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On the choice of immediate monetary targets
Journal of Banking and Finance 2 (1978) 1-13. © North-Holland Publishing Company
ON THE CHOICE O F IMMEDIATE MONETARY TARGETS* Florin AFTALION ESSEC, Cergy, France
Lawrence J. WHITE New York University, New York, NYIO006, U.S.A. This paper studies the control problem of a stochastic monetary system. The Central Bank has the choice of two targets: the size of its portfolio of assets or the level of interest rate on that class of assets. If its objective is to minimize the variance of a monetary aggregate or of private sector interest rate, the conditions under which one or the other of the targets should be used are analyzed. However, if the ultimate objective is a minimum variance in a desired GNP level the effect of the immediate target is the same whether the central bank employs a money stock or a private interest rate target.
1. Introduction In the discussion concerning the choice of targets for monetary authorities (usually interest rates versus monetary aggregates), only a modest amount of attention has been given to the fact that the monetary authorities do not in fact have direct control over the stock of money or over the interest rates that directly affect the private economy. 1 Instead, the monetary authorities usually have direct control-in the sense of having immediate (daily or even hourly) information and an ability to affect immediate changes-over two things: the size of their portfolio of assets (this is frequently government debt, but in many European countries it is primarily commercial bank borrowings from the Central Bank) and the level of interest rates on that general class of assets. Thus, in their day-to-day dealings, the monetary authorities can only have a portfolio target or an interest rate target for their portfolio. This must be *This paper was presented at the ESSEC Conference (Cergy, France, June 9-11, 1977), which was financially supported by FNEGE. It has been subjected to the normal reviewing process of the Journal. 1See Davis (1971, 1974), Kareken et al. (1971), Holbrook and Shapiro (1970) and Poole and Lieberman (1972). As Craine and Havenner point out, there are three sources of error in monetary determination: equation errors, errors in forecasting exogenous variables, and errors in estimating the structural coefficients. We focus largely on the first, though the second is included. The third is beyond the scope of this paper.
2
F. Aftalion and L.J. White, Monetary targets
true even if the Central authorities would prefer to pursue a monetary aggregate or private sector interest rate target. The figures for the latter
targets are always reported with some delay, and hence they cannot be immediate targets. This is clearly true for the monetary aggregates, and, though corporate bond interest rates are quickly and widely known, commercial bank lending terms are not. 2. Since there are random fluctuations in the monetary behavior of the private sector and the banking sector vis-a-vis each other, the achievement of an immediate portfolio or interest rate target cannot guarantee the exact achievement of the intermediate monetary aggregates or private sector interest rate target. These intermediate targets can only be achieved in an average sense (provided the correct behavioral relationships are known); there will still be variance in the actual levels of the monetary aggregates or private interest rates achieved. There is an important consequence of this variance for the conduct of monetary policy. Suppose we have a world such as that described by Poole (1970), in which errors in the I S curve and in the L M curve combine with the coefficients in each relationship to determine whether a monetary aggregates target or a private interest rate target would achieve the least variance in achieving a G N P goal. Suppose further that the parameters of that model are such that, for example, a monetary aggregate target is the chosen vehicle. If we add the possibility of disturbances in the monetarybanking sector, we then must face a choice of an immediate portfolio target or an interest rate target as the means of achieving that monetary aggregates target. And there are quite reasonable circumstances under which the monetary authorities would choose an immediate interest rate target as a way of achieving the monetary aggregates target with the least variance. In effect, the added disturbances in the monetary-banking sector and the consequent variance in the monetary aggregates would make the pursuance of an immediate interest rate target the superior strategy. Similarly, even if the monetary authorities had decided on an intermediate private sector interest rate target, there are circumstances under which an immediate portfolio target strategy would be the least variance way of achieving that eventual target. And, finally, it is worth remembering that if the true ultimate goal is the achievement of a desired G N P level with the least amount of variance, the full effects of the stochastic elements in the private and banking sectors' monetary behavior must be included in the determination of targets. This paper will demonstrate these propositions using a simple model of the monetary-banking sector. The next section will provide the model and the basic results. The following section will relate our model to that of Poole's. And the final section will offer some conclusions from our model. 2Though changes in the prime rate in the U.S. are widely publicized, there is some slippage between the publicized prime rate and the actual terms charged to customers.
F. Aftalion and L.J. White, Monetary targets
3
2. The model Our model of the monetary-banking sector can be characterized as follows: Commercial banks are required by the monetary authorities to hold as reserves a specified fraction of their deposits. The banks can obtain reserves by obtaining advances from the Central Bank. These advances, along with holdings of foreign exchange, constitute the assets of the Central Bank. (This kind of arrangement is broadly descriptive of at least one European country, France. It is not too different conceptually from American procedures, in which the Central Bank's assets are mainly in the form of government debt which is bought from and sold to the commercial banks, rather than in the form of commercial bank debt.) The commercial banks make loans to the public; they are the only source of loanable funds to the public. The public holds its money in the form of deposits in the commercial banks and cash. Finally, the monetary authorities have a choice between two immediate targets: they can set a limit on the advances from the Central Bank by the commercial banks (in effect, a target on the Central Bank's portfolio) and in effect auction off these borrowings at a market clearing interest rate; or they can specify an interest rate on the advances and let the commercial banks borrow as much as they wish at that interest rate. We can now establish a set of definitions: S = gold and foreign exchange and treasury obligations held by the Central Bank, A = advances by the Central Bank to commercial banks ('refinancing'), R = reserves held by commercial banks against deposits, C = currency held by the public, D = total deposits at commercial banks e
= C/D,
.V/ = money ( = D q- C), L =loans by commercial banks to the public, X --a set of exogenous variables influencing the public demand for loans, e.g. income, expectations, rL = the interest rate on loans, 2 = the required reserve ration on deposits, ro = t h e interest rate on deposits (this includes any bank costs in servicing deposits; this interest rate is assumed to be set by the Central Bank), rA = t h e interest rate paid by commercial banks on their advances from the Central Bank, u = a disturbance element in the public's demand for loans from the commercial banks, with mean zero and a finite variance, v = a disturbance element in the public's desired currency holdings, with mean zero and a finite variance,
4 W
F. Aftalion and L.J. White, Monetary targets
= a disturbance element in the interest rate on loans by the commercial banks, with mean zero and a finite variance. Formally, the Central Bank's balance sheet must be
S+A=R+C.
