A three asset determination of the transactions demand for money

RICHARD W. DOUGLAS, JR. Bowling

Green

State

Bowling

University Green,

Ohio

A Three-Asset Determination of the Transactions Demand for Money This paper uses the idea behind the well-known Baumol-Tobin model to develop a three-asset model of the transactions demand for money. A number of monetary issues that are not easily discussed in the two-asset framework can be discussed in the context of a three-asset model. Two versions of the model are employed. An LM curve is derived from both versions so that several issues can be explored using IS/LM analysis and the three-asset model can be more easily compared with the two-asset model.

1. Introduction Monetary policy in the late 1980s presents a number of challenges. Two of the most interesting problems arise from the continuing controversy over the appropriate definition of money and from the deregulation of depository institutions, which has allowed interest to be paid even on narrowly defined money (M,). The first problem has led the Fed to target more than one money aggregate and at various times to emphasize one aggregate over others. The second problem has led to account switching by the public that has produced great instability in the velocity of M1 and finally caused the Fed to abandon its Ml target in 1987. Unfortunately, neither issue is easily analyzed using the time honored IS/LM model, which despite its shortcomings remains a standard feature in macroeconomic texts and continues to make regular appearances in journal articles. The problem is that the IS/LM model admits only two assets (money and bonds) and therefore does not allow discussion of multiple money aggregates or interest differences on the components of a single aggregate. This paper proposes to build a framework for discussion of these and other issues by building a three-asset model that can easily be adapted to the IS/LM model. Of course, three-asset models are not new. The models developed by Tobin (1969), B runner and Meltzer (1963, 1972), Hendershott (1976), and Meyer (1980), among others, focus on a third asset that provides a vehicle for storing wealth (that is, equities) JoumaE of Macroeconomics, Winter 1989, Vol. Copyright 0 1989 by Louisiana State University 0164-0704/89/$1.50

11, No. Press

1, pp.

95-108

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Richard

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rather than one that has transactions properties. Orr (1971), on the other hand, has developed a three-asset transactions model. Like Orr, we consider the transactions motive for holding a third asset. But our approach is to extend the well-known models of Baumol (1952) and Tobin (1956) to three assets.’ This kind of model is, unlike the Orr model, nonstochastic, so it is simpler and more easily adaptable to the IS/LM framework. We present two versions of the model since each allows the analysis of a different set of issues. From the money demand equations we develop, we can derive LM curves and discuss problems of financial deregulation and monetary policy that are not amenable to standard IS/LM analysis. The “cost” of this exercise is that the standard macro model is made more complex with the addition of a third asset and a second rate of interest. The “benefit” is the additional insight provided on a number of issues. Since the basic simplicity of IS/LM analysis is retained, hopefully the reader will conclude the benefit exceeds the cost.

2. Three-Asset

Model

I: Currency,

NOW Accounts,

and Bonds

In the Baumol-Tobin model, a person receives income at the beginning of each period. Some of the income is used to buy bonds and some of it is held as money. Regular expenditures are assumed to exhaust gradually the money balance at which point some of the bonds are sold to rebuild it. The new money balance is spent, more bonds are sold, and so on, until at the end of the period, all of the income is spent and bond and money holdings are zero. The person determines how many times to sell bonds (and thus the size of his average money balance) by minimizing the cost of managing his portfolio. That cost is the opportunity cost of holding money (the interest rate times the average money balance) plus a transactions cost which results from a brokerage fee that is incurred whenever bonds are bought or sold. The well-known result is that the average money balance has an income elasticity of l/2 and an interest elasticity of -I/2. ‘The effect of credit cards model has been explored by ever, the inclusion of credit model, for credit cards merely the desired amount of bonds provide a third alternative to

96

on money demand in the context of the Baumol-Tobin Akhand and Milbourne (1986), among others. Howcards does not turn Baumol-Tobin into a three-asset allow deferred payment. While this might increase and reduce the desired amount of money, it does not bonds and money.

