Welfare Evaluation of Monetary Policy Rules in a Model with

Nominal Rigidities
Jangryoul Kim∗
Federal Reserve Bank of Minneapolis

April, 2003

Abstract
This paper examines the welfare implications of monetary policy rules in a business cycle
model with nominal rigidities. Rules are compared in terms of a utility-based welfare metric,
and the welfare effects of the non-linear dynamics of the model are captured by a quadratic
approximate solution method. Rules for fixed paths of nominal income and money outperform
the fixed inflation rule, and the former rules, in turn, rank below variants of the Taylor rule.
Long run deflationary rules increase welfare. The welfare maximizing rule among a class of
Taylor-style rules is characterized by i) super-inertial adjustments in interest rates; ii) strong
short run anti-inflation coupled with long run deflation, and iii) increasing interest rates in
response to higher real output level and growth.

Keywords: Monetary Policy rules, Nominal Rigidites, Welfare Evaluations, Quadratic Approx-
imate Solutions.

JEL Classification: E3, E5

∗
Banking and Policy Studies, Banking Supervision Department, Federal Reserve Bank of Minneapo-
lis, 90 Hennepin avenue, Minneapolis, MN 55480-0291; phone (612) 204-5146; fax (612) 204-5114; e-mail:
jangryoul.kim@mpls.frb.org. I thank Christopher Sims for advice and guidance. My thanks also go to William
Brainard, George Hall, and Preston Miller for helpful comments and discussions. All errors are mine.

1
1. Introduction

Since the work of Taylor (1993), the effects of the monetary regimes on welfare and business

cycles have been a key issue in monetary economics. The interest in this topic is best reflected

in the plethora of research over the past decade: in the quest for “good” monetary policy rules,

researchers have either proposed optimal policy rules (e.g., Clarida et al., 1999, 2001; Ireland, 1996;

Khan et al., 2002; King and Wolman, 1999; Rotemberg and Woodford, 1999; Woodford, 1999) in

the context of specific models, or compared the performances of alternative rules under different

scenarios of the aggregate economy to provide guidelines for desirable policy rules (e.g., Henderson

and Kim, 1998; McCallum and Nelson, 1999; Rudebusch and Svensson, 1999). Focusing on the

overall performances of alternative rules, others (e.g., Levin et al., 1999) have examined how robust

the performances of popular rules are across a wide range of structural models.

A common, if not omnipresent, limitation in the existing literature is that monetary policy

rules are evaluated in terms of ad hoc or non-structural loss functions usually constructed from

variabilities in output gap and inflation (or price level). For example, Levin et al. (1999) use

the weighted average of the variances in quarterly output gap and inflation rate, and Rudebusch

and Svensson (1999) in addition consider the variabilities of the changes in nominal interest rates.

To address this arbitrariness, some in the literature opt to use the utility of agents as a natural

metric. To name a few, Rotemberg and Woodford (1997) derive an economically interpretable

loss function from the deep structure of the model economy. More specifically, in a model with

price stickiness à la Calvo (1983), the authors approximate the unconditional expectation of the

households’ utility by a weighted average of the variabilities in aggregate inflation and output gap.

Amato and Laubach (2003) and Erceg et al. (2000) extend the work of Rotemberg and Woodford

(1997) in a model with both price and wage stickiness, and develop welfare measures in terms of

the variabilities in aggregate price inflation, aggregate wage inflation, and output gap. Instead of

2
approximating the expected utility with aggregate volatilities, Ghironi (2000) directly evaluates the

exact unconditional expectation of the households’ utility under the assumption that the system

variables have lognormal distributions.

An addition to the recent literature on normative monetary economics, this paper examines

the welfare implications of monetary policy rules. To that aim, this paper posits and estimates

a monetary business cycle model with price and wage rigidities, subjects the estimated model to

alternative monetary policy rules, and compares them in terms of a utility-based welfare metric. For

correct welfare evaluations, I construct the welfare metric using a quadratic approximate solution

of the model instead of the conventional log-linear approximate solutions. Despite their widespread

use and good performance in fitting the model to data, log-linear approximate solution methods

have been criticized by recent papers (e.g., Kim and Kim, 2002), which point out the pitfalls in

measuring welfare level correctly. To capture the welfare effects of nonlinear dynamics, I use a

solution method based on a quadratic approximation of the equilibrium conditions, developed by

Sims (2000) and applied to a small open economy model by Kollmann (2002).

The estimated model features a higher degree of nominal rigidities in the labor market than in

the goods market, and a reasonable amount of utility (or equivalently, transaction costs saved) from

holding money. Given the estimated snapshot of the economy, comparison of welfare levels under

different rules suggests that adopting strict inflation targeting would hardly be optimal because that

rule puts the whole burden of adjustments to shocks on the wrong (i.e., stickier) variable. Fixed

nominal income targeting and money growth targeting show comparable welfare performance, and

are superior to fixed inflation targeting. This suggests that, given the trade-off in stabilizing price

inflation and wage inflation, targeting neutral nominal anchors is a good firsthand strategy. I

also find that variants of Taylor rules perform better than rules targeting fixed paths of nominal

anchors, which confirms the desirability of gradual rather than instantaneous adjustments of target

3
variables.

Based on the insights from the comparison results, I find the welfare maximizing rule among

a class of Taylor-type rules for endogenous responses of nominal interest rate. The optimized rule

requires i) super-inertial adjustments of nominal rate, ii) a strict anti-inflation reflected in aggressive

responses to the increases in nominal anchors such as inflation, wage inflation, and money growth

rate, iii) long-run deflation in the spirit of Friedman (1969), and iv) increasing interest rate in

response to higher real output, both in levels and growth rates.

The paper is organized as follows. Section 2 presents and estimates the model with nominal

rigidities. In section 3, I describe how to construct a natural welfare measure for policy rules

evaluation. Section 4 considers the welfare features of the economy estimated in section 2. Section

5 compares the performance of three fixed targeting rules and two variants of the Taylor rule.

Section 6 examines the welfare maximizing rule among a class of Taylor-style rules. Section 7

concludes the paper.

2. The Model
The economy consists of four types of agents: households, firms, government, and the aggre-

gator. Firms produce differentiated products using capital and labor supplied by households and

the aggregator, respectively. Households purchase output from the aggregator for consumption

and investment purposes, and supply capital and differentiated labor to firms and the aggregator,

respectively. Households and firms are obliged to satisfy demands at the wages and prices they

set. Nominal rigidities in the labor and goods market take the form of adjustment costs. The

aggregator uses a CRS technology of Dixit and Stiglitz to combine the differentiated goods (and

labor service) into a homogenized output (and labor), and sells the resulting output (and labor)

to households (and firms). Subject to its own period-by-period budget constraint, the government

4
manages monetary policy.

2.1 Households and Wage Setting

An individual household i ∈ [0, 1] carries Mi,t−1 units of nominal money, Bi,t−1 units of govern-

ment bond, and Kit units of physical capital from the previous period. In period t, the household

i earns factor income Wit Lit + Qt Kit from renting capital Kit and labor service Lit , where Wit and

Qt denote the nominal wage rate and nominal rental rate for capital, respectively. The interest

income from government bond holding is (Rt−1 − 1)Bi,t−1 where Rt−1 is the gross nominal interest
R
rate between period t-1 and t, and the dividend income from firms is sij Γijt dj where sij and Γijt

are household i’s fixed share of firm j and the profit of firm j, respectively. The household also

receives a lump-sum nominal transfer payment Tit from the government.

The household uses its funds to purchase the final good from the aggregator at the price of

Pt , and divide its purchase into consumption Cit and investment Iit . In order to make new capital

operational, the household needs to purchase additional materials in the amount

· ¸2
φK Iit I
ACitk = − Kit (1)
2 Kit K

where Iit = Ki,t+1 − (1 − δ t )Kit is the real investment spending, φK > 0 is the scale parameter for

I
the capital adjustment costs, and K
is the steady state ratio of investment to existing capital stock.

Dubbed the depreciation shock, δ t denotes the stochastic decay rate of capital stock. Its stochastic

properties are specified later. The household then carries Mit units of nominal money, Bit units of

government bond, and Ki,t+1 units of capital into period t + 1. Therefore, the household is subject

to the budget constraint

Mit Mi,t−1 Bit Bi,t−1

Cit + Ki,t+1 − (1 − δ t )Kit + − + − + ACitk
Pt Pt Pit Pt
R
Wit Lit Qt Kit sij Γijt dj (Rt−1 − 1)Bi,t−1
≤ + + Tit + + , t ≥ 0. (2)
Pt Pt Pt Pt

5
 The household maximizes lifetime utility

∞
X
E0 [ β t U (Cit∗ , Lit , Mit /Pt )], 0 < β < 1 (3)
t=0

where the instantaneous utility function U (·) has the form

Mit 1 h© ∗a ª1−σ i
U (Cit∗ , Lit , )= Cit (1 − Lit )1−at −1 , 0 < σ, 0 < a < 1, ν < 0. (4)
Pt 1−σ
1
In equation (3) and (4), Cit∗ = (Citν + bt (Mit /Pt )ν ) ν is the CES bundle of consumption Cit and

real money balance Mit /Pt . The stochastic properties of the money demand shock bt and the labor

supply shock at will be specified later.

Each household sells its differentiated labor service to the aggregator, who in turn uses a CRS

technology
Z 1
θ
Lt = ( LitL di) θL , θL ∈ [0, 1] (5)

to transform the differentiated labor service into a single labor index Lt . The implied demand for

household i’s labor service and the aggregate wage rate Wt are
Z θL
Wit θ 1−1 θ L −1 θ L −1
Ldit =( ) L Lt , Wt = ( Wit di) θL . (6)
Wt

Each household is subject to quadratic costs of adjusting its nominal wage

µ ¶2
Φw Wit Wt
ACitw = − Πw
t−1 (7)
2 Wi,t−1 Pt

where Πw
t−1 = Wt−1 /Wt−2 is the gross wage inflation rate at period t − 1, and Φw > 0 is the scale

parameter for the degree of nominal rigidity in the labor market. Note that the presence of lagged

wage inflation in equation (7) renders sticky both aggregate wage rate and wage inflation.1

The first order conditions for (Cit , Mti , Ki,t+1 , Bit , Wit ) are given by

∂Uit
= Λit (8)
∂Cit
1
This specification generalizes Rotemberg (1982), and addresses the claim of Fuhrer and Moore (1995) that sticky
price models should be able to generate sufficient inertia in inflation as well as in price level.

