This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research

Volume Title: Annals of Economic and Social Measurement, Volume 4, number 2

Volume Author/Editor: NBER

Volume Publisher: NBER

Volume URL: http://www.nber.org/books/aesm75-2

Publication Date: April 1975

Chapter Title: A Comparison of Three Control Algorithms as Applied to the Monetarist-Fiscalist Debate

Chapter Author: Andrew B. Abel

Chapter URL: http://www.nber.org/chapters/c10396

Chapter pages in book: (p. 239 - 252)

La.a..mae

4unu/s of !conoinn cult1 Sun-/al .!eucsuuu-iflunl. 4 2. '175

A (UMI'AR!SON Ot l'HRI:± CONtROL ALGORIFHMS ASAPPlIED TO THE MONETARIST-FISCALIST I)EBATE

nv ANDREW B. AIIEI.*

Ibis pajier employs an optunal control tramcicu,rk to ili?cJli-:e thu ru/ui lit-u el/ectireness ui ?nunlc'tar ci 'tul/isuui policies. The econuuniv is mu;uleled (ui a lust- .snnph' rn-u-equation hilt-Or ci i-nuiii:iu iuushl uuiili tinmactIc suppli and golerumeni ex;emlitnres as two inslrurnents. Three control algorithms. whirl, dif/erin their lreatnu'nts of unierlainic all'! h'uirning_ are app//i'd I,, this nuudiI to cuihuhite optima! puilieius and,nu,i,pnuni expected wi//curt- cOsts. With nunimuni expected welfare test .serring as the u-riuuriunu of c//itt/ui-iu'ss. ii appears iliac fiscal polo v is somewhat nuuurt' c//taut-c than ,nonetuurv pu/ui it-tub ru's putt 0' thuquadratic ice//ore than/un coup/u ved in this pulper, although using hot/i policies ru gut/rut is much inturu'

IJ cci tic' tiiaii Usitug cit/icr pohic alone.

1. INrRoDuc-noN

In this paper we shall use an optimal control framework to examine the relativeeffectiveness of monetary and fiscal policies for the purpose of controlling themajor macroeconomic aggregates. In Section 2. we shall present a very simplelinear macroeconomic model with additive Gaussian disturbances. After brieflydescribing three control algorithms in Section 3. we shall, in Section 4. applythese three control algorithms to our simple model. Monetary policy will berepresented by the money supply and fiscal policy will be represented by govern-ment expenditures. In evaluating the effectiveness of a given instrument, we shalldesignate that instrument as a discretionary instrument and the other instrumentas a passive instrument, and then solve an optimal control problem. The valuesof the discretionary instrument are determined subject to feedback control, butthe values of the passive instrument are constrained to change at a constant rateover the planning horizon. In addition, we solve the control problem with bothinstruments assumed to be discretionary. Comparison of the expected welfarecosts in the three situations serves to evaluate the effectiveness of each discrctionarinstrument.

Three different algorithms will be used to perform our analysis of the relativeeffectiveness of monetary and fiscal policies in order to determine whether ourresults are sensitive to the choice of control algorithm employed. The threealgorithms presented in this paper differ in their treatments of !eaining Method Iis a certainty equivalence control algorithm formulated by Chow (1972). Theassumptions of certainty equivalence preclude the possibility of learning byassuming that the parameters ol the linear model are known with ccrlaintvMethod II. which is presented by Chow ( l973a) recognizes the uncertainties inthe parameters of the model but ignores the possibility ol learning. Method Ill.which is a dual adaptive control algorithm presented by Chow (197k). anticipatesthat learning will occur through the re-estimation of the unknown parameters ofthe linear model as additional observations are obtained with the passage of time.Method Ill is closest to being optimal and contains method I and method II asspecial cases.

* ! would like to express ray most sincere thanks to Professor Grcguirv C. Chow. ni; thcis uduisorfor generously sharing his time and his ideas with me.

239

2. A Sr.%1I'l.F MACROP('ONOMR MoiiFor the policy analysis of this paper, we shall employ a very simple aggregative

model. It is based on real quarterly data covering [he pci iod fro,ii 1954/I to 1963f1Vwhich corresponds roughly to the period between the end of the Korean Warand the beginning of heavy United States involvement in Vietnam. It Consists ofonly two cndogenous target variables, consumption (C,) and investment (I,), andtwo instruments, government expenditures (Er) and the money supply (M,).We assume that in the short-run, government authorities can control E, and M,in real terms since prices do not change rapidly enough to seriously offset theiractions. Over the time period covered by our data, the rate of inflation was lowenough to make this assumption plausible.

