Several recent papers have usefully emphasized the inefficiency that arises from discretionary monetary policymaking, relative to optimal policy from a 'timeless perspective,' in macroeconomic models with forward-looking private behavior. The inefficiency in question is in terms of average outcomes of the conditional expectation of a policy objective that reflects the discounted present value of current and future period losses (which involve squared deviations of inflation and output from specified target levels). In the literature, most of the analysis has been conducted in an optimizing model that features a Calvo-Rotemberg price adjustment equation that includes a 'cost-push' shock term. This literature suggests that policy, which keeps inflation equal to a negative multiple of the change in the output gap, is optimal with respect to the criterion mentioned above -- the unconditional expectation of the policymaker's objective function. Results reported here show, however, that this is not the case -- that an alternative policy rule, suggested by the approach of 'policy design' rather than by 'optimal control,' delivers superior results.

Bennett C. McCallum

Christian Jensen

English

Research Papers in Economics

NBER WORKING PAPER SERIES
THE NON-OPTIMALITY OF PROPOSED MONETARY POLICY RULES UNDER
TIMELESS-PERSPECTIVE COMMITMENT
Christian Jensen
Bennett T. McCallum
Working Paper 8882
http://www.nber.org/papers/w8882
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2002
The authors are indebted to Edward Nelson, Lars Svensson, and Alexander Wolman for helpful suggestions.
The views expressed herein are those of the authors and not necessarily those of the National Bureau of
Economic Research.
© 2002 by Christian Jensen and Bennett T. McCallum.  All rights reserved.  Short sections of text, not to
exceed two paragraphs, may be quoted without explicit permission provided that full credit, including ©
notice, is given to the source.The Non-Optimality of Proposed Monetary Policy Rules Under
Timeless-Perspective Commitment
Christian Jensen and Bennett T. McCallum
NBER Working Paper No. 8882
April 2002
JEL No. E52, E58, E30
ABSTRACT
Several recent papers have usefully emphasized the inefficiency that arises from discretionary
monetary policymaking, relative to optimal policy from a “timeless perspective,” in macroeconomic
models with forward-looking private behavior.  The inefficiency in question is in terms of average
outcomes of the conditional expectation of a policy objective that reflects the discounted present value
of current and future period losses (which involve squared deviations of inflation and output from
specified target levels).  In the literature, most of the analysis has been conducted in an optimizing model
that features a Calvo-Rotemberg price adjustment equation that includes a “cost-push” shock term. This
literature suggests that policy, which keeps inflation equal to a negative multiple of the change in the
output gap, is optimal with respect to the criterion mentioned above—the unconditional expectation of
the policymaker’s objective function.  Results reported here show, however, that this is not the case—that
an alternative policy rule, suggested by the approach of “policy design” rather than by “optimal control,”
delivers superior results.
Christian Jensen Bennett C. McCallum
Graduate School of Industrial Administration Graduate School of Industrial Administration
Carnegie Mellon University Carnegie Mellon University
Pittsburgh, PA 15213 Pittsburgh, PA 15213
(412) 421-6064 and NBER
cjensen@andrew.cmu.edu (412) 268-2347
bm05@andrew.cmu.edu 1
  Several recent papers, some quite prominent,
1 have usefully emphasized the 
inefficiency that arises from discretionary monetary policymaking, relative to optimal policy 
from a “timeless perspective,” in macroeconomic models with forward-looking private 
behavior.  The inefficiency in question is in terms of average outcomes of the conditional 
expectation of a policy objective that reflects the discounted present value of current and 
future period losses (which involve squared deviations of inflation and output from specified 
target levels).  “Forward-looking” in the statement above means that expectations of future 
variables (e.g., inflation) appear in structural relations representing private behavior.  In the 
literature in question, most of the analysis has been conducted in an optimizing model that 
includes a price adjustment equation of the Calvo-Rotemberg type, often referred to as a New 
Keynesian Phillips Curve, that includes a “cost-push” shock term. 
  Policy from a timeless perspective reflects a type of commitment, on the part of the 
optimizing monetary policymaker, that avoids influences from the conditions that happen to 
prevail at the date at which the posited type of policy behavior begins. It is therefore arguably 
more credible than policy behavior that has the central bank planning to behave differently in 
the policy’s initial period than in those to follow (as with ordinary commitment choices).  It 
has the feature of being time-consistent from its own perspective, although not from the 
viewpoint of Kydland and Prescott (1977).
2 The literature seems to suggest that policy 
satisfying stated conditions—exemplified by (5) below—is optimal with respect to the 
criterion mentioned above, the unconditional expectation of the policymaker’s objective 
function.  It has recently been shown by Jensen (2001a), however, that this is not the case—
                                                 
