 4.3 Conclusion  . 67 ^86  Chapter 5: Concluding Remark ^ Reference ^  88 .90  Appendices ^  ^... 96  Appendix Al ^  96  Appendix A2 ^  99  iv  List of Tables Table 3.1 Decomposition of Interest Elasticity under Various 0 and a ^ 60 Table 4.1 Benchmark Parameter Values ^ Table 4.2 Estimates of AR(p) Model ^  72 .75  Table 4.3 Estimates and Test Statistics for the Whole Sample Period ^ 76 Table 4.4 Estimates and Test Statistics for the First Sample Period ^ 78 Table 4.5 Estimates and Test Statistics for the Second Sample Period Table 4.6 New Estimates under Alternative pt^  ^ ..81 85  Table 4.7 New Estimates under Alternative Structural Money Demand Functions ^. .86  v  List of Figures Figure 1.1 Saw-toothed Cash Holdings of the Household ^ Figure 2.1 Firm's Marginal Cost of Cash Management  .2  ^ 25  Figure 2.2 Saw-toothed Cash Holdings of the Firm^  26  Figure 2.3 Household's Inter-Pay-Period Cash Withdrawals ^  30  Figure 2.4 Household's Cash Holding Periods ^  31  Figure 2.5 Saw-toothed Cash Holdings of the Household^  .34  Figure 4.1 Actual and Predicted Ml Velocity from 1960-2000 [Lucas 1988] ^. .70 Figure 4.2 Money Transaction Costs from 1960-2000^  73  Figure 4.3 Actual vs. Predicted Ml Velocity from 1960-2000^  77  Figure 4.4 Actual vs. Predicted Ml Velocity from 1960-1981 ^  79  Figure 4.5 Actual vs. Predicted M1 to GDP ratio from 1960-1981 ^ 80 Figure 4.6 Actual vs. Predicted Ml Velocity from 1982-2000^  82  Figure 4.7 Actual vs. Predicted Ml to GDP ratio from 1982-2000 ^.. 82 Figure 4.8 Long-run Estimates of Money Transaction Cost [Lucas (2000) vs. Mine]. ^ .84  vi  Acknowledgements I am extremely grateful to my advisors Angela Redish and Patrick Francois for their invaluable advice and guidance. I am also grateful to Paul Beaudry, Henry Siu, Michael Devereux, Amartya Lahiri and Jacob Wong for helpful comments and suggestions. All errors are mine.  vii  Chapter 1: Literature Review and Thesis Overview 1.1 Introduction This thesis contributes to the literature on the general equilibrium inventory theoretic approach and on empirical money demand. As part of this introductory chapter, Section 2 reviews the existing literature, including various partial equilibrium, general equilibrium inventory theoretic frameworks and empirical long-run money demand studies. The main purpose of the literature review is in providing the motivation for the current work of this thesis. Section 3 presents a thesis overview, and summarizes the main theoretical and empirical results of the following chapters.  1.2 Literature Review 1.2.1 Inventory Theoretic Approach The inventory theoretic approach, first developed by Baumol (1952) and Tobin (1956), served as one of the most important theoretical frameworks in monetary economics. The influential idea implied by this approach is that people hold money to reduce the resource costs of money transaction. This idea is behind most of the existing money demand frameworks. One of the most appealing insights of the approach is that the income and interest elasticity of money demand are constant and both equal to 0.5. Income elasticity and interest elasticity are the key parameters for determining the demand for money. As pointed out by Lucas (1988), the values of the interest elasticity and income elasticity of a money demand function answer two important questions pertaining to economic policy. When long-run real output growth is fairly predictable and insensitive to changes in monetary policy, the value of income elasticity provides the answer to the question: what rate of growth of money is consistent with long-run price stability? The interest elasticity is the key parameter needed to answer the question: what are the welfare costs to society for deviation from long-run price stability? Therefore, this review focuses on the implications from the existing inventory theoretic frameworks on these two parameters. 1  Partial Equilibrium Inventory Theoretic Frameworks Baumol (1952) and Tobin (1956) Early work by Baumol (1952) and Tobin (1956) (B-T) first applied the inventory-theoretic approach to the transactions demand for money. In these models, a consumer is paid an amount at the beginning of a period and spends this amount uniformly over the period. As cash earns no interest, the consumer only carries a small portion of cash holdings for immediate spending, and keeps the rest in the form of an interest-bearing asset (either bank deposits or bond). Because a brokerage fee or a fixed transaction cost is involved per money withdrawal, the consumer holds inventories of cash and makes currency conversion only infrequently. This leads to the famous square-root formula with average cash holdings given by I2BT  M  (1.2.1.1)  where T is the amount of consumption expenditure that the individual spends over the period; i is the interest rate on bank deposits; and B is the fixed brokerage fee. Equation (1.2.1.1) is obtained by minimizing the sum of interest forgone on total money holdings and total transaction costs. It indicates that the consumer operates on the intensive cash management margin (chooses the cash holding period) to equate the marginal interest forgone on cash holdings to the fixed transaction cost. As illustrated in Figure 1.1, (1.2.1.1) suggests that the consumer's cash holdings exhibit the so-called saw-toothed pattern: reaching the peak amount after every cash withdrawal, while gradually returning to zero before the next one. In Figure 1.1, f denotes the number of cash withdrawals by the consumer and T/f is the amount of each cash withdrawal. Figure 1.1 Saw-toothed Money Holdings A  T f  2 ^  2  Lastly, but most importantly, (1.2.1.1) suggests that money demand has constant income elasticity and interest elasticity which are both equal to 0.5. Miller and Orr (1966) (M 0) -  Another early application of inventory theory to the transactions demand for money is the work of Miller and On (1966) (M-0), which emphasized a firm's cash management problem. M-0 also introduced the precautionary motive into the demand for money, as the firm must pay a penalty when its money holding balance is below a minimum level. In their model, two assets are present - money and bonds. A fixed transaction cost is incurred per conversion between the two assets. The crucial feature is that the firm's cash flows are stochastic in a fixed time period (say 1/f of a day). In the simplest case, cash flows are assumed to follow a random walk without drift, with an equal probability of a positive or negative cash flow of m dollars. The firm's optimal cash management decision consists of an upper bound of money balances, H, and a return level, S. When the lower bound of money balances is reached, S dollars of bonds are converted to cash; when the upper bound is reached, (H-S) dollars of cash are converted to bonds. By minimizing the sum of expected daily transaction costs and interest forgone on money balances, M-0 showed that the optimal size of average money balances is given by M*, where 4 3B 3 4i  2  M * = - [ - a ] 3 ,c 2 ---= M 2 f ,  (1.2.1.2)  where m is the average size of each cash flow; f is the average daily frequency of transactions; and o 2 denotes the daily variance of changes in money balance. Like the approach of Baumol-Tobin, Miller and Orr's model yields constant interest elasticity, yet with a value of 1/3 rather than 1/2. The income elasticity is somewhat ambiguous as there is no direct scale variable corresponding to income. If one thinks in terms of the size of each cash flow, m, then the income elasticity is 2/3; however, if one thinks in terms of f, the frequency of transactions, then the scale elasticity is 1/3. This ambiguity regarding the scale variable made the M-0 framework ill-suited for empirical estimation.  3  Karni (1973) Both B-T and M-0 frameworks have the implication of constant elasticity, yet these partial equilibrium models assume that the variables entering the money demand function are mutually independent. In a comparative statistics study of the partial equilibrium inventory theoretic approach, Karni (1973) investigated the impact of possible inter-dependence among the variables entering the BT square root formula (1.2.1.1) on income and interest elasticity. The inter-dependence comes from the assumption that the money transaction involves costs in pecuniary terms and in terms of time (forgone earnings). Under this assumption a change in the marginal value of time affects both the transaction cost of assets conversion and total income (through changing wage income). The production technology of money transactions takes the following form: b  =  p'X  +  7w,  (1.2.1.3)  where b denotes the fixed brokerage fee measured by units of commodities; X denotes a vector of commodities used as inputs in the production of the exchange at a unit level; p' is the corresponding deflated price vector; y is the amount of time spent in producing the exchange at a unit level; and w is the real hourly wage rate. T  =  Vi  +  Lw.  (1.2.1.4)  In the simplest form, income is composed of labor income, denoted by wL, where L is the number of hours worked, and interest income, denoted by Vi, where V is the amount of interest bearing assets, which is assumed to be independent of the use of time. Taking logs on both sides of (1.2.1.1) — (1.2.1.4) and then manipulating the terms, with constant p'X, y and V (assumed to be independent of the real wage rate W and the interest rate i), we can derive the expressions for partial income elasticity and partial interest elasticity, respectively:  a log M a log M a log T 1—S x 1 , a logT alogW a logW 2(1—SV)(1+eL) + 2 a log M = —1 (S v —1), a logi^2  (1.2.1.5) (1.2.1.6)  where S v (=Vi/T) denotes the share of non-labor income to total income; S x (=p'X/b) denotes the 4  share of resource costs to total transaction costs; and e L (.3loglialogW) denotes labor supply elasticity. Two implications can be drawn from these two expressions. First, the income elasticity suggested by the square root formula depends on the nature of transaction costs. When the money transaction costs are only composed of the resource costs measured in units of consumption goods, i.e. S,=1, the income elasticity reaches 0.5 as the lower limit; but when the money transaction costs involve part of the time costs, the income elasticity is greater than 0.5. Second, (1.2.1.6) suggests that the interest elasticity of money demand is always negative, as 0^S v ^1. The inventory theoretic approach of the B-T and M-0 frameworks yields constant interest and income elasticity. Nevertheless, these models do not allow the inter-dependence among the variables entering the money demand functions. Although Karni (1973) went some way towards relaxing this assumption, the core of his analysis still rests on a partial equilibrium framework; for example, in his analysis the interest rate and the real wage are taken to be exogenously given. The partial equilibrium approach is not adequate to account for monetary equilibrium, as this approach implicitly assumes that the conduct of monetary policy (the money supply) is exogenous to the first order effects of a shift in the money demand curve. Moreover, due to lacking of microeconomics foundation, partial equilibrium frameworks cannot be used to examine the interactions between monetary (prices) and real phenomena (real allocations). Hence, it is important for researchers to examine the inventory theoretic approach in general equilibrium settings. General Equilibrium Inventory Theoretic Frameworks The existing general equilibrium inventory theoretic frameworks that can be used to study the demand for money are from Jovanovic (1982) and Romer (1986). These authors developed steady state versions of general equilibrium inventory theoretic models.  5  Jovanovic (1982) People hold two assets--physical capital and money. Agents are endowed with capital and have access to a productive storage technology for the capital. The capital can only be consumed if it is supplied to a market at a fixed transfer cost. The fixed cost is a resource cost measured in units of goods. Agents need to consume continuously, so money is needed to finance consumption between the dates when the capital is liquidated. Agents are almost identical except that they supply capital at different dates, but at the same uniform rate, so on each date a distribution always occurs across people, of money and of capital. Minimizing the sum of output forgone on capital and total transfer costs of resource yields approximately the same square root formula as (1.2.1.1). This result is not surprising, as suggested by Karni (1973), when the fixed cost is a resource cost, and the B-T square root formula approximately holds.  Romer (1986) Romer (1986) considered an over-lapping generation model (OLG). In the model, two assets are considered - money and bank deposits. There are N generations which are of equal size. At each moment, a new generation with size 1/N is born and lives for N period. Each generation is only endowed at birth, with an endowment of amount E of a non-storable consumption good and a lump sum transfer of real amount S of fiat money, but wishes to consume throughout its life. At the beginning of its life the individuals sell their endowment for money and then keep some wealth in the form of bank deposits. It costs agents' a fixed utility cost, which can be thought of as a time cost, to convert bank deposits into money. Like the approach of Jovanovic (1982), as every generation is exactly the same as others except for their dates of birth, at each instant a distribution always exists across generations, of goods and of money. Under the assumptions of no time discount for the agent and the log utility function, Romer's model predicts that the income elasticity of money is no longer 0.5, but 1. Romer argued that this can be explained by the fixed money transaction cost being a time cost instead of a resource cost. The result on the interest elasticity of money demand depends on the agent's preference over the disutility of money transaction costs. Since Romer did not specify how agents value their time 6  (the wage rate) in the model, the disutility time cost by construction directly enters and affects the agent's optimal cash management decision rule. With a linear disutility cost of money transactions, Romer derived a similar square root formula, which takes the following form: 2b  M = T.\/- , i  (1.2.1.7)  where b is the real fixed cost measured in units of the agent's utility. Equation (1.2.1.7) suggests that the interest elasticity of money is 0.5. Nevertheless, one critical drawback of Romer's model is that it can only be solved under special assumptions. With heterogeneous agents and non-recursive OLG structure, the model becomes impossible to solve with alternative specifications of the utility function and time discount rates. The frameworks of Jovanovic (1982) and Romer (1986) have serious limitations. The results of both models depend on the assumption that the economy always stays at one steady state (without the ability to transit to other steady states). 1 Hence, these models can at best be described as incomplete versions of the general equilibrium inventory theoretic models. Other General Equilibrium Inventory Theoretic Frameworks The liquidity literature, such as Grossman and Weiss (1983), Rotemberg (1984), Christiano and Eichenbaum (1995) and Alvarez, Atkeson and Kehoe (1998), can be viewed as variants of the general equilibrium inventory models. These authors assumed that agents are heterogeneous in their ability to access the asset market and at some point, only a fixed fraction of agents are free to access the asset market, while others face an infinitely large cost to do so. The endogenous segmented market literature, such as Chatterjee and Corbae (1992), Alvarez, Atkeson and Kehoe (2002) and Khan and Thomas (2006), are also variants of the general equilibrium inventory models. These papers assumed that agents receive heterogeneous shocks in the short-run. Those 'In the models of Jovanovic (1982) and Romer (1986), monetary policy shocks would disturb the uniform distribution of money across people, so that monetary equilibrium can not be reached.  7  active agents who can access the asset market must pay a fixed cost to exchange bonds with money. The fraction of active traders in the asset market among all heterogeneous agents is endogenous. In Chatterjee and Corbae (1992) and Alvarez, Atkeson and Kehoe (2002), agents differ in their initial endowment. Individuals who have high or low endowment, which is sold for money holding, are active traders in the asset market (either saving or withdrawing), while others are inactive. In both models, the income velocity of money is constant. Khan and Thomas (2006) assumed that agents face heterogeneous transfer costs to access the asset market. Those who face higher transfer costs carry money holdings across periods, while others trade assets in the current period (Cash-in-Advance constraint holds). Both the liquidity literature and the endogenous segmented market literature have been developed to investigate the short-run effects of monetary policy shocks. In the long-run when no shocks are present (all the agents in the models are identical), all these models become the Cash-In-Advance model with the constant velocity of money, and thus they cannot be used to account for the empirical long-run demand for money. 2  1.2.2 Empirical Money Demand A major concern in monetary economics is the stability of money demand. A stable demand function for money, linking real balances, real income and interest rates, has been long recognized as a requirement for the use of monetary aggregates in the conduct of policy. Empirical Evidence for Money Demand from the Partial Adjustment Approach An early test of the stability of money demand employed the so-called single equation partial adjustment model, which was first proposed by Goldfeld (1973). By including distributed lags into the money demand regression, this method was intended to obtain close fits to short-run data (quarterly), but it was not designed for estimating the interest and income elasticity of long run money demand. The basic model takes the following form:  2 A11 these authors used short-run M2 velocity in their calibration exercises, since, in these models the maximum velocity of money is unity by design (possibly less than one). This feature is suitable for short-run M2 data, while in US long-run data, both M l velocity and M2 velocity are greater than one.  8  login, =1) 0 + b l log y t +b 2 logi c + b 3 log m t _ t + b 4 n„^  (1.2.2.1)  where n t is the aggregate inflation rate of some price index. As explained by Goldfeld and Sichel (1990), the money demand specification (1.2.2.1) can be motivated by cost minimizing behavior, where the costs of disequilibrium are balanced against adjustment costs. Within this framework, the stability of money demand can be tested against the alternative that a structural change exists in the coefficient estimates of money demand. Goldfeld (1973) examined data for the period from 1952 to 1972. The stability of money demand is supported both by the formal tests of stability, conditional on an assumed sample split, and by the observation that the same specification estimated through the end of 1962 and simulated until 1972 did a good job of tracking the actual data of real balances. Nevertheless, when the data period is extended to the 1980's, it deteriorated in its fit with data. Specifically, Goldfeld and Sichel (1990) found that (1.2.2.1) systematically over-predicts the actual real balances for the late 1970s and under-predicts the actual figures for the 1980's. This represents evidence against the stability of the money demand function for the 1970's and the 1980's. Following works after Goldfeld (1973), to a large extent, were committed to the improvement of Goldfeld's money demand specification. While observing substantive financial deregulation accompanied by numerous financial innovations at the time, many researchers attempted to incorporate the impact of financial innovations into the specification of the money demand function. Since it is difficult to measure financial transaction technology directly, most researchers adopted indirect measurement for this purpose. Duesenberry (1963) argued that high interest rates create incentives to incur fixed costs necessary to introduce a new technology but, once interest rates decline, the technology remains in place. Using this idea, Goldfeld (1976) and Enzler, Johnason and Paulus (1976) added a previous peak interest rate to (1.2.2.1) to capture the effect of financial transaction technology. As suggested by Ireland (1995), these studies treated financial innovations as investment projects which are endogenous choices made by the private financial sector. Alternatively, Lieberman (1977) included the time trend to capture the effect of exogenous financial technological changes. In contrast to the previous peak variable approach and the time trend approach, Dotsey (1985) used the number of electronic fund transfers over the Fed wire as a proxy for cash management innovations. Although the inclusion of the indirect 9  measure of financial transaction technology improves the fit of money demand functions, the resulting estimates in most of these studies do not appear to be very robust, either to small changes in the specification or to the use of additional data. Porter and Offenbacher (1982) constructed a more direct measure of transaction costs —the brokerage fee proxies in the money demand specification, using Miller-Orr's setup, however, given data difficulties even the authors were cautious about pushing this approach. The single equation partial adjustment frameworks are vulnerable to the Lucas critique. This is because in this framework the money demand function is estimated in isolation, with implicit assumptions that the conduct of monetary policy (the money supply process) is exogenous to the variation of the money demand process. Nowadays the single equation partial adjustment method has been largely abandoned.  Empirical Evidence for Money Demand from the Structural Approach Lucas (1988; 2000) Lucas (1988; 2000) presented the evidence that a stable long-run relationship, though not exactly a conventional money demand relationship, is present, linking the logarithm of Ml, GDP and nominal interest rates for the period from 1900 to 1985, with an income elasticity estimate close to 1 and an interest elasticity estimate close to 0.5. He supported earlier findings of Meltzer (1963) that a stable long-run money demand function has an income elasticity of around unity. Moreover, Lucas argued that a stable long-run money demand function should be theoretically defined as the long-run balanced equilibrium relationship in a general equilibrium model, rather than a traditional money demand relationship. Specifically, Lucas (2000) used a McCallumGoodfriend shopping time model to derive such a long-run money demand function whose interest and income elasticity are consistent with the empirical estimates. By calibrating the model, he showed that the elasticity estimates require the assumption of linear shopping technology.  