Journal of Molietary
11 (1983) 337 -349. North-Holland
ASSET PMCES, ASSET STOCKS AND RATIONAL
Carl E. WALSH Princeton University, Princeton, NJ 08544, USA This paper explores the relationship between asset return covariances and the impact of asset stock changes on asset prices. In the process the paper reconciles recent contradictory results on the effect of changes in the stock of government debt on equity prices. A solution for asset prices in a rational expectations equilibrium is also derived. This solution has the property that clt the prices implied by the solution the demand for each asset equals its supply and the distribution of future asset prices impbed by the solution is identical to the distribution upon which asset holders base their demands.
I. Introduction In a recent paper, Roley (1979) derives a rather remarkable result; he shows that within the framework of a mean-variance portfolio choice model the assumption thal all assets are gross substitutes is a sufficient condition for an increase in government debt to have a contractionary effect on aggregate demand by reducing equity prices. This result f;ollows from (1) Roley’s demonstratilon that the effect on equity prices of an increase in the stock of government debt is proportional (with opposite sign) to the covariance between the returns on government debt and equity, and (2) the proof by Blanchard. and Plantes (1977) that a necessary condition for all assets to be gross substitutes is that all asset returns be positively correlated. This result on the impact of changes in the stock of government debt stands in contrast to the conclusions reached by Tobin (1963), Friedman (1978), and Cohen and McMenamin (1978). In the models used by these authors, the assumption that all assets are gross substitutes is not sufficient to determine the direction en which equity prices will change in response to a change in the stock of government debt. Even ignoring wealth effects, these authors find that the effect of debt on the return to equity is ambiguous. This theoretical ambiguity is unfortunate since the effects of changes in the stock of government debt on interest rates, asset prices, and aggregate demand are important for an analysis of debt financed versus money financed fiscal policy and for studying open market operations. The impact on aggregate demand of an increase in government debt is also important for stability analysis and is relevant for discusions of crowding out. *An earlier version of some of thi, material appeared as N.B.E.R. working paper no. 566 (Oc’t. 1980), ‘Asset Prick, Substitution Elects, and the Impacts of Changes in Asset Stocks’. I would like to thank Sumner LaCtoix and an anonymous referee for helpful comments. 0~04_~9~~,/8~/$cF3.0(3 4.1 Elsevier Science Publishers
B-V. t North-Holland)
C.E. Walsh, Asset prices, stocks and rational ex,pectations
Roley’s result:, would seem to imply that these issues can be resolved by simply examining the covariance between the returns to debt and equity. ‘11nisis what makes his conclusion so remarkable. Strong conclusions generally require strong assumptions. The first purpose of this paper is to reconcile the results of Roley with those of Tobin, Friedman, and Cohen and McNIenamin. This is done by showing that Roley, ard and Plantes, rely on the assumption that money is a as asset in order to derive their results.’ If money’s return is also subject ris to risk due to uncertain future inflation, then the impact of government debt uity prices depends upon the difference between two asset return on rices.. -Thus., the assumption that the covariances are all positive is not CQV uffrcient to determine the direction in which equity prices will move. This result, that the sign of the effect depends upon a comparison of two variances, parallels the result, for the models of Friedman and Cohen and enamin which incorporate money, debt, and equity, that the elffect on ’ prices of a change in the stock of debt depends upon the difference n two substitution effects. This is generalized to the case in which ere are n assets in Walsh (1982a) using Hick’s (19%) q-substitution effects. suming all mats to be gross substitutes i&plies that the individual tion effects (defined with respect to changes in asset prices) are all srtive but does not allow differences between substitution effects to be In all the papers so far cited, the analysis is carried out under the sumption that the distribution of future asset returns is fixed. In the meanpee framework, this means that expected future asset prices are assumed unaffected when the current stock of debt is varied. Since current asset prices depend, in these models, upon current asset stocks, and a change in an asset stock generally would imply a change in the stock of the asset in the future, changes in current stocks should produce changes in expected future ore generally, lin macroeconomic models asset stocks are normally ed as exogenous for the purpose of short-run analysis. In the models nchard and Plantes and Roley in which current asset prices n asset stocks and the distribution (means and covariances) ces, the distribution of future asset prices will, under the mption of rational expectations, be determined by the stochastic process e evolution of asset stocks. nd purpose of this paper is to show that when expectations are rationally we cannot generally determine the effect of an asset ange by knowing abtit return covariances even if money is viewed as at is, Roley’s results no longer hold if expectations are allowed to sume the existence of a riskless asset. Since Roley uses this asset as (and futures ptice set equal to 1, it is referred to here as money.