Our model is based on a closed economy. Desired consumption is a linearfunction of GNP, and the realized period-to-period adjustment in consumptionis subject to a partial adjustment factor:

(2.1) C, = (IC,_1 + bi, + hE, + d.

The structural equation for investment is based L1Ofl a modification olSamuelson'sprivate consumption accelerator. We posit that the desired level of the capitalstock is a linear function of consumption and that the realized adjustment of thecapital stock is subject to a partial adjustment factor. Since gross investment, I,,is defined as K, - (I - D)K,_ where D is the depreciation rate of the capitalstock, we have

(2.2) I, = eC, - (1 - D)eC,. + fl, + g.In addition, we assume that the level of gross investment is linearly related to themoney supply in order to capture some of the effects of interest rates upon invest-ment:

(2.3) I, cC, - (1 - D)eC,.. + fl, + liM, + g.The estimated reduced form equations corresponding to the structural equationsare

R2 = 0.8749

D W = 1.7582.Note that each of these estimated equations has a high value of R2. In addition.the Durbin-Watson statistic, although biased toward 2.0 because of the laggedendogenous variable, does not suggest significant serial correlation in eitherequation.

240

(2.4) C, 0.9266C,1 - 0.0203l, -- 0.3190E, + 0A206M, - 63.2386:(0.0534) (0.0916) (0.1389) (0.1863) (25.7719)

R2 = 0.9958

D W = 1.7084(2.5) 1, = 0.1527C,_, + 0.38061, - 0.0735E, - l.5389M, 210.8994:

(0.0781) (0.1339) (0.2031) (0.2724) (37.6899)

A criticism that may he raised against the above model is that it includesonly the current values of M, and E, among the explanatory variables. 1-lowever.concerning the lagged or delayed effects of M and E,. our model implicitlyassumes a lag structure with geometrically declining weights for M, and E,because the lagged endogenous variable appears as an explanatory variable ineach equation. In this paper, we do not explore more complicated lag structures.

3. THE AuIoRI111ls I

The three algorithms presented in this paper are applicable to linear stochasticdiscrete-time econometric models with unknown parameters and additiveGaussianerrors. We shall write the model as a first-order linear difference equation

(3.1) y7 = Ay.1 + Cx, ± h, 4- 'rwhere A, C. and b, are random parameters whose values will be estimated usingthe Bayesian techniques presented by Chow (1973a). The vector i'7 is a stackedvector containing values of the endogenous variables and the instruments, x, is avector of instruments, b, is a vector which models the effects of the noncontrollableexogenous variables, and e, is a vector of random variables such that e, N(O, E).The e, are assumed to be serially uncorrelated and uncorrelated with the randomparameters A, C, and b,.

The objective of each of our three control algorithms is to minimize theexpected value of the following qudratic welfare cost function

(3.2) W = (v7 a,)'K,(y - a,),

where T is the length of the planning horizon, a, is the target value of v7. and K,is a weighting matrix. Observe that (3.2) may be rewritten as

T T

(3.3) W = v7'K,y ± vk, ± constant,

where k, = - K,a,. We solve this problem using the method of dynamic program-rning. L.et E, it', denote the expected welfare cost from period up to and includingperiod T. with the subscript t - I indicating that the expectation is conditionalon information available at the end of period t - I. We first minimize ET IttTwith respect to IT. Letting H = K and h = k1. we obtain

(3.4) ET 1i- = 1-_ 1(-yHV + VT/IT) + constant.

Substituting (3.1) into (3.4) and partially differentiating with respect to xi., weobtain the following feedback control equation, which yields the optimal valueof

.*- T - T)T-1 -- T.

A more complete discussion of these algorithms is presented in Andrew B. Abel, "A ComparL;iof Three Optimal Control ,-llgorithnis as Applied to the Alotie1arisi-fi,aljsi Debate.'' Senior Thesis.Princeton University. Department of Economics, 1974.

24!

