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Uncertainty and the demand for liquid assets
JOURNAL
OF ECONOMIC
Uncertainty
THEORY
2,
368-382 (1970)
and The Demand
for Liquid
Assets
K. DIXIT
AVINASH
Balliol College, Oxford, England AND STEVEN M.
GOLDMAN
University of Califbrnia, Berkeley, Carifrnia
94720
Received March 26, 1970
1. INTRODUCTION Monetary theory is concerned with the choice among assetsof differing degrees of liquidity. Were it not for the essential characteristic of uncertainty, such a choice would be straightforward. Conventional analysis in monetary theory (e.g., Patinkin [4]) only implicitly accounts for uncertainty. Portfolio selection theory, while considering uncertainty explicitly, makes no provision for liquidity whatever (e.g., Levhari and Srinivasan [3]). Our attempt here is to provide a simple analysis which incorporates both of these ingredients-uncertainty and liquidity-into a theory of portfolio selection. We consider the case of a representative consumer who begins his infinite life with a specified endowment of wealth. At each moment in time, he can consume out of this wealth and allocate the remainder between cash balancesand a real asset.The real rate of return to the real asset is presumed known with certainty; this might be visualized as the productivity of capital in the simplest linear model of production. Since the rate of change in the price level is uncertain, so, therefore, is the real rate of return to cash balances. For simplicity, we assumethat the consumer’s perception of the structure of his expectations regarding the rate of inflation is, at each moment in time, projected as stationary. A full dynamic treatment would involve the endogenous explanation of the perceived distribution of the rates of inflation. Cash balances, by virtue of their liquidity, provide services which, together with the flow of consumer goods, provide “utility.” For analytical convenience, we have represented these services by the stock of real cash 368
UNCERTAINTY
AND LIQUID ASSETS
369
balances (see also Douglas [l], Sidrauski [6], and Uzawa 181, but also Hahn [2] and Tobin [7]). There is no conceptual difficulty as regards defining such a derived utility function, but it is only under rather stringent conditions that it will have the desired concavity. As we shall also assume a utility function of a specific form, the analysis is, doubtless, too narrow in this respect. We shall begin the next section by outlining the basic model and discussing the existence and uniqueness of the consumer’s equilibrium. Section 3 will be concerned with various comparative results describing the effects upon the demands for money, consumption, and real assets arising out of a change in the level of uncertainty. Section 4 will employ some of these results to examine the interaction between monetary expansion, uncertainty, and the rate of inflation.
2. CONSUMER EQUILIBRIUM Suppose the consumer has an initial endowment a, of real assets. Of this he consumes c,, in period 0, holds m, in the form of cash balances, and invests the rest, k, = a, - c,, - m, ,
(1)
in the only available real asset. For this section, we shall suppose that the real asset, k, , yields an uncertain one-period return, r,, , which is positive with probability one. (Thus the rate of return is lOO(r, - 1) per cent). Cash balances carry a real return yO. (If the rate of inflation is lOOi,, per cent and no interest is paid on cash balances, then y0 = (1 + i&l). Thus, the nominal rate of return to the real asset is (r,,/yo) - 1, and we shall suppose this positive with probability one. Otherwise cash balances may dominate the real asset by having both a higher rate of return and greater liquidity. At the beginning of period 1, then, the real value of the consumer’s assets will be al = Yom0 + roko .
The consumer’s decision will be made so as to maximize temporally additive and stationary utility function z. P&
(2)
an inter-
y mt),
where ct is the flow of consumer goods to him in period t and m, the stock of cash balances held by him. The discount factor /3 is less than one. We 642/2/4-4
370
DIXIT
AND
GOLDMAN
shall further suppose that the instantaneous utility function is strictly concave and homogeneous of constant degree in its arguments. This implies unit income elasticity of demand for consumption and real cash balances. This assumption of a constant ratio between real cash balances and expenditures (except for changes in the “constant” resulting from changes in the rates of return) is a familiar generalization of the quantity theory of money. With only minor losses in generality (indicated in the conclusion of Section 3), we may additionally suppose that the utility function is of the constant elasticity form,
Now suppose the maximum sum of discounted utilities the asset position a, is defined to be
starting from
subject to the feasibility conditions like (1) and (2) for every t; the expectation being taken with respect to the distributions of rt and yt . Provided these distributions are stationary over time, i.e., expectations currently held assume stationarity over the future, we can write the usual two-stage equation resulting from the principle of optimality,
where E, is expectation over the distribution of r. and y, . Note that this does not require unchanged formation of expectations: The expectation structure may change when the consumer considers his decision at time 1, but then his expectations for periods 1,2, 3,... must all be the same. We shall confine ourselves to an interior maximum; the conditions for its existence will be commented upon later. The necessary conditions can be obtained by equating to zero the partial derivatives with respect to co and m, of the expression within braces in (5). This gives (cOmi-a)-R w;-lmi-a (cOmi-Tn
- ~E[V’(a,)
(1 - CX)co”m;” - f3E[ V(ul)(ro
ro] = 0,
- y,)] = 0.
(6)
Now V’(u) is the increment in the maximum utility which would result from a unit increase in initial assets. When the programme (ct , mt)
UNCERTAINTY
AND
LIQUID
ASSETS
371
is arranged optimally the allocation of this extra unit will be immaterial. In particular, consuming all of it would yield V(u) = uc(c, m).
(7)
A rigorous proof can be built along the lines given by Levhari and Srinivasan [3]; we merely ask the reader to note the similar situation in demand theory where the marginal utility of income equals that of each good divided by its price. The corresponding equation for m is more difficult as it involves changes in utilities in subsequent periods, but we shall not need it. Using (7) in (6), we find
~(co~m~-FVco= BJ3u,(c,, ml>rol, (1 - or)(c,“m~-YVm, = B~[~,(c, , mlPo - Y&I.
(8)
Our assumptions concerning stationarity and homotheticity will ensure that the proportions of assets consumed and allocated to the two categories will remain constant over time. Let us define X to be the fraction consumed, and 6 to be the fraction of the savings (1 - h) a held in cash balances. Then, using (2), we get after some simplification in (5), (1 - A)>”= f%!qr(Sy + (1 - S) r)+], l--LX - a
x I--x
(9)
(1 - x>n = /3E[fs(r - y)(Sy + (1 - 8) r)-“I.
