Economic Quarterly—Volume ——Pages 1–Please run LaTeX on this file again to get end page number!
Policy Implications of the
New Keynesian Phillips
Curve
Stephanie Schmitt-Groh´e and Mart´ın Uribe
T
he theoretical framework within which optimal monetary policy was
studied before the arrival of the New Keynesian Phillips curve (NKPC),
but after economists had become comfortable using dynamic, opti-
mizing, general equilibrium models and a welfare-maximizing criterion for
policy analysis, was one in which the central source of nominal nonneutrality
was a demand for money. At center stage in this literature was the role of
money as a medium of exchange (as in cash-in-advance models, money-in-
the-utility-function models, or shopping-time models) or as a store of value (as
in overlapping-generations models). In the context of this family of models a
robust prescription for the optimal conduct of monetary policy is to set nomi-
nal interest rates to zero at all times and under all circumstances. This policy
implication, however, found no fertile ground in the boardrooms of central
banks around the world, where the optimality of zero nominal rates was dis-
missed as a theoretical oddity, with little relevance for actual central banking.
Thus, theory and practice of monetary policy were largely disconnected.
The early 1990s witnessed a profound shift in monetary economics away
from viewing the role of money primarily as a medium of exchange and
toward viewing money—sometimes exclusively—as a unit of account. A key
insight was that the mere assumption that product prices are quoted in units
of fiat money can give rise to a theory of price level determination, even if
money is physically nonexistent and even if fiscal policy is irrelevant for price
Stephanie Schmitt-Groh´e is affiliated with Columbia University, CEPR, and NBER. She can
in this paper do not necessarily reflect those of the Federal Reserve Bank of Richmond or the
Federal Reserve System. The authors would like to thank Andreas Hornstein and Alexander
Wolman for many thoughtful comments.
2 Federal Reserve Bank of Richmond Economic Quarterly
level determination.
1
This theoretical development was appealing to those
who regard modern payment systems as operating increasingly cashlessly.
At the same time, nominal rigidities in the form of sluggish adjustment of
product and factor prices gained prominence among academic economists.
The incorporation of sticky prices into dynamic stochastic general equilibrium
models gave rise to a policy tradeoff between output and inflation stabilization
that came to be known as the New Keynesian Phillips curve.
The inessential role that money balances play in the New Keynesian liter-
ature, along with the observed actual conduct of monetary policy in the United
States and elsewhere over the past 30 years, naturally shifted theoretical in-
terest away from money growth rate rules and toward interest rate rules: In
the work of academic monetary economists, Milton Friedman’s celebrated
k-percent growth path for the money supply gave way to Taylor’s equally
influential interest rate feedback rule.
In this article, we survey recent advancements in the theory of optimal
monetary policy in models with a New Keynesian Phillips curve. Our survey
identifies a number of important lessons for the conduct of monetary policy.
First, optimal monetary policy is characterized by near price stability. This
policy implication is diametrically different from the one that obtains in models
in which the only nominal friction is a transactions demand for money. Second,
simple interest rate feedback rules that respond aggressively to price inflation
deliver near-optimal equilibrium allocations. Third, interest rate rules that
respond to deviations of output from trend may carry significant welfare costs.
Taken together, lessons one through three call for the adherence to an inflation
targeting objective. Fourth, the zero bound on nominal interest rates does not
appear to be a significant obstacle for the actual implementation of low and
stable inflation. Finally, product price stability emerges as the overriding goal
of monetary policy even in environments where not only goods prices but also
factor prices are sticky.
Before elaborating on the policy implications of the NKPC, we provide
some perspective by presenting a brief account of the state of the literature on
optimal monetary policy before the advent of the New Keynesian revolution.
1. OPTIMAL MONETARY POLICY PRE-NKPC
Within the pre-NKPC framework, under quite general conditions, optimal
monetary policy calls for a zero opportunity cost of holding money, a result
known as the Friedman rule. In fiat money economies in which assets used
for transactions purposes do not earn interest, the opportunity cost of holding
money equals the nominal interest rate. Therefore, in the class of models
1
This is the case, for instance, when the monetary stance is active and the fiscal stance is
passive, which is the monetary/fiscal regime most commonly studied.
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 3
commonly used for policy analysis before the emergence of the NKPC, the
optimal monetary policy prescribed that the riskless nominal interest rate—the
return on federal funds, say—be set at zero at all times.
In the early literature, a demand for money is motivated in a variety of
ways, including a cash-in-advance constraint (Lucas 1982), money in the
utility function (Sidrauski 1967), a shopping-time technology (Kimbrough
1986), or a transactions-cost technology (Feenstra 1986). Regardless of how
a demand for money is introduced, the intuition for why the Friedman rule is
optimal in this class of model is straightforward: A zero nominal interest rate
maximizes holdings of a good—real money balances—that has a negligible
production cost. Another reason why the Friedman rule is optimal is that
a positive interest rate can distort the efficient allocation of resources. For
instance, in the cash-in-advance model with cash and credit goods, a positive
interest rate distorts the allocation of private spending across these two types of
goods. In models in which money ameliorates transaction costs or decreases
shopping time, a positive interest rate introduces a wedge in the consumption-
leisure choice.
To illustrate the optimality of the Friedman rule, we augment a neoclassical
model with a transaction technology that is decreasing in real money holdings
and increasing in consumption spending. Specifically, consider an economy
populated by a large number of identical households. Each household has
preferences defined over processes of consumption and leisure and described
by the utility function
E
0
∞
t=0
β
t
U(c
t
,h
t
), (1)
where c
t
denotes consumption, h
t
denotes labor effort, β ∈ (0, 1) denotes
the subjective discount factor, and E
0
denotes the mathematical expectation
operator conditional on information available in period 0. The single period
utility function, U, is assumed to be increasing in consumption, decreasing in
effort, and strictly concave.
Final goods are produced using a production function, z
t
F(h
t
), that takes
labor, h
t
, as the only factor input and is subject to an exogenous productivity
shock, z
t
.
A demand for real balances is introduced into the model by assuming
that money holdings, denoted M
t
, facilitate consumption purchases. Specif-
ically, consumption purchases are subject to a proportional transaction cost,
s(v
t
), that is decreasing in the household’s money-to-consumption ratio, or
consumption-based money velocity,
v
t
=
P
t
c
t
M
t
, (2)
where P
t
denotes the nominal price of the consumption good in period t. The
transaction cost function, s(v), satisfies the following assumptions: (a) s(v)
4 Federal Reserve Bank of Richmond Economic Quarterly
is nonnegative and twice continuously differentiable; (b) there exists a level
of velocity, v
> 0, to which we refer as the satiation level of money, such
that s(v
) = s
(v) = 0; (c) (v − v)s
(v) > 0 for v = v; and (d) 2s
(v) +
vs
(v) > 0 for all v ≥ v. Assumption (b) ensures that the Friedman rule (i.e.,
a zero nominal interest rate) need not be associated with an infinite demand
for money. It also implies that both the transaction cost and the distortion
it introduces vanish when the nominal interest rate is zero. Assumption (c)
guarantees that in equilibrium money velocity is always greater than or equal
to the satiation level. Assumption (d) ensures that the demand for money is
decreasing in the nominal interest rate.
Households are assumed have access to risk-free pure discount bonds,
denoted B
t
. These bonds are assumed to carry a gross nominal interest rate
of R
t
when held from period t to period t + 1. The flow budget constraint of
the household in period t is then given by
P
t
c
t
[1 + s(v
t
)] + P
t
τ
L
t
+ M
t
+
B
t
R
t
= M
t−1
+ B
t−1
+ P
t
z
t
F(h
t
), (3)
where τ
L
t
denotes real lump sum taxes. In addition, it is assumed that the
household is subject to a borrowing limit that prevents it from engaging in
Ponzi-type schemes. The government is assumed to follow a fiscal policy
whereby it rebates any seigniorage income it receives from the creation of
money in a lump sum fashion to households.
A stationary competitive equilibrium can be shown to be a set of plans {c
t
,
h
t
, v
t
}, satisfying the following three conditions:
v
2
t
s
(v
t
) =
R
t
− 1
R
t
, (4)
−
U
h
(c
t
,h
t
)
U
c
(c
t
,h
t
)
=
z
t
F
(h
t
)
1 + s(v
t
) + v
t
s
(v
t
)
, and (5)
[1 + s(v
t
)]c
t
= z
t
F(h
t
), (6)
given monetary policy {R
t
}, with R
t
≥ 1, and the exogenous process {z
t
}.