(3.5)

0

whereC, (I, ('!11C) 'tL, (' lI,,.I)

and

= (!T ,o) '[(1T Lj1T6I) + (L ()h.1 1.Note that the feedback control equation is not linear in v since the parameters

G1 and g. are functions of the posterior density of .4. C. and b1 at the end of period

T 1. which is a function of t' - . - 2 After substituting into (3.4) tobtain theoptimal expected welfare cost for the last period, we then approximatethe function 1T by a modified second-order Taylor series expansion to obtain

T T

(3.6) .'Q,* +37(1,' +

Using Bellmans principle of optimality. we minimize F7. n' with respect tounder the assumption that the optimal value of x1. i.e,.\1.. will he selected

in period T. Hence. we seek to minimize

(3.7) 13T - = E. 7(tT K., iYT -- + k, i ± iT) + constant.

Substituting the Taylor series approximation for i' and combining like terms in- within the expectation operator. we obtain

(3.8) ET. zT- -= E1 _,(4v' 1H, - - + v'_ ,h, ) + constant''.where "T - I = + QL. 1 1- q . and constant" absorbs those termsin (3.6) which arc not dependent upon x1 - and v- . It should he observed that(3.8) is identical in form to (3.4). Hence, we may solve for ST.. in the same mannerthat we solved for 'T' This backward induction procedure is repeated until weobtain values for . and . This algorithm, which we shall call method 111. is adual adaptive control algorithm. It gives only approximate solutions since itinvolves a quadratic approximation about a somewhat arbitrary tentative path.3

In the certainty equivalence algorithm (method I). it is assumed that the valuesof the random parameters. A. C. and h, are equal. with certainty, to their respecLieconditional expectations at time 0. i.e..A. C'. and h,. 1-lence. the parameters ofthe feedback control equations are G7 = (C'H.,.Cy '(C'H.4) and g1 =tC'HT( '[(C'I-iTbTI + C/IT]. The feedback control equation is strictly lineatin ' and the function aT is truly quadratic in . Therefore, there is no needto approximate ti' by a second-order Taylor series expansion and thus thecertainty equivalence solution to the control problem is exact.

Method II. which is another special case of method Ill, takes account ofuncertainty in the parameters but does not anticipate future learning. In method II.all conditional expectations are evaluated at time 0 so that the coeflicients of the

2 In the calculations summariicd later we employ a standard '1a br series with cross-partials. and3.2) includes terms of the form ft* -- a,)'F., -- a,). }Iowecr. se let K, , = 0 for ill t

In our calculations we obtain the tentative path by using the certainI equis alence lgorithni todetermine .,. and then apply the esiimated model, without random disturbance, to .,. to generate y.fori = I '1-

242

feedback control equation are 6, = - (E((111C) EOCHT.* and g,(L0C'H1C) 'I(EoC'JJThT) + (E0C')hT]. As in method I, the parameters of thefeedback control equation are independent of v .. and hence T is a linearfunction of y . Thus ItT is truly quadratic in r and method II yields anexact solution to the modified control problem.

4. POLICY ANALYSIS USING TIlE Si\1iI.E MoI)It.

Studies of the relative effectiveness of monetary and fiscal policy often focuson the size of long-run and short-run multipliers of the monetary and fiscalinstruments, e.g., Kmenta and Smith (1973). However, Brainard (1967) arguesthat an examination of the minimum expected welfare cost attainable with agiven set of instruments is a more meaningful approach to the question of theeffectiveness of the instruments than is an examination of the multipliers of theseinstruments. Indeed, from the point of view of maximizing social welfare, anyrelevant features of the multipliers will be reflected in the minimum expectedvalue of the we!fare cost function and hence multiplier analysis is unnecessary,if not misleading. In this paper we shall compare the minimum expected welfarecost attainable using only a discretionary monetary instrument with the minimumexpected welfare cost attainable using only a discretionary fiscal instrument.However, before solving the control problem using only one discretionaryinstrument at a time, we solve the control problem using both M, and E, asdiscretionary instruments subject to feedback control. We rewrite the reducedformequations(2.4)and(2.S)asv7 = Ay'_1 + Cx, + b, -i- e,,where

(4.1) r7' = (C,. I,. E,. M,).

is a 4 x 4 matrix containing the 2 x 2 matrix A1.

Co--I

is a 4 x 2 matrix containing the 2 x 2 matrix C0.

= (E,. M,)

is the vector of instruments.

is a 4-vector containing the 2-vector ,.