1 Among the more prominent papers are those of Clarida, Gali, and Gertler (1999), H. Jensen (1999), and 
Woodford (1999a, 1999b, 2000, 2001).  Other significant items include Dennis (2001), Gianonni (2001), 
McCallum and Nelson (2000), Steinsson (2000), Svensson and Woodford (1999), Vestin (2000), and Walsh 
(2001).  2
that an alternative policy rule, suggested by the approach of “policy design” rather than by 
“optimal control,” delivers superior results.
3  The magnitude of improvement is not large, for 
realistic parameter values, but is distinctly non-zero.  The purpose of the present note is to 
provide a compact description and demonstration of this particular result of Jensen’s (2001a). 
  Following Clarida, Gali, and Gertler (CGG, 1999), H. Jensen (1999), Woodford 
(1999a, 1999b), and others, suppose that price adjustment behavior is given by 
(1)  πt = βEtπt+1 + αyt + ut,                                                             α > 0, 0 < β < 1, 
where πt is inflation, yt is the output gap, and ut is a stochastic shock term that is assumed to 
be autoregressive of order one with AR parameter ρ and innovation variance σ
2.  By the 
output gap we mean the fractional difference between realized output and the flexible-price 
or natural-rate level of output.   
  The policymaker’s objective at an arbitrary time t = 1 is to minimize 
(2) E1 ∑
∞
=1 t
β
t-1 (πt
2 + ω yt
2), 
where ω ≥ 0 reflects the relative importance of output-gap variability in policymaker 
preferences.
4  The macroeconomic model that we have in mind also includes an optimizing 
IS-type relationship of the form 
(3) yt = Etyt+1 + b(Rt − Etπt+1) + vt,                                                    b < 0, 
where Rt is the central bank’s interest rate instrument and vt is a shock that pertains to 
preferences, government spending, and the exogenous natural-rate value of output.  But we 
shall suppose, as in much of the literature under discussion, that the central bank (CB) can 
                                                                                                                                                       
2 See Woodford (1999b, pp. 293-4) for further details on the timeless perspective. 
3 Policy design is the term used by Prescott (1977) for a procedure that involves search for optimal policy-rule 
parameters after solving the model with a policy rule that includes all relevant state variables.  The procedure 
has been used by Taylor (1979), Rotemberg and Woodford (1999), and others.  3
directly control πt as an instrument—an assumption that is innocuous for the purposes of this 
note (though not for all issues).  Then relation (3) becomes irrelevant, and the policy problem 
is to minimize (2) subject to the constraint in (1) for the current and all future periods. 
  The optimality conditions proposed by CGG (1999, p. 1703) and Woodford (1999a, 
p. 24; 1999b, pp. 305-6), followed by McCallum and Nelson (2000), may be written as 
follows:
5 
(4a) 2  ω yt +  α λt = 0                                                  t = 1, 2, … 
(4b) 2  πt −  λt + λt-1 = 0                                                t = 2, 3, … 
 
(4c) 2  πt − λt = 0                                                           t = 1. 
Here it is arbitrarily assumed that the policy is being initiated (started up) in period t = 1.  But 
to adopt the timeless perspective, the CB ignores (4c) and applies (4b) in period 1 as well as 
in 2, 3, ….  Substituting out the Lagrangian multiplier yields 
(5)  πt = −(ω/α)(yt − yt-1). 
Thus the behavior of πt and yt is governed, under the proposed timeless perspective 
commitment policy, by relations (1) and (5) for periods t = 1, 2, ….   
  The minimum-state-variable (MSV) solution
6 for this system is of the form 
(6)  πt = φ11yt-1 + φ12 ut 
(7) yt = φ21yt-1 + φ22 ut, 
and the coefficients can straightforwardly be found to equal φ11 = (ω/α)(1−δ), φ12 = (γ−βδ)
-1, 
φ21 = δ, and φ22 = −(α/ω)(γ−βδ)
-1, where δ = [γ−(γ
2 − 4β)
0.5]/2β with γ = 1 + β + (α
2/ω).
7  
                                                                                                                                                       
4 For notational simplicity, we assume zero to be the target values of inflation and the output gap. 
5Here λt denotes the Lagrangian multiplier attached to constraint (1) for period t.    
6 See McCallum (1999) for an extensive discussion of this solution concept. 
7 See McCallum and Nelson (2000, pp. 7-8).  4
This solution agrees with those of CGG (1999) and Woodford (1999b, pp. 295, 307), and is 
dynamically stable (so Eπt = 0 and Eyt =0). 
  To summarize policy performance, CGG (1999), Woodford (1999a, 1999b), and 
McCallum and Nelson (2000) report average values of the loss function (2), i.e., values of the 
unconditional expectation of (2). Because of the law of iterated expectations, that expression 
equals (2) with E replacing E1.  Then taking E inside the summation operator, we find that 
the result equals (1 − β)
-1 times the unconditional expectation of the single-period loss,         
E(πt
2 + ω yt
2).  That equality is used here only for evaluation purposes, however; it is not 
utilized in the derivation (which is not discussed here) of the proposed conditions (4).  In the 
cited papers, the values of the average loss criterion just described is reported for outcomes 
with policy rule (5) and compared with values resulting when (5) is replaced with the optimal 
discretionary policy condition, which is 
 (8)  πt = −(ω/α) yt , 
as shown by CGG (1999), Woodford (1999a, 1999b), or McCallum and Nelson (2000).  In 
all of their reported cases, the average loss with (5), henceforth denoted L(5), is smaller than 
that obtained with rule (8). 
  What Jensen (2001a) demonstrates, however, is that (5) does not yield the smallest 
average loss, even if attention is restricted to rules (i.e., conditions for πt) including the same 
variables as (5).  Specifically, if policy is conducted according to 
(9)  πt = −(ω/α)(yt − β yt-1), 
then average values of (2) are smaller.
8  Table 1 below reports some representative results for 
L(5) and (analogously-defined) L(9) for various values of the parameters ω, β, and ρ, given 
                                                 
