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Optimal macroeconomic policy adjustment under conditions of risk
JOURNAL
OF ECONOMIC
Notes,
THEORY 4, 58-71
Comments,
Optimal
and Letters
Macroeconomic under
of Political
Economy,
Policy
Conditions
W.
DALE Department
(1972)
to the
Editor
Adjustment
of Risk
HENDERSON
University
of Pennsylvania,
Philadelphia,
PA 19104
AND STEPHEN Department
of Economics,
TURNOVSKY
University Received
1.
J.
of Toronto,
Toronto,
Canada
July 7, 1970
INTRODUCTION
AND
SUMMARY
This paper investigates optimal paths of adjustment for macroeconomic policy instruments under conditions of risk when adjustment costs are present. The problem of how to adjust policy instruments so as to attain desired values for target variables under conditions of less than perfect certainty has received increasing attention in the literature, and the formulation adopted in the present paper attempts to synthesize the existing approaches to this questi0n.l In Section 2 we consider the one instrument-one target case. First, the optimal stationary value for the instrument is obtained; then we derive the path of optimal adjustment to this value. The stationary optimum is essentially the same as Brainard’s [I] static optimum, so that our model can be regarded as a dynamic extension of his work. The effects of changes in various parameters, particularly those pertaining to risk, on both the 1 At least three approaches to the problem can be identified. First there is the approach originally suggested by Phillips [8] and later utilized by Mundell [7] and Cooper [2] in a somewhat different context. These authors assume that the parameters of the constraints relating the target and instrument variables are nonstochastic but unknown. They then determine whether or not arbitrary, but plausible, rules of adjustment for the instrument variables lead to the attainment of desired values for the target variables. For a summary of this work see Lancaster [5]. Second, there is the formulation adopted by Theil and others [9, 111, who employ an explicitly stochastic formulation and whose main result is the “certainty equivalence theorem.” Finally, Brainard [l] has recently applied mean-variance portfolio analysis to the problem within a static context. 58 0
1972
by Academic
Press, Inc.
POLICY ADJUSTMENT UNDER RISK
59
long run equilibrium and the path of adjustment are considered. The main results are as follows. Under plausible assumptions, the effect of increased risk is usually to reduce the optimal stationary value of the instrument, although important exceptions may be found. Changes in the expected value of the coefficient of the policy variable may raise or lower the optimal stationary value of the policy variable, depending upon the size of the coefficient of variation of this parameter. The optimal adjustment rule requires the policy maker to adjust the instrument so as to close a constant proportion of the gap between the desired and actual values of the instrument variable. Increased risk raises the speed at which a gap of given size between the desired and actual values of the policy instrument should be closed. However, the effect of increasing risk on the rate at which the instrument is actually changed is more ambiguous, and one case is analyzed in detail. In Section 3 the model is extended to the case of two instruments and one target. It is shown that even under certainty a policy maker pursuing a single target must adjust both instruments toward determinate optimal stationary values when confronted with adjustment costs. In accordance with Brainard’s [I] static optimum results, determinate stationary optimum values can also be derived for both instruments when risk is present. We show that in general optimal adjustment under risk requires that each instrument be changed in response to discrepancies between actual and optimal values for both instruments. We report and discuss the effects of increases in risk upon the rates at which the instruments are adjusted toward their respective optima in an important special case.
2. THE ONE TARGET, ONE INSTRUMENT CASE
In this section the macroeconomic policy maker is assumed to have one target, say the level of national income y, and one instrument, taken to be the amount of fiscal stimulus g. We assume a closed economy where monetary authorities act passively to keep the rate of interest constant. The policy maker’s preferences are described by the quadratic utility function2
w = J,”hY -
&a2y2 + b, g - +b2g2 -
$cg2] e-Tt dt,
(1)
a The assumption of a quadratic utility function is, of course, restrictive and must be regarded merely as an approximation. Its use is justified largely on grounds of convenience. Not only does it lead to optimality conditions which are linear in all variables, but it also implies that only the means, variances, and covariances of the stochastic
60
HENDERSON
AND
TURNOVSKY
where r is the social rate of discount and the coefficients ai , bi , and c are positive.3 Increases in y are desired up to a point (y* = al/a,) because of decreasing unemployment, but beyond this point further increases in y are undesirable because they result in “too much” inflation. Likewise there is a value of g, (g* = b,/b,), preferable to all others with deviations from this value leading to reductions in welfare. The saturation level of g is presumably determined by cost-benefit calculations.” An essential feature of our formulation is that changing the level of the instrument is assumed to involve administrative and deliberative costs which can be described by a quadratic cost function5 If administrative and deliberative skills and energies are absorbed in achieving a change in g they will not be available for other tasks. The elaborate justifications and lengthy discussions associated with major changes in stabilization instruments result in less time and resources being devoted to the consideration of other important social goals. Of course, entering these other social costs through 2, as we do here, means that we have chosen a somewhat special utility function, but one we believe to be suggestive and useful. Note also that the coefficients appearing in (1) are all assumed to be independent of time and that by writing our objective function in the form of an integral of discounted social benefits we are assuming that benefits are additive over time.6 We assume that national income y is linearly related to the contemporaneous value of the fiscal stimulus g by the equation
Y(f) = Wig(t) + 4th An equation
(2)
such as (2) can be readily derived from a simple national
elements are relevant. Of course, this is not true with more general utility functions. Recently, the quadratic utility function has fallen into disrepute in problems related to portfolio theory under uncertainty, where it is shown to have rather absurd implications (such as implying that all risky assets are inferior). However, this should not mean that the quadratic utility function is illegitimate in all stochastic contexts. Since it does lead to reasonable conclusions for our model, it is felt that its use can be justifiedat least as an approximation. 3 Both g and y are functions of 1, although the functional dependence on t will usually be omitted. 4 The analysis could be extended by adding an interaction term since the costs and benefits from public projects may depend upon the level of employment. 5 This kind of approach has recently been used by Lucas [6] and Gould [3] to justify distributed lags in investment functions. As a reasonable approximation we are assuming that changing the instrument either upwards or downwards by the same absolute amount costs the same. Consequently, there is no linear term in @in the cost function. 6 Working with the one instrument-one target case, Brainard [I] proposes maximizing the utility function, W = -(y - Y*)~, where y* is the desired level of the target.
POLICY ADJUSTMENT
61
UNDER RISK
income determination model in which all expenditures depend only upon current income. The parameter 0 represents the fiscal stimulus multiplier, while u is ~9 times the total amount of nongovernment autonomous expenditure.’ Both variables are assumed to be random functions with8 E 6(r) = 6,
(34
E u(t) = I’,
(3b)
where for convenience both 8 and 0 are taken to be positive. Furthermore, we assume that both variables are independently distributed through time, that contemporaneous values are correlated with correlation coefficient p, while noncontemporaneous values are uncorrelated. Hence var O(t) = o&
E[{e(t) - O} . (e(s) - Lq] = 0 s # t;
W-4
var u(t) = a,$
E[(u(t) - 6} ’ {u(s) - E}] = 0 s # t;
WI
E[(Qt) -e> . {u(t) -C}] = pu#J,; E[{B(t) -CT} *(u(s) -V}] = 0
s # t.