C.E. Walsh, Asset prices, stocks and rational
adjust. As one would expect, under the assumpiion of rational expectations an unanticipated shock to the stock of government debt has a very diffexnt effect than does a change in the underlying stochastic process determining the stock of debt. The latter change will result in changes in both expected future asset prices and the covariances between asset prices. Under rational expectations, the covariance matrix of asset prices is determined endogenously as a function of the stochastic properties of asset stocks. This implies that in analyzing the effects of government debt it is inappropriate to begin by assuming that all assets are gross substitutes. The implications of such an assumption (covariances are positive when money is riskless) may be inconsistent with the underlying behavior of asset stocks. In the next section, th,: contradictory result!; discussed above are reconciled. The distribution of future asset prices is taken to be fixed in this section, in keeping vith the literature already cited. In section 3 this assumption is dropped and replaced with the assumption of rational expectations. Under the assumption that asset stocks follow a stationary, moving average process, an exact solution for asset prices is derived. This solution has the property that the distribution of future asset prices implied by the solution is identical to the distribution upon which asset demands are based and the demand for e,lch asset equals its supply. Numerical examples for a model with two risky ,lssets are presented to illustrate thle solution. A brief summary of the paper’s results is contained in section 4.
2. Asset stock changes with fixed expectations Following Roley, a mean-variance framework is adopted to analyze an asset holder’s portfolio choice problem. Let xJr= (_Xljr.. ., _Ykjr) be the jth individual’s portfolio at time t where Xijr equals the units held of asset i. Define pi=(Plt,..., pkt) as the: vector of current asset prices, and ,pi + t as the expected value, formed at time t, of pi+ r. Let C = [Oij] be the k x k covariance matrix of pr+ 1. Throughout we assume for convenience that the only return on an asset is its capital gain and that all investors hold the same beliefs about ,pi+ 1 and C. The portfolio -Yjris chosen to maximize expected utility which we assume is given by’
with the maximization carried out subject to the budget restriction p;-Yj,s (:)jr+ ‘Roley assumes utility is given by U&pi+ IXj,, _x,r.‘~.~j,). He assumes, however. that - ZL’2 II, 1~ a constant where Ui is the partial derivative of U(a) with respect to its ith argument. This IS equivalent to assuming a utality function of the form given in (1).
CE. Wdsh, Asset prices, stocks and rational expectations
ndividual i’s current wealth. In (1). ,p;+ 1Xj, is the expected value wealth, and X;,CXjris its variance. The parameter yj measures jth individual’s risk aversion and is assumed to be positive. From the der condition for the maximization of (1) subject to the budget e can obtain the following relationship between the individual’s rtfolio and current and expected future asset prices:
E.j is the Lagrangian multiplier associated with the individual’s budget the macroeconomic asset demand equation approach of Friedman enamin and the mean-variance framework of Roley, the ination of current asset prices, pr given expectations future prices, & + 19 the covariance matrix, Z, and the vector of ously given aggregate asset stocks, x,. Such an aggregate expression can be obtained by dividing the first order condition by yj and viduals to obtain
tting 7 - ’ = Cj yi ’ and P = & ;liyi ‘, this equation can be written as ,p; i- 1
YPPi = 0.
ltiplying by x, we can solve for p: P = CA + ix, - Y~%l/Y~~9 re CO, =& is the aggregate market value of the asset vector x,. inating p from (2) yields an aggregate relationship between pr, tpt + 1, E,
ce o! asset i *wethen have rCfPi+ lxf -YxJxf19
on in the numerator of asset J3 r, Xi,
is now over assets, and Xjt refers to
ii refer to the aggregate stock of asset j.