The first equilibrium condition can be interpreted as a demand for money
or liquidity preference function. Given our maintained assumptions about
the transactions technology, s(v
t
), the implied money demand function is
decreasing in the gross nominal interest rate, R
t
. Further, our assumptions
imply that as the interest rate vanishes, or R
t
approaches unity, the demand
for money reaches a finite maximum level given by c
t
/v
. At this level of
money demand, households are able to perform transactions costlessly, as the
transactions cost, s(v
t
), becomes nil. The second equilibrium condition shows
that a level of money velocity above the satiation level, v
, or, equivalently, an
interest rate greater than zero, introduces a wedge between the marginal rate of
substitution of consumption for leisure and the marginal product of labor. This
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 5
wedge, given by 1 + s(v
t
) + v
t
s
(v
t
), induces households to move away from
consumption and toward leisure. The wedge is increasing in the nominal
interest rate, implying that the larger is the nominal interest rate, the more
distorted is the consumption-leisure choice. The final equilibrium condition
states that a positive interest rate entails a resource loss in the amount of s(v
t
)c
t
.
This resource loss is increasing in the interest rate and vanishes only when the
nominal interest rate equals zero.
We wish to characterize optimal monetary policy under the assumption
that the government has the ability to commit to policy announcements. This
policy optimality concept is known as Ramsey optimality. In the context of
the present model, the Ramsey optimal monetary policy consists in choosing
the path of the nominal interest rate that is associated with the competitive
equilibrium that yields the highest level of welfare to households. Formally,
the Ramsey policy consists in choosing processes R
t
, c
t
, h
t
, and v
t
to maximize
the household’s utility function given in equation (1) subject to the competitive
equilibrium conditions given by equations (4) through (6).
When one inspects the three equilibrium conditions above, it is clear that if
the policymaker sets the monetary policy instrument, which we take to be the
nominal interest rate, such that velocity is at the satiation level, (v
t
= v), then
the equilibrium conditions become identical to an economy without the money
demand friction, i.e., c
t
= z
t
F(h
t
) and −U
h
(c
t
,h
t
)/U
c
(c
t
,h
t
) = z
t
F
(h
t
).
Because the real allocation in the absence of the monetary friction is Pareto
optimal, the proposed monetary policy must be Ramsey optimal. By a Pareto
optimal allocation we mean a feasible real allocation (i.e., one satisfying c
t
=
z
t
F [h
t
]) with the property that any other feasible allocation that makes at least
one agent better off also makes at least one agent worse off. It follows from
equation (4) that setting the nominal interest rate to zero (R
t
= 1) ensures
that v
t
= v. For this reason, optimal monetary policy takes the form of a zero
nominal interest rate at all times.
Under the optimal monetary policy, the rate of change in the aggregate
price level varies over time. Because, to a first approximation, the nominal
interest rate equals the sum of the real interest rate and the expected rate of
inflation, and because under the optimal monetary policy the nominal interest
rate is held constant, the degree to which the inflation rate fluctuates depends
on the equilibrium variations in the real rate of interest. In general, opti-
mal monetary policy in a model in which a role for monetary policy arises
solely from the presence of money demand is not characterized by inflation
stabilization.
A second important consequence of optimal monetary policy in the context
of the present model is that inflation is, on average, negative. This is because,
with a zero nominal interest rate, the inflation rate equals, on average, the
negative of the real rate of interest.
6 Federal Reserve Bank of Richmond Economic Quarterly
2. THE NKPC AND OPTIMAL MONETARY POLICY
The New Keynesian Phillips curve can be briefly defined as the dynamic
output-inflation tradeoff that arises in a dynamic general equilibrium model
populated by utility-maximizing households and profit-maximizing firms—
such as the one laid out in the previous section—and augmented with some
kind of rigidity in the adjustment of nominal product prices. The foundations
of the NKPC were laid by Calvo (1983) and Rotemberg (1982). Woodford
(1996) and Yun (1996) completed its development by introducing optimizing
behavior on the part of firms facing Calvo-type dynamic nominal rigidities.
The most important policy implication of models featuring a New Key-
nesian Phillips curve is the optimality of price stability (see Goodfriend and
King [1997] for an early presentation of this result). We will discuss the price
stability result in a variety of theoretical models, including ones with a realistic
set of real and nominal rigidities, policy instruments and policy constraints,
and sources of aggregate fluctuations. We start, however, with the simplest
structure within which the price stability result can be obtained. To this end,
we strip the model presented in the previous section from its money demand
friction and instead introduce costs of adjusting nominal product prices. In the
resulting model, sticky prices represent the sole source of nominal friction.
To introduce sticky prices into the model of the previous section, assume
that the consumption good, c
t
, is a composite good made of a continuum of
intermediate differentiated goods. The aggregator function is of the Dixit-
Stiglitz type. Each household/firm unit is the monopolistic producer of one
variety of intermediate goods. In turn, intermediate goods are produced using
a technology like the one given in the previous section. The household/firm
unit hires labor,
˜
h
t
, from a perfectly competitive market.
The demand faced by the household/firm unit for the intermediate in-
put that it produces is of the form Y
t
d(
˜
P
t
/P
t
), where Y
t
denotes the level
of aggregate demand, which is taken as exogenous by the household/firm
unit;
˜
P
t
denotes the nominal price of the intermediate good produced by the
household/firm unit; and P
t
is the price of the composite consumption good.
The demand function, d(·), is assumed to be decreasing in the relative price,
˜
P
t
/P
t
, and is assumed to satisfy d(1) = 1 and −d
(1) ≡ η>1, where η
denotes the price elasticity of demand for each individual variety of interme-
diate goods that prevails in a symmetric equilibrium. The restrictions on d(1)
and d
(1) are necessary for the existence of a symmetric equilibrium. The
monopolist sets the price of the good it supplies, taking the level of aggregate
demand as given, and is constrained to satisfy demand at that price, that is,
z
t
F(
˜
h
t
) ≥ Y
t
d(
˜
P
t
/P
t
).
Price adjustment is assumed to be sluggish, `a la Rotemberg (1982). Specif-
ically, the household/firm unit faces a resource cost of changing prices that is
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 7
quadratic in the inflation rate of the good it produces:
Price adjustment cost =
θ
2
˜
P
t
˜
P
t−1
− 1
2
. (7)
The parameter θ measures the degree of price stickiness. The higher is θ ,
the more sluggish is the adjustment of nominal prices. When θ equals zero,
prices are fully flexible. The flow budget constraint of the household/firm unit
in period t is then given by
c
t
+ τ
L
t
≤ (1 − τ
D
t
)w
t
h
t
+
⎡
⎣
˜
P
t
P
t
Y
t
d
˜
P
t
P
t
− w
t
˜
h
t
−
θ
2
˜
P
t
˜
P
t−1
− 1
2
⎤
⎦
,
where τ
D
t
denotes an income tax/subsidy rate. We introduce this fiscal instru-
ment as a way to offset the distortions arising from the presence of monopo-
listic competition. We restrict attention to a stationary symmetric equilibrium
in which all household/firm units charge the same price for the intermediate
good they produce. Letting π
t
≡ P
t
/P
t−1
denote the gross rate of inflation,
the complete set of equilibrium conditions is then given by
π
t
(π
t
− 1) =
ηc
t
θ
w
t
z
t
F
(h
t
)
−
η − 1
η
+ βE
t
U
c
(c
t+1
,h
t+1
)
U
c
(c
t
,h
t
)
π
t+1
(π
t+1
− 1),
(8)
−
U
h
(c
t
,h
t
)
U
c
(c
t
,h
t
)
= (1 − τ
D
t
)w
t
, and (9)
z
t
F(h
t
) −
θ
2
(π
t
− 1)
2
= c
t
. (10)
The above three equations provide solutions for the equilibrium processes of
consumption, c
t
, hours, h
t
, and the real wage, w
t
, given processes for the rate
of inflation, π
t
, and for the tax rate, τ
D
t
, which we interpret to be outcomes of
the monetary and fiscal policies in place, respectively.
The first equilibrium condition, equation (8), represents the NKPC, to
which the current volume is devoted. It describes an equilibrium relation-
ship between current inflation, π
t
, the current deviation of marginal cost,
w
t
/z
t
F
(h
t
), from marginal revenue, (η − 1)/η, and expected future inflation.