A Simple Shopping Time Model The household's objective function depends on goods consumption C t only:  10  ^  `.*^C^1-G • I3 j E, 11-j^> O.^ J  =^1— a  (1.2.2.2)  In period t, each household has one unit of time endowment, which can be allocated either to goods production or to goods transaction. Let g t denote the fraction of time committed to goods transaction. The goods production technology is: C t =(1—g t )y t .^  (1.2.2.3)  The cash constraint expressed in real terms is : ( 1 + nt+i)mt+i =m t +J t +(1—g t )y t —C t .^  (1.2.2.4)  In (1.2.2.4), nt+1=Pt+i/Pt, where P t is the time t aggregate price level; and J t is the government money transfer in real terms. Lucas assumed that the technology to carry out transactions is of the following form: C, = m t F(g t ).^  (1.2.2.5)  where F(.) specifies the shopping technology which only depends on the fraction of time devoted to transacting, g t . Substituted (1.2.2.3) into (1.2.2.5), the velocity of money, Z t defined as Y t/m t , is given by : (1.2.2.6)  Z = “g t ) . 1— g  (1.2.2.6) indicates that Z, only depends on the fraction of goods transaction time, g t , but does not depend on the real income y t ; therefore, a quantity exchange equation suggests that the real balances m t is homogeneous of degree one in the real income y t . Hence, (1.2.2.6) is consistent with a unit income elasticity of money. Along the balanced growth path, the inter-temporal Euler equation suggests the following condition: 3 F(g)^i  (1.2.2.7)  F'(g)^Z •  3  I drop the time script t , as real allocations except consumption are time independent along a balanced growth path.  11  Given the nominal interest rate i, (1.2.2.6) and (1.2.2.7) together determine the long-run balanced growth equilibrium solutions for the model. Lucas argued that when the shopping technology is linear, i.e. F(g)=kg, where k is a constant, and k is large relative to the nominal interest rate i, solving (1.2.2.6) and (1.2.2.7) yields approximately the following square root formula: m 1 \I 1 Y Z ik  (1.2.2.8)  Equation (1.2.2.8) predicts that the steady state interest elasticity of long-run money demand is 0.5. In his calibration exercise, Lucas (2000) selected k so that the money to income ratio curve (1.2.2.8) passes through the geometric means of the data pairs of nominal interest rates and the money to income ratio. For the data period from 1900-1994, the value of k is approximately 400. Empirical Evidence for Money Demand from the Cointegration Approach Nowadays the cointegration approach is the main-stream approach for testing the stability of money demand. The idea of the cointegration approach is that, although money, income and interest rates are each found to be integrated by order one in the data, if these variables are cointegrated, then a stationary process could be thought of as a long-run equilibrium relationship among these variables. In this case, the task of characterizing the long-run equilibrium relationship becomes the problem of estimating cointegrating vectors. The basic model has an error correction representation which is given by: log m r — log m r _, = c o (log y r — log y,_, ) — bo (log m,_, — e log y r _, — e, log i r _, ) + d o (log i t —^), (1.2.2.9) = log m t _, — e y log^— e,log^.^  (1.2.2.10)  In (1.2.2.9), e y denotes the income elasticity, and e, denotes the interest elasticity of money. In (1.2.2.10), Q t _i denotes the last period deviation of money from its long-run monetary equilibrium. When Q t _i =0, (1.2.2.10) describes the long-run monetary equilibrium relation. If one does not expect this relationship to hold in every period, then at least Q t which is called the error correction term should be stationary. In this case (1.2.2.9) describes the short-run adjustment 12  dynamics of the model. If mt ,  yt  , and i t are not cointegrated (a linear combination is not  stationary), then they do not satisfy a long-run relationship. Unlike the structural approach that imposes theoretical restrictions on the key parameter values of money demand, the cointegration approach not only can identify the existence of a long run stable money demand relationship, but also can estimate these key parameters of money demand directly from data. Using the cointegration approach, Hafer and Jansen (1991) and Breuer and Lippert (1996) found that US aggregate money demand (M1 or M2) is not cointegrated with real income and longterm interest rates. Using short-term interest rates, Hafter and Jansen (1991) found that the long run money demand relationship is empirically supported by M2 but not for Ml. Friedman and Kuttner (1992) adopted the cointegration approach and found that a structural change is seen in the long-run behavior of the money demand equation, when the period 1970 to 1990 is considered; i.e. the cointergration relationship is satisfied for the period from 1970 to 1979, but it disappears for periods after the 1980's. Stock and Watson (1993) applied the cointegration approach to US long-run money demand. Based on various constructed estimation methods, they obtained estimates that were close to Lucas' results, and yet, also argued that the longer span of the data set (1900 to 1987) is necessary for the parameter estimates of money demand to be precise and that postwar data alone (1945-1987) yields imprecise estimates due to the low frequency of data and the colinearity between income and interest rates. Ball (2001) made another contribution to this line of research by arguing that extending the data period to 1996 yields the precise parameter estimates of money demand. Using the same method as Stock and Watson (1993), Ball estimated a linear relationship between the logarithm of M1 and GDP, and the nominal interests. He found that postwar data (1946-1996) suggested an income elasticity estimate of 0.5 and an interest semi-elasticity estimate of 0.05. 4 Both of these values are only half the size of the previous estimates by Lucas (1988; 2000) and Stock and 4  ^i nterest nterest elasticity of 0.5 is approximately equal to the interest semi-interest elasticity of 0.1.  13  Watson (1993). The significant difference between the prewar and postwar parameter estimates of money demand led Ball to believe that the stability of money demand is no longer satisfied. Yet, Ball (2001) also found that a trend in the transaction technology and low income elasticity are nearly observationally-equivalent interpretations of US postwar M1 money demand data for the period from 1946 to 1994. Even when Ball imposed the time trend and unitary income elasticity, the estimate of semi-interest elasticity is still 0.05, and thus the unique money demand function is not supported as well. Finally, Ball argued that the major problem of testing the time trend in a money demand equation is that a trend associated with financial transaction technology is highly collinear with income, making it difficult to disentangle their effects. Motivating This Thesis The theory of money demand had been one of the central topics in monetary economics and in the analysis of monetary policy especially in 1970's and 1980's. For a number of years, a stable long-run aggregate demand for real balance has been the central proposition and an important building block for many theoretical models, including monetarist models [Friedman (1956)], New Classical models [Sargent and Wallace (1975)], New Keynesian analyses [Mankiw (1991)] and the monetary approach to exchange-rate determination [Dornbusch (1976)]. Nowadays, however, the theory of money demand has been losing its popularity. The main reason is the empirical finding of an unstable money demand function beginning from 1980's , which not only presents a significant challenge to existing money demand theories, but also inhibits monetary authorities from implementing a monetary target policy to maintain a stable price target. A theory without empirical support might be outdated; however, there is the other possibility that the theory itself needs to be advanced to accommodate the empirical findings. This thesis takes the second option. Among many existing theoretical money demand frameworks, the one adopted in this thesis is the inventory theoretic approach because of two reasons. Firstly, the cause of the instability of money demand is traditionally attributed to financial deregulation and numerous financial innovations occurred in 1980's and 1990's. The impact of financial innovations is to lower financial transaction costs. In this aspect, the inventory theoretic 14  approach provides a natural framework to begin with, and to study the relationship between the transaction cost and the demand for money. Secondly, following the tradition of Lucas (1988) and (2000), it is important to rely on a general equilibrium model that recognizes the endogeneity of the income velocity of money. In this aspect, despite its theoretical importance in monetary economics, the inventory theoretic approach has been unsuccessful, as both explained by the previous literature review and pointed out by Lucas (2000): the construction of an explicit general equilibrium model in which agents solve Baumol-like cash management problems has not been carried out. Hence, this thesis is devoted to the general equilibrium theory of inventory approach and its empirical application to the issue on the instability of money demand. Finally, I want to point out that, following the tradition in Lucas (1988, 2000), in this thesis I call the general equilibrium relationship between real income, real balances and interest rates a `demand function for money'. This seems to be a violation of the traditional definition of the Marshallian money demand function. However, money demand functions are estimated on data that are generated in a general equilibrium world, and are therefore the empirical counterpart to the theoretical construct that I am calling a 'demand function for money'.  1.3 Thesis Overview The main theoretical contribution of this thesis is to develop a general equilibrium inventory theoretic framework that can account for long-run money demand. The point of this exercise is to show that in general equilibrium, the elasticity results from partial equilibrium framework no longer hold. Specifically, Chapter 2 applies the inventory approach into a new partial equilibrium framework. Compared with the B-T framework, this partial equilibrium framework has richer micro-founded structures. In the B-T framework, one type of agent—the consumer has a cash management problem in choosing the timing of cash transfers, to minimize the total interest forgone on money holdings and total fixed brokerage fees. By contrast, in the model I present here, two types of agents -- the consumer and the firm-each must solve their own cash management problem. By allowing the firm to choose the frequency of wage payments, this model endogenizes not only 15  the timing of cash transfers of the consumer and the firm, but also their timing of wage transfers, I show that when the firm must borrow working capital and pay wages by cheque, the cash holding periods of the firm and the household coincide, i.e. both agents make cash and wage transfers (withdrawals and deposits) only every payday, as they have the symmetric marginal cost structures of cash management. Their cash holding period balances marginal interest forgone on cash holdings and wage transfers with the fixed money transaction cost. In terms of the demand for money, I show that along the Baumol-Tobin intensive margin, this partial equilibrium framework keeps the basic insight of the B-T framework that both the income and interest elasticity of money demand is 0.5. Chapter 3 extends this partial equilibrium model into a general equilibrium model, where the fixed money transaction cost is measured by the unit of the broker's hours of work. I analyze a structural money demand function that links real income, real balances and nominal interest rates in general equilibrium. Following Lucas (2000), such a money demand function, which can be used to describe a long-run theoretical equilibrium relationship, is therefore more suitable for studying long-run money demand, compared to the use of a conventional money demand function. In terms of elasticity implications, first, I show that in the general equilibrium setting, when real income, the interest rate and the wage rate are endogenous, the income elasticity is no longer half, but unity. Second, the interest elasticity of money demand can have two values depending on interest rates. When the interest rates are below a threshold value, the model becomes the Cash-In-Advance (CIA) model where the income velocity of money is constant, and the interest elasticity of money demand is zero; otherwise the velocity fluctuates in response to variations in the interest rates, and the interest elasticity largely obeys the Baumol-Tobin intensive margin, with small deviation from 0.5 caused by the general equilibrium effects of inflation. I also demonstrate that these inflation effects are quantitatively small. Finally, I show that, in contrast to the results of Romer (1986), the general equilibrium elasticity results in my model are unaffected by alternative specifications of the agent's utility, suggesting that these elasticity results are mainly derived from the agent's optimal cash management behavior. Chapter 4 uses this general equilibrium model as a basis for a calibration exercise. The contribution of this calibration exercise is to provide a unified framework that can account for 16  the last 40 years of US long-run money demand for M 1. A benefit of this approach is that it is able to explain this pattern of money demand without appealing to an exogenous structural break in the money demand function as was the case in Ball (2001). To show this in specific terms, Chapter 4 makes use of a structural money demand function which describes a general equilibrium relationship in the model in Chapter 3. Because of the robustness of the elasticity results, and the functional form of the structural money demand relationship, I can use this money demand function to construct the empirical 'residual' measure of money transaction costs (or financial transaction technology) in a way similar to how others construct the measure of total factor productivity—the Solow residual, which is the part of growth that cannot be explained through capital accumulation. The money transaction costs are defined as the unexplained part of money (M1) through income (GDP) and nominal interest rates. I then analyze the time series behavior of long run money transaction costs with an autoregressive process and a linear time trend. First, I find a structural change in the money transaction costs occurring in 1981; for the period from 1981 to 2000, a constant fall in the trend value of money transaction costs was seen at a quarterly rate of 1.37%, indicating systematic and exogenous shifts might have occurred in the long-run money demand relationship; since little variation in the money transaction costs is seen, no exogenous shifts occur in the long-run money demand for the period from 1960 to 1981. Second, I show that, given the identified structural change, the structural money demand function can successfully track actual demand data for Ml. The finding that a unique long-run money demand function is still present raises the question whether it is time to reconsider the re-adoption of the monetary target policy. A stable long-run money demand function is a necessary condition for implementing the monetary target policy, yet it is not a sufficient condition. This is because the stable long-run demand function cannot explain the high volatility of short-run money velocity observed in US data. This could violate the policy goal of price stabilization by the use of this kind of policy, as the high volatility of velocity would transmit to the nominal sector according to a quantity exchange equation, with a k-percent constant money growth rule. Hence, I believe that in an environment with high volatility of velocity, the sufficient and necessary condition for the use of monetary target policy is both long-run and short-run stable money demand functions that fit the data closely. 17  Chapter 2: The Frequency of Pay-Periods and Transactions Demand for Money 2.1 Introduction Agents hold transaction balances for cash to bridge the time gaps between income receipts and expenditure. The degree of synchronization between income receipts and expenditure influences the amount of cash holdings needed for a given volume of goods transactions. In most macroeconomic literature, the frequency of income receipts is fixed. In this chapter, I develop an inventory money demand model, which allows the frequency of income receipts to be endogenous, to study the transactions demand for cash. The model developed here is closely related to the classical partial equilibrium models of cash management such as Baumol (1952), Tobin (1956) and Miller (1966). Baumol (1952) and Tobin (1956) studied the consumer's cash management problem under a discrete and continuous time settings, respectively; Miller (1966) carried out the analysis of the firm's cash management problem. By contrast, this paper investigates the household's and the firm's cash management decisions jointly; i.e. how their decisions endogenously affect each other. The chapter provides an alternative partial equilibrium inventory framework to study the transactions demand for money. This framework will serve as the basis for developing a general equilibrium model of inventory money demand in the next chapter. The key agents in the economy are a household and a firm. The household supplies a large number of workers to the firm which produces non-storable consumption goods. The consumption goods can only be purchased by cash. The household consumes the consumption goods at a constant rate. Hence, the goods must be produced and sold at the same rate every period. There are two types of financial assets: bank deposits (CDs) and cash. The bank deposits earn daily interest, while cash does not. A fixed brokerage fee is charged for every cash transfer except at the beginning and the end of the period. The brokerage fee, which is independent of the  18  size of the cash transfer and whether the transfer is a deposit or/and a withdrawal, is paid to the bank at the end of every period. One of the most important features in the model is that the firm endogenously chooses the number of pay periods (paydays) within a fixed time, 'the period'. Each pay period, defined as the interval between two successive paydays, is a subset of the period and of equal length; 5 e.g. if the workers are paid weekly, the paydays are every Monday. Two key assumptions ensure the firm to choose the frequency of pay periods endogenously. The first assumption is that the firm faces a wage in advance constraint in every pay period. It means that the firm must pay workers in advance every payday before the subsequent production begins. This assumption follows Christiano and Eichenbaum (1995)'s idea that the firm must borrow working capital in advance, but in their model there is only one payday which is the first day of the period, so the firm must borrow a large amount of wage loans before it collects revenue from goods sales. The interest cost of the wage loans directly affects the firm's marginal cost of production. By contrast, in my model, by increasing the frequency of pay periods, the firm is able to reduce the size of loans and thereby the interest of the loans. Since the wage loans are costly and the firm's sales revenue accrues continuously during the period, the firm will only borrow for the initial pay period and subsequent wage payments can be financed by the ongoing cash flow. This assumption implies that the firm has the incentive to pay workers as frequently as possible so as to reduce the interest cost of the loans. To develop the tradeoff in the firm's decision on the number of pay periods, I introduce costs of wage payments, which lead to the second assumption that the firm pays wages by cheque. This assumption is consistent with the empirical observation that the payroll cheque is one of the most common used means of income payment in U.S. 6 This presumption can be justified by assuming that a firm who is of large scale faces a large cost when paying wages in cash. That is, when the firm who hires a large number of workers pays wages in cash, it must employ a costly  5  The equal length of pay periods in the period is imposed for the technical tractability of the model. More discussion on this assumption will be given in later paragraphs. 6 Recently the physical paycheck has been increasingly replaced by electronic direct deposit.  19  technology that can be thought to either verify that the recipients of cash payment are the actual workers, or secure a large amount of cash for making wage payments. To avoid this large cost of cash payments, the firm could simply deposit all cash into a bank by paying a small brokerage fee, and then write a paycheck to each worker. With the fixed brokerage fee, the firm has the incentive to pay workers infrequently, since the more frequently the firm pays workers, the more fixed brokerage fees it must pay. The theoretical finding with respect to the agents' cash management problems is that both the firm and the household choose to transfer cash only every payday. The household's cash management decision coincides with that of the firm; this is because they have the symmetric marginal cost structures of cash management. Baumol (1952) showed that with two sorts of costs of cash management- the interest forgone due to holding cash for purchasing consumption goods and the fixed money transaction cost, the consumer chooses the cash holding period to marginally balance the one cost with the other. This sort of behavior leads to the famous square root formula, which predicts that the elasticity of money is -0.5 with respect to the interest rate and 0.5 with respect to income. In my model, besides these two types of costs, agents confront another type of cost of cash management, which results from the friction in wage transfers. Specifically, since the firm must make wage payments every payday and cannot withhold them till the end of the period, it forgoes the interest cost on wage transfers; similarly, the household receives their wage payments on separate paydays, unlike the agent in the Cash-in-Advance model [as the one in Christiano and Eichenbaum (1995)] who receives and deposits all wages at the beginning of the period; so the household in this world suffers from additional interest cost due to delay in their wage receipts. In comparison to the conclusion of Baumol (1952), this model shows that, to minimize the total costs of cash management, both the household and the firm choose the cash holding period to balance the total marginal interest forgone on cash transfers and on wage transfers to the fixed brokerage fee.' Moreover, this model keeps the basic insight of Baumol (1952); i.e. along the intensive cash management margin, the income elasticity is 0.5 and the interest elasticity is -0.5.  