(1)
The commercial banks' balance sheet must be
L+R=D+A.
(2)
The required reserves are R =2D.
(3)
The public's demand for loans is
L = a l +a2X +a3rL +u,
(4)
and the public's desired currency holdings are
C=eD+v.
(5)
By substituting (2) into (1), we get
S+L=C+D=M,
(6)
and by substituting (4) and (5) into (6) and rearranging, we get 1 D =~ IS +al + a 2 X + a 3 r L + u -- v]. l+e
(7)
We now assume that the Central Bank has chosen a money supply target M* as its means of achieving an ultimate income goal, in the spirit of the Poole model. But in an operational sense, it can choose only between a strategy of setting an immediate interest rate target r~ or an immediate advances target A. And the Central Bank is interested in minimizing the variance in achieving its money supply target. (In the remainder of this section we will be assuming that the covariance between the random elements involved in determining the money supply and the other random elements in the economy helping to determine the variance of GNP are small and unimportant; thus we can focus on a strategy that simply minimizes the variance in the money supply. In the following section we will return to the question of these covariances.)
F. Aftalion and L.J. White, Monetary targets
5
First, let us suppose it chooses a strategy of an immediate interest rate target, so it pegs rA = r~, and allows all the advances that the commercial banks wish at that rate. As we have demonstrated elsewhere [Aftalion and White (1978)], the equilibrium interest rate on loans r* that will be set by the commercial banks under this kind of arrangement will depend on the market structure of the banking system. A profit-maximizing competitive banking system would arrive at 3
rE=r * ,
(8)
while a profit-maximizing monopoly banking system would arrive at
rE = (2 +e)r~ 1 -be
12 ~3rL ~L"
(9)
For our purposes, it is sufficient to represent (8) or (9) as
r * = f (r~),
(10)
but it is reasonable to assume that this relationship is not deterministic but also has stochastic elements. In effect, though the banking sector is trying to achieve the profit maximizing outcome, it too may have information lags or other impediments to the exact and instantaneous achievement of its desired outcome. Hence, we represent the lending rate function as 4
r E = f (r*) + w.
(11)
3Both (8) and (9) assume that there are no administrative costs to making loans. Also, we have assumed that the banking system is neutral in its attitude toward risk and hence effectively ignores the stochastic element in the demand for its loans. To assume risk averse behavior by the banking system would needlessly complicate the model yet further without appreciably affecting the results. For further discussion of risk averse behavior by profit-maximizing firms, see Lintner (1970), Sandmo (1971), Leland (1972), Hawawini (1978) and White (1977). 4The formulation of. this stochastic element is slightly tricky. It is easiest to suppose that the banking system tiehaves as a price setter, setting the interest rate and then providing all that the market requires at that interest rate. In this case, the stochastic element in the quantity demand for loans (u) by itself would not create a stochastic element in the interest rate, since ex ante the banking system would have to make its best guess about the level of loan demand and that guess should be u=O. Hence the stochastic element in the interest rate (w) must come from other aspects of the banking system's behavior. Alternatively, if the banking system were to behave as a quantity setter, then it sets a .specific quantity and allows the price to clear at whatever level the demand curve determines. In this case, the stochastic element in the demand for loans would be completely translated into a stochastic element in interest rates, and we would not observe any variance in the quantity of loans. To reintroduce this quantity variance, we would have to replace equation (11) with one that stated that the quantity of loans provided by the banking system contained a stochastic element varying around the profit-maximizing level. We could then achieve basically the same results as are achieved in the text. For ease of exposition, we have chosen the first alternative.
6
F. Aftalion and L.J. White, Monetary targets
If the monetary authorities choose to peg r~ as their immediate target, the actual money supply achieved at any given time will be .TVI= C + D = S + L = S +al +a2X +aar~ + u ,
(12)
and the variance in the ultimate money target achieved will be var M, A= a22 var X + a ] var w + var u + 2a2a3 c o v X w + 2a2 c o v X u + 2a3 c o v uw.
(13)
This variance in the money supply will be compared with an alternative measure developed below. Suppose instead that the Central Bank adopts a strategy of setting an immediate portfolio target. In our model, this is equivalent to the Central Bank setting A to some target A and then letting the banking system bid for these advances until some interest rate ~A just clears the market for these advances. To determine the monetary consequences of this strategy, we must work back through the system of equations, starting with A to determine the level of loans L that can be made by the banking system from those A advances. By substitution and manipulation, we find 5 l+e S1-2 v(1-2) L='4"~+-ee+ ~+--ee- 2 + e
(14)
The actual money supply achieved at any given time again will be
M=C+D=S+L,
(6)
5The determination of ~A in this case is straightforward. By substituting (14) into (4) and rearranging, we can determine the market clearing ~L as a function of ,4, represented in (18) below. In a competitive banking system, the commercial banks will continue to bid up the interest rate on advances to the point where f,4 = f z (plus a possible stochastic term). If the banking system is monopolistic with respect to its lending functions but acts competitively when bidding for Central Bank advances, then the marginal revenue curve of the aggregate loan demand becomes the basis for the commercial bank demand curve for advances. The intersection of that curve and the vertical supply of advances ,4 determines the interest rate on advances. This can be determined by rearranging the profit maximizing result (9) so that f~t is on the left-hand side. By then substituting (14) for L and the market clearing value for rL, we get the appropriate expression for fA as a function of ,4 (again, plus a possible stochastic term). This latter kind of behavior could arise from incomplete collusion among a group of banks. It could also arise if each bank had a legally protected loan market in a geographic area or industrial sector into which no other bank could enter.
F. Aftalion and L.J. White, Monetary targets
7
but now by substitution from (14), M=C+D=S+L=
l+e [A+S]2+e
1-2 v. 2+e
(15)
And the variance in the money supply will now be
var
[1-2]2 MA=[_-~-+--~e_I vary.