Under full price flexibility, the firm would always set marginal revenue equal
to marginal cost. However, in the presence of price adjustment costs, this
practice is costly. To smooth out price changes over time, firms set prices to
equate an average of current and future expected marginal costs to an average
of current and future expected marginal revenues. This optimal price-setting
behavior gives rise to a relation whereby, given expected inflation, current in-
flation is an increasing function of marginal costs. Intuitively, this relation is
steeper the more flexible are prices (i.e., the lower is θ), the more competitive
8 Federal Reserve Bank of Richmond Economic Quarterly
are product markets (i.e., the higher is η), and the higher is the current level
of demand (i.e., the larger is c
t
). At the same time, given marginal cost, cur-
rent inflation is increasing in expected future inflation. This is because, with
quadratic costs of changing nominal prices, a firm expecting higher inflation
in the future would like to smooth out the necessary price adjustments over
time by beginning to raise prices already in the current period.
We have derived the New Keynesian Phillips curve in the context of the
Rotemberg (1982) model of price stickiness. However, a similar relation-
ship emerges under other models of nominal rigidity, such as those due to
Calvo (1983), Taylor (1993), Woodford (1996), and Yun (1996). For instance,
in the Calvo-Woodford-Yun model, price stickiness arises because firms are
assumed to receive an idiosyncratic random signal each period indicating
whether they are allowed to reoptimize their posted prices. A difference be-
tween the Rotemberg and the Calvo-Woodford-Yun models is that the latter
displays equilibrium price dispersion across firms even in the absence of ag-
gregate uncertainty. However, up to first order, the NKPCs implied by the
Rotemberg and Calvo-Woodford-Yun models are identical. Indeed, much of
the literature on the NKPC focuses on a log-linear approximation of this key
relationship, as in equation (11).
The second equilibrium condition presented in equation (9) states that
the marginal rate of substitution of consumption for leisure is equated to the
after-tax real wage rate. The third equilibrium condition, equation (10), is
a resource constraint requiring that aggregate output net of price adjustment
costs equal private consumption.
It is straightforward to establish that, in this economy, the optimal mone-
tary policy, that is, the policy that maximizes the welfare of the representative
household, is one in which the inflation rate is nil at all times. Formally, the
optimal monetary policy must be consistent with an equilibrium in which
π
t
= 1
for all t ≥ 0. This result holds exactly provided the fiscal authority sub-
sidizes labor income to a point that fully offsets the distortion arising from
the existence of imperfect competition among intermediate goods producers.
Specifically, the income tax rate, τ
D
t
, must be set at a constant and negative
level given by
τ
D
t
=
1
1 − η
for all t ≥ 0.
To see that the proposed policy regime is optimal, we demonstrate that it
implies a set of equilibrium conditions that coincide with the one that arises
in an economy with fully flexible prices (θ = 0) and perfect competition in
product markets (η =∞), such as the one analyzed in Section 1 in the absence
of a money-demand distortion. In effect, when π
t
= 1 for all t, equilibrium
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 9
condition (10) collapses to c
t
= z
t
F(h
t
). In addition, under zero inflation
the NKPC (equation [8]) reduces to w
t
= z
t
F
(h
t
)(η − 1)/η. Using this
expression along with the proposed optimal level for the income tax rate in
the equilibrium labor supply, equation (9), yields the efficiency conditions
−U
h
(c
t
,h
t
)/U
c
(c
t
,h
t
) = z
t
F
(h
t
), which, together with the resource con-
straint c
t
= z
t
F(h
t
), constitute the equilibrium conditions of a perfectly
competitive flexible-price economy. As we show in Section 1, the resource
allocation in this economy is Pareto optimal.
3. THE OPTIMAL INFLATION RATE
At this point, it is of interest to summarize and compare the results in this
section and in previous ones. We have shown that when prices are fully
flexible and the only nominal friction is a demand for money, then optimal
monetary policy takes the form of complete stabilization of the interest rate at
a value of zero (R
t
= 1 for all t). We have also established that in a cashless
economy in which the only source of nominal friction is given by product
price stickiness, optimal monetary policy calls for full stabilization of the rate
of inflation at a value of zero (π
t
= 1 for all t). Under optimal policy in the
monetary flexible-price economy, inflation is time varying and equal to the
negative of the real interest rate on average, whereas in the cashless sticky-
price economy, inflation is constant and equal to zero at all times. Also, in the
monetary flexible-price economy, optimal policy calls for a constant nominal
interest rate equal to zero at all times, whereas in the cashless sticky-price
economy, it calls for a time-varying nominal interest rate equal to the real
interest rate on average.
These results raise the question of what the characteristics of optimal
monetary policy are in a more realistic economic environment in which both
a demand for money and price stickiness coexist. In particular, in such an
environment a policy tradeoff emerges between the benefits of targeting zero
inflation—i.e., minimizing price-adjustment costs—and the benefits of deflat-
ing at the real rate of interest—i.e., minimizing the opportunity cost of holding
money. In the canonical economies with only one nominal friction studied in
this and previous sections, the characterization of the optimal rate of inflation
is relatively straightforward. As soon as both nominal frictions are incorpo-
rated jointly, it becomes impossible to determine the optimal rate of inflation
analytically. One is therefore forced to resort to numerical methods.
The resolution of the Friedman-rule-versus-price-stability tradeoff was
studied by, among others, Khan, King, and Wolman (2003) and Schmitt-Groh´e
and Uribe (2004a, 2007b). As one would expect, when both the money demand
and sticky-price frictions are present, the optimal rate of inflation falls between
zero and the one called for by the Friedman rule. The question of interest,
however, is where exactly in this interval the optimal inflation rate lies. Khan,
10 Federal Reserve Bank of Richmond Economic Quarterly
King, and Wolman find, in the context of a stylized model calibrated to match
aspects of money demand and price dynamics in the postwar United States,
that the optimal rate of inflation is −0.76 percent per year. By comparison, in
their model the Friedman rule is associated with a deflation rate of 2.93 percent
per year. Thus, in the study by Khan, King, and Wolman, the optimal policy
is closer to price stability than to the Friedman rule. Taking these numbers at
face value, one might conclude that price stickiness is the dominant friction
in shaping optimal monetary policy. However, Khan, King, and Wolman
(2003) and Schmitt-Groh´e and Uribe (2004a, 2007b) show that the resolution
of the tradeoff is quite sensitive to plausible changes in the values taken by
the structural parameters of the model.
In Schmitt-Groh´e and Uribe (2007b), we find that a striking character-
istic of the optimal monetary regime is the high sensitivity of the welfare-
maximizing rate of inflation with respect to the parameter governing the de-
gree of price stickiness for the range of values of this parameter that is em-
pirically relevant. The model underlying the analysis of Schmitt-Groh´e and
Uribe (2007b) is a medium-scale model of the U.S. economy featuring, in
addition to money demand by households and sticky product prices, a number
of real and nominal rigidities including wage stickiness, a demand for money
by firms, habit formation, capital accumulation, variable capacity utilization,
and investment adjustment costs. The structural parameters of the model are
assigned values that are consistent with full- as well as limited-information
approaches to estimating this particular model.
In the Schmitt-Groh´e and Uribe (2007b) model, the degree of price stick-
iness is captured by a parameter denoted α, measuring the probability that a
firm is not able to optimally set the price it charges in a particular quarter. The
average number of periods elapsed between two consecutive optimal price ad-
justments is given by 1/(1 − α). Available empirical estimates of the degree
of price rigidity using macroeconomic data vary from two to five quarters, or
α ∈ [0.5, 0.8]. For example, Christiano, Eichenbaum, and Evans (2005) esti-
mate α to be 0.6. By contrast, Altig et al. (2005) estimate a marginal-cost-gap
coefficient in the Phillips curve that is consistent with a value of α of around
0.8. Both Christiano, Eichenbaum and Evans (2005) and Altig et al. (2005)
use an impulse-response matching technique to estimate the price-stickiness
parameter α. Bayesian estimates of this parameter include Del Negro et al.
(2004), Levin et al. (2006), and Smets and Wouters (2007), who report pos-
terior means of 0.67, 0.83, and 0.66, respectively, and 90 percent posterior
probability intervals of (0.51,0.83), (0.81,0.86), and (0.56,0.74), respectively.