C.E. Walsh, Asset prices, stocks and ratimal espectatinns
According to (3), the current price of asset i, relative to current wealeh, is equal to the expected price of asset i minus a cor;zction for the risk of asset i, all relative to expected future wealth minus y times the variance of future
wealth. The risk factor for asset i, Ci~ii~jt9 is simply the covariance between Pit+1 and the market portfolio evaluated at pt+ 1. If we view current wealth, w,, as equal to the market value of the existing stock of assets, a normalization needs to be imposed in order to determine the vector of current asset prices. Letting money be asset 1, the natural normalization is obtained by setting pnt= 1 and interpreting pit as the price in terms of money today of asset i. Under this normalization, we can rewrite eq. (3) as Pit =
CPit+ 1 [
j I/[ i 1-
rPlt+ 1 -YC”lj-xjt
Our normalization does not require that tplt+ l = 1 since this is the expected price, in terms of money today, of a unit of money next period. From eq. (3’) then, the current price, in terms of money, of asset i is equal to the risk corrected expected future price of asset i relative to the risk corrected expected future price of money. Consider now a change in the stock of asset j. This produces a change in the composition of the aggregate portfolio, and asset prices must adjust so that the new portfolio of assets is willingly held. The necessary change in the price of asset i can be found by differentiating eq. (3’): dPit/dx
j 1c clj-xjf
Comparing this equation with Roley’s eq. (5) (1979, p. 917) in which, in the notation of this paper, dpit/dXjr = -“~a~~,it is clear that Roley’s results hold for the special case in which money is riskless (a,j = 0 for all j) and p1t + l = 1. When the price of money in the future is subject to uncertainty, the sign of dpit/dxjt is given by the sign of Pitdlj - aij. This cannot be signed simply from the assumption that Oij is positive for all i and j. If Sj is the stock of government debt and pi is the price of equity, an increase in the stock of debt will be expansionary (dpit/dxj, >O) if the covariance of the price of debt with the price of money is greater than one over the price of equity times the covariance between the price of debt and the price of equity. If aij>O for all i and j, debt is expansionary if, in somewhat loose terms, the covariance of the price of debt with the price of money is greater than the covariance between the price of debt argd the price of equity. This parallels the results of T&in, Friedman, and Cohen and McMenamin who showed that debt is expansionary, in a model with money, debt, and equity, if debt is a closer
C.E. Walsh,Asset prices, stocks and rational expectations
~~t~t~for money than it is for equity. This result is generalized to an n ode1 in Walsh (1982a) where it is shown that the efifect on equity change in the stock of government debt depends only upon a of the substitutability oi debt and money on the one hand and sltty on the other even when there are other assets. r rational ex
malysis of section 2, current asset prices were assumed to adjust in to changes in asset stocks while expected future asset prices were to remain fixed. Pn this section this assumption will be dropped. it is assumed that asset holders are aware that eq. (3) must be in eqoilibrium, and they use (3), along with forecasts of future asset their expectations. This implies that pt can be as ii function oi X and current and expected future asset stocks. xprr:ssion also implies a matrix of covariances among asset prices this must agree with C. The distribution of plr+1 upon which asset are based should, in a rational expectations equilibrium, equal the implied by the assumption of market clearing. tional expectations equilibrium solution for pt will be derived in this r a special case of the model discusseld in section 2. The special case in which money is riskless, and, for convenience, p1t + s = 1 for all1s 2 0. oley has shown that dpi,/dXjt = -yoij in this case. tations are determined rationally within the model it is again the n 2, that the observed covariance between pi and ine the sign of dpi,/dxjts a.U j= l,...,k aEd plr=tplt+f =P~~+~ = 1, current
i=2,. . ., k.
fine p, C. and x to refer only to the k - 1 risky assets. Then, since pt = -‘/Cc, is a function of current asset stocks, pf + 1 will depend upon and ar,y uncertainty about pl+ t (and hence C) must therefore clepend stochastic properties of x,, f. Given a process generating x,, it possible to express J+ + 1 and C, and therefore pt, in terms of the eter.5 of the X, process if it is assumed that expectations are rational.4 ume then that X, follows a stationary process which can be expressed as stic component plus a moving average process:
of the demand
cy rule for just this reason.