(16)
We have thus arrived at a position in which we can compare the efficacy of the two alternative immediate strategies. If the Central Bank chooses an immediate interest rate target, the variance in the money supply will be (from (13)) a22 varX + a 2 var w + var u + 2a2a3 covXw + 2a2 covXu + 2a 3 cov uw, and if it chooses an immediate portfolio target, the variance in the money supply will be (from 16)) 1 --/~l 2 2 + e d vary.
The first measure involves the variances in the loan demand and supply; the second involves the variance in the way the public divides its money holdings between currency and demand deposits. W e have assumed that the Central Bank is interested in choosing the immediate strategy which yields the smallest variance in its money supply target. A priori, it is difficult to make any statements about which immediate target strategy will yield the smallest variance. It clearly depends on the error variances and the various coefficients. But one interesting and paradoxical possibility is that, if the loan market is relatively stable in the short run compared to the currency and demand deposit behavior of the public, the Central Bank would prefer to pursue an immediate interest rate target strategy as a way of pursuing its intermediate money supply target policy. The same logic applies if the Central Bank wishes to pursue an intermediate private sector interest rate policy and is interested in minimizing the variance in achieving its target. Again, it can choose between an immediate strategy of a portfolio target or an interest rate target. If it chooses an immediate interest rate target, then the actual private sector
8
F. Aftalion and L.J. White, Monetary targets
interest rate achieved at any given time will be represented by r~=f(r*)+w,
(11)
and the variance in the private sector interest rate will be simply var r ~ = var w,
(17 )
where ot=r a. If instead the Central Bank pursues an immediate portfolio target strategy, the quantity of loans forthcoming is represented by (14) above. If we substitute (14) into (4) and rearrange terms, we find that the actual private sector interest rate at any time is now
~L=lI~ (1 +e'~+ 1-2 \2--~e] S ( 2 ~ e ) - a l - a 2 X - ( 1 2 - ~ + 2 e ) V - U ] '
(18)
and the variance in the private sector interest rate is ^ 2 + (1 --~"~ 2 varrL~=a2varX \ 2 + e ] v a r v + v a r u
1-2 + 2a2 (__f__~e)COv Xv + 2a2 cov Xu + 2 (~_~e)COVUV. 1-2
(19)
Again, if the Central Bank wishes to minimize the variance in pursuit of its interest rate target, the choice of immediate strategies will depend on the error variances and the coefficients. But, again, we could get the paradoxical result that the Central Bank would prefer to pursue an immediate portfolio strategy as its means of achieving its intermediate interest rate target policy. Finally, it should be noted that an immediate interest rate strategy does have one inherent advantage over a portfolio strategy, irrespective of relative variance magnitudes. Some economists argue that short-term, random, and reversible shifts in the demand for money should be accommodated by monetary policy, since such accommodation prevents these random influences from having any effect on real variables [Davis (1974)]. 6 An immediate interest rate strategy provides such accommodation; a portfolio target strategy does not. 6The crucial problem, of course, is to be able to differentiate such short-term reversible shifts from a secular shift in the demand for money schedule.
F. Aftalion and L.J. White, Monetary targets
9
3. Linking the present model with the Poole model The previous section assumed that the monetary authorities had already settled on a target policy but had to choose an immediate strategy which would yield the minimum variance in achieving that policy. It is worth examining, however, the larger issue of how best to achieve the ultimate objective of a minimum variance in a desired GNP level. And, in this context, integrating our model with the Poole (1970) model is probably the most straightforward way of doing this. If we take our equation (12) and substitute income or GNP (Y) for X on the right hand side (and roll S and any other elements in X, such as expectations, into the error term), we have Poole's LM equation: (20)
M = a x +aeY +a3rL +u,
and then the equation is rearranged to express income as a stochastic function of a controllable money supply. Poole's IS curve is expressed simply as
(21)
Y = b 1 + bzrL + z.
In the Poole model, ~he income variance from a private sector interest rate target that can be achieved with certainty is simply var Y~L= var z.
(22)
The income variance from a money supply target is achieved by solving (21) for rL, substituting into (20), and solving that for Y:
Y =
" [ a 3 b I - a 1b 2 + b E M - b2 u +
a3z],
b2a 2 d- a 3
(23)
and with a money supply target that can be achieved with certainty we get
var YM= b2a£+a3
"[b~varu+a~varz-2b2a3covuz].
(24)
But, as we argued in the previous section, the monetary authorities cannot achieve either the money supply target or the private sector interest rate target exactly. Each has an error component. Hence, the true income
F. Aftalion and L.J. White, Monetary targets
10
v a r i a n c e s are var Y~L= b2 var rL + v a r z + 2b2 cov rLz,
(25)
and
var Yu=
b2a£-+ a3
• [b~ var u + a~ var z + b~ var M
- 2bzaa c o y u z - 2b~ coy Mu + 2bza 3 c o y M z ] .
(26)
A n d we m u s t n o w s u b s t i t u t e t h e a p p r o p r i a t e variances f r o m the a p p r o p r i a t e i m m e d i a t e targets. If the m o n e t a r y a u t h o r i t i e s c h o o s e an i m m e d i a t e interest r a t e strategy, t h e n we m u s t s u b s t i t u t e (17) into (25) a n d s u b s t i t u t e (13) i n t o (26). W e t h u s get V a r Y~ = b~ var w + v a r z + 2b2 cov wz,
(27)
w h e r e fl = r L a n d ~ = ra, a n d
b2a£ + a 3
VarYu=
"[b~ varu+a32var z
+ b~ (a~ var Y + a~ var w + var u + 2a2a3 c o y Yw + 2a 2 c o v Yu + 2a3 cov u w ) - 2bza3 coy uz - 2b2a3 c o y uz - 2b~ coy Mu + 2bEa3 c o y M z ] .
(28)
It is easily s h o w n t h a t c o y Mu = a2 c o y Yu + v a r u +
a 3 coy
wu,
c o y M z = a2 c o y Yz + c o y uz + a3 coy wz,
(29)
and coy
Yu = c o v
uz + b 2 coy
c o y Yz = var z +
b 2 coy
uw,
wz,
COV Yw = b 2 var w + c o v wz.
