- Email: [email protected]
The individual's transactions demand for money
Journal of Monetary Economics 2 (1976) 237-256. 0 North-Holland Publishing Company
THE INDIVJDUAL’S TRANSACTIONS
DEMAND FOR MONEY
A utility maximization approach Robert E. ANDERSON* United States Department of the Interior, Washington, DC, U.S.A. This model of the transactions demand shows how an individual may simultaneously choose patterns of consumption, money holdings, and bond holdings over time that maximize utility when faced with a wide variety of possible cyclical patterns in his flow of income. Interesting conclusions about the transactions demand and the real balance effect are derived. For example, there is no theoretical reason to believe that a single individual’s demand for money is proportional to his income, and a small excess stock of money is likely to cause a large increase in his level of consumption.
1. Introduction is now over twenty years since Baumol first presented his analysis of the transactions demand for money based on the classic lot-size model from inventory theory. Though this model has been presented in somewhat different ways by Tobin and then Johnson (1969), it has remained the basic model of the transactions demand for money. This is perhaps not surprising even though the lotsize model has at least two serious shortcomings when it is used to describe a consumer or household’s demand for money. The basic motive for holding transactions balances of money has long been recognized to be the lack of synchronization between receipts and expenditures over time. But until comparatively recently the techniques available to solve optimization problems that explicitly included the time dimension have been very limited. The first shortcoming of the lot-size model when it is used to describe a consumer or household’s demand for money is the assumption of a highly arbitrary pattern of receipts and expenditures.’ An individual’s income is assumed to be received in equal lump-sum amounts at the beginning of periods of unit length. This income is then spent at a constant rate on consumption, creating the now famous ‘sawtooth’ pattern of surpluses. Though this assumed pattern of income It
+I would like to thank Jiirg Niehans for his initial encouragement to do research in this area and for a number of helpful comments and suggestions. Part of this research was undertaken while I was a recipient of the Harold Stonier Fellowship from the American Bankers Association. ‘The lot-size model can also be used to describe the transactions demand for money by economic units other than households (business firms, financial institutions, etc.), but in this paper I am only analyzing the demand by htiuseholds.
238
R.E. Anderson, The indi,vidual’s transactions demand
may bc close to an individual’s pattern of wage or salary income, there are a number of other sources of income tha.t may be received in an entirely different pattern -for example, financial income, overtime pay, bonuses, and income from seasonal employment. Instead of a constant rate of consumption, an individual may also have a definite pattern of expenditures - for example, high rates of expenditures in August for a vacation, in December for Christmas, and at the beginning of each month because that is when bills are due. Can the ad hoc techniques of the lot-size model be used to analyze models with more general patterns of income and expenditures? My own efforts have convinced me that this is not possible and an entirely new approach is necessary. The second shortcoming of the lot-size model is the assumption of an exogenously’ given pattern of consumption. In other words, an individual’s decisions about consumption are independent of his decisions about money or bond holdings. In the lot-size model the rate of consumption is given exogenously, and the individual then determines the optimal pattern of money and bond holdings to finance :he resulting surpluses. Hewever, this is at odds with one of the oldest goals of economic theory, which is to explain how an individual makes decisions about consumption. In the model of the transactions demand for money to be presented here, I will show how an individual might simultaneously choose the pattern of money holdings, the pattern of bond holdings, and the pattern of consumption over time that maximize utility when faced with a wide variety of possible patterns in his 3ow of income. From this model, one can gain interesting insights into both Lhe transactions demand for money and the real balance effect that are not possible from the lot-size model because of its restrictive assumptions. However, this more general and sophisticated model is not superior to the lotsize model in all respects. One of the major advantages of the lot-size model is that it can be used to analyze the case of fixed or lump-sum transactions costs. Because of the still inadequate mathematical optimization techniques, transactions costs in the analysis presented here will have to be assumed to be proportional to the amount of bonds bought or sold, 2. Basic assumptions A basic motive for an individual to hold stocks of any asset is the lack of synchronization over his lifetime between his pattern of income and his desired pattern of consumption. On the one extreme, an individual must decide what assets to hold when he may keep them for almost his entire life-time, for example, saving for old age or retirement. On the other extreme, he must decide in what form to hold his monthly salary until ir is spent for consumption during the subsequent four weeks. It is the daily, weekly, or monthly pattern in consumption and income that creates the transactions demand for assets. In order to separate the problem of the transactions demand for assets from
R.E. Anderson, The individrcal’stransactions demand
239
the more general problem of life-cycle saving and portfolio composition, assume that there is a pattern in an individual’s nonfinancial or endowment income that lasts for a period of unit length and repeats itself unchanged period after period. For ease of reference, call this unit period a ‘year’. Denote the flow of income by the continuous function x(f), and thus the assumption is that x(t+ 1) = x(t). The particular function x(t) which would approximate the pattern of income used in the lot-size model would have periodic peaks during the year and then quickly drop to zero for the intervals between the peaks. In a more general model than that presented here, annual income would also be assumed to vary over an individual’s lifetime. Such a model should explain how an individual determines his lifetime pattern of consumption, saving, and portfolio composition in addition to his transactions demand for assets. The theory of portfolio composition based on uncertainty about interest rates pioneered by Tobin and Markowitz would have to be an essential part of the model. To avoid the considerabie complexity of such a model, annual income (but not daily, weekly, or monthly income) is assumed to be constant throughout this paper. Given this cyclical pattern of income, the individual’s assumed maximization problem is to choose the pattern of consumption and the pattern of asset holdings over time subject to constraints, to be specified later, that maximize
1,”exp(-t&(c)
dr ,
(1)
where 0 is a positive discount rate, c(t) is the flow of consumption over time, and u(c) is a differentiable and strictly concave instantaneous utility function or (as some authors prefer) felicity function. There are reasons why one may object to discounting future instantaneous utility, but this is a very common assumption and is also necessary in order to guarantee that an optimum exists in the infinite horizon case. 3. The single-asset case Initially assume an individual may only hold one asset, money In the lotsize model, this assumption would create a trivial problem since an individual would simply hold all of his exogenously given surpluses as cash. However, the goal of the analysis presented here is to show how an individual simultaneously determines his pattern of money holdings, his pattern of consumption, and consequently his pattern of surpluses. Later on more complex models will be considered where the individual may hold interest earning bonds in addition to money. The individual’s maximization problem can be stated as follows. Choose the pattern of consumption that maximizes the functional given by (1) subject to
R.E. Anderson, The individual’stransactions demand
TAO
the following three constraints: any discrepancy between income and consumption must be financed by increasing or decreasing the stock of money, M(t); the stock of’money must be nonnegative; and the individual is c.ssumed to begin the first year of his planning horizon with the stock of money, MO,
n;r =p(x-c),
@a)
M 1
0,
CW
M(0) = MO.
w
MO is the individual’s entire stock of financial wealth at t = 0, and he receives no interest or other financial income. The price level, p, is for the present assumed to be constant. 2 Conditions which are sufficient for an optimal solution to t‘his problem are the following. If there exist continuous functions c(t) and M(t) satisfying relations (2a-c) and a continuous function A(t), where fi and l are integrable such that for each value oft, 0 s t < 00, I = exp (- &)24’(c),
(3)
AM-O,
(44
L < 0,
WO
and in addition satisfy a transversality condition, lim A(t) 2 0,
t+a,
lim I(t)M(t)
r+m
= 0,
(5)
then c(t) and M(t) maximize the functional (1) subject to the constraints (2a-c).3 The proof of the sufficiency of these conditions is a variation of the argument used by Mangasarisn for problems with a finite time horizon. Denote the path of consumption and money holdings that satisfy the above conditions for an optimum by an asterisk. What must be proven is that for any other feasible and contin.uous paths of consumptii?n and moneyholdings,
s,”exp
(- &>u(c*) dt 2 10”exp (-
Bt)u(c)
dt.
ZStrictis speaking, the level of consumption should also be constrained to be nonnegative though this constraint is not likely to be binding except in very rare situations. 3From the results derived by Gamkrelidze in his joint work with Pontryagin rt al. (1962, p. 301), conditions (3) and (4) can also be shown to be necessar!rfor an optimum. The variable A is equal to the adjoint variable in the jargon of optimal control if M > 0 and is equal to the adjoint variable plus a Lagrangian multiplier if A4 = 0.