6
 1 − Rt −1 = bt (Cit Pt /Mit )1−ν (9)
· ¸ " Ã !#
k
∂ACitk ∂ACi,t+1
Λit 1 + = βEt Λi,t+1 Qt+1 /Pt+1 + 1 − δ t+1 − (10)
∂Ki,t+1 ∂Ki,t+1
· ¸
Pt
Λit = βRt Et Λi,t+1 (11)
Pt+1
½ · w ¸¾
∂Uit dLit ∂ACitw ∂ACi,t+1 1 ∂ (Wit Lit )
= Λit + β Et Λi,t+1 − Λit (12)
∂Lit dWit ∂Wit ∂Wit Pt ∂Wit

where Λit the Lagrangian multiplier on the household i’s budget constraint, interpretable as the

shadow value of an additional unit of consumption.

Using equations (4), (5), and (7), I rewrite equation (12) as

· ¸ 1
−1 · ¸ 1 · ¸
Wit 1−θ L Wit Wt 1
1−θ L Wit w Wt
M RSit = θL + (1 − θL )φW − Πt−1
Wt Wt Pt Lt Wi,t−1 Pt
" µ ¶2 #
β(1 − θL )φW Λi,t+1 Wi,t+1 Wt+1 Wt
+ Et
Lt Λit Wit Pt+1 Wit
· ¸
β(1 − θL )φW Λi,t+1 w Wt+1 Wi,t+1 Wt
− Et Πt (12’a)
Lt Λit Pt+1 Wit Wit

where
∗a(1−σ)
M RSit = (1 − at )Cit (1 − Lit )(1−at )(1−σ)−1 Λ−1
it (12’b)

is the household i0 s marginal rate of substitution between leisure and consumption. In the analysis

that follows, equations (12’a) and (12’b) will be substituted for equation (12).

2.2 Firms and Price Setting

During period t, an individual firm j ∈ [0, 1] hires Kjt units of physical capital (from households)

and Ljt units of aggregate labor service (from the aggregator), and produce Yjt units of its own

product. All firms have the identical CRS production technology

αt t
Yjt = At Kjt (g Ljt )1−αt , g ≥ 1 . (13)

The stochastic properties of the aggregate productivity shock At and the capital share shock αt are

detailed later.

7
 As in the labor market, the demand function for the firm j’s output Yjt is

Pjt θ 1−1
Yjtd = ( ) Y Yt , θY ∈ [0, 1] (14)
Pt

where the aggregate demand Yt and the aggregate price level Pt are defined as

Z Z θ θY−1 θ Y −1
θY 1 Y
Yt = ( Yjt dj) θY , Pt = ( Pjt di) θY . (15)

Nominal rigidities in the goods market take the form of price adjustment costs

µ ¶2
Φp Pjt
ACitp = − Πt−1 Yt (16)
2 Pj,t−1

where Pjt is the price of the firm j set in period t, and Πt−1 = Pt−1 /Pt−2 is the inflation rate

prevailing in period t − 1. Equation (16) implies that both the price level and the inflation rate are

sticky.

The firm j is assumed to solve its profit maximization problem through two steps. First, given

aggregate price level and factor prices, the firm solves the cost minimization problem. Second,

given the cost function thus derived, it determines the optimal price Pjt to charge by solving the

following profit maximization problem

·∞ t µ ¶¸
P β Λt Pjt Yjt M Ct p
max E0 − Yjt − ACjt (17)
t=0 Λ0 Pt Pt

β t Λt R
where Λ0 is the discount factor for its real profit between period 0 and t, and Λt = [0,1] Λit di

is the average marginal utility of consumption across all households.2 The marginal cost M Ct ,

common to all firms, is independent of the output level due to the CRS production function.

The FOCs for (Kjt , Ljt, Pjt ) require

Ljt Qt /Pt 1 − αt
= (18)
Kjt Wt /Pt αt
2
If all households are identical and have the same shares Γijt of firm j ∈ [0, 1], the assumption of complete markets
t
establishes the unique market discount factor βΛΛ0 t between period 0 and t.

8
 M Ct Wt /Pt
= (19)
Pt M P Lt
( p
" p
#)
1 ∂ (Pjt Yjt ) M Ct ∂Yjt ∂ACjt βΛt+1 ∂ACjt+1
= + + Et . (20)
Pt ∂Pjt Pt ∂Pjt ∂Pjt Λt ∂Pjt

where M P Lt is the marginal productivity of labor.

Using equations (14), (16), and (19), I rewrite the equation (20) as
· ¸ θ θY−1
¸ 1 · · ¸
Pjt Pjt θY −1 Wt /Pt
Y Pjt Pjt
θY − + (1 − θY )ΦP − Πt−1
Pt Pt M P Lt Pj,t−1 Pj,t−1
" µ ¶2 #
Λt+1 Pj,t+1 Yt+1
= β(1 − θY )ΦP Et
Λt Pjt Yt
· ¸
Λt+1 Yt+1 Pj,t+1
−β(1 − θY )ΦP Et Πt (20’a)
Λt Yt Pjt

αt
M P Lt = (1 − αt )At Kjt Ljt −αt g t(1−αt ) (20’b)

In what follows, I replace equations (19) -(20) with equations (20’a) and (20’b).

2.3 Government

The government maintains a balanced budget every period by equating the total lump-sum

payment to households with the sum of seigniorage gain and the increase in net debt

Tt = Mt − Mt−1 + Bt − Rt−1 Bt−1 (21)

R1 R1 R1
where Tt = 0 Tit di, Mt = 0 Mit di, Bt = 0 Bit di. 3,4 Subject to the condition (21), the government

conducts monetary policy by adjusting the short-term nominal interest rate Rt according to the
3
To offset the effects of monoploistic distortions on the steady state output and/or labor , Rotemberg and Woodford
(1997, 1999) and other researchers (e.g., Amato and Laubach, 2003; Erceg et al., 2000) assume government subsidies
on sales revenue and/or labor income. I consider the “as-is” economy without such subsidies, because I believe such
schemes belong in principle to the realm of fiscal policies.

4
Although the model exhibits Ricardian Equivalence, fiscal considerations are in order for the equilibrium to exist
and be unique. For example, if the growth rate of nominal bonds is higher than the inflation rate, which is possible
under a deflationary policy, real government debts will explode. Implicitly, I assume fiscal policy is specified as
Bt−1
Tt = gt T − τ
Pt
where T and τ are some constants, so that increasing government debts can be financed by negative transfer payments.

9
rule

· ¸
Rt Rt−1 Πt Yt M Gt
log = ρR log + (1 − ρM ) γ π log + γ y log + γ m log + εMt , 0 < ρR < 1 (22)
R R Π Yt MG

where R is the gross nominal interest rate, M G is the growth rate of nominal money, R is the

steady state gross nominal interest rate, all in the steady state. Πt is the gross inflation rate

between period t − 1 and t, and Y t is the deterministic level of output in period t, respectively. Π

is the long-run “reference” level of inflation rate.5 The monetary policy disturbance εMt is a white

noise with mean 0 and variance σ 2ε , independent of all other random shocks in the model. The

rule (22) is a generalization of Taylor (1993) in that it allows policy to respond to the variations in

money growth in addition to output and inflation.

2.4 Equilibrium

I take the economy to be subject to six structural disturbances. In addition to the monetary

policy disturbance εRt , the model is driven by stochastic evolution of five structural disturbances

(At , αt , δ t , bt , at ) , each of which follows a logarithmic AR(1) of the form

χt χ
log = ρχ log t−1 + εχt (23)
χ χ

where χ is the steady state level of χt , and εχt is a white noise with mean 0 and variance σ 2χ .

The autoregressive coefficients are constrained within the stationary region. Innovations in the

disturbances are not correlated with one another, except that the two errors (εAt , εαt ) in the

production function are allowed to be correlated. 6

5
I use the term “reference” rate of inflation to denote the long-run inflationary stance of a rule. The term “target”
rate probably is used more frequently in the literature. The reason for the unfamiliar nomenclature is to prevent
possible confusions around the usage of “targets”or “targeting” observed in the literature.

6
Christiano et al. (2001) assume variable capital utilization, under which positive productivity shock will increase
the effective marginal productivity of labor, and leads to a higher amount of labor employed relative to capital.
Allowing for the correlation between At and αt in the present model is intended as a reduced form to capture such
dynamics.

10
 An equilibrium of the economy (under the benchmark monetary policy rule) is given by a set of

decision rules {Cit , Ki,t+1 , Mit , Bit } and a wage rule Wit of household i; a capital demand rule Kjt ,

a labor demand rule Ljt and a price rule Pjt of firm j, the monetary policy rule of government;

and a price vector {Pjt , Wit , Qt , Rt } such that: i) {Cit , Ki,t+1 , Mit , Bit } maximizes lifetime utility

(3) subject to the budget constraint (2), adjustment costs (1) and (7), and labor demand in (6);

ii) Kjt , Ljt , and Pjt maximizes profit stream (17) subject to the production technology (13) and

adjustment cost (16); iii) {Rt , Mt } evolve according to (22) subject to the government’s budget

constraint (21); iv) {Pjt , Wit , Qt , Rt } clear the goods market, the labor market, capital market, and

the money market.

In what follows, I focus on a particular symmetric equilibrium in which all firms and house-

holds make identical decisions. Since most of the real and nominal variables in the model exhibit

deterministic trends due to the constant rate of labor-augmenting technical progress (g) and the

reference inflation rate, I deflate variables by their deterministic trends to transform the system

into a stationary one.

2.5 Estimation7

The transformed system, described in more detail in the appendix, is cast into the form

G1 (zt , zt−1 , εt ) = 0N1 ×1 (24a)

G2 (zt , zt−1 , εt ) + η t = 0N2 ×1 (24b)

where εt is the vector of the six innovations, and η t is a vector of endogenous errors satisfying

0 , z 0 )0 , where
Et−1 η t = 0, for all t. The N -dimensional system vector zt is decomposed as zt = (z1t 2t

z2t denotes the N2 -dimensional auxiliary variables used to denote conditional expectation terms
7
The estimation results are mostly taken from Kim (2003), who estimates a model with identical first order
hahavior.