(30)
F. Aftalion and L.J. White, Monetary targets
11
By substituting (29) and (30) into (28) and appropriately simplifying, we find that Var YM,= Var Y~ = Var Y~ = b22var w + var z + 262 cov wz.
(31 )
And this is exactly what we should expect to find: The effect on the variance of income from choosing an immediate interest rate strategy should be identical, regardless of whether we envisage it as working through the money supply or working through the private sector interest rate. Similarly, if the monetary authorities choose an immediate portfolio strategy, we must substitute (19) into (25) and substitute (16) into (26). We thus get Var Yt~A= b22 a2 var Y + \ 2 + e / var v + var u + 2a 2
cov
Yv
+ 2{1-2~covuv+ 2a2 cov Yu]+ var z \2+eJ + 2 b 2 [ - a 2 coy
Yz-(12-~+2e)COVVZ-COVUZ],
]21
(1-2~ 2
(32)
and Var YMA =
[
1
bza2+a 3
b2varu+a2varz+b2\-2---~e / vary
- 262a 3 cov
uz - 2b 2 cov Mu + 2b2a 3 cov Mz 1.
(33)
We must now replace (29) and (30) with: 1-~, coy M u = - (2--~e) cov uv, 1-2
COV Yu = b 2 c o v
rLU +
(34)
COV
UZ,
coy
Yz = b2 cov rLZ + vat z,
COY
Yv =
b 2 cov
rLV +
COV
VZ,
(35)
F. Aftalion and L.J. White, M o n e t a r y targets
12
1-2 coy
rLU
=
-- a 2 COV
Yu - ~. + e
COV
rLZ
=
-- a 2 COV Y2 -
COY
fLY
=
-- a 2 coy
2+e
cov
uv -
var u,
COV
VZ-
COV UZ,
1-2 var v - coy uv. 2+e
(36)
Yv - - -
By substitution and simplification we discover that
var Y~. = var YM. = var YA=
• 2 { 1 - ~\2
+/~2 ~,2--~-~)
[1 b2a£+ a3
b~ var u + a 2 var z
varv-2b2a3co.vuz 1-~,
+ 2b2 ( l~+2e)COVUV- 2b2a3 (-~e) COVVZ1
(37)
If the ultimate goal of the monetary authorities is to minimize the variance in GNP around its desired level, then the choice of immediate strategies should depend on the comparison between (31) and (37), since these take into account the full variance effects. If the covariance terms cov rLz, cov Mu, and cov Mz are small, then the procedure we described in the previous section-settling on an intermediate money supply or private sector interest rate target in the context of the Poole model and then choosing the immediate strategy which allows the intermediate target to be reached with the least variance-will achieve the desired result. Once the intermediate target to minimize GNP variance has been chosen, the immediate target that minimizes the variance in the intermediate target also minimizes the variance in GNP. But if those covariances are large, then the two step procedure is not adequate, and the explicit comparison of the ultimate effects on GNP variance of the immediate strategy choice must be made, using (31) and (37). In effect the different immediate strategies cause different time paths for the money supply and the private sector interest rate. Because those covariances (by assumption) are large and are a component in the overall variance in GNP, those different ~time paths matter. Hence the full comparison of (31) and (37) must be made in this case.
F. Aftalion and L.J. White, Monetary targets
13
4. Conclusions In the choice of monetary targets, variances and possibly covariances matter. This has been clear from the Poole model. We have tried to extend this logic to the case in which the money supply and the private sector interest rate are not under the direct control of the monetary authorities but are themselves random variables and subject to delay in reporting. We believe this characterizes reality in virtually all monetary systems. In these circumstances, the monetary authorities realistically face a choice between two immediate targets which they can control with certainty: a portfolio target or an interest rate target. We have tried within the context of a simple model to show the nature of the choices. And we wish again to emphasize that if the ultimate objective is to minimize the variance in income, the full consequences on that variance of the choice between the two immediate targets must be examined. If the covariances are large, an immediate strategy which only minimizes the variance of an intermediate target may not be optimal. References Aftalion, F. and L.J. White, 1977, A study of a monetary system with a pegged discount rate under different market structures, Journal of Banking and Finance 1, no. 4, 349-371. Craine, R. and A. Havenner, 1977, The optimal monetary instrument: An empirical assessment, Special Studies Paper No. 100, Division of Research and Statistics, Federal Reserve Board, July 20. Davis, R.G., 1971, Short-run targetsfor open market operations, in Board of Governors of the Federal Reserve System, Open Market Policies and Operating Procedures-Staff Studies, 3769. Davis, R.G., 1974, Implementing open market policy with monetary aggregate objectives, in Federal Reserve Bank of New York, Monetary Aggregates and Monetary Policy, 7-19. Hawawini, G., 1978, A mean-standard deviation exposition of the theory of the firm under uncertainty, American Economic Review (forthcoming). Holbrook, R. and H. Shapiro, 1970, The choice of optimal intermediate economic targets, American Economic Review, 40-46, May. Karaken, J., T. Muench, T. Supel and N. Wallace, 1971, Determining the optimum monetary instrument variable, in Board of Governors of the Federal Reserve System, Open Market Policies and Operating Procedures-Staff Studies, 85-96. Leland, H.E., 1972, Theory of the firm facing uncertain demand, American Economic Review, 277-291, June. Lintner, J., The impact of uncertainty on the 'traditional' theory of the firm: Price setting and tax shifting, in J.W. Markham and G.F. Papanek, eds., Industrial Organization and Economic Development. Poole, W., 1970, Optimal choice of monetary policy instruments in a simple stochastic macro model, Quarterly Journal of Economics, 197-216, May. Poole, W. and C. , Lieberman, 1972, Improving monetary control, Brookings Papers on Ecofiomic Activity (No. 2), 293-335. Sandmo, A., 1971, On the theory of the competitive firm under price uncertainty, American Economic Review, 65-73, March. White, L.J., 1977, Price~luantity decisions under uncertainty: An interpretation using familiar price theory geometry. New York University, Graduate School of Business Administration, Working Paper No. 77-43, June.