8 Since writing this note, we have learned that a very recent paper by Blake (2001) also reports this result.  5
α = 0.02 and E(ut
2) = (1−ρ
2)
-1σ
2 = (0.005)
2.
9  Since our calibration implicitly assumes 
quarter-year time periods, and outcomes are reported in fractional units, the value of 0.0625 
for ω represents equal weights on πt
2 and yt
2 in the objective function.  It is clear from the 
numbers in Table 1 that policy rule (9) provides smaller losses than does (5) over a wide 
range of parameter magnitudes.  The difference is greater when β is smaller, of course, and 
when ρ is large. 
A few words are needed concerning optimization conditional upon initial conditions.  
It is widely recognized that condition (5) fails to minimize (2), given y0 and u1, if applied in t 
= 1 as well as t = 2, 3, ….
10   Jensen (2001a) points out that, in addition, (5) is not generally 
optimal within the class of rules—conditions applied in all periods 1, 2, …—of  the same 
form.  With specified values of y0 and u1, for example, one can find a rule including the same 
variables as (5) that yields a lower value of (2) than does (5), with optimal coefficient values 
that depend on the initial conditions.  As an example, suppose that y0 = 0.03 and u1 = −0.01 
with ω = 0.0625, ρ = 0.5 and β = 0.99.  Then (5) yields a loss value for (2) of 0.008878 
whereas the use of  
 (10)  πt = -2.617 yt +2.502 yt-1  
results in a loss of 0.008658. 
11  This example illustrates that (5) does not minimize (2) when 
the same condition must be used in all periods, at least not in general.
 12 
13   
                                                 
9 This last magnitude is of no importance; changing it would scale all the values in Table 1 up or down 
proportionately. 
10 See, among others, Woodford (1999b), Svensson and Woodford (1999), King and Wolman (1999), and 
Dennis (2001). 
11 Calculation of these values is discussed by Jensen (2001b).  All reported results use the MSV solution. 
12 Condition (5) is optimal in this case if y0 = 0, but not otherwise. 
13 No table is provided since it would require extensive computation and a single example suffices to make the 
point at issue.  6
  Overall, our point is not that there is anything conceptually wrong with the timeless 
perspective type of policy making, in which the initial period policy reaction function is 
constrained to be the same as in all succeeding periods, but that (5) is not the optimal 
condition from this perspective, even if conditions (4) minimize (2).  Jensen (2001a) argues 
that the fashion in which optimality conditions (4) of the unconstrained problem are modified 
to produce (5) does not give the optimality conditions for the constrained problem.  Here our 
objective is not to put forth any explanation, however, but merely to demonstrate the 
superiority of rule (9) over the previously-utilized (5) from the perspective of the average 
value of the loss function (2).  We also point out the non-optimality of (5) with respect to 
objective (2) on a conditional basis, when credible commitment requires that the same policy 
rule be used in all periods including the one in which the policy regime is introduced. 
 
 
  7
 
Table 1 
 
Losses with Policy Rules (5) and (9) 
 
[Reported values are losses times 10
3: L(5) / L(9)] 
 
Value of 
β and ρ 
ω = 0.01  ω = 0.0625  ω = 0.10  ω = 1.00 
0.99, 0.0  2.0659 / 2.0656 
= 1.0002 
2.330 / 2.329 
= 1.0004 
2.369 / 2.368 
= 1.0006 
2.473 / 2.469 
= 1.0017 
0.99, 0.5  5.916 / 5.914 
= 1.0004 
8.029 / 8.019 
= 1.0013 
8.407 / 8.393 
= 1.0016 
9.500 / 9.452 
= 1.0051 
0.99, 0.9  29.721 / 29.678 
= 1.0015 
81.010 / 80.550 
= 1.0057 
97.952 / 97.194 
= 1.0078 
176.52 / 171.47 
= 1.0295 
0.98, 0.0  1.042 / 1.041 
= 1.0007 
1.176 / 1.174 
= 1.0018 
1.196 / 1.193 
= 1.0022 
1.247 / 1.240 
= 1.0059 
0.98, 0.5  2.980 / 2.974 
= 1.0018 
4.046 / 4.025 
= 1.0050 
4.235 / 4.208 
= 1.0064 
4.772 / 4.690 
= 1.0174 
0.98, 0.9  14.904 / 14.817 
= 1.0059 
40.610 / 39.711 
= 1.0227 
49.055 / 47.591 
= 1.0308 
86.910 / 78.892 
= 1.1016 
  8
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The Non-Optimality of Proposed Monetary Policy Rules Under Timeless-Perspective Commitment

The Non-Optimality of Proposed Monetary Policy Rules Under Timeless-Perspective Commitment

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