I assume that the frequency of pay periods and the number of cash transactions are continuous variables. This is in line with Baumol (1952).  20  The remainder of this chapter is organized as follows. In section 2 and 3, I analyze the optimal cash management decisions of the firm and the household in turn. In section 4, the properties of money demand are analyzed. Section 5 concludes.  2.2 The Firm's Cash Management Problem In this section, I define the firm's costs of cash management first, and then investigate the firm's optimal cash management decisions, i.e. the frequency of wage payments, the timing and the size of cash transfers. 8  The Firm's Costs of Cash Management Within a fixed time period, of which the length is normalized to 1, a firm collects a steady stream of cash revenue from goods sales. The firm is assumed to pay wage bills on the first day of every pay period, of which the length is identical. Let W denote the total amount of wage payments within the period and Z denote the number of paydays in the period. Since goods are non-storable and the household purchases them at a constant rate, the rate of cash revenue must be equal to the rate of goods production. With identical length of pay periods, it follows that the firm must hire equal amount of labor during every pay period, so the wage bills paid every payday amount to W/Z. With the first payday starting on the first day of the period, the firm borrows wage loans from a bank for the initial pay period, and repays the interest and loans to the bank at the end of the period; subsequent wages can be financed by the firm's ongoing cash flow throughout the period. The loan market is assumed to open only on the first payday. On one hand, unless the accumulated cash revenue is converted to bank deposits, the firm forgoes interest on the cash holdings. One the other hand, each conversion costs the firm a fixed brokerage fee. Here, these two types of cost of cash management are defined in the same way as  8 Since the framework developed here is a partial equilibrium model of money, the structure of the goods market will not be specifically modeled in this chapter.  21  those in Baumol (1952). Besides these two sorts of costs of cash management, the firm also sacrifices another type of cost of cash management--the interest cost given up for paying wages. This cost arises because this model assumes that the firm cannot withhold workers' wage payments to the end of the period. In Christiano and Eichenbaum (1995), since the firm pays workers full amount of wages all at once at the beginning of the period, the interest cost on wage payments (the initial borrowing costs of wage loans) reaches the maximum; while in this model, workers periodically withdraw their wages out of the firm's bank deposits; the more frequently the firm pays workers, the smaller are the size of every wage withdrawal and the smaller the interest cost on wage payments. I define each of these three types of costs of cash management specifically in the following paragraphs. Interest Forgone on Cash Holdings Let T denote the total amount of cash revenue that the firm collects during the period, f-1 denote the total number of cash deposits, hk denote the cash holding period for the kth deposit (k=1,... f1) and hf denote the cash holding period for the period end sales revenue, which the firm will not deposit. The following constraint must hold. h = 1.^  (2.2.1)  k  k=1  With a positive interest rate i, the firm deposits all the cash holdings into the bank during each visit to the bank. The amount of the kth cash deposit is hk T, which earns interest for (1– ^h i ) .=,  time period long. Thus, the firm's total interest returns on cash deposits can be written as follows: f t^k^  Ti/ [h (1 k  —  ft  Ti –[TiE(h Eh ) I h i )] = — 2^  k=1^i=1^  k=1^i=1  Ti  (2.2.2)  2  In (2.2.2), the terms in the bracket represent the firm's interest forgone on its cash holdings, and when the firm makes cash transfers infinitely frequently, its interest returns from cash revenue reach the maximum value of Ti/2. Interest Cost on Wage Payments Given that workers redeem their wage cheques every payday, the firm has to forgo a portion of interest cost for paying wages, which is calculated as the accumulated sum of the interest cost on 22  each wage payment. Wi^Z –1 k 1Wi Wi [1+ ^ +.—..+— ]=—+— Z 2 2Z . Z^ Z Z  (2.2.3)  In (2.2.3), k/Z is the interest cost per dollar on the kth wage payment and 1/Z is the length of one pay period. Here, the interest cost on wage payments depends merely on the number of pay periods Z, but does not depend on the number of cash deposits (f-1) or the timing of cash deposits {111,...hf}. (2.2.3) suggests that when the firm makes wage payments infinitely often, the interest cost on wage payments reaches the minimum value of Wi/2; when Z=1, the interest costs reach the maximum equal to Wi.  Besides the interest costs given up for both holding cash and making wage payments, the firm must also pay the brokerage fees, which are equal to (f-1) B in total. Therefore, the firm's total costs of cash management, which are the sum of these three sorts of costs of cash management, are given by Ti Wi +(f –1)B.–(— – ) k=1^1,1^2Z^2^2  (2.2.4)  k  The firm's problem is to choose a sequence of {h 1 ,...hf}, Z and f to minimize its total costs of cash management. I will derive the firm's optimal cash management decisions in two steps. In the first step, the firm chooses the optimal timing of cash deposits {h 1 ,...h f } to minimize its interest forgone on cash holdings, given the number of cash deposits f-1. In the second step, I will determine the firm's optimal number of cash deposits, f-1, and wage payments Z, given the optimal timing decisions of {h1,• . • ,hf} •  Step 1: Determine the Timing of Cash Deposits In this step, the firm's cash management problem is to choose a sequence of {h 1 , h 2 ,..., hf} to minimize its interest forgone on cash holdings, as expressed by (2.2.2), subject to the constraint (2.2.1), given {i, T, W , B}.  The first order condition with respect to hk implies the following result:  23  1 = —Vk =1,....f.  (2.2.5)  (2.2.5) states that the firm equalizes the cash holding periods across cash deposits. The minimized total interest forgone on cash holdings is calculated by substituting (2.2.5) into (2.2.2). r  .„.„ f —1^1, Ti^Ti .^ f^f^2 2f  (2.2.6)  Step 2: Determine the Number of Cash Deposits and Pay Periods In this step, the firm's problem is to choose Z and f to minimize its costs of cash management, taking I i, T, W, B1 as given. Substitute (2.2.6) into (2.2.4), and rewrite the firm's objective cost function of cash management Cf(f, Z) as a function of Z and f : min zf Cf (Z) = .  W(1+ Z) .^.  1+ (f — 1)B + — T—1 . 2Z^2f  (2.2.7)  Since the firm only needs to borrow for one pay period, each pay period is of identical interval, and the firm makes cash deposits across identical intervals, it follows that the number of cash deposits must be at least as great as the number of pay periods; otherwise the firm is unable to pay workers on time on the second payday. This implies the following constraint: (2.2.8)  f > Z >1 .^ The first order conditions with respect to f and Z yield the following result: f = Z .^  (2.2.9)  (2.2.9) states that the firm equates the number of pay periods to the number of cash deposits. The intuition for this result is very simple: With any given number of cash deposits f , the firm's interest forgone on cash holdings and the total brokerage fees are unaffected by the firm's decision on the number of pay periods Z; but the more frequently the firm pays workers, the less interest cost on wage payments is incurred, and thus it is optimal for the firm to set the number of pay periods to the maximum [Z= f] to lower the third type of cost of cash management—the interest cost on wage payments to the minimum. Substitute (2.2.9) into (2.2.7) and rewrite the firm's objective cost function of cash management as follows: W1± Z) T Cf (Z) =(1±(Z-1)B F -'-  (2.2.10)  2Z^2Z  24  The firm's marginal cost of cash management MCf (Z*) is defined as the additional change in Cf (Z)  when one more cash deposit is made.  ^MC f Z)  -  ac f (z) az  2Z2  1  2Z 2  ^(2.2.11)  1.  According to (2.2.11), the effect of adding one more cash deposit on Cf(Z) is to increase the brokerage fee payment by B, reduce the interest cost on wage payments by Wi/(2Z 2 ) and reduce the interest forgone on cash holdings by Ti/(2Z 2 ). When MCf(Z)>O, the firm can reduce the costs of cash management by cutting down the number of trips to the bank, and vice versa. When MCf(Z)=0, the firm cannot further reduce its costs of cash management, it follows: Z* 2 = W±T 2B i•  (2.2.12)  The result of (2.2.12) is illustrated in Figure 2.1. The marginal interest forgone, which is represented by the curve IC, is a decreasing function of the number of pay periods Z, but the marginal fixed cost, which is described by a horizontal line-- FC, is fixed at a constant--B. When MCf(Z)=0, the two curves intersects at Z=Z* and the firm minimizes its total costs of cash management. Figure 2.1 Firm's Marginal Cost of Cash Management •^IC'^IC  ^+T)  ^(W +T) ^ 2Z 2^2Z2  B^  FC  ^Z Z=1^Z*^Number of cash deposits Throughout the previous analysis, the firm takes the interest rate i as given. According to (2.2.12), the firm's interest forgone depends positively on the interest rate, while the fixed brokerage fee does not. As shown in Figure 2.1, when the nominal interest rate is small enough, the marginal interest forgone curve IC' might intersect with the FC curve at the point where Z<1. However, in this model, the minimum number of pay periods is Z=1 in Figure 2.1, where the marginal interest forgone is less than the fixed brokerage fee; as a result, the firm will not make 25  any cash deposits during the period and the number of pay periods Z takes the corner solution. Define i as the threshold interest rate at which IC' intersects with FC at Z*=1. At i, the firm is indifferent between making a cash deposit and not doing so. It is equal to: 2B i= ^ T+W  (2.2.13)  I now focus on the interior solutions for Z* when i>i. The firm's optimal cash management decisions are stated in Proposition (2.1). Proposition (2.1): For a sufficiently large interest rate i>i, the firm's total costs of cash T  management are minimized when it deposits an equal amount of money y across identical intervals 7 each payday; the number of pay periods Z is given by  z  (T+ W)i  (2.2.14)  2B  According to Proposition (2.1), the firm's cash holding displays a "saw-toothed" pattern as shown in Figure 2.2. Its cash holding accrues until reaching T/Z and then declines to zero after every payday. Figure 212 Saw-toothed Money Holdings of the Firm  T  1 st^2nd ^ Zth pay day  Discussion  To simplify the analysis of the firm's cash management problem, I have assumed that (i) the pay periods are of equal length in the period; (ii) the firm needs to repay the wage loans and interest at the end of the period and (iii) the loan market opens on the first payday only. Here, I will  26  briefly discuss the relevance of these assumptions to the firm's optimal cash management decision. On one hand, the assumption (i) is crucial for deriving the result in Proposition (2.1). As suggested by that Proposition, given the symmetric length of each pay period and the constant rate of cash revenue, the firm's optimal timing of cash deposits is symmetric too. However, the firm's cost structure of cash management is asymmetric across time, since the wage loan is only needed for the initial pay period, but not for subsequent pay periods. If the firm can freely choose the length of each pay period besides the timing and the number of cash deposits, the firm's optimal timing of cash deposits will be asymmetric too. In that case, the model would be too complicated to solve. Hence, I impose this assumption for the technical tractability of the model. On the other hand, the assumptions (ii) and (iii) are not mandatory to drive the final result. If assumption (ii) is not imposed, i.e. the firm is allowed to repay the interest and loans during the period, the firm will still make cash deposits only on every payday. The reason is that in the model the interest of loans is identical to the interest rate on savings (no interest spread), so the firm's decision to maximize the interest returns on savings can be alternatively understood as the equivalent way to minimize the interest costs of wage loans. If the assumption (iii) is not imposed, i.e. if the loan market remains open during the period, one possibility for the firm is to borrow wage loans on subsequent paydays to finance the wage bills. The firm might benefit from this option and alter its cash management decision, especially when the interest rate is low and the firm is unwilling to incur the fixed brokerage fee. In this case, I need another assumption that the firm must pay the same brokerage fee for borrowing each amount of wage loans, in order to derive the result in Proposition (2.1). With this assumption, the firm always prefers depositing its cash flow for subsequent wage payments to borrowing loans; otherwise the firm suffers from additional interest forgone on cash holdings (since the firm does not deposit its cash revenue), while paying the same interest costs and brokerage fees. I think the assumption that there is a brokerage fee for borrowing is reasonable, as mentioned by Baumol (1952): the brokerage fee which is not meant to be taken literally covers all non-interest costs of borrowing or making one cash withdrawal.  27  2.3 The Household's Cash Management Problem The representative household is endowed with the initial nominal wealth M, which will be divided into its initial cash holding, denoted by I' , and initial bank deposits , denoted by (M-I 1 ), at the beginning of period. The household spends its cash on consumption at a constant rate. The amount of its consumption expenditure is equal to the firm's sales revenue T. When the household exhausts its cash holding, it will make cash withdrawals subject to paying a fixed brokerage fee. The household, which works for the firm, takes the number of pay periods Z as given and receives a wage cheque from the firm every payday. The wage cheque can not be used to purchase goods, as goods sellers have no ways of ensuring that the writers of cheques have the funds to match them, and thus the household must either cash it or deposit it at a bank, subject to paying the same fixed brokerage fee. The bank, however, has the role of verifying a payment by cheque. The household chooses the timing and the number of cash withdrawals (either from its own savings or from the wage cheque). Its optimal cash management decisions are summed up in three propositions, which are briefly reviewed here. Proposition (2.2) and (2.3) demonstrate the household's optimal timing decision of cash withdrawals and Proposition (2.4) summarizes the household's optimal number and timing of cash withdrawals. In particular, Proposition (2.2) implies that the household's cash holding period is no shorter than every pay period. This is because even though the household can reduce the interest forgone on cash holdings with a shorter cash holding period than the pay period, it pays more brokerage fees than the interest savings. Proposition (2.3) indicates that the household makes cash withdrawals symmetrically across payday(s), and Proposition (2.4) then concludes that just like the firm, the household makes cash withdrawals only every payday. To gain some intuition on Proposition (2.4), it is easier to consider a scenario in which the household will always deposit its wage cheque into the bank and always withdraw cash from its own savings only. As in the standard Baumol-Tobin cash management environment, the household's decision on cash holding periods affects its total interest forgone on cash holdings. 28  In addition, the household must also decide how often to deposit its wage cheque(s) into the bank. The frequency of wage deposits affects the interest cost on wage receipts. If the household makes payday visits only to the bank, it can bundle the cash withdrawal and the cheque deposit together, yet with one payment of brokerage fee only, as I assumed here that the bank only charges a client the brokerage fee once every time when he accesses his own bank account, regardless whether it is a deposit or/and a withdrawal, and regardless of the size of the cash transfer. Moreover, the more payday visits the household makes, the less interest forgone on both its cash holdings and wage withdrawals incurred, but the more brokerage fees paid. The household faces exactly the same trade off as the firm does in its cash management problem, though they make their cash management decisions independently. As a result, when the household's cash holding period is equal to the time length of the pay period, it also equates the marginal benefit of making an additional cash transfer to the marginal cost—the fixed brokerage fee. Just like the firm's cash management problem, the household's cash management problem can be solved in two steps. First, given a fixed number of cash withdrawals, the household chooses the optimal timing of cash withdrawals, and then given the optimal timing decision, the household determines the optimal number of cash withdrawals and the optimal cash holding periods. Determine the Timing of Cash Withdrawals Proposition (2.2): The household makes no inter-pay-period cash withdrawals, given the firm's optimal frequency of pay periods Z*.  Proof: Suppose that, on the contrary, the household makes cash withdrawals within the nth pay period (n=1,or,2...,or Z). First, I rule out the household's "lazy" choices that it does not redeem the wage cheque on some payday, but does so later in the same pay period instead. Note that I do not refer to "some payday" as the first payday when the household can freely convert cash. It is obvious that the rational household will not choose this kind of behavior, as there is no additional benefit to the household except incurring the unnecessary interest cost on the wage withdrawals.  29  Next, if the household makes cash withdrawals during the nth pay period, we can conclude that without this kind of lazy choices, the rational household must have made cash withdrawals on the nth pay day as well. In addition, since every pay period is symmetric and the household spends cash at a constant rate, the household's optimal cash management behavior must recur within each pay period. This indicates that the household must visit the bank not only within each pay period but also every payday. Within the nth pay period, let us consider the household's marginal decision of making an additional inter-pay-period cash withdrawal. Figure 2.3 exemplifies the points of time when cash withdrawals occur. The nth payday is labeled as point E, and the time points of the following two inter-pay-period cash transfers are labeled as F and G, respectively. Figure 2.3 Household's Inter-Pay-Period Cash Withdrawals h^h'  ^  Ar"---111har'',116--  1  ^■  F^G  nth payday  In Figure 2.3, h denotes the interval between E and F and h' denotes the interval between F and G. Note the cash withdrawal at point G must occur on any days no later than the n+1 payday, since the rational household will not skip the cash withdrawal on the n+1 th pay day. This implies that h + h' ^  1 —  , where 1/Z is the time length of one pay period.  Let us now evaluate if it is worthwhile for the household to make the cash withdrawal at point F. The household will not incur any brokerage fees unless its cash holdings are exhausted. If the household does not make the cash withdrawal at point F, then from point E to G, the household must hold an additional Th' amount of cash for an extra h period. This amount is used to finance the consumption purchase within the interval FG. As a cost, the household must give up the extra interest forgone which is equal to Th'ih , but as a benefit, the household avoids 30  paying the fixed brokerage charge B. However, using (2.2.14), it can be shown that the following inequality must hold. Th' ih Ti  (h + h' )2  Ti  (T + W)i  ^ = B. < 4^4Z2^2Z2 <  (2.3.1)  (2.3.1) demonstrates that the extra interest forgone on cash holdings is less than the brokerage fee B, and thus the rational household will not make the cash withdrawals at point F. Q.E.D Proposition (2.2) implies that the household makes cash withdrawals on paydays only. It leads to two possibilities. That is, the household can either visit the bank every payday or on some paydays only. The second scenario could occur when the brokerage fee is too costly and the rational household might be unwilling to redeem the wage cheque as often as it does every payday. Based on the previous analysis, now let us consider that the household redeems its wage cheque according to the following plan: For a given number of cash withdrawals S, the jth visit to the bank is qj pay periods ahead of the (j+1)th visit, during which the household converts amount of bank deposits from its own savings besides its wage withdrawals from the cheque(s). Let us call the interval between the jth and j+1 th visit to the bank the qj th sub-period. It follows: ±q 2 + qs)/Z=1,q' ^ 1Vj=1,...S.,^  (2.3.2)  where q] /Z is the time length of the jth cash holding period. (2.3.2) states that the total time length of the household's cash holding periods is equal to the time length of the period--one. If qj=1, Vj, it indicates that the household visits the bank every payday; if q 3 >1, for some j, it suggests that the household skips some payday visit(s) to the bank during the q' th sub-period; in this case, S<Z. The definitions of and can be exemplified by Figure 2.4, in which the household visits the bank every two pay periods. Figure 2.4 Household's Cash Holding Periods ^q'  q2  X^qS-1  ^  q  s^  4  2 nd^3rd s Payday I S- ' ^ Is  The pattern of the household's wage withdrawals can be revealed as follows. On the first payday, the amount of wage withdrawal is equal to W/Z; on the second visit, the amount of wage 31  withdrawal is q'W/Z; the amount of wage withdrawal on the jth visit is qj -1 W/Z, Vj=2,...5. Note that when q s >l, it suggests that the household misses the firm's wage payments for the last (q s -1)/Z length of time on its last visit to the bank. There is a minus one here, because on the Sth visit, the household is able to obtain the wage payments for the next pay period, but not for the ones afterwards. As a result, the household has to postpone the wage receipts for this length of time to the end of the period. The amount of consumption expenditure during the qi sub-period is Tqj/Z. Given the pattern of the household's wage withdrawals, it follows: (2.3.3)  I' = (Tq l /Z)—W/Z,^ = (Tq' /Z)—qJ I W/Z^Vj= 2,....S.