ON THE CHOICE O F IMMEDIATE MONETARY TARGETS* Florin AFTALION ESSEC, Cergy, France
Lawrence J. WHITE New York University, New York, NYIO006, U.S.A. This paper studies the control problem of a stochastic monetary system. The Central Bank has the choice of two targets: the size of its portfolio of assets or the level of interest rate on that class of assets. If its objective is to minimize the variance of a monetary aggregate or of private sector interest rate, the conditions under which one or the other of the targets should be used are analyzed. However, if the ultimate objective is a minimum variance in a desired GNP level the effect of the immediate target is the same whether the central bank employs a money stock or a private interest rate target.
1. Introduction In the discussion concerning the choice of targets for monetary authorities (usually interest rates versus monetary aggregates), only a modest amount of attention has been given to the fact that the monetary authorities do not in fact have direct control over the stock of money or over the interest rates that directly affect the private economy. 1 Instead, the monetary authorities usually have direct control-in the sense of having immediate (daily or even hourly) information and an ability to affect immediate changes-over two things: the size of their portfolio of assets (this is frequently government debt, but in many European countries it is primarily commercial bank borrowings from the Central Bank) and the level of interest rates on that general class of assets. Thus, in their day-to-day dealings, the monetary authorities can only have a portfolio target or an interest rate target for their portfolio. This must be *This paper was presented at the ESSEC Conference (Cergy, France, June 9-11, 1977), which was financially supported by FNEGE. It has been subjected to the normal reviewing process of the Journal. 1See Davis (1971, 1974), Kareken et al. (1971), Holbrook and Shapiro (1970) and Poole and Lieberman (1972). As Craine and Havenner point out, there are three sources of error in monetary determination: equation errors, errors in forecasting exogenous variables, and errors in estimating the structural coefficients. We focus largely on the first, though the second is included. The third is beyond the scope of this paper.
2
F. Aftalion and L.J. White, Monetary targets
true even if the Central authorities would prefer to pursue a monetary aggregate or private sector interest rate target. The figures for the latter
targets are always reported with some delay, and hence they cannot be immediate targets. This is clearly true for the monetary aggregates, and, though corporate bond interest rates are quickly and widely known, commercial bank lending terms are not. 2. Since there are random fluctuations in the monetary behavior of the private sector and the banking sector vis-a-vis each other, the achievement of an immediate portfolio or interest rate target cannot guarantee the exact achievement of the intermediate monetary aggregates or private sector interest rate target. These intermediate targets can only be achieved in an average sense (provided the correct behavioral relationships are known); there will still be variance in the actual levels of the monetary aggregates or private interest rates achieved. There is an important consequence of this variance for the conduct of monetary policy. Suppose we have a world such as that described by Poole (1970), in which errors in the I S curve and in the L M curve combine with the coefficients in each relationship to determine whether a monetary aggregates target or a private interest rate target would achieve the least variance in achieving a G N P goal. Suppose further that the parameters of that model are such that, for example, a monetary aggregate target is the chosen vehicle. If we add the possibility of disturbances in the monetarybanking sector, we then must face a choice of an immediate portfolio target or an interest rate target as the means of achieving that monetary aggregates target. And there are quite reasonable circumstances under which the monetary authorities would choose an immediate interest rate target as a way of achieving the monetary aggregates target with the least variance. In effect, the added disturbances in the monetary-banking sector and the consequent variance in the monetary aggregates would make the pursuance of an immediate interest rate target the superior strategy. Similarly, even if the monetary authorities had decided on an intermediate private sector interest rate target, there are circumstances under which an immediate portfolio target strategy would be the least variance way of achieving that eventual target. And, finally, it is worth remembering that if the true ultimate goal is the achievement of a desired G N P level with the least amount of variance, the full effects of the stochastic elements in the private and banking sectors' monetary behavior must be included in the determination of targets. This paper will demonstrate these propositions using a simple model of the monetary-banking sector. The next section will provide the model and the basic results. The following section will relate our model to that of Poole's. And the final section will offer some conclusions from our model. 2Though changes in the prime rate in the U.S. are widely publicized, there is some slippage between the publicized prime rate and the actual terms charged to customers.
F. Aftalion and L.J. White, Monetary targets
3
2. The model Our model of the monetary-banking sector can be characterized as follows: Commercial banks are required by the monetary authorities to hold as reserves a specified fraction of their deposits. The banks can obtain reserves by obtaining advances from the Central Bank. These advances, along with holdings of foreign exchange, constitute the assets of the Central Bank. (This kind of arrangement is broadly descriptive of at least one European country, France. It is not too different conceptually from American procedures, in which the Central Bank's assets are mainly in the form of government debt which is bought from and sold to the commercial banks, rather than in the form of commercial bank debt.) The commercial banks make loans to the public; they are the only source of loanable funds to the public. The public holds its money in the form of deposits in the commercial banks and cash. Finally, the monetary authorities have a choice between two immediate targets: they can set a limit on the advances from the Central Bank by the commercial banks (in effect, a target on the Central Bank's portfolio) and in effect auction off these borrowings at a market clearing interest rate; or they can specify an interest rate on the advances and let the commercial banks borrow as much as they wish at that interest rate. We can now establish a set of definitions: S = gold and foreign exchange and treasury obligations held by the Central Bank, A = advances by the Central Bank to commercial banks ('refinancing'), R = reserves held by commercial banks against deposits, C = currency held by the public, D = total deposits at commercial banks e
= C/D,
.V/ = money ( = D q- C), L =loans by commercial banks to the public, X --a set of exogenous variables influencing the public demand for loans, e.g. income, expectations, rL = the interest rate on loans, 2 = the required reserve ration on deposits, ro = t h e interest rate on deposits (this includes any bank costs in servicing deposits; this interest rate is assumed to be set by the Central Bank), rA = t h e interest rate paid by commercial banks on their advances from the Central Bank, u = a disturbance element in the public's demand for loans from the commercial banks, with mean zero and a finite variance, v = a disturbance element in the public's desired currency holdings, with mean zero and a finite variance,
4 W
F. Aftalion and L.J. White, Monetary targets
= a disturbance element in the interest rate on loans by the commercial banks, with mean zero and a finite variance. Formally, the Central Bank's balance sheet must be
S+A=R+C.