are shown to depend
where a, is the deterministic component of x,, A(L) = I + A 1L-t A,L? +. . . is a (k - 1) x (k - 1) matrix of polynomials in the lag operator L. and E, is a k - 1 vector of random variables with a multivariate normal distribution, Es, =0, and Es,&:= Sz for l ==s and the zero matrix otherwise. ut+i is known for all iz0. Substituting (5) into (4) expresses pt as a function of t~I + 1, C, and Ed.For a given C, assume that the solution obtained for pr, when I~t +1 is eliminated by substitution using the assumption of rational expectations, takes the form pt =
where b, is the deterministic component of pt, and B(L) = B0 + B,L + . . . is a matrix of polynomials in L. From (6), the vector of asset prices at time t + 11 is given by bt+ 1 + B(L)&,+1 so that, with rational expectations, rPt+1 =
4 +1+ ww,
+ =b,+l+Ble,+B2E,_1 1
Substituting (5x7) into (4) yields b, + B(L)&,= b, + 1 + Blct + B2ct_ l + . . . - yCa, - ;C4( L)E,
which must, in equilibrium, hold for all realizations of E. This will only be the case if the coeficients on each E,+ are equal on both sides of eq. (8). This implies B,=B,-yC, Bi=Bi+l-yZ’Ai,
In addition, the deterministic components must satisfy (9cl
AS is clear from eq. (4), the asset pricing :nodel can only determine pI + 1 p,; it cannot determine the absolute levels of pr or pt + 1. From (9~). if /+. b r+l, bt+2.. . is a solution sequence for b,+;, then so is b, + n, h,, a + n,. . . for any arbitrary constant ro. If A(L) is of f”lnPteorder 4, eq. (9a) and (9b) do not yield a unique solution for B(L). This rroblem is common in rational expectations models [Taylor (197?)]. Since Ai =0 for I > q, eq. (9b) only requires that Bi = Bi+ I for i > q5 5The cond!ltion that the unconditional variance crf r be minimized i > q and woc Id thus yield a ur,ique solution. See Ta!*lo, ( 197?).
would require that B, = 0.
C.E. Walsh, Asset prices, stocks and rational expectations
pointed out by McCafferty and Driskill (1980) with reference to Muth’s ory speculation model, the covariances of prices will also depend itibrium solution for prices. Hence, in the present model, Z will
are two relationships connecting B(L) and C. One is given by (9), the by (IO). Since (10) is non-linear in the elements of BO, there may be ” fij There are in fact two. One is given by Bi abbe solutions bo (9) and (lb,. and Z =O. In this solution, pr + f - pt = 0 for all t, and risky ts are never held. In Walsh (1982b) the rational expectations equilibrium d for a two asset, two period, overlapping generations model, and, one solution involves asset prices identically equal to zero. second solution for B(L) and 6, substitute (10) into (9a):
) to recursively substitute for Biyi= 1,2,. B,= -~B,Q&(l+A,
fine k4= (I+ A 1 + A2 + . . .); it is assumed that H is finite and non-singular. itiplying both sides of (11) by -(!/y)Q- IB, ’ and postmultiplying by
hich expresses B, in terms of the properties of the exogenous stochastic rties of X, (H and a) and the properties of the utility functions of asset holders (7). From ( lO),
e from (9bJ6
Also, the restrictLans imposed by eq. (10) imply d to II0 and still y:eld a solution.
C.E. Walsh, Asset prices, stocks and rutional expectations
Eqs. (6) and (12)-( 15) provide a complete description of equilibrium asset prices, expressing pt as a function of the underlying process generating x, and the aggregate risk parameter y. With this solution it is now possible to study the impact of a change in an asset stock under the assumption of rational expectations. As with most models which incorporate the assumption of rational expectations, it is necessary to distinguish between anticipated changes in x, represented by changes in a,, changes in x, produced by realizations of E,, given A(L) in (5), and changes in X, produced by changes in A(L). The latter type of change will, as is clear from (12j(15), result in shifts in the matrix B(L), the covariance matrix Z, and the unconditional expected return, br+l -ht.