11
in equations (10), (11), (12’a) and (20’a), and z1t is the (N − N2 ) dimensional vector of all other

variables including all exogenous and endogenous state variables.8 When the system (24) is log-

linearized around its steady state, the method of Sims (2002) can be applied to obtain a solution

of the form

d log zt = g1 d log zt−1 + g2 εt (25)

where g1 and g2 are complicated matrix functions of the model parameters. Since the solution (25)

takes the form of a state-space model driven by innovations εt , maximum likelihood estimates of the

parameters can be obtained by an application of Kalman filtering, using data on the observables

in zt .9

The raw data used in this study are extracted from DRI BASIC economic series for the sample

period 1959:Q1-1999:Q4.10 Since two main features of the model are i) the nominal rigidities

in goods and labor markets; and ii) the interest rate feedback rule for monetary policy, it is

imperative to use the data on monetary aggregates as well as prices and quantities in goods and

labor markets. Therefore, the following six series are used for the actual estimation purpose: per

capita output (Y ), per capita labor hours (L), rate of price inflation (Π), the growth rate of per
8
In the conventional rubric, z1t and z2t correspond to the “state” and “jump” variables, repectively. As summarized
in the appendix, the number of conditional expectations (or endogenous errors) in the present model is 6: one in the
capital adjustment equation (10), one in the Fisherian equation (11), two in wage equation (12’a), and two in price
equation (20’a).

9
Maximization of the likelihood function over the parameter set requires one to cope with the parameter regions
in which i) the candidates of estimates yield nonsensical (mainly negative) steady state values of the variables; and ii)
the model does not have a unique equilibrium. As in Leeper and Sims (1994), I assign an arbitrary very low likelihood
value to parameters in such bad regions, and the resulting discontinuity in the likelihood function is addressed by a
“cliff-robust” optimization routine csminwel.m written by the latter author.

10
All raw series, except for interest rate and wage, are seasonally adjusted.

output : gross domestic products, billions of 1992 dollars.

employment : average weekly hours of production workers in manufacturing sector.
price : implicit price deflator for gross national products.
money : M2 stock, billions of current dollars.
interest rate : federal funds rate, per annum.
wage : index of compensation per hour in nonfarm business sector, 1982=100.
population civilian population, in thousands.

12
capita money balance (M G), interest rates (R), and wage inflation rates (Πw ). To express the

data series conformable with the theoretic counterparts in the model, per capita output and money

balance series are obtained by dividing GDP and M2 balance, respectively, by population size. Per

capita labor hours are obtained by dividing weekly working hours by 120, under the assumption

that each worker is endowed with 5 working days per week. The resulting series imply households

devote 33.8% of their time endowment to working. Since federal funds rates are measured in annual

percentage rates, I transform them into quarterly rates by dividing by 400 and adding one. Price

and wage inflations are obtained by log-differencing the price and wage series.

Since the dataset bears little information about some structural parameters, a few parameters

are fixed before estimation: steady state values of capital share α and depreciation δ are fixed at

1/3 and 0.025, respectively. The market power θY in the goods market is fixed at the conventionally

calibrated value of 0.9, because only two of (A, θY , θL ) are identified from the series on output and

labor. Assuming the Fed has successfully managed the inflation rate around the intended reference

level, I fix the steady state inflation rate Π at its actual average 1.01005 over the sample period. The

CRRA parameter σ is fixed at 1, which amounts to the logarithmic instantaneous utility function.

Two parameters (ν, b), crucial to the form of money demand and welfare calculations, are esti-

mated by running calibration and estimation jointly. More specifically, at each step of maximizing

the likelihood function I determine b as

−1
b = (1 − R )[Vd ]ν−1 (9’)

given all other candidate parameters, where the empirical velocity Vd is set equal to the actual

average over the sample period.11

The estimated parameters are reported in Table 1 along with corresponding functional forms

of the structural equations. Asymptotic standard errors are in the parentheses, computed from the
11
Therefore, the estimates are obtained under the constraint that the estimated velocity is equal to the actual.

13
Hessian of the maximized likelihood function. The estimate of growth rate g is 1.0056, which is

higher than the actual average growth rate 1.0052 of per capita GDP over the sample period. The

estimate of discount factor β is 0.9986, falling between the estimate 0.9974 for post-79 era in Ireland

(2001) and 0.9999 in Kim (2000) for 59:Q1- 95:Q1, although higher than the usually calibrated value

of 0.99. The share a of consumption bundle Ct∗ in the instantaneous utility function is 0.4681, which

is higher than the usually calibrated value of 0.4. The estimates of (ν, b) are (-22.7561, 0.0008),

which shows that the indifference curves on the (C, RM ) plain are highly convex to origin in the

C
estimated steady state: one percent increase in M/P ratio results in 23.7561 percent decrease in the

marginal rate of substitutions between C and RM.12 These estimates also imply that an interest-

semi-elasticity of money demand is about 0.04, which is well below the usual empirical estimates.13

The estimate of θL is 0.6888, lower than the calibrated value 0.75 in Huang and Liu (1999).

The real rigidity parameter for capital adjustment cost (φK =16.8456) shows a considerable de-

gree of real rigidity in the economy: when the economy is initially at the estimated steady state,

transforming one unit of consumption good into the same unit of operational capital involves an

additional 0.0668 units of output as adjustment costs.

The parameters for the monetary policy rule, used as the benchmark rule in the following

analysis, show the systematic evolution of nominal interest rates in response to inflation and money

growth, but none to output over the sample period.14 The estimate ρM = 0.1395 implies a modest
12
As discussed in Feenstra (1986), one can construct an isomorphic model in terms of transaction costs, by redefining
C ∗ in the utility function (4) as the usual consumption and replacing C in the budget constraint (2) with
· µ ¶ν ¸ 1
M/P ν
C ∗∗ = C ∗ × 1 − b ∗
C
where C ∗∗ represents the gross spending on consumption inclusive of (multiplicative) transaction costs. The ratio
C ∗∗ / C ∗ (evaluated at the estimated steady state) is 1.0006, implying transaction costs are a reasonably small fraction
of consumption C ∗ .

13
This seems to arise because the model makes the demand for money adjust instantaneously, whereas empirical
work usually allows lags or uses longer frequency data.

14
Also observed in Ireland (1999) are the small responses of nominal interest rate to output in the presence of

14
degree of policy inertia.

Regarding the structural disturbances, the estimated AR(1) coefficients show the economy has

been subject to highly persistent structural shocks. Except for the labor supply shock, the half-

lives of the aggregate shocks are around 6 years. The labor supply shock at exhibits negative

serial correlations. Finally, the innovations in the shocks At and αt are negatively correlated with

correlation coefficient -0.9755.

The estimates of (Φw , Φp ) =(20.0341,10.0970) show the degree of nominal rigidities is higher

in the labor market than in the goods market. Those parameters are precisely estimated with

respective standard errors 0.7025 and 0.7393.

3. Welfare Metric
I now construct a natural utility-based metric, to be used for welfare evaluation of alternative

monetary policy rules. Since the instantaneous utility function U (·) has a deterministic trend due

to the labor-augmenting technological growth, I first transform Ut = U (Ct∗ , Lt , Mt /Pt ) to achieve

stationarity:

Ut = g a(1−σ)t × ut
µ ¶
a(1−σ)t 1 h ν ν a (1−at )
i1−σ
= g (ct + bt rmt ) ν (1 − Lt ) −1 (26)
1−σ

where ct and rmt are the stationary transformed consumption and real balance, respectively. Using

the second order Taylor expansion of ut around the deterministic steady state, I get the present

discount value of a representative household’s expected utility conditional on an initial condition

inflation and money growth as policy indicators.

15
Ω0 :

hX∞ i
EW = E β t∗ ut | Ω0
t=0
· ¸0 X ³ h i´
1 dut (ζ) ∞
' u(ζ) + ⊗ζ β t∗ E bζ t | Ω0
1 − β∗ dζ t t=0
µ h i · ¸¶
1 X∞ t b d2 ut (ζ) 0
+ tr β ∗ V ar ζ t | Ω0 ⊗ (ζζ ) (27)
2 t=0 dζ 2t

where β ∗ = βg a(1−σ) , and tr(·) is the trace of a square matrix.15 The symbol ⊗ denotes the matrix

operator of element by element multiplication.

h i
The presence of conditional expectation E b ζ t | Ω0 in equation (27) highlights the need to use

higher order approximate solution methods for correct welfare evaluation. For a simple illustration,

suppose that the exact solution of ζ t can be represented as a function ζ(·) of εt , so that the second

order Taylor expansion of ζ t around the steady state is dζ t = ζ + ζ 0 εt + 12 ζ 00 ε2t . If a first order

approximate solution of ζ t is inserted into EW, the expectation of the third term in dζ t is ignored,

which is in general the same size as the other second order terms that appear in the last term in

EW.

To get a solution of the system (24) with the accuracy of up to the second order, I use the

method by Sims (2000) based on a quadratic Taylor expansion of (24) around the deterministic

steady state z. Under a set of regularity conditions, a unique and stationary second order accurate

solution to (24) is obtained of the form

zb1it = F1ij zb1j,t−1 + F2ij εjt + F3i (28a)

+0.5 (F11ijk zb1j,t−1 zb1k,t−1 + 2F12ijk zb1j,t−1 εkt + F22ijk εjt εkt ) ,

zb2it = Si zb1it + Ti M11ijk zb1jt zb1kt + Ti M2i . (28b)

where zbt = log zt − log z denotes the % deviation of zt from its deterministic steady state, and
15
Many researchers (e.g., Clarida et al.,1999; Rotemberg and Woodford, 1997,1999; and Erceg et al., 2000; Koll-
mann, 2002) have used unconditional expectation of utility, which corresponds to using E [ut | Ω0 ] with t = ∞.
Acknowledging the importance of transitional dynamics, this paper uses the discounted stream of utility.

16
S, T, F 0 s, and M 0 s are matrix functions of the deep parameter of the model.16 In particular, the

terms F3 and M2 represent the degree of certainty non-equivalence. Note that equations in (28)

utilize the tensor notation for the simplicity of exposition. For example, the term F11ijk zb1j,t−1 zb1k,t−1

can be interpreted as the quadratic form in terms of lagged zbt for the ith equation, constructed by

the lag of zb1t . By using equation (28a) recursively, I can compute {µ1t , Σ1t : t ≥ 0}, the conditional

first and second moments of zb1t , from which the welfare measure EW is constructed. The details

involved are given in the appendix.