A1 0.4= --I---

0 0

is a 4-vector containing the 2-vector ,. and

I;,

= o

243

= ( 0

Before proceeding with the application of the control algorithms, we mustspecify the following parameters of the welfare function: (1) 1. the number ofperiods in the planning horizon; (2) a1, the target values of v,*; and (3) K,, theweighting matrices. We shall solve the control for a 6-period planning horizon,i.e., T = 6. In order to select appropriate target growth rates for C, and I, weexamine the historical percentage growth rates shown in Table i. It should henoted that the growth rate for 1, for the 11 quarters ending with 1963/tv is muchhigher than the growth rate for the 40 quarters ending with 1963/tV becauseinvestment was near a cyclical low in 1961/Il. With these historical growth ratesin mind, we somewhat arbitrarily choose target growth rates of 1.25 percent perquarter for C, and !.

TABLE IQUARTERLY GRowTh RATES ()

For the weighting matrices in the welfare function, we set

i ;0K,

= ,= 1,.., T,

where I is the 2 x 2 identity matrix. Note that since the instruments are assignedzero weight in each K1, they do not explicitly appear as arguments of the welfarefunction. Since the ultimate objective of our analysis is to compare the relativeeffectiveness of monetary and fiscal policy in reaching given targets over time, weshall examine the welfare cost net of the costs directly associated with the instru-ments. We will, however, examine the stability of the instruments in each solutionto make sure that they do not fluctuate excessively.

Method I is applied to the two-instrument, two-target control problem, with

and

Co =0.3190

- 0.0735Then method H and method III are applied to the same problem. For the quadraticapproximation of method III, a deterministically generated tentative path derivedfrom the solution of method I is used. The solutions to this problem by the threealgorithms are presented in Table 4.

In order to investigate the effectiveness of one instrument alone, we shallassume that the instrument under consideration is a discretionary instrument

Since E, and At, hae zero weight in the welfare function, it is not necessary to specify targets forthese variables.

244

D.4206

1.5389

Period C, I

l954l to 1963 1V1961/Il to 1963/tV

0.91

1.10

1.14

2.61

0.9266 0.0203 b= 63.23860.1527 0.3806, 210.8994

the values of which are chosen by the policy maker subject to feedback control.It is assumed that the values of the other instrument are determined by a passivepolicy of a constant percentage change per quarter. In the notation of (3.1), thediscretionary instrument is represented by the scalar x, and the passive instrumentis modeled as a noncontrollable exogenous variable which is absorbed in the valueof b1. The solution to the control problem will be sensitive to the values of b1,

1.....T, and hence the values of the passive instrument must be chosenjudiciously. To determine the values of the passive policy variable, we solve thesix-period, two-instrument control problem in which each of the instruments isconstrained to change at a constant rate throughout the planning horizon Thesolution to this problem, calculated under the assumption of certainty equivalence,is shown in Table 2. There is no a priori reason to believe that the growth rateobtained in this manner for each instrument will be optimal when the values ofthe other instrument are chosen subject to feedback control. A better approachto selecting an optimal growth rate for the passive instrument would be to solvethe control problem repeatedly with one discretionary instrument and one passiveinstrument, allowing the growth rate for the passive instrument to vary in succes-sive computations of the solution. The optimal growth rate for the passive instr'i-ment is the growth rate for which the optimal expected welfare cost is minimized.

TABLE 2CERTAINrY EQuIVALENCE SoruTioN TO CONTROL PROHIESI WHEN BOTH lNTRUSIENrS ARE

PASSIVE

Note: This solution was obtained using the OPTCDIAG option of the certainty equi-valence program described in Douglas R. Chapman and Gregory C. Chow. "Optimal ControlPrograms: User's Guide," Econometric Research Program. Princeton University. ResearchMemorandum No. 141, May. 1972. Slight inconsistencies may appear above as a result ofrounding since the program used a percentage growth rate with 6 decimal places. Also notethat i, = 68.3074.

In lieu of performing an extensive search to determine the optimal growth ratefor each instrument when the other instrument is discretionary,we merely examinetwo other growth rates for each instrument to check whether the growth ratesshown in Table 2 appear to he approximately optimal. The optimal welfare Costs

c shown in Table 3 were obtained from the solution, by method I. to the controlproblem in which the passive instrument grows at the given rate and the discretion-

e ary instrument is determined by feedback control. Note that for each instrument.the value of obtained using the growth rate from Table 2, is smaller than thevalues of s' obtained using growth rates 0.1 percent larger and 0.1 percent smallerthan the growth rate in Table 2. This result lends some credence to the assertionthat, for each instrument, the growth rate in Table 2 reasonably approximatesthe optimal rate when the other instrument is determined by feedback control.