R.E. Anderson, The individual’s transactions demand
241
The string of equalities and inequalities given in appendix A will prove this result. In these conditions for an optimum, the variable I is the shadow price of consumption and is equal to the present discounted value of the marginal utility of consumption. The variable 3, can also be interpreted as the shadow price on the real stock of money, M/p. A decision to increase the real stock of money at time t by one unit must mean that real consumption at time t is reduced by one unit for a loss in utility equal to 1. The planning horizon can be divided into two distinct types of subintervals. In the first type of subinterval, the stock of money is positive. From condition (4a), L must be constant; and thus from condition (3), u’(c)
i=o--,
u”(c)
(7)
Since U(C)is strictly concave, i < 0; and thus whenever the stock of money is positive, consumption must decline over time. In the second type of subinterval, the stock of money is zero. From condition (2a), x(t) = c(t); from condition (4b), x 2 0; and from condition (3),
lieu’.
u”(x)
(8)
This last condition implies that whenever the level of income is declining very fast, the individual must be using stocks of money to smooth out his pattern of consumption. The optimal strategy of consumption and money holdings consists of alternating these two types of subintervals. First, the stock of money is positive for some subinterval, and consumption must satisfy the differential equation (7). Next, the stock of money is reduced to zero, and consumption just equals income for some subinterval. Then the stock of money again becomes positive and so forth. It is not possible for the first type of subinterval to last indefinitely far into the future. The stock of money must eventually be reduced to zero. If M > 0 for all t > t*, E. must be constant; from relation (7) the level of consumption must continue to decline; eventually annual consumption will be less than annual income; the stock of money will increase without bound; and the transversality condition (5) will not be satisfied. It is possible for the second type of subinterval to last indefinitely far into the future if the pattern of income is so smooth that condition (8) is always satisfied. In this case there is no need to use positive stocks of money to further smooth out the pattern of consumption. Given an annual cycle in income, it is not difficult to convince oneself that the optimal patterns of consumption and money hfJldings will also eventually
242
RX. Anderson, The individual’s transacrions demand
develop unique year-long cycles regardless of what the intial stock of money, M,, happens to be. I have termed these long-run cycles in consumption and money holdings the cyclical steady state (CSS) patterns. If c(t), M(t), and J(t) satisfy the sufficiency conditions then identical values of consumption and the stock of money at t+ 1 and L(t+ 1) = exp [-011(t) will also satisfy these conditions. In fig. 1, I have given examples of optimal solutions for a simple income pattern and for various initial stocks of money, Maa
Fig. 1
The CSS pattern of consumption in fig. 1 has an interesting property that needs to be explored further. The level of consumptioli reaches a peak about the same time the peak in income occurs, and then declines slowly until just before the next peak in income. This pattern of income stems from the assumption that the individual discounts the future utility from consumption. If an individual shows a preference for present utility over future utility (which is a very common assumption), then it should not be surprising to observe, for example, a high rate ofconsumption immediately following each paycheck and a somewhat lower rate of consumption just before each paycheck. In section 5 it is shown that this phenomenon disappears if money earns a rate of return approaching the individual’s discount rate. A question that is always of interest when examining a model of the demand for money is whether or not the demand for money is proportional to income. For example, in the lot-size model, the demand for money will increase less than in proportion to real income if there are Lixed transactions costs in buying and selling bonds, and will increase strictly in proportion if there is no fixed component to transactions costs. If the change in nominal income is due solely to a change in the price level, then in the model presented here there is no doubt that the change in the nominal demand for money will be proportional. However,
RX. Anderson, The individual’s transactions demand
243
there is no guarantee that the change in the demand for money will be proportional if real income instead is changed, even assuming that there is no fixed component to transactions costs - a result that contradicts the conclusion from the lot-size model. The explanation for this is that a change in real income can cause the pattern of consumption to change in many ways. Thus the average stock of money can increase faster, slower, or at the same rate as real income, depending on the particular type of utility function. If the utility function is homothetic, i.e. z&c) = H(a)u(c) and u’(ac) = [H(u)/a]u’(c), where H(a) > 0, then it can be shown that the average stock of money will increase strictly in proportion to real income; but I can see no compelling reasons why utility functions must be homothetic. For a single individual, it is impossible to predict how his demand for money will change with income unless one has detailed information about his utility function. The characteristics of the aggregate demand for money can only be determined by econometric techniques. 4. The real balance effect
The real balance effect is the effect on consumption of a discrepancy between an individual’s desired and actual stock of money. There are a number of possible reasons why an individual might discover that his actual stock of money, for example, is greater than his desired stock. These include a fall in his real income which lowers his desired stock of transactions balances, a fall in prices and wages which again lowers his desired stock, or some transitory income which increases his actual stock. In most existing theories or models of the real balance effect, including those by Liviatan, Marty, and Patinkin, an individual is assumed to have a utility function with consumption and real balances as the arguments. This utility function is the basis for analyzing the trade-off between the present utility of consumption and the future utility of holding money balances. In the analysis of the real balance effect presented here, there is no need for such an artifice. Initially assume that the individual’s actual stock of money is greater th,an his desired stock, and assume that this occurs at the initial time point, t = 0. In other words, M0 is assumed to be greater than the CSS stock of money for that point during the year. Relation (7) says that whenever the stock of money is positive, the rate of consumption must be declining. As is illustrated in fig. 1, the individual must choose an initial rate of consumption, c(O), large enough so that by the time his intial stock of money is exhausted the rate of consumption will be reduced to a level just equal to his rate of income. At this point, the paths of consumption and money holdings will again follow the CSS patterns. Alternatively, assume that the individual’s actual stock of money is less than the CSS stock. As is illustrated in fig. 1 by MA, this will temporarily reduce the level of consumption below the CSS level ; but this reduction cannot last longer than the unit period.
244
RX. Anderson, The individwl’s transactions demand
There are two important questions about the real balance effect I would like tlo explore using this model of the transactions demand. First, by what amount will the rate of consumption be increased above its normal CSS level when the stock of money is increased above its normal CSS level? Secondly, how long will it be before the rate of consumption and the stock of money are reduced to their CSS levels? Some insights into these questions can be obtained by looking at a specific example. In this example, illustrated in fig. 2, the rate of income is assumed to be constant over time. This example will not be a bad approximation to more general cases if the actual rate of income is relatively constant during the initial period of adjustment created by the excess stock of money or if the initial stock of money is quite large compared to the CSS stock. Secondly, it is assumed that the individual’s utility function is of the constant elasticity variety, i.e. u(c:) = (1 - g)-lcl -uf where 0 2 0. For G = 1, U(C) = In c. Thirdly, the price level is equal to one. For this specific example, fig. 2 shows the paths of consumption and money holdings during the period of adjustment to the CSS, 0 $ t 5 7’, and after the CSS is reached, t >. T. The CSS stock of money is equal to zero, and the CSS rate of consumption is equal to the rate of income. In many models of adjustment to a long-run equilibrium, variables approach their long-run values asymptotically. For example, this is the assumed behavior in econometric stock adjustment models where the change in a stock is some constant proportion of the differeylce between the actual and desired stock. In the example illustrated in fig. 2, such asymptotic behavior is impossible. As was argued earlier, the stock of money may not remain positive indefinitely. For this specific example it is possible to derive the length of the period of adjustment, T, and the intial level of consumption relative to the CSS level, c(O)/x, as functions of three parameters: (i) the individual’s rate of discount, 0; (ii) the elasticity of the utility function, 1 -a; and (iii) the excess stock of money relative to annual income, M,/x. From the assumption of a constant elasticity utility function and condition (3), 3, = exp (-Ot)cmu. Since 3.is a constant during the period of adjustment,
c(t) = A-‘/” exp [ - (0/a)t],
for
t 6
T,
(9)
and
Atr = T, consumption x =
must equal income, and thus
%-riu exp [-(0/cr)T],
(10)
R.E. Anderson, The individual’s transactions demand
245
Fig. 2
and finally ,l
c(O)/x = exp WMTI.
W
At any point in time during the adjustment pe!:iod, the stock of money is defined bY M(t) =
J;x dz- J;c dt+MO.
M(t)
ts=t(a~o)r
From (9), =
““{exp
[ - (O/a)t]
-
1) + MO.
Since M(T) = 0,
MO= (apl)PU exp f - (fl/c)Tl+ (a/O)A-l’u- TX. From (lo),
MO/X = (o/O){exp [(O/a)T]- 1 } -T.
(12)
R.E. Anderson, The individuals transacticns demand
246
Thus from (12), we can find the value of T if we know the values of 44,/x, (T,and 6; and from (11) we can then find the value of c(O)/x. Table 1 The percentage increase in consumption, %dc, and the length of the period of adjustment, T, for eight values of u (or IO0 O/c&and two values of the excess stock of money as a percent of annual income, %M/x (0 = 0.05). _..-._ __.. --..___..._. - _.._.____I_-__ ---_PP _. 2.5 5.0 0.1 ti; :;3j :d.23) pdhf;‘x (“00 (I/a) Kj t&j (50): (2) (1) --.------____ __..__ __ ..-_-._--_.__ -._ -----11.7 4.6 3.3 2.6 2.0 1.4 0.8 0.7 1.0 Y’ 0.20 0.64 1.43 2.45 2.9 0.45 0.78 1.0 5,@
-__
T %Ac ______
24.0 0.43 ._._ -_-
7.3 1.4 5.8 1.7 10.3 0.98 -_-_.-. ---_-
4.5 2.2 __----
3.2 5.5 1.8 .---._-.------.-_~
::; ._. _. -.___
In table 1 are given values for the initial increase in consumption stated as a percent of the CSS level and for the period of adjustment. All of the excess stock of money will eventually be used for consumption, but a smaller value for Q or a larger value for 6 will cause a larger initial increase in consumption and will shorten the period before consumption returns to normal. In table 1, the rate of time preference (0) is assumed to be 0.05. Since it is the ratio between tr and 0 that determines the individual’s behavior, a simple calculation using table 1 will give the results for other values of 0. An important conclusion from this table is that an excess stock of money equal to a rather small percentage of annual income can have a significant effect on consumption. From the first row of the table, an excess stock of money equal to one percent of income can cause an increase in consumption of between eleven percent and about one percent. 5. Inflation and deflation Thus far the price level has been assumed to be constant over time. A more general case would be to assume that the price level at time t = 0 is equal to one and then changes continuosuly at a constant rate 4. Now define the real stock of money, m(t>, as the nominal stock divided by the price level, m(t) = PM(t)/&t) = M(t) exp
[-&I.