04
Since we are dealing with a situation of risk, the decision maker is assumed to know the values of all parameters appearing in (3) and (4), all of which are assumed to remain constant over time.s Substituting for (2) in (1) and taking expected values, we obtain that expected utility is given by E(W) = jm [a,(Og + 6) - &l,[(B”- + q?) g” + 2~wJ,p
+ fiti>g + fi2+ %Y
0
+ b,g - &b,g” - +$] eert dt,
(5)
which is expressed as a function of the control variable g. The decision maker’s problem is thus to select g(t) so as to maximize (5). Hence the 7 Equation (2) can also be interpreted differently. For example, if g(f) is the money supply, assumed to be a government instrument, the equation may be regarded as a reduced form relationship between money supply and national income. 8 For a formal definition of a random function, see, e.g., Yaglom 612, p. 91. More informally, a random variable x(t), say, which is continuous over time, is defined to be a random function. BThus the probability distributions governing e(t) and v(t) are assumed to be stationary.
62
HENDERSON AND TURNOVSKY
optimal path for g(t) must satisfy the following Euler equation, which is a necessary condition for a maximumlo &a, - u,i;) + b, - a2ueo,p- [a,(@ $ CT;“)+ b,]g = -cg + crj.
(6)
This is a second-order differential equation in g, and the optimal stationary solution for g, denoted by g, is given byll
s = [~(~l - a,fi) + bl - w,%pl/[~,(~2 + d) + &I
(7)
An immediate question of interest is to determine how g responds to changes in the various parameters-particularly those pertaining to risk. In studying this question we shall assume that a, - a,6 > 0. Since the “saturation” level of y-that is, the level where the marginal utility of y drops to zero-is given by ul/uz , this condition asserts that the utility maximizing level of y exceeds what the expected value of y would be in the absence of policy action. This assumption seems reasonable, but it is possible to have a, - a& < 0, in which case some of our conclusions would have to be appropriately modified. Differentiating (7) with respect to ug2, uu2, p we obtain
ag/auo2= --a,PG + %p1/bTb2@2 + uo2)+ b21, ag/au,2 = - a.,42%Ca2(8” + 9”) + &I, agjap = -uzasu,/[u2(~2 -t us”) + b,].
(8)
(9) (10)
As can readily be seen, agjap is unambiguously negative, while the sign of ag/ao,2 varies inversely with the sign of p, so that the effect of risk in the exogenous expenditure depends upon how this is correlated with the multiplier. In general, the effect of ug2 on g is ambiguous, although some light is shed by considering the special case where p = 0. In that case our previous assumptions imply that ag/au,2 is unambigously negative. Since an increase in oe2 makes the marginal expected utility of g negative at the old g, g must be lowered to keep the marginal expected utility of g equal to zero. If p # 0, the results become somewhat more complicated. When p is lDBecause of the fact that the target variable y and the instrument g are related contemporaneously, we can use the simple calculus of variations approach. Had the relationship involved the derivative of y-as it would if the economic model underlying (2) involved an accelerator relationship-the problem would have had to be solved using the more recently developed methods of stochastic control theory. I1 Our result is consistent with the result Brainard obtains from maximizing the expected value of the objective function in footnote 6 if we replace a,/aa by y* and set b, = b, = 0.
POLICY
ADJUSTMENT
UNDER
RISK
63
positive, ag/ao,2 is negative so long as g is positive. The reasoning is the same as in the previous case. However, when p is negative, raising ue2 has an ambiguous effect on the marginal expected utility of g even if g is positive. So, for example, if p is sufficiently negative, an increase in co2 makes the marginal expected utility of g positive at the old g, so g must be raised to reestablish optimality. Other possibilities including the case where a, - a& < 0 can be similarly worked out. It is also of interest to consider @jag. Under certainty (assuming b, = 0, a, - a,C > 0) this expression is clearly negative, since an increase in the multiplier means that the instrument need be operated less intensively in order to achieve the optimal income. However, when risk is present this conclusion may not hold as the following expression indicates:
ag -z at7
(a, - a$) a,82(cz -
1) + 2a,28a,ovp + b&a, - a.$) - 2b,a,B
[a,(@+ ~2) + b,l”
;w
where c = se/e = coefficient of variation of 19. Considering the case where p = 0, and b, = b, = 0 we see immediately that ag/ie 2 0, according as c2 2 1. Increasing 0 has an ambiguous effect on the marginal expected utility of g. Thus if c < l-so that the instrument is relatively efficient-an increase in the instrument’s effectiveness will lead to an decrease in its use just as under certainty; the opposite is true when c > 1. A positive p increases the chances of ag:g/at?being positive; a negative p reduces them. Extension to the case of positive b, and b, is straightforward but not particularly instructive. Solving Eq. (6) and taking account of the fact that the decision maker wants g to approach g as t --j cc we arrive at the following adjustment rule for g:12 ii = &T
- E>
(12)
[c2r2 + 4c[a2(02 + oo2) + b2]]lj2 < o 2c
(13)
where h = rc 1
Hence along the optimal path, the instrument is continually adjusted so as to close some proportion of the gap between its present value and its long run optimal value. Equation (12) is very similar to the kind of adjustment proposed as plausible by Phillips and others [2,7, 81, although in our model it is a consequence of optimizing behavior. I2 This “transversality condition” teristic equation A, is relevant.