Recent empirical studies have documented the frequency of price changes
using microdata underlying the construction of the U.S. consumer price index.
These studies differ in the sample period considered, in the disaggregation
of the price data, and in the treatment of sales and stockouts. The median
frequency of price changes reported by Bils and Klenow (2004) is four to five
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 11
Figure 1 Price Stickiness, Fiscal Policy, and Optimal Inflation
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
CEE
ACEL
ACEL
CEE
Optimal Distortionary
Ta xes
Lump-Sum Taxes
-o-o-
Notes: CEE and ACEL indicate, respectively, the values for the parameter, α, estimated
by Christiano, Eichenbaum, and Evans (2005) and Altig et al. (2005).
months, the one reported by Klenow and Kryvtsov (2005) is four to seven
months, and the one reported by Nakamura and Steinsson (2007) is eight to
11 months. However, there is no immediate interpretation of these frequency
estimates to the parameter, α, governing the degree of price stickiness in Calvo-
style models of price staggering. Consider, for instance, the case of indexation.
In that case, even though firms change prices every period—implying the
highest possible frequency of price changes—prices themselves may be highly
sticky, for they may be only reoptimized at much lower frequencies.
Figure 1 displays with a solid line the relationship between the degree of
price stickiness, α, and the optimal rate of inflation in percent per year, π ,
implied by the model studied in Schmitt-Groh´e and Uribe (2007b). When α
equals 0.5, the lower range of the available empirical evidence using macro-
data, the optimal rate of inflation is −2.9 percent, which is the level called
for by the Friedman rule. For a value of α of 0.8, which is near the upper
range of the available empirical evidence using macrodata, the optimal level
of inflation rises to −0.4 percent, which is close to price stability.
12 Federal Reserve Bank of Richmond Economic Quarterly
Besides the uncertainty surrounding the estimation of the degree of price
stickiness, a second aspect of the apparent difficulty in establishing reliably the
optimal long-run level of inflation has to do with the shape of the relationship
linking the degree of price stickiness to the optimal level of inflation. The
problem resides in the fact that, as is evident from Figure 1, this relationship
becomes significantly steep precisely for that range of values of α that is
empirically most compelling. It turns out that an important factor determining
the shape of the function relating the optimal level of inflation to the degree
of price stickiness is the underlying fiscal policy regime.
Fiscal considerations fundamentally change the long-run tradeoff between
price stability and the Friedman rule. To see this, we now consider an economy
where lump-sum taxes are unavailable (τ
L
= 0). Instead, the fiscal authority
must finance government purchases by means of proportional capital and la-
bor income taxes. The social planner jointly sets monetary and fiscal policy
in a welfare-maximizing (i.e., Ramsey-optimal) fashion.
2
Figure 1 displays
the relationship between the degree of price stickiness, α, and the optimal
rate of inflation, π . The solid line corresponds to the case discussed earlier
featuring lump-sum taxes. The dash-circled line corresponds to the economy
with optimally chosen distortionary income taxes. In stark contrast to what
happens under lump-sum taxation, under optimal distortionary taxation the
function linking π and α is flat and very close to zero for the entire range of
macrodata-based empirically plausible values of α, namely 0.5 to 0.8. In other
words, when taxes are distortionary and optimally determined, price stability
emerges as a prediction that is robust to the existing uncertainty about the
exact degree of price stickiness. Even if one focuses on the evidence of price
stickiness stemming from microdata, the model with distortionary Ramsey
taxation predicts an optimal long-run level of inflation that is much closer to
zero than to the level called for by the Friedman rule.
Our intuition for why price stability arises as a robust policy recommenda-
tion in the economy with optimally set distortionary taxation runs as follows.
Consider the economy with lump-sum taxation. Deviating from the Friedman
rule (by raising the inflation rate) has the benefit of reducing price adjustment
costs. Consider next the economy with optimally chosen income taxation and
no lump-sum taxes. In this economy, deviating from the Friedman rule still
provides the benefit of reducing price adjustment costs. However, in this econ-
omy, increasing inflation has the additional benefit of increasing seigniorage
revenue, thereby allowing the social planner to lower distortionary income tax
rates. Therefore, the Friedman rule versus price stability tradeoff is tilted in
favor of price stability.
2
The details of this environment are contained in Schmitt-Groh´e and Uribe (2006). The
structure of this economy is identical to that studied in Schmitt-Groh´e and Uribe (2007b), except
for the inclusion of fiscal policy.
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 13
It follows from this intuition that what is essential in inducing the opti-
mality of price stability is that, on the margin, the fiscal authority trades off the
inflation tax for regular taxation. Indeed, it can be shown that if distortionary
tax rates are fixed, even if they are fixed at the level that is optimal in a world
without lump-sum taxes, and the fiscal authority has access to lump-sum taxes
on the margin, the optimal rate of inflation is much closer to the Friedman rule
than to zero. In this case, increasing inflation no longer has the benefit of re-
ducing distortionary taxes. As a result, the Ramsey planner has less incentives
to inflate.
We close this section by drawing attention to the fact that, quite indepen-
dently of the precise degree of price stickiness, the optimal inflation target
is below zero. In light of this robust result, it is puzzling that all countries
that self-classify as inflation targeters set inflation targets that are positive. In
effect, in the developed world inflation targets range between 2 and 4 percent
per year. Somewhat higher targets are observed across developing countries.
An argument often raised in defense of positive inflation targets is that neg-
ative inflation targets imply nominal interest rates that are dangerously close
to the zero lower bound on nominal interest rates and, hence, may impair the
central bank’s ability to conduct stabilization policy. In Schmitt-Groh´e and
Uribe (2007b) we find, however, that this argument is of little relevance in the
context of the medium-scale estimated model within which we conduct policy
evaluation. The reason is that under the optimal policy regime, the mean of
the nominal interest rate is about 4.5 percent per year with a standard devi-
ation of only 0.4 percent. This means that for the zero lower bound to pose
an obstacle to monetary stabilization policy, the economy must suffer from
an adverse shock that forces the interest rate to be more than ten standard
deviations below target. The likelihood of such an event is practically nil.
4. THE OPTIMAL VOLATILITY OF INFLATION
Two distinct branches of the existing literature on optimal monetary policy
deliver diametrically opposed policy recommendations concerning the cycli-
cal behavior of prices and interest rates. One branch follows the theoretical
framework laid out in Lucas and Stokey (1983). It studies the joint determina-
tion of optimal fiscal and monetary policy in flexible-price environments with
perfect competition in product and factor markets. In this strand of the litera-
ture, the government’s problem consists of financing an exogenous stream of
public spending by choosing the least disruptive combination of inflation and
distortionary income taxes.
Calvo and Guidotti (1990, 1993) and Chari, Christiano, and Kehoe (1991)
characterize optimal monetary and fiscal policy in stochastic environments
with nominal nonstate-contingent government liabilities. A key result of these
papers is that it is optimal for the government to make the inflation rate highly
14 Federal Reserve Bank of Richmond Economic Quarterly
volatile and serially uncorrelated. For instance, Schmitt-Groh´e and Uribe
(2004b) show, in the context of a flexible-price model calibrated to the U.S.
economy, that under the optimal policy the inflation rate has a standard devi-
ation of 7 percent per year and a serial correlation of −0.03. The intuition for
this result is that, under flexible prices, highly volatile and unforecastable in-
flation is nondistorting and at the same time carries the fiscal benefit of acting
as a lump-sum tax on private holdings of government-issued nominal assets.
The government is able to use surprise inflation as a nondistorting tax to the
extent that it has nominal, nonstate-contingent liabilities outstanding. Thus,
price changes play the role of a shock absorber of unexpected innovations in
the fiscal deficit. This “front-loading” of government revenues via inflationary
shocks allows the fiscal authority to keep income tax rates remarkably stable
over the business cycle.
However, as discussed in Section 2, the New Keynesian literature, aside
from emphasizing the role of price rigidities and market power, differs from
the earlier literature described above in two important ways. First, it assumes,
either explicitly or implicitly, that the government has access to (endogenous)
lump-sum taxes to finance its budget. An important implication of this as-
sumption is that there is no need to use unanticipated inflation as a lump-sum
tax; regular lump-sum taxes take on this role. Second, the government is
assumed to be able to implement a production (or employment) subsidy to
eliminate the distortion introduced by the presence of monopoly power in
product and factor markets.