245

riodInstrument

1 2 4 5 6 RateCh:ngeper Period

E, (billions)M1 (billions)

110.9

143.6112.4143.4

113.9143.3

115.4143.1

116.9

142.9118.4142.7

+ 1.313--0.l22°

$

TABLE 3OPTItAL Exp1:crI:D \Vui EARL Cosis Method I) Wiit' Oa ksiRiMFNI

is PASSIVE AND ONE INSTRUMENF is I)IssIuioNARY

E, is passive

Orowth rate of E,

is pas'Ive

(rosOh rtIc of

The control problem is now solved using methods I. II, and Ill under theassumption that E1 is a discretionary instrument and M, is a passive instrumentexogenously set equal to the values given in Table 4.2. This procedure is thenrepeated with M as the discretionary instrument and E as the passive instrumentThe results of the control computations for period I arc presented in Table 4Let w(P) be the optimal expected welfare cost function from period I to period Twhere i I. II. 1lt refers to the algorithm employed and P P = {E. M,refers to the set of discretionary policy variables used in the application of thealgorithm. Note that for each ie{l, 11, IH}, wP) = w1({E. M1}), whichis an illustration of the well-known fact that in a control problem with two targets.a lower optimal expected welfare cost is attainable using two instruments subjectto feedback control than by using only one ofthese instruments subject to feedbackcontrol. More significant for our economic analysis. however, is the result that foreach I. w1({EJ) < w1( Ma). Therefore, assuming that the economy of the UnitedStates is appropriately modeled by (2.4) and (2.5), fiscal policy as represented byE1 is somewhat more effective with respect to the given welfare function than ismonetary policy represented by M:. We note, however, that this difference issmall, especially when we allow for uncertainty in method II.

To study our solution more closely, the value of w,(P) can be decomposedinto a deterministic welfare cost and a stochastic welfare cost. To compute thedeterministic welfare cost, we first generate a deterministic time patti for each ofthe endogenous variables by assuming that each parameter of the linear model(3.1) is equal to its point estimate at time 0. and that e1 = 0. for t = I.....T.The deterministic welfare cost is weighted sum of squared deviations of the deter-ministic time paths of the endogenous variables from their respective targets.The stochastic welfare cost is due to the randomness in v and results from theadditive stochastic disturbance e,. Substituting the optimal value of x1 from (3.3)into (3.1). we obtain

(4.2) = (A -t- CG,)1 + Cg, + h1 + e,.

Assuming that the covariance matrix of e, is E for > 0. it can he shown thatthe covariance matrix ofy is given by the recursive formula

(4.3) = (A + It/I ± CG1)' + ,..246

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The stochastic welfare cost is equal to

(4.4) = tr (K,;),

where 1' is the estimate of E based on the estimate of ,. at the current time.In Figure 1 we present the target time path for C and the deterministic time

paths for C, using the instrument sets {E,) and { M,}. In addition to the deterministictime path for each instrument set, we present the values of C, one standarddeviation above and below the deterministic time path of C,. For each time period,

38

382

380

378

5(6

a)372

370

566

3 6

362

36o

1=i

i.rcjL-- 1n LruncnL set (I: )

1nstrumnt set {1j

1 2 3 5 Ti t.Figure 1 Expected Time Paths of C, with Standard Deviation Bands (Obtained from Certainty

Equiva'ence SoIutios)

248

S

the vertical distance between the deterministic time path of C, fora given instrumentset and the target time path of C, essentially measures the square root of thc deter-rninistic cost attributable to C, . Similarly, for each period, thc standard deviationof C, around its deterministic time path reflects the stochastic cost attributableto C, in that period. Note that the deterministic time path of C, for the instrumentset {E,} is generally above the target time path whereas for the instrument set{M,} it is generally below the target time path. If the deterministic cost compriseda major portion of the expected welfare cost, we might have to consider the givenquadratic welfare cost function to be inappropriate because it assigns costs to theoverachievement of the targets for C, through the use oitlie instrument set E, aswell as to the underachievement of the targets for C, through the use of {M,.However, we note that for each instrument set, the standard deviation of thestochastic variation around the deterministic time path of C, far outweighs thedeterministic "standard deviation" of the deterministic time path around thetarget time path. Hence, the adverse effects of assigning deterministic costs toexpected positive deviations from the target values may be neglected since theyappear to be unimportant. We also note that the standard deviation band aroundthe deterministic time path obtained using {E,} lies within the standard deviationband around the deterministic time path obtained using {M,}, except for period 1.Hence, the stochastic cost attributable to C, is smaller for {F,} than for {M,}.Figure 2 is analogous to 1 except that the endogenous target variable is I,. As inFigure 1, we observe that the stochastic welfare cost is much larger than thedeterministic welfare cost.