^  (2.3.4)  The household's total interest returns, which are equal to the interest returns on its initial wealth, M, less the accumulated sum of the interest forgone on every cash withdrawal, are given by S^S n i  Mi^(ELI—)A k=1^j=k Z  ,  (2.3.5)  where I q' / Z is the length of the period for which I k could earn interest otherwise, and J =k /Z is the interest forgone on the k-th cash withdrawal. pi<  Substitute out and I k in (2.3.5) using (2.3.3) and (2.3.4) respectively, and rearrange terms: „I w S '^s^nk ^W S n j Mi^ — q k -)(E11— )] Z Z^ Z Z j,kZ j=1^k=2^ Ti^S^S^wi^S^S^S 2 [(E q j)2 + Z( q j)2] ± ^ [2qj ± (E02 Z(02] . = Mi ^ 2Z^ 2Z 2 j=1^j=1^j=1 j= 1^j=1^  (2.3.6)  Determine the Timing of Cash Withdrawals In this stage, the household chooses qJ to maximize its interest earnings, as expressed by (2.3.6), subject to constraint (2.3.2). The first order condition with respect to qj is given by:  32  wi^s Ti q' +i(q )]+ ^ 2 [2+I(q )-2q1= 2Z—,Vj=1,....,S , Z2^ i=1  (2.3.7)  where 8 is the shadow price for the cash holding period constraint (2.3.2). It follows: Z q' = q = —,Vj =1,....,S .  (2.3.8)  According to (2.3.8), the household equalizes its cash holding periods across cash withdrawals. This condition ensures that given a fixed number of cash withdrawals S, the household minimizes its total interest forgone and maximizes its total interest returns. The household's optimal decision on the timing of cash withdrawals is summarized in the following proposition. Proposition (2.3): The household's cash holding periods are spanned symmetrically across paydays. Determine the Number of Cash Withdrawals  Substitute out qi terms in (2.3.7) using (2.3.8), and then subtract the total brokerage fees—(S-1)B from (2.3.7) to obtain the following objective function of cash management—Ch(S) of the household, which chooses the number of cash withdrawals S to minimize Ch(S), Min s C h (S) =  T(S +1)i  + (S 1)B +  W(S –1)i [  W  W  + (— – — )i]– Wi .  2S^2S^S Z  (2.3.9)  In (2.3.9), given S, the terms in the square brackets represent the household's interest cost on wage withdrawals. When 1/S>1/Z, the household makes cash withdrawals every q(=Z/S) paydays. The terms W(1/S-1/Z) denote the amount of wage payments that the household misses after its last visit to the bank. If 1/S=1/Z, the household redeems its wages on time every payday and these terms go to zero. The household's marginal cost of cash management MCh (S) is given by  ac (s)^W^T ^B _^2 .  (2.3.10)  MC h (S)^h  as^2S 2^2S  MC h (S) denotes the additional change in Ch (5), when the household makes one more cash withdrawal. In (2.3.10), Wi/(2S 2 ) captures the marginal reduction in the interest cost on wage withdrawals and Ti/(2S 2 ) captures the marginal reduction in the interest cost on cash holdings. If these interest cost savings are greater than the brokerage fee, i.e. MCh(S)<O, then the household 33  is able to reduce the costs of cash management by increasing the number of cash withdrawals. When MCh(S)=0, the household cannot further reduce its costs of cash management. MCh(S) has exactly the same marginal cost structure as MCf(Z). This implies that the household's optimal number of cash withdrawals S is identical to Z*, so that MCh(S)= MCf(Z)=0. This result leads to the following Proposition. Proposition (2.4): The household makes cash withdrawals only every payday. Based on Proposition (2.4), the household's cash holding displays a saw-toothed shape as shown in Figure 2.5. The household withdraws T/Z (where T/Z?W) amount of cash from the bank every payday, and its cash holding declines to zero at the end of each pay period. Figure 2.5 Saw-toothed Cash Holdings of the Household  T  2 ^  2.4 Money Demand In this economy, cash initially flows from a household to a firm through consumption purchases; every payday cash indirectly transfers to the household from the firm via a bank. In this sense, the velocity of money is identical to the number of pay periods Z. In equilibrium, the quantity exchange equation holds. Expressed in real terms, it is given by y  =  mZ  .  (2.4.1)  In (2.4.1), y(=T/P where P is the aggregate price level) denotes real income and m denotes real balances. The money demand function, expressed in real terms, is derived by substituting out Z in (2.4.1) using (2.2.14). 34  2by m = li [1+W/T]i  (2.4.2)  In (2.4.2), b is the real fixed brokerage fee measured in units of goods.  Properties of the money demand function The interest elasticity of money is defined as: the percentage change in real balances in response to a one percentage deviation in the interest rate, ceteris paribus. The income elasticity of money is defined as: the percentage change in real balances in response to a one percentage deviation in real income, ceteris paribus. The income elasticity and the interest elasticity depend upon the firm's production technology. With a constant return to scale (CRS) production function, i.e. y=aN, where a is a constant productivity measure, W/T is a function of the real wage rate. Since real income, the interest rate, the fixed brokerage fee and the real wage are mutually independent in the partial equilibrium model, changing one variable has no impact on others. This implies that the income elasticity of money is 0.5 and the interest elasticity of money is -0.5 in this model. These elasticity results are consistent with those predicted by the original (B-T) square root formula as in (1.2.1.1), though (2.4.2) has a different functional form from (1.2.1.1). am ai / — = —0.5, m i' am ay n i ^= m y  (2.4.3)  VI  2.5 Conclusion This chapter develops a partial equilibrium inventory model, where a firm and a household must pay a fixed cost to transfer cash from bank deposits. When the firm must borrow working capital in advance and make wage payments by means of cheque, and first, I show that both the firm and the household choose the same pattern of cash transfers, i.e. making cash transfers only every payday, since their marginal cost structures of cash management are identical. Their cash holding periods balance the marginal interest forgone on cash holdings and wage transfers with the fixed cost. Second, I show that, along the Baumol-Tobin intensive cash management margin, the income elasticity of money demand is 0.5 and the interest elasticity of money demand is -0.5. 35  Chapter 3: A General Equilibrium Model of Baumol-Tobin Money Demand 3.1 Introduction What properties should a money demand function possess? In his influential essays, Lucas (1988; 2000) argued that a long-run money demand relationship is characterized by an interest elasticity close to 0.5 and an income elasticity close to 1, according to annual US money demand data for the last century. The interest elasticity of 0.5 is reminiscent of the celebrated inventory-theoretic studies of money demand such as those by Baumol (1952), Tobin (1956) and Miller and On (1966). These studies, which investigated the agent's cash management problem in partial equilibrium frameworks, serve as the corner stone for many existing money demand frameworks. Despite the interest elasticity of 0.5, however, these earlier inventory studies predict that the income elasticity of money demand is 0.5, not 1. The partial equilibrium analysis is not enough to study monetary equilibrium, as this approach implicitly assumes that the conduct of monetary policy is exogenous to the first order effects of a shift in the money demand curve. More importantly, owing to lacking of microeconomics foundation, the partial equilibrium analysis cannot explore the interactions between prices and real allocations; hence it is important to examine the implication of the inventory theoretic approach to the demand for money in a general equilibrium setting. Specifically, the elasticity results of the classical inventory theoretic approach were built on the assumption that real income, the interest rate, and the cost of cash transfers are mutually independent. In a general equilibrium model, however, this assumption does not apply, since real income and the interest rate may be determined jointly. In this chapter, I develop a general equilibrium model of inventory money demand, where agents must pay a fixed cost to exchange cash with bank deposits. The purpose for developing this model is to show that the partial equilibrium elasticity results do not hold in the general equilibrium setting.  36  The model is an extension of the partial equilibrium model of cash management that I developed in the second chapter. Both models borrow a majority of their structure from the models of Christiano and Eichenbaum (1995) and Baumol (1952). The model economy is populated with infinitely lived identical households, a continuum of monopolistic competitive final goods firms, a monetary authority, and a competitive bank. As in the model of Christiano and Eichenbaum (1995), the bank initially combines funds received from the households with lump-sum injections of money from the monetary authority and makes loans to the firms. These firms need the loans because they must pay wages before they sell their output. Households supply labor to both the firms and to the bank. As in the model of Baumol (1952), the consumption goods are non-storable and have to be purchased by cash. Every period, households make consumption purchases at a constant rate and consequently the firms produce the goods at the same rate. Two types of financial assets are present: bank deposits (CDs) and cash. The bank deposits earn daily interest, while cash does not. Agents must pay a fixed brokerage fee each time to transfer cash, except at the beginning of every period. 9 The brokerage fee is independent of the size of the cash transfer and whether the transfer is a deposit and/or a withdrawal. The brokerage fees are paid to the bank at the end of every period. As in the model in the previous chapter, this model allows the firms to endogenously choose the frequency of pay periods within a fixed length of time - the period. In the standard CIA model, such as described in Christiano and Eichenbaum (1995), money initially flows from households to firms through goods sales, and then circulates back to the households through the income payment made by the firms. Because the frequency of pay periods is unchangeable and fixed at one, the income velocity of money is fixed, too, which leads to zero interest elasticity in the money holding period. This implication is inconsistent with the empirical evidence that the demand for money is interest elastic. In my model, however, as the frequency of pay periods is endogenous, the income velocity of money can vary and thus, the cash holding period is sensitive to changes in the interest rate. Two key assumptions are used to motivate the idea of the variable frequency of pay periods. The first assumption is that firms must borrow working capital in advance. As explained earlier, with  9  This assumption permits the derivation of the Cash-In-Advance model as a special case.  37  this assumption, firms have the incentive to pay workers as frequently as possible to reduce the loan interest to zero. The second assumption is that the firms pay wages by cheque. Based on this assumption, the firms have the incentive to pay workers only infrequently, as the firm must pay a fixed cost to deposit cash on time for the workers to cash their wage cheques. In the general equilibrium model, I first show that the firms' optimal cash management decision is the same as in the partial equilibrium model. I then show that the household, which now seeks to maximize inter-temporal asset returns, also makes the same decision to maximize the one period asset returns as in the partial equilibrium model. That is, both agents transfer cash only every payday. From this result, the income velocity of money can be interpreted as the frequency of pay periods in this model. I derive a structural money demand function that links real income, real balances and nominal interest rates in general equilibrium. Following Lucas (2000), this structural money demand function can be used to describe a long-run theoretical equilibrium relationship that is more suitable for studying long-run money demand, compared to the use of a conventional money demand function. This structural money demand function has several implications for income and interest elasticity. The income elasticity of money is 1, but the interest elasticity of money has two values, depending on the cut-off value of the interest rate. When interest rates are below the cut-off, the fixed brokerage fee is larger than the total interest forgone and the firms endogenously set the number of pay periods to one as in the CIA model. In this case, the interest elasticity is zero. When interest rates are large enough, the frequency of pay periods fluctuates in response to the variation in the interest rates and the interest elasticity is close to 0.5. Finally, in a quantitative experiment, I show that these elasticity results are robust across alternative specifications of the agent's utility, to suggest that they are mainly derived from the agents' optimal cash management behavior. The result of a unitary income elasticity is in sharp contrast to the elasticity result from the classical Baumol-Tobin framework which predicts that both the income elasticity and the interest elasticity of money demand is 0.5. In the partial equilibrium model, the elasticity results depend on the assumption that real income, the rate of interest, and the cost of cash withdrawals are 38  mutually independent. An increase in real income produces a proportional change in the interest forgone on cash holdings, so with a constant brokerage fee, people will reduce the frequency of money transactions (the income velocity of money). In the general equilibrium model, however, an exogenous increase in the real income caused by the aggregate technological change leads to a proportional change in real wages and the real brokerage fee. The increase in interest forgone (due to the rise in the real income) and in the real brokerage fee has offsetting effects on the income velocity of money, leaving it unaffected. This suggests that an exogenous change in the real income generates a proportional change in real balances, leading to the unitary income elasticity. The general equilibrium result for the interest elasticity is, in large part, due to the Baumol-Tobin intensive cash management margin. As shown in the previous chapter, when real income and real wages are exogenously given in the partial equilibrium inventory model, the Baumol-Tobin intensive cash management margin leads to an interest elasticity of precisely 0.5. Nevertheless, in general equilibrium, the value of interest elasticity has a small deviation that is caused by two separate effects of inflation. On one hand, inflation (or the opportunity cost of money) represents a tax on the cash goods - the consumption goods. Higher inflation induces the household to demand more leisure and less consumption goods. This effect of inflation reduces the labor supply and real income, and further lowers the demand for money below what the Baumol-Tobin intensive margin predicts, leading to an increase in the interest elasticity. On the other hand, higher inflation raises brokerage fees and the relative wage dispersion between the bank sector and the production sector, since the brokerage fees which are paid at the end of the period must be compensated with the higher opportunity costs of money. Higher brokerage fees induce agents to demand more money, thus, reducing the interest elasticity of money. In the quantitative experiments, however, I show that both effects tend to offset each other and are quantitatively negligible. The general equilibrium model of inventory money demand developed here is related to the existing general equilibrium inventory theoretic frameworks. As reviewed in Chapter 1, Jovanovic (1982) and Romer (1986) developed steady state versions of general equilibrium inventory models, which assumed that the economy always stays in one steady state, without the 39  ability to transit to other steady states. Thus, their models can be at best described as incomplete versions of the general equilibrium models of inventory money demand. The liquidity literature, such as Grossman and Weiss (1983), Rotemberg (1984), Christiano and Eichenbaum (1995) and Alvarez, Atkeson and Kehoe (1998), and the endogenous segmented market literature, such as Chatterjee and Corbae (1992), Alvarez, Atkeson and Kehoe (2002) and Khan and Thomas (2006), can be viewed as variants of the general equilibrium inventory models. In all these papers, agents receive heterogeneous shocks in the short-run. However, since agents are identical in the long-run when no shocks are present; these model frameworks become the standard Cashin-Advance model with the constant income velocity of money. The heterogeneity is necessary for these models to study the short run impacts of monetary shocks, but not necessary to account for long-run money demand. The remainder of this chapter is organized as follows. Section 2 outlines the setup of the model. Section 3 defines the competitive equilibrium and analyzes the income and interest elasticity of the structural money demand function in the general equilibrium. Section 4 concludes.  3.2 The Representative Firm's Problem In this section, I define the firms' profit maximization problem. Firms maximize their profits by minimizing the costs of cash management and production. Outline of the Representative Firm's Problem In this economy, there are a continuum of final good producers, indexed by j E (0,1). Each of them is a monopolist, who employs a large number of workers. Firms are identical except that each of them produces a differentiated final good. Since the final goods are non-storable and households consume them at a constant rate, firms must produce them at the same rate every period. Each firm is owned by a household. At time t, the consumption aggregator technology is an integral of differentiated goods:  40  ^  ^Pt  1^1 C t =[ iC t (j)Pdjr ,f.te [1 9 .),^ 0  (3.2.1)  as in Dixit and Stiglitz (1977). In this expression, C t (j) denotes the quantity of the good produced by the jth firm in period t and C t is the time t aggregate output. (3.2.1) suggests that each firm faces a downward sloping demand curve with a constant elasticity which is equal to IA 11-1 ). The constant elasticity demand for each final good takes the form: P rt-l_ (j) IL C [ ^t ^t (j)^Ct  (3.2.2)  where P t (j) denotes the time t price of the jth final good and P t denotes the time t aggregate price index which is given by I^1 Pt ={1Pt(1) 1-P dit" 0  (3.2.3)  The jth firm produces the good using the following CRS (constant returns to scale) production technology: C, (j) = A i N, (j),^  (3.2.4)  where N t (j) denotes the time t jth firm's demand for labor input and A t is the labor productivity which is identical across all firms. There are two perfectly competitive labor markets; one is for the production sector, and the other is for the banking sector. W t f and Wtb denote the time t competitive wage rates for the production sector and the financial sector respectively. Production firms rent labor in a sector-specific labor market. Firms choose how frequently to pay workers, who are paid in advance by cheque every payday. In period t, the first payday is assumed to be the first day and paydays are set across identical intervals. For example, if the length of period t is one month and wages are paid on a weekly  41  basis, firms pay workers every Monday, or if the length of period t is one year and wages are paid on a monthly basis, firms do so on the first day of every month. Since goods sales have not begun on the first payday, firms must borrow their initial wage bills from the bank at the beginning of the period. Since loans are costly and firms can steadily accumulate cash flows through ongoing goods sales, they only need wage loans for the first pay period. Let Zt (j) denote the number of pay periods chosen by the jth firm in period t, where Zt(j)>1. The jth firm's demand for loans is W t fN t (j)/ Zt (j). Repayment of loans and interest take place at the end of period t. Firms pay the subsequent wages by cheque after depositing their accumulated cash revenue into the bank. After workers withdraw their wages, the monopolistic profits from production remain in the firms' bank account to earn interest. These profits and interest earnings are distributed back to the firms' owner, the representative household, at the end of the period. Since the j-th firm's profit maximization problem is repeatedly and independently solved in every period, its optimal cash management decisions which are summarized by Proposition (2.1) in the previous chapter are unchanged in the general equilibrium setting. Hence, I will not repeat the derivation of this proposition here. Let f t (j)-1 denote the jth firm's number of cash deposits As shown earlier, Proposition (2.1) has demonstrated the firm's optimal decision on the timing and number of cash deposits, which states that (1) the firm equalizes its cash holding period , for a given number of cash deposits 0)-1; (2) the firm will deposit cash only every payday, i.e. ft=4. The first result ensures that the firm minimizes the interest forgone on its cash holdings and the second result ensures that the firm's marginal interest costs given up for both holding cash and making wage payments are equal to the fixed brokerage fee. In the language of this model these results are summarized by  zt (i) = Z t ( j) =1  [Pt (I)C t^+ W t f N t^t  2B t If i t <i*.  If i t >i*; (3.2.5)  42  In (3.2.5), i* is the threshold nominal interest rate, at which the firm's interest cost incurred by making one cash deposit is exactly the same as the fixed brokerage fee. When the nominal interest rate goes below i*, the firm will not visit the bank during the period, i.e. f t (j)=Zt (j)=1.  The j-th firm's cash holding period is 1/Z t (j). Its total interest returns on cash deposits, the interest cost given up for paying wages and total fixed brokerage fees are calculated by Pt (K t (Di t  Zt ( j)1  Zt (j) —1 + ^i = Pt (DC t (Di t 2Zt ( j) ^Z (D Zt (i)^  (3.2.6)  Z(j)-1 1^Z(j) +1 W f tN t (j)i t (1+ ^ +....+^]— W f tN t (j)i„ Zt (j)^Zt (j)^2Zt (j)  [Z t (j) —1]113 1  .  (3.2.7)  (3.2.8)  In (3.2.6), total interest returns are calculated as the accumulated sum of the interest returns on every one of the kth cash deposit [k=1,..f t (j)-1] . (3.2.7) sums up the accumulated sum of the interest cost given up for each one of the kth wage payment, k=1,..Z t (j). k/ Zt (i) is the per dollar interest cost on the kth wage payment. The j-th firm's profit function can be written by: (j)+1.^f^1+ Z Max Pt (j)C t ( j){1 + ^ it^W t N E (j){1 + ^ it ] [Z t (j) 2Z t (j)^2Zt (j)  —  1]13 .