Download Optimality of Inflation and Nominal Output Targeting

Optimality of Inflation and Nominal Output Targeting
Julio Garı́n∗
Robert Lester†
Department of Economics
Department of Economics
University of Georgia
University of Notre Dame
First Draft: January 7, 2015
Please Do Not Cite or Distribute: Incomplete and Preliminary
Abstract
Is nominal income targeting superior to inflation targeting? We address this question
within a DSGE model with nominal price and wage rigidities. In the case of a model
without capital accumulation, the answer depends entirely on the relative rigidity of
prices and wages. If prices are sufficiently stickier than wages, inflation targeting is
preferred; otherwise the optimal policy calls for nominal income targeting. After going
over the intuition in this stylized case, we estimate a medium scale DSGE model and
show how the optimal rule changes based on the parameter values and frictions in the
economy.
JEL Classification: E31; E47; E52; E58.
Keywords: Optimal Policy; Nominal Targeting; Monetary Policy.
∗
†
E-mail address: [email protected].
E-mail address: [email protected].
1
Introduction
What rule should the central bank follow in the formation of monetary policy? Despite
extensive research about this topic, it remains an open question. In this paper we compare
two often proposed rules: nominal GDP targeting and inflation targeting. Both rules adjust
the interest rate to a nominal anchor, but the nominal GDP targeting rule implicitly includes
real activity as well. We explore this question within the context of a relatively standard
dynamic stochastic general equilibrium (DSGE) model.
Although there is some disagreement on which type of rule central banks should follow,
economists agree on several principles in the design of monetary policy. First, rules are
preferred to discretion. Rules allow households to anchor expectations which improves the
inflation-output gap tradeoff. This is true in models of either ad hoc Phillips curves as
in Barro and Gordon (1983) or in microfounded Phillips curves as in Woodford (2003).
Second, the central bank faces information constraints which should be taken into account
in the formation of monetary policy. Responding to precisely measured variables is superior
to responding to imprecisely measured variables or variables that are hypothetical constructs
of a model. Finally, the policy objectives of central banks should be understandable to the
public. As argued by Bernanke and Mishkin (1997), this requires the central bank to be
more accountable. Furthermore, even if monetary policy follows some strict rule, forming
expectations is difficult if households do not understand the rule.
Nominal GDP targeting and inflation targeting satisfy all three of these conditions. By
definition, they are both rules. While it is difficult to argue that nominal GDP and inflation
are measured with precision, they are both found in the data rather than being hypothetical
constructs. Finally, both concepts are easy to explain to the public.
Bernanke and Mishkin (1997) argue that inflation targeting is superior to nominal GDP
targeting on the last two principles. However, we are not aware of any work that systematically studies the precision of inflation versus nominal GDP measurement. On the other
hand, there are theoretical reasons to believe that nominal GDP targeting dominates inflation targeting. Sumner (2014) outlines the basic logic in an aggregate demand - aggregate
supply framework. After a negative aggregate demand shock, the price level and real output
decline. Monetary policy in either regime lowers the nominal interest rate to combat the
shock. Hence, if aggregate demand shocks were the only shocks driving the dynamics of the
economy, then the choice of monetary policy rule would not matter. However, if there is a
negative aggregate supply shock, the price level increases and real output decreases. The
interest rate rises under the inflation targeting regime, reducing the price level and inflation.
If nominal wages are sticky, real wages rise above their flexible level and unemployment rises.
2
The interest rate under the nominal GDP targeting regime may either rise or fall depending
on the details of the model, but in any case unemployment will not rise by as much. While
an exact variance decomposition of shocks to the economy will depend on the details of the
data and model used, it is plausible that productivity and oil price shocks (aggregate supply
shocks) and shocks to velocity (aggregate demand shocks) play prominent roles over the
business cycle. One of our contributions is to compare the two rules in an estimated model
that includes several structural shocks which map into reduced form aggregate demand or
aggregate supply shocks.