The key result of the New Keynesian literature, which we presented in
Sections 2 and 3, is that the optimal monetary policy features an inflation
rate that is zero or close to zero at all times (i.e., both the optimal mean and
volatility of inflation are near zero). The reason price stability is optimal in
environments of the type described there is that it minimizes (or completely
eliminates) the costs introduced by inflation under nominal rigidities.
Together, these two strands of research on optimal monetary policy leave
the monetary authority without a clear policy recommendation. Should the
central bank pursue policies that imply high or low inflation volatility? In
Schmitt-Groh´e and Uribe (2004a), we analyze the resolution of this policy
dilemma by incorporating in a unified framework the essential elements of
the two approaches to optimal policy described above. Specifically, we build
a model that shares two elements with the earlier literature: (a) The only
source of regular taxation available to the government is distortionary income
taxes. As a result, the government cannot implement production subsidies
to undo distortions created by the presence of imperfect competition, and (b)
the government issues only nominal, one-period, nonstate-contingent bonds.
At the same time, the setup shares two important assumptions with the more
recent body of work on optimal monetary policy: (a) Product markets are
imperfectly competitive, and (b) product prices are assumed to be sticky and,
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 15
hence, the model features a New Keynesian Phillips curve. Schmitt-Groh´e
and Uribe (2004a) introduce price stickiness as in the previous section by
assuming that firms face a convex cost of price adjustment (Rotemberg 1982).
In this environment, the government faces a tradeoff in choosing the path
of inflation. On the one hand, the government would like to use unexpected
inflation as a nondistorting tax on nominal wealth. In this way, the fiscal
authority could minimize variations in distortionary income taxes over the
business cycle. On the other hand, changes in the rate of inflation come at a
cost, for firms face nominal rigidities.
When price changes are brought about at a cost, it is natural to expect
that a benevolent government will try to implement policies consistent with a
more stable behavior of prices than when price changes are costless. However,
the quantitative effect of an empirically plausible degree of price rigidity on
optimal inflation volatility is not clear a priori. In Schmitt-Groh´e and Uribe
(2004a), we show that for the degree of price stickiness estimated for the U.S.
economy, this tradeoff is overwhelmingly resolved in favor of price stability.
The Ramsey allocation features a dramatic drop in the standard deviation of
inflation from 7 percent per year under flexible prices to a mere 0.17 percent
per year when prices adjust sluggishly.
3
Indeed, the impact of price stickiness on the optimal degree of inflation
volatility turns out to be much stronger than suggested by the numerical results
reported in the previous paragraph. Figure 2, taken from Schmitt-Groh´e and
Uribe (2004a), shows that a minimum amount of price stickiness suffices to
make price stability the central goal of optimal policy. Specifically, when the
degree of price stickiness, embodied in the parameter θ (see equation [7]), is
assumed to be ten times smaller than the estimated value for the U.S. economy,
the optimal volatility of inflation is below 0.52 percent per year, 13 times
smaller than under full price flexibility.
A natural question elicited by Figure 2 is why even a modest degree of
price stickiness can turn undesirable the use of a seemingly powerful fiscal
instrument, such as large revaluations or devaluations of private real financial
wealth through surprise inflation. Our conjecture is that in the flexible-price
economy, the welfare gains of surprise inflations or deflations are very small.
Our intuition is as follows. Under flexible prices, it is optimal for the central
bank to keep the nominal interest rate constant over the business cycle. This
means that large surprise inflations must be as likely as large deflations, as
variations in real interest rates are small. In other words, inflation must have a
near-i.i.d. behavior. As a result, high inflation volatility cannot be used by the
Ramsey planner to reduce the average amount of resources to be collected via
distortionary income taxes, which would be a first-order effect. The volatility
3
This price stability result is robust to augmenting the model to allow for nominal rigidities
in wages and indexation in product or factor prices (Schmitt-Groh´e and Uribe 2006, Table 6.5).
16 Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Degree of Price Stickiness and Optimal Inflation Volatility
0246810
0
1
2
3
4
5
6
7
Degree of Price Stickiness,
θ
Std. Dev. of
π
←
←
Flexible Prices
Baseline
Notes: The parameter, θ, governs the cost of adjusting nominal prices as defined in
equation (7). Its baseline value is 4.4, in line with available empirical estimates. The
standard deviation of inflation is measured in percent per year.
of inflation primarily serves the purpose of smoothing the process of income
tax distortions—a second-order source of welfare losses—without affecting
their average level.
Another way to gain intuition for the dramatic decline in optimal
inflation volatility that occurs even at very modest levels of price stickiness
is to interpret price volatility as a way for the government to introduce real
state-contingent public debt. Under flexible prices, the government uses state-
contingent changes in the price level as a nondistorting tax or transfer on private
holdings of government assets. In this way, nonstate-contingent nominal pub-
lic debt becomes state-contingent in real terms. So, for example, in response
to an unexpected increase in government spending, the Ramsey planner does
not need to increase tax rates by much because by inflating away part of the
public debt he can ensure intertemporal budget balance. It is, therefore, clear
that introducing costly price adjustment is the same as if the government were
limited in its ability to issue real state-contingent debt. It follows that the larger
the welfare gain associated with the ability to issue real state-contingent pub-
lic debt—as opposed to nonstate-contingent debt—the larger the amount of
price stickiness required to reduce the optimal degree of inflation volatility.
Aiyagari et al. (2002) show that indeed the level of welfare under the Ramsey
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 17
policy in an economy without real state-contingent public debt is virtually the
same as in an economy with state-contingent debt. Our finding that a small
amount of price stickiness is all it takes to bring the optimal volatility of in-
flation from a very high level to near zero is thus perfectly in line with the
finding of Aiyagari et al. (2002).
If this intuition is correct, then the behavior of tax rates and public debt
under sticky prices should resemble that implied by the Ramsey allocation in
economies without real state-contingent debt. Indeed, in financing the budget,
the Ramsey planner replaces front-loading with standard debt and tax instru-
ments. For example, in response to an unexpected increase in government
spending, the planner does not generate a surprise increase in the price level.
Instead, he chooses to finance the increase in government purchases partly
through an increase in income tax rates and partly through an increase in pub-
lic debt. The planner minimizes the tax distortion by spreading the required
tax increase over many periods. This tax-smoothing behavior induces near-
random walk dynamics into the tax rate and public debt. By contrast, under
full price flexibility (i.e., when the government can create real-state-contingent
debt), tax rates and public debt inherit the stochastic process of the underlying
shocks.
An important conclusion of this analysis is, thus, that the Aiyagari et al.
(2002) result, namely, that optimal policy imposes a near-random walk be-
havior on taxes and debt, does not require the unrealistic assumption that the
government can issue only nonstate-contingent real debt. This result emerges
naturally in economies with nominally nonstate-contingent debt—clearly the
case of greatest empirical relevance—and a minimum amount of price rigid-
ity.
4
However, if government debt is assumed to be state contingent, the
presence of sticky prices may introduce no difference in the Ramsey real allo-
cation, depending on the precise specification of the demand for money (see
Correia, Nicolini, and Teles 2008). The reason for this result is that, as shown
in Lucas and Stokey (1983), if government debt is state-contingent and prices
are fully flexible, the Ramsey allocation does not pin down the price level
uniquely. In this case, there is an infinite number of price-level processes (and
thus of money supply processes) that can be supported as Ramsey outcomes.
4
It is of interest to relate the near-random walk in taxes and debt that emerges as the optimal
policy outcome in a model featuring a New Keynesian Phillips curve with the celebrated tax-
smoothing result of Barro (1979). In Barro’s formulation, the objective function of the government
is the expected present discounted value of squared deviations of tax rates from a target or desired
level. The government minimizes this objective function subject to a sequential budget constraint,
which is linear in debt and tax rates. The resulting solution resembles the random walk model of
consumption with taxes taking the place of consumption and public debt taking the place of private
debt. The analysis in Schmitt-Groh´e and Uribe (2004a) departs from Barro’s ad hoc loss function
and replaces it with the utility function of the representative optimizing household inhabiting a
fully-articulated, dynamic, stochastic, general-equilibrium economy. In this environment, the random
walk result obtains from a more subtle channel, namely, the introduction of a miniscule amount
of nominal rigidity in product prices.
18 Federal Reserve Bank of Richmond Economic Quarterly
Loosely speaking, the introduction of price stickiness simply “uses this degree
of freedom” to pin down the equilibrium process of the price level without
altering other aspects of the Ramsey solution.