It is not difficult to show that the constraints (2~4 of the real stock of money as
can now be written in terms
ni = x-c-&l,
U3a)
0,
Wb)
m(O)= M,.
(13c)
112 2
R.E. Anderson, The individual’s transactions demand
247
It is often said that inflation has an effect analogous to a tax on money balances. In this model of the transactions demand, the effect is identical. The solution to exactly the same optimization problem as the one above would give the optimal paths of consumption and nominal money ballrnces if a tax at the rate 4 were collected continuously on an individual’s stock of money. Similarly if # were a negative number, the above model would describe an individual’s behavior when confronted with a constant rate of deflation equal to -4 or when interest was paid continuously on money balances at the rate -#* The conditions for an optimum analogous to conditions (3) through (5) are
A = exp [-0+‘(c),
(14)
(L c-jJa)m= 0,
(15a;
x--c@ 5 0,
(15b)
and the transversality
condition becomes
lim 1.(r) 2 0,
lim E_(r)m(l) = 0.
I+=
t-m
(16)
The proof of the sufficiency of these conditions follows the same pattern as that used in the case of a constant price level with (A--@) replacing 1 where necessary. It is not difficult to show from conditions (14) and (15a) that whenever the stock of money is positive, (17) Thus an increase in the rate of inflation will cause an increase in the steepness of the path of consumption Curing those periods when the stock of money is positive. As one can see from fig. 1, an increase in the steepness of the path of consumption will reduce the intervals when the stock of money is positive, will reduce the stock of real money balances at each point during the interval, and will shift consumption backwards in time. Also, an increase in the rate of inflation will shorten the period of adjustment to the CSS caused by some initial excess stock of money and will increase the intial level of consumption. Thus the higher the rate of inflation, the larger will be the real balance effect. These effects of inflation have often been described but this model rigorously shows how they come about. A number of eronomists, including Friedman (1969), Johnson (1969) and Samuelson (1969), have argued for various reasons that it would be desirable to
R.E. Anderson, The individual’stransactions demand
248
pay a rate of interest on money, or equivalently, to have a constant rate of deflation. Though it cannot be determined here whether or not such a scheme is desirable from the point of view of the entire economy, this model of the demand for money will at least indicate how an individual’s behavior would change if such a proposal were carried out. If there is either deflation or a positive interest yield on money (4 < 0), then from relation (I ‘7) the path of consumption must become flatter, consumption is shifted forward in time, the intervals when the stock of money is positive must become longer, and the stock of real money balances at each point becomes larger. If the rate of return on money approaches the individual’s rate of discount, the rate of consumption will become constant and equal to the average rate of income over the year. He will not attempt to conserve on the use of money
_-_ i
t
)_/
a__-
I f
JX
c\ -w-w
__.B-,
I
1 I
E
__-_
, w-w-
\
i
I
__
I ‘*t
o~~~~y+JJj~+ Fig. 3
.
balances by adopting the cyclical pattern of consumption illustrated in fig. 1. The CSS patterns of consumption and money holdings for this limiting case of .A = - 8 are illustrated in fig. 3. Y 6. The two-asset case Assume now that an individual may hold stocks of an interest-bearing asset called a bond in addition to stocks of money. For an individual to hold any money at all, since it normally does not earn interest, there must be transactions costs in buying and selling bonds which in at ieast some situations more than offset the gains from the interest income. Transactions costs are assumed to be proportional to the amount of bonds hought or sold. Any discrepancy betw’een the individual’s desired rate of consumption and his rate of income must be financed by a change in either his stock of money or his stock of bonds. Bonds, B(t), are measured in dollars, interest is paid continuously at a constant r:&.: Y,and thus interest income at time t is d?(t). For mathematical convenience, &tinges ;G the stock of bonds are separated into
RX. Anderson, The individual’stransactions demand
249
additions to the stock, A(t), and subtractions from the stock, S(t), i.e. B = A - S. Transactions costs at time t are some fixed proportion of the value of the bonds bought or sold at that time, i.e. uA +uS, where 0 < u < 1. In order to guarantee that an optimum exists, it must be assumed that the individuai’s rate of discount is at least slightly greater than the rate of interest. The mathematical problem that now arises is that the time paths of money and bond holdings may be discontinuous. It is possible that the individual may suddenly sell part of his stock of bonds and hold the proceeds as money or use part of his stock of money to suddenly buy bonds. Arrow and Kurz (1970) using techniques developed by Vind (1967) prove if the functional and the constraints satisfy certain concavity properties then a jump in a state variable can never be optimal, except possibly at the initial time point. For these jumps to occur at any later point in time would imply the individual has either made a mistake in planning at some point in the past or there has been a sudden unexpected change in the interest rate, the pattern of income, or some other parameter which would cause him to completely plan anew his paths of consumption, bond holdings, and money holdings.4 In order to allow for the possibility that the stock of bonds may suddenly change in value at t = 0, let A^be any initial increase in the stock and let g be any initial decrease. This also means there will be a matching decrease in the initial stock of money equal to (1 + u)J or an increase equal to (1 - u)s‘ since transactions costs must be paid. After any initial change in value, let the new stock of bonds be denoted by B(O+) and the new stock of money by M(0’). The maximization of relation (1) is now subject to the following constraints: ni= p(x-c)+rB-(1
+v)A+(l
-U)S,
(lga)
B = A-S,
(W
M(O+) = 14(O)-(1 +v)AI+(l-1))S,
(18~)
B(O+) = B(O)-:-A-
s,
(184
.M,B, A, S,2, s 2
0.
Wf)
Though optimal control techniques that can cope with discontinuities in the state variables are still in their infancy, I have been able to derive sufficient conditions for an optimum to this particular problem. The sufficient conditions JHowever, if there are fixed transactions costs then it will undoubtedly be optimal to buy or sell bonds in large discrete amounts. This indicates one reason why optimization problems with fixed transactions costs are so difficult to solve.
R.E. Anderson, l%e individual’stransactionsdemand
250
are the following. If there exist functions c(t), A(i), L,(t), and A,(r) continuous for t 2 0, funcc!ions B(t) and M(t) continuous for t > 0, where B, A8, d,, and x2 are integrable, and values for A^ and ,T?such that conditions (18a-f) are satisfied, such that for each value oft, 0 5 t 5 00,
A1 = exp [ - Bt]u’(c),
(19)
A,M = 0,
(2W
(A,+r%,)B= 0,
(214
A,+&
W)
5 0,
{A,-(l+u)A,}If %,-(l+u)R,
= 0,
(22a)
GW
5 0,
@a)
such that at t = 0, (%2(O)-(1 +Q;(o))A
= 0,
Wa)
Wb) ((l-4&(0)-
MO))s = 0,
(1 -z~)n,(o)-&(o)
5 0,
@a) (2W
and in addition satisfy the transversality conditions lim A,(i) 2 0, 1-Q lim A,(t) 2 0,
t-+03
lim A,(t)M(r) = 0, I-ta,
@a)
lim A,(#)B(t) = 0,
(2W
f-+00
then c(t), B(f), M(t), A(t), A, and s maximize the functional (1) subject to the constraints (18a-f7 The proof of the sufficiency of these conditions can be found in appendix B.
R.E. Anderson, The individual’s transactions demand
251
The variable 1, can be thought of as the shadow price of consumption or the shadow price of the real stock of money. The variable & can be thought of as the shadow price of the real stock of bonds. As can be seen from conditions (22) through (25), the exact value of ilr will differ from 1, because of transactions The planning horizon can be divided into four different types of subintervals. During the first type of subinterval, the stock of money is positive; and the path of consumption must again satisfy the differential equation (7). During the second type of subinterval, the stock of bonds is increasing. From conditions (19), (21a), and (22a), the path of consumption during this subinterval must satisfy
i=
r
u’(c)
( > 0-w
-.
1 + U u”(c)
(27)
During the third type of subinterval, the stock of bonds is decreasing. From conditions (19) (2 1a), and (23a), the path of consumption during this subinterval must satisfy
During the fourth type of subinterval, both the stock of money and the stock of bonds are zero, thus c = s, and condition (8) must again be satisfied. The first step in finding the optimal patterns of consumption, bond holdings, and money holdings to is determine the optimal sequence of these four types of subintervals. The best way to see the characteristics of an optimal solution is to look again at an example. The upper half of fig, 4 shows a simple path of nonfinancial income with its year-long cycle and the optimal patterns of consumption. The curve in the lower half of fig. 4 shows the total stock of financial assets. The distance below the horizontal line represents the maximum stock of bonds held during that part of the cycle. The distance between the horizontal line and the curve represents the stock of money. The optimal psths are shown for :hree different values of the initial stock of money, assuming for simplicity that the initial stock of bonds is sero. No matter what the initial stock of money or bonds happens to bc, eventually the time paths of consumption, bond holdings, and money holdings will follow unique CSS patterns. The maximum length of time that the stock of money will remaiu positive is 2rs;r. If an asset is to be held for a period of time less than 20/r, the interest income from bonds is not enough to compensate for the transactions costs, and the optimal asset to hold is money. Only for surpluses lasting longer than 217”~will part of the surplus be held as bonds. To show this result more rigorously look at the interval (tI, t2) in fig. 4. At t, the stock of bonds ceases to
252
K.E. Anderson, The individual’s transactions demand
Fig. 4
increase, and tke surplus of income over consumption is now added to the stock of money. At tz the stock of money is exhausted, and the deficit of consumption over income is now paid for by selling bonds. Since L, and AZ are continuous, conditions (22a) and (23a) require that
and
Since both the stock of money and the stock of bonds ar positive during this subinterval, conditions (20a) and (21a) require that ,I, be constant and that 8 I.2 = - r).,. Therefore integrating 1, over the interval (t, , t2) produces (l-U)+(I+Uj;Lr
= -r(t,--t,)%,,
or tz-fl
=
2u/r.