In the next section, we discuss a basic linearized New Keynesian model which includes
nominal price and wage rigidity as in Erceg et al. (2000). Our results confirm Sumner (2014)’s
intuition. If nominal wages are very sticky compared to nominal prices, nominal GDP
targeting is preferred to inflation targeting. If nominal prices are sufficiently sticky relative
to nominal wages, inflation targeting is preferred. Following this, we go to a more realistic
model that includes capital accumulation and several other variables that are empirically
important. We then perform the same exercise as in the simpler model.
Our paper is related to several strands in the literature. The relative merits of nominal GDP targeting versus inflation targeting has been recently revived by Billi (2014) and
Woodford (2012) who discuss the rules within the context of the zero lower bound. Cecchetti (1995) and Hall and Mankiw (1994) show in counterfactual simulations that nominal
GDP targeting would lower the volatility of real and nominal variables. Neither of these
latter two papers have a structural model to conduct the welfare analysis, which limits their
ability to make definitive judgments. Jensen (2002) compares the two rules in a linearized
New Keynesian model with price stickiness and Kim and Henderson (2005) compare the two
rules in a model of wage and price stickiness. While both of these papers include structural
models similar to our own, they conduct their analysis within a stylized environment, while
ours is more empirically realistic.
Our paper is also related to Schmitt-Grohe and Uribe (2007) who compare simple and
implementable rules in a New Keynesian model with capital accumulation. However, they
do not explicitly consider nominal income targeting. Finally, since we estimate a DSGE
model, out paper is related to Smets and Wouters (2007) and Christiano et al. (2005) who
estimate New Keynesian models with U.S. data.
The paper proceeds as follows. Section 2 describes a basic New Keynesian and Section 3
presents a medium scale model.
3
2
The Basic New Keynesian Model
This section presents the New Keynesian model with nominal wage and price rigidity.
In comparing the welfare results of nominal GDP targeting versus inflation targeting, we
follow Erceg et al. (2000) in taking a quadratic approximation of the utility function and
linearizing the policy functions.
The forward looking “IS” equation results from log linearizing the household’s Euler
equation and is given by
1
p
Ỹt = Et Ỹt+1 − (ĩt − Et π̃t+1
).
(1)
σ
Output today, Ỹt , is an increasing function of expected future output and a decreasing
function of the real interest rate, ĩt − Et π˜p t+1 . The parameter σ is the coefficient of relative
risk aversion from a utility function that is separable in consumption and leisure. The price
and wage inflation Phillips curves are
p
π̃tp = κp ω̃t + β Et π̃t+1
(2)
w
π̃tw = κw ((σ + η) X̃t + ω̃t ) + β Et π̃t+1
.
(3)
Price and wage inflation, given by π̃tp and π̃tw respectively, are forward looking functions of
expected future inflation. The price Phillips curve is also a function of the real wage gap, ω̃t ,
which is the gap between the equilibrium real wage and the real wage in the flexible price
and wage economy. Similarly, wage inflation is a function of the real wage gap and the real
output gap, X̃t , which is the difference between equilibrium real output and real output in
the flexible price and wage economy. η is the inverse elasticity of substitution of labor supply
(1−θp )(1−θp β)
and
and β is the household’s discount factor. Under Calvo (1983) pricing, κp =
θp
(1−θw )(1−θw β)
κw = θw (1+w η) where θp is the probability a firm cannot adjust their price and θw is
the probability a worker cannot adjust their price. Also, w is the elasticity of substitution
between workers. Observing equations (2) and (3) show that there is a tradeoff between
balancing the two gaps and the two inflation rates. In general circumstances, they cannot
all be simultaneously eliminated.
The wage setting process leads to an evolution of the real wage gap given by
ω̃t = ω̃t−1 + π̃tw − ãt + ãt−1 − π̃tp
(4)
where ãt is TFP which follows the exogenous stochastic process ãt = ρãt−1 + t . One can
1+σ
show analytically that the output gap is defined by X̃t = Ỹt − σ+η
ãt . The model is closed by
specifying a nominal interest rate rule. In the next section, the interest rate is the solution
4
to the Ramsey problem. In the following sections we compare this optimal interest rate to
Taylor rules that target price inflation and nominal GDP.