5. IMPLEMENTATION OF OPTIMAL POLICY
We established that in the simple New Keynesian model presented in Section
2, the optimal policy consists of setting the inflation rate equal to zero at all
times (π
t
= 1) and imposing a constant output subsidy (τ
D
t
= 1/[1 − η]).
The question we pursue in this section is how to implement the optimal policy.
Because central banks in the United States and elsewhere use the short-term
nominal interest rate as the monetary policy instrument, it is of empirical
interest to search for interest rate rules that implement the optimal allocation.
Using the Ramsey-Optimal Interest Rate Process as a
Feedback Rule
One might be tempted to believe that implementation of optimal policy is
trivial once the interest rate associated with the Ramsey equilibrium has been
found. Specifically, in the Ramsey equilibrium, the nominal interest rate can
be expressed as a function of the current state of the economy. Then, the
prescription would be simply to use this function as a policy rule in setting
the nominal interest rate at all dates and under all circumstances. It turns
out that conducting policy in this fashion would, in general, not deliver the
intended results. The reason is that although such a policy would be consistent
with the optimal equilibrium, it would at the same time open the door to
other (suboptimal) equilibria. It follows that the solution to the optimal policy
problem is mute with respect to the issue of implementation of such policy. To
see this, it is convenient to consider as an example a log-linear approximation
to the equilibrium conditions associated with the cashless, sticky-price model
presented in Section 2. It can be shown that the resulting linear system is
given by
5
ˆπ
t
= βE
t
ˆπ
t+1
+ κ ˆc
t
− γ ˆz
t
(11)
and
−σ ˆc
t
=
ˆ
R
t
− σE
t
ˆc
t+1
− E
t
ˆπ
t+1
, (12)
where σ,κ,and γ>0 are parameters. Hatted variables denote percent devi-
ations of the corresponding nonhatted variables from their respective values
5
The log-linearization is performed around the nonstochastic steady state of the Ramsey equi-
librium. In performing the linearization, we assume that the period utility function is separable in
consumption and hours and that the production function is linear in labor.
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 19
in the deterministic steady state of the Ramsey equilibrium. Equation (11)
results from combining equations (8), (9), and (10) and is typically referred to
as the New Keynesian Phillips curve.
6
Equation (12) is a linearized version
of an Euler equation that prices nominally risk-free bonds, where R
t
denotes
the gross nominal risk-free interest rate between periods t and t + 1. This
equation is typically referred to as the intertemporal IS equation.
Substituting the welfare-maximizing rate of inflation, ˆπ
t
= 0, into the
intertemporal IS curve (12) implies that the nominal interest rate is given by
ˆ
R
t
= σE
t
( ˆc
t+1
−ˆc
t
), which states that under the optimal policy the nom-
inal and real interest rates coincide. Suppose that the central bank adopted
this expression as a policy feedback rule for the nominal interest rate. The
question is whether this proposed rule implements the Ramsey equilibrium
uniquely. The answer to this question is no. To see why, consider a solu-
tion of the form ˆπ
t
=
t
, where
t
is i.i.d. normal with mean zero and an
arbitrary standard deviation σ
≥ 0. Notice that for all positive values of
σ
, the proposed solution for inflation is different from the optimal one. It
is straightforward to see that the proposed solution satisfies the intertemporal
IS equation (12). The solution for consumption can be read off the NKPC as
being ˆc
t
= γ/κˆz
t
+ (1/κ)
t
. We have, therefore, constructed a competitive
equilibrium in which a nonfundamental source of uncertainty, embodied in the
random variable
t
, causes stochastic deviations of consumption and inflation
from their optimal paths. Notice that in this example there exists an infinite
number of different equilibria indexed by the parameter, σ
, governing the
volatility of the nonfundamental shock,
t
.
One possible objection against the interest rate feedback rule proposed in
the previous paragraph is that it is cast in terms of the endogenous variable,
c
t
. In particular, one may wonder whether this endogeneity is responsible
for the inability of the proposed rule to implement the Ramsey equilibrium
uniquely. This concern is indeed unfounded. For, even if the interest rate
feedback rule were cast in terms of exogenous fundamental variables, the
failure of the strategy of using the Ramsey solution for R
t
as an interest rate
feedback rule remains. Specifically, substituting the optimal rate of inflation,
ˆπ
t
= 0, into the New Keynesian Phillips curve (11) yields ˆc
t
= γκ
−1
ˆz
t
.
In turn, substituting this expression into the intertemporal IS curve (12) im-
plies that in the optimal equilibrium the nominal interest rate is given by
ˆ
R
t
=ˆr
n
t
≡ σγκ
−1
E
t
(ˆz
t+1
−ˆz
t
). The variable r
n
t
denotes the risk-free real (as
well as nominal) interest rate that prevails in the Ramsey optimal equilibrium
and is referred to as the natural rate of interest. Using this expression, equa-
tions (11)–(12) become a system of two linear stochastic difference equations
6
For detailed derivations of this expression, see, for instance, Woodford (2003). This linear
expression is the NKPC studied in the papers by Nason and Smith (2008) and Schorfheide (2008)
that appear in this issue.
20 Federal Reserve Bank of Richmond Economic Quarterly
in the endogenous variables ˆπ
t
and ˆc
t
. This system possesses one eigen-
value inside the unit circle and one outside. It follows that the solution for
inflation and consumption is of the form ˆπ
t+1
= ζ
ππ
ˆπ
t
+ ζ
πz
ˆz
t
+
t+1
and
ˆc
t
= ζ
cπ
ˆπ
t
+ζ
cz
ˆz
t
, where ˆπ
0
is arbitrary,
t
is a nonfundamental shock such as
the one introduced above, and the parameter ζ
ππ
is less than unity in absolute
value. Clearly, the competitive equilibrium that the proposed rule implements
displays persistent and stochastic deviation from the optimal solution. We
conclude that the use of the Ramsey-optimal interest rate process as a policy
feedback rule fails to implement the desired competitive equilibrium.
Can the Taylor Rule Implement Optimal Policy?
A Taylor-type rule is an interest rate feedback rule whereby the nominal interest
rate is set as an increasing linear function of inflation and deviations of real
output from trend, with an inflation coefficient greater than unity and an output
coefficient greater than zero. Formally, a Taylor-type interest rate rule can be
written as
ˆ
R
t
= α
π
ˆπ
t
+ α
y
ˆy
t
, where α
π
> 1 and α
y
> 0 are parameters
and ˆy
t
represents the percent deviation of real output from trend. Taylor’s
rule has been widely studied in monetary economics since the publication of
Taylor’s (1993) seminal article. It has been advocated as a desirable policy
specification and is considered by some to be a reasonable approximation
of actual monetary policy in the United States and many other developed
countries. For this reason, we now consider the question of whether a Taylor
rule can implement the optimal allocation. The answer is that, in general, an
interest rate feedback rule of the type proposed by Taylor is unable to support
the Ramsey optimal equilibrium. This issue was first analyzed by Woodford
(2001).
To establish whether the Taylor rule presented above can implement the
optimal allocation, we set the inflation rate at its optimal value of zero ( ˆπ
t
= 0)
and combine the intertemporal IS equation (12) with the Taylor rule. This
yields α
y
ˆc
t
= σ(E
t
ˆc
t+1
−ˆc
t
). Now, replacing ˆc
t
with its optimal value of
γ/κˆz
t
, we obtain E
t
ˆz
t+1
= (1 + α
y
/σ )ˆz
t
. This expression represents a con-
tradiction because the productivity shock, ˆz
t
, is assumed to be an exogenous
stationary process with a law of motion independent of the policy parameter,
α
y
, and the preference parameter, σ . We have established that the proposed
Taylor rule fails to implement the optimal equilibrium. One can show that
this result also obtains when the nominal interest rate is assumed to respond
to deviations of output from its natural level, which is defined as the level of
output associated with the optimal equilibrium and is given by y
n
t
≡ γ/κˆz
t
.
However, the optimal allocation can indeed be implemented by a modified
Taylor rule of the form
ˆ
R
t
=ˆr
n
t
+ α
π
ˆπ
t
as long as α
π
> 1. In this rule, ˆr
n
t
denotes the natural rate of interest defined earlier. The first term in the modified
Taylor rule makes the Ramsey allocation feasible as an equilibrium outcome.