implies that only the negative root of the charac-
64
HENDERSON
AND
TURNOVSKY
There are two important questions pertaining to policy adjustment under risk. First one would like to know how the presenceof risk affects the speedat which an initial gap of a given size between actual and desired g is closed. In other words, how is X, affected by risk? Secondly, it is of interest to determine how risk affects the rate at which the instrument is actually changed for given values of g. To answer this question we must analyze the expression for g. First let us consider the effects of changes in parameters on the speed with which a given gap is closed. Differentiating h, , with respect to 8, og2,and r we obtain &I,/~0
= -2a,B[c2r2
+ 4c[a,(8”
+ ffB2) + b2]]-1’2
< 0,
ah,/&J@2 = -u2[c2r2 -t 4c[a2(d2 + ue2> + b211-1~2 < 0, ax,-
59 -
[C2r2
$
4C[a2(~2
+
uo2)
$-
6~11~‘~
2[c2r2 + 4c[a2(e2 + us”) +
-
cr
Z72111i2
>
(14) (15)
o
'
(16)
Note that all these responseshave definite signs as indicated. Recalling that h, < 0, expression (14) assertsthat the speedof adjustment increases in responseto an increase in the expected multiplier. As the instrument becomes more effective, any discrepancy between actual and desired g can be eliminated more rapidly. An increasein variability has a similar effect, although for a somewhat different reason. As risk increases, the probability of random errors causingy to take on someextreme undesirable values increases. By approaching g more rapidly, the variability in y obviously still remains, but now it is about a value nearer its optimum. In other words, if there is variability in the system, it is better to fluctuate about an optimum value than about some other point. From Eq. (16) we see that an increase in the social rate of discount has the opposite effect on X, , i.e., it tends to slow down the rate of adjustment. Hence the lessfuture deviations from the optimum matter, the slower will the adjustment proceed-again a perfectly plausible result. In addition, we can show that a given gap is closed more rapidly the higher the disutility of having g away from g and the lower the cost of changing g. In particular, if changing g is costless,then g will always be at g. It should also be pointed out that the speedof adjustment is independent of parameters describing the probability distribution of ZIand the correlation coefficient. Parameters such asp, uv2and Vhave been shown to affect g; however, they do not influence h, . We now consider how risk affects the rate at which the instrument g is changed for given values of g. At any point of time, the actual level of g is a given datum and hence is not affected by decisions affecting g. The
POLICY
ADJUSTMENT
UNDER
RISK
65
effect of risk on g is studied by differentiating the dynamic Eq. (12) with respect to the various parameters. The effect of raising go2 is shown by the following: ag/acQ = (ahl/aa,2)(g - g) - hl(ag/au,g2),
(17)
and analogous expressions are obtained for the other parameters. We see immediately that risk affects g on two counts. First, it affects the long run equilibrium [see Eq. (S)]; second, it affects the rate at which a given gap is closed [see Eq. (15)]. We can obtain an expression for 8~/&~,~ in terms of g and the parameters; however, it is extremely cumbersome, and it is not worthwhile to report it here. Instead we analyze an important special case by means of a phase diagram. This same approach could easily be applied to other cases. Consider the case where g is positive, ig/30s2 is negative, and imagine two adjustment paths of low and high risk, respectively. For simplicity we shall designate parameters relevant to the path of low risk by a subscript c for “certain” and for a path of high risk by a subscript u for “uncertain.” Figure 1 plots the dynamic Eq. (12). Since Z~/&02 < 0, gU lies to the
FIGURE
1
left of gc . ah1/%,2 < 0, so A,, has a steeper slope than A,, . It can be shown that the two lines intersect at a point g* < 0, so that the partial derivative ag/&r,2 clearly changes sign as g moves through g* from left to right. 642/4/I-5
66
HENDERSON
AND
TURNOVSKY
The following conclusions can be drawn from Fig. 1. As long as g is positive, the algebraic magnitude of fC exceeds the algebraic magnitude of gU . However, it is possible that g may be negative (in the case of a strong government surplus) and if it lies to the left of g*, the above conclusion is reversed. Notice that the instrument should not always be applied more gently under conditions of increased risk. For example, if g initially lies to the right of EC, then the absolute change in g under increased risk will exceed that under certainty. Since g is farther from its equilibrium value in the “uncertain” case, it must be subjected to a greater (negative) change. Finally, we see from the figure that if g happens to lie between gU and ge , then the policy maker will decrease g if he is acting under “uncertainty” and increase it under “certainty”. The responses of g to other parameters are easier to analyze. Thus for example,
(18) so that ;Ig/8u2 2 change in the rate risk in the amount of the correlation
0 according to whether p 5 0. Hence the algebraic of adjustment of the instrument in response to increasing of exogenous expenditure varies inversely with the sign coefficient between 13and 21.
3. THE ONE TARGET-TWO
INSTRUMENT
CASE
Perhaps the most basic result in the theory of policy adjustment under certainty is the proposition, due to Tinbergen [lo], that, when the relationships between instruments and targets are linear, policy makers need only use as many instruments as they have targets in order to achieve the desired values of all target variables. Other instruments are redundant and may be set at arbitrary levels. Brainard showed that under risk a policy maker should use all the instruments at his disposal even if he has only one target variable, since by doing so he can reduce the variance of the target for given levels of the expected value. In this section we demonstrate that when adjustment costs are present a policy maker should use all the instruments at his disposal even under certainty. We then explore the effects of risk on optimal adjustment. For simplicity we shall assume that welfare does not depend upon the levels of the policy instruments and that there is no time discounting. This simplification is only made for convenience and does not lead to any significant alteration of the results. Hence our objective function becomes w = J,” [a,y - hazy2 - frqj12 -
B&Cl dt.
(19)
POLICY
ADJUSTMENT
UNDER
67
RISK
The two policy variables g, , g, may be regarded as measures of fiscal stimulus and of monetary stimulus, respectively. National income is linearly related to the contemporaneous values of both the policy instruments and, again for reasons of simplicity, we assume that v is nonstochastic and equal to zero. Thus we have
y = 4g1 + &g, .
(20)
We assume that 8, and 19~are independently distributed through time and that, while contemporaneous values are correlated, noncontemporaneous values are uncorrelated. Substituting for y in (19) and taking expected values leads to the following expression:
where EOi = 6ii , var Bi = ui2 and p now measures the correlation between o1 and 8, . The decision maker’s problem is thus to select the values of the control variables g, and g, so as to maximize (21). The necessary conditions for optimality are given by the following pair of Euler equations: dl
-
%lgl
-
"12g2
=
-c&
d2
-
012181
-
a,,g,
=
-@2,
,
(2% (22b)
where
First, let us consider the case of certainty (g12 = uZ2 = p = 0). In that case Eqs. (22a) and (22b) define the same relationship between the optimal stationary values of the two instruments. This result seems to imply that there is an infinity of pairs of optimal stationary values lying on a straight line, but as we now show, the optimal stationary pair is uniquely determined by the initial conditions.
To see this we must solve Eqs. (22). Using the fact that the instruments
68
HENDERSON
AND
TURNOVSKY
are required to converge to finite optimal must take the formI
stationary values, the solution
g1(t) -- gl = k,e?,
(234
g?(t) - En -= k,(&c/O,d) A, = -[u,(B,4
enIf,
(2%)
+ 8,%f)/Cd]1~~,
gs = (al - a,8,&)/a,8,
.
(234 (234
(23d) is implied by (22a) and (22b). Substituting the expression for gZ into (23b) and letting t = 0 we can use (23a) and (23b) to determine & and k, . Thus, in order to assure that the optimal path for each instrument passes through its initial value, we must set gI , and therefore gZ , appropriately. The adjustment equations themselves can be written in an infinite number of ways, once gI and gZ are determined: A5
=
4,(g,
-
‘5)
+
A&g,
-
&),
Wa
d?