^(3.2.9)  The firm is subject to the constraints (3.2.2) and (3.2.4). The firm's optimal labor demand schedule is given by Zt(j)+l i 1+ wtf^2Zt(j)it N t (1) :Pt (i) ^At 1+ Z t (j) —1  (3.2.10)  2Z t (1) The jth firm equates the marginal benefit of hiring one unit of labor,  Pt(j)[1+0.5it-0.5it/Za],  a  constant markup over the marginal cost, W ft (j)[1+0.5i t +0.5i t/Zt (j)]. In a symmetric market equilibrium, Ct(j)=Ct, Nt(j)=Nt, Pt(j)=Pt, and Zt(j)=Zt for all j.  43  3.3 The Representative Household's Problem There are infinite numbers of identical households. At time t, the representative household seeks to maximize 00  E t {Zru(Ct + ,,Lt^t )} , 0 <13 <1 .  (3.3.1)  j=0  Here C t denotes the time t units of consumption, L t denote the time t hours of work and u(C„ L t ) = log(C t ) — y o L t  > 0,  (3.3.2)  where L t = N, +(Z, +S t — 2)7 .  (3.3.3)  In (3.3.3), N t and (St+4-2)7 denote the time t hours of work for the production sector and the banking sector, respectively; y denotes the constant fixed broker's hours of work for producing one money transaction; and (S t -1) and (Z t -1) are the total number of cash transactions conducted by brokers for the household and firms in period t, respectively. In (3.3.2), the fixed money transaction cost behaves as a fixed utility cost. This assumption is inline with the formation of the utility function in Romer (1986). The linear disutility of work implies that the household's labor supply schedule is perfectly inelastic. Hence, the equilibrium employment in both sectors is determined by labor demand. In the quantitative analysis, I also consider alternative specifications of the household's utility. At the beginning of period t, the household carries its bank deposits M t from period t-1 into period t. The household can withdraw some cash at no cost at the beginning of period t. The rest of savings earn interest. The initial cash withdrawal is used to finance a steady stream of consumption purchases for the time period before the next cash withdrawal. Since there is a fixed brokerage fee charged for every cash withdrawal during the period, the household would only visit the bank infrequently. The household takes the firms' frequency of pay periods as given and receives wage payments by cheque periodically. Cheques can not be used to purchase goods directly, so the household must redeem these cheques at the bank subject to paying the fixed  44  brokerage fee every time. Brokers are paid at the competitive wage rate W bt and receive these brokerage fees from the bank at the end of period t. The household solves the cash management problem by choosing the number of cash withdrawals S t and the timing of these withdrawals. In addition to them, the household chooses the consumption C t and the labor supply to the production sector, N t , to maximize the expected utility (3.3.1). 10  The Household's Cash Management Problem In the partial equilibrium model in Chapter 2, the household maximizes its asset returns in one period and the household's optimal cash management decision is to transfer cash only every payday. In the general equilibrium, however, the household makes the inter-temporal savings decision to maximize the asset returns for multiple periods. Therefore, I re-evaluate the household's optimal cash management decisions in an inter-temporal utility maximization framework. I will first evaluate the household's optimal decision on the timing of cash withdrawals, given a fixed number of cash withdrawals S t , and then I will determine the optimal S t.  Determine the Timing of Cash Withdrawals (I) Evaluate the inter-pay period cash withdrawal  As shown in Proposition (2.2) in the previous chapter, the household makes no inter-pay-period cash withdrawals, given the firm's optimal frequency of pay periods 4*. This result is unchanged in the general equilibrium setting. Please refer to the proof of Proposition (2.2) for details. When firms endogenously choose one pay period, Proposition (2.2) implies that the household transfers cash on the first payday only. In this case the velocity of money is constant and the model becomes the standard CIA model. I restate this implication in the following lemma.  lo I assume that the household takes the constant fixed time cost of money transaction 7 as given, but that can vary its labor supply to the banking sector by choosing the number of cash transactions S t .  45  Lemma (2.1): When firms endogenously choose one pay period, the household makes no cash withdrawal within the period and the velocity of money is constant.  (II) Evaluate the payday cash withdrawal As explained earlier, Proposition (2.2) suggests that the household can either visit the bank every payday or on some paydays only. Now suppose that the household's jth visit to the bank is qt pay periods ahead of the (j+l)th visit. It follows: ±qt 2 ±...cit s )/zt  1, q t J  ^ 1Vj =1,...S, ,^  (3.3.4)  where qt/Z t denotes the jth cash holding period. (3.3.4) indicates that the total length of cash holding periods is one.  The amount of wages claimed on the jth visit is q tj-i W tNat , when combined with the extra amount of cash withdrawal from its own bank deposits Ii t , are used to finance the consumption purchase for the next qt pay periods. Let's call the interval between the jth and j+lth household's visit to the bank the qt th sub-period. The household's cash constraints for the first sub-period and the following sub-periods are given by "I f m 1 Pt C t^"t ^+Tit , qt^  (3.3.5)  Zt  PtCt^wtfNt  ±Ft,v j= 2,....,st  qt^^- qt j^(3.3.6) Z t^Zt In (3.3.5), the household obtains the wages for one pay period on the first payday. For a positive interest rate i t , the utility maximizing household will not have any unspent cash left at the end of each sub-period and the cash constraint must be binding.  The household's time t budget constraint is given by s^s I(L'Ich k ) ^ 1),C, +M (+1 =1■4,(1+0— J=1^k= ' i +NAT,'N, +7,(1+0+7E,^ +W, b y(S, +Z, —2)—B t (S, —1).  46  (3.3.7)  In (3.3.7), WEI denotes remaining bank deposits at the end of period t, after the household spends the rest of income in purchasing consumption. The right hand side of (3.3.7) summarizes the household's sources of income in period t, including the wage income W t fN t received from firms during the period and the remaining one received at the end of the period. The sum of the remaining income is equal to the wage income W tby(S t +Zt -2) from the bank, dividend income itt from firms, J t (l+i t ) from the bank, and the interest returns [M t (l+i t )-^E qt k / Zt )i t J on jz-1^k=j  bank deposits, less brokerage fee payments B t (S t -1). Here J t denotes the time t money injection from the government to the bank, M,(1+i t ) are total interest returns on initial bank deposits as if s^s no cash withdrawals take place during the period, and^q t k )i t /Z t are the sum of the t=1^k=j S^k  interest forgone on both cash holdings and wage receipts, where i t Iq, /Z t is the interest k=, forgone on every dollar of the jth withdrawal. Just like the firms' problem, we need to determine the optimal timing of cash transfers (determine and ql t ) at first, given a fixed number of cash withdrawals S t ; and then given the optimal timing decision, we can solve S t , M t.„, , C t , and N t . The first order conditions for are given as follows: zqt  k (3.3.8)  , t i t k=j^ =8jt, j=1...S t Zt  :^Xtit  Xi -t- t  E  it k  k=1 ^+81- t  Pt C t  81+It  Wt f N t  Z t^Zt^Zt^Zt  Ei  (3.3.9)  tk  k= 1  Zt  1,...S, —1,^  =^.  (3.3.10)  Zt  47  ^  In (3.3.9) and (3.3.10),  Et  denotes the time t shadow price of cash holding period ; k t denotes the  time t marginal utility of one dollar worth of income; and O l t and &i t are the time t marginal utility of one dollar worth of cash for constraints (3.3.5) and (3.3.6) respectively. Substitute out & i t in (3.3.9) and (3.3.10) using (3.3.8), keep iterating (3.3.9) for all j, and then write the first difference equation for q tj as follows: Nt , , Pt PC^• t +ch i Wt f ^_0,vi= Z t^, Z  29...st  .  (3.3.11)  Comparing (3.3.11) with the binding cash constraint (3.3.6), it is straightforward to show that: 1^2^s^Zt = qt = •• = q t = St  I t2 =..  (3.3.12)  ^ = Ist = I t .  (3.3.13)  The household equalizes its cash holding periods across cash withdrawals. This condition ensures that the household minimizes its total interest forgone (interest cost on the I t terms), given a fixed number of cash withdrawals S t . The result in (3.3.12) confirms that Proposition (2.3) in the previous chapter still holds in the general equilibrium setting.  Determine the Number of Cash Withdrawals Substitute (3.3.12) into (3.3.5), (3.3.6) and (3.3.7), and then rewrite these expressions as follows: 13,C,^W f IN t ±Ilt, St  (3.3.14)  P,C, < W f tN, ±if,v . j^2,....,S t ,^ S t^St  (3.3.15)  „  +M, + , = + Wt h y,(S, +Z,  —  2) B, (S, —  it +  —  I,(S, —1)i  '1+ W, I N, +J,(1+1,)+ rc, 2^  (3.3.16)  1).  Given i t , Pt, Wt b ,Wt f Zt , and 7 , the household's first order conditions with respect to and N t are expressed by: 48  Mt+1, St,  Ct,  ^  M t+1 : X t = P E t X t+1 [1+ i ti- J 9  (3.3.17)  I I t .^Xtit = K ' t ,  (3.3.18)  It :  x (S t —1)i t = K 2 t ,  (3.3.19)  2  i^PtCt^2 P C OW S t :—T o y+A,,(Wt b^ 7—I t —B t )+ K't ^ +lc t t t^t = 0 2 2^S2^S, t 1^K l t^K 2 t C t : — := P [X t + — + —]  C,^'^St^S t  (3.3.20)  9  ,  (3.3.21)  ^,i^_2 N t : To = wtf R t + '-+ 't__] . z t^S t  (3.3.22)  x l t and K 2 , denote the time t marginal utility of one dollar worth of cash for constraints (3.3.14) and (3.3.15) respectively. The household equalizes the returns of work between the banking sector and production sector: (3.3.23) = xt wb t.^ To  The optimal S t * is derived from (3.3.20) by substituting out for It ,  K ' t ,K  2  t,  and yo using the  binding cash constraint (3.3.15), (3.3.18), (3.3.19) and (3.3.23) respectively. 5  * , li^ Pt C t + W t f N t i 2B t  (3.3.24)  Comparing (3.3.24) with (3.2.5), it is straightforward to show that when i t >i*; S t *=Zt *; otherwise S t *. Zt * =1, as suggested by Lemma 1. In summary, the result in (3.3.24) justifies that  Proposition (2.4) derived from the partial equilibrium model still holds in the general equilibrium setting. As explained earlier, both firms and the household choose the same timing and number of cash transfers, because they manage the same marginal cost structures of cash management. Intertemporal Optimization Substituting out for K' t , x 2 t and Xt in (3.3.21) using (3.3.18),(3.3.19) and (3.3.20) yields the Euler equation for savings 49  Pt+i  Ct+i ( 1 + Zt+1 2Z t++l , it,' )  131),C t (1+ Z ' +1 i t ) =E t ^ 2Z t^1+i t-I-1 Z +1  1+ In (3.3.25), IM^  t  2Zt  (3.3.25)  i t ] denotes the time t average cost of spending one dollar on  consumption. (3.3.25) states that the household equates the marginal utility of a dollar spent at time t to the expected marginal utility of the returns from spending that dollar at time t+1. The intratemporal condition for the labor supply N t is given by (PoCt  Wf = nt •  (3.3.26)  rt  Finally, (3.3.23) and (3.3.26) imply the following intra-temporal labor supply condition for the two sectors. Wtb  =1+ (Z t +1)i t  (3.3.27)  Wtf^2Zt^.  (3.3.27) states that the relative wage of brokers to workers is compensated by the average opportunity cost of money, so that the household is indifferent between working in the production sector and working in the banking sector. Monetary Authority At the beginning of period t, a monetary authority injects J t amount of free cash into the bank. J t — M t+1 M t •  (3.3.28)  ^  Bank At the beginning of period t, a perfectly competitive bank receives the lump sum money injections J t from the monetary authority and initial saving deposits (M t -It ) from households prior to their first wage receipt. The bank supplies loans to a competitive market, at the nominal interest rate i t . Firms, who seek wage loans for the first pay period, deposit the wage loans at the  50  bank to meet households' wage withdrawals on the first payday. The loan market closes after the first payday. The loan market clearing condition requires: Wt^t N ^± Z  jt  (3.3.29)  Every following payday, the bank receives the firms' cash deposits, which are equal to Prat , and then supplies the funds to the households. At the end of the period, firms reimburse the loans and the loan interest, (1+it)W t fN t /Zt , to the bank. The bank reimburses the interest payments to the firms and the households. The bank's total payments of interest are calculated as follows: It Z t +^It (Z, —1) . (M t -Z,I t )i t + ^ + ^ 2^2 (  In (3.3.30),  (Mt-Ztit)it  (3.3.30)  represent the households' interest returns on term deposits. The second  term and third term represent the households' and the firms' time t interest returns on their cash deposits, respectively. Finally, the bank distributes J t (l+i t) to the households as profits at the end of period t. At time t, brokers produce banking services according to the following production technology: B t = W t b y.^  (3.3.31)  (3.3.31) implies that the perfectly competitive bank earns zero profit from providing cash exchange services.  3.4 Market Equilibrium and Money Demand Balanced Growth Equilibrium Definition of Market Equilibrium:  The competitive market equilibrium is a set of allocations {Ct, Nt, Lt, Ct(j),  Nt(j),  Mt+t, St, Zt (j)},  and prices Pt, Wt f, Wt b ,^it},with the following properties: {  (i)^Given the prices, the allocations solve the optimization problem of the household and each jth final goods firm.  51  ^l  (ii)^The goods, labor, money and loan market clear. I will only study the symmetric equilibrium, where P t =Pt (j),C t =C t (j), Nt,=Nt(j) and Zt= Zt (j) for all j. Computation of Balanced Growth Equilibrium Along a balanced growth path, the real income C t and the aggregate technology A t grows at the same constant rate, and the other real allocations {N t , Lt, Zt, S t } remain stable. Given the unit time cost 7 and the aggregate technology level A t , they are jointly determined by the following system of six equations. C, = A, N, ^ ±  i  (3.4.1)  t^i t  2 2Z, 11(PoNt =^l.^.^,^ 1+ t + 1t 2 2Z,  (3.4.2)  L, –N, = (Z, +S, –2)y,^  (3.4.3)  S, = Z„^  (3.4.4)  i  i  2  1  When i t>i*, 2y(1+ -1- + --)Z, = (- + N,)i, ; 2 2Z ,^(Po  (3.4.5)  Otherwise: 4=1. 1+i t = p ( 1 +I-tnit)( 1 +at),  (3.4.6)  where at =(A t+ i/A t -1) is the growth rate of aggregate technology. Equation (3.4.1) is the goods market clearing condition. Equation (3.4.2), which is derived by substituting out C t and W t f/Pt in (3.3.26) using (3.4.1) and (3.2.10), is the labor market equilibrium condition for the production sector. (3.4.3) is the labor market equilibrium condition for the banking sector. (3.4.4) and (3.4.5) summarize the agents' optimal cash management decisions. (3.4.4) indicates that both the firms and the household choose the same frequency of cash transfers. (3.4.5) is the decision rule for the number of paydays, and is derived by substituting out C t , B t and W t f/Pt in (3.2.5) using (3.4.1), (3.3.31), (3.3.23) and (3.2.10). (3.4.5) indicates that the equilibrium number of paydays Z depends on the nominal interest rate; when  52  the nominal interest rate is small enough, both the firms and the household endogenously choose not to transfer cash during the period, so that Z=1; otherwise they do so according to (3.4.5). The familiar Fisher Equation (3.4.6) states that, along the balanced growth path, the nominal interest rate equates to the sum of the real interest rate and the growth rate of money u mt . It is the balanced growth version of the Euler Equation (3.3.25) , and suggests that money is neutral in this economy, i.e. the change in the level of money stock generates a proportional change in the price level and has no effect on real allocations. In balanced growth equilibrium, the growth rate of money u mt is equal to the inflation rate n. Consequently, given the growth rate of aggregate technology a t , the long-run nominal interest rate only depends on the money growth rate umt, which is specified exogenously by the monetary authority. In other words, the determination of nominal interest rates is independent of real allocations in the balanced growth equilibrium. Now consider the corner solutions when Z*=1 and i<i*. In the equilibrium, substituting Z=1 into (3.4.2), we can express N as a function of i. The threshold nominal interest rate i* is that at which the firm is indifferent between adding another pay period and not doing so. It suggests that N(i*), Z=1 and i* should satisfy (3.4.6). Thus, the solutions for N(i*) and i* are given by N=L= i* =  1  (3.4.7)  ^ !No ( 1 + i * ) '  110.t +1) 2 — 8nap o — 0.1 +1) + 41i_up o  21_1(1 — 2yp o )  ^  (3.4.8)"  In (3.4.7), N=L is because no labor demand is coming from the banking sector. In the general case when i t >i*, Zt is a function of i t only. This may be seen by substituting out N t in (3.4.5), using (3.4.2), to obtain the following equilibrium condition for Z:  z  f 2+i t \z 3 + 2 r k- 2i t ^t^t^L  i t^ii, +1  lz ^it (µ —1)^. 0  4 + 2i t^2[two t 4.790(2 + i t )  (3.4.9)  Note that (3.4.9) holds both in and out of the balanced growth equilibrium, since the Fisher Equation (3.4.6) has not been imposed in deriving (3.4.9). Given the nominal interest rate i t ,  II  In U.S. annual Ml data, the calibrated parameter value of i* is approximately 0.2%  53  Equation (3.4.9) is the only equilibrium condition needed to solve the model. This is because once Zt (i) becomes known, Ct, Nt, Lb and S t , can be solved by combining (3.4.1) - (3.4.4). Generally speaking, a cubic function such as (3.4.9) has three roots. In this case, however, as shown in the appendix, (3.4.9) has two negative roots and one positive root. The positive root serves as the unique solution for 4, which leads to the following proposition. Proposition (3.1): Given the long-run nominal interest rate i, and the exogenous growth process of aggregate technology  At,  one unique set of solutions can be found to { Z(i), N(i),  L(i), C t (i, A t)} that determines the balanced growth equilibrium of real allocations. Please see the proof in Appendix Al. Derivation of the Quantity Exchange Equation Having examined the equilibrium real allocations of the model, we now turn to the analysis of monetary equilibrium. According to Walras's law, the money market clears, if and only if, the goods market and the labor market clear. The money market clearing condition is represented by the quantity exchange equation, which is derived by substituting out for W t fNt /Zt in the loan market clearing condition (3.3.29) using the cash constraint (3.3.14) : Pt C t ^ =It +(Mt It) + Jt = Mt+1 •^ Zt  (3.4.10)  Equation (3.4.10) can be interpreted as follows: every pay period, cash circulates from the household to the firms and then back to the household. In this sense, the frequency of pay periods Z, is also the income velocity of money. Income elasticity In this model, real balances are determined by C, 11.1 t+1 =^  (3.4.11)  z, '  where m t+ t denotes the time t period end real balances. In (3.4.11), Z t (i t ) is the positive root of (3.4.9) when i t >i*; otherwise Z t (i t )=1, for all i t ^i*.  54  Along a balanced growth path, since Zt (i) summarizes the long-run equilibrium condition in the real sectors, (3.4.11) describes a long-run equilibrium relationship between money, income, and nominal interest rates and thus, is more suitable for studying long-run money than is a conventional money demand function, following Lucas (1988; 2000). In this setting, the income elasticity of money is defined as the percentage change in the period end real balances, mt+i, [in (3.4.11)] with respect to the percentage change in real income, Ct, given the nominal interest rate. As explained before, given the nominal interest rate, the equilibrium income velocity of money Z t does not depend on real income. Intuitively, this result can be understood by substituting B t by W tby into (3.2.5). An exogenous increase in the real income C t is induced by the aggregate technological change in A t , as suggested by (3.4.1). This leads to a proportional increase in real wages. In (3.2.5), on one hand, an increase in real income yields a higher interest forgone on cash and wage transfers, leading to a higher income velocity of money; on the other hand, an proportional increase in the brokerage fees tends to lower the velocity, so these changes jointly have no impact on the income velocity of money. Therefore, (3.4.11) suggests that real balances are homogeneous of degree one in real income and that the income elasticity of money is 1 in this model. This implication is in sharp contrast to the partial equilibrium result. In a partial equilibrium model, an exogenous change in real income is assumed to be independent of variation in real wages, leading to an increase in the income velocity of money. Interest elasticity Similarly, in (3.4.11), the interest elasticity of money is defined as the percentage change in the period end real balances with respect to the percentage change in the nominal interest rate, given the real income C t . With constant real income C t , the interest elasticity of money equates to the negative interest elasticity of income velocity, i.e.: drnt+1/ mt-ti dZ, 1 Z, e di, I i^di, I i  =  55  (3.4.12)  (3.2.5) indicates that when i t^i*, Zt =1 for all i t , and the interest elasticity of money is zero. When i t >i*, the property of the interest elasticity is then summarized in Proposition (3.2). Proposition (3.2): Given a small positive nominal interest i t , the interest elasticity of money holdings is close to 0.5. Please refer to the detailed calculations in Appendix A2. The interest elasticity ei is close to 0.5 mainly because of the Baumol-Tobin intensive cash management margin. To see this, substitute (3.2.5) into (3.4.11) and then express the real balances mt+i by the following Ct Z.  —=  W C, ) =A t i ^1 ,C +W 'N tb 2 (^ t t 3 t t  PtCt^Wtb {NJ^f^+1]^,^(3.4.13) Wt N,^Wt it  Z +l t it) wb Z P ^2Zt t ,C t = where and t =1+ l i 2Zt it Wt ' 1+ Z t —1 . Wt f 2Z t p.