A Three-Asset

Determination

Now add a third asset to the model by assuming that there are two kinds of money-an interest-bearing checking account (NOW account) and currency.2 Suppose at the beginning of the period the person decides to spend a certain proportion of his income by using currency and the rest by writing checks on his NOW account. In order to use currency, he goes through a two-step conversion process that proceeds as follows. The portion of income ultimately to be spent in currency is initially invested in each of the three assets-currency, bonds, and a NOW account. Regular expenditures exhaust the initial currency balance, but instead of selling bonds to replace the currency, a check is cashed. Each time the currency balance falls to zero another check is cashed, until eventually the NOW balance is also exhausted. At this point bonds are sold to restore the initial currency and NOW balances. Then more currency is spent, more checks are cashed, and so forth, until once again the currency and NOW balances are exhausted. More bonds are sold and the process is repeated until all of the income the person planned to spend in cash is spent by the end of the period. As in the two-asset model, a brokerage fee is incurred whenever bonds are bought or sold. But in addition, a check cashing fee is incurred whenever some of the NOW balance is converted to currency. Let Y = A= N = C = rg = r, = b2 = b, = Kx =

income received at the beginning of the period, the proportion of income ultimately spent in currency, the initial NOW balance that will be converted to currency, the initial currency balance, the interest rate on bonds, the interest rate on NOW accounts, the brokerage fee, the check cashing fee, and the cost of managing the XY portfolio.

‘According to Orr (1971). the Baumol-Tobin type model is better suited to describing household behavior, while the stochastic model he develops is better suited to describing the liquidity management problem faced by firms. Since firms are not presently allowed to own NOW accounts and do not typically spend currency, the three assets chosen here, as well as the model itself, seem more appropriate for the household than the firm.

97

Richard

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Jr.

As before, there is an opportunity cost and a transactions cost, but these are more complex than in the two-asset model. The opportunity cost of a dollar of currency is rs; the opportunity cost of a dollar in a NOW account is ra - r,. The brokerage fee is incurred XY/(N + C) times (including once for buying bonds) and the check cashing fee is incurred N/C times after each bond sale and N/C times after the initial bond purchase.

(1) c -= 2

F-1 b&Y 2r,

‘I2

(2)

’

(3) Differentiation of (1) with respect to C and N yields the costminimizing average balances shown in (2) and (3). Equation (3) shows the average money balance for the portion of income ultimately spent as currency. But (2) and (3) are inappropriate for extreme parameter values, where the model degenerates to a two-asset problem. For example, if the brokerage and check cashing fees are the same and the interest rate on bonds is higher than the NOW rate, bonds are always preferred to a NOW account. ‘If the NOW rate is zero, currency is always preferred because checks are costly to cash. In both cases, the person would have to make a bonds-currency decision, but would not bother with a NOW account.3 To complete the model and determine the total transactions demand for money, we must account for the rest of the income, which is spent by check. Here the person makes a two-asset decision involving bonds and the NOW account.

zero

98

31n the three-asset model, as (b, - b,)/b, approaches

(4) and (5) suggest (r, - rJ/rN.

that

the

NOW

balance

approaches

A Three-Asset

Determination

Let 1 -

h = the proportion of income spent by check, fi = the initial NOW balance that is spent directly rather than converted to currency, and K,-, = the cost of managing the (1 - A)Y portfolio. (4)

(5) The average money balance for transactions by check is found by adding (3) and (5). M -=2

N+C 2

and

fi +i

= w2 - m11’2 + [bz(l 3. Implications

in currency

of Three-Asset

- W’“~

Model

.

(6)

I

As in the two-asset model, an increase in the brokerage fee or a decrease in the interest rate on bonds increases the demand for money. Changes in the demand for money associated with financial deregulation and other factors may be explained by the terms in (6) that do not appear in the two-asset model. Two products of deregulation are the NOW accounts, which have been available nationwide since 1981, and automatic teller machines, which have become more widespread in part because laws that limit branch banking have been weakened. In our model, these events increase the NOW rate (from zero) and decrease the check cashing fee (since it is easier to reach a banking facility). Both an increase in the NOW rate and a decrease in the check cashing fee increase the demand for money, according to Equation (6).” An increase in the proportion of income spent as currency (A), which might result from growth in the “underground economy” or 4Here and in the subsequent come spent as currency is not be the case.

discussion, we assume that affected by the NOW interest

the proportion rate, which

may

of innot

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Richard

W. Douglas,

Jr.

from increased use of automatic tellers, might increase or decrease money demand. Differentiation of (6) with respect to A shows that money demand is maximized when h = (bz - b,)/(2b, - b,). This critical value for A is close to l/2 if the check cashing fee is small relative to the brokerage fee. Since Federal Reserve surveys suggest X may in fact be near l/2, it is unclear whether an increase in h would increase or decrease money demand. A change in money demand might also result from a change in the relative values of the bond rate and the NOW rate. To analyze this problem in the context of the three-asset model, it is useful to derive an LM curve where the NOW rate is assumed to depend on the bond rate.

j@= YYk 6-B- riJh.

(7)

r, = gr, + 2 .