These two equations define the optimal proportions h and 6. This, however, is still not the most convenient form for us. It remains to define 8 = r - y, which is positive with probability one since r/y exceeds unity with probability one. Making use of 0, and subtracting the second equation from the first, we get the pair (1 - h)” = j%E[r(r - 8(9-n],
(10) (1 - h)lz [1 - +
&]
= /3E[(r - se)l-n].
To obtain the optimum pair (6, X), we examine in the (6, h) space the loci defined by these two equations. In each, it is easy to see that the left side is a decreasing function of X. Since n > 0, the right side is an increasing function of 6. Now take a point (6, h) on the curve and increase 6. This increases the right side, so h must decrease to increase the left side and regain the curve. Thus the first curve, say h,(6), slopes downward. In the second equation, the right side is an increasing function of 6 if n > 1 and a decreasing function if n < 1. Thus that curve &(6) slopes upward if
372
DIXIT
AND
GOLDMAN
n < 1 and downward if n > 1. In case n < 1, then, uniqueness of the intersection is assured. If n > 1, we can show that at any intersection h,(6) has a gentler slope than h,(6). Since inequalities between slopes must reverse at successive intersections, there can be only one point of intersection. Figure 1 illustrates the two cases.
FIGURE
1.
To prove the assertion concerning the relative magnitudes of the slopes in case n > 1, we first calculate the slopes by implicit differentiation. This gives u1 = -(l - h) E[rB(r - SO)-+l]/E(r(r
- SO)-n],
u’2 = -(I - h)cn - I) ~[e(r - sep+~[~
-
hep]
+ +
(1 - q-l].
At an intersection point, u1 2 o2 as c1 ? c2 , where ci are the elasticities of 1 - Xi with respect to 6, El - Ez = >
E[ro(r - S@+i] cn - 1) E[ecr - sepq ’ [ E[r(r- se)-n] - nE[(r_ ae)l-n]1 ’ ; O1(1-h)%-lI E(rS8(r - se)-+l]- E[SB(r - 28-y I
=
E[r(r - Sfl-n]
E[(r - S@-y
E[rz(r - St9)-+1] _ E[r(r - Se)-%] E[r(r - S6)-n] E[(r - iW)l-%] I ’
I
UNCERTAINTY
313
AND LIQUID ASSETS
since r#(r - 68)+-l = r2(r - 88)-,-l - (r - 8@n, etc. Now define two functions on the joint probability space of r and 19byf = r(r - 68)-(n+1)/2 and g = (r - 80)(1-n)/2. Note the Cauchy-Schwarz inequality
W2) *E(g2) 2 Lwk)12. This gives c1 > l 2 , i.e., u1 < u2 . As both u1 and u2 are negative, 1 u1 / > / u2 1 as asserted. To investigate the existence of a solution, put 6 = 0 in (10) to get the corresponding values h,, and h,, of h defined by (1 - A,$
= /3E(P-ln) = (1 - h2Jn (1 - +
1 h’uh,, ).
This shows (1 - A,,,)” < (1 - X2,Jn, i.e., h,, > h,, . Now put 6 = 1 to get values h,, and h,, . For an interior solution to exist, we clearly need h,, > h,, . The condition which can be derived from this is not particularly illuminating, so we shall not spell it out. If it is violated, the optimum 6 will equal one, i.e., all of the assets saved will be held as cash balances. Thus the condition is a rather more stringent form of the earlier requirement ruling out a higher return as well as a higher liquidity for cash balances. The possibility of an optimum with zero cash balances is ruled out by the assumption of infinite marginal utility of zero cash balances.
3. COMPARATIVE
DYNAMICS
In this section we shall consider the comparative dynamic properties of the optimal allocation variables h and 6, concentrating on the effects of a change in uncertainty. The analysis is difficult enough to warrant several simplifications. First, we assume that the rate of return r to the real asset is known with certainty. Then the only uncertainty concerns the rate of return to cash balances, and hence the rate of inflation. We define a new random variable x = 0/r, which is positive with probability one. Then the equation (10) becomes (1 - h)” = j3r1-“E[( 1 - 6x)-“] (1 - A)” [ 1 - e
(11) A]
= /3r1-mE[(l - 6x)l-“1.
The second simplification restricts the kinds of changes allowed in the distribution of x. The most general sensible definition of an increase in
374
DIXIT
AND
GOLDMAN
uncertainty would be any mean-preserving shift in the distribution which lowers the expected value of any concave function of the random variable, i.e., which makes any risk-averse individual worse off. Such definitions and their consequences are studied in detail by Rothschild and Stiglitz [5]. We are concerned with shifts in both equations in (11) resulting from a change in uncertainty in x, and this involves the relative convexity of (1 - 6x)-” and (1 - Sx)l-,. This needs stronger general theorems, and we feel that the special definition we consider allows enough specific and interesting answers. We suppose that x can assume a fixed finite number of values xi > 0 with probabilities pi , pi 3 0, C pi = 1. The mean X = C pixi is also fixed. A decrease in uncertainty will be achieved simply by shifting every xi towards X in the same proportion, i.e., by moving xi to xi(P)
=
Px
+
(l
-
p)
Xi
3
O
Thus an increase in p decreases uncertainty. We shall then find the response of 6 and X to changes in p, calculating the appropriate derivatives and evaluating them at p = 0. For brevity, define zi = (1 - 6x,), and arrange zi in increasing order. Write the two expectations involved in (11) as El = c piz;“,
E, = c p,z;-“.
We shall make use of several elementary transformations and inequalities concerning expectations involving xi and zi . These are set out in Appendix A, and are referred to in the text by number. We shall need the following results on partial derivatives of Ei . aEJa8 = n C piz;“-lxi
> 0,
aE,/as = (n -
1) cpaz;“xi
aE,lap
piz,+yz
=
n8 c
-
= n c piz,-+yz, 1 n Cp,z;” [
by
1: 8 j: xi)
- 2) --Z c piz;n-ll
L4,
; = ;:
(12’)
UNCERTAINTY
AND
LIQUID
375
ASSETS
Turning to less simple expressions, 1 aE,
1 aE, ----_ E, as
El
as
=- 1 n C Pizi” s I. ( c p,z,l-”
< 0
by
L3
_ ~~P;~~~-‘) ,z, n
and
+ (1
P&yJ pi,;-,
L2.