=
Mg,
-
8,)
f
&,(g,
-
&),
Wb)
)
We are free to choose arbitrarily one of the A’s in each of the Eqs. (24a) and (24b). If we choose A,, = A,, z 0, policy adjustment is “separable”. By this we mean that we can give the policy maker in charge of adjusting each instrument a rule which involves only deviations between actua1 and optimal values for his own instrument and X, . This type of separability is of limited value, however, because we must use information about the whole system to determine h, , El , and gZ . Thus in contrast to the case where there are no adjustment costs, both instruments must be adjusted during the transition to equilibrium. The reason for this result can be stated in a somewhat more intuitive way. Although there is only one target variable, we really have two objectives: attaining the optimal value for the target variable and minimizing adjustment costs. Since adjustment costs vary with the square of the change in 13The characteristic equation for the pair of differential Eqs. (22) is cdP - a,@,*d + @,3c)A2= 0, the only negative root of which is A, given in the text.
POLICY
ADJUSTMENT
UNDER
69
RISK
each of the instruments, it is better to adjust many instruments a little rather than one instrument more intensively at each instant of time. Thus in the presence of adjustment costs, the policy maker is better off when many instruments are available even under certainty. Returning to the case of risk, the optimal stationary values for g, and g, can be obtained by setting & = gz = 0 in (22a) and (22b). The fact that the resulting equations can be solved for unique, determinate values of g1 and gz , implies that in the long run all available instruments will be used in pursuit of a single target when risk is present. However, unlike the deterministic case, the long run equilibrium is independent of initial conditions. Since Brainard’s [l] analysis of the effects of risk on his static optimum values can be applied directly to our long-run stationary values, gl and gz , we proceed directly to the dynamic analysis. In the general case, Eqs. (22) can be shown to imply the following set of adjustment equations:l* & = &kl
- El) + Bl2k2 - g2>,
22 = B2lkl
- 21) + B22(g2 - Zz),
where B
11
=
c&h2
+
4%
+
0111
<
0,
0,
A,)-
B21 = d(Ala:
A,=-[ B =
a,,d
p + (p” 2cd - 4cdy)lp +
A,)
11129 x2=-[/3 -
a22c,
Y =
%a22
-
(p” 2cd - 4cdy)W I 1122 %X21.
The terms A1and h, are the two negative roots of the characteristic equation cd4
-
Cd+
~224
X2 +
(~11~22
-
al201211
=
0,
and are both real. As before, negative roots are chosen so as to ensure ultimate convergence to the long-run optima. It can be readily verified that choosing these two roots ensures that the adjustment system (25) is indeed stable. In this more general case, both instruments must be adjusted to I4 For details see, for example, Kaplan [4, p, 2471.
70
HENDERSON
AND
TURNOVSKY
discrepancies between optimal and actual values for both instruments.15 Assessing the effects of risk on the adjustment parameters in the general case is very tedious, so we shall only deal with the case where the two instruments are independent. When p = 0, all the Bij’s are negative, and it can be shown that the effect of increasing or2 or uz2 is to make all of the Bij’s more negative. If both instruments are on the same side of their optimal values, the effect of increases in o12 or az2 is to increase the rate at which g, and g, should be adjusted toward their optimal values for given gaps between the optimal and actual values for the two instruments. This result is analogous to the conclusion reached in the one target-one instrument case above. Since y will be subjected to random disturbances even when g = El and g, = g2 and since these random disturbances cost less if g, and g, are at their optimal values, there is an incentive to get them to their optimal values more quickly the greater the amount of expected variation. If the instruments are on opposite sides of their optimal values -a situation which is quite likely to arise at some point in the adjustment process-the results are less clear-cut but just as intuitive. The flavor of these results can be obtained by considering the case where g, = & and g, > gz . An increase in o12 or u22 means that g, should be adjusted more rapidly toward its optimal value and that g, should be adjusted more rapidly assay from its optimal value. Since g, is above its optimal value, random disturbances which make y “too high” are likely. Reducing g, below its optimal value is, therefore, beneficial because this action lowers the likelihood of random disturbances which make y “too high”. Even under the assumption that p = 0, the net effect of an increase in variability on & and g2 for given values of g, and g, is ambiguous. We can get some feel for why this is so by considering an example:
If both instruments are above their optimal values, the first three terms are IS There is one special case in which the policy system is “separable” in a more important sense than that discussed in the text above. In this case the optimal long-run stationary value and short-run adjustment of each instrument can be determined independently on the basis of parameters pertaining to that instrument alone without taking account of any parameters relating to the other instrument. Intuition might lead us to expect that this kind of separability would occur when p = 0, but this is not so. Instead, separation of the adjustment system requires 01~~= as1 = 0 or p = -&&/u102. If this condition holds, the effects of both g, and g, on Ey and Ey2 are independent of all parameters pertaining to the other instrument. Clearly, this condition will only be satisfied by chance, so that, in general, each policy maker will require information on both instruments.
POLICY ADJUSTMENT
UNDER RISK
71
negative while the last is positive and could outweigh the first three. Other cases lead to similar ambiguities. We have considered more complicated cases, but we will not deal with them here. It is straightforward, but tedious, to study parameter responses to risk when p # 0. The case of two targets and two instruments is another possible extension, and we have pursued this enough to determine that the optimal adjustment system is very similar to Eqs. (25) where both instruments are adjusted to discrepancies between optimal and actual values for both instruments.
ACKNOWLEDGMENTS We wish to thank Oliver Williamson, Paul Taubman, and particularly the referee for their comments on an earlier draft. We also appreciate suggestions by the editor which helped us to improve the exposition.
1. W. BRAINARD, Uncertainty and the effectiveness of policy, Amer. Econ. Rev. Proc. 57 (1967), 411-425. 2. R. N. COOPER, Macroeconomic policy adjustment in interdependent economies, Quart. J. Econ., 83 (1969), l-24. 3. J. GOULD, Adjustment costs in the theory of investment of the firm, Rev. Econ. Studies 35 (1968), 47-55. 4. W. KAPLAN, “Ordinary Differential Equations,” Addison-Wesley, Reading, MA, 1953. 5. K. LANCASTER, “Mathematical Economics,” Macmillan, New York, 1968. 6. R. LUCAS, “Optimal Investment Policy and the Flexible Accelerator,” Inter. Econ. Rev. 3 (1967), 78-85. 7. R. A. MUNDELL, “International Economics,” Macmillan, New York, 1968. 8. A. W. P~LLTPS, Stabilization policy in a closed economy, Econ. J. 64 (1954), 290-323. 9. H. THEIL, “Optimal decision rules for government and industry,” Rand-McNally, Chicago, 1964. 10. J. TINBERGEN, “On The Theory of Economic Policy,” North-Holland, Amsterdam, 1952. 11. C. VAN DE PANNE, Optimal strategy decisions for dynamic linear decision rules in feedback form, Econometrica 33 (1965), 307-348. 12. A. M. YAGLOM, “An Introduction to the Theory of Stationary Random Functions,” Prentice-Hall, Englewood Cliffs, NJ, 1962.