(8)

Equation (7) is an LM equation for an exogenous money supply (fi) and a money demand equation similar to (6). The variables A, bi, and b, in (6) are assumed to be constant and are captured by the constant (y) in (7). The income and interest elasticities are specified as k and h rather than I/2 for the sake of generality. In Equation (8), g and Z are constants, where 0 c g < 1, and Z may be either positive or negative. If Z is positive (negative), a given percentage change in the bond rate causes a smaller (larger) percentage change in the NOW rate. By substituting (8) into (7) and differentiating, we can calculate the elasticity of the bond rate with respect to income along the LM curve. r,=

[;]I’“[-$--]

drB/rB -= a/y

[ I[ -k h

1

1 + Z/(r/ii3)“hYk’h

The elasticity of the LM curve derived transactions model is k/h, but (10) sh ows that three-asset LM curve may be greater than or pending on the sign of Z. For example, if Z is loo

(9)

y’/^t&.

1 .

from the two-asset the elasticity of the less than k/h, denegative, the NOW

A Three-Asset

Determination

rate to bond rate ratio increases as interest rates rise, so the LM curve’s elasticity exceeds k/h and is relatively steep. (The limiting case occurs if g = 1 and Z is negative, so there is a constant gap between the two rates. This produces a vertical LM curve.) Figure 1 illustrates the opposite case. LM, assumes no interest is paid on checkable deposits, which yields the standard twoasset LM curve. LMa allows interest on NOW accounts and assumes 2 is positive, so LMr, is relatively flat. An increase in the money supply causes an equal rightward shift in both LM curves, but as Figure 1 indicates, the increase in income is not as great for LMB since the interest rate on bonds does not fall as far. As the money supply increase reduces the bond rate, the difference in the rate of return on bonds compared to money is reduced much faster than if money did not pay interest. Consequently, the demand for money increases more rapidly and the bond rate does not have to fall as far to restore equilibrium in the money market. The preceding analysis provides a nice textbook explanation for the changing impact of monetary policy. That macro texts are aware that interest on money can make a difference is evidenced by Gordon (1987), but Gordon reaches the conclusion that interest on money should always make the LM curve steeper. This is because he arbitrarily includes interest on money in a linear money demand function without providing the sort of justification that we have for a nonlinear function in light of our three-asset transactions model. The flatter LM curve of Figure 1 seems to be supported by the events of late 1985 through early 1987.5 Rapid money growth produced a decline in interest rates but little real growth or inflation. The rate on checkable deposits fell by a much smaller proportion than other interest rates, and the consequent large increase in money demand reduced the impact of money on spending so much that the Fed concluded it should abandon its MI target altogether.

4. Three-Asset

Model

II: Money,

Near Money,

and Bonds

The three assets used in developing our previous transactions model were currency, NOW accounts, and bonds. By renaming the

ship

‘See Darby, et al. (1987) for between money and income

a comprehensive through 1986.

review

of the

changing

relation-

101

Richard

W. Douglas, Jr.

Figure

1.

assets and considering the same two-stage transactions process, we can gain some additional insight into monetary issues and policy. Suppose that at the beginning of the period, income is invested in money, near money, and bonds. In this model, it is assumed that money does not pay interest. Near money, which does pay interest, includes items such as savings acounts and money market deposit accounts. After the initial money balance is spent, some of the near money is converted to money, more spending occurs, and so on, until the near money is exhausted. Then bonds are sold to replace the money and near money, and the process is repeated until all the income is spent by the end of the period. Following the same logic of cost minimization as before, we can determine the demand for two kinds of money. Let Y = income received at the beginning of the period, S = the initial near money balance, Mi = the initial money balance, rate on bonds, rB = the interest rs = the interest rate on near money, b2 = the brokerage fee, b, = the fee for converting near money to money, K = the cost of managing the portfolio, and M2 = money plus near money. 102

A Three-Asset

Determination

(11) Ml -= 2

[1

b,Y lk2 2rs ’

M2 -=-= S + Ml 2

2

(12)

[

(b, - b,)Y 2(rB - rJ

1 1’2

’

(13)

Equations (12) and (13) are similar to (2), (3), and (6) in the previous model, except that A does not appear in (12) and (13) since all transactions are assumed to be carried out in Ml. Equation (12) suggests that the demand for money depends only on the interest rate on near money (that is, the short-term rate) rather than the interest rate on bonds (the long-term rate). This is different from the prediction of Orr’s (1971) three-asset model, but it is consistent with the landmark empirical study by Goldfeld (1973).