Similarly 1 aE, ----Es ap
1 a& > 0 El ap .
The comparative dynamics of 6 need some simple manipulations to eliminate A, yielding
in (I I)
1-a n = @l-“E1 . 1 - cxE,IE, 1
(13)
This can be differentiated totally treating 8 as a function of p, to get
(14)
>o
using earlier results.
Thus a decrease in uncertainty leads to an increase in the proportion of assets held in cash balances-a sensible result. Now we can use the first equation in (1 l), differentiating it totally with respect to p using the expression (14) for dS/dp. This gives dh -=--
4
a/3r1-n Cl - ‘In-l
aE, aE, ---
ap as
--aE, aE,
ap as
!$k (1 - 01E2/El) _ ncuE, ($ !?j$ - i
-. 2)
(15)
376
DIXIT
Since the denominator the sign of
is positive, the sign of dh/dp is the opposite of
--aE, aE,
AND
GOLDMAN
aE, aE,
ap as -Zap
= n(n- 1)6%[cPiZi”*CpiXiZin--l - Cp&“-l ’ 1 Pix,‘cn] = n(n - 1) x [c pizF+l
- 1 piz:-n - (C piz;“ja]
>O
L3.
if if
n>l ntl
by
by
L1
Thus
A decrease in uncertainty, therefore, implies a higher propensity to consume if n < 1, which is precisely the opposite of the result obtained by Levhari and Srinivasan [3] in the case where there is only one asset. We are not able to pinpoint the precise cause of this, but it clearly relates to the liquidity property of our uncertain asset. Two further results involving variation in the degree of uncertainty are derived in Appendices B and C: 1. A reduction in uncertainty causes an increase in the demand for real cash balances. 2. A reduction in uncertainty causes a decrease in the demand for real assets. Summarizing the results of this section then, a decrease in the uncertainty of the quotient of the nominal rate of interest on real assets divided by one plus that rate (or simply x) implies: 1. An increase in 6, the proportion of assets held as money; 2. An increase (decrease) in h, the average propensity to consume, when n < l(n > 1); 3. An increase in 6(1 - h), the demand for real cash balances; 4. A decrease in (1 - 6)(1 - h), the demand for real assets. For the analysis of the more general homogeneous utility function, the results 1, 3, and 4 above are unaltered. Result 2 is modified to read 2’. An increase in h, when n ,( 1 and u‘(b) < 0; A decrease in X, when n > 1 and u’(b) > 0;
UNCERTAINTY
AND LIQUID ASSETS
377
An increase (decrease) in X, when n < l(n > 1) and o’(b) = 0, where b = m/c
and u(b) = mu,@, 4 cu,(c, m> *
4. THE RATE OF INCREASE IN THE MONEY SUPPLY In this section we shall relate the rate of increase in the money supply to the rate of inflation and describe the qualitative effects of uncertainty upon this relationship. We must first, however, specify the manner by which money enters the private sector. Without a completely described financial structure, our analysis can merely be suggestiveand, indeed, is offered in that spirit. Let us suppose that the government introduces new money through the purchase of real assetsfrom the private sector. Further, let the future income from these assetsbe distributed uniformly to all consumers who are, in turn, presumed identical in all respects. This consumer will follow the behavioral rules described in Sections 2 and 3. So long as the per capita governmental holdings of real assetsare smaller than those desired by the consumer, it is a matter of indifference to him where the title of ownership resides.That is, he has chosen to opt for a stream of real income and publicly owned resources provide a perfect substitute for individual assetsso long as he desiresthat income stream over the alternatives of additional consumption and/or money holdings. Some limitations imposed on government policy by this attitude are examined in Appendix D. Let p,, - 1 denote the rate of increasein the money supply, -Ml = 6,(1 - X,) a, ) PI
y. = r. - e. ,
Pl
=Po/Yo~
a, = (1 - ho) ao(ro- ~9~8,).
Thus x1 =
po - (1 - h1)81’80 /ho- (1 - h,) 6, .
(16)
In this manner we are able to assessthe implications of monetary expansion upon the rate of inflation. Pleasenote, however, that h, and 6, depend upon the consumer’s reaction to an observed x0 . Equation (16) provides only an implicit solution for x0 . Rewritten below, we may
378
DIXIT
AND
GOLDMAN
express the required rate of increase in the money supply as a function of the desired level of inflation and given expectations,
do= (1- wwo - %x0) (1- x0) ’
(17)
In the absence of uncertainty, the same rate of inflation produced with * _ (1 - A”)(1 - 6*x,) P (1 - x0) ’
would be (18)
where A* and S* are determined from (11) by the pair of equations (1 - A*), = /3rl-“( 1 - s*X&n, s*x, = (J+)(
1 TA*
).
(19)
Let us now suppose that x,, = X0, i.e., policy is to generate the consumer’s expected value of x. If the observation of such an x0 does not alter his future expected x and does not increase his level of uncertainty (dp 2 01, then p. > P *. Phrased simply, the presence of uncertainty implies that a higher rate of monetary expansion is required to generate the consumer’s mean expectations regarding x. The proof is offered below: ho = (1 - u@l/~o l--Z0
- S800) ’
p* = (1 - A*)(1 - s*zo) l-Z0 * p. - p* is equal in sign to (1 - h,)(S,/S, - &X0) - (1 - x*)(1 - 6*x0) > (1 - Ar)(l - 6,x,) - (1 - A’)(1 - 6*x0) since dp 2 0, and therefore 6, > So. But E[(l - A,)(1 - 6,x,)]-”
= [(l - A*)(1 - s*z,)]-”
and n > 0 imply (1 - A,)(1 - S,Z~) > (1 - A*)( 1 - S*zo), and therefore PO > P** We note one further relationship between the rate of monetary expansion and the level of inflation. From (19) we observe that A* is independent
UNCERTAINTY
AND
LIQUID
379
ASSETS
of x0 . Thus 6*x, is also independent of x0, and ~~(1 - x,,) depends only upon p, r, and n. An increase in the rate of monetary expansion therefore requires an increase in x0 , which in turn implies an increase in the rate of inflation. A
APPENDIX
L1: c pixiz;“/c p&k = f lz PiZYk/CPie-” - 11. Simply substitute xi = (1 - zJ6: L2:
c piz;” - c p&”
= c p,z,“(l
L3:
Cp&”
> (&J,z;“)2
. cpiz;l-”
- 1 + 6X$) > 0;
by the Cauchy-Schwarz inequality; L4:
Define Z = 1 - 6X, then z > c p&“/C
RHS=Qg
PiZLk,
zi = C qizi , say. 35
Then C pi = x qi = 1, and qi/pi = z;“lC pjzJ:” is a decreasing function of zi . Since zi are arranged in increasing order, 3, such that
and $
(4i
Pi) = t (Pi - 4i) = S, ie+l
-
say.