(1)
The commercial banks' balance sheet must be
L+R=D+A.
(2)
The required reserves are R =2D.
(3)
The public's demand for loans is
L = a l +a2X +a3rL +u,
(4)
and the public's desired currency holdings are
C=eD+v.
(5)
By substituting (2) into (1), we get
S+L=C+D=M,
(6)
and by substituting (4) and (5) into (6) and rearranging, we get 1 D =~ IS +al + a 2 X + a 3 r L + u -- v]. l+e
(7)
We now assume that the Central Bank has chosen a money supply target M* as its means of achieving an ultimate income goal, in the spirit of the Poole model. But in an operational sense, it can choose only between a strategy of setting an immediate interest rate target r~ or an immediate advances target A. And the Central Bank is interested in minimizing the variance in achieving its money supply target. (In the remainder of this section we will be assuming that the covariance between the random elements involved in determining the money supply and the other random elements in the economy helping to determine the variance of GNP are small and unimportant; thus we can focus on a strategy that simply minimizes the variance in the money supply. In the following section we will return to the question of these covariances.)
F. Aftalion and L.J. White, Monetary targets
5
First, let us suppose it chooses a strategy of an immediate interest rate target, so it pegs rA = r~, and allows all the advances that the commercial banks wish at that rate. As we have demonstrated elsewhere [Aftalion and White (1978)], the equilibrium interest rate on loans r* that will be set by the commercial banks under this kind of arrangement will depend on the market structure of the banking system. A profit-maximizing competitive banking system would arrive at 3
rE=r * ,
(8)
while a profit-maximizing monopoly banking system would arrive at
rE = (2 +e)r~ 1 -be
12 ~3rL ~L"
(9)
For our purposes, it is sufficient to represent (8) or (9) as
r * = f (r~),
(10)
but it is reasonable to assume that this relationship is not deterministic but also has stochastic elements. In effect, though the banking sector is trying to achieve the profit maximizing outcome, it too may have information lags or other impediments to the exact and instantaneous achievement of its desired outcome. Hence, we represent the lending rate function as 4
r E = f (r*) + w.
(11)
3Both (8) and (9) assume that there are no administrative costs to making loans. Also, we have assumed that the banking system is neutral in its attitude toward risk and hence effectively ignores the stochastic element in the demand for its loans. To assume risk averse behavior by the banking system would needlessly complicate the model yet further without appreciably affecting the results. For further discussion of risk averse behavior by profit-maximizing firms, see Lintner (1970), Sandmo (1971), Leland (1972), Hawawini (1978) and White (1977). 4The formulation of. this stochastic element is slightly tricky. It is easiest to suppose that the banking system tiehaves as a price setter, setting the interest rate and then providing all that the market requires at that interest rate. In this case, the stochastic element in the quantity demand for loans (u) by itself would not create a stochastic element in the interest rate, since ex ante the banking system would have to make its best guess about the level of loan demand and that guess should be u=O. Hence the stochastic element in the interest rate (w) must come from other aspects of the banking system's behavior. Alternatively, if the banking system were to behave as a quantity setter, then it sets a .specific quantity and allows the price to clear at whatever level the demand curve determines. In this case, the stochastic element in the demand for loans would be completely translated into a stochastic element in interest rates, and we would not observe any variance in the quantity of loans. To reintroduce this quantity variance, we would have to replace equation (11) with one that stated that the quantity of loans provided by the banking system contained a stochastic element varying around the profit-maximizing level. We could then achieve basically the same results as are achieved in the text. For ease of exposition, we have chosen the first alternative.
6
F. Aftalion and L.J. White, Monetary targets
If the monetary authorities choose to peg r~ as their immediate target, the actual money supply achieved at any given time will be .TVI= C + D = S + L = S +al +a2X +aar~ + u ,
(12)
and the variance in the ultimate money target achieved will be var M, A= a22 var X + a ] var w + var u + 2a2a3 c o v X w + 2a2 c o v X u + 2a3 c o v uw.
(13)
This variance in the money supply will be compared with an alternative measure developed below. Suppose instead that the Central Bank adopts a strategy of setting an immediate portfolio target. In our model, this is equivalent to the Central Bank setting A to some target A and then letting the banking system bid for these advances until some interest rate ~A just clears the market for these advances. To determine the monetary consequences of this strategy, we must work back through the system of equations, starting with A to determine the level of loans L that can be made by the banking system from those A advances. By substitution and manipulation, we find 5 l+e S1-2 v(1-2) L='4"~+-ee+ ~+--ee- 2 + e
(14)
The actual money supply achieved at any given time again will be
M=C+D=S+L,
(6)
5The determination of ~A in this case is straightforward. By substituting (14) into (4) and rearranging, we can determine the market clearing ~L as a function of ,4, represented in (18) below. In a competitive banking system, the commercial banks will continue to bid up the interest rate on advances to the point where f,4 = f z (plus a possible stochastic term). If the banking system is monopolistic with respect to its lending functions but acts competitively when bidding for Central Bank advances, then the marginal revenue curve of the aggregate loan demand becomes the basis for the commercial bank demand curve for advances. The intersection of that curve and the vertical supply of advances ,4 determines the interest rate on advances. This can be determined by rearranging the profit maximizing result (9) so that f~t is on the left-hand side. By then substituting (14) for L and the market clearing value for rL, we get the appropriate expression for fA as a function of ,4 (again, plus a possible stochastic term). This latter kind of behavior could arise from incomplete collusion among a group of banks. It could also arise if each bank had a legally protected loan market in a geographic area or industrial sector into which no other bank could enter.
F. Aftalion and L.J. White, Monetary targets
7
but now by substitution from (14), M=C+D=S+L=
l+e [A+S]2+e
1-2 v. 2+e
(15)
And the variance in the money supply will now be
var
[1-2]2 MA=[_-~-+--~e_I vary.