OF ECONOMIC
Notes,
THEORY 4, 58-71
Comments,
Optimal
and Letters
Macroeconomic under
of Political
Economy,
Policy
Conditions
W.
DALE Department
(1972)
to the
Editor
Adjustment
of Risk
HENDERSON
University
of Pennsylvania,
Philadelphia,
PA 19104
AND STEPHEN Department
of Economics,
TURNOVSKY
University Received
1.
J.
of Toronto,
Toronto,
Canada
July 7, 1970
INTRODUCTION
AND
SUMMARY
This paper investigates optimal paths of adjustment for macroeconomic policy instruments under conditions of risk when adjustment costs are present. The problem of how to adjust policy instruments so as to attain desired values for target variables under conditions of less than perfect certainty has received increasing attention in the literature, and the formulation adopted in the present paper attempts to synthesize the existing approaches to this questi0n.l In Section 2 we consider the one instrument-one target case. First, the optimal stationary value for the instrument is obtained; then we derive the path of optimal adjustment to this value. The stationary optimum is essentially the same as Brainard’s [I] static optimum, so that our model can be regarded as a dynamic extension of his work. The effects of changes in various parameters, particularly those pertaining to risk, on both the 1 At least three approaches to the problem can be identified. First there is the approach originally suggested by Phillips [8] and later utilized by Mundell [7] and Cooper [2] in a somewhat different context. These authors assume that the parameters of the constraints relating the target and instrument variables are nonstochastic but unknown. They then determine whether or not arbitrary, but plausible, rules of adjustment for the instrument variables lead to the attainment of desired values for the target variables. For a summary of this work see Lancaster [5]. Second, there is the formulation adopted by Theil and others [9, 111, who employ an explicitly stochastic formulation and whose main result is the “certainty equivalence theorem.” Finally, Brainard [l] has recently applied mean-variance portfolio analysis to the problem within a static context. 58 0
1972
by Academic
Press, Inc.
POLICY ADJUSTMENT UNDER RISK
59
long run equilibrium and the path of adjustment are considered. The main results are as follows. Under plausible assumptions, the effect of increased risk is usually to reduce the optimal stationary value of the instrument, although important exceptions may be found. Changes in the expected value of the coefficient of the policy variable may raise or lower the optimal stationary value of the policy variable, depending upon the size of the coefficient of variation of this parameter. The optimal adjustment rule requires the policy maker to adjust the instrument so as to close a constant proportion of the gap between the desired and actual values of the instrument variable. Increased risk raises the speed at which a gap of given size between the desired and actual values of the policy instrument should be closed. However, the effect of increasing risk on the rate at which the instrument is actually changed is more ambiguous, and one case is analyzed in detail. In Section 3 the model is extended to the case of two instruments and one target. It is shown that even under certainty a policy maker pursuing a single target must adjust both instruments toward determinate optimal stationary values when confronted with adjustment costs. In accordance with Brainard’s [I] static optimum results, determinate stationary optimum values can also be derived for both instruments when risk is present. We show that in general optimal adjustment under risk requires that each instrument be changed in response to discrepancies between actual and optimal values for both instruments. We report and discuss the effects of increases in risk upon the rates at which the instruments are adjusted toward their respective optima in an important special case.
2. THE ONE TARGET, ONE INSTRUMENT CASE
In this section the macroeconomic policy maker is assumed to have one target, say the level of national income y, and one instrument, taken to be the amount of fiscal stimulus g. We assume a closed economy where monetary authorities act passively to keep the rate of interest constant. The policy maker’s preferences are described by the quadratic utility function2
w = J,”hY -
&a2y2 + b, g - +b2g2 -
$cg2] e-Tt dt,
(1)
a The assumption of a quadratic utility function is, of course, restrictive and must be regarded merely as an approximation. Its use is justified largely on grounds of convenience. Not only does it lead to optimality conditions which are linear in all variables, but it also implies that only the means, variances, and covariances of the stochastic
60
HENDERSON
AND
TURNOVSKY
where r is the social rate of discount and the coefficients ai , bi , and c are positive.3 Increases in y are desired up to a point (y* = al/a,) because of decreasing unemployment, but beyond this point further increases in y are undesirable because they result in “too much” inflation. Likewise there is a value of g, (g* = b,/b,), preferable to all others with deviations from this value leading to reductions in welfare. The saturation level of g is presumably determined by cost-benefit calculations.” An essential feature of our formulation is that changing the level of the instrument is assumed to involve administrative and deliberative costs which can be described by a quadratic cost function5 If administrative and deliberative skills and energies are absorbed in achieving a change in g they will not be available for other tasks. The elaborate justifications and lengthy discussions associated with major changes in stabilization instruments result in less time and resources being devoted to the consideration of other important social goals. Of course, entering these other social costs through 2, as we do here, means that we have chosen a somewhat special utility function, but one we believe to be suggestive and useful. Note also that the coefficients appearing in (1) are all assumed to be independent of time and that by writing our objective function in the form of an integral of discounted social benefits we are assuming that benefits are additive over time.6 We assume that national income y is linearly related to the contemporaneous value of the fiscal stimulus g by the equation
Y(f) = Wig(t) + 4th An equation
(2)
such as (2) can be readily derived from a simple national
elements are relevant. Of course, this is not true with more general utility functions. Recently, the quadratic utility function has fallen into disrepute in problems related to portfolio theory under uncertainty, where it is shown to have rather absurd implications (such as implying that all risky assets are inferior). However, this should not mean that the quadratic utility function is illegitimate in all stochastic contexts. Since it does lead to reasonable conclusions for our model, it is felt that its use can be justifiedat least as an approximation. 3 Both g and y are functions of 1, although the functional dependence on t will usually be omitted. 4 The analysis could be extended by adding an interaction term since the costs and benefits from public projects may depend upon the level of employment. 5 This kind of approach has recently been used by Lucas [6] and Gould [3] to justify distributed lags in investment functions. As a reasonable approximation we are assuming that changing the instrument either upwards or downwards by the same absolute amount costs the same. Consequently, there is no linear term in @in the cost function. 6 Working with the one instrument-one target case, Brainard [I] proposes maximizing the utility function, W = -(y - Y*)~, where y* is the desired level of the target.
POLICY ADJUSTMENT
61
UNDER RISK
income determination model in which all expenditures depend only upon current income. The parameter 0 represents the fiscal stimulus multiplier, while u is ~9 times the total amount of nongovernment autonomous expenditure.’ Both variables are assumed to be random functions with8 E 6(r) = 6,
(34
E u(t) = I’,
(3b)
where for convenience both 8 and 0 are taken to be positive. Furthermore, we assume that both variables are independently distributed through time, that contemporaneous values are correlated with correlation coefficient p, while noncontemporaneous values are uncorrelated. Hence var O(t) = o&
E[{e(t) - O} . (e(s) - Lq] = 0 s # t;
W-4
var u(t) = a,$
E[(u(t) - 6} ’ {u(s) - E}] = 0 s # t;
WI
E[(Qt) -e> . {u(t) -C}] = pu#J,; E[{B(t) -CT} *(u(s) -V}] = 0
s # t.