Then C (Pi -
4i)
zi
=
izl
(pi
-
qi)
Zi
-
> s(z*,+1 - z&J > 0. Therefore Z = Cpizi
> C qizi .
z
(qi
-
pi)
Z
380
DIXIT
AND
GOLDMAN
B
APPENDIX
&3(1 - A)/ap is, by (1 l), (14), and (15), equal in sign to
which, by sbustituting equals T
+ na(n - l)(Ezl-” = [ =
[
41-Q -4 ‘(1;
1
I
aE,/a&
aE,pp,
from (12’)
aE,jS
- ZEZP-~)(OI(~ - n) - 1) Ez-”
[n(Ez-”
r-7
for aE,pp,
- ZEz-“)(Ez-“-I] + a( 1 - n)(Ez-“Ez-”
[E(Z - z) z-“-lEz-”
‘) ] [a(1 -n) -~][Ez-~Ez-”
- Ezl-“Ez-“-1
- Ezl-“E~r”-~)] + Ezrn-%(Z
- z) zP],
which is positive by Cauchy-Schwarz and (12’).
C
APPENDIX
a(1 -X)(1 - 8) = _---%(l -A) + aP
ah n (l$) aP [
x [Ez-“Ez+
- Ez~-“Ez-“-~]
][[l
- a(1 - n)]
+ Ez-“-lE(z
- 2) z-“1
+q!$L][$$-~~] =
t
{[[l - ~(1 - t~)][Ez-~Ez-~ - Ez~-~Ez-~-‘]}
+ Ez-“-lE(z
- 5) z-“1
- [a( 1 - n)] r+] (l-4 zzz n ___ I El x [Ez-“Ez-”
I
[Ez-“Ez-”
- Ez-“-~Ez~-“]
{[ol(l - n)(Z - 1) + l] - Ez-+~Ez~-“]
+ EzP-~E(z - 2) z-“} < 0,
since 1 + 01(1 - n)(T - 1) > 0 for X < 1, (12’), and the Cauchy-Schwarz inequality.
UNCERTAINTY
AND LIQUID ASSETS
APPENDIX
381
D
As noted, our treatment of government requires that the public holdings of real assets be less than the real assets that the private sector wishes held. Were this condition violated then individual planning would include, by necessity, projections of future governmental asset accumulation. In the following, we turn our attention from uncertainty to examine the possible monetary policies which render private and public planning consistent under our formulation. We shall consider governmental policy under conditions of long run equilibrium, i.e., policies which exactly satisfy consumer expectations where those expectations are held with certainty. Supposing that initially the government’s per capita holdings of real assets are less than the private sector’s demand, i.e., k,(O) -=ck(O), we may relate the rate of inflation to the expansion of the money supply through (18) as c1 = (1 - A*)(1 - 6”X) l-x or
-=1 +g l+i
whereg=p-
ptn,vn
(20)
1.
The rate of growth in the demand for real assets is then Ak -Yzz k
B
The change in the money supply is given by the government’s purchases of assets, i.e., (p + Ap) Ak, = AM = gM, and thus
4k -
Iti
*
(22)
In order to avoid the total governmental ownership of the economy’s real assets, the rate of monetary expansion (or, alternately, the rate of
382
DIXIT AND GOLDMAN
inflation) must be limited. (This, of course, is a consequence of the “inside” nature of money in our model). Specifically Ak, < Ak, or g [-&I then, letting $ = plnrlln may write
< (1 + g)(l - *) and substituting
= g - i; from (lg), (19), and (20) we
gal - “I@ - 41 + 1 - 41 < 44 - l)(r - 4).
(24)
If (1 - ol)(r - $) + 1 - 4 > 0 then (24) may be interpreted as a limitation on the maximum rate of growth of money. However, when (1 - a)(r - 4) + 1 - C#J< 0, then (24) has the interpretation of a lower bound on the rate of growth. The explanation of this latter, counterintuitive result lies in the observation that an increase in g increases Ak and will decrease [a/(1 - S)].
ACKNOWLEDGMENTS The authors gratefully acknowledge their debt to beneficial discussion with Professor D. McFadden and to research support from the Ford Foundation and the National Science Foundation. However, the conclusions, opinions, and other statements in this study are those of the authors and are not necessarily those of the foundations.
REFERENCES
1. A. DOUGLAS, A theory of saving and portfolio selection, Rev. Econ. Stud. 35 (1968). 2. F. HAHN, Walras Bowley Lecture, Econometric Society Meetings, December, 1969. 3. D. LEVHARI AMD T. N. SRINIVASAN, Optimal saving under uncertainty, Rev. Econ. Stud. 36 (1969). 4. D. PATINKIN, “Money, Interest and Prices,” Row-Petersen, Evanston, Ill. 1956. 5. M. ROTHSCHILD AND J. E. STIGLITZ, “Increasing Risk: A Definition and its Economic Consequences,” Cowles Foundation Discussion Paper No. 275, 1969. 6. M. SIDRAUSKI, Rational choice and patterns of growth, J. Polit. Econ. 77 (1969). 7. J. TOBIN, Money and economic growth, Econometrica (1965). 8. H. UZAWA, Time Preference, the Consumption Function, and Optimum Asset Holdings, in “Value, Capital and Growth” (J. N. Wolfe, Ed.), pp. 485-504, Edinburgh University Press, Edinburgh, 1968.