2.1
Interest Rate Rules
As a baseline, we compute the optimal interest rate under commitment. The quadratic
loss function is a weighted sum of squared values of the output gap, price inflation, and wage
inflation. The objective function of the policy maker is to choose π̃tp , π̃tw , and X̃t to minimize
p p 2 w w 2
1 ∞ t
(π̃ ) +
(π̃ ) ]
∑ β [(σ + η) X̃t2 +
2 t=0
κp t
κw t
subject to equations (2)–(4). The first order conditions for the problem are given by
X̃t = −κw ξ2,t
p
ξ1,t − ξ1,t−1 = ξ3,t + π̃tp
λp
w w
π̃ .
ξ2,t − ξ2,t−1 = −ξ3,t +
λw t
(5)
(6)
(7)
If either wages or prices were flexible, the flexible price equilibrium could be restored.
When both are sticky, the Ramsey rule balances the welfare losses due to price stickiness and
those due to wage stickiness. Given equations (5) through (7), the Ramsey interest rate can
be inferred from equation (1). The Ramsey interest rate serves as the benchmark we make
welfare comparisons to. However, since the Ramsey rule relies on the central bank observing
theoretical constructs like the output gap, it is more realistic to consider some variation on
the Taylor rule (Taylor (1993)). In particular, we consider two versions of the Taylor rule.
One strictly targets price inflation and the other targets nominal GDP. They are given by
ĩt = ρi ĩt−1 + (1 − ρi )φπ π˜p t
(8)
nom
).
ĩt = ρi ĩt−1 + (1 − ρi )φy (Ỹtnom − Ỹt−1
(9)
and
Note that equation (9) says that the nominal interest rate responds to the the growth
rate of nominal GDP. An alternative is to make the nominal rate respond to level differences
instead of growth rates. Although we consider both of these rules in the paper, we choose
(9) as a baseline.
5
2.2
Quantitative Analysis
While we estimate the model with capital in the next section, we consider a standard
parametrization for now. We set β = 0.99 implying an annual risk free interest rate of
approximately four percent. Preferences are log over consumption and the Frisch elasticity
is one implying σ = 1 and η = 1. The elasticities of substitution for intermediate goods and
workers, p and w , are set equal to 10, implying a little more than a ten percent price and
wage markup in steady state. We consider various parameter values for the price and wage
stickiness parameters, θw and θp . Since these parameters are probabilities, they are between
zero and one. Prices (wages) are stickier as θp (θw ) approaches one.
We consider the following quantitative experiments. For a given choice of θw and θp we
find the Ramsey interest rate rule and the welfare loss associated with it. Next we simulate
the model twice using the inflation targeting rule, equation (8), in the first and nominal
GDP targeting, equation (9), in the second. For each of these rules we choose the value of
φπ and φy that minimizes the loss function. We experiment with different values of ρ which
governs the exogenous persistence of the interest rate.
Table 1 shows the results when ρ = 0. With the exception of the case when θw = 0.1 and
θp = 0.9, nominal GDP targeting results in a smaller welfare loss than inflation targeting.
The exception corresponds to a case when wages are close to fully flexible and prices are quite
sticky. This makes sense in light of Clarida et al. (1999) and Blanchard and Gal? (2007) who
show that strict inflation targeting implements the first best in the basic New Keynesian
model with only sticky prices and technology shocks. The idea is that an interest rate rule
that responds sufficiently strongly will eliminate price inflation and the output gap. Since
those are the only two sources of welfare loss in that simpler model, the first best is restored.
As emphasized by Sumner (2014), when the economy is hit with an aggregate supply shock
and nominal wages are sticky, unemployment increases and output decreases. Over the long
run, this results in a welfare loss and that is exactly what the results in Table 1 show.