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 21
The second term makes it unique. The key difference between the standard
and modified Taylor rules is that the latter features a time-varying intercept
that allows the nominal interest rate to accommodate movements in the real
interest rate one-for-one without requiring changes in the price level. More
generally, the optimal competitive equilibrium can be implemented via rules
of the form
ˆ
R
t
=ˆr
n
t
+ α
π
ˆπ
t
+ α
y
( ˆy
t
−ˆy
n
t
), with policy parameters α
π
and
α
y
satisfying the restrictions imposed by the definition of a Taylor-type rule
given above.
Applying this type of rule can be quite impractical, for it would require
knowledge on the part of the central bank of current and expected future values
taken by all of the shocks that affect the real interest rate, as well as of the
function mapping such values to the natural rate of interest. This difficulty
raises the question of how close a less sophisticated interest rate rule would
get to implementing the optimal equilibrium. We turn to this issue next.
Optimal, Simple, and Implementable Rules
In this subsection, we analyze the ability of simple, implementable interest
rate rules to approximate the outcome of optimal policy. We draw from our
previous work (Schmitt-Groh´e and Uribe 2007a), where we evaluate policy in
the context of a calibrated model of the U.S. business cycle featuring monop-
olistic competition, sticky prices in product markets, capital accumulation,
government purchases financed by lump-sum or distortionary taxes, and with
or without a transactional demand for money.
7
In the model, business cycles
are driven by stochastic variations in the level of total factor productivity and
government consumption. We impose two requirements for an interest rate
rule to be implementable. First, the rule must deliver a unique rational expecta-
tions equilibrium. Second, it must induce nonnegative equilibrium dynamics
for the nominal interest rate. For an interest rule to be simple, we require that
the interest rate be set as a function of a small number of easily observable
macroeconomic indicators. Specifically, we study interest rate feedback rules
that respond to measures of inflation, output, and lagged values of the nominal
interest rate. The family of rules we consider is of the form
ln(R
t
/R
∗
) = α
R
ln(R
t−1
/R
∗
) + α
π
E
t
ln(π
t−i
/π
∗
) + α
y
E
t
ln(y
t−i
/y
∗
);
i =−1, 0, 1, (13)
where y
∗
denotes the nonstochastic Ramsey steady-state level of aggregate
demand, and R
∗
, π
∗
, α
R
, α
π
, and α
y
are parameters. The index, i, can take
three values: 1, 0, and −1. When i = 1, we refer to the interest rate rule as
backward-looking, when i = 0 as contemporaneous, and when i =−1as
7
See also Rotemberg and Woodford (1997).
22 Federal Reserve Bank of Richmond Economic Quarterly
Table 1 Evaluating Interest Rate Rules
α
π
α
y
α
R
Welfare Cost σ
π
σ
R
Ramsey Policy — — — 0.000 0.01 0.27
Optimized Rules
Contemporaneous (i =0)
Smoothing 3 0.01 0.84 0.000 0.04 0.29
No Smoothing 3 0.00 — 0.001 0.14 0.42
Backward (i = 1) 3 0.03 1.71 0.001 0.10 0.23
Forward (i = −1) 3 0.07 1.58 0.003 0.19 0.27
Nonoptimized Rules
Taylor Rule (i = 0) 1.5 0.5 — 0.522 3.19 3.08
Simple Taylor Rule 1.5 — — 0.019 0.58 0.87
Inflation Targeting — — — 0.000 0.00 0.27
The interest rate rule is given by ln(R
t
/R
∗
) = α
R
ln(R
t−1
/R
∗
) + α
π
E
t
ln(π
t−i
/π
∗
) +
α
y
E
t
ln(y
t−i
/y
∗
); i =−1, 0, 1. In the optimized rules, the policy parameters α
π
,α
y
,and
α
R
are restricted to lie in the interval
[
0, 3
]
. The welfare cost is defined as the percent-
age decrease in the Ramsey-optimal consumption process necessary to make the level of
welfare under the Ramsey policy identical to that under the evaluated policy. Thus, a
positive figure indicates that welfare is higher under the Ramsey policy than under the
alternative policy. The standard deviation of inflation and the nominal interest rate is
measured in percent per year.
forward-looking. The optimal simple and implementable rule is the simple
and implementable rule that maximizes welfare of the representative agent.
Specifically, we characterize values of α
π
, α
y
, and α
R
that are associated with
the highest level of welfare of the representative agent within the family of
simple and implementable interest rate feedback rules defined by equation
(13). As a point of comparison for policy evaluation, we also compute the real
allocation associated with the Ramsey optimal policy.
The first row of Table 1 shows that under the Ramsey policy inflation is
virtually equal to zero at all times.
8
The remaining rows of Table 1 report
policy evaluations. The welfare associated with each interest rate feedback
rule is compared to the level of welfare associated with the Ramsey-optimal
policy. Specifically, the welfare cost is defined as the fraction, in percentage
points, of the consumption stream an agent living in the Ramsey economy
would be willing to give up to be as well off as in an economy in which
8
In the deterministic steady state of the Ramsey economy, the inflation rate is zero. One may
wonder why, in an economy featuring sticky prices as the single nominal friction, the volatility
of inflation is not exactly equal to zero at all times under the Ramsey policy. The reason is
that we do not follow the standard practice of subsidizing factor inputs to eliminate the distortion
introduced by monopolistic competition in product markets. Introducing such a subsidy would
result in a constant Ramsey-optimal rate of inflation equal to zero.
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 23
monetary policy takes the form of the respective interest rate feedback rule
shown in the table.
We consider seven different monetary policies: Four constrained-optimal
interest rate feedback rules and three nonoptimized rules. In the constrained-
optimal rule labeled no-smoothing, we search over the policy coefficients, α
π
and α
y
, keeping α
R
fixed at zero. The second constrained-optimal rule, labeled
smoothing in the table, allows for interest rate inertia by setting optimally all
three coefficients, α
π
, α
y
, and α
R
.
We find that the best no-smoothing interest rate rule calls for an aggressive
response to inflation and a mute response to output. The inflation coefficient of
the optimized rule takes the largest value allowed in our search, namely 3. The
optimized rule is quite effective as it delivers welfare levels remarkably close
to those achieved under the Ramsey policy. At the same time, the rule induces
a stable rate of inflation, a feature that also characterizes the Ramsey policy.
Taking together this finding and those obtained in the previous subsection, we
conclude that although a Taylor rule cannot exactly implement the Ramsey
allocation, it delivers outcomes that are so close to the optimum in welfare
terms that, for practical purposes, it can be regarded as implementing the
Ramsey allocation.
We next study a case in which the central bank can smooth interest rates
over time. Our numerical search yields that the optimal policy coefficients are
α
π
= 3, α
y
= 0.01, and α
R
= 0.84. The fact that the optimized rule features
substantial interest rate inertia means that the monetary authority reacts to
inflation much more aggressively in the long run than in the short run. The
finding that the interest rule is not superinertial (i.e., α
R
does not exceed
unity) means that the monetary authority is backward-looking. So, again, as
in the case without smoothing, optimal policy calls for a large response to
inflation deviations in order to stabilize the inflation rate and for no response
to deviations of output from the steady state. The welfare gain of allowing
for interest rate smoothing is insignificant. Taking the difference between the
welfare costs associated with the optimized rules with and without interest
rate smoothing reveals that agents would be willing to give up less than 0.001
percent of their consumption stream under the optimized rule with smoothing
to be as well off as under the optimized policy without smoothing.
The finding that allowing for optimal smoothing yields only negligible
welfare gains spurs us to investigate whether rules featuring suboptimal de-
grees of inertia or responsiveness to inflation can produce nonnegligible wel-
fare losses at all. PanelA of Figure 3 shows that, provided the central bank does
not respond to output, α
y
= 0, varying α
π
and α
R
between 0 and 3 typically
leads to economically negligible welfare losses of less than 0.05 percent of
consumption. In the graph, crosses represent combinations of α
π
and α
R
that
are implementable and circles represent combinations that are implementable
24 Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 The Cashless Economy
3.0
a
p
a
R
a
Y
=0
a
y
Welfare Cost
Implementable Rule
Welfare Cost <0.05%
2.5
2.0
1.5
1.0
0.5
0.0
0.5
1.0 1.5
2.0
2.5 3.0
Panel A: Implementability and Welfare
Panel B: The Importance of Not Responding to Output
0.0
0.25
0.2
0.15
0.1
0.05
0.1 0.2 0.3 0.4 0.5 0.6
0.7
(l
u
x 100)
and that yield welfare costs less than 0.05 percent of consumption relative to
the Ramsey policy.