Table 5 summarizes the results presented in Figures 1 and 2. In this table,the deterministic welfare cost is expressed as an average over the six-periodplanning horizon. The stochastic welfare cost for each target variable is theaverage variance of that variable around its deterministic time path. The lasttwo columns of Table 5 present the square roots of the corresponding values inthe first two columns of the table and represent deviations in terms of 1958 dollars.It is clear from Table 5 that the total cost attributable to 1, is greater than the totalcost attributable to C,. and the stochastic cost is much greater than the deter-ministic cost for each instrument set.

Since the instruments receive zero weights in the welfare function, we shallbriefly examine the dynamic characteristics of the time paths of the instruments(derived from method 1)10 determine whether they are highly volatile. We notethat for the instrument set {E,, M,}, the period-to-period fluctuations of E, alongits deterministic time path are all less than 1.3 billion 1958 dollars, and for {E,all of the deterministic changes are less than 2.5 billion 1958 dollars. Furthermorefor each instrument set, the standard deviation of E, around its deterministictime path remains fairly stable and is less than 5.3 billion 1958 dollars in eachperiod. For each of the instrument sets {E,, M,} and {M,}, the deterministicpeiiod-to-piiod fluctuatiotis of M, are all less than 0.1 billion 1958 dollars.The standard deviation of M, remains fairly stable at about 0.9 billion 1958 dollarsfor {E,, M,} and about 1.1 billion 1958 dollars for {M,}. Hence, it appears thatfor each instrument, neither the deterministic period-to-period fluctuations nor

This expected welfare Cost ignores ihe faior of in 13.2).

249

96

95

914

95

92

91

90

89

88

8

Trqet

- - -- Instuwent set

-- IflstrIJ:!eflt. set

86 (1 2 6 Tirset

Figure 2 Expected time paths ofI with standard deviation bands jobtained from certainly euuivalencesolutions)

TABLE 5AVRAE VEI.FARI Cosis PIR Pt RIOt)

Note: Costs exclude factor of which appears in (2.4 and 4.6).

250

Variance (Welfare Cost)(billions of 1958 dollars)2

Standard Deviations(billions of 1958 dollars)

Instrument TargetSet Deterministic Stochistic Total Deterministic Stochastic

)E,} C 0.028 3.816 3.844 0(67 I 9531, 0.696 10.178 10.874 0.834 3.190

Total 0.724 13.994 14.718

C,I,

0.072 7.2730,016 8.656

7.3458.672

0268 2.6970.126 2.942

Total 0.088 15.929 16(17

)E. Af( C,1,

0 3.7750 8.074

.7758.074

0 1.9430 2.841

Total 0 11.849 11.849

the stochastic variation arotind the deterministic time path is large enough topresent serious problems of implementation.

We observe in Table 4 that for a given instrument set, the coefficients of thefeedback control equatioti at e subject to considerable variation across algorithmswith the introduction of uncertainty and the anticipation of learning. However,it should be noted that the optimal values of the instrument do not appear to hevery sensitive to the presence of uncertainty or to the anticipation of learning.In Table 6. we present the percentage variation across the three algorithms of theoptimal first-period settings of the instruments for each of the three controlproblems. Note that when the policy maker treats both E1 and M1 as discretionaryinstruments, there is an extremely small percentage variation in . across thethree algorithms. This result suggests that for the purpose of determining theoptimal values of E1 and M1. it makes little difference whether the effects ofuncertainty and learning are considered.

TABLE 1,iIR(rtAGI ARiA 1105 iS X A(R0 I III 1111(11

.Ar(

restrictive assumptions of certainty equivalence. We noted that although theintroduction of uncertainty may significantly change the coefficients of the feed-back control equation, the optimal first-period policy is rather insensitive touncertainty in the parameters of the linear model. The implication of this resultfor policy formulation is that we may fairly accurately determine the optimalvalues of E1 and M by any of the three algorithms discussed in this paper.

Mossaduiseits institute' 0/ Technoiogi'

BlItI.IOGRAPFIY

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