04
Since we are dealing with a situation of risk, the decision maker is assumed to know the values of all parameters appearing in (3) and (4), all of which are assumed to remain constant over time.s Substituting for (2) in (1) and taking expected values, we obtain that expected utility is given by E(W) = jm [a,(Og + 6) - &l,[(B”- + q?) g” + 2~wJ,p
+ fiti>g + fi2+ %Y
0
+ b,g - &b,g” - +$] eert dt,
(5)
which is expressed as a function of the control variable g. The decision maker’s problem is thus to select g(t) so as to maximize (5). Hence the 7 Equation (2) can also be interpreted differently. For example, if g(f) is the money supply, assumed to be a government instrument, the equation may be regarded as a reduced form relationship between money supply and national income. 8 For a formal definition of a random function, see, e.g., Yaglom 612, p. 91. More informally, a random variable x(t), say, which is continuous over time, is defined to be a random function. BThus the probability distributions governing e(t) and v(t) are assumed to be stationary.
62
HENDERSON AND TURNOVSKY
optimal path for g(t) must satisfy the following Euler equation, which is a necessary condition for a maximumlo &a, - u,i;) + b, - a2ueo,p- [a,(@ $ CT;“)+ b,]g = -cg + crj.
(6)
This is a second-order differential equation in g, and the optimal stationary solution for g, denoted by g, is given byll
s = [~(~l - a,fi) + bl - w,%pl/[~,(~2 + d) + &I
(7)
An immediate question of interest is to determine how g responds to changes in the various parameters-particularly those pertaining to risk. In studying this question we shall assume that a, - a,6 > 0. Since the “saturation” level of y-that is, the level where the marginal utility of y drops to zero-is given by ul/uz , this condition asserts that the utility maximizing level of y exceeds what the expected value of y would be in the absence of policy action. This assumption seems reasonable, but it is possible to have a, - a& < 0, in which case some of our conclusions would have to be appropriately modified. Differentiating (7) with respect to ug2, uu2, p we obtain
ag/auo2= --a,PG + %p1/bTb2@2 + uo2)+ b21, ag/au,2 = - a.,42%Ca2(8” + 9”) + &I, agjap = -uzasu,/[u2(~2 -t us”) + b,].
(8)
(9) (10)
As can readily be seen, agjap is unambiguously negative, while the sign of ag/ao,2 varies inversely with the sign of p, so that the effect of risk in the exogenous expenditure depends upon how this is correlated with the multiplier. In general, the effect of ug2 on g is ambiguous, although some light is shed by considering the special case where p = 0. In that case our previous assumptions imply that ag/au,2 is unambigously negative. Since an increase in oe2 makes the marginal expected utility of g negative at the old g, g must be lowered to keep the marginal expected utility of g equal to zero. If p # 0, the results become somewhat more complicated. When p is lDBecause of the fact that the target variable y and the instrument g are related contemporaneously, we can use the simple calculus of variations approach. Had the relationship involved the derivative of y-as it would if the economic model underlying (2) involved an accelerator relationship-the problem would have had to be solved using the more recently developed methods of stochastic control theory. I1 Our result is consistent with the result Brainard obtains from maximizing the expected value of the objective function in footnote 6 if we replace a,/aa by y* and set b, = b, = 0.
POLICY
ADJUSTMENT
UNDER
RISK
63
positive, ag/ao,2 is negative so long as g is positive. The reasoning is the same as in the previous case. However, when p is negative, raising ue2 has an ambiguous effect on the marginal expected utility of g even if g is positive. So, for example, if p is sufficiently negative, an increase in co2 makes the marginal expected utility of g positive at the old g, so g must be raised to reestablish optimality. Other possibilities including the case where a, - a& < 0 can be similarly worked out. It is also of interest to consider @jag. Under certainty (assuming b, = 0, a, - a,C > 0) this expression is clearly negative, since an increase in the multiplier means that the instrument need be operated less intensively in order to achieve the optimal income. However, when risk is present this conclusion may not hold as the following expression indicates:
ag -z at7
(a, - a$) a,82(cz -
1) + 2a,28a,ovp + b&a, - a.$) - 2b,a,B
[a,(@+ ~2) + b,l”
;w
where c = se/e = coefficient of variation of 19. Considering the case where p = 0, and b, = b, = 0 we see immediately that ag/ie 2 0, according as c2 2 1. Increasing 0 has an ambiguous effect on the marginal expected utility of g. Thus if c < l-so that the instrument is relatively efficient-an increase in the instrument’s effectiveness will lead to an decrease in its use just as under certainty; the opposite is true when c > 1. A positive p increases the chances of ag:g/at?being positive; a negative p reduces them. Extension to the case of positive b, and b, is straightforward but not particularly instructive. Solving Eq. (6) and taking account of the fact that the decision maker wants g to approach g as t --j cc we arrive at the following adjustment rule for g:12 ii = &T
- E>
(12)
[c2r2 + 4c[a2(02 + oo2) + b2]]lj2 < o 2c
(13)
where h = rc 1
Hence along the optimal path, the instrument is continually adjusted so as to close some proportion of the gap between its present value and its long run optimal value. Equation (12) is very similar to the kind of adjustment proposed as plausible by Phillips and others [2,7, 81, although in our model it is a consequence of optimizing behavior. I2 This “transversality condition” teristic equation A, is relevant.