OF ECONOMIC
Uncertainty
THEORY
2,
368-382 (1970)
and The Demand
for Liquid
Assets
K. DIXIT
AVINASH
Balliol College, Oxford, England AND STEVEN M.
GOLDMAN
University of Califbrnia, Berkeley, Carifrnia
94720
Received March 26, 1970
1. INTRODUCTION Monetary theory is concerned with the choice among assetsof differing degrees of liquidity. Were it not for the essential characteristic of uncertainty, such a choice would be straightforward. Conventional analysis in monetary theory (e.g., Patinkin [4]) only implicitly accounts for uncertainty. Portfolio selection theory, while considering uncertainty explicitly, makes no provision for liquidity whatever (e.g., Levhari and Srinivasan [3]). Our attempt here is to provide a simple analysis which incorporates both of these ingredients-uncertainty and liquidity-into a theory of portfolio selection. We consider the case of a representative consumer who begins his infinite life with a specified endowment of wealth. At each moment in time, he can consume out of this wealth and allocate the remainder between cash balancesand a real asset.The real rate of return to the real asset is presumed known with certainty; this might be visualized as the productivity of capital in the simplest linear model of production. Since the rate of change in the price level is uncertain, so, therefore, is the real rate of return to cash balances. For simplicity, we assumethat the consumer’s perception of the structure of his expectations regarding the rate of inflation is, at each moment in time, projected as stationary. A full dynamic treatment would involve the endogenous explanation of the perceived distribution of the rates of inflation. Cash balances, by virtue of their liquidity, provide services which, together with the flow of consumer goods, provide “utility.” For analytical convenience, we have represented these services by the stock of real cash 368
UNCERTAINTY
AND LIQUID ASSETS
369
balances (see also Douglas [l], Sidrauski [6], and Uzawa 181, but also Hahn [2] and Tobin [7]). There is no conceptual difficulty as regards defining such a derived utility function, but it is only under rather stringent conditions that it will have the desired concavity. As we shall also assume a utility function of a specific form, the analysis is, doubtless, too narrow in this respect. We shall begin the next section by outlining the basic model and discussing the existence and uniqueness of the consumer’s equilibrium. Section 3 will be concerned with various comparative results describing the effects upon the demands for money, consumption, and real assets arising out of a change in the level of uncertainty. Section 4 will employ some of these results to examine the interaction between monetary expansion, uncertainty, and the rate of inflation.
2. CONSUMER EQUILIBRIUM Suppose the consumer has an initial endowment a, of real assets. Of this he consumes c,, in period 0, holds m, in the form of cash balances, and invests the rest, k, = a, - c,, - m, ,
(1)
in the only available real asset. For this section, we shall suppose that the real asset, k, , yields an uncertain one-period return, r,, , which is positive with probability one. (Thus the rate of return is lOO(r, - 1) per cent). Cash balances carry a real return yO. (If the rate of inflation is lOOi,, per cent and no interest is paid on cash balances, then y0 = (1 + i&l). Thus, the nominal rate of return to the real asset is (r,,/yo) - 1, and we shall suppose this positive with probability one. Otherwise cash balances may dominate the real asset by having both a higher rate of return and greater liquidity. At the beginning of period 1, then, the real value of the consumer’s assets will be al = Yom0 + roko .
The consumer’s decision will be made so as to maximize temporally additive and stationary utility function z. P&
(2)
an inter-
y mt),
where ct is the flow of consumer goods to him in period t and m, the stock of cash balances held by him. The discount factor /3 is less than one. We 642/2/4-4
370
DIXIT
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shall further suppose that the instantaneous utility function is strictly concave and homogeneous of constant degree in its arguments. This implies unit income elasticity of demand for consumption and real cash balances. This assumption of a constant ratio between real cash balances and expenditures (except for changes in the “constant” resulting from changes in the rates of return) is a familiar generalization of the quantity theory of money. With only minor losses in generality (indicated in the conclusion of Section 3), we may additionally suppose that the utility function is of the constant elasticity form,
Now suppose the maximum sum of discounted utilities the asset position a, is defined to be
starting from
subject to the feasibility conditions like (1) and (2) for every t; the expectation being taken with respect to the distributions of rt and yt . Provided these distributions are stationary over time, i.e., expectations currently held assume stationarity over the future, we can write the usual two-stage equation resulting from the principle of optimality,
where E, is expectation over the distribution of r. and y, . Note that this does not require unchanged formation of expectations: The expectation structure may change when the consumer considers his decision at time 1, but then his expectations for periods 1,2, 3,... must all be the same. We shall confine ourselves to an interior maximum; the conditions for its existence will be commented upon later. The necessary conditions can be obtained by equating to zero the partial derivatives with respect to co and m, of the expression within braces in (5). This gives (cOmi-a)-R w;-lmi-a (cOmi-Tn
- ~E[V’(a,)
(1 - CX)co”m;” - f3E[ V(ul)(ro
ro] = 0,
- y,)] = 0.
(6)
Now V’(u) is the increment in the maximum utility which would result from a unit increase in initial assets. When the programme (ct , mt)
UNCERTAINTY
AND
LIQUID
ASSETS
371
is arranged optimally the allocation of this extra unit will be immaterial. In particular, consuming all of it would yield V(u) = uc(c, m).
(7)
A rigorous proof can be built along the lines given by Levhari and Srinivasan [3]; we merely ask the reader to note the similar situation in demand theory where the marginal utility of income equals that of each good divided by its price. The corresponding equation for m is more difficult as it involves changes in utilities in subsequent periods, but we shall not need it. Using (7) in (6), we find
~(co~m~-FVco= BJ3u,(c,, ml>rol, (1 - or)(c,“m~-YVm, = B~[~,(c, , mlPo - Y&I.
(8)
Our assumptions concerning stationarity and homotheticity will ensure that the proportions of assets consumed and allocated to the two categories will remain constant over time. Let us define X to be the fraction consumed, and 6 to be the fraction of the savings (1 - h) a held in cash balances. Then, using (2), we get after some simplification in (5), (1 - A)>”= f%!qr(Sy + (1 - S) r)+], l--LX - a
x I--x
(9)
(1 - x>n = /3E[fs(r - y)(Sy + (1 - 8) r)-“I.