Consider first the effects of an anticipated change in x,, interpreted as a change in a,. The effect of ajt on (Pit+ 1 - pi,) is given by the 0th element of (llr)(HQH’) - l = $5 Therefore,
&Jit + 1 - PiJldajt = Wij*
Since Roley treats tPit+ 1 as fixed, his analysis implies that d(,p, + 1 - pi,)~d.~j, =YQ. It follows that Roley’s results are correct for anticipated changes in the deterministic component of asset stocks. I: will be shown below that his results do not hold for unanticipated changl:s in x,. Even for anticipated changes the results are not quite equivalent. In one case, the conditional expected return on asset i, rPit+1 -pit, changes by >‘aii; in the other, it is the unconditional expected return, b,, 1 -b,, which changes. Also, Roley interprets the change in tPit+1 -pit as a change in Pit since tPil+ 1 is assumed to be fixed. In (I 6), the effect on Pit cdimot be determined. The current asset price could fall by Yaij as Roley assumes, or all future prices of asset i could rise by yaij, leaving the current price of asset i unchanged. Thus, even for anticipated changes in xjt, it cannot be concluded that dp,,/dxjl= - Taij. The effects of anticipated changes in asset stocks depend on Z:, the covariance matrix of asset prices. Despite the optimizing behavior of asset holders and the assumption of rational expectations, unconditional expected returns are functions of the deterministic component of asset stocks. Anticipated changes matter because of the assumption that all investors are risk averse. Since no! all assets are equally risky, a predictable change in an asset stock affects the riskiness of the aggregate market portfolio which must, in equilibrium, be held by in*,‘stcrs as a who!,=. Expected returns must adjust to compensate for this change in risk. From (13), C depends upon hi and Q. The effects of anticipated changes in X, depend, therefore, on the stochastic structure of the unanti, ipated changes in x,. For a given A(L), the effect on asset prices of the current innovation in the stock of assets can be fcund using (6) and (12); the effect of 1, on pt is given J.Mon
C.E. W~bh, Asset prices, stocks and rational qwctatians
= =-(1 Q#SZH’)- *. The impact on the price of asset i of the realization the innovation in the stock of asset j, is given by the ijth element of e elements of B,, can be more easily interpreted by using (8.3) to write
l fixed, it was shown in section 2 that dpi,,‘d_xjt = - yaij. When J+ +1 to
vary, the coefficient on E,;~in the equililbrium expression for Pit is
is the jkth element of A, ke the fixed expectations case, eq. (16) shows that, even when money is E, the effect on Pit of an unanticipated change in the stock of asset j is ndent not -just on aii, but also on all dim’s to the extent that a shock to results in subsequent changes in the stocks of other assets or in changes in asset j. These effects are measured by the A,,,j’s. tations are held fixed, the sign of dpit/dxj,, and hence the t prices of a change in the stock of government debt, can be hned from the assumptions that all iassets are gross substitutes and oney is riskiess. When expectations are instead determined under the ption of rational expectations, eq. (18) shows that, in general, the sign ~~~d~~~for unanticipated changes in Xi? cannot a priori be signed. Asset adjustments depend upon the stochastic process generating asset stocks. o illustrate these results, consider a model in which there are three assets; t 1 (money) is assume-d to be riskless and its price each period is normdiztxi to eyual one. Suppose that the stocks of assets 2 and 3, the risky ts, are given by
X2*= E2t+ 0.5&,,_ I+ 0.2583,_1, X3t = E3t
?hat O~=&=l, CO23 = -0.5 where a$ is the variance of Ei anld covariance between s2 and s3. The deterministic components are to zero for simplicity. Their effects can easily be found once C is
eqs. (6), ( 12), and (14), the equilibrium prices of assets 2 and 3 are )[ - 0.8982,-0.44&,, - 0.30&z,-
1 + 0.05&, t - 11,
0.25&,,_1 t 1s5E3r-J*
The prices determined by eqs. (21) and (22) are consistent with individual portfolio choice [the maximization of (l)] and market clearing (asset demand equals asset supply). From (13), the covariance matrix of (pzr+ 1, p3t + 1) is
Asset returns are positively correlated, and the two risky assets are gross substitutes.’ An increase in the price of asset i increases the demand for asset
J From (21) and (22), unanticipated increases in either asset reduce the price of the other asset. They are thus Hicksian gross q-substitutes [Walsh (1982a)] increasing the stock of (say) x2 in individual’s portfolios reduces their demand for x3 since x2 can substitute for x3. Since the stock of .x3 is unchanged, market equilibrium requires that p3 fall. However. an increase in x3t, while causing a fall in pzt, will result in a rise in p2t + 1. In this example, innovations in the x2 and x3 processes are negatively correlated. Unanticipated changes in x2 tend to be associated with unanticipated changes in x3 in the opposite direction. Suppose. instead, that the sign of this covariance is changed so the ~23 =OS. With only this change, asset prices are now given by pzt = (I/y) [ - 0.89&,,+ 0.44&,,- 0.30&,,_ 1 - 0.54&,,_ 1], 2.‘b,,
0.59 -0 99 .