implies that only the negative root of the charac-
64
HENDERSON
AND
TURNOVSKY
There are two important questions pertaining to policy adjustment under risk. First one would like to know how the presenceof risk affects the speedat which an initial gap of a given size between actual and desired g is closed. In other words, how is X, affected by risk? Secondly, it is of interest to determine how risk affects the rate at which the instrument is actually changed for given values of g. To answer this question we must analyze the expression for g. First let us consider the effects of changes in parameters on the speed with which a given gap is closed. Differentiating h, , with respect to 8, og2,and r we obtain &I,/~0
= -2a,B[c2r2
+ 4c[a,(8”
+ ffB2) + b2]]-1’2
< 0,
ah,/&J@2 = -u2[c2r2 -t 4c[a2(d2 + ue2> + b211-1~2 < 0, ax,-
59 -
[C2r2
$
4C[a2(~2
+
uo2)
$-
6~11~‘~
2[c2r2 + 4c[a2(e2 + us”) +
-
cr
Z72111i2
>
(14) (15)
o
'
(16)
Note that all these responseshave definite signs as indicated. Recalling that h, < 0, expression (14) assertsthat the speedof adjustment increases in responseto an increase in the expected multiplier. As the instrument becomes more effective, any discrepancy between actual and desired g can be eliminated more rapidly. An increasein variability has a similar effect, although for a somewhat different reason. As risk increases, the probability of random errors causingy to take on someextreme undesirable values increases. By approaching g more rapidly, the variability in y obviously still remains, but now it is about a value nearer its optimum. In other words, if there is variability in the system, it is better to fluctuate about an optimum value than about some other point. From Eq. (16) we see that an increase in the social rate of discount has the opposite effect on X, , i.e., it tends to slow down the rate of adjustment. Hence the lessfuture deviations from the optimum matter, the slower will the adjustment proceed-again a perfectly plausible result. In addition, we can show that a given gap is closed more rapidly the higher the disutility of having g away from g and the lower the cost of changing g. In particular, if changing g is costless,then g will always be at g. It should also be pointed out that the speedof adjustment is independent of parameters describing the probability distribution of ZIand the correlation coefficient. Parameters such asp, uv2and Vhave been shown to affect g; however, they do not influence h, . We now consider how risk affects the rate at which the instrument g is changed for given values of g. At any point of time, the actual level of g is a given datum and hence is not affected by decisions affecting g. The
POLICY
ADJUSTMENT
UNDER
RISK
65
effect of risk on g is studied by differentiating the dynamic Eq. (12) with respect to the various parameters. The effect of raising go2 is shown by the following: ag/acQ = (ahl/aa,2)(g - g) - hl(ag/au,g2),
(17)
and analogous expressions are obtained for the other parameters. We see immediately that risk affects g on two counts. First, it affects the long run equilibrium [see Eq. (S)]; second, it affects the rate at which a given gap is closed [see Eq. (15)]. We can obtain an expression for 8~/&~,~ in terms of g and the parameters; however, it is extremely cumbersome, and it is not worthwhile to report it here. Instead we analyze an important special case by means of a phase diagram. This same approach could easily be applied to other cases. Consider the case where g is positive, ig/30s2 is negative, and imagine two adjustment paths of low and high risk, respectively. For simplicity we shall designate parameters relevant to the path of low risk by a subscript c for “certain” and for a path of high risk by a subscript u for “uncertain.” Figure 1 plots the dynamic Eq. (12). Since Z~/&02 < 0, gU lies to the
FIGURE
1
left of gc . ah1/%,2 < 0, so A,, has a steeper slope than A,, . It can be shown that the two lines intersect at a point g* < 0, so that the partial derivative ag/&r,2 clearly changes sign as g moves through g* from left to right. 642/4/I-5
66
HENDERSON
AND
TURNOVSKY
The following conclusions can be drawn from Fig. 1. As long as g is positive, the algebraic magnitude of fC exceeds the algebraic magnitude of gU . However, it is possible that g may be negative (in the case of a strong government surplus) and if it lies to the left of g*, the above conclusion is reversed. Notice that the instrument should not always be applied more gently under conditions of increased risk. For example, if g initially lies to the right of EC, then the absolute change in g under increased risk will exceed that under certainty. Since g is farther from its equilibrium value in the “uncertain” case, it must be subjected to a greater (negative) change. Finally, we see from the figure that if g happens to lie between gU and ge , then the policy maker will decrease g if he is acting under “uncertainty” and increase it under “certainty”. The responses of g to other parameters are easier to analyze. Thus for example,
(18) so that ;Ig/8u2 2 change in the rate risk in the amount of the correlation
0 according to whether p 5 0. Hence the algebraic of adjustment of the instrument in response to increasing of exogenous expenditure varies inversely with the sign coefficient between 13and 21.
3. THE ONE TARGET-TWO
INSTRUMENT
CASE
Perhaps the most basic result in the theory of policy adjustment under certainty is the proposition, due to Tinbergen [lo], that, when the relationships between instruments and targets are linear, policy makers need only use as many instruments as they have targets in order to achieve the desired values of all target variables. Other instruments are redundant and may be set at arbitrary levels. Brainard showed that under risk a policy maker should use all the instruments at his disposal even if he has only one target variable, since by doing so he can reduce the variance of the target for given levels of the expected value. In this section we demonstrate that when adjustment costs are present a policy maker should use all the instruments at his disposal even under certainty. We then explore the effects of risk on optimal adjustment. For simplicity we shall assume that welfare does not depend upon the levels of the policy instruments and that there is no time discounting. This simplification is only made for convenience and does not lead to any significant alteration of the results. Hence our objective function becomes w = J,” [a,y - hazy2 - frqj12 -
B&Cl dt.
(19)
POLICY
ADJUSTMENT
UNDER
67
RISK
The two policy variables g, , g, may be regarded as measures of fiscal stimulus and of monetary stimulus, respectively. National income is linearly related to the contemporaneous values of both the policy instruments and, again for reasons of simplicity, we assume that v is nonstochastic and equal to zero. Thus we have
y = 4g1 + &g, .
(20)
We assume that 8, and 19~are independently distributed through time and that, while contemporaneous values are correlated, noncontemporaneous values are uncorrelated. Substituting for y in (19) and taking expected values leads to the following expression:
where EOi = 6ii , var Bi = ui2 and p now measures the correlation between o1 and 8, . The decision maker’s problem is thus to select the values of the control variables g, and g, so as to maximize (21). The necessary conditions for optimality are given by the following pair of Euler equations: dl
-
%lgl
-
"12g2
=
-c&
d2
-
012181
-
a,,g,
=
-@2,
,
(2% (22b)
where
First, let us consider the case of certainty (g12 = uZ2 = p = 0). In that case Eqs. (22a) and (22b) define the same relationship between the optimal stationary values of the two instruments. This result seems to imply that there is an infinity of pairs of optimal stationary values lying on a straight line, but as we now show, the optimal stationary pair is uniquely determined by the initial conditions.
To see this we must solve Eqs. (22). Using the fact that the instruments
68
HENDERSON
AND
TURNOVSKY
are required to converge to finite optimal must take the formI
stationary values, the solution
g1(t) -- gl = k,e?,
(234
g?(t) - En -= k,(&c/O,d) A, = -[u,(B,4
enIf,
(2%)
+ 8,%f)/Cd]1~~,
gs = (al - a,8,&)/a,8,
.
(234 (234
(23d) is implied by (22a) and (22b). Substituting the expression for gZ into (23b) and letting t = 0 we can use (23a) and (23b) to determine & and k, . Thus, in order to assure that the optimal path for each instrument passes through its initial value, we must set gI , and therefore gZ , appropriately. The adjustment equations themselves can be written in an infinite number of ways, once gI and gZ are determined: A5
=
4,(g,
-
‘5)
+
A&g,
-
&),
Wa
d?