These two equations define the optimal proportions h and 6. This, however, is still not the most convenient form for us. It remains to define 8 = r - y, which is positive with probability one since r/y exceeds unity with probability one. Making use of 0, and subtracting the second equation from the first, we get the pair (1 - h)” = j%E[r(r - 8(9-n],
(10) (1 - h)lz [1 - +
&]
= /3E[(r - se)l-n].
To obtain the optimum pair (6, X), we examine in the (6, h) space the loci defined by these two equations. In each, it is easy to see that the left side is a decreasing function of X. Since n > 0, the right side is an increasing function of 6. Now take a point (6, h) on the curve and increase 6. This increases the right side, so h must decrease to increase the left side and regain the curve. Thus the first curve, say h,(6), slopes downward. In the second equation, the right side is an increasing function of 6 if n > 1 and a decreasing function if n < 1. Thus that curve &(6) slopes upward if
372
DIXIT
AND
GOLDMAN
n < 1 and downward if n > 1. In case n < 1, then, uniqueness of the intersection is assured. If n > 1, we can show that at any intersection h,(6) has a gentler slope than h,(6). Since inequalities between slopes must reverse at successive intersections, there can be only one point of intersection. Figure 1 illustrates the two cases.
FIGURE
1.
To prove the assertion concerning the relative magnitudes of the slopes in case n > 1, we first calculate the slopes by implicit differentiation. This gives u1 = -(l - h) E[rB(r - SO)-+l]/E(r(r
- SO)-n],
u’2 = -(I - h)cn - I) ~[e(r - sep+~[~
-
hep]
+ +
(1 - q-l].
At an intersection point, u1 2 o2 as c1 ? c2 , where ci are the elasticities of 1 - Xi with respect to 6, El - Ez = >
E[ro(r - S@+i] cn - 1) E[ecr - sepq ’ [ E[r(r- se)-n] - nE[(r_ ae)l-n]1 ’ ; O1(1-h)%-lI E(rS8(r - se)-+l]- E[SB(r - 28-y I
=
E[r(r - Sfl-n]
E[(r - S@-y
E[rz(r - St9)-+1] _ E[r(r - Se)-%] E[r(r - S6)-n] E[(r - iW)l-%] I ’
I
UNCERTAINTY
313
AND LIQUID ASSETS
since r#(r - 68)+-l = r2(r - 88)-,-l - (r - 8@n, etc. Now define two functions on the joint probability space of r and 19byf = r(r - 68)-(n+1)/2 and g = (r - 80)(1-n)/2. Note the Cauchy-Schwarz inequality
W2) *E(g2) 2 Lwk)12. This gives c1 > l 2 , i.e., u1 < u2 . As both u1 and u2 are negative, 1 u1 / > / u2 1 as asserted. To investigate the existence of a solution, put 6 = 0 in (10) to get the corresponding values h,, and h,, of h defined by (1 - A,$
= /3E(P-ln) = (1 - h2Jn (1 - +
1 h’uh,, ).
This shows (1 - A,,,)” < (1 - X2,Jn, i.e., h,, > h,, . Now put 6 = 1 to get values h,, and h,, . For an interior solution to exist, we clearly need h,, > h,, . The condition which can be derived from this is not particularly illuminating, so we shall not spell it out. If it is violated, the optimum 6 will equal one, i.e., all of the assets saved will be held as cash balances. Thus the condition is a rather more stringent form of the earlier requirement ruling out a higher return as well as a higher liquidity for cash balances. The possibility of an optimum with zero cash balances is ruled out by the assumption of infinite marginal utility of zero cash balances.
3. COMPARATIVE
DYNAMICS
In this section we shall consider the comparative dynamic properties of the optimal allocation variables h and 6, concentrating on the effects of a change in uncertainty. The analysis is difficult enough to warrant several simplifications. First, we assume that the rate of return r to the real asset is known with certainty. Then the only uncertainty concerns the rate of return to cash balances, and hence the rate of inflation. We define a new random variable x = 0/r, which is positive with probability one. Then the equation (10) becomes (1 - h)” = j3r1-“E[( 1 - 6x)-“] (1 - A)” [ 1 - e
(11) A]
= /3r1-mE[(l - 6x)l-“1.
The second simplification restricts the kinds of changes allowed in the distribution of x. The most general sensible definition of an increase in
374
DIXIT
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uncertainty would be any mean-preserving shift in the distribution which lowers the expected value of any concave function of the random variable, i.e., which makes any risk-averse individual worse off. Such definitions and their consequences are studied in detail by Rothschild and Stiglitz [5]. We are concerned with shifts in both equations in (11) resulting from a change in uncertainty in x, and this involves the relative convexity of (1 - 6x)-” and (1 - Sx)l-,. This needs stronger general theorems, and we feel that the special definition we consider allows enough specific and interesting answers. We suppose that x can assume a fixed finite number of values xi > 0 with probabilities pi , pi 3 0, C pi = 1. The mean X = C pixi is also fixed. A decrease in uncertainty will be achieved simply by shifting every xi towards X in the same proportion, i.e., by moving xi to xi(P)
=
Px
+
(l
-
p)
Xi
3
O
Thus an increase in p decreases uncertainty. We shall then find the response of 6 and X to changes in p, calculating the appropriate derivatives and evaluating them at p = 0. For brevity, define zi = (1 - 6x,), and arrange zi in increasing order. Write the two expectations involved in (11) as El = c piz;“,
E, = c p,z;-“.
We shall make use of several elementary transformations and inequalities concerning expectations involving xi and zi . These are set out in Appendix A, and are referred to in the text by number. We shall need the following results on partial derivatives of Ei . aEJa8 = n C piz;“-lxi
> 0,
aE,/as = (n -
1) cpaz;“xi
aE,lap
piz,+yz
=
n8 c
-
= n c piz,-+yz, 1 n Cp,z;” [
by
1: 8 j: xi)
- 2) --Z c piz;n-ll
L4,
; = ;:
(12’)
UNCERTAINTY
AND
LIQUID
375
ASSETS
Turning to less simple expressions, 1 aE,
1 aE, ----_ E, as
El
as
=- 1 n C Pizi” s I. ( c p,z,l-”
< 0
by
L3
_ ~~P;~~~-‘) ,z, n
and
+ (1
P&yJ pi,;-,
L2.