Table 1: TFP Shock
φp = 0.1
φp = 0.75
φp = 0.75
φp = 0.9
φw = 2/3
φw = 0.1
φw = 0.9
φw = 0.75
φw = 0.75
Ramsey Rule
0.0001
0.0008
0.0011
0.0011
0.0002
Inflation
0.0149
0.0055
0.0050
0.0036
0.0006
1.10
4.60
4.50
3.80
5.00
0.0001
0.0011
0.0015
0.0016
0.0037
2.30
2.70
3.00
4.40
3.00
Response
NGDP
Response
φp = 2/3
To explain.
6
To obtain some intuition for exactly what is going on in the model, consider an unexpected
negative technology shock. In the language of aggregate demand - aggregate supply, this
shock shifts aggregate supply to the left. Figure 1 shows the impulse responses following this
shock when wages and prices are equally sticky (θp = θw = 0.75). An inflation targeting rule
sharply raises the nominal interest rate which results in a large decrease in real GDP from
its flexible price level (a decrease in the output gap). Since price inflation is also negative
on impact, the real rate rises.
−3
RGDP
0
2.5
−0.005
2
−0.01
1.5
−0.015
1
−0.02
0.5
−0.025
0
−4
Wage Inflation
x 10
4
Price Inflation
x 10
2
0
−2
−0.03
0
5
−3
6
10
15
20
−0.5
0
5
−3
Eq Real Rate
x 10
−4
8
10
15
20
0
5
Nom rate
x 10
10
15
20
Output Gap
0.01
0.005
6
4
−6
0
4
2
−0.005
2
−0.01
0
−2
0
0
5
10
15
20
−2
Ramsey
Inf−Targ
NGDP−Targ
−0.015
0
5
10
15
20
−0.02
0
5
10
15
20
Figure 1: Wages and Price Equally Sticky (θp = θw = 0.75)
On the other hand, the Ramsey rule increases the nominal rate only slightly and real
output actually exceeds the flexible price/wage level of output. Note that a fall in real output
need not imply that it falls below its flexible price/wage level. The reason is that after a
contraction in technology, output would fall even if all prices and wages were flexible. When
carrying out the welfare calculations, it is deviations from the flexible price/wage level that
matters. Monetary policy can try and correct for this inefficiency, but once it is removed the
equilibrium is Pareto optimal.
The nominal GDP rule actually keeps the nominal interest rate fixed following the shock.
One can see why this is algebraically by substituting the lagged value of equation (1) into
(9):
ĩt = φy (p̃t − p̃t−1 + Ỹt − Ỹt−1 )
= φy (p̃t − p̃t−1 + ĩt−1 + Et−1 π˜p t )
= φy (π̃t − Et−1 π˜p t + it−1 ) .
To the extent the forecast error is small, this term is zero (don’t actually know why/if this
7
is true). The result is that real GDP, inflation, and the output gap rise by more than under
the Ramsey rule. On the whole, the nominal GDP rule produces impulse responses that
are closer to the Ramsey rule. Consequently, the impulse response analysis corroborates the
findings of Table 2.
To gain some intuition, Figures 2 and 3 show the results in two extreme cases. Prices
are extremely sticky and wages are nearly flexible (φp = 0.1, φw = 0.9) in Figure 2. Both the
nominal GDP rule and the Ramsey rule minimize price inflation and the output gap. The
optimal coefficient in the inflation target is very small since most of the nominal rigidity and
welfare loss is coming from wage inflation. On the other hand, the Calvo parameters are
reversed in Figure 3 and inflation targeting performs better than nominal GDP targeting.
Inflation targeting produces less wage and price inflation and a smaller output gap. This
visually confirms the “divine coincidence” result discussed earlier.