(1+  In partial equilibrium, the change in the opportunity cost of money does not have any impact on the real income C t , the labor supply N t , and the real wages W t b and W t f, i.e. the terms in the brackets of (3.4.13) are holding constant. In this case, the interest elasticity of money obeys the Baumol-Tobin intensive margin and is precisely 0.5. Yet, in general equilibrium, the change in the opportunity cost of money has two additional effects, which are captured by the terms in the brackets, on the interest elasticity. These two effects are the cause of the small deviation of interest elasticity from 0.5. The first effect is the inflation tax effect which is captured by additional changes in the terms N t ( 1 +PtCt/W ftNt) -1 caused by the variation in inflation. That is, higher inflation (higher nominal interest rates) increases the tax on the cash goods- the consumption goods and the financial costs of production, i.e., as seen in the labor demand relation (3.2.10), causing P t C t/W ft N t to rise; in the basic C.I.A model of Christiano and Eichenbaum (1997a), this term is just equal to [t(l+i t ), which is the firms' interest cost of wage loans scaled by the markup. This induces the household to substitute goods demand for leisure demand, and also induces the firms to reduce the demand for labor and goods production, i.e., both N t and C t fall. As a result, the inflation tax effect lowers the demand 56  for money and also increases the interest elasticity beyond the Bamoul-Tobin intensive margin. The second effect of inflation is on the relative wage dispersion between the banking sector and the production sector. This is captured by the variation in W tb/W t f. A higher opportunity cost of money raises the broker's relative wage, according to (3.3.27), and also raises the brokerage fee according to (3.3.31). The resulting effect is that agents increase their demand for money, which leads to a smaller value of the interest elasticity less than 0.5. The two effects tend to offset each other. To see the size of these effects, I construct the measure of interest elasticity variation caused by the inflation tax effect, and the measure of interest elasticity variation caused by the relative wage dispersion. The inflation tax effect measure, denoted by einf , is defined as the percentage change in [N t (1+Prt/W ftNt) -1 ] with respect to a 1% change in the interest rate; the relative wage dispersion measure, denoted by e w , is defined as the percentage change in W tb/W t f with respect to a 1% change in the interest rate. According to these definitions, it follows that: e i = –0.5 + e inf + e w ,^  (3.4.14)  where 0.5 is the interest elasticity resulted from the Baumol-Tobin intensive margin. Note that the derivation of (3.4.13) does not rely on any specifications of the household's utility function. Substituting the expressions for W t b/W t f and P tC t/W ftNt in (3.4.13) into the definitions of eh- i f and e w respectively yields the following expressions: _1 a log[N( wPC + 1) ] fN  a log N^a log(pe z +1) ^ 0.5 0.5 a log i^a logi^a log i  e^ mf =  — (1+ e ) ^a= 0.5 ^ log N^ z 0.5 ^  (3.4.15)  '  alogi  due z + 1  1+ z^1+ I z _a log(1+ 02 ^i ^(1 ± e )i 2Z '( 2Z i) = + ----• 0.5 ^'^ e w^a log i^a log i^4^4Z '  (3.4.16)  In the derivation of these expressions, I use the approximation rule that log(l+r)rzr for a small positive number, r, as well as the definition for the interest elasticity [ e, = —(i / z )( az /ai)]. 57  On the right hand side of (3.4.15), the first term captures the inflation tax effect on the labor supply and the second term captures the inflation tax effect on the financial cost of production, i.e. a higher inflation tax raises the interest costs of wage loans and the interest forgone on cash holdings and thus reduces the firms' demand for labor. According to these expressions, ei n f is negative and e w is positive. Moreover, for small values of interest rate, both the second term and e w are small. Given a linear production technology A, the first term,  a log N la log i (= a log C/ a log i) , in  (3.4.15) denotes the traditional measure of the welfare cost of inflation, which is defined by Lucas (1988) as the percentage income compensation needed to leave the agent indifferent between a higher inflation rate ( usually 10%) and zero inflation rate. In the welfare cost literature, the magnitude of the welfare cost of inflation might be varied under different specifications of the utility function and general equilibrium model structures, but the general conclusion is that the welfare cost is found to be very small, e.g. Cooley and Hansen (1989) measured the welfare costs within a C.I.A model, and their estimates for welfare costs range from 0.11% to 0.94%, for reducing steady state inflation rate from 10% to 0%; and in the shopping time model, Lucas (2000) found that the welfare cost is less than 1% for reducing inflation from 10% to 0%. These findings suggest that the size of  a log Nia log i is likely to be  small, so is e in f. In the later quantitative experiments, I further verify that the measures of inflation effects are quantitatively small. As a result, the conclusion in Proposition (3.2) is that the interest elasticity of money is close to 0.5 in the general equilibrium setting. Discussion Mulligan and Sala-i-Martin (1996) applied data from the 1989 Survey of Consumer Finances to test the hypothesis that there is a fixed cost of having positive amount of the interest bearing assets; at low interest rates, a large fraction of households are unwilling to incur this cost and thus only hold cash. According to the data, Mulligan and Sala-i-Martin (1996) found that the interest elasticity is very small for small interest rates, but close to 0.5 for moderate interest rates. Their finding is consistent with the interest results derived here.  58  Alternative Assumption for the Agent's Utility In this section, I check the robustness of the elasticity results across alternative specifications of the representative household's preferences. In this structural model, the income elasticity result does not depend on the household's utility function, since, in any system of five equations like (3.4.1) - (3.4.5), with five unknown variables {C t , N t , L,, Zt , St} , we can always solve the equilibrium income velocity Zt by a function of the given nominal interest rate, i t only. In turn, (3.4.11) suggests that the real balances are always homogeneous of degree one in real income, regardless of the household's preferences. To check the robustness of the interest elasticity result, I modify the preferences of the representative household as follows: C'6 u(Ct,Lt)=  t  ^ 1_, 13+1 ^(4)0 ^t  1 - a^0 +1  cp o > O.^  (3.4.17)  In (3.4.17), a is the agent's risk aversion coefficient and 0 denotes the inverse of agent's labor supply elasticity with respect to the real wage. The CES (constant elasticity of substitution) utility function (3.4.17) is a more general representation of the household's preference than in (3.3.2), where a=1 and 0=0. Different choices for a and 0 affect the interest elasticity result by affecting the magnitude of the inflation effects. Yet, by numerically solving the model, I find that changing these parameters has a negligible impact on the interest elasticity because of the small inflation effects. Table 3.1 records the numerical calculations for the arc elasticity for e m , emf and e w under various values of a and 0, when the nominal interest rate increases from 1% to 10%. These calculations confirm that the interest elasticity e m is close to 0.5 for all pairs of a and 0. Either the inflation tax effect or the relative wage dispersion effect is very small, and their joint effects account for far less than even 1% of the total interest elasticity of money e t , or the Baumol-Tobin intensive margin can accounts for more than 99% of the total interest elasticity. These numerical calculations indicate that the interest elasticity result, stated in Proposition (3.2), is robust across the alternative CES specifications of the representative agent's preference.  59  Table 3.1 Decomposition of Interest Elasticity under Various 0 and a a=1, 0 =0  a=1, 0 =1  a=1, 0 =5  a=5, 0 =5  ei  -0.495  -0.497  -0.499  -0.495  einf  -0.007  -0.009  -0.011  -0.007  ew  0.012  0.012  0.012  0.012  The sensitivity analysis presented above suggests that the elasticity results in the general equilibrium setting are derived from the agent's optimal cash management behavior rather than from structural assumptions about the agent's utility function.  Discussion Romer (1986) presented a steady state version of general equilibrium inventory model, in which the nature of the fixed transfer cost is a utility cost that can be explained as a time cost. Romer's model predicted that the income elasticity of money is 1, but that the interest elasticity result depends on the specification of the household's preference. Since Romer did not exactly model the value of time- the wage rate, alternative specifications of the disutility cost, which enters the agent's optimal cash management decision rule, directly alters this decision. By contrast, in this model, it is the wage rate rather than the disutility cost entering the agent's cash management decision rule [as shown in (3.2.5)]. With small inflation effects, alternative specifications of the household's preference do not generate a significant impact on the real wage rate, as a result, the interest elasticity result is unaffected.  3.5 Conclusion This chapter develops a general equilibrium inventory theoretic model, where agents explicitly solve their cash management problem. I show that the elasticity results predicted by the partial equilibrium Baumol-Tobin framework no longer hold in the general equilibrium setting. The income elasticity of money is 1, in contrast to 0.5 as predicted by the Baumol-Tobin framework; interest elasticity can have two values. In particular, when interest rates are below a threshold value, the interest elasticity is zero and the model is the CIA model. In the general case when the interest rates exceed this threshold value, the interest elasticity is close to the partial equilibrium 60  result,0.5. The small deviation comes from the general equilibrium inflation tax effects. I also show that the general equilibrium elasticity results are robust across alternative specifications of the household's preference. The money demand function derived from the general equilibrium satisfies Lucas (2000)'s criteria of the "best money demand function', so that this chapter provides a promising framework to study empirical long-run money demand.  61  Chapter 4: A Stable Money Demand Function in a General Equilibrium Inventory Theoretic Model 4.1 Introduction The relation between the demand for money and its determinants is an essential component in most theories of macroeconomic behavior and a critical element in the formulation of monetary policy. A stable demand function for money, linking real balances, real income and interest rates has been long recognized as a requirement for the use of monetary aggregates in the conduct of policy. As mentioned in Chapter 1, previous authors, such as Lucas (1988; 2000) and Stock and Watson (1993), discovered that the U.S. time series data of the logarithm of M1, GDP and short term market interest rates for the period from 1900 to the late 1980's supports a stable long-run money demand function, with an interest elasticity estimate close to 0.5 and an income elasticity estimate close to 1. After extending the data period to 1996, Ball (2001), however, found that the coefficient estimates of the money demand function were subject to a structural change. According to his estimates, for the postwar period (1946-1996), the income elasticity of money demand is 0.5 and interest semi-elasticity is 0.05. Both estimates are significantly lower than those of Lucas (1988; 2000) and Stock and Watson (1993). The finding of Ball (2001) suggests that the long-run money demand function might have become 'flatter', so that money has become less sensitive to changes in income and nominal interest rates. This represents strong. evidence against a stable long-run money demand function. A similar finding was also derived by Friedman and Kuttner (1992). Conventionally, the cause of the structural change in the money demand function is attributed to financial deregulation and numerous financial innovations. Nevertheless, as pointed out by Goldfeld and Sichel (1990), Lieberman (1977) and Judd and Scadding (1982a), conventional econometric specifications of money demand often omit the measure of financial transaction technology or financial transaction costs due to the difficulty in measuring it. Based on the 62  transaction theory of money demand, in following the Baumol-Tobin inventory approach, financial technological changes, which lower the money transaction costs, lead to exogenous shifts in a money demand relationship, without modifying the shape of the money demand function. If indeed this is the case, then an econometric specification, which misinterprets the exogenous effect of financial innovations as endogenous adjustment across different variables in the money demand relationship, will yield inconsistent parameter estimates across different time periods. This chapter aims to examine whether U.S. M1 money demand data could suggest that a longrun money demand relationship still exists, but financial technological changes cause long-run systematic and exogenous shifts in this money demand relationship. This question is examined through calibrating a structural money demand function to U.S. quarterly time series data of M1, GDP and interest rates for the period from 1960 to 2000. The money demand function used for calibration is not exactly a conventional money demand relationship, but rather, as defined in Lucas (1988; 2000), is a theoretical equilibrium relationship linking real income, real balances, interest rates and the fixed time cost of money transaction. According to Lucas, such a relationship derived from general equilibrium is theoretically more appealing to analyze the empirical long-run demand for money than a conventional money demand function. In following this suggestion, the structural money demand function used here is also derived from a general equilibrium model of inventory money demand, which was presented in the previous chapter. It has an interest elasticity close to 0.5 and an income elasticity of 1 and these two parameter values match closely with the empirical estimates by Lucas (1988; 2000) and Stock and Watson (1993). As shown in the previous chapter, the elasticity results, or the functional form of the structural money demand, do not require any theoretical restrictions imposed on the specifications of the agent's preference, but are mainly derived from the agent's optimal cash management behavior. In my model, the fixed time cost of money transaction serves as a measure of financial transaction technology and thus, as a measure of improvement in financial transaction technology that corresponds to a reduction in the fixed time cost of money transaction. This implication is in-line with the argument of Guidotti (1993) who maintained that  63  the main effect of numerous financial innovations is to lower the time-cost of financial transaction. The calibration strategies are as follows. Based on this structural money demand function, I define the fixed time cost of money transaction as being the unexplained 'residual' part of money (M1) by the measure of income (GDP) and by the measure of the opportunity cost of money (the nominal interest rate). This empirical measure of money transaction costs is defined analogously to the measure of total factor productivity - the Solow residual, which is the part of growth that cannot be explained through capital accumulation. I can then obtain a time series for money transaction costs by computing the residuals of the money demand function. This allows me to use standard econometric techniques to analyze the long-run time series behavior of money transaction costs and to detect the possible structural change in it by applying a rolling Chow test. Specifically, I apply an autoregressive model with a linear time trend (AR(p)) to the time series of the logarithm of money transaction costs. The testable null hypothesis is that the time series of money transaction costs is trend stationary. Non-rejection of the null hypothesis would suggest that a long-run money demand relationship exists, but might be shifted by the exogenous long run financial technological change. On the other hand, if the time series of money transaction costs is not trend stationary, i.e., containing a unit root, then it suggests that long-run money demand is affected by the short-run non-stationary forces, and thus cannot be stable. The main findings of my calibration approach are as follows. A structural break in the time series behavior of money transaction costs occurred in 1981. This is precisely one year before the US Federal Reserve began to deemphasize M1 target policy, and also one year after the announcement of the Depository Institutions Deregulation and Monetary Control Act, which opened the prologue of financial deregulation. In particular, (1) for the period from 1960 to 1981, the time series data of the logarithm of money transaction costs follows a stationary process and little variation was seen in money transaction costs. This suggests that no exogenous shift occurred in long-run money demand. (2) By contrast, the time series data from 1982 to 2000 follows a trend stationary process and the quarterly descending rate of money transaction costs is 1.37%; therefore, systematic and exogenous shifts might occur in the long-run money demand relationship. (3) Both the estimates of the AR(P) coefficient and the intercept are almost identical 64  between the two sample periods. For the first sub-period, the intercept can be interpreted as the stable long-run money transaction cost, of which the estimate is approximately 40$ (USD2002) per money transaction. (4) Given the identified growth process of money transaction costs, the structural money demand function can successfully predict the trend of actual M1 velocity for both sample periods. These findings suggest that the structural change in the long-run trend growth in financial transaction technology is the key factor causing the divergence in the time series behavior of the last 40 years of US money demand. As it is implicitly adopted in the present chapter, the idea of including a time trend in the specification of money demand to measure the long-run financial transaction technological change has been long available in the literature. It serves as one way to capture the effect of financial innovations on money demand. Since empirically it is extremely difficult to construct a direct measure of financial transaction technology, most researchers instead seek indirect measures of financial transaction technology instead. For example, the impact of financial transaction technology on the money demand function was measured by previous peak interest rates in Goldfeld (1976) and Enzler, Johnason, and Paulus (1976), and by the number of electronic fund transfers over the Fed wire in Dotesy (1985). 12 Lieberman (1977) utilized a time trend in the money demand function to capture the effect of financial technological changes, and Porter and Offenbacher (1982) constructed a more direct measure of transaction costs — the brokerage fee proxies in the money demand specification. Most of these studies estimated the reduced form specifications of the money demand function, for which the identification implicitly requires the assumption that the nominal money supply process is exogenous to variation of the money demand process. A time trend in the money demand function was also examined by Ball (2001). He found that a trend in the transaction technology and low income elasticity are nearly observationallyequivalent interpretations of US postwar M1 money demand data for the period from 1946 to 1994. By imposing the time trend and the unitary income elasticity, Ball still obtained a semiinterest elasticity estimate of 0.05, which is much lower than the value of interest elasticity 12  As argued by Ireland (1995), these authors treated financial innovations as investment projects which are endogenous choices made by the private financial sector.  65  proposed in this chapter. This estimate of interest elasticity does not support a unique long run money demand relationship as well. Ball also argued that the major problem of testing the time trend directly in a money demand function is that a trend associated with financial transaction technology is highly collinear with income, so that their effects can not be disentangled. Related approaches that examined the issue on the stability of money demand in the literature are the single equation partial adjustment model and the cointegration approach. Earlier tests employed the single equation partial adjustment model and identified whether or not the coefficient estimates of money demand were subject to a structural change. The works of Goldfeld (1973; 1976), Enzler, Johnason and Paulus (1976), Lieberman (1977), and Porter and Offenbacher (1982) are examples of this approach. Such a money demand specification, which, however, systematically over-predicts actual M1 figures for the late-1970s and under-predicts the actual figures for the 1980's [Goldfeld and Sichel (1990) ], has been almost abandoned. The cointegration approach, as adopted by Hafer and Jansen (1991), Breuer and Lippert (1996), Friedman and Kuttner (1992), Stock and Watson (1993) and Ball (2001), has become the mainstream approach in the literature." This approach not only can identify the existence of a long run stable money demand relationship, but also can directly estimate the key parameters of money demand. The general conclusion in this literature is that neither M1 nor M2 is cointegrated with real income and long-term interest rates. When short term interest rates are considered, Hafter and Jansen (1991) and Breuer and Lippert (1996) found that the long run money demand relationship is empirically supported by M2 but not by M1. Friedman and Kuttner (1992) found a structural change in the long-run behavior of the money demand equation. Stock and Watson (1993) found a stable long run demand relationship for M1 with the income and interest elasticity estimates close to those of Lucas (2000). However, Ball (2001) found that postwar data (1946-1996) suggested an income elasticity estimate of 0.5 and an interest semielasticity estimate of 0.05. Both of these values are only half the size of the previous estimates by Lucas (1988; 2000) and Stock and Watson (1993), and thus suggest that the stability of long run money demand is no longer satisfied.  13  See Chapter 1 for a discussion of the cointegration approach.  66  Unlike these approaches, the structural model approach used in this chapter does not estimate the parameter values of money demand directly from data, but derives them, based on theoretical predictions. In this aspect, it is closely related to the structural model approach adopted by Lucas (1988; 2000). The remainder of this chapter is organized as follows. Section 2 calibrates the structural money demand function to US M1 money demand data for the period from 1960 to 2000. The time series of money transaction costs is then constructed and analyzed in the AR (p) model. Section 3 concludes.  4.2 Calibration of Money Demand Structural Money Demand Function The general functional form of money demand, as suggested by the transaction theory of money demand, can be described by log(m t ) = e y log(y t ) — e, log(i t ) + F[y, ] .^  (4.2.1)  In (4.2.1), e y denotes the income elasticity; y t denotes the time varying unit money transaction cost; and F(.) is a function of y t , which specifies financial transaction technology. The specific structural money demand function used here is given by (3.4.9), which represents a monetary equilibrium relationship in the general equilibrium model in Chapter 3. 