Similarly 1 aE, ----Es ap
1 a& > 0 El ap .
The comparative dynamics of 6 need some simple manipulations to eliminate A, yielding
in (I I)
1-a n = @l-“E1 . 1 - cxE,IE, 1
(13)
This can be differentiated totally treating 8 as a function of p, to get
(14)
>o
using earlier results.
Thus a decrease in uncertainty leads to an increase in the proportion of assets held in cash balances-a sensible result. Now we can use the first equation in (1 l), differentiating it totally with respect to p using the expression (14) for dS/dp. This gives dh -=--
4
a/3r1-n Cl - ‘In-l
aE, aE, ---
ap as
--aE, aE,
ap as
!$k (1 - 01E2/El) _ ncuE, ($ !?j$ - i
-. 2)
(15)
376
DIXIT
Since the denominator the sign of
is positive, the sign of dh/dp is the opposite of
--aE, aE,
AND
GOLDMAN
aE, aE,
ap as -Zap
= n(n- 1)6%[cPiZi”*CpiXiZin--l - Cp&“-l ’ 1 Pix,‘cn] = n(n - 1) x [c pizF+l
- 1 piz:-n - (C piz;“ja]
>O
L3.
if if
n>l ntl
by
by
L1
Thus
A decrease in uncertainty, therefore, implies a higher propensity to consume if n < 1, which is precisely the opposite of the result obtained by Levhari and Srinivasan [3] in the case where there is only one asset. We are not able to pinpoint the precise cause of this, but it clearly relates to the liquidity property of our uncertain asset. Two further results involving variation in the degree of uncertainty are derived in Appendices B and C: 1. A reduction in uncertainty causes an increase in the demand for real cash balances. 2. A reduction in uncertainty causes a decrease in the demand for real assets. Summarizing the results of this section then, a decrease in the uncertainty of the quotient of the nominal rate of interest on real assets divided by one plus that rate (or simply x) implies: 1. An increase in 6, the proportion of assets held as money; 2. An increase (decrease) in h, the average propensity to consume, when n < l(n > 1); 3. An increase in 6(1 - h), the demand for real cash balances; 4. A decrease in (1 - 6)(1 - h), the demand for real assets. For the analysis of the more general homogeneous utility function, the results 1, 3, and 4 above are unaltered. Result 2 is modified to read 2’. An increase in h, when n ,( 1 and u‘(b) < 0; A decrease in X, when n > 1 and u’(b) > 0;
UNCERTAINTY
AND LIQUID ASSETS
377
An increase (decrease) in X, when n < l(n > 1) and o’(b) = 0, where b = m/c
and u(b) = mu,@, 4 cu,(c, m> *
4. THE RATE OF INCREASE IN THE MONEY SUPPLY In this section we shall relate the rate of increase in the money supply to the rate of inflation and describe the qualitative effects of uncertainty upon this relationship. We must first, however, specify the manner by which money enters the private sector. Without a completely described financial structure, our analysis can merely be suggestiveand, indeed, is offered in that spirit. Let us suppose that the government introduces new money through the purchase of real assetsfrom the private sector. Further, let the future income from these assetsbe distributed uniformly to all consumers who are, in turn, presumed identical in all respects. This consumer will follow the behavioral rules described in Sections 2 and 3. So long as the per capita governmental holdings of real assetsare smaller than those desired by the consumer, it is a matter of indifference to him where the title of ownership resides.That is, he has chosen to opt for a stream of real income and publicly owned resources provide a perfect substitute for individual assetsso long as he desiresthat income stream over the alternatives of additional consumption and/or money holdings. Some limitations imposed on government policy by this attitude are examined in Appendix D. Let p,, - 1 denote the rate of increasein the money supply, -Ml = 6,(1 - X,) a, ) PI
y. = r. - e. ,
Pl
=Po/Yo~
a, = (1 - ho) ao(ro- ~9~8,).
Thus x1 =
po - (1 - h1)81’80 /ho- (1 - h,) 6, .
(16)
In this manner we are able to assessthe implications of monetary expansion upon the rate of inflation. Pleasenote, however, that h, and 6, depend upon the consumer’s reaction to an observed x0 . Equation (16) provides only an implicit solution for x0 . Rewritten below, we may
378
DIXIT
AND
GOLDMAN
express the required rate of increase in the money supply as a function of the desired level of inflation and given expectations,
do= (1- wwo - %x0) (1- x0) ’
(17)
In the absence of uncertainty, the same rate of inflation produced with * _ (1 - A”)(1 - 6*x,) P (1 - x0) ’
would be (18)
where A* and S* are determined from (11) by the pair of equations (1 - A*), = /3rl-“( 1 - s*X&n, s*x, = (J+)(
1 TA*
).
(19)
Let us now suppose that x,, = X0, i.e., policy is to generate the consumer’s expected value of x. If the observation of such an x0 does not alter his future expected x and does not increase his level of uncertainty (dp 2 01, then p. > P *. Phrased simply, the presence of uncertainty implies that a higher rate of monetary expansion is required to generate the consumer’s mean expectations regarding x. The proof is offered below: ho = (1 - u@l/~o l--Z0
- S800) ’
p* = (1 - A*)(1 - s*zo) l-Z0 * p. - p* is equal in sign to (1 - h,)(S,/S, - &X0) - (1 - x*)(1 - 6*x0) > (1 - Ar)(l - 6,x,) - (1 - A’)(1 - 6*x0) since dp 2 0, and therefore 6, > So. But E[(l - A,)(1 - 6,x,)]-”
= [(l - A*)(1 - s*z,)]-”
and n > 0 imply (1 - A,)(1 - S,Z~) > (1 - A*)( 1 - S*zo), and therefore PO > P** We note one further relationship between the rate of monetary expansion and the level of inflation. From (19) we observe that A* is independent
UNCERTAINTY
AND
LIQUID
379
ASSETS
of x0 . Thus 6*x, is also independent of x0, and ~~(1 - x,,) depends only upon p, r, and n. An increase in the rate of monetary expansion therefore requires an increase in x0 , which in turn implies an increase in the rate of inflation. A
APPENDIX
L1: c pixiz;“/c p&k = f lz PiZYk/CPie-” - 11. Simply substitute xi = (1 - zJ6: L2:
c piz;” - c p&”
= c p,z,“(l
L3:
Cp&”
> (&J,z;“)2
. cpiz;l-”
- 1 + 6X$) > 0;
by the Cauchy-Schwarz inequality; L4:
Define Z = 1 - 6X, then z > c p&“/C
RHS=Qg
PiZLk,
zi = C qizi , say. 35
Then C pi = x qi = 1, and qi/pi = z;“lC pjzJ:” is a decreasing function of zi . Since zi are arranged in increasing order, 3, such that
and $
(4i
Pi) = t (Pi - 4i) = S, ie+l
-
say.