(16)
We have thus arrived at a position in which we can compare the efficacy of the two alternative immediate strategies. If the Central Bank chooses an immediate interest rate target, the variance in the money supply will be (from (13)) a22 varX + a 2 var w + var u + 2a2a3 covXw + 2a2 covXu + 2a 3 cov uw, and if it chooses an immediate portfolio target, the variance in the money supply will be (from 16)) 1 --/~l 2 2 + e d vary.
The first measure involves the variances in the loan demand and supply; the second involves the variance in the way the public divides its money holdings between currency and demand deposits. W e have assumed that the Central Bank is interested in choosing the immediate strategy which yields the smallest variance in its money supply target. A priori, it is difficult to make any statements about which immediate target strategy will yield the smallest variance. It clearly depends on the error variances and the various coefficients. But one interesting and paradoxical possibility is that, if the loan market is relatively stable in the short run compared to the currency and demand deposit behavior of the public, the Central Bank would prefer to pursue an immediate interest rate target strategy as a way of pursuing its intermediate money supply target policy. The same logic applies if the Central Bank wishes to pursue an intermediate private sector interest rate policy and is interested in minimizing the variance in achieving its target. Again, it can choose between an immediate strategy of a portfolio target or an interest rate target. If it chooses an immediate interest rate target, then the actual private sector
8
F. Aftalion and L.J. White, Monetary targets
interest rate achieved at any given time will be represented by r~=f(r*)+w,
(11)
and the variance in the private sector interest rate will be simply var r ~ = var w,
(17 )
where ot=r a. If instead the Central Bank pursues an immediate portfolio target strategy, the quantity of loans forthcoming is represented by (14) above. If we substitute (14) into (4) and rearrange terms, we find that the actual private sector interest rate at any time is now
~L=lI~ (1 +e'~+ 1-2 \2--~e] S ( 2 ~ e ) - a l - a 2 X - ( 1 2 - ~ + 2 e ) V - U ] '
(18)
and the variance in the private sector interest rate is ^ 2 + (1 --~"~ 2 varrL~=a2varX \ 2 + e ] v a r v + v a r u
1-2 + 2a2 (__f__~e)COv Xv + 2a2 cov Xu + 2 (~_~e)COVUV. 1-2
(19)
Again, if the Central Bank wishes to minimize the variance in pursuit of its interest rate target, the choice of immediate strategies will depend on the error variances and the coefficients. But, again, we could get the paradoxical result that the Central Bank would prefer to pursue an immediate portfolio strategy as its means of achieving its intermediate interest rate target policy. Finally, it should be noted that an immediate interest rate strategy does have one inherent advantage over a portfolio strategy, irrespective of relative variance magnitudes. Some economists argue that short-term, random, and reversible shifts in the demand for money should be accommodated by monetary policy, since such accommodation prevents these random influences from having any effect on real variables [Davis (1974)]. 6 An immediate interest rate strategy provides such accommodation; a portfolio target strategy does not. 6The crucial problem, of course, is to be able to differentiate such short-term reversible shifts from a secular shift in the demand for money schedule.
F. Aftalion and L.J. White, Monetary targets
9
3. Linking the present model with the Poole model The previous section assumed that the monetary authorities had already settled on a target policy but had to choose an immediate strategy which would yield the minimum variance in achieving that policy. It is worth examining, however, the larger issue of how best to achieve the ultimate objective of a minimum variance in a desired GNP level. And, in this context, integrating our model with the Poole (1970) model is probably the most straightforward way of doing this. If we take our equation (12) and substitute income or GNP (Y) for X on the right hand side (and roll S and any other elements in X, such as expectations, into the error term), we have Poole's LM equation: (20)
M = a x +aeY +a3rL +u,
and then the equation is rearranged to express income as a stochastic function of a controllable money supply. Poole's IS curve is expressed simply as
(21)
Y = b 1 + bzrL + z.
In the Poole model, ~he income variance from a private sector interest rate target that can be achieved with certainty is simply var Y~L= var z.
(22)
The income variance from a money supply target is achieved by solving (21) for rL, substituting into (20), and solving that for Y:
Y =
" [ a 3 b I - a 1b 2 + b E M - b2 u +
a3z],
b2a 2 d- a 3
(23)
and with a money supply target that can be achieved with certainty we get
var YM= b2a£+a3
"[b~varu+a~varz-2b2a3covuz].
(24)
But, as we argued in the previous section, the monetary authorities cannot achieve either the money supply target or the private sector interest rate target exactly. Each has an error component. Hence, the true income
F. Aftalion and L.J. White, Monetary targets
10
v a r i a n c e s are var Y~L= b2 var rL + v a r z + 2b2 cov rLz,
(25)
and
var Yu=
b2a£-+ a3
• [b~ var u + a~ var z + b~ var M
- 2bzaa c o y u z - 2b~ coy Mu + 2bza 3 c o y M z ] .
(26)
A n d we m u s t n o w s u b s t i t u t e t h e a p p r o p r i a t e variances f r o m the a p p r o p r i a t e i m m e d i a t e targets. If the m o n e t a r y a u t h o r i t i e s c h o o s e an i m m e d i a t e interest r a t e strategy, t h e n we m u s t s u b s t i t u t e (17) into (25) a n d s u b s t i t u t e (13) i n t o (26). W e t h u s get V a r Y~ = b~ var w + v a r z + 2b2 cov wz,
(27)
w h e r e fl = r L a n d ~ = ra, a n d
b2a£ + a 3
VarYu=
"[b~ varu+a32var z
+ b~ (a~ var Y + a~ var w + var u + 2a2a3 c o y Yw + 2a 2 c o v Yu + 2a3 cov u w ) - 2bza3 coy uz - 2b2a3 c o y uz - 2b~ coy Mu + 2bEa3 c o y M z ] .
(28)
It is easily s h o w n t h a t c o y Mu = a2 c o y Yu + v a r u +
a 3 coy
wu,
c o y M z = a2 c o y Yz + c o y uz + a3 coy wz,
(29)
and coy
Yu = c o v
uz + b 2 coy
c o y Yz = var z +
b 2 coy
uw,
wz,
COV Yw = b 2 var w + c o v wz.