As in the previous single asset case, the CSS holdings of money and bonds are always proportional to the price level. They are again proportional to real income if income at each point in time is increased by a uniform percentage and if the titility function is homothetic. The transactions demand for money now depends on the interest rate on bonds, and this is a considerably more complicated problem than in the lot-size inventory model. In the lot-size model, an increase in the interest rate reduces the proportion of the surplus held as
R.E. Anderson, The individuall’s transactions demand
253
money at each point in time but has no effect on the size of the surplus5 Consequently, there is no doubt in the lot-size model that interest rates and the demand for money move inversely. However, in the mode1 of the transactions demand presented here, an increase in the interest rate on bonds has at least two separate effects on an individual’s behavior. First, it tends to flatten the pattern of consumption whenever the stock of bonds is being increased or decreased. From relations (27) and (29, c will on the average not be such a large negative number when r is increased. This tends to increase the total stock of money and bonds at each point in time as these assets are used to a greater extent to smooth out the pattern of consumption. The second and perhaps most important effects of an increase in interest rates is to reduce the value of 2+, the maximum period of time that money should be held. This tends to reduce the proportion of total assets held as money at each point in time. On the one hand the total stock of transactions assets is increased, but on the other hand the proportion held as money is reduced. Consequently, the net effect on the transactions demand for money is uncertain. Though I am convinced that the demand for money is normally, or on the average, reduced by an increase in the rate of interest, I cannot prove that the stock of money is reduced at all points in time, for all conceivable patterns of income, and for all utility functions. The real balance effect described above is now made more complicated by the introduction of an interest-earning asset. If the initial stocks of assets, M0 and B,, are different from their CSS values, the individual’s behavior may be altered in two ways. First, he may find it optimal to rearrange his initial portfolio of assets. If the individual has a large initial stock of money, it may be optimal to use part of the stock to buy bonds if the bonds are to be held for a period of time greater than 2u/r. However, it would never be optimal to sell some of the initial stock of bonds and instead hold money. To do so would sacrifice interest income with no saving of transactions costs. Secondly, the individual’s path of consumption will be different than the CSS path during some initial period of adjustment. In order to illustrate the effect on consumption of an initial stock of money greater than the CSS stock, let us consider two situations depicted in fig. 4. In the first situation, the intial stock (A@ is only modestly greater than the CSS stock. Because the stock of money will be spent on consumption in a relatively short period of time (less than 20/r), there is no advantage in using pari of the initial stock to buy bonds. This initial stock of money will only cause a temporary increase in consumption. In the second situation, the initial stock of money is quite large, I@ in fig. 4; and it is now profitable to use part of the initial stock to buy bonds since the bonds will be held for a period of time greater than Sin a later work, Johnson (1970) does attempt to modify the lot-size model SO that net income from managing the portfolio of transactions assets (interest income minustotal transactions costs) is included as part of total income.
R.E. Anderson, The individual’s transactions demand
254
2v/r. In concluGon, a small excess stock of money will only cause an increase in consumption. L*the excess stock is large, it will cause an increase in both the level of consumption and the demand for bonds. 7. Concluding remarks The basic motive for an individual to hold transactions balances of money or of other assets is the lack of synchronization between his pattern of expenditures and his pattern of receipts. The lot-size inventory model developed by Baumol, Tobin, and others shows the optimal patterns of money and bond holdings for an exogenously given a5d a very arbitrary pattern of receipts and expendiWres, fn the mode! of tke transactions demand for money presented here, I have shown how a:? individual might simultaneously choose the patterns of money holdings, bond holdings, and consumption expenditures over time that maximize utility when faced with a wide variety of possible income patterns. The pattern of consumption expenditures, like the patterns of money and bond holdings, depends on such parametl:rs as the individual’s rate of time preference, the rate of deflation or inflation, the interest rate on bonds, and the transactions costs in buying and selling bonds. The single most important defect of the model of the transactions demand for money presented here is the assumption that transactions costs are proportional to the size of the transaction. Also, if there are fixed costs or economies of scale in buying consumption goods, an individual is likely to find it optimal to hold inventories of commodities in addition to money and bonds.(’ However, the inclusion of these generalizations into this model sf the transactions demand must await the further development of optimization techniques for problems with fixed costs or integer variables. Another unrealistic assumption in this model is that the individlual is assumed to know his future pattern of income and his future needs for consumption with complete certainty. In other words, the precautionary motive for holding money has been ex’cluded. An important area for future research is to extend the model developed above to include uncertainty about the pattern of income and about preferences for consumption. Appendix A
p-‘jr exp [-Bt]{rr(c*)-U(C)) >
p-’ Jr exp
[-
Ot]~‘(c*)(c~
2 s,” exp [-6t]l*(A&-lh*)dt,
dt -
C)
dt
,
(b) (c)
Teige and Parkin (1971) have extended the lot-size model to include the possibility of commodity inventories.
R.E. Anderson, The individual’s transactions demand
=
+Io” i*(M* -M) dt,
n*(~-iv*)/;
2 A*(M-M*)l;
(d)
,
(d
5 ~*Gw*(o) - iz*(o)M(o), =
255
(0
0.
(s)
Explanation. ( a) rewriting (6); (b) since u(c) is assumed to be differentiable and concave and by Taylor’s formula; (c) from (2a) and (3); (d) integrating by parts; (e) by relations (4a), (4b) and (2b); (f) by the transversality condition (5); (g) by relation (2~). Appendix B
The proof of the sufficiency of these conditions is the same as that used in appendix A through line (b). = jr n:(n;r-n;l*+rB-(l -rB+(l
+f+t*+(l
+u)A -(l-v)S}
+s;(1;+rl;)B* -
dt,
(e)
dt,
2 ATMI,” -A:M*lo” +I;BI; 2
-U)S*
(4 -lw;B*lo”
m
,
R:(O+)M(O+) + Rr(O)M*(O+) - i.;(O) B(O+)
+ Iz;(O)B*(O+), = If(O){M*(O,-(1 +(I +u+-(1
(g) +t’)A*+(l
-@*-M(O)
-u)S)
+$(O){B*(O)+if*-S*-B(O)-A+S}, = -{$(O)-(1
+n)if(O)}R-((1
(h) -tl)J$(O)--Az(O)]S
+ {AZ(O) - (1 + u)l:(o)}AI* + {(1 2 0.
P)A:(o)
-
i;co>)S* .
256
R.E. Anderson, The individual’s tranJ;actionsdemand
Explanation. (c) fram (19j and (Isa); (d) from (2?a), (23a) and (1 Sb); (e) integrating by parts; (f) from (20a) through (21 b), and (1 Sf); (g) from (26a-b); (h) from (18c-d); (i) from (1 Se); (j) from (24a) through (25b). References Arrow, K.J. anld M. Kurz, 1970, Public investment, the rate of return and optimal fiscal policy (The Johns Hopkins Press, Bahimore). Baumol, W.J., 1952, The transactions demand for money: An inventory theoretic approach, Quarterly Journal of Economics 56,54556. Feige, E.L. and M. Parkin, 1971, The optimal quantity of money, bonds, commodity inventories, and capital, American Economic Review 61,335-49. Friedman. M., 1969, The optimum quantity of money and other essays (Aldine, Chicago). Johnson, H.G., 1969, Inside money, outside money, income wealth and welfare in monetary theory, Journal of Moc;y, Credit and Banking 1,30-45. Johnson, H.G., 1970, A note on the theory of the transactions demand for cash, Journal of Money, Credit and Banking 2,383-4. Johnson, H.G., 1969, Notes on the theory of transactions demand for cash, in: Essays in Monetary Economics (London). Liviatan, N., 1965. The long-run theory of consumption and real balances, Oxford Economic Papers 17,205-l 8. Mangasarian, O.L., 1966, Sufficient conditions for the optimal control of nonlinear systems, Journal of SIAM Control 4,139-52. Marty, A.L., 1964, The real balance effect: An exercise in capital theory, Canadian Journal of Economics and Political Science 30,360.-67. Niehans, J., 1971. Money and barter in general equilibrium with transactions costs, American Economic Review 6 I, 773-83. Patinkin, D., 19G2, Money, interest and prices, 2nd ed. (Harper and Row, London) ch. 3. Pontryagin, L. et al., 1962, The mathematical theory of optimal processes (John Wiley, New York). Samuelson, P.A., 1969, Nonoptimality of money holding under laissez faire, Canadian Journal of Economics 2,303-g. Vind, K., 1967, Control systems with jumps in the state variables, Econometrica 35,273-7.