When the metric EW is applied, it is essential to use the same initial condition Ω0 for all rules

being compared. This in turn requires one to use the same pair of (µ0 , Σ0 ) or the same distribution

of the initial state for every policy rule.17 In the following analysis, I set both µ0 and Σ0 to be 0,

under the assumption that the economy has been at the estimated steady state until the initial

period.18

For interpretational convenience, the relative performances of alternative rules are measured by

consumption compensations defined as follows: suppose that the benchmark rule (22) (say, rule 0)

and another rule (say, rule 1) deliver EW 0 and EW 1 level of expected welfare, respectively. The

consumption compensation for the rule 1 (relative to rule 0) is defined as

£ ¤ £ ¤
1 EW 0 − EW 1 1 EW 0 − EW 1
dc = + (29)
2 λ0 2 λ1

where λ0 and λ1 are the marginal utility of consumption evaluated at steady state under the rule 0

and rule 1, respectively.19 The interpretation of the relative measure dc is straightforward: if EW 0

16
The original codes of Sims (2000) give a slightly different (but essentially equivalent) solution, in which the
solutions for the “jump” variables z2t are indirectly given by linear combinations of the whole system vector in
log-deviations. The routine for transforming the original solutions into those in (28) is available upon request.
17
If each policy rule’s own implied unconditional mean and variance are used, the ranking is identical to what
results when the non-discounted unconditional expectation of ut is used.

18
Since welfare calculations vary depending on the initial conditions, I also use the conditioning information set
which comprises the unconditional mean and variance under the benchmark rule (22). The qualitative results, which
remain the same, are available upon request.

19
Since the monetary authority manipulates the “reference” rate of inflation, the steady state level of marginal

17
is higher than EW 1 , the representative household under the rule 1 should be compensated with

dc amount of one time consumption to have (approximately) the same level of lifetime expected

welfare as under the benchmark rule.

Before further analysis, it is meaningful to gauge the accuracy gains from using a higher order

approximate solution method. Equation (28b) shows a key difference between the linear and the

quadratic approximate solution methods: the latter method gives a quadratic parametrization of

the conditional expectations as on the RHS of (28b), while the former only gives the first term

for linear parametrization. Therefore, I compare the accuracy of the two solution methods in the

spirit of parametrized expectation approach (PEA) as follows: i) setting off from the estimated

deterministic steady state, I generate a very long path of exogenous disturbances of the sample

size 100,000; ii) the linear and quadratic parametrizations of the conditional expectations z2t are

combined with the original subsystem (24a) for z1t ; iii) for each parametrized version of the whole

system thus constructed, I solve forwardly for the whole system variable zt given the path of

exogenous disturbances, again starting from the deterministic steady state; iv) the solved paths of

zt are substituted in (24b) to generate the simulated paths of endogenous errors η t under the two

solution methods; and iv) I check the accuracy of the solutions by regressing the simulated errors

on state variables: the R2 s should come out very small for both solution methods, and even smaller

for the quadratic approximation method.20

utility of consumption is also policy-dependent. Taking arithmetic average is analogous to the same convention in
calculating arc elasticities.

20
The state variables are categorized into

x1t = (mgt , kt+1 , Πt , rwt , rmt , Rt , Πw

t ), x2t = (At , αt , δ t , bt , at )

where the first and second sets denote endogenous and exogenous state variables, respectively. The resulting simulated
endogenous errors are regressed on the following 25 “explanatory” variables for corresponding solution method:

i) constant (1 term)
ii) (log x1t − log x1 ) , and (log x2t − log x2 ) (12 terms)
iii) square terms in (log x1t − log x1 ) and (log x2t − log x2 ) (12 terms)

18
 Table 2 reports the R2 s and F statistics for the null of “all-zero coefficients” from the two

regressions. The left panel is for the conventional first order solution, and the right panel is for

the second order solution. The F statistics show that the null is rejected for all endogenous errors

regardless of the solution methods. The null is eventually rejected with as large a simulation sample

size as 100,000, however, because both solution methods are only approximate. The results in Table

2 also indicate that the inaccuracy in the approximate solutions will not be detected with sample

size as large as the historical data, because the rejection of null for the sample size of 160 requires

R2 be 0.242 or higher. At any rate, the results in Table 2 demonstrate higher accuracy of the

second order approximate solution: for all of the six endogenous errors, the quadratic approximate

solution generates smaller R2 s and F statistics, and the improvement is most conspicuous for the

errors in the pricing equation.

4. Insights from the Benchmark Rule

Table 3 reports the performance of BM, the estimated benchmark rule (22) implemented with-

out εMt .21 The first column of the upper panel shows that, at the current stance Π = 1.01005

of long-run inflation, the welfare measure EW amounts to 607.8661. Evaluated at the estimated

steady state, this level of welfare translates into 17.5563 units of consumption each period for an

eternal life.

As shown in (27), EW comprises three terms on the RHS representing i) steady state utility;

ii) utility from the first order deviations from steady state; and iii) utility from the second order

deviations from steady state. Had the welfare metric EW been constructed naively from the

conventional first order approximate solution, the second term would not have come into play. In

that case, the naive welfare calculation under BM would have been lower by 5.1690 than the correct
21
The omission of policy mistakes εM t is for fair comparison with other rules, which are assumed to be implemented
exactly in the later section.

19
level. In terms of the consumption compensation measure, this “measurement error” amounts to

107.4934 units of one-time consumption, or equivalently 0.8370 units each period for life which is

6.22% of output in the deterministic steady state.22

Two features of the present model are suggestive of ways to improve upon the benchmark

rule in terms of welfare. The first one is the nature of nominal rigidities in the model: both

price and wage are sticky, not only in levels but also in the rates of changes. One striking

and frequently criticized implication of many New Keynesian models is that, when there is a

single nominal variable whose level is sticky, Pareto optimum is attainable in the absence of other

distortions because there is no trade-off between stabilization of inflation and stabilization of the

output gap. This rosy implication is an inherent artifact of price adjustment processes that involve

only forward looking expectations of private sector.23 Roughly speaking, purely forward looking

expectations implies a Phillips curve in which inflation depends only on the current and expected

future output gap, without any lagged dependence on past inflation. This being the case, the

monetary authority manipulates the parameters of the policy rule to stabilize output gap, achieving

inflation stabilization as a serendipity.

In a recent paper with both price and wage level rigidities, however, Erceg et al. (2000) establish

that even if expectations are purely forward looking, monetary authorities cannot achieve Pareto

optimum unless either prices or wages are perfectly flexible. In the present model, the trade-off

that the monetary authority faces is all the more serious because the aggregate price and wage
22
For ABM, the underprediction of welfare due to using the first order approximate solution amounts to 110.3507
units of one-time consumption.

23
If the price adjustment cost is specified as
µ ¶2
P ΦP Pjt
ACit = − Π Yt
2 Pj,t−1

where Π is the steady state rate of inflation, then inflation is driven by purely forward-looking expectations. Similar
results hold for the wage adjustment costs.

20
are dependent upon both their past history as well as expectations on future levels. Therefore,

the monetary authority in the present model is required to find the best compromise between the

stabilization of prices and wages, while keeping low volatility of output gap as well. This in turn

requires wage inflation to be another nominal anchor to which monetary instruments are adjusted.

The second one, closely related to Friedman (1969), is the explicit consideration of money in

the model. The dictum of Friedman (1969) is that the optimal inflation policy is what makes the

private cost of holding money equal to the social cost, or a policy achieving zero nominal interest

rate via long-run deflation. Equipped with an optimization based money demand function invariant

to the changes in policies, the present model can be used to measure the costs/benefits in terms

of welfare of changing long-run inflation, so long as the money demand shock bt is not abstracted

away.24

At this point, the two possibilities of welfare improvement are addressed informally in the

framework of the benchmark policy rule by considering how much welfare metric varies i) if the

reference inflation rate varies, and ii) if wage inflation is used as another indicator variable in the

benchmark rule BM.

The lower panel of Table 3 summarizes the changes in welfare under ABM, the benchmark

rule augmented with wage inflation. Evidently, the welfare gains from using another indicator are

uniformly positive for all three reference inflation rates. At the current rate of reference inflation,

the augmented benchmark rule ABM yields slightly higher welfare level equivalent to 5.4643 units

of one-time consumption. For higher and lower reference rates, the consumption gains over the
24
Rotemberg and Woodford (1997,1999) and Erceg et al. (2000) exclude money from the models on the implicit
assumption that money is additively separable in the period utility function so that the behavior of the model will
be invariant to adding money. However, the welfare implications of different monetary policies are not invariant,
because the welfare costs of positive nominal interest rates are ignored under that assumption. The sensitivity of
welfare measure EW with respect to the inclusion/exclusion of money will be examined in more detail later in
section 5.

21
benchmark rule amount to 2.6183 and 8.3205 units, respectively.

The last two columns of Table 3 report the performances of BM and ABM for roughly

symmetrically higher (Π =1.011) and lower (Π =1.0091) levels of reference inflation rates. As

expected, more inflationary policy yields lower welfare level. For example, when the reference

inflation rate is increased to 4.473% per annum, households demand 2.9042 additional units of one-

time consumption to be as happy as ever, which amounts to 21.6% of steady state output. When

the reference inflation rate is lowered, households are willing to forego 2.9019 units of one-time

consumption.

As will be shown later in section 6, the welfare maximizing rate of long-run inflation is negative

at least among a class of simple endogenous rules adjusting short term rates. The legacy of Friedman

(1969) in the present model is in fact in contrast with a few papers in the literature (e.g., King and

Wolman, 1999; Wolman, 2001) which favor zero or slightly positive long-run inflation depending on

whether the policy objective is present value of welfare or steady state. It is therefore worthwhile

to briefly discuss why I reach the opposite end of the policy spectrum. In models with staggered

contracts and transaction costs, steady long run inflation has two offsetting welfare effects. On

the one hand, positive long run inflation affects the distribution of relative prices, decreasing the

markups of firms whose prices were set in the previous periods. This erosion of relative prices (and

increased output) of those firms can be welfare improving, because monopolistically competitive

firms are obliged to satisfy all demands at their individual prices posted. On the other hand, it is

desirable to have long run deflation so that the nominal interest rate is zero, as long as there exists

deadweight losses (i.e., the “shoe leather costs” of inflation) under the money demand curve. In the

present model where nominal rigidities are imposed via adjustment costs, there are no adjustment

costs under steady inflation and therefore only the second negative effect of long run inflation comes

into play. Equipped with a simple quantity equation and staggered prices à la Taylor, however, the

22
models in King and Wolman (1999) and Wolman (2001) allow only the first welfare channel of long

run inflation to work, deviating from the dictum of Friedman.