=
Mg,
-
8,)
f
&,(g,
-
&),
Wb)
)
We are free to choose arbitrarily one of the A’s in each of the Eqs. (24a) and (24b). If we choose A,, = A,, z 0, policy adjustment is “separable”. By this we mean that we can give the policy maker in charge of adjusting each instrument a rule which involves only deviations between actua1 and optimal values for his own instrument and X, . This type of separability is of limited value, however, because we must use information about the whole system to determine h, , El , and gZ . Thus in contrast to the case where there are no adjustment costs, both instruments must be adjusted during the transition to equilibrium. The reason for this result can be stated in a somewhat more intuitive way. Although there is only one target variable, we really have two objectives: attaining the optimal value for the target variable and minimizing adjustment costs. Since adjustment costs vary with the square of the change in 13The characteristic equation for the pair of differential Eqs. (22) is cdP - a,@,*d + @,3c)A2= 0, the only negative root of which is A, given in the text.
POLICY
ADJUSTMENT
UNDER
69
RISK
each of the instruments, it is better to adjust many instruments a little rather than one instrument more intensively at each instant of time. Thus in the presence of adjustment costs, the policy maker is better off when many instruments are available even under certainty. Returning to the case of risk, the optimal stationary values for g, and g, can be obtained by setting & = gz = 0 in (22a) and (22b). The fact that the resulting equations can be solved for unique, determinate values of g1 and gz , implies that in the long run all available instruments will be used in pursuit of a single target when risk is present. However, unlike the deterministic case, the long run equilibrium is independent of initial conditions. Since Brainard’s [l] analysis of the effects of risk on his static optimum values can be applied directly to our long-run stationary values, gl and gz , we proceed directly to the dynamic analysis. In the general case, Eqs. (22) can be shown to imply the following set of adjustment equations:l* & = &kl
- El) + Bl2k2 - g2>,
22 = B2lkl
- 21) + B22(g2 - Zz),
where B
11
=
c&h2
+
4%
+
0111
<
0,
0,
A,)-
B21 = d(Ala:
A,=-[ B =
a,,d
p + (p” 2cd - 4cdy)lp +
A,)
11129 x2=-[/3 -
a22c,
Y =
%a22
-
(p” 2cd - 4cdy)W I 1122 %X21.
The terms A1and h, are the two negative roots of the characteristic equation cd4
-
Cd+
~224
X2 +
(~11~22
-
al201211
=
0,
and are both real. As before, negative roots are chosen so as to ensure ultimate convergence to the long-run optima. It can be readily verified that choosing these two roots ensures that the adjustment system (25) is indeed stable. In this more general case, both instruments must be adjusted to I4 For details see, for example, Kaplan [4, p, 2471.
70
HENDERSON
AND
TURNOVSKY
discrepancies between optimal and actual values for both instruments.15 Assessing the effects of risk on the adjustment parameters in the general case is very tedious, so we shall only deal with the case where the two instruments are independent. When p = 0, all the Bij’s are negative, and it can be shown that the effect of increasing or2 or uz2 is to make all of the Bij’s more negative. If both instruments are on the same side of their optimal values, the effect of increases in o12 or az2 is to increase the rate at which g, and g, should be adjusted toward their optimal values for given gaps between the optimal and actual values for the two instruments. This result is analogous to the conclusion reached in the one target-one instrument case above. Since y will be subjected to random disturbances even when g = El and g, = g2 and since these random disturbances cost less if g, and g, are at their optimal values, there is an incentive to get them to their optimal values more quickly the greater the amount of expected variation. If the instruments are on opposite sides of their optimal values -a situation which is quite likely to arise at some point in the adjustment process-the results are less clear-cut but just as intuitive. The flavor of these results can be obtained by considering the case where g, = & and g, > gz . An increase in o12 or u22 means that g, should be adjusted more rapidly toward its optimal value and that g, should be adjusted more rapidly assay from its optimal value. Since g, is above its optimal value, random disturbances which make y “too high” are likely. Reducing g, below its optimal value is, therefore, beneficial because this action lowers the likelihood of random disturbances which make y “too high”. Even under the assumption that p = 0, the net effect of an increase in variability on & and g2 for given values of g, and g, is ambiguous. We can get some feel for why this is so by considering an example:
If both instruments are above their optimal values, the first three terms are IS There is one special case in which the policy system is “separable” in a more important sense than that discussed in the text above. In this case the optimal long-run stationary value and short-run adjustment of each instrument can be determined independently on the basis of parameters pertaining to that instrument alone without taking account of any parameters relating to the other instrument. Intuition might lead us to expect that this kind of separability would occur when p = 0, but this is not so. Instead, separation of the adjustment system requires 01~~= as1 = 0 or p = -&&/u102. If this condition holds, the effects of both g, and g, on Ey and Ey2 are independent of all parameters pertaining to the other instrument. Clearly, this condition will only be satisfied by chance, so that, in general, each policy maker will require information on both instruments.
POLICY ADJUSTMENT
UNDER RISK
71
negative while the last is positive and could outweigh the first three. Other cases lead to similar ambiguities. We have considered more complicated cases, but we will not deal with them here. It is straightforward, but tedious, to study parameter responses to risk when p # 0. The case of two targets and two instruments is another possible extension, and we have pursued this enough to determine that the optimal adjustment system is very similar to Eqs. (25) where both instruments are adjusted to discrepancies between optimal and actual values for both instruments.
ACKNOWLEDGMENTS We wish to thank Oliver Williamson, Paul Taubman, and particularly the referee for their comments on an earlier draft. We also appreciate suggestions by the editor which helped us to improve the exposition.
1. W. BRAINARD, Uncertainty and the effectiveness of policy, Amer. Econ. Rev. Proc. 57 (1967), 411-425. 2. R. N. COOPER, Macroeconomic policy adjustment in interdependent economies, Quart. J. Econ., 83 (1969), l-24. 3. J. GOULD, Adjustment costs in the theory of investment of the firm, Rev. Econ. Studies 35 (1968), 47-55. 4. W. KAPLAN, “Ordinary Differential Equations,” Addison-Wesley, Reading, MA, 1953. 5. K. LANCASTER, “Mathematical Economics,” Macmillan, New York, 1968. 6. R. LUCAS, “Optimal Investment Policy and the Flexible Accelerator,” Inter. Econ. Rev. 3 (1967), 78-85. 7. R. A. MUNDELL, “International Economics,” Macmillan, New York, 1968. 8. A. W. P~LLTPS, Stabilization policy in a closed economy, Econ. J. 64 (1954), 290-323. 9. H. THEIL, “Optimal decision rules for government and industry,” Rand-McNally, Chicago, 1964. 10. J. TINBERGEN, “On The Theory of Economic Policy,” North-Holland, Amsterdam, 1952. 11. C. VAN DE PANNE, Optimal strategy decisions for dynamic linear decision rules in feedback form, Econometrica 33 (1965), 307-348. 12. A. M. YAGLOM, “An Introduction to the Theory of Stationary Random Functions,” Prentice-Hall, Englewood Cliffs, NJ, 1962.