+ [email protected]
_ 1 +
2.02~~~ _ J,
Here, asset prices are negatively correlated, and the two risky assets are gross complements; an increase in pi reduces the demand for _~j.8In addition, a positive value of increases Pit. Since the asset returns are negatively correlated, an increase in the stock of asset j increases the demand for asset i since, to reduce risk, individuals wish to increase their holdings of an asset whose return Is negatively correlated with the return to asset j. Since the stock of asset i is fixed, its price rises. In both of these examples, the sign of dpi,/d.Yj, for an unanticipated change in .4;j was the same as the sign of -j)olj. as was the case in R&y’s model Ejt
‘Asset demand equations are given by X, =( l,‘y)C- ‘&p#+, -p,) so that C:Xi~~~P,r z= - gii 7, where is the ijth element of the inverse of Z. For this example, 2xz,/Sp3, = ?_Yq,,~C:p2, = 0.307. *axlt‘/?1;131 - ax ,,/%$,, = - 0 SOJJ< 0.
C.E. Wdsh, Asset prices, stocks and rational expectations
itk fixed expectations. Eq. (18) showed that this need not always be the As a Gina1illustration, suppose (19) and (20) are replaced by x2, := E& --
0.50&~r- 1+ OSO&,,-
and again w~=wJ=~,
~023~ -0.5. Equilibrium asset prices can be found as (12), and (14).’ For this example, an unanticipated reduces the price of asset 2. However, 023 is also negative; the f the two assets are negatively correlated. In this case, the sign of is the same as y623, contrary to the model’s prediction when
In macroeconomic models of the financial sector, the assumption that all are gross substitutes has been found to be insufficient for signing the ts of changes in the stock of government debt on the prices of other w. Using a mean-variance framework, Roley found such an assumption saf%ient. These results were reconciled by showing that Roley also treated maney as a riskless asset. If the return to holding money is uncertain, the imph.ct on asset prices of a change in the stock of debt depends upon a comparison of asset return covariances, just as in the analysis of Tobin it et! upon a comparison of substitution effects. r the assumption of rational expectations, equilibrium asset price functions were derived. These functions express asset prices in terms of the ropcrt& of the representative individual’s utility function and the stochastic rties of asset stocks. Even when money is riskless, it was shown that is nc.!simple relationship between. the effects of an unanticipated change in an asset stock and the covariances between asset prices. “For &is example, the solutions for p2 and p3 are pzr = I Qf,f - 2.6%,, - l-33&,,+ 2.67&,,_ 1- 4.00~~~ _ J,
pjr :=(il_i)[l.33&,, - 1.33& 3r Im
ition, a$ = .5.33,$i2 = OS,and
- I+ 4.00&,,- 1-J.
A note on grms substitutability of financial assets, cMena.min, 1978, The role of fiscal policy in a financially ournal of Money, Credit, and Banking 10, no. 3, Aug.,
Friedman, Benjamin &I., 1978, Crowding out or crowding in? Economic consequences of financing goverEinenc deficits, Brooking Papers on Econon$c Activity, no. 3, 595 641. Hicks, JR., 1956, A revision of demand theory (Oxford University Press, Oxford). McCafferty, S. and M. Driskill, 1980, Problems of existence and uniqueness in ronlinear rational expectations models, Ecogometrica 48. no. 5, July, 13 13- 13 17. Muth, J., 1961, Rational e:rpectations and the theory of price movements. Econometrica 39. no. 3, July, 315-335. Roley, V. Vance, 1979, A theory of federal debt management, American Economic Review 69, no. 5, Dec., 915-926. Taylor, J., 1977, Conditions for unique solutions in stochastic macroeconomic models with rational expectations, Econometrica 45, no. 5, Sept., 1377-l 386. Tobin, J., 1963, An essay on the principles of debt management, in: Fiscal and debt manrtgement policies: Commission on money and credit (Prentice- Hall, Englewood-Cliffs, NJ). and monetary policy: An alternative Walsh, Carl E., 1982a, Asset substitutability characterization, Journal of Monetary Economics 9, no. 1, Jan.. 59-71. Walsh, Carl E., 1982b, The demand for money under uncertainty and the role of monetary policy, Mimeo.