At any rate, the findings from Tables 3 are supportive of a welfare-improving deflationary policy

rule equipped with wage inflation as another policy indicator. The analyses in the two subsequent

sections, where I examine alternative policy rules and an optimized rule among a restrictive class

of rules, are based upon this insight.

5. Welfare Analysis: a Performance Derby

In this section, I consider two types of alternative monetary policy rules in which nominal

interest rate is used as the policy instrument. The first type of rules, dubbed “targeting” rules,

postulate that the monetary authority maintains pre-specified deterministic paths of three variables:

price, money stock, and nominal income.25 The second type of rules are basically generalizations of

the Taylor rule, in which monetary instruments are adjusted in response to the endogenous indicator

variables such as output gap and inflation rate. The insight developed in the previous section is

further assayed by considering versions of each alternative rule with lower reference inflation rates

or endogenous responses to wage inflation.

An important yet frequently ignored issue in the literature is that a candidate rule should be

supported by the economy. In particular, the zero bound on the nominal interest rate should be

accounted for: given that a solution method involves approximations around a deterministic steady

state with inflation rate close to 0, the nominal interest rate would be negative a nonnegligible

portion of time under a rule implying highly volatile nominal rates. Therefore, I focus on the set
25
Usually, as in McCallum and Nelson (1998), the expression “X-targeting” or “having a target level X ∗ for variable
X” describes a regime in which the monetary authority sets its instrument according to a rule involving responses
to deviation of X from its desired path. Alternatively, in a series of papers, Svensson (1999) and Rudebusch and
Svensson (1999) identify X-targeting as a regime in which the monetary authority sets a level for the variable X and
use all available information to bring X in line with that level. According to the terminology of Svensson, the Taylor
rule is a rule “responding” to inflation and output gap, while according to McCallum and Nelson (1998) it is a rule
that “targets” both variables. In this paper, the term “targeting” is used to denote the usage advocated by Svensson.

23
of f easible monetary rules, i.e., rules under which the 2.55 times standard deviation confidence

intervals for the nominal interest rate, constructed around its unconditional expectation, do not

contain zero.26 Crude as it is, this apparatus does impose a condition that too aggressive an activist

rule yielding too volatile nominal interest rate is not compatible with low reference inflation rate,

a generic implication of models such as the present one.

5.1 Alternative Policy Rules

5.1.1 Inflation Targeting (PHIT)

If the monetary authority sets the nominal interest rate to keep inflation rate Πt at a fixed level,

the corresponding path of nominal interest rate (up to the first order accuracy) can be found from

the money demand function (9). Taking the first difference of the log-linearized version of (9) and

d
ct = M
using ∆rm b t , one gets
Gt − Π

1 b 1 b
Rt = Rt−1 + ∆bbt + (1 − ν)∆b d
ct − (1 − ν)M Gt . (30)
R−1 R−1

Therefore, the strict inflation targeting dictates the change in interest rate be positively related

with consumption growth, and negatively related with nominal money growth. This rule will be

labelled PHIT for future references.27 If the rule (30) is implemented with Π = 1, it is equivalent

to strict price targeting.

It is worth noting that, again, up to the first order, fixed inflation targeting is equivalent to

targeting a constant markup in the goods market that would result if nominal prices were fully
26
In a quarterly model like the present one, if nominal interest rate is normally distributed, such a restriction
1
alows zero interest rate once every 4(1−0.9946) = 46.3 years. The threshold of 2.55 is higher than the empirical
mean-standard deviation ratio 2.0248 during the etimation period.
27
Note that all three targeting rules considered in this section are not policy recipes in a strict sense. Instead of
giving a functional form for a policy instrument to follow, it describes how the instrument evolves with the other
variables on the RHS if inflation is somehow kept constant.

24
flexible. This can be seen from the log-linearized version of the pricing equation:

θY h i h i
Π b t−1 +
bt = Π d t + βg a(1−σ) Et Π
c t − mpl
rw bt
b t+1 − Π (31)
2
(1 − θY )Φp Π

b t at zero implies rw
where keeping Π d for all t.
c t = mpl t

5.1.2 Nominal Income Targeting (NIT)

Taking the first difference of the log-linearized version of (9) and using ∆pc b
t y t = Πt + ∆b
yt , I get

the nominal income targeting rule of the form

1 b 1 b
Rt = Rt−1 + ∆bbt + (1 − ν)∆b d
ct − (1 − ν)M Gt − (1 − ν)∆b
yt . (32)
R−1 R−1

This rule requires the current nominal rate increase over the previous level in response to positive

consumption growth and negative growth in nominal money stock and real output. This rule is

labelled as NIT for future reference

5.1.3 Money Growth Targeting (MGT) When the central bank stabilizes the aggregate

money growth rate, (9) implies

1 b 1 b
Rt = Rt−1 + ∆bbt + (1 − ν)∆b bt
ct + (1 − ν)Π (33)
R−1 R−1

Under the strict money growth targeting, therefore, the current interest rate relative to previous

rate is positively related with consumption growth and inflation rate. This rule will be labelled as

MGT for future reference.

5.1.4 Variants of Taylor Rules (TR, ATR) I consider two versions of rules that share the

spirit of Taylor (1993). The first one is an endogenous rule by which the central bank adjusts the

nominal rate gradually in response to price inflation and GDP gap:

bt = ρR
R bt−1 + γ 1 Π
b t + γ 2 ybt . (34)

25
Clarida et al. (1999) estimate the rule (34) over the tenure of Volker and Greenspan 1979:Q3-

1996:Q4, and obtain the estimates (ρ, γ 1 , γ 2 )=(0.79,0.4915,0.1953). When the rule (34) is imple-

mented with those estimates, however, the feasibility condition is violated. To obtain a feasible

solution of the model, I put (ρ, γ 1 , γ 2 )=(0.8,1.7,0.01) implying very small responses to current

output gap and aggressive response to inflation rate. This feasible rule is labelled TR for future

reference.

Also considered is another version of (34), augmented with wage inflation as another indicator:

R bt−1 + β 1 Π
bt = ρR b t + β 2 ybt + β 3 Π
bw
t (35)

labelled as ATR for future reference. I put (ρ, β 1 , β 2 , β 3 )=(0.8,1.7,0.01,0.17) in the following anal-

ysis.

5.2 Performances of Alternative Rules

Table 4 summarizes how the five alternative rules compare with regard to the current (i.e.,

Π = 1.01005), modest (i.e., Π = 1.0063), and low (i.e., Π = 1.0025) rates of reference inflation.

NIT and MGT achieve comparably higher welfare than PHIT does. In fact, at the current stance

of long-run inflation, adopting PHIT instead of the BM actually lowers welfare, while gains from

moving toward NIT or MGT is tantamount to about 2.2 units of additional one-time consumption.

The two versions of Taylor rules, TR and ATR, outperform all of the three fixed targeting rules

at all rates of reference inflation. This finding reflects the emphasis in the nearly entire literature

that a policy should bring target variables to designated levels only gradually. Because monetary

policy effects on nominal anchors are usually smooth and delayed, strict targeting rules like the

ones examined here necessarily lead to instability, which is penalized by the third term in the

welfare measure EW. Also observed in Table 4 is that augmenting TR with wage inflation leads

to additional welfare gains amounting to 2.14 units of consumption compensation for both current

26
and modest inflation stance.

In addition to the instability due to targeting a fixed path of aggregate price, there is another

reason for the poor performance of PHIT in the present model: the estimated structure of the

economy imposes a much higher degree of nominal rigidities in the labor market than in the goods

market. Intuitively, when monetary authority faces a trade-off in stabilizing output, price inflation,

and wage inflation, it is a better strategy to put more weight on stabilizing the more rigid nominal

variable (i.e., wage inflation in the present setup), thus letting the other more flexible one (i.e.,

price inflation in this model) account for a larger share of the adjustment process.28 Simply put, if

wages are not fully flexible, holding prices stable prevents the real wage from adjusting as it should.

Hence, assuming the estimated structure of the economy as granted, PHIT is chasing the wrong

variable. In view of this intuition, the good performance of MGT (relative to PHIT) is explained

as follows: in an economy where output is demand determined, monetary authority can stabilize

the economy by controlling a neutral (i.e., money growth) nominal anchor. In doing so, monetary

policy implicitly allows the more flexible one (i.e., price) between the two sticky nominal variables

to adjust more to the changes in economic environment.

5.3 Sensitivity Analysis

The results in the previous subsection hinge on two aspects of substitutions in the model: First, in

the intra-temporal context, the welfare measure EW is dependent upon the elasticity of substitution

between consumption and real balance (or, equivalently, the transaction technology) implied by

the utility function (4). Second, in the inter-temporal context, the attitude of households toward

cyclical variations also affects the welfare measure. In terms of a sensitivity checkup, the former

is related to the pair of parameters (ν, b) governing transaction technology, and the latter to the
28
This insight is provided by Aizenman and Frenkel (1986) in a static model, and by Erceg et al. (2000) in a
dynamic setting.

27
parameter σ measuring the degree of risk aversion. In particular, the money demand parameter b

is central to measuring the welfare costs incurred by the need to economize on non-interest-bearing

money.

Tables 5A-5E report how the initial ranking ATRÀTRÀNITÀMGTÀPHIT observed in

Table 4 changes for different values of the three parameters (σ, ν, b). In those tables, the reference

inflation rate is fixed at the benchmark level Π = 1.01005.

Table 5A compares the alternative rules for higher degrees of risk aversion, where the overall

ranking is quite different from that under σ = 1. In particular, the two versions of Taylor rules show

less impressive performance: when σ is raised to 1.3, for example, the once second best performer

TR finishes barely ahead of PHIT, and ATR steps down from the top to third place. The relative

performance MGT shows the most conspicuous improvement with higher degrees of risk aversion,

finishing first for higher degrees of risk aversion. PHIT is consistently the worst performer.

When it comes to the transaction technology, Tables 5B and 5C show that the initial ranking

is almost entirely preserved for higher values of ν and b alike. The only exceptions are when either

parameter is much higher than the benchmark value, in which cases NIT and MGT trade their

places.