THE INDIVJDUAL’S TRANSACTIONS
DEMAND FOR MONEY
A utility maximization approach Robert E. ANDERSON* United States Department of the Interior, Washington, DC, U.S.A. This model of the transactions demand shows how an individual may simultaneously choose patterns of consumption, money holdings, and bond holdings over time that maximize utility when faced with a wide variety of possible cyclical patterns in his flow of income. Interesting conclusions about the transactions demand and the real balance effect are derived. For example, there is no theoretical reason to believe that a single individual’s demand for money is proportional to his income, and a small excess stock of money is likely to cause a large increase in his level of consumption.
1. Introduction is now over twenty years since Baumol first presented his analysis of the transactions demand for money based on the classic lot-size model from inventory theory. Though this model has been presented in somewhat different ways by Tobin and then Johnson (1969), it has remained the basic model of the transactions demand for money. This is perhaps not surprising even though the lotsize model has at least two serious shortcomings when it is used to describe a consumer or household’s demand for money. The basic motive for holding transactions balances of money has long been recognized to be the lack of synchronization between receipts and expenditures over time. But until comparatively recently the techniques available to solve optimization problems that explicitly included the time dimension have been very limited. The first shortcoming of the lot-size model when it is used to describe a consumer or household’s demand for money is the assumption of a highly arbitrary pattern of receipts and expenditures.’ An individual’s income is assumed to be received in equal lump-sum amounts at the beginning of periods of unit length. This income is then spent at a constant rate on consumption, creating the now famous ‘sawtooth’ pattern of surpluses. Though this assumed pattern of income It
+I would like to thank Jiirg Niehans for his initial encouragement to do research in this area and for a number of helpful comments and suggestions. Part of this research was undertaken while I was a recipient of the Harold Stonier Fellowship from the American Bankers Association. ‘The lot-size model can also be used to describe the transactions demand for money by economic units other than households (business firms, financial institutions, etc.), but in this paper I am only analyzing the demand by htiuseholds.
238
R.E. Anderson, The indi,vidual’s transactions demand
may bc close to an individual’s pattern of wage or salary income, there are a number of other sources of income tha.t may be received in an entirely different pattern -for example, financial income, overtime pay, bonuses, and income from seasonal employment. Instead of a constant rate of consumption, an individual may also have a definite pattern of expenditures - for example, high rates of expenditures in August for a vacation, in December for Christmas, and at the beginning of each month because that is when bills are due. Can the ad hoc techniques of the lot-size model be used to analyze models with more general patterns of income and expenditures? My own efforts have convinced me that this is not possible and an entirely new approach is necessary. The second shortcoming of the lot-size model is the assumption of an exogenously’ given pattern of consumption. In other words, an individual’s decisions about consumption are independent of his decisions about money or bond holdings. In the lot-size model the rate of consumption is given exogenously, and the individual then determines the optimal pattern of money and bond holdings to finance :he resulting surpluses. Hewever, this is at odds with one of the oldest goals of economic theory, which is to explain how an individual makes decisions about consumption. In the model of the transactions demand for money to be presented here, I will show how an individual might simultaneously choose the pattern of money holdings, the pattern of bond holdings, and the pattern of consumption over time that maximize utility when faced with a wide variety of possible patterns in his 3ow of income. From this model, one can gain interesting insights into both Lhe transactions demand for money and the real balance effect that are not possible from the lot-size model because of its restrictive assumptions. However, this more general and sophisticated model is not superior to the lotsize model in all respects. One of the major advantages of the lot-size model is that it can be used to analyze the case of fixed or lump-sum transactions costs. Because of the still inadequate mathematical optimization techniques, transactions costs in the analysis presented here will have to be assumed to be proportional to the amount of bonds bought or sold, 2. Basic assumptions A basic motive for an individual to hold stocks of any asset is the lack of synchronization over his lifetime between his pattern of income and his desired pattern of consumption. On the one extreme, an individual must decide what assets to hold when he may keep them for almost his entire life-time, for example, saving for old age or retirement. On the other extreme, he must decide in what form to hold his monthly salary until ir is spent for consumption during the subsequent four weeks. It is the daily, weekly, or monthly pattern in consumption and income that creates the transactions demand for assets. In order to separate the problem of the transactions demand for assets from
R.E. Anderson, The individrcal’stransactions demand
239
the more general problem of life-cycle saving and portfolio composition, assume that there is a pattern in an individual’s nonfinancial or endowment income that lasts for a period of unit length and repeats itself unchanged period after period. For ease of reference, call this unit period a ‘year’. Denote the flow of income by the continuous function x(f), and thus the assumption is that x(t+ 1) = x(t). The particular function x(t) which would approximate the pattern of income used in the lot-size model would have periodic peaks during the year and then quickly drop to zero for the intervals between the peaks. In a more general model than that presented here, annual income would also be assumed to vary over an individual’s lifetime. Such a model should explain how an individual determines his lifetime pattern of consumption, saving, and portfolio composition in addition to his transactions demand for assets. The theory of portfolio composition based on uncertainty about interest rates pioneered by Tobin and Markowitz would have to be an essential part of the model. To avoid the considerabie complexity of such a model, annual income (but not daily, weekly, or monthly income) is assumed to be constant throughout this paper. Given this cyclical pattern of income, the individual’s assumed maximization problem is to choose the pattern of consumption and the pattern of asset holdings over time subject to constraints, to be specified later, that maximize
1,”exp(-t&(c)
dr ,
(1)
where 0 is a positive discount rate, c(t) is the flow of consumption over time, and u(c) is a differentiable and strictly concave instantaneous utility function or (as some authors prefer) felicity function. There are reasons why one may object to discounting future instantaneous utility, but this is a very common assumption and is also necessary in order to guarantee that an optimum exists in the infinite horizon case. 3. The single-asset case Initially assume an individual may only hold one asset, money In the lotsize model, this assumption would create a trivial problem since an individual would simply hold all of his exogenously given surpluses as cash. However, the goal of the analysis presented here is to show how an individual simultaneously determines his pattern of money holdings, his pattern of consumption, and consequently his pattern of surpluses. Later on more complex models will be considered where the individual may hold interest earning bonds in addition to money. The individual’s maximization problem can be stated as follows. Choose the pattern of consumption that maximizes the functional given by (1) subject to
R.E. Anderson, The individual’stransactions demand
TAO
the following three constraints: any discrepancy between income and consumption must be financed by increasing or decreasing the stock of money, M(t); the stock of’money must be nonnegative; and the individual is c.ssumed to begin the first year of his planning horizon with the stock of money, MO,
n;r =p(x-c),
@a)
M 1
0,
CW
M(0) = MO.
w
MO is the individual’s entire stock of financial wealth at t = 0, and he receives no interest or other financial income. The price level, p, is for the present assumed to be constant. 2 Conditions which are sufficient for an optimal solution to t‘his problem are the following. If there exist continuous functions c(t) and M(t) satisfying relations (2a-c) and a continuous function A(t), where fi and l are integrable such that for each value oft, 0 s t < 00, I = exp (- &)24’(c),
(3)
AM-O,
(44
L < 0,
WO
and in addition satisfy a transversality condition, lim A(t) 2 0,
t+a,
lim I(t)M(t)
r+m
= 0,
(5)
then c(t) and M(t) maximize the functional (1) subject to the constraints (2a-c).3 The proof of the sufficiency of these conditions is a variation of the argument used by Mangasarisn for problems with a finite time horizon. Denote the path of consumption and money holdings that satisfy the above conditions for an optimum by an asterisk. What must be proven is that for any other feasible and contin.uous paths of consumptii?n and moneyholdings,
s,”exp
(- &>u(c*) dt 2 10”exp (-
Bt)u(c)
dt.
ZStrictis speaking, the level of consumption should also be constrained to be nonnegative though this constraint is not likely to be binding except in very rare situations. 3From the results derived by Gamkrelidze in his joint work with Pontryagin rt al. (1962, p. 301), conditions (3) and (4) can also be shown to be necessar!rfor an optimum. The variable A is equal to the adjoint variable in the jargon of optimal control if M > 0 and is equal to the adjoint variable plus a Lagrangian multiplier if A4 = 0.