5. Implications

of Three-Asset

Model II

One important event associated with financial deregulation is the increased liquidity of assets outside the Ml definition of money. In our model, this is represented by a decline in the cost of converting near money to money, which can be seen in (12) and (13) to decrease the demand for Ml and to increase the demand for M,. This is exactly what happened during what Gordon (1987) calls “the early stages of financial deregulation” (1974-1980). The demand for Ml fell as repurchase agreements and money market mutual funds became more popular. The increase in income that occurred during this period without a corresponding increase in money is sometimes called “the case of the missing money.” To see how our second model illuminates other monetary issues, it is useful, as before, to derive an LM curve. In our previous LM derivation, we assumed money (NOW accounts plus currency) was exogenous and that the NOW rate could be eliminated from the LM equation by assuming it was functionally related to the bond rate. Here we assume Ml and M2 (money plus near money) are 103

Richard W. Douglas, Jr. exogenous and that both the bond rate and the near money rate are endogenous. The money market equilibrium equations are derived from money demand equations (12) and (13).

Ml = YIYk’ ?-$I.

h-4,=

YJk2 (r, - ?-JhP .

05)

As in our previous LM derivation, the conversion fees are included in the constant terms y1 and yz. And again, for generality, we assume the income and interest elasticities are kr, kz, hl, and h2, rather than l/2. The LM curve is derived by substituting (14) into (15): V-3) The elasticity by (17). -dr,/r, a/y

of the bond rate with respect to income is given

= (kz/h,)(y,/iG,)“hz Ykelhz+ (kl/hl)(yl/til)“h’ (r2/n;Iz)llhz ykzlh + (n/~l)‘lh yhlh

Ykllhl .

(17)

Equation (17) indicates that the elasticity of our new three-asset LM curve is the same as the two-asset LM curve if k, = k2 = k and h, = h, = h. In both cases the elasticity is k/h. To determine the effect of monetary policy, we must make some assumptions regarding the relationship between ti, and ti,. Suppose the Fed wants to maintain a constant proportion of R, to it?r, such that & = q&i,. (Such a policy could be achieved by appropriate adjustment of the reserve requirement on money or near money if one of the aggregates grew too rapidly, though this would not be necessary if the public’s preference for currency and near money compared to checkable deposits remains constant.) The horizontal shift in the LM curve that results from a proportionate increase in tir and Xf’, is found by substituting for ti, in (16), rearranging terms, and differentiating dY/Y

(l/h,)(y,/qti,)“hp

Ykn’hz+ (l/hJ(~Jfil)“h’

d&2,/h%, = (kz/hz)(y2/qA21)1’h2 Ykzlhz+ (kl/hl)(yI/n;il)“h’ 104

Ykl’hl Ykl’hl *

(18)

A Three-Asset

Determination

Again, assuming kl = k, = k and h, = h2 = h, the elasticity of Y with respect to M is l/k, the same as it is for the two-asset LM CXU-Ve.

So far we have restricted the model so that our new threeasset LM curve has the same properties as the two-asset LM curve. If the income elasticities of the two aggregates are the same, if the interest elasticities are also the same, and if the ratio of B, to ti, is constant, then the addition of i@, to the model makes no difference. This is an interesting conclusion in itself, but of course these conditions may not be satisfied in the real world. Suppose, for example, we consider other motives for holding money besides the transactions motive. A precautionary and/or a speculative demand would, when combined with the transactions demand, increase the income and interest elasticities to some value greater than I/2 in the two-asset model.” However, in the three-asset model, it seems reasonable that a nontransactions demand would increase the elasticities of near money but not of narrowly defined money. (Why hold money for precautionary and speculative reasons when near money can be used just as easily and pays interest as well?) A proportionate increase in k and h weakens the effect of monetary policy in the two-asset model. In the three-asset model, a proportionate increase in k, and h, (holding k, and h, constant) also weakens the effect of monetary policy, but it is difficult to make a precise comparison between the models on this point for we would need to know the magnitude of the k, and h, increase that would correspond to a given k and h increase. Suffice it to say that the three-asset model has more predictive power than the two-asset model, assuming accurate estimates of kl, k,, hl, and h, can be obtained. Another issue amenable to discussion in the context of the three-asset model is the issue of multiple money targets. Equation (16) suggests that if the Fed either does not want or cannot maintain a constant proportion between h?, and ti2, it may have trouble stabilizing income. Since both 1Li, and ti, appear in (16), an increase in either one increases the demand for bonds and lowers the bond rate (shifts the LM curve rightward). This provides a justification for multiple money targets, which 6As Hall and Taylor (1986) point out, the empirical evidence money is consistent with the notion that the transactions demand part of the total demand. For a good summary of the literature,

on the demand for for money is only see Laidler (1985).