The cases with b very close to 0 are worthy of consideration, since these cases correspond to the

abstraction of money. It is evident from Table 5C that ignoring money leads to the overprediction

of welfare level. In fact, higher values of b are associated with lower welfare levels for each rule

considered. Observing the near-invariance with respect to b of the ranking in Table 5C, however, I

interpret that the relative performances of rules are not affected by the size of welfare gains from

deflating the economy.

Noting that the benchmark value of ν = −22.7561 implies that the transaction technology is

highly insensitive to b, I also report the sensitivity with respect b for two higher values of ν =-12 and

28
-5 in Tables 5D and 5E, respectively. Generally, the initial ranking for the benchmark parameters

is preserved.

The analyses thus far have been in the context of the “structural” metric EW . Here, I consider

briefly how robust the previous results are when a conventional ad-hoc loss function evaluated at

the first order approximate solutions is employed as a performance criterion. More specifically, I

consider the following expected loss function constructed from the conditional variances:

hX∞ i
EL = E β t∗ Lt | Ω0 (36a)
t=0

where the period loss function Lt is constructed as

b t ) + λ1 var(b
Lt = var(Π bt − R
yt ) + λ2 var(R bt−1 ) (36b)

The expected loss function above is similar to that used in Rudebusch and Svensson (1999).

Tables 6A-C repeat the sensitivity analyses with Π = 1.01005 for different combinations of (λ1 , λ2 ).

Since the actual values of the expected loss in (36) are hard to interpret in economic terms, I only

report the ranking of the five alternative rules. The initial “non-structural” ranking evaluated at the

estimates in Table 1 are shown in the square brackets, displaying NITÀMGTÀATRÀTRÀPHIT

for all different weights (λ1 , λ2 ).

It is striking that the performance of alternative rules is highly sensitive to welfare measures

employed. In Tables 6A-6C, the new ranking under EL shows the two variants of Taylor rules are

now running behind the two fixed targeting rules NIT and MGT. In fact, as shown in Table 6C,

ATR and TR reclaim their thrones only when the non-structural measure EL is so constructed

that i) interest rate changes are severely penalized, and ii) transaction technology is more sensitive

to b due to higher values of ν.

PHIT, however, is the poorest performer under the new measure as well despite the fact that

the measure in (36) explicitly incorporates price inflation (not wage inflation) as an argument. It

29
ranks fourth under the current inflationary regime of Π = 1.01005. This finding corroborates the

non-optimality of strict price inflation targeting in an economy with a higher degree of nominal

rigidities in the labor market, as discussed in the previous subsection.

In section 4, it was demonstrated that the naive welfare measure based on first order approximate

solutions would have resulted in considerable distortions in welfare calculations. What directly

follows is whether such distortions are serious enough to reverse the ranking of policy rules and

result in the wrong policy implications for the monetary authority. Table 7 reports how the five

rules would compare with one another if they were evaluated in terms of the first order solutions.

What is striking is that the two decent fixed targeting rules, NIT and MGT, are now running

ahead of the two versions of Taylor rules. In fact, TR and ATR are worse than the benchmark

rule, requiring positive consumption compensation!

The intuition behind the (spurious) dominance of the two fixed targeting rules is as follows. As

was discussed above, fixed targeting rules incur inherent instabilities trying to force target variables

on track every period. This “behind the scene” welfare diminishing feature of fixed targeting rules
h i
is not fully captured unless the first order bias terms E b
ζ t | Ω0 in (27) are taken into account by

using second order approximate solutions in the construction of the welfare measure.

The fact that even the relative ranking of alternative rules is highly dependent upon the per-

formance measures suggests that the monetary authority should be cautious in choosing a proper

metric. Formulating loss function grounded upon the utility function of households is a justifiable

way to resolve this criterion-dependency. Of course, this is the case only insofar as the assumed

structure of the economy is free of controversy.

6. Welfare Analysis: Toward the Optimal Policy Rule

In this section, I construct an endogenous interest rate rule of the form

bt = ρR
R bt−1 + β 1 Π
b t + β 2M
d bw
Gt + β 3 Π bt + γ 2 ybt−1
t + γ1y (37)

30
which maximizes the welfare metric EW. On the RHS of (37), the money growth rate and lagged

real output are included as policy indicators in order to exploit the good performance of two

fixed targeting rules MGT and NIT shown in the preceding analysis.29 Having observing the

performance of ABM and ATR, I also include wage inflation as an essential indicator. In additions

to the coefficients on the RHS of (37), the reference inflation rate is also another, possibly the most

important, policy variable for monetary authority.

To find the optimized coefficients for the class of the rules in (37), one has to contend with

complicated boundaries defined by i) the need for the solution of the model to exist and be unique,

and ii) the need for policy rules to be feasible. At each maximization step, I check whether the

candidate parameters fall outside such boundaries, and assign an artificially large negative value

of EW to those cases. The resulting discontinuity at such boundaries is addressed again by the

optimization routine used for estimating the model.

Given the estimates of non-policy parameters in Table I, the optimized rule is

bt = 2.0156R
R bt−1 + 0.8429b
yt − 0.8463b b t + 0.6584Π
yt−1 + 0.8134Π bw d
t + 0.1791M Gt (38)

with the optimal reference inflation rate of Π = 0.9952.

The optimized rule (38), dubbed OPT, exhibits many features advocated in the literature as

what good monetary rules should have. First, it is a deflationary policy rule: the corresponding

annual rate of deflation is 1.91% which falls between the 2.93% of Friedman and 0.76% of Khan

et al. (2002). Second, nominal interest rate is adjusted with “super-inertia” in the terminology

of Woodford (1999). The coefficient 2.0156 on the lagged nominal rate is much higher than the

estimate 0.79 of Clarida et al. (1999) over the tenure of Volker and Greenspan 1979:Q3- 1996:Q4,

and 0.795 in Levin et al. (1999). According to Woodford (1999), the virtue of inertial adjustments
29
The estimated policy rule in Levin et al. (1999) shows that historical US monetary policy over 1980:Q1- 1996:Q4
responded to not only the level but also the recent growth rate of output. In their work, the estimated coefficients
on current and lagged real output are around 1 and -1, respectively.

31
in nominal rate is that they signal how serious monetary authority is about stabilizing its goal

variables even in the distant future, exploiting the forward looking behavior of the private sector

in forming their expectations. In fact, the optimal interest rate rules in Rotemberg and Woodford

(1997,1999) also exhibit super-inertia: for example, in Rotemberg and Woodford (1997), the largest

root 1.33 of the autoregressive polynomial for nominal rate is greater than one.

Third, OPT exhibits strong anti-inflationary adjustments of nominal rate. The coefficients on

the nominal anchors sum up to 1.6509, which is per se higher than the 1.5 in the simple rule of

Taylor (1993). With super-inertia, the long run degree of aggressiveness under the optimized rule is

literally explosive. It is particularly noteworthy that OPT embodies “targeting” for wage inflation

and money growth in the sense of McCallum and Nelson (1998), with coefficients 0.6584 and 0.

1791, respectively.

Fourth, the coefficients on current and real output in OPT show the monetary authority needs

to “lean against the wind”, by increasing the nominal rate in response to the increase in growth

rate of real output as well as its current levels. Coupled with the inflation coefficient of comparable

magnitude, those coefficients translates into a rule responding to the nominal income growth.

Hence, the optimized rule (38) has the feature of nominal income “targeting,” again in the usual

sense.

Table 8A reports the performance of the optimized rule. Implemented with the optimal degree of

deflation, the rule (38) yields considerable welfare gain over the benchmark rule with Π = 1.01005:

the households are willing to make 50.9776 units of one time sacrifice in consumption, which is

more than five times the steady state consumption.

It would be interesting to see if the welfare dominance of OPT is still preserved in economies

with higher long-run inflationary stance. Furthermore, it is in order to net out the effect of defla-

tionary stance to see how the other features of OPT contribute to welfare improvement. Therefore,

32
I compare in Table 8B the performance of OPT and the other five rules for different reference rates

of inflation, which shows an almost consistent dominance of OPT for both inflationary and defla-

tionary economies. The only exception is when Π = 1.01005, under which TR outperforms OPT

by an equivalent of 0.5593 units of one-time additional consumption.

Since the rule (37) is designed to mimic the two good targeting rules (NIT, MGT), it is worth

asking what common features of those rules contribute to the welfare improvement by OPT. To

get some insight, I plot the impulse responses of some key variables under four rules PHIT, NIT,

MGT, and OPT toward one unit of favorable technology shock in Figure 1 to 4, respectively.30

It is evident that even a casual eyeball test for choosing stabilizing policies would reject PHIT:

as displayed in Figure I, the costs of targeting the wrong variable appear as “boom-bust” responses

in money growth, real money, and output, and higher volatilities in wage inflation and nominal

interest rate.

Compared with PHIT, NIT generates much smoother or dampened responses (except for

inflation) as displayed in Figure II: real output shows monotone initial increases, and subsequent

adjustments are distributed over a very long horizon. Money growth also shows deviations of

considerably smaller magnitudes and shorter length. The initial below-zero responses of inflation

show that the goal of maintaining constant (up to a deterministic trend) nominal income is achieved

by suppressing price level. It is worth noting that, unlike under PHIT, almost all the burden of

nominal adjustments falls on the right variable (i.e., wage rate).

Figure III displays impulse responses under MGT. In comparison with NIT, MGT shows a

similar degree and longevity of responses in output and inflation, coupled with slightly smaller

volatilities in monetary variables such as money growth and interest rate. These findings strongly

support that controlling volatilities in the monetary sector also contributes to welfare improvements,
30
In those figures, the reference rate of inflation is set at 1.0.

33
which is evident from the fact that, as long as other things are held constant, the welfare metric

EW decreases in the volatility of real balance (or nominal interest rate in view of money demand

function (8)). Furthermore, as under NIT, the nearly constant responses of wage inflation suggest

that chasing the “neutral” nominal variable (i.e., money growth) in effect prevents the problem in

targeting the “wrong” variable (i.e., price).

The responses of the economy under OPT are displayed in Figure IV, where the impulse

responses under OPT show striking resemblance to those under NIT and MGT. In particular,

the two latter rules share with OPT i) the hump-shaped responses of output and real money; ii)

very small and smooth responses of money growth and interest rate; and, most of all, iii) near

constancy of wage inflation.

As alluded by the coefficients in OPT, the resemblance of impulse responses under OPT and

(NIT, MGT) is not a coincident: OPT inherits the virtues of MGT and NIT, by incorporat-

ing money growth as an indicator and by effectively “targeting” the growth in nominal income,

respectively. Also, by not putting the whole weight on price inflation in adjusting nominal rates to

the short run inflationary pressure, the optimized rule shuns the undesirable feature in PHIT of

favoring the wrong variable.