R.E. Anderson, The individual’s transactions demand
241
The string of equalities and inequalities given in appendix A will prove this result. In these conditions for an optimum, the variable I is the shadow price of consumption and is equal to the present discounted value of the marginal utility of consumption. The variable 3, can also be interpreted as the shadow price on the real stock of money, M/p. A decision to increase the real stock of money at time t by one unit must mean that real consumption at time t is reduced by one unit for a loss in utility equal to 1. The planning horizon can be divided into two distinct types of subintervals. In the first type of subinterval, the stock of money is positive. From condition (4a), L must be constant; and thus from condition (3), u’(c)
i=o--,
u”(c)
(7)
Since U(C)is strictly concave, i < 0; and thus whenever the stock of money is positive, consumption must decline over time. In the second type of subinterval, the stock of money is zero. From condition (2a), x(t) = c(t); from condition (4b), x 2 0; and from condition (3),
lieu’.
u”(x)
(8)
This last condition implies that whenever the level of income is declining very fast, the individual must be using stocks of money to smooth out his pattern of consumption. The optimal strategy of consumption and money holdings consists of alternating these two types of subintervals. First, the stock of money is positive for some subinterval, and consumption must satisfy the differential equation (7). Next, the stock of money is reduced to zero, and consumption just equals income for some subinterval. Then the stock of money again becomes positive and so forth. It is not possible for the first type of subinterval to last indefinitely far into the future. The stock of money must eventually be reduced to zero. If M > 0 for all t > t*, E. must be constant; from relation (7) the level of consumption must continue to decline; eventually annual consumption will be less than annual income; the stock of money will increase without bound; and the transversality condition (5) will not be satisfied. It is possible for the second type of subinterval to last indefinitely far into the future if the pattern of income is so smooth that condition (8) is always satisfied. In this case there is no need to use positive stocks of money to further smooth out the pattern of consumption. Given an annual cycle in income, it is not difficult to convince oneself that the optimal patterns of consumption and money hfJldings will also eventually
242
RX. Anderson, The individual’s transacrions demand
develop unique year-long cycles regardless of what the intial stock of money, M,, happens to be. I have termed these long-run cycles in consumption and money holdings the cyclical steady state (CSS) patterns. If c(t), M(t), and J(t) satisfy the sufficiency conditions then identical values of consumption and the stock of money at t+ 1 and L(t+ 1) = exp [-011(t) will also satisfy these conditions. In fig. 1, I have given examples of optimal solutions for a simple income pattern and for various initial stocks of money, Maa
Fig. 1
The CSS pattern of consumption in fig. 1 has an interesting property that needs to be explored further. The level of consumptioli reaches a peak about the same time the peak in income occurs, and then declines slowly until just before the next peak in income. This pattern of income stems from the assumption that the individual discounts the future utility from consumption. If an individual shows a preference for present utility over future utility (which is a very common assumption), then it should not be surprising to observe, for example, a high rate ofconsumption immediately following each paycheck and a somewhat lower rate of consumption just before each paycheck. In section 5 it is shown that this phenomenon disappears if money earns a rate of return approaching the individual’s discount rate. A question that is always of interest when examining a model of the demand for money is whether or not the demand for money is proportional to income. For example, in the lot-size model, the demand for money will increase less than in proportion to real income if there are Lixed transactions costs in buying and selling bonds, and will increase strictly in proportion if there is no fixed component to transactions costs. If the change in nominal income is due solely to a change in the price level, then in the model presented here there is no doubt that the change in the nominal demand for money will be proportional. However,
RX. Anderson, The individual’s transactions demand
243
there is no guarantee that the change in the demand for money will be proportional if real income instead is changed, even assuming that there is no fixed component to transactions costs - a result that contradicts the conclusion from the lot-size model. The explanation for this is that a change in real income can cause the pattern of consumption to change in many ways. Thus the average stock of money can increase faster, slower, or at the same rate as real income, depending on the particular type of utility function. If the utility function is homothetic, i.e. z&c) = H(a)u(c) and u’(ac) = [H(u)/a]u’(c), where H(a) > 0, then it can be shown that the average stock of money will increase strictly in proportion to real income; but I can see no compelling reasons why utility functions must be homothetic. For a single individual, it is impossible to predict how his demand for money will change with income unless one has detailed information about his utility function. The characteristics of the aggregate demand for money can only be determined by econometric techniques. 4. The real balance effect
The real balance effect is the effect on consumption of a discrepancy between an individual’s desired and actual stock of money. There are a number of possible reasons why an individual might discover that his actual stock of money, for example, is greater than his desired stock. These include a fall in his real income which lowers his desired stock of transactions balances, a fall in prices and wages which again lowers his desired stock, or some transitory income which increases his actual stock. In most existing theories or models of the real balance effect, including those by Liviatan, Marty, and Patinkin, an individual is assumed to have a utility function with consumption and real balances as the arguments. This utility function is the basis for analyzing the trade-off between the present utility of consumption and the future utility of holding money balances. In the analysis of the real balance effect presented here, there is no need for such an artifice. Initially assume that the individual’s actual stock of money is greater th,an his desired stock, and assume that this occurs at the initial time point, t = 0. In other words, M0 is assumed to be greater than the CSS stock of money for that point during the year. Relation (7) says that whenever the stock of money is positive, the rate of consumption must be declining. As is illustrated in fig. 1, the individual must choose an initial rate of consumption, c(O), large enough so that by the time his intial stock of money is exhausted the rate of consumption will be reduced to a level just equal to his rate of income. At this point, the paths of consumption and money holdings will again follow the CSS patterns. Alternatively, assume that the individual’s actual stock of money is less than the CSS stock. As is illustrated in fig. 1 by MA, this will temporarily reduce the level of consumption below the CSS level ; but this reduction cannot last longer than the unit period.
244
RX. Anderson, The individwl’s transactions demand
There are two important questions about the real balance effect I would like tlo explore using this model of the transactions demand. First, by what amount will the rate of consumption be increased above its normal CSS level when the stock of money is increased above its normal CSS level? Secondly, how long will it be before the rate of consumption and the stock of money are reduced to their CSS levels? Some insights into these questions can be obtained by looking at a specific example. In this example, illustrated in fig. 2, the rate of income is assumed to be constant over time. This example will not be a bad approximation to more general cases if the actual rate of income is relatively constant during the initial period of adjustment created by the excess stock of money or if the initial stock of money is quite large compared to the CSS stock. Secondly, it is assumed that the individual’s utility function is of the constant elasticity variety, i.e. u(c:) = (1 - g)-lcl -uf where 0 2 0. For G = 1, U(C) = In c. Thirdly, the price level is equal to one. For this specific example, fig. 2 shows the paths of consumption and money holdings during the period of adjustment to the CSS, 0 $ t 5 7’, and after the CSS is reached, t >. T. The CSS stock of money is equal to zero, and the CSS rate of consumption is equal to the rate of income. In many models of adjustment to a long-run equilibrium, variables approach their long-run values asymptotically. For example, this is the assumed behavior in econometric stock adjustment models where the change in a stock is some constant proportion of the differeylce between the actual and desired stock. In the example illustrated in fig. 2, such asymptotic behavior is impossible. As was argued earlier, the stock of money may not remain positive indefinitely. For this specific example it is possible to derive the length of the period of adjustment, T, and the intial level of consumption relative to the CSS level, c(O)/x, as functions of three parameters: (i) the individual’s rate of discount, 0; (ii) the elasticity of the utility function, 1 -a; and (iii) the excess stock of money relative to annual income, M,/x. From the assumption of a constant elasticity utility function and condition (3), 3, = exp (-Ot)cmu. Since 3.is a constant during the period of adjustment,
c(t) = A-‘/” exp [ - (0/a)t],
for
t 6
T,
(9)
and
Atr = T, consumption x =
must equal income, and thus
%-riu exp [-(0/cr)T],
(10)
R.E. Anderson, The individual’s transactions demand
245
Fig. 2
and finally ,l
c(O)/x = exp WMTI.
W
At any point in time during the adjustment pe!:iod, the stock of money is defined bY M(t) =
J;x dz- J;c dt+MO.
M(t)
ts=t(a~o)r
From (9), =
““{exp
[ - (O/a)t]
-
1) + MO.
Since M(T) = 0,
MO= (apl)PU exp f - (fl/c)Tl+ (a/O)A-l’u- TX. From (lo),
MO/X = (o/O){exp [(O/a)T]- 1 } -T.
(12)
R.E. Anderson, The individuals transacticns demand
246
Thus from (12), we can find the value of T if we know the values of 44,/x, (T,and 6; and from (11) we can then find the value of c(O)/x. Table 1 The percentage increase in consumption, %dc, and the length of the period of adjustment, T, for eight values of u (or IO0 O/c&and two values of the excess stock of money as a percent of annual income, %M/x (0 = 0.05). _..-._ __.. --..___..._. - _.._.____I_-__ ---_PP _. 2.5 5.0 0.1 ti; :;3j :d.23) pdhf;‘x (“00 (I/a) Kj t&j (50): (2) (1) --.------____ __..__ __ ..-_-._--_.__ -._ -----11.7 4.6 3.3 2.6 2.0 1.4 0.8 0.7 1.0 Y’ 0.20 0.64 1.43 2.45 2.9 0.45 0.78 1.0 5,@
-__
T %Ac ______
24.0 0.43 ._._ -_-
7.3 1.4 5.8 1.7 10.3 0.98 -_-_.-. ---_-
4.5 2.2 __----
3.2 5.5 1.8 .---._-.------.-_~
::; ._. _. -.___
In table 1 are given values for the initial increase in consumption stated as a percent of the CSS level and for the period of adjustment. All of the excess stock of money will eventually be used for consumption, but a smaller value for Q or a larger value for 6 will cause a larger initial increase in consumption and will shorten the period before consumption returns to normal. In table 1, the rate of time preference (0) is assumed to be 0.05. Since it is the ratio between tr and 0 that determines the individual’s behavior, a simple calculation using table 1 will give the results for other values of 0. An important conclusion from this table is that an excess stock of money equal to a rather small percentage of annual income can have a significant effect on consumption. From the first row of the table, an excess stock of money equal to one percent of income can cause an increase in consumption of between eleven percent and about one percent. 5. Inflation and deflation Thus far the price level has been assumed to be constant over time. A more general case would be to assume that the price level at time t = 0 is equal to one and then changes continuosuly at a constant rate 4. Now define the real stock of money, m(t>, as the nominal stock divided by the price level, m(t) = PM(t)/&t) = M(t) exp
[-&I.