The blank area in the figure identifies α
π
and α
R
combinations that are
not implementable either because the equilibrium fails to be locally unique
or because the implied volatility of interest rates is too high. This is the case
for values of α
π
and α
R
such that the policy stance is passive in the long run,
that is,
α
π
1−α
R
< 1. For these parameter combinations the equilibrium is not
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 25
locally unique. This finding is a generalization of the result that, when the
inflation coefficient is less than unity (α
π
< 1), the equilibrium is indetermi-
nate, which obtains in the absence of interest rate smoothing (α
R
= 0). We
also note that the result that passive interest rate rules render the equilibrium
indeterminate is typically derived in the context of models that abstract from
capital accumulation. It is, therefore, reassuring that this particular abstraction
appears to be of no consequence for the finding that (long run) passive policy
is inconsistent with local uniqueness of the rational expectations equilibrium.
Similarly, we find that determinacy obtains for policies that are active in the
long run,
α
π
1−α
R
> 1.
More importantly, Panel A of Figure 3 shows that virtually all parameter-
izations of the interest rate feedback rule that are implementable yield about
the same level of welfare as the Ramsey equilibrium. This finding suggests
a simple policy prescription, namely, that any policy parameter combination
that is irresponsive to output and active in the long run, is equally desirable
from a welfare point of view.
One possible reaction to the finding that implementability-preserving vari-
ations in α
π
and α
R
have little welfare consequences may be that in the class
of models we consider, welfare is flat in a large neighborhood around the
optimum parameter configuration, so that it does not really matter what the
government does. This turns out not to be the case. Recall that in the wel-
fare calculations underlying Panel A of Figure 3, the response coefficient on
output, α
y
, was kept constant and equal to zero. Indeed, interest rate rules
that lean against the wind by raising the nominal interest rate when output is
above trend can be associated with sizable welfare costs. Panel B of Figure 3
illustrates the consequences of introducing a cyclical component to the interest
rate rule. It shows that the welfare costs of varying α
y
can be large, thereby
underlining the importance of not responding to output. The figure shows the
welfare cost of deviating from the optimal output coefficient (α
y
≈ 0) while
keeping the inflation coefficient of the interest rate rule at its optimal value
(α
π
= 3) and not allowing for interest rate smoothing (α
R
= 0). Welfare costs
are monotonically increasing in α
y
. When α
y
= 0.7, the welfare cost is over
0.2 percent of the consumption stream associated with the Ramsey policy.
This is a significant figure in the realm of policy evaluation at business-cycle
frequency.
9
This finding suggests that bad policy can have significant welfare
costs in our model and that policy mistakes are committed when policymakers
are unable to resist the temptation to respond to output fluctuations.
9
A similar result obtains if one allows for interest rate smoothing with α
R
taking its opti-
mized value of 0.84.
26 Federal Reserve Bank of Richmond Economic Quarterly
It follows that sound monetary policy calls for sticking to the basics of
responding to inflation alone.
10
This point is conveyed with remarkable sim-
plicity by comparing the welfare consequences of a simple interest rate rule
that responds only to inflation with a coefficient of 1.5 to those of a standard
Taylor rule that responds to inflation as well as output with coefficients 1.5 and
0.5, respectively. Table 1 shows that the Taylor rule that responds to output
is significantly welfare inferior to the simple interest rate rule that responds
solely to inflation. Specifically, the welfare cost of responding to output is
about half a percentage point of consumption.
11
The Ramsey-optimal monetary policy implies near complete inflation sta-
bilization (see Table 1). It is reasonable to conjecture, therefore, that inflation
targeting, interpreted to be any monetary policy capable of bringing about zero
inflation at all times (π
t
= 1 for all t), would induce business cycles virtually
identical to those associated with the Ramsey policy. We confirm this con-
jecture by computing the welfare cost associated with inflation targeting. The
welfare cost of targeting inflation relative to the Ramsey policy is virtually nil.
An important issue in monetary policy is determining what measures of
inflation and aggregate activity the central bank should respond to. In par-
ticular, a question that has received considerable attention among academic
economists and policymakers is whether the monetary authority should re-
spond to past, current, or expected future values of output and inflation.
Here we address this question by computing optimal backward- and forward-
looking interest rate rules. That is, in equation (13) we let i take the values −1
and +1. Table 1 shows that there are no welfare gains from targeting expected
future values of inflation and output as opposed to current or lagged values of
these macroeconomic indicators. Also, a muted response to output continues
to be optimal under backward- or forward-looking rules.
Under a forward-looking rule without smoothing (α
R
= 0), the rational
expectations equilibrium is indeterminate for all values of the inflation and
output coefficients in the interval [0,3]. This result is in line with that obtained
by Carlstrom and Fuerst (2005). These authors consider an environment sim-
ilar to ours and characterize determinacy of equilibrium for interest rate rules
that depend only on the rate of inflation. Our results extend the findings of
Carlstrom and Fuerst to the case in which output enters in the feedback rule.
We close this section by noting that most of the results presented here,
extend to a model economy with a much richer battery of nominal and real
rigidities. In Schmitt-Groh´e and Uribe (2007b), we consider an economy
featuring four real rigidities: habit formation, variable capacity utilization,
10
Other authors have also argued that countercyclical interest rate policy may be undesirable
(e.g., Ireland 1997 and Rotemberg and Woodford 1997).
11
The simple interest rate rule that responds solely to inflation is implementable, whereas
the standard Taylor rule is not, because it implies too high a volatility of nominal interest rates.
S. Schmitt-Groh´e and M. Uribe: Policy Implications of the NKPC 27
investment adjustment costs, and monopolistic competition in product and
labor markets. The economy in that study also includes four nominal fric-
tions, namely, sticky prices, sticky wages, money demand by households, and
money demand by firms. Finally, the model features a more realistic shock
structure that includes permanent stochastic variations in total factor produc-
tivity, permanent stochastic variations in the relative price of investment, and
stationary stochastic variations in government spending. The values assigned
to the structural parameters are based on existing econometric estimations of
the model. These studies, in turn, argue that the model explains satisfactorily
observed short-term fluctuations in the postwar United States. We find that
the Ramsey policy calls for stabilizing price inflation. More importantly, a
simple interest rate rule that responds only to inflation (with mute responses
to wage inflation or output) attains a level of welfare remarkably close to that
associated with the Ramsey optimal equilibrium.
6. CONCLUSION
In this article, we present a selective account of recent developments on the
policy implications of the New Keynesian Phillips curve. The main lesson
derived from our analysis is that price stability emerges as a robust policy pre-
scription in models with product price rigidities. In fact, a minimum amount
of price stickiness suffices to make inflation stabilization the overriding goal
of monetary policy.
The desirability of price stability obtains in several variations of the stan-
dard New Keynesian framework that include expanding the set of nominal
and real rigidities to allow for government spending financed by distortionary
taxes, a transactional demand for money by households and firms, nominal
wage rigidity, habit formation, variable capacity utilization, and investment
adjustment costs.
A second important message that emerges is that a simple interest rate
feedback rule that responds aggressively only to a measure of consumer price
inflation delivers outcomes that are remarkably close to the Ramsey opti-
mal equilibrium. In particular, to emulate optimal monetary policy it is not
necessary that in setting the nominal rate the monetary authority respond to
deviations of output from trend or past values of the interest rate itself. In
this sense, the policy implications of the NKPC identified in this survey are
consistent with a pure inflation targeting objective.
We have left out a number of important issues in the theory of inflation
stabilization. For example, we limit attention to monetary policy under com-
mitment. There is an active literature exploring the policy implications of the
NKPC when the government is unable to commit to future actions. A central
theme in this literature is to ascertain whether lack of commitment gives rise to
an optimal inflation bias. A second omission in the present analysis concerns
28 Federal Reserve Bank of Richmond Economic Quarterly
models with asymmetric costs of price adjustment. Here again, the central
question is whether in the presence of downwardly rigid prices or wages, the
policymaker should pursue a positive inflation target. Finally, our article does
not discuss the recent literature on optimal monetary policy in models with
credit constraints. An important focus of this literature is whether this type of
friction introduces reasons for the central bank to respond to financial variables
in setting the short-term interest rate.
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