14 This money demand relationship predicts that the income elasticity is 1, and that the interest elasticity of money is close to 0.5. These elasticity results suggest that (4.2.1) can be approximated by the following specification: Yt  ^—0.5log(i, ) + F[y, ] . ^  (4.2.2)  According to a quantity exchange equation, the money to income ratio, m t/Y t , is the inverse of the income velocity of money. In the previous chapter, I showed the derivation of (4.2.2), and that the transaction function F[.] does not require any theoretical restrictions imposed on the  14  Refer to Chapter 3 for the detailed description of the model.  67  functional form of the agent's utility, though it directly comes from the agent's optimal cash management behavior. With log preference over consumption and linear preference over the disutility of work, the explicit functional form of (4.2.2) can be written as follows: ( 2+i, )zt 3 + zt 2  ±r  it^µ+1  7t ^it(p-1)  2i,^4 + 2i,^21.ty,y 0^2iwy,y0(2 + i ,) or y, = Lu(1+  ^=0,^  (4.2.3)15  Z, —1 Z, +1 , i, )+ (1+ i, Ai, /[2Z,2vo,u(1+ i, )1.^(4.2.4) 2Z,^2Z,^2Z,  Z, +1  In these two expressions, 4 denotes the income velocity of money; ju denotes the markup parameter which dictates the firms' profit margin; and yo is the coefficient for the hours of work in the representative agent's utility function. In the sensitivity analysis, I also consider the Lucas-type of money demand function, which has an income elasticity of 1, and an interest elasticity of precisely 0.5, and other structural money demand functions derived from alternative assumptions about the agent's utility function. My calibration exercises will proceed as follows. After briefly describing the data set, I will adopt the estimation method of Lucas (1988) to demonstrate the instability of the long-run demand for M1 in the data. With this method, the money transaction cost is implicitly assumed to be constant in a money demand specification such as (4.2.2). Then, the financial deregulation occurred in the 1980's and 1990's is briefly reviewed. Subsequently, I use a US time series for MI, GDP and the short term market interest rate for the period from 1960 to 2000 to calibrate the money demand function (4.2.3) and to construct the time series data for money transaction costs y t . Finally, I analyze the long-run time series behavior of money transaction costs by the autoregressive process with a linear time trend and then test the implicit hypothesis that a long run money demand relationship exists.  15 Based on the previous analysis in Chapter 3, given a constant money transaction cost y, there is a threshold interest rate i* below which the structural money demand function takes the functional form: log(m u/y t )= 0 in the special case. As shown in equation (3.4.8) in Chapter 3, the value of i* depends on that of y. Here, I can compute the threshold i* for each y, conditional on (4.2.3) being the correct functional form for the structural money demand function. By comparing the actual interest rates with the threshold i*, I find that the interest rates used in this calibration exercise are always beyond these threshold values, and thus I can omit this special case  68  Data Description The set of time series data contains quarterly US short term market interest rates, M1 velocity and GDP, for the period from the first quarter of 1960 (1960.Q1) to the fourth quarter of 2000 (2000.Q4). 16 The short term market interest rate is the weighted average of the monthly one year treasury constant maturity rate. M1 is in billions of dollars, period end data for each quarter and seasonally adjusted. GDP is in billions of dollars and seasonally adjusted at an annual rate. M1 velocity is defined as the ratio of GDP to Ml. Instability of Money To predict the demand for money in the long-run, Lucas (2000) adopted the following structural money demand function. log( -mFt ) = —0.5log(i f )+ H^  (4.2.5)  In (4.2.5), H is a constant term, which is chosen so that the money to income ratio curve (4.2.5) passes through the geometric means of the data pairs of nominal interest rates and money to income ratio. For the period from 1960 to 2000, H is approximately equal to -2.5. Then, the predicted velocity of money is calculated by the right hand terms of (4.2.5) - the annual nominal interest rate and H. Lucas (2000) demonstrated that the long-run trend of US M1 data ending in the late-1980's can be well approximated by this structural money demand function. 17 Nevertheless, such a money demand function fails for the period after the late-1980's. Both phenomena are displayed in Figure 4.1, which plots the time series of actual M1 velocity (described by the solid line) and the predicted M1 velocity. As shown in Figure 4.1, although the predicted time series is more volatile than the actual one, the trends of the two series are very similar until 1990. After that the two time series begin to diverge, with the trend of M1 velocity going up, and nominal interest rates moving down.  16 Data are taken from the US Federal Reserve Bank's web-site: http://research.stlouisfed.org/fred2/ Interest rate-GS1M; MI-M1SL.  17  See Figure 4 on page 253 in Lucas (2000).  69  Figure 4.1 Actual and Predicted Ml Velocity from 1960-2000 [Lucas 2000] 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005  The divergence between the two time series after the late 1980's indicates that the money demand function proposed by Lucas (1988; 2000) alone can not explain the long-run trend of time series data for the whole sample period from 1960 to 2000. To further verify this, I cut the sample data set into two sub-groups at the fourth quarter (Q4) of 1989, and following Lucas (1988), estimate a linear relationship between the logarithm of M1 to GDP ratio and nominal interest rates for the sample period 1960.Q1 to 1989. Q4 and 1990.Q1 to 2000 Q4, respectively.  18  The OLS regression yields quite different interest rate coefficients for each sub-period. The estimate of interest elasticity is 0.45 with the standard error of 0.022 for the first sub-period and 0.19 with the standard error of 0.077 for the second one. The Wald test cannot reject the hypothesis that interest elasticity is 0.5 for the first sample period, at a 5% significance level, but rejects the same hypothesis at any conventional significance level for the second sample period. Although the choice of break date is arbitrary, the point of this exercise is to show that the parameter estimates of interest and income elasticity, between the two sub-periods, fails the parameter constancy test in an obvious way.  18  In this econometric specification, the income elasticity of 1 is imposed.  70  OLS Regression with quarterly data:  GM1P)  1960.Q1 - 1989.Q4 : log( D = - 2.21— 0.45 log i t GDP ^(0.092)^(0.022)  M1  1990.Q1 - 2000.Q4 : log(GDP ) — 1.40 — 0.19 log i ^(0.335)^(0.077)  Financial Deregulation and Financial Innovations The enactment of the Depository Institutions Deregulation and Monetary Control Act of 1980, which abolished most of the interest ceilings imposed on deposit accounts, symbolized the beginning of the financial deregulation. Since then, a number of major deregulation acts have been passed. They included: the Federal Deposit Insurance Improvement Act of 1991, which introduced some risk sensitivity to deposit insurance premiums, and the Neal-Riegle Interstate Banking Act of 1994, which created financial holding companies, and ended the artificial separation of insurance companies and commercial and investment banks. The deregulation of the financial sector creates increasing competition in the banking industry and was accompanied by numerous financial innovations, fueled by the rapid improvements in data processing and telecommunications. The major financial innovations are the development of Internet banking, electronic payments, as well as information exchanges. For example, ATMs first came into wide use during the early- to mid-1980's. Both the adoption of new cash management techniques and increasing competition among commercial banks likely brought about significant gains to consumers and considerably lowered money transaction costs (Berger, 2003). Benchmark Parameter Values The model structural parameters are f3,90 and g. The benchmark parameter values are assigned to match those commonly adopted in the literature. The discount parameter f3 is set to (1.031 0 - 25 for quarterly data. The markup parameter 11, is set to 1.2, following Hornstein(1993) and Christiano and Eichenbaum (1997). Since the value of 90 only affects the size of C t , , Li and other variables without affecting their distribution in balanced growth equilibrium, I assign y o to 71  normalize the aggregate employment to one in the balanced growth equilibrium when the interest rate is zero, given the values of other parameters. In this case, it is easy to verify that the value of 90 is 1/ pt. Table 4.1 records the benchmark parameter values. Table 4.1 Benchmark Parameter Values  Parameter  (3  Value  0.9926  11 1.2  (Po 0.8333  Time Series Data of Money Transaction Cost  As explained before, the fixed time cost of money transaction y t is defined as the part of money demand unexplained by income and by nominal interest rates. Based on this definition, the fixed time cost of money transaction y t is an implicit function of the nominal interest rate i t and Ml velocity, according to equation (4.2.3). Given the benchmark parameter values and the actual time series data of i t and 4, I can compute the time series of y t . The value of y t is interpreted as the ratio of the unit time cost of one bank transaction to the total quarterly hours of work. The calculated values of y t range from 0.0028 to 0.014 with a mean of 0.0078. According to Vissing (2002), the average U.S. household devotes approximately 0.2 of their total time endowment to work, and the average hourly wage is $10 (2002 USD), so the calculated dollar values of y t range approximately from $12 to $62 with a mean value of $34. Figure 4.2 shows plots of the hp-trend of log(y t )(described by the dotted line), actual log(y t ) (described by the solid line) and the approximated F(y t ) (described by the dashed line), which is defined as log(m t/yt )+0.5log(i t ) according to (4.2.2). 19 In the graph, there is a strong downward hp-trend for y t for the period after the early 1980's, while there is no obvious sign of trend movement before that. This phenomenon suggests that there might be a structural change in money transaction costs around this time. In addition, as shown in the figure, the time series behavior of F(y t ) replicates that of log(y t ); as a matter of fact, if I multiply F(y t ) by the average of log(y t ) to that of F(y t ), the two series almost coincides with each other. This fact reinforces the previous result that the income and interest elasticity of (4.2.3) is consistent with the Lucas  19  In other words, F(y,) is defined as the residual from the Lucas type of money demand function.  72  type of money demand function. Moreover, it suggests that F(.) is approximately a linear function of log(70. Figure 4.2 Money Transaction Costs from 1960-2000 -3  -3.5  -4  -4.5  -5  -5.5  -6 ^ ^ ^ ^ ^ ^ ^ ^ ^ 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005  Econometric Model: AR(p) To gain a better understanding of this phenomenon, I use an econometric model to identify the time series behavior of money transaction costs. I assume that the data generating process for logy t ) is an autoregressive process with a lag order of p [AR(p)] and a linear time trend. The AR(p) process is expressed as follows: log(y t ) = log(y) + bt + e t , P  Et  =Ep k E t _ k +u t , u t — iid(0,a 2 ) Epk <1. k=1  (4.2.6)  k=1  In (4.2.6), u t denotes the shock to the current growth rate of money transaction costs; b captures the quarterly trend growth rate of y t . (4.2.6) is equivalent to the following OLS model (4.2.8), which allows me to test if the time series of log(y t ) is trend stationary, and , if so, estimate the long-run deterministic trend values of log(7 t ) according to the following:  73  (4.2.7)  1og(7, ) = bt + log(y) ,^ P^  P  k=1^  k=1  logy t ) = (1-Ep k )log(y)+c+gt +Ep k log(y t _ k )+u t , u t - iid(0,a 2 ),^(4.2.8) where g = b(1-  P^  Ep  k  P  ), and c = bE kp k .^  (4.2.9)  k=1^ k=1  The best order of lag p is selected by three different procedures and criteria. In the first procedure which I call the OLS procedure, I alternate the number of lag p in a descending order from 10 to 1 in the regression model of (4.2.8), until the OLS coefficient p k for the last lag variable-- log(y t _k) is found to be statistically significant at the conventional significance level. Alternatively, I also employ the so-called Akaike Information Criterion (AIC) and Schwartz Bayesian Criterion (SBC) to determine optimal lag length in the AR process. These criteria that serve as the measures of the goodness of fit of an estimated statistical model are the two of the most widely adopted lag length selection criteria in a AR(p) model. Specifically, AIC and SBC statistics compute the log of some likelihood function of the AR(p) model with the fixed lag length. 2° The optimal lag length is reached when the corresponding test statistics are the lowest among those with alternative lag length. In the current analysis, I consider the case when the number of lag p is less than or equal to 10. In (4.2.8), the null hypothesis that the time series of log(7 t ) is trend-stationary can be tested against the alternative that it contains a unit root. The approval of the null hypothesis would lead to a straightforward economic interpretation of the data. The trend component of log(y t ) captures the long-run growth of financial transaction technology. For example, it captures the continuous diffusion process of adopting the new cash management technique such as ATM machines. 21 The temporary shock u t captures the short-run disturbance to the current growth rate of financial transaction technology. Since its impact gradually dies out, eventually log(y t ) would converge to  20  The formula for computing the AIC and SBC: AIC = 2L(1 &)+ 2K and SBC = 2L(E a2)+ K ln(n) , where k —  —  is the number of parameters in the AR(p) model, at is the residual at time t ,L(  E (1,2 ) is the log of the likelihood -  function of the AR(p) model and n is the number of observations. 21 Rosenberg (1972) argued that the diffusion of many innovations has two important characteristics: its apparent overall slowness on the one hand, and the wide variations in the rates of acceptance of different inventions, on the other.  74  its long-run trend growth path. On the other hand, the rejection of the null hypothesis would suggest that the determinant of the time series of log(y t ) is stochastic both in the short-run and in the long-run, and thus it would reject the existence of the long-run deterministic monetary equilibrium, as indicated by (4.2.3), in the general equilibrium model.  Estimation Results The key test results and estimates of AR(p) specification are reported in Table 4.2.  Table 4.2 Estimates of AR(p) Model 1960 -2000  1960.01-1981.01  1960.01-1981.01  1981.02-2000.04  Time Trend g  -0.000716  -0.00015  No trend  -0.0021  (t-statistics)  ( -2.88)  (-0.32)  AR(p) lag  4  4  4  2  Unit Root Test  Reject at 5%  Reject at 10%  Reject at 10%  Reject at 5%  DW statistics  2.03  2.11  2.10  2.02  R-square  0.95  0.80  0.77  0.96  b  -0.0074  -0.001  0  -0.0137  57$  41$  40$  40$  0.90  0.83  0.84  0.85  y(2002USD) iPk l k =I  The whole  ( -2.54)  sample period  At first, I examine the time series data for the whole sample period. There are total 164 observations. Both the AIC and SBC statistics are the lowest when the lag length equates to 4. The OLS procedure reports that the lag length of the last statistically significant coefficient is also 4, and thus the optimal number of lag for the whole sample period is undoubtedly 4. Durbin Watson statistic rejects the null hypothesis that error terms are serially correlated at 5% significance level. The time trend coefficient g is statistically significantly different from zero, so the quarterly descending rate of money transaction costs b is 0.0074 according to (4.2.9) and the annual rate is approximately 2.9%. The augmented Dickey Fuller test on the de-trended series 75  rejects the null hypothesis of unit root at 5% significance level and thus supports that the time series of log(7 t ) is trend stationary. Detailed AR(p) estimates and test statistics including the AIC - and SBC (under various number of lag) for the whole sample period are reported in the following table 4.3.  Table 4.3 Estimates and Test Statistics for the Whole Sample Period log(7, ) = -0.43 — 0.000716t +1.20log(7  ti  ) - 0.53log(7,_ 2 ) + 0.47log(7 t _ 3 ) - 0.2410g(7, 4 )  (-2.98)^(-2.88)^(15.38)^(-4.45)^(3.97)^(-3.13)  lag  1  2  3  4  5  6  7  8  9  10  AIC  -4.65  -4.70  -4.74  -4.78  -4.77  -4.76  -4.74  -4.72  -4.72  -4.71  SBC  -4.61  -4.65  -4.66  -4.69  -4.69  -4.62  -4.58  -4.55  -4.52  -4.49  Predicted long run M1 velocity -  The model predicted long-run M1 velocity, denoted by Z , can be calculated by log( t) 1111 = —log(Z t ) = —e i log(i t ) + F(7 t^(4.2.10)  In (4.2.10), the trend value of money transaction cost-- t is given by (4.2.7) and I use both the hp-filtered trend of the interest rate and the actual interest rate as the proxy for the long-run interest rate, respectively. The model predicted long-run M1 velocity Z t is computed by substituting y t and the i t h P into equation (4.2.4). Based on the previous analysis in Chapter 3, Z t is the unique positive root of the cubic function (4.2.4). In Figure 4.3, the star points represent the actual M1 velocity; the dotted line represents the predicted long-run M1 velocity computed by the hp trend of the interest rates and the solid line represents the predicted M1 velocity computed by the actual interest rates. It demonstrates that the predicted M1 velocity fails to capture the trend of actual M1 velocity. There are systematic forecast errors for the whole sample period. In particular, the predicted time series of M1 velocity (based on the hp trend of i t ) over-predicts the actual time series of 60s, and does a 76  relatively better job of tracking the actual ones for the period of 70s, but tend to underestimate them for the period after 1980. Figure 4.3 Actual vs. Predicted Ml Velocity from 1960-2000 2.6 2.4 2.2 2 1.8 1.6 1.4 1_2 1 * Actual Predictec1(1-hp) — Preclicted(i)  0.8 0_6 1960^1965  1970^1975^1980  1985^1990^1995 2000^2005  The mismatch between the predicted Ml velocity and the actual one might be due to the structural change in money transaction costs. If the financial transaction technology grew faster in the period before the financial deregulation than the one afterwards, the predicted trend values of money transaction costs based on the whole sample period would understate the actual values of yt prior to the 1980's and overstate them afterwards. This might lead to the mismatch and systematic forecast errors between the predicted M1 velocity and the actual one. Test for Structural Break To test the possibility of a structural change in money transaction costs, I use a rolling Chow test—variously called Quandt likelihood ratio statistic (QLR) — to identify whether and when there is a structural break in the OLS model (4.2.8). The QLR statistic is defined as the largest one among all the Chow statistics for each tested break date. I compute QLR over a 15% trimming of the whole sample period. Since the total number of observations is 164, I set the initial testing break date to 1966.Q1 which is the 25 th observation and the last testing break date to 1994.Q1, which is the 137 th observation. The Chow statistics are computed for each date in 77  between. The maximum F-statistic is 7.24 at the date of 1981.Q1. It supports the hypothesis of the structural break at this date at 1% critical value. First Sub Sample Period from 1960.Q1 to 1981.Q1 -  In this sample period, the SBC statistics and the OLS procedure indicate that the optimal lag length is 4, yet the AIC statistics suggests the lag length is 5. Here, I still choose 4 as the optimal lag length for the AR model. In a AR(4) regression model with the time trend, the DurbinWatson test rejects that error terms are serially correlated. The t-test rejects that the time trend coefficient is statistically significant at any conventional significance level (see Table 4.2). Therefore, I drop the time trend and re-estimate the AR(p) model. Detailed regression estimates and the AIC and SBC statistics are presented in table 4.4.  Table 4.4 Estimates and Test Statistics for the First Sample Period log(y t ) = -0.77 +1.77log(y t _ i ) - 0.711og(y t _ 2 ) + 0.681og(y t _ 3 ) - 0.311og(y t _ 4 ) (-2.61)^(10.74)^(-4.51)^(4.29)^(-2.83)  lag  1  2  3  4  5  6  7  8  9  10  AIC  -4.44  -4.47  -4.56  -4.62  -4.62  -4.61  -4.57  -4.55  -4.51  -4.47  SBC  -4.38  -4.38  -4.44  -4.47  -4.44  -4.40  -4.33  -4.27  -4.20  -4.13  In this case, OLS procedure, the AIC and SBC statistics suggest that the best order of lag is 4. The augmented Dickey fuller test rejects the hypothesis of unit root in time series data of logy, at 10% significance level, supporting that the time series of log(y t ) is stationary for this time period. The long-run mean of y t or the constant term in (4.2.7) is 0.0091. Without the time trend, this term could be interpreted as the stable long-run money transaction cost. The accumulated sum of estimated lag coefficients -- I k=I p k is 0.84, which implies that the transitory shocks to the growth rate of money transaction costs are highly persistent. By this estimate, it takes about 4 years for the impact of a 1% transitory shock on the growth rate of y t to die out.  78  Predicted long-run M1 velocity The predicted velocity of money is calculated by (4.2.11)  log( f i i t ) = —log(Z, ) = —e log(i, ) + F (y) .^  In comparison to (4.2.10), to predict M1 velocity, (4.2.11) uses the constant long-run money transaction cost y for this time period, instead of the time varying trend values (7, ) for the whole sample period. In Figure 4.4, I plot the predicted MI velocity computed by the hp trend of interest rates (dotted line) and by the actual interest rates (solid line) respectively, along with the actual M1 velocity (star points). As shown in Figure 4.4, the time series of the predicted M1 velocity successfully captures the trend of the actual M1 velocity. Since the variations in the predicted M1 velocity base on the variations in the nominal interest rates according to (4.2.11), this suggests that the nominal interest rate might be the key factor determining the long-run trend of MI velocity in this sub-period. Figure 4.4 Actual vs. Predicted Ml Velocity from 1960-1981 2.2  2  1.8  t6 ••  5  1.4  1.2 * Actual Predicted(i-hp) — Predicted(i)  1  0.8 1960  1965  1975  1970  1980  1985  year  In Figure 4.5, I sort the data pairs of the interest rate and M l to GDP ratio by the hp trend values of the interest rates and plot the actual figures of M1 to GDP ratio (described by star points) as 79  well as the model predicted M1 to GDP ratio curve (described by solid line) respectively. Figure 4.5 illustrates that the actual data of M1 to GDP ratio can be well approximated by the single predicted M1 to GDP curve.  