Then C (Pi -
4i)
zi
=
izl
(pi
-
qi)
Zi
-
> s(z*,+1 - z&J > 0. Therefore Z = Cpizi
> C qizi .
z
(qi
-
pi)
Z
380
DIXIT
AND
GOLDMAN
B
APPENDIX
&3(1 - A)/ap is, by (1 l), (14), and (15), equal in sign to
which, by sbustituting equals T
+ na(n - l)(Ezl-” = [ =
[
41-Q -4 ‘(1;
1
I
aE,/a&
aE,pp,
from (12’)
aE,jS
- ZEZP-~)(OI(~ - n) - 1) Ez-”
[n(Ez-”
r-7
for aE,pp,
- ZEz-“)(Ez-“-I] + a( 1 - n)(Ez-“Ez-”
[E(Z - z) z-“-lEz-”
‘) ] [a(1 -n) -~][Ez-~Ez-”
- Ezl-“Ez-“-1
- Ezl-“E~r”-~)] + Ezrn-%(Z
- z) zP],
which is positive by Cauchy-Schwarz and (12’).
C
APPENDIX
a(1 -X)(1 - 8) = _---%(l -A) + aP
ah n (l$) aP [
x [Ez-“Ez+
- Ez~-“Ez-“-~]
][[l
- a(1 - n)]
+ Ez-“-lE(z
- 2) z-“1
+q!$L][$$-~~] =
t
{[[l - ~(1 - t~)][Ez-~Ez-~ - Ez~-~Ez-~-‘]}
+ Ez-“-lE(z
- 5) z-“1
- [a( 1 - n)] r+] (l-4 zzz n ___ I El x [Ez-“Ez-”
I
[Ez-“Ez-”
- Ez-“-~Ez~-“]
{[ol(l - n)(Z - 1) + l] - Ez-+~Ez~-“]
+ EzP-~E(z - 2) z-“} < 0,
since 1 + 01(1 - n)(T - 1) > 0 for X < 1, (12’), and the Cauchy-Schwarz inequality.
UNCERTAINTY
AND LIQUID ASSETS
APPENDIX
381
D
As noted, our treatment of government requires that the public holdings of real assets be less than the real assets that the private sector wishes held. Were this condition violated then individual planning would include, by necessity, projections of future governmental asset accumulation. In the following, we turn our attention from uncertainty to examine the possible monetary policies which render private and public planning consistent under our formulation. We shall consider governmental policy under conditions of long run equilibrium, i.e., policies which exactly satisfy consumer expectations where those expectations are held with certainty. Supposing that initially the government’s per capita holdings of real assets are less than the private sector’s demand, i.e., k,(O) -=ck(O), we may relate the rate of inflation to the expansion of the money supply through (18) as c1 = (1 - A*)(1 - 6”X) l-x or
-=1 +g l+i
whereg=p-
ptn,vn
(20)
1.
The rate of growth in the demand for real assets is then Ak -Yzz k
B
The change in the money supply is given by the government’s purchases of assets, i.e., (p + Ap) Ak, = AM = gM, and thus
4k -
Iti
*
(22)
In order to avoid the total governmental ownership of the economy’s real assets, the rate of monetary expansion (or, alternately, the rate of
382
DIXIT AND GOLDMAN
inflation) must be limited. (This, of course, is a consequence of the “inside” nature of money in our model). Specifically Ak, < Ak, or g [-&I then, letting $ = plnrlln may write
< (1 + g)(l - *) and substituting
= g - i; from (lg), (19), and (20) we
gal - “I@ - 41 + 1 - 41 < 44 - l)(r - 4).
(24)
If (1 - ol)(r - $) + 1 - 4 > 0 then (24) may be interpreted as a limitation on the maximum rate of growth of money. However, when (1 - a)(r - 4) + 1 - C#J< 0, then (24) has the interpretation of a lower bound on the rate of growth. The explanation of this latter, counterintuitive result lies in the observation that an increase in g increases Ak and will decrease [a/(1 - S)].
ACKNOWLEDGMENTS The authors gratefully acknowledge their debt to beneficial discussion with Professor D. McFadden and to research support from the Ford Foundation and the National Science Foundation. However, the conclusions, opinions, and other statements in this study are those of the authors and are not necessarily those of the foundations.
REFERENCES
1. A. DOUGLAS, A theory of saving and portfolio selection, Rev. Econ. Stud. 35 (1968). 2. F. HAHN, Walras Bowley Lecture, Econometric Society Meetings, December, 1969. 3. D. LEVHARI AMD T. N. SRINIVASAN, Optimal saving under uncertainty, Rev. Econ. Stud. 36 (1969). 4. D. PATINKIN, “Money, Interest and Prices,” Row-Petersen, Evanston, Ill. 1956. 5. M. ROTHSCHILD AND J. E. STIGLITZ, “Increasing Risk: A Definition and its Economic Consequences,” Cowles Foundation Discussion Paper No. 275, 1969. 6. M. SIDRAUSKI, Rational choice and patterns of growth, J. Polit. Econ. 77 (1969). 7. J. TOBIN, Money and economic growth, Econometrica (1965). 8. H. UZAWA, Time Preference, the Consumption Function, and Optimum Asset Holdings, in “Value, Capital and Growth” (J. N. Wolfe, Ed.), pp. 485-504, Edinburgh University Press, Edinburgh, 1968.