It is not difficult to show that the constraints (2~4 of the real stock of money as
can now be written in terms
ni = x-c-&l,
U3a)
0,
Wb)
m(O)= M,.
(13c)
112 2
R.E. Anderson, The individual’s transactions demand
247
It is often said that inflation has an effect analogous to a tax on money balances. In this model of the transactions demand, the effect is identical. The solution to exactly the same optimization problem as the one above would give the optimal paths of consumption and nominal money ballrnces if a tax at the rate 4 were collected continuously on an individual’s stock of money. Similarly if # were a negative number, the above model would describe an individual’s behavior when confronted with a constant rate of deflation equal to -4 or when interest was paid continuously on money balances at the rate -#* The conditions for an optimum analogous to conditions (3) through (5) are
A = exp [-0+‘(c),
(14)
(L c-jJa)m= 0,
(15a;
x--c@ 5 0,
(15b)
and the transversality
condition becomes
lim 1.(r) 2 0,
lim E_(r)m(l) = 0.
I+=
t-m
(16)
The proof of the sufficiency of these conditions follows the same pattern as that used in the case of a constant price level with (A--@) replacing 1 where necessary. It is not difficult to show from conditions (14) and (15a) that whenever the stock of money is positive, (17) Thus an increase in the rate of inflation will cause an increase in the steepness of the path of consumption Curing those periods when the stock of money is positive. As one can see from fig. 1, an increase in the steepness of the path of consumption will reduce the intervals when the stock of money is positive, will reduce the stock of real money balances at each point during the interval, and will shift consumption backwards in time. Also, an increase in the rate of inflation will shorten the period of adjustment to the CSS caused by some initial excess stock of money and will increase the intial level of consumption. Thus the higher the rate of inflation, the larger will be the real balance effect. These effects of inflation have often been described but this model rigorously shows how they come about. A number of eronomists, including Friedman (1969), Johnson (1969) and Samuelson (1969), have argued for various reasons that it would be desirable to
R.E. Anderson, The individual’stransactions demand
248
pay a rate of interest on money, or equivalently, to have a constant rate of deflation. Though it cannot be determined here whether or not such a scheme is desirable from the point of view of the entire economy, this model of the demand for money will at least indicate how an individual’s behavior would change if such a proposal were carried out. If there is either deflation or a positive interest yield on money (4 < 0), then from relation (I ‘7) the path of consumption must become flatter, consumption is shifted forward in time, the intervals when the stock of money is positive must become longer, and the stock of real money balances at each point becomes larger. If the rate of return on money approaches the individual’s rate of discount, the rate of consumption will become constant and equal to the average rate of income over the year. He will not attempt to conserve on the use of money
_-_ i
t
)_/
a__-
I f
JX
c\ -w-w
__.B-,
I
1 I
E
__-_
, w-w-
\
i
I
__
I ‘*t
o~~~~y+JJj~+ Fig. 3
.
balances by adopting the cyclical pattern of consumption illustrated in fig. 1. The CSS patterns of consumption and money holdings for this limiting case of .A = - 8 are illustrated in fig. 3. Y 6. The two-asset case Assume now that an individual may hold stocks of an interest-bearing asset called a bond in addition to stocks of money. For an individual to hold any money at all, since it normally does not earn interest, there must be transactions costs in buying and selling bonds which in at ieast some situations more than offset the gains from the interest income. Transactions costs are assumed to be proportional to the amount of bonds hought or sold. Any discrepancy betw’een the individual’s desired rate of consumption and his rate of income must be financed by a change in either his stock of money or his stock of bonds. Bonds, B(t), are measured in dollars, interest is paid continuously at a constant r:&.: Y,and thus interest income at time t is d?(t). For mathematical convenience, &tinges ;G the stock of bonds are separated into
RX. Anderson, The individual’stransactions demand
249
additions to the stock, A(t), and subtractions from the stock, S(t), i.e. B = A - S. Transactions costs at time t are some fixed proportion of the value of the bonds bought or sold at that time, i.e. uA +uS, where 0 < u < 1. In order to guarantee that an optimum exists, it must be assumed that the individuai’s rate of discount is at least slightly greater than the rate of interest. The mathematical problem that now arises is that the time paths of money and bond holdings may be discontinuous. It is possible that the individual may suddenly sell part of his stock of bonds and hold the proceeds as money or use part of his stock of money to suddenly buy bonds. Arrow and Kurz (1970) using techniques developed by Vind (1967) prove if the functional and the constraints satisfy certain concavity properties then a jump in a state variable can never be optimal, except possibly at the initial time point. For these jumps to occur at any later point in time would imply the individual has either made a mistake in planning at some point in the past or there has been a sudden unexpected change in the interest rate, the pattern of income, or some other parameter which would cause him to completely plan anew his paths of consumption, bond holdings, and money holdings.4 In order to allow for the possibility that the stock of bonds may suddenly change in value at t = 0, let A^be any initial increase in the stock and let g be any initial decrease. This also means there will be a matching decrease in the initial stock of money equal to (1 + u)J or an increase equal to (1 - u)s‘ since transactions costs must be paid. After any initial change in value, let the new stock of bonds be denoted by B(O+) and the new stock of money by M(0’). The maximization of relation (1) is now subject to the following constraints: ni= p(x-c)+rB-(1
+v)A+(l
-U)S,
(lga)
B = A-S,
(W
M(O+) = 14(O)-(1 +v)AI+(l-1))S,
(18~)
B(O+) = B(O)-:-A-
s,
(184
.M,B, A, S,2, s 2
0.
Wf)
Though optimal control techniques that can cope with discontinuities in the state variables are still in their infancy, I have been able to derive sufficient conditions for an optimum to this particular problem. The sufficient conditions JHowever, if there are fixed transactions costs then it will undoubtedly be optimal to buy or sell bonds in large discrete amounts. This indicates one reason why optimization problems with fixed transactions costs are so difficult to solve.
R.E. Anderson, l%e individual’stransactionsdemand
250
are the following. If there exist functions c(t), A(i), L,(t), and A,(r) continuous for t 2 0, funcc!ions B(t) and M(t) continuous for t > 0, where B, A8, d,, and x2 are integrable, and values for A^ and ,T?such that conditions (18a-f) are satisfied, such that for each value oft, 0 5 t 5 00,
A1 = exp [ - Bt]u’(c),
(19)
A,M = 0,
(2W
(A,+r%,)B= 0,
(214
A,+&
W)
5 0,
{A,-(l+u)A,}If %,-(l+u)R,
= 0,
(22a)
GW
5 0,
@a)
such that at t = 0, (%2(O)-(1 +Q;(o))A
= 0,
Wa)
Wb) ((l-4&(0)-
MO))s = 0,
(1 -z~)n,(o)-&(o)
5 0,
@a) (2W
and in addition satisfy the transversality conditions lim A,(i) 2 0, 1-Q lim A,(t) 2 0,
t-+03
lim A,(t)M(r) = 0, I-ta,
@a)
lim A,(#)B(t) = 0,
(2W
f-+00
then c(t), B(f), M(t), A(t), A, and s maximize the functional (1) subject to the constraints (18a-f7 The proof of the sufficiency of these conditions can be found in appendix B.
R.E. Anderson, The individual’s transactions demand
251
The variable 1, can be thought of as the shadow price of consumption or the shadow price of the real stock of money. The variable & can be thought of as the shadow price of the real stock of bonds. As can be seen from conditions (22) through (25), the exact value of ilr will differ from 1, because of transactions The planning horizon can be divided into four different types of subintervals. During the first type of subinterval, the stock of money is positive; and the path of consumption must again satisfy the differential equation (7). During the second type of subinterval, the stock of bonds is increasing. From conditions (19), (21a), and (22a), the path of consumption during this subinterval must satisfy
i=
r
u’(c)
( > 0-w
-.
1 + U u”(c)
(27)
During the third type of subinterval, the stock of bonds is decreasing. From conditions (19) (2 1a), and (23a), the path of consumption during this subinterval must satisfy
During the fourth type of subinterval, both the stock of money and the stock of bonds are zero, thus c = s, and condition (8) must again be satisfied. The first step in finding the optimal patterns of consumption, bond holdings, and money holdings to is determine the optimal sequence of these four types of subintervals. The best way to see the characteristics of an optimal solution is to look again at an example. The upper half of fig, 4 shows a simple path of nonfinancial income with its year-long cycle and the optimal patterns of consumption. The curve in the lower half of fig. 4 shows the total stock of financial assets. The distance below the horizontal line represents the maximum stock of bonds held during that part of the cycle. The distance between the horizontal line and the curve represents the stock of money. The optimal psths are shown for :hree different values of the initial stock of money, assuming for simplicity that the initial stock of bonds is sero. No matter what the initial stock of money or bonds happens to bc, eventually the time paths of consumption, bond holdings, and money holdings will follow unique CSS patterns. The maximum length of time that the stock of money will remaiu positive is 2rs;r. If an asset is to be held for a period of time less than 20/r, the interest income from bonds is not enough to compensate for the transactions costs, and the optimal asset to hold is money. Only for surpluses lasting longer than 217”~will part of the surplus be held as bonds. To show this result more rigorously look at the interval (tI, t2) in fig. 4. At t, the stock of bonds ceases to
252
K.E. Anderson, The individual’s transactions demand
Fig. 4
increase, and tke surplus of income over consumption is now added to the stock of money. At tz the stock of money is exhausted, and the deficit of consumption over income is now paid for by selling bonds. Since L, and AZ are continuous, conditions (22a) and (23a) require that
and
Since both the stock of money and the stock of bonds ar positive during this subinterval, conditions (20a) and (21a) require that ,I, be constant and that 8 I.2 = - r).,. Therefore integrating 1, over the interval (t, , t2) produces (l-U)+(I+Uj;Lr
= -r(t,--t,)%,,
or tz-fl
=
2u/r.