(30)
F. Aftalion and L.J. White, Monetary targets
11
By substituting (29) and (30) into (28) and appropriately simplifying, we find that Var YM,= Var Y~ = Var Y~ = b22var w + var z + 262 cov wz.
(31 )
And this is exactly what we should expect to find: The effect on the variance of income from choosing an immediate interest rate strategy should be identical, regardless of whether we envisage it as working through the money supply or working through the private sector interest rate. Similarly, if the monetary authorities choose an immediate portfolio strategy, we must substitute (19) into (25) and substitute (16) into (26). We thus get Var Yt~A= b22 a2 var Y + \ 2 + e / var v + var u + 2a 2
cov
Yv
+ 2{1-2~covuv+ 2a2 cov Yu]+ var z \2+eJ + 2 b 2 [ - a 2 coy
Yz-(12-~+2e)COVVZ-COVUZ],
]21
(1-2~ 2
(32)
and Var YMA =
[
1
bza2+a 3
b2varu+a2varz+b2\-2---~e / vary
- 262a 3 cov
uz - 2b 2 cov Mu + 2b2a 3 cov Mz 1.
(33)
We must now replace (29) and (30) with: 1-~, coy M u = - (2--~e) cov uv, 1-2
COV Yu = b 2 c o v
rLU +
(34)
COV
UZ,
coy
Yz = b2 cov rLZ + vat z,
COY
Yv =
b 2 cov
rLV +
COV
VZ,
(35)
F. Aftalion and L.J. White, M o n e t a r y targets
12
1-2 coy
rLU
=
-- a 2 COV
Yu - ~. + e
COV
rLZ
=
-- a 2 COV Y2 -
COY
fLY
=
-- a 2 coy
2+e
cov
uv -
var u,
COV
VZ-
COV UZ,
1-2 var v - coy uv. 2+e
(36)
Yv - - -
By substitution and simplification we discover that
var Y~. = var YM. = var YA=
• 2 { 1 - ~\2
+/~2 ~,2--~-~)
[1 b2a£+ a3
b~ var u + a 2 var z
varv-2b2a3co.vuz 1-~,
+ 2b2 ( l~+2e)COVUV- 2b2a3 (-~e) COVVZ1
(37)
If the ultimate goal of the monetary authorities is to minimize the variance in GNP around its desired level, then the choice of immediate strategies should depend on the comparison between (31) and (37), since these take into account the full variance effects. If the covariance terms cov rLz, cov Mu, and cov Mz are small, then the procedure we described in the previous section-settling on an intermediate money supply or private sector interest rate target in the context of the Poole model and then choosing the immediate strategy which allows the intermediate target to be reached with the least variance-will achieve the desired result. Once the intermediate target to minimize GNP variance has been chosen, the immediate target that minimizes the variance in the intermediate target also minimizes the variance in GNP. But if those covariances are large, then the two step procedure is not adequate, and the explicit comparison of the ultimate effects on GNP variance of the immediate strategy choice must be made, using (31) and (37). In effect the different immediate strategies cause different time paths for the money supply and the private sector interest rate. Because those covariances (by assumption) are large and are a component in the overall variance in GNP, those different ~time paths matter. Hence the full comparison of (31) and (37) must be made in this case.
F. Aftalion and L.J. White, Monetary targets
13
4. Conclusions In the choice of monetary targets, variances and possibly covariances matter. This has been clear from the Poole model. We have tried to extend this logic to the case in which the money supply and the private sector interest rate are not under the direct control of the monetary authorities but are themselves random variables and subject to delay in reporting. We believe this characterizes reality in virtually all monetary systems. In these circumstances, the monetary authorities realistically face a choice between two immediate targets which they can control with certainty: a portfolio target or an interest rate target. We have tried within the context of a simple model to show the nature of the choices. And we wish again to emphasize that if the ultimate objective is to minimize the variance in income, the full consequences on that variance of the choice between the two immediate targets must be examined. If the covariances are large, an immediate strategy which only minimizes the variance of an intermediate target may not be optimal. References Aftalion, F. and L.J. White, 1977, A study of a monetary system with a pegged discount rate under different market structures, Journal of Banking and Finance 1, no. 4, 349-371. Craine, R. and A. Havenner, 1977, The optimal monetary instrument: An empirical assessment, Special Studies Paper No. 100, Division of Research and Statistics, Federal Reserve Board, July 20. Davis, R.G., 1971, Short-run targetsfor open market operations, in Board of Governors of the Federal Reserve System, Open Market Policies and Operating Procedures-Staff Studies, 3769. Davis, R.G., 1974, Implementing open market policy with monetary aggregate objectives, in Federal Reserve Bank of New York, Monetary Aggregates and Monetary Policy, 7-19. Hawawini, G., 1978, A mean-standard deviation exposition of the theory of the firm under uncertainty, American Economic Review (forthcoming). Holbrook, R. and H. Shapiro, 1970, The choice of optimal intermediate economic targets, American Economic Review, 40-46, May. Karaken, J., T. Muench, T. Supel and N. Wallace, 1971, Determining the optimum monetary instrument variable, in Board of Governors of the Federal Reserve System, Open Market Policies and Operating Procedures-Staff Studies, 85-96. Leland, H.E., 1972, Theory of the firm facing uncertain demand, American Economic Review, 277-291, June. Lintner, J., The impact of uncertainty on the 'traditional' theory of the firm: Price setting and tax shifting, in J.W. Markham and G.F. Papanek, eds., Industrial Organization and Economic Development. Poole, W., 1970, Optimal choice of monetary policy instruments in a simple stochastic macro model, Quarterly Journal of Economics, 197-216, May. Poole, W. and C. , Lieberman, 1972, Improving monetary control, Brookings Papers on Ecofiomic Activity (No. 2), 293-335. Sandmo, A., 1971, On the theory of the competitive firm under price uncertainty, American Economic Review, 65-73, March. White, L.J., 1977, Price~luantity decisions under uncertainty: An interpretation using familiar price theory geometry. New York University, Graduate School of Business Administration, Working Paper No. 77-43, June.