Figure 4.5 Actual vs. Predicted Ml to GDP ratio from 1960-1981 0.03  0.025  0.02  0.015  0.01  0.005 ^^ ^ 1 1.1 1.2 0 5^0.6^0.7^0.8^0.9 M1 to GDP ratio  Second Sub-Sample Period from 1981.Q2 to 2000.Q4 In the second sub-period, the SBC and OLS procedure report the optimal lag length is 2, but the AIC statistics suggests that the optimal lag length is 3. Here, my democratic choice for the best order of lag p is 2. The Durbin-Watson test rejects that the error terms u t are serially correlated. The coefficient estimate of the time trend g is -0.003 and the t-test supports the hypothesis that it is statistically significantly different from zero at 2% significance level. The quarterly trend growth rate of financial transaction technology b is calculated to be 1.37% according to (4.2.8) and the implied annual trend growth rate is approximately 5.5%. The augmented Dickey Fuller test on the de-trended time series data of log(y,) rejects the null hypothesis of unit root at 5% critical value. This evidence supports that the time series data of log y t is trend stationary in this sub period. The regression estimates and the AIC and SBC statistics are given in the following table.  80  Table 4.5 Estimates and Test Statistics for the Second Sample Period log(y , ) = -0.74 — 0.00216t + 1.18log(y ,_, ) - 0.34log(y t _ 2 ) (-3.13)^(-2.54)^(10.81)^(-3.08)  lag  1  2  3  4  5  6  7  8  9  10  AIC  -5.02  -5.13  -5.14  -5.16  -5.12  -5.09  -5.18  -5.15  -5.21  -5.17  SBC  -4.97  -5.04  -5.02  -5.0  -4.93  -4.87  -4.93  -4.90  -4.90  -4.82  The estimate of long-run de-trended mean of money transaction cost  -- y  is 0.0092, which is very  close to the estimate for the previous sample period. Likewise, the accumulated sum of estimated lag coefficients-- 1 2  1  pu  is 0.85, which is also quite close to the previous estimate.  The close match between these estimates for different sub-periods lends support to the AR(P) specification used in this chapter. Moreover, these estimates suggest that the long-run trend growth in financial transaction technology is the key factor driving the difference in the time series behavior of money transaction costs between the two sample periods. Predicted Ml velocity The predicted Ml velocity is calculated in the same way as it is in the whole sample-period. Figure 4.6 shows plots of the predicted M1 velocity [computed by the hp trend of the interest rates (dotted line) and by the actual interest rates (solid line), respectively] and the actual M1 velocity (star points). As illustrated in this graph, the time series data of the predicted M1 velocity can successfully track the trend of the actual M1 velocity. Since the predicted M1 velocity is calculated by both  yt  and  i t hP  according to (4.2.10) in the second sub-period, it  suggests that both the long-run financial technological change and the variations in the nominal interest rates play determining roles in shaping the long-run M1 velocity. The inventory theoretic approach implies that the structural money demand function (4.2.10) shifts continuously and systematically in response to the exogenous trend growth in financial transaction technology. Figure 4.7 plots the sorted data pairs of actual M1 to GDP ratio and the nominal interest rates as well as the exogenous shifts in the predicted M1 to GDP ratio curves (dashed line). 81  Figure 4.6 Actual vs. Predicted Ml Velocity from 1982-2000 2.6  2.4  * Actual Predicted(i-hp) — Predicted@  2_2  2  1.8  1.6  1.4  1.2 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002  Figure 4.7 Actual vs. Predicted Ml to GDP ratio from 1982-2000 0.035  0.03  * Actual ^ Predicted exogenous shift  0.025  0.02  0.015  0.01  0.005 0 4^0.45^0.5^0.55^0.6 M1 to GDP ratio  0.65  0.7  0.75  Discussion on the Structural Break Date Based on the present estimation approach, the identified structural break date of the structural money demand function is ahead of the late 1980's that was noted by Lucas (2000). This is because the ways of estimating the long-run money transaction costs are different for the two approaches. In the calibration exercise of Lucas (2000), the "residual' term H in the structural money demand relationship (4.2.5) is a constant. In the language of my approach, it is equivalently saying that for the whole sample period, a stable long-run money transaction cost is present, for which the logarithm is approximately the long-run average of all log(y t ). This stable long-run money transaction cost, denoted by yt og , is calculated to be approximately 0.0072. By contrast, based on the present approach, the estimate of the stable long-run  y is -  equal to 0.0091  for the first sub-period, and for the second sub-period, the trend estimate of log(y t ) is given by log(y, ) = —0.0137t + log(0.0092) , ^  (4.2.12)  where 0.0092 is the long-run mean of the de-trended y t , and -0.137 is the quarterly trend growth rate of money transaction costs. If we plot my estimate of the long-run y t along with the estimate of Lucas (2000) in Figure 4.8, the estimate of Lucas (2000) is clearly lower than my estimate for the period from 1960s to 1986, while it is greater for the period from 1987 to 2000. As is also clearly seen in Figure 4.1, because of this, the time series of the predicted M1 velocity of Lucas (2000) tends to overestimate the actual M1 velocity before 1986 and underestimate the actual Ml velocity after that. Figure 4.1 and Figure 4.8 also suggest that, for the period from 1960 to the late-1980s, when Lucas (2000)'s estimate of the long-run y t is not far away from my estimate, his predicted M1 velocity seems to work fine in terms of tracking the actual Ml velocity; but after 1990 when his estimates start to deviate further away from the trend value of the long-run log(y t ), his predicted M1 velocity fails to match the actual M1 velocity. Sensitivity Analysis In this section, I check the sensitivity of my estimates to the changes in the benchmark parameter values and in the specifications of the representative agent's utility. The key estimates governing the structural change in the long-run money transaction costs are the structural break date, the 83  stable long-run money transaction cost for the first sub-period, denoted by 7t, the constant term in (4.2.7), denoted by 72, the time trend b , and the sum of AR(p) coefficients (the slope). Figure 4.8 Long-run Estimates of Money Transaction Cost [Lucas (2000) vs. Mine] X 10  3  10 ^  ^ Lucas (2000) — - My estimates 3 1960 1965 1970 1975 1980^1985 1990 1995 2000 2005  Changing the markup parameter .t In the general equilibrium model in Chapter 3, higher markup implies that firms have a higher profit margin and a lower wages to total revenue ratio. It leads to lower marginal interest forgone on wage payments, given the firms' total revenue from goods sale. Consequently, given the actual velocity of money (the frequency of cash deposits) observed in the data, the suggestion is that the firms must face a higher fixed money transaction cost, to match (equate itself to) the lower marginal interest forgone. It turns out that the variation in the markup parameter pt, only affects the size of the predicted long-run log(7 t ), without affecting its distribution. As a result, the estimates of the structural break date, the time trend b, and the AR(p) coefficients for each sub-period are unchanged. To show this, I select another two values for j.t: 1.11 and 1.15, and re-do the same calibration exercise as before. The new estimates are reported in Table 4.6.  84  Table 4.6 New Estimates under Alternative  pi  Benchmark ii=1.2  p=1.5  ia=1.11  Break Date  1981.Q1  1981.Q1  1981.Q1  (QLR statistics )  (7.24)  (7.24)  (7.24)  71  0.0091  0.0103  0.0087  'Y2  0.0092  0.0105  0.0088  -1.37%  -1.37%  -1.37%  0.84  0.84  0.84  0.85  0.85  0.85  b AR(p)1960-1981  AR(D) 1982-2000 -  ,  Alternative Structural Money Demand Functions As suggested by the analysis in the previous chapter, although the exact functional form of a structural money demand function might vary slightly under different assumptions for the representative agent's utility, the general properties of this class of structural money demand functions are that the income elasticity is 1 and the interest elasticity is close to 0.5. The small variation in the interest elasticity is caused by inflation effects. Here, I also check the robustness of my estimates across alternative structural money demand functions. First, I consider a Lucastype of structural money demand function, for which the interest elasticity is precisely 0.5. In the absence of the inflation tax and relative wage dispersion effects, the general equilibrium model in Chapter 3 would imply the Lucas-type of structural money demand function. Specifically, the derivation of such a function requires the assumptions that firms have a constant profit margin and that the relative wage between the banking sector and the production sector is constant. The structural money demand function (3.4.13) in the previous chapter, can then be approximated by the following Lucas-type of money demand function: wb N PC t +1) -1 ^ ) = At = A^{ Mt^ -IA i t^Wf tN t^W'^  2y id-19001+1)  (4.2.13)  Second, I also consider the structural money demand function which is derived from the general CES utility specification, as in (3.4.14), with various assigned values of the risk aversion  85  coefficient, a, and the labor supply elasticity coefficient, 0. It can be shown that the implicit structural money demand function of the CES utility specification takes the following form: 0 A 7 -I- 1 2-^1^ j0+09^ 7t = [ 1 ]' ° il"ei^ (p 0^B,7^B,7 + 2(Z, —1)  where B ,^ 7=  (4.2.14)  2Z, 2 [1+(Z, +1)i, /2Z,] A y = 1+ (Z, - 1)1 1 / 2Z t i [1+4 1+ (Z, +1)i, / 2Z t ] ' '^111+ (Z t +1)i t / 2Z t ] '^1+ (Z t —1)i t / 2Z t  Table 4.7 records the new estimates along with the previous estimation results that are based on the assumption that 6=1 and that 0=0, for comparison. The table shows that, even with the alternative values of a and 0, the estimates of the structural break date, the stable long-run log(y) (the intercept), the time trend b, and the sum of AR(p) coefficients (the slope) are almost identical to the previous estimates, indicating that the estimation results are robust with respect to alternative specifications of the structural money demand functions. Table 4.7 New Estimates under Alternative Structural Money Demand Functions Benchmark  Lucas-type  0=1,a=1  0=5,a=1  0=0,a=5  0=0,a=1 Break Date  1981.Q1  1981.Q1  1981.Q1  1981.Q1  1981.Q1  (QLR statistics )  (7.24)  (7.36)  (7.22)  (7.20)  (7.28)  71  0.0091  0.0093  0.0091  0.0091  0.0092  72  0.0092  0.0095  0.0092  0.0092  0.0093  -1.37%  -1.38%  -1.37%  -1.37%  -1.37%  0.84  0.83  0.84  0.84  0.84  0.85  0.84  0.85  0.85  0.84  b mi(p),960-,981 AR(P)1960-I981  4.3 Conclusion This chapter examines U.S. M1 money demand data for the period from 1960 to 2000 by calibrating a structural money demand function which is derived from a general equilibrium model of inventory money demand. The structural money demand function has an interest 86  elasticity close to 0.5 and an income elasticity of 1. By defining the fixed time cost of money transaction as being the unexplained residual part of money (M1) by the measure of income (GDP) and by the measure of the opportunity cost of money (short term market interest rates), I compute a time series of money transaction costs using this money demand function. Then, I analyze the time series behavior of the long-run money transaction costs by an autoregressive process with the lag order of p and a linear time trend. The main findings are as follows. By applying the rolling Chow test, I find a structural break in the time series behavior of money transaction costs occurring in 1981. For the period from 1960 to 1981, the long-run money transaction costs were stable, and consequently long-run money demand was stable too. For the period from 1982 to 2000, the money transaction costs were found to have a linear descending trend at a quarterly rate of 1.37%; consequently the money demand function might have shifted consistently. Given the identified growth process of money transaction costs, this structural money demand function can successfully track the last 40 years of the US long-run money demand for M 1 , without appealing to an exogenous structural break in the money demand function.  87  Chapter 5: Concluding Remarks This thesis has investigated the theoretical and empirical implications of the general equilibrium inventory theoretic approach to the demand for money. Specifically, it provides a positive theory to explain the key parameters that shape the demand for money — the income and interest elasticity of money; more importantly, it provides a unified framework to explain the pattern of the last 40 years of US demand data for M1. I conclude this thesis by underlining some limitations of the current work and potential extensions for future research. Firstly, the calibration approach of Chapter 4 gives rise to the importance of identifying the long-run trend growth of financial transaction technology. This approach, however, cannot explain why there was a structural break in the financial technological growth and what the fundamental determinants of the financial technology are. To answer these questions and also to further test the robustness of this approach, a more direct measure of financial transaction cost needs to be constructed. This is one empirical area which awaits future research. One indirect way to test the robustness of this approach is to compare and examine data for countries with similar experience. Secondly, the finding of a unique long-run money demand function in Chapter 4 lends support to the re-adoption of the monetary target policy. Yet, a stable long-run money demand function is only a necessary condition, but not a sufficient condition for the use of this sort of policy. Such a long-run function cannot account for the observed high volatility of short-run money velocity in US data, which could worsen the implementation of the monetary target policy. For example, if monetary authority adopts a k-percent constant money growth rule, then the high volatility of velocity would transmit to the nominal sector according to a quantity exchange equation. This would violate the price stabilization goal of the monetary authority. In this environment, the sufficient and necessary condition for the use of monetary target policy is both short-run and long-run stable money demand functions that fit data closely. Thirdly, one concern for the current calibration approach is related to the choice of monetary aggregate--M1, which has been commonly used in time series studies, especially over periods of 88  time when demand deposits become interest-bearing. As suggested by Lucas (2000), one response to this concern is to formulate a model of the banking system in which currency, reserves, and deposits have distinct roles. In this direction, Chari, Christiano, and Eichenbaum (1995) extended the basic Cash-in-Advance model as in Christiano and Eichenbaum (1995)] with the bank system and the roles of different monetary aggregates distinctly modeled. The model was used to investigate the responses of various monetary aggregates to monetary policy shocks and real shocks. Following their footsteps, the current model framework of Chapter 3, which extends the basic Cash-in-Advance model by endogenizing the income velocity of money, can serve as a promising framework in this direction to further explore the link among different monetary aggregates, the velocity of money and monetary policy shocks. At last, another extension is to apply the general equilibrium framework of Chapter 3 to the evaluation of the welfare cost of inflation, especially under low interest rates. As pointed out by Lucas, the money holding behavior at very low interest rates is central for estimating welfare costs. The general equilibrium model makes the prediction that the interest rate is zero at low interest rates, but close to 0.5 when interest rates are larger. As suggested by Mulligan and S alai-Martin (1996), this prediction might lead to the interesting implication that a lowering of the interest rate from a very small rate would have much smaller welfare gains than that from higher interest rates.  89  Reference Aiyagari, S.R., Gertler, M., (1991) "Asset returns with transactions costs and uninsured individual risk" Journal of Monetary Economics 27(3), Pages 311-331. 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Tobin, James (1956): "The Interest Elasticity of the Transactions Demand for Cash," Review of Economics and Statistics, 38(3), Pages 241-247. Vissing-Jorgensen, Annette (2002): "Towards an Explanation of Household Portfolio Choice Heterogeneity: Non-financial Income and Participation Cost Structures." National Bureau of Economic Research , Working Paper # 8884 Walsh, Carl E.(1998): "Monetary Theory and Policy." MIT Press, Chapters 2-4 White, Lawrence J. and Horvitz, P (2000):" Technological Change, Financial Innovation, and Financial Regulation in the U.S.: The Challenges for Public Policy", in P. Harker and S. Zenios, eds., the Performance of Financial Institutions, Cambridge University Press.  95  Appendices Al. Proof of Proposition (3.1) Proposition (3.1): Given the long-run nominal interest rate i, and the exogenous growth process of aggregate technology A t, one unique set of solutions can be found to { Z(i), N(i), L(i), C t (i, A t)) that determines the balanced growth equilibrium of real allocations.  2+  r  3  2 (^ )Z + Z ± 2i, 4 + 2i,  ,u+1^i(,u- 1) 2/dm  2,urcoo (2 + ) — lZ^  0  (3.4.9)  Given any constant , (po and 7, I will show that the cubic function (3.4.9) has a unique feasible, or positive root Z t (i t), which is a function of the interest rate i t . According to intermediate value theorem, every cubic equation with real coefficients has at least one real root. The proof of Proposition (3.1) proceeds as follows: first, I will show that there is at least one positive root for (3.4.9), and then I will show that there is only one positive root only. If a cubic equation has the following form: a3 x 3 + a2 x 2 + (X / X ao = 0  ^  (A.1.1)  where al, a2 and a 3 are real numbers. Let A denote the discriminant of the cubic equation (A.1.1). A = 4a 1 3 a3 —a, 2 a2 2 + 4a0 a2 3 —18a0 a 1 a2 a3 + 27a0 2 a3 2 A>0, one real root and two complex conjugate roots; A<0, three distinct real roots; A=0, at least two roots coincide. In (3.4.9): 2+i,^ ,u +1^i, Cu —1) ^ ; a2 = 1; = ac, = ^ 2i,^4 + 2i, 2,umo^2,unoo (2 + i ) ,  Thus, the discriminant of (3.4.9) is the following:  96  2+i^i^p+1 3^i i t^4U + 1 ) 2 _ 4i, (p —1) Cu — 1) it ^p +1 ^)^( A = 4()(^ ' +9 ( 2i, 4 + 2i, 2,u7rp o^4 + 2i, 2,uno0^pygoo (2 + i, )^2pyg 4 + 2i, 2,unoo +  )  — l) ] 2  27  [ (,u 4 2,u yvo  27(p —1) +9(p 2 —1)^7i,(11-1) < p +1 0^(A.1.2) I + 2i, 2pyg 2,unoo^4(2,unoo ) 2^4,urcoo (2 + i, ) A<0 Vp>1; Ti  it^/1+1  )2  ^^  That A < 0 suggests that (3.4.9) has three distinct real roots. Let Z1, Z2, and Z3 denote the three real roots of (3.4.9). If r is the root for the cubic function (A.1.1), the cubic equation can be always rewritten as follows: (Z t - r)[Z t 2 + (r + c )Z, +^+^r + r 2 ]. 0 a3^a3 a 3  (A.1.3)  Similarly, (3.4.9) can be also rewritten as follows: (Z t - Z 3 )[Z,2+ (Z 3 +  2i,^2i,^ )z, +^[^ p +1^2i,^ ^I+ ^ Z3 +Z 3 2 ] = 2+i,^2+i, 4+2i, 2pyg^2+i,  (A.1.4)  Now suppose there are no positive roots for (3.4.9). In this case, it follows:  Z, + Z 2 -(Z3 +  2 +t ) <  0^  (A.1.5)  2i, p +1 Z 1 Z 2 = 24 [ 4 ^/1+1 ]+ 4^2 Z 3 + Z 3 = ( 4 +Z 3 ) ^ > 0^(A.1.6) 2 + i, 4 + 2/, 2,uyvo^2 +^2+/,^2+ i, 2,uyvo According to (3.4.9): (p + 1)(1 + —+ —)Z 3 2 2i^+1^ g^21/2^2^3 2Z  ^2+i,  2 /-0V0 0^+1— —(1+ i + i ) 2^Z3^2 2Z 3  (A.1.7) ']  Substitute (A.1.7) into (A.1.6):  97  ^i^  i^i  (1+I+ ^) 2^(p+1)(1+ + ^)  Z 1 Z 2 = Z 3 2 [^  2 2Z 3^2  2Z 3  ^ i (1+ 1- ) 2^(1+ —i )[p+1 i (1+ ^ i + ' ) -1 ] 2^2^Z3^2 2Z 3  ]  (p-1)(1+ + ^)i  2 2Z 3 = i^ ^ i^i ^ 2+i (1+—i )[g+1— —(1+ +^) ] 2^Z3^2 2Z 3  £1  it ) 2 ^11  2 +^(p +1),uy(po Z 3  (A.1.8)  ^< OVZ 3 < 0„u > 1  This contradicts to (A.1.6). Thus (3.4.9) must have at least one positive root. Let Z3 denote one positive root of (3.4.9). According to (A.1.5) and (A.1.6), Z 1 + Z 2 = -(Z3 + if^ ) < 0;^ 2+ Z1Z2  1t µ - 1^>0^ 2 +i t^(la +1)p)q)0 Z 3  (A.1.9) (A.1.10)  (A.1.9) and (A.1.10) imply that Zt<0 and Z2<0. Therefore, there is only one positive root Z3(it) for (3.4.9). Once Z, becomes known, the rest equilibrium solutions for {N t , C t , Lt } can be solved from (3.4.2), (3.4.3) and (3.4.1) respectively. Q.E.D  98  ^ ^  A2. Calculation of Proposition (3.2) Proposition (3.2): Given a small positive nominal interest i t , the interest elasticity of money is close to 0.5.  The money demand function (3.4.9) can be rewritten as follows: i^ 21.11wpc,(1+- +-- -) 2 Z, 2 =[(g+1)(1+--L)+(-1) ]i, (A.2.1) 2 2iZ ,^2^, 2Z Take logs on both sides of (A.2.1): ^i log(2ptocoo ) + 2log(1+-i +--)+ 2logZ, = logRpt +1)(1+^logic^(A.2.2) 2 2Z,^ 2^2Z, Take total derivative with respect to 4 and i t on both sides of (A.4.2) and then rearrange terms:  11+1^. g-1 it^ it ± it^(^ + ^)1, (µ1) i t^ 2^2Z, di , Z,^2Z,^dZ,^Z ^ [ 2 ^ ^ l ^ = H. ^ + ]^ i i i,^Z,^ it + it ^i l) ^ ^1+1` + 1+ t^ + i t^41+1)(1+ t)+(g 1)^ it 2 2Z,^2^2Z,^2 2Z,^2^2Z, (A.2.3) Recall the definition of the interest elasticity: dm, + ,/m, + ,^dZ,/Z, e =^- ^ di,/i^di,/i  (A.2.4)  (A.2.3) implies:  i ^11+1^tt -1 .^ .+ i^ ^(^ it i + ^)1 (I-1^1 )^t dZ /Z 2^2Z 2Z, -em = ]/[ 2 ^. Z ' .^+ ^ ' ' = [1^ Z '. +^ .^1 1 ^ i ' 1 1 i m^di t iii 1+ ` + 1 '^(g+1)(1+ ' )+(tt 1) ^1+ ' + ' ^(g+1)(1+ ')+(.t 1) t ` 2 2Z,^2^2Z,^2 2Z,^2^2Z, :  i^.^ii, -1 .^ p.-1. it +21, i , +(tt+l)i, 1 Z^Z^ 2Z, ^ 2Z, ' ^0.5 [1-0.5 ` ^+0.5 ^ +0.5 ^ . ] i^i^i^I ^It i 1+ ' +^t '^ ^+ I '^ 01+1)(1+ ' )+0.i. 1) `^01+1)(1+ ' )+ (p, 1) ` 2 2Z,^2 2Z,^2^2Z,^2^2Z, (-1-1 ) ^ Z^ 2Z , 2 ^ ` ^+ ^  i i i i 1+ ' + t^(1+1)(1+ ' )+(.i. -1) 2 2Z,^2^2Zt ,  i, i 11+1 + ii -1 . (^ ^,^ i + i '^ )1 (µ -) ' 2^4Z,^2Z,^Z,^ 2Z, ] = 0.5+{ ^ }/[ 2 ^ + ^ i i ^i i (g +1)(1+ )+ (It 1) it ^1+ ' + ^1+i + ^ (1+1)(1+ ^)+(g. 1) ^ 2^2Z ^2 2Z,^2 2Z,^2^2Z,  (A.2.5) 99  ^ ^  Since: -1i , it + 11^-r ^t^ +1 2Z,^2Z, 1+t` + — 1 i, ^ 1+ i t + i t 2 lit +1 2Z,^2 2Z,  (A.2.6)  Hence: 1i . +µ -1 0.5^' 11+1 2Z, —0.5+  i, Cst 1+1L 1+ + ^ 2 g+1 2Z,  > e in >-0.5+  1+^ 0.5^2Z, Cst 1+  2 2Z,  — 1) ^  Z,^2Z, Cst = 2 ^ +^ i 1+ + ^ (1+1)(1+ )+(µ-1) 2 2Z,^2^2Z,  The Cst term is close to 2 and both terms in inequality (A.2.6) are very small and close to zero for small and positive interest rate i t , thus the interest elasticity of money is close to 0.5.  1 00  

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