The responses of money growth and nominal rates suggest that OPT improves upon MGT

by mimicking NIT. As shown in Figure 2, NIT procyclically accommodates the initial increase in

real output by lowering nominal rates when a positive technology shock occurs.31 This feature is

present under OPT as well, although with very small magnitude. The cost of accommodation is

higher volatility in money growth under OPT than under MGT.

7. Conclusion
In this paper, I have applied an estimated monetary business cycle model with nominal rigidities
31
Ireland (1996) also argues optimal policy should be procyclical to supply shock, in the sense that positive tech-
nology shock is followed by an increase in money growth.

34
to evaluating performances of monetary policy rules. Performances are measured in terms of a

natural metric based on the utility function of agents, and the task of accurate welfare evaluation

is achieved owing to the second order approximate solution method.

The results suggest that in the presence of a higher degree of nominal rigidities in the labor mar-

ket than in the goods market, strict inflation targeting cannot be an optimal policy. The optimized

rule has a strict anti-inflation stance requiring that the nominal interest rate be aggressively ad-

justed to the increases in nominal anchors such as inflation, wage inflation, and money growth rate.

Furthermore, the optimized rule is deflationary in the spirit of Friedman, with long-run deflation

of 1.51% per annum. It also features a high degree of inertia, and the countercyclical adjustment

of nominal rates with respect to higher real output growth.

All of these results are highly dependent upon the estimated model and the performance measure

constructed from it. The price and wage adjustment scheme, degree of nominal rigidities in markets,

and the way nominal money enters the present model are particularly critical features that limit

the sense in which the optimized rule is really welfare improving. In particular, the instability

of money demand coupled with the advent of the evermore increasing interest bearing monetary

assets is an important issue to be addressed before taking the results in this paper as warranted.

One suggestion by Lucas (2000) of applying Divisia monetary index is a promising way for further

research to resolve this difficulty. I hope in the future work to extend the methodological framework

used in this paper to examine how robust the findings in this paper are in light of such critical

model aspects.

8. Appendix

8.1 Stationary Transform of the System Three different transform schemes are used to

make the system stationary in a symmetric equilibrium. First, all occurrences of deflated nominal

35
variables (Mt /Pt , Qt /Pt , Wt /Pt ) are re-defined as real variables:

RMt = Mt /Pt , RQt = Qt /Pt , RWt = Wt /Pt .

Second, real variables (Yt , Ct , Kt , Λt , M RSt , RMt , RQt , RWt ) are transformed using respective de-

terministic trend growth rates. For example:

yt = Yt /g t , ct = Ct /g t , kt = Kt /g t , λt = Λt /g [−1+a(1−σ)]t , rmt = RMt /g t .

Finally, occurrences of (Pt /Pt−1 , Wt /Wt−1 , Mt /Mt−1 ) are replaced by growth rates:

Πt = Pt /Pt−1 , Πw
t = Wt /Wt−1 , M Gt = Mt /Mt−1 .

For notational simplicity, I define

· ¸
gkt − (1 − δ t−1 )kt−1
xt = − δ g , δ g = g − 1 + δ, β g = βg a(1−σ)−1 .
kt−1

8.1.1 Household Block The stationary-transformed version of the households’ block of the

system is given below.

a−aσ−ν
λt = a[cνt + bt (rmt )ν ] ν (1 − Lt )(1−at )(1−σ) cν−1
t (A1)

1 − 1/Rt = bt (ct /rmt )1−ν (A2)

λt [1 + φK xt+1 ] = β g Z1t (A3)

· ¸
gkt+1 φK 2
Z1,t−1 = λt 1 − δ t + rqt + φK xt+1 − x + η1t (A4)
kt 2 t+1

λt = β g Rt Z2t (A5)

Z2,t−1 = λt Π−1
t + η 2t (A6)

£ ¤ w −1
mrst = θL rwt + (1 − θL )Φw Πw w
t − Πt−1 rwt Πt Lt

−βg a(1−σ) Φw (1 − θL )Z3t + βg a(1−σ) Φw (1 − θL )Z4t (A7)

36
 ∗a(1−σ)
mrst = λ−1
t ct (1 − at )(1 − Lt )(1−a)(1−σ)−1 (A8)

λt
Z3,t−1 = [Πw ]2 rwt L−1
t−1 + η 3t (A9)
λt−1 t
λt w w
Z4,t−1 = Π Π rwt L−1
t−1 + η 4t (A10)
λt−1 t t−1

where η0t s are the martingale difference expectational errors.

8.1.2 Firms Block The equations for the decision problems of firms are given by

Lt rqt 1 − αt
= (A11)
kt rwt αt

· ¸
rwt
λt θ Y − + (1 − θY )Φp (Πt − Πt−1 )Πt (A12)
mplt

= (1 − θY )βg a(1−σ) Φp Z5t − (1 − θY )βg a(1−σ) Φp Z6t

mplt = At (1 − αt )ktαt L−α

t
t
(A13)

yt
Z5,t−1 = λt [Πt ]2 + η5t (A14)
yt−1
yt
Z6,t−1 = λt Πt Πt−1 + η 6t (A15)
yt−1

8.1.3 Other Equations Combining the budget constraint of households, aggregate profit of

firms, and the government budget constraint, we get the resource constraint :
· ¸2
φ gkt+1 − (1 − δ t )kt
ct + gkt+1 − (1 − δ t )kt + K − δg kt = yt (A16)
2 kt

The aggregate real wage and real money stock evolve following

rwt Πω rmt M Gt
g = t, g = (A17)
rwt−1 Πt rmt−1 Πt

For the benchmark economy, the monetary policy rule is transformed into

h i
b b b d
Rt = ρR Rt−1 + (1 − ρR ) γ π Πt + γ y ybt + γ m M Gt + εMt (A18)

The exogenous shocks, stationary themselves, do not need to be transformed.

37
8.3 Recursive Calculations of (µt , Σt ) For the sake of second order accuracy, all terms of

orders higher than two may be dropped out: accordingly, only the first two terms in equation (28a)

describe the evolution of the conditional variances of zb1t :

Σ1t = F1 Σ1,t−1 F10 + F2 Σε F20 (A19)

where F1 and F2 are the matrices of the coefficients on zb1,t−1 and εt , respectively, representing the

first order parts of the solution.

Recursive calculations of µt are more involved. The subsystem (28a) may be re-written in an

expanded form as

zb1t = F1 zb1,t−1 + F2 εt + F3
     
0 (1) 0 0 (1) (1)
zb1,t−1 F11 zb1,t−1 zb1,t−1 F12 εt εt F22 εt
1  ..   ..  1 .. 
+   .
+
  .
+ 
 2 .  (A20)
2 (N1 ) 0 (N1 ) (N1 )
0
zb1,t−1 F11 zb1,t−1 0
zb1,t−1 F12 εt εt F22 εt

(i)0
where F3 is a N1 × 1 column vector, and Fjk s are the matrices constructing quadratic terms for

the ith equation in the second order solution (28a).

Taking expectation of (A20) conditional on Ω0 , I get

µ1t = F1 µ1,t−1 + F3 (A21)

 ³ ´   ³ ´ 
(1) (1)
tr Σ1,t−1 F11 tr Σε F22
1 ..
 1
  ..


+  . +  
2 ³ ´  2 ³ . ´ 
(N ) (N )
tr Σ1,t−1 F11 1 tr Σε F22 1

where tr(·) is the trace of a square matrix. One can calculate {µ1t , Σ1t : t ≥ 1} recursively by using

(A19) and (A21) jointly given some initial condition Ω0 = (µ10 , Σ10 ) .

38
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42
 Table 1: Estimation Results
Functional Forms Estimates and Standard Deviations

Yt = At Ktαt (g t Lt )1−αt A=5.5668(0.0718), g=1.0056(8.8×10−5 )

β t U (Ct , Lt , M
Pt£)
t
¤ β = 0.9986(0.0003),
t
= β log Ct∗a (1 − Lt )1−at a = 0.4681(0.0016)
1
Ct∗ = (Ctν + bt (Mt /Pt )ν ) ν ν=-22.7561(0.4765), b =0.0008(5.5×10−5 )
1
Lit = ( W
Wt )
it θ L −1
Lt θL = 0.6888(0.0087)
k φK It I 2
ACt = 2 ( Kt − K ) Kt Φk = 16.8456(1.5501)

Rt Rt−1
log
h R = ρR log R + (1 − ρM )× i ρR = 0.1395(0.0112), γ π =0.8042(0.0045)
γ π log ΠΠt + γ y log YYt + γ m log MG
MG
t
γ y =4.4×10−6 (4.5×10−5 ), γ m =0.4276(0.0187)
t
+εMt σ 2M = 4.3 × 10−5 (5.2 × 10−6 )
³ ´2
ΦP Pjt
ACitP = 2 Pj,t−1 − Πt−1 Yt Φp = 10.0970(0.7393)
³ ´2
Φw Wi Wt
ACitW = 2 Wi,t−1 − Πw
t−1 Pt Φw = 22.0341(0.7025)

log AAt = ρA log At−1A + εAt ρA =0.9761(0.0002), σ 2A =0.0012(9.9×10−5 )

log ααt = ρα log αt−1
α + εαt ρα =0.9690(0.0015), σ 2α =0.0003(2.7×10−5 )
At δ t−1
log δ = ρδ log δ + εδt ρδ =0.9563(0.0012), σ 2δ =0.0129(0.0021)
log AAt = ρb log bt−1
b + εbt ρb =0.9450(0.0022), σ 2b =0.0716(0.0078)
log aat = ρa log at−1
a + εat ρa =-0.4573(0.0482), σ 2a =0.2080(0.0302)
cov(εAt , εαt ) = −0.0006(5.0×10−5 )

43
 Table 2: Accuracy of the Solution
Endogenous First Order Solution Second Order Solution
Errors in R2 F − stat. R2 F − stat.
∂K 0.0649 289.1125 0.0648 288.6361
∂B 0.0092 38.6796 0.0088 36.9829
∂W [1] 0.0046 19.2504 0.0031 12.9536
∂W [2] 0.0037 15.4701 0.0021 8.7662
∂P [1] 0.0138 58.3900 0.0030 12.5345
∂P [2] 0.0272 116.4731 0.0032 13.3728