As in the previous single asset case, the CSS holdings of money and bonds are always proportional to the price level. They are again proportional to real income if income at each point in time is increased by a uniform percentage and if the titility function is homothetic. The transactions demand for money now depends on the interest rate on bonds, and this is a considerably more complicated problem than in the lot-size inventory model. In the lot-size model, an increase in the interest rate reduces the proportion of the surplus held as
R.E. Anderson, The individuall’s transactions demand
253
money at each point in time but has no effect on the size of the surplus5 Consequently, there is no doubt in the lot-size model that interest rates and the demand for money move inversely. However, in the mode1 of the transactions demand presented here, an increase in the interest rate on bonds has at least two separate effects on an individual’s behavior. First, it tends to flatten the pattern of consumption whenever the stock of bonds is being increased or decreased. From relations (27) and (29, c will on the average not be such a large negative number when r is increased. This tends to increase the total stock of money and bonds at each point in time as these assets are used to a greater extent to smooth out the pattern of consumption. The second and perhaps most important effects of an increase in interest rates is to reduce the value of 2+, the maximum period of time that money should be held. This tends to reduce the proportion of total assets held as money at each point in time. On the one hand the total stock of transactions assets is increased, but on the other hand the proportion held as money is reduced. Consequently, the net effect on the transactions demand for money is uncertain. Though I am convinced that the demand for money is normally, or on the average, reduced by an increase in the rate of interest, I cannot prove that the stock of money is reduced at all points in time, for all conceivable patterns of income, and for all utility functions. The real balance effect described above is now made more complicated by the introduction of an interest-earning asset. If the initial stocks of assets, M0 and B,, are different from their CSS values, the individual’s behavior may be altered in two ways. First, he may find it optimal to rearrange his initial portfolio of assets. If the individual has a large initial stock of money, it may be optimal to use part of the stock to buy bonds if the bonds are to be held for a period of time greater than 2u/r. However, it would never be optimal to sell some of the initial stock of bonds and instead hold money. To do so would sacrifice interest income with no saving of transactions costs. Secondly, the individual’s path of consumption will be different than the CSS path during some initial period of adjustment. In order to illustrate the effect on consumption of an initial stock of money greater than the CSS stock, let us consider two situations depicted in fig. 4. In the first situation, the intial stock (A@ is only modestly greater than the CSS stock. Because the stock of money will be spent on consumption in a relatively short period of time (less than 20/r), there is no advantage in using pari of the initial stock to buy bonds. This initial stock of money will only cause a temporary increase in consumption. In the second situation, the initial stock of money is quite large, I@ in fig. 4; and it is now profitable to use part of the initial stock to buy bonds since the bonds will be held for a period of time greater than Sin a later work, Johnson (1970) does attempt to modify the lot-size model SO that net income from managing the portfolio of transactions assets (interest income minustotal transactions costs) is included as part of total income.
R.E. Anderson, The individual’s transactions demand
254
2v/r. In concluGon, a small excess stock of money will only cause an increase in consumption. L*the excess stock is large, it will cause an increase in both the level of consumption and the demand for bonds. 7. Concluding remarks The basic motive for an individual to hold transactions balances of money or of other assets is the lack of synchronization between his pattern of expenditures and his pattern of receipts. The lot-size inventory model developed by Baumol, Tobin, and others shows the optimal patterns of money and bond holdings for an exogenously given a5d a very arbitrary pattern of receipts and expendiWres, fn the mode! of tke transactions demand for money presented here, I have shown how a:? individual might simultaneously choose the patterns of money holdings, bond holdings, and consumption expenditures over time that maximize utility when faced with a wide variety of possible income patterns. The pattern of consumption expenditures, like the patterns of money and bond holdings, depends on such parametl:rs as the individual’s rate of time preference, the rate of deflation or inflation, the interest rate on bonds, and the transactions costs in buying and selling bonds. The single most important defect of the model of the transactions demand for money presented here is the assumption that transactions costs are proportional to the size of the transaction. Also, if there are fixed costs or economies of scale in buying consumption goods, an individual is likely to find it optimal to hold inventories of commodities in addition to money and bonds.(’ However, the inclusion of these generalizations into this model sf the transactions demand must await the further development of optimization techniques for problems with fixed costs or integer variables. Another unrealistic assumption in this model is that the individlual is assumed to know his future pattern of income and his future needs for consumption with complete certainty. In other words, the precautionary motive for holding money has been ex’cluded. An important area for future research is to extend the model developed above to include uncertainty about the pattern of income and about preferences for consumption. Appendix A
p-‘jr exp [-Bt]{rr(c*)-U(C)) >
p-’ Jr exp
[-
Ot]~‘(c*)(c~
2 s,” exp [-6t]l*(A&-lh*)dt,
dt -
C)
dt
,
(b) (c)
Teige and Parkin (1971) have extended the lot-size model to include the possibility of commodity inventories.
R.E. Anderson, The individual’s transactions demand
=
+Io” i*(M* -M) dt,
n*(~-iv*)/;
2 A*(M-M*)l;
(d)
,
(d
5 ~*Gw*(o) - iz*(o)M(o), =
255
(0
0.
(s)
Explanation. ( a) rewriting (6); (b) since u(c) is assumed to be differentiable and concave and by Taylor’s formula; (c) from (2a) and (3); (d) integrating by parts; (e) by relations (4a), (4b) and (2b); (f) by the transversality condition (5); (g) by relation (2~). Appendix B
The proof of the sufficiency of these conditions is the same as that used in appendix A through line (b). = jr n:(n;r-n;l*+rB-(l -rB+(l
+f+t*+(l
+u)A -(l-v)S}
+s;(1;+rl;)B* -
dt,
(e)
dt,
2 ATMI,” -A:M*lo” +I;BI; 2
-U)S*
(4 -lw;B*lo”
m
,
R:(O+)M(O+) + Rr(O)M*(O+) - i.;(O) B(O+)
+ Iz;(O)B*(O+), = If(O){M*(O,-(1 +(I +u+-(1
(g) +t’)A*+(l
-@*-M(O)
-u)S)
+$(O){B*(O)+if*-S*-B(O)-A+S}, = -{$(O)-(1
+n)if(O)}R-((1
(h) -tl)J$(O)--Az(O)]S
+ {AZ(O) - (1 + u)l:(o)}AI* + {(1 2 0.
P)A:(o)
-
i;co>)S* .
256
R.E. Anderson, The individual’s tranJ;actionsdemand
Explanation. (c) fram (19j and (Isa); (d) from (2?a), (23a) and (1 Sb); (e) integrating by parts; (f) from (20a) through (21 b), and (1 Sf); (g) from (26a-b); (h) from (18c-d); (i) from (1 Se); (j) from (24a) through (25b). References Arrow, K.J. anld M. Kurz, 1970, Public investment, the rate of return and optimal fiscal policy (The Johns Hopkins Press, Bahimore). Baumol, W.J., 1952, The transactions demand for money: An inventory theoretic approach, Quarterly Journal of Economics 56,54556. Feige, E.L. and M. Parkin, 1971, The optimal quantity of money, bonds, commodity inventories, and capital, American Economic Review 61,335-49. Friedman. M., 1969, The optimum quantity of money and other essays (Aldine, Chicago). Johnson, H.G., 1969, Inside money, outside money, income wealth and welfare in monetary theory, Journal of Moc;y, Credit and Banking 1,30-45. Johnson, H.G., 1970, A note on the theory of the transactions demand for cash, Journal of Money, Credit and Banking 2,383-4. Johnson, H.G., 1969, Notes on the theory of transactions demand for cash, in: Essays in Monetary Economics (London). Liviatan, N., 1965. The long-run theory of consumption and real balances, Oxford Economic Papers 17,205-l 8. Mangasarian, O.L., 1966, Sufficient conditions for the optimal control of nonlinear systems, Journal of SIAM Control 4,139-52. Marty, A.L., 1964, The real balance effect: An exercise in capital theory, Canadian Journal of Economics and Political Science 30,360.-67. Niehans, J., 1971. Money and barter in general equilibrium with transactions costs, American Economic Review 6 I, 773-83. Patinkin, D., 19G2, Money, interest and prices, 2nd ed. (Harper and Row, London) ch. 3. Pontryagin, L. et al., 1962, The mathematical theory of optimal processes (John Wiley, New York). Samuelson, P.A., 1969, Nonoptimality of money holding under laissez faire, Canadian Journal of Economics 2,303-g. Vind, K., 1967, Control systems with jumps in the state variables, Econometrica 35,273-7.