105

RichardW. Douglas,Jr. is often overlooked. It is often claimed that the Fed targets multiple aggregates because it cannot decide which one is most closely related to income, or because it wants to confuse the public and avoid accountability for its actions. In fact, as our analysis shows, no single target can be used to stabilize income unless there is a constant (or at least predictable) relationship between the aggregates. If, for example, the Fed thinks it knows the relationship between Mi and income and focuses on MI target to achieve its income goal, income will increase too rapidly if Mz increases faster than the Fed expects. (If the Fed has taken no account whatever of M,, income will increase too rapidly if M2 grows faster than MI.) The Fed has the same problem if it targets M2and MI grows too rapidly. This result, while nicely demonstrated by the three-asset model, should not be too surprising. If MI is constant while M2 increases, near money must be increasing. If M2 is constant while MI increases, people must be switching from near money to money. In either case, the liquidity of the public increases, so spending and income should rise.

6. Conclusion The addition of a third asset to the standard IS/LM model allows one to explore a number of questions that are difficult to treat in the two-asset framework. This paper has considered several of them. In addition, either of the three-asset models we have developed could be used in other ways that might interest the reader. For example, in Model I, one might consider the effect of making the proportion of income spent as currency endogenous through allowing it to depend on the NOW account interest rate. In Model II, the effects of allowing ii!f, to depend on A?, could be explored more fully. Profit-maximizing behavior by banks might suggest that the supply of near money depends on the supply of MI, the bond rate, and the money rate in a more complex way than we have assumed.7 There are, no doubt, other possibilities for extension of the model. Receiued: June 1987 Final version: March

1988

‘For example, the MS/M, ratio may a negative function of the near money

106

be a positive rate since

function of the bond rate and banks might tend to be more

A Three-Asset

Determination

References Akhand, H., and R. Milboume. “Credit Cards and Aggregate Money Demand.” Journal of Macroeconomics 8, no. 4 (Fall 1986): 47178. Avery, It., G. Elliehausen, A. Kennickell, and P. Spindt. “Changes in the Use of Transactions Accounts and Currency From 1984 to 1986.” Federal Reserve Bulletin 73, no. 3 (March 1987): 179-96. Baumol, W. “The Transactions Demand for Cash: An Inventory Theoretic Approach.” Quarterly Journal of Economics 66 (November 1952): 545-56. Brunner, K., and A. Meltzer. “The Role of Financial Intermediaries in the Transmission of Monetary Policy.” American Economic Review 53 (May 1963): 372-82. -. “Money, Debt, and Economic Activity.” Journal of Political Economy 80, no. 5 (September/October 1972): 951-77. Darby, M., W. Poole, D. Lindsey, M. Friedman, and M. Bazdarich. “Recent Behavior of the Velocity of Money.” Contemporary Policy Issues 5, no. 1 (January 1987): l-33. Goldfeld, S. “The Demand for Money Revisited.” Brookings Papers on Economic Activity 3 (1973): 577-646. Gordon, R. “The Demand for Money and the Effects of Financial Deregulation” (chap. 13). “Federal Reserve Monetary Control and Its Limitations” (chap. 14). In Macroeconomics. 4th ed. Boston: Little, Brown, and Co., 1987. Hall, R., and J. Taylor. “The Monetary System.” In Macroeconomics: Theory, Performance, and Policy. New York: W.W. Norton and Co., 1986. Hendershott, P. “A Tax Cut in a Multiple Security Model: Crowding-out, Pulling-In, and the Term Structure of Interest Rates.” Journal of Finance 31 (September 1976): 1185-99. Laidler, D. The Demand for Money: Theories and Evidence. 3d ed. New York: Harper and Row, 1985. Meyer, L. “The Supply and Demand for Bonds (A Three-Asset Model).” In Macroeconomics: A Model Building Approach. Cincinnati, OH: South-Western Publishing Co., 1980. Orr, D. “Three Asset Models of the Demand for Money” (chap. 6). “The Household Demand for Cash” (chap. 7). In Cash Management and the Demand for Money. New York: Praeger, 1971. aggressive creased.

in competing

for

funds

by

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CDs,

etc.,

if the

“spread”

were

in-

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W. Douglas, Jr.

Tobin, J. “The Interest Elasticity of the Transactions Demand for Cash.” Review of Economics and Statistics 38 (August 1956): 241-

47. -.

“A General

Equilibrium

Journal of Money, Credit,

Approach

and Banking

to Monetary Theory.” (February 1969): 15-29.