44
 Table 3: Performance of Benchmark Rule32
Ref. Inf . 1.01005 1.011 1.0091

I. BM

bt =
R bt−1 +
0.1395R b t+
0.6920Π 3.8×10−6 ybt d
+0.3680M Gt

EW 607.8661 607.7265 608.0061∗

dC - 2.9042 -2.9109∗

II. ABM

bt =
R bt−1 +
0.1395R b t+
0.6920Π 3.8×10−6 ybt d
+0.3680M Gt bw
+0.3460Πt

EW 608.1288 607.9916 608.2603

dC -5.4633 -2.6103 -8.3205

32
Result with asterisk is feasible with a 2-SD bound around the unconditional expectation of nominal interest rate.

45
 Table 4: Performances of Alternative Rules
Ref. Inf. 1.01005 1.0063 1.0025

I. PHIT 1 b
R=
R−1 t
R +∆bbt
1 b
R−1 t−1
+(1-ν) ∆b
ct d
-(1-ν) M Gt

EW 607.5599 607.8834 607.8954

dC 6.3672 -0.3595 -0.6082

II. NIT 1 b
R=
R−1 t
R +∆bbt
1 b
R−1 t−1
+(1-ν) ∆bct
d
-(1-ν) M Gt -(1-ν)∆b
yt

EW 607.9744 608.5208 609.0646

dC -2.2527 -13.6072 -24.8974

III. MGT 1 b
R=
R−1 t
R +∆bbt
1 b
R−1 t−1
+(1-ν) ∆b
ct bt
+(1-ν) Π

EW 607.9720 608.5304 609.1142

dC -2.2014 -13.8818 -25.9267

IV. TR bt =
R bt−1
0.8R bt
+ 1.7Π + 0.01b
yt

EW 608.0639 608.6387∗ Unfeasible

dC -4.1124 -16.0591∗

V. ATR bt =
R bt−1 +
0.8R b t+
1.7Π 0.01b
yt bw
+0.17Πt

EW 608.1667 608.7401∗ Unfeasible

dC -6.2504 -18.1836∗

46
 Table 5A: Sensitivity Analysis

σ 1.1 1.2 1.3 1.4 1.5

PHIT 510.5009 [5] 439.8141 [5] 386.0093 [5] 343.6682 [5] 309.4715 [5]
NIT 510.8381 [1] 440.0892 [2] 386.2328 [2] 343.8482 [3] 309.6140 [4]
MGT 510.7469 [3] 440.1430 [1] 386.5505 [1] 343.4870 [1] 310.5966 [1]
TR 510.6630 [4] 439.8604 [4] 386.0635 [4] 343.7981 [4] 309.7150 [3]
ATR 510.7599 [2] 439.9529 [3] 386.1526 [3] 343.8851 [2] 309.7788 [2]

Table 5B: Sensitivity Analysis (Cont.)

ν -20 -15 -12 -8 -5
PHIT 607.6681 [5] 607.8090 [5] 607.8333 [5] 607.7123 [5] 607.3538 [5]
NIT 608.1090 [3] 608.3410 [3] 608.4654 [3] 608.5986 [3] 608.6517 [4]
MGT 608.1027 [4] 608.3239 [4] 608.4432 [4] 608.5901 [4] 608.7473 [3]
TR 608.2178 [2] 608.4858 [2] 608.6354 [2] 608.8099 [2] 608.9067 [2]
ATR 608.3201 [1] 608.5869 [1] 608.7357 [1] 608.9086 [1] 609.0035 [1]

Table 5C: Sensitivity Analysis (Cont.)

b 10−10 10−5 10−4 10−3 10−2
PHIT 609.0876 [5] 608.0913 [5] 607.8300 [5] 607.5430 [5] 607.2280 [5]
NIT 609.2143 [3] 608.4072 [3] 608.1946 [3] 608.9607 [3] 608.7033 [4]
MGT 609.1882 [4] 608.3967 [4] 608.1881 [4] 608.9584 [4] 608.7059 [3]
TR 609.3986 [2] 608.5312 [2] 608.3018 [2] 608.0490 [2] 608.7703 [2]
ATR 609.4957 [1] 608.6320 [1] 608.4036 [1] 608.1518 [1] 609.8744 [1]

Table 5D: Sensitivity Analysis (Cont.)33

b 10−10 10−5 10−4 10−3 10−2
PHIT 609.8504 [4]* 608.6501 [5] 608.2625 [5] 607.8053 [5] 607.2666 [5]
NIT 609.9169 [3] 609.0610 [3] 608.7796 [3] 608.4448 [3] 608.0462 [4]
MGT 609.7902 [5] 608.9970 [4] 608.7357 [4] 608.4240 [4] 608.0524 [3]
TR 610.1601 [2] 609.2627 [2] 608.9666 [2] 608.6136 [2] 608.1926 [2]
ATR 610.2538 [1] 609.3603 [1] 609.0655 [1] 608.7140 [1] 609.2948 [1]

Table 5E: Sensitivity Analysis (Cont.)34

b 10−10 10−5 10−4 10−3 10−2
PHIT 610.4774 [5] 609.0797 [5] 608.3393 [5] 607.2833 [5] 605.7927 [5]
NIT 610.3910 [3] 609.6310 [3] 609.2164 [3] 608.6106 [3] 607.7259 [4]
MGT 609.8523 [4] 609.3766 [4] 609.1123 [4] 608.7204 [4] 608.1361 [2]
TR 610.6599 [2] 609.8956 [2] 609.4774 [2] 608.8652 [2] 607.9687 [3]
ATR 610.7512 [1] 609.9893 [1] 609.5725 [1] 608.9621 [1] 609.0684 [1]
33
ν is set at -12. Results with asterisk are feasible with a 2-SD bound around the unconditional expectation of
nominal interest rate.

34
ν is set at -5.

47
 Table 6A: Non-structural measures: (λ1 , λ2 ) = (1, 0.5)
σ ν b= 10−x
1.1 1.2 1.3 1.4 1.5 -20 -15 -12 -8 -5 -10 -5 -4 -3 -2
PHIT[5] 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
NIT[1] 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1
NGT[2] 2 2 2 1 1 2 2 2 2 2 2 2 2 2 2
TR[4] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
ATR[3] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Table 6B: Non-structural measures: (λ1 , λ2 ) = (0.2, 0.5)

σ ν b= 10−x
1.1 1.2 1.3 1.4 1.5 -20 -15 -12 -8 -5 -10 -5 -4 -3 -2
PHIT [5] 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
NIT [1] 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1
NGT [2] 2 2 2 1 1 2 2 2 2 4 2 2 2 2 2
TR [4] 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4
ATR [3] 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3

Table 6C: Non-structural measures: (λ1 , λ2 ) = (1, 1)

σ ν b= 10−x
1.1 1.2 1.3 1.4 1.5 -20 -15 -12 -8 -5 -10 -5 -4 -3 -2
PHIT [5] 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
NIT [1] 1 1 1 2 2 1 1 1 4 4 1 1 1 1 1
NGT [2] 2 2 2 1 1 2 2 2 3 3 2 2 2 2 2
TR [4] 4 4 4 4 4 4 4 4 2 2 4 4 4 4 4
ATR [3] 3 3 3 3 3 3 3 3 1 1 3 3 3 3 3

48
 Table 7: Performances of Alternative Rules:
First Order Approx. Solutions35
Ref. Inf. 1.01005 1.0063 1.0025

I. PHIT
EW 602.4210 603.0270 603.6425
dC 5.7420 -6.8564 -19.6390

II. NIT
EW 602.9524 603.5398 604.1416
dC -5.3098 -17.5160 -30.0074

III. MGT
EW 602.9539 603.5397 604.1408
dC -5.3397 -17.5145 -29.9919

IV. TR
EW 602.5488 603.1506* Unfeasible
dC 3.0849 -9.4263*

V. ATR
EW 602.5512 603.1521* Unfeasible
dC 3.0339 -9.4564*
35
Results with asterisk are feasible with a 2-SD bound around the unconditional expectation of nominal interest
rate. dC measures are relative to the benchmark rule, which is also evaluated via the first oreder approximate
solutions.

49
 Table 8A: Performance of Optimized Rules
Ref. Infla 0.9952 1.01005 1.0063 1.0025

OPT

bt =
R bt−1 +
2.0156R 0.8429ybt - 0.8463ybt−1 + b t+
0.8139Π bw
0.6584Πt +
d
0.1791M Gt

EW 610.3226 608.1398 608.6909 609.2489

dC -50.9756 -5.6928 -16.7500 -28.6143

Table 8B: Comparison of OPT and Alternative Rules36

Ref. Inf. 0.9952 0.9975 1 1.01005 1.025 1.05

OPT 610.3226 [1] 609.9634 [1] 609.5681[1] 608.1398 [2] 606.1331 [1] 603.0531[1]

PHIT Unfeasible 604.0447 [4] 607.1164[4] 607.5599 [6] 605.7222 [6] 602.5542[6]

NIT 609.5016 [2] 609.6620 [3] 609.3711[3] 607.9744 [4] 605.9756 [4] 602.8985[4]

MGT Unfeasible 609.9330 [2] 609.4821[2] 607.9720 [5] 605.9584 [5] 602.8871[5]

TR Unfeasible Unfeasible Unfeasible 608.0639 [3] 605.9836 [3] 602.7932[3]

ATR Unfeasible Unfeasible Unfeasible 608.1667 [1] 606.0886 [2] 602.9022[2]

36
b is set at the benchmark estimate 0.0008.

50
 Output Inflation
5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(a) (b)

Real Balance Wage Inflation

5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(c) (d)

Money Growth Interest Rate

5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(e) (f)

Figure 1: Impulse Responses under PHIT

51
 Output Inflation
5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(a) (b)

Real Balance Wage Inflation

5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(c) (d)

Money Growth Interest Rate

5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(e) (f)

Figure 2: Impulse Responses under NIT

52
 Output Inflation
5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(a) (b)

Real Balance Wage Inflation

5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(c) (d)

Money Growth Interest Rate

5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(e) (f)

Figure 3: Impulse Responses under MGT

53
 Output Inflation
5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(a) (b)

Real Balance Wage Inflation

5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(c) (d)

Money Growth Interest Rate

5 0.6

0.4

0.2

0 0

-0.2

-0.4

-5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
(e) (f)

Figure 4: Impulse Responses under OPT

54