−3
2
−3
RGDP
x 10
10
0
−2
−4
Wage Inflation
x 10
5
8
4
6
3
4
2
2
1
Price Inflation
x 10
−4
−6
−8
−10
0
0
−12
−2
−1
0
5
−3
10
10
15
20
12
Ramsey
Inf−Targ
NGDP−Targ
8
6
5
−3
Eq Real Rate
x 10
0
10
15
20
5
−3
Nom rate
x 10
0
10
10
10
15
20
15
20
Output Gap
x 10
8
8
6
6
4
4
4
2
2
2
0
0
0
−2
−2
−2
0
5
10
15
20
0
5
10
15
20
0
5
10
Figure 2: Sticky Prices and Nearly Flexible Wages (θp = 0.1, θw = 0.9)
8
−4
RGDP
0
12
−0.002
10
−3
Wage Inflation
x 10
6
4
8
−0.004
Price Inflation
x 10
2
6
−0.006
0
4
−0.008
2
−0.01
0
−0.012
−2
0
5
−3
1
−2
10
15
20
0
5
−4
Eq Real Rate
x 10
−4
10
10
15
20
−6
10
8
0.5
5
−3
Nom rate
x 10
0
10
20
Output Gap
x 10
Ramsey
Inf−Targ
NGDP−Targ
8
6
15
6
4
0
4
2
−1
2
0
−0.5
0
5
10
15
20
−2
0
−4
−2
0
5
10
15
20
0
5
10
15
20
Figure 3: Sticky Wages and Nearly Flexible Prices (θp = 0.1, θw = 0.9)
3
Medium Scale Model
While the previous section allows one to understand some of the intuition in the inflation
versus nominal GDP, it did so within the context of a very simplified model. In this section,
we use model with capital accumulation, variable utilization, investment adjustment costs,
and several other features in addition to nominal wage and price rigidities. Such a model has
been shown to capture the dynamic effects of monetary policy and the most salient business
cycle facts.1
3.1
Production
Production takes place in two phases. A representative final goods firm buys a continuum of inputs produced by intermediate good firms distributed on the unit interval. Each
intermediate good firm produces one unique input. The inputs are imperfect substitutes and
are combined with a constant elasticity of substitution (CES) production function with an
elasticity of substitution equal to p . Indexing the inputs with j, the profit maximization
problem for the final goods firm is
max pt (∫
{yj,t }
1
0
1
p
p −1
p −1
p
yj,t dj)
−∫
1
0
pj,t yj,t .
See Christiano et al. (2005) as an example of the former and Smets and Wouters (2007) for the latter.
9
The demand equation for input j is given by
yj,t = yt (
p −1
where yt =
1
(∫0 yj,tp
pj,t −p
)
pt
(10)
p /(p −1)
dj)
. Therefore, the output of firm j is increasing in total output
and decreasing in it’s Using the assumption of perfect competition, substituting equation
1
(10) into the objective function shows that the aggregate price index ispt = (∫0 pj,t p ) 1−p .
Intermediate good firms hire utilization adjusted capital and labor. Because they produce
differentiated goods, intermediate good firms have some pricing power. We solve their profit
maximization problem in two steps. The first step is to minimize costs. The constrained
optimization problem is
min wt nj,t + Rt k̂j,t
1
1−
{nj,t ,k̂j,t }
subject to
α 1−α
yj,t ≤ At k̂j,t
nj,t .
Here wt and Rt are the rental rates for labor, nj,t , and utilization adjusted capital, k̂j,t ,
respectively. Denoting the multiplier by mcj,t , the first order conditions are:
α
k̂j,t
)
wt = mcj,t At (
nj,t
Rt = mcj,t At (
nj,t
k̂j,t
1−α
)
.
Combining the first order conditions show firms hire capital and labor in the same ratio and
that marginal costs are constant across firms.
Turning to the pricing problem, each firm has a probability of 1 − θp of updating its price.
ζp
A firm who last updates in period t can charge a price of Πt−1,t+s−1
pj,t . The parameter ζp ,
governs the degree of indexation. If ζp = 0 there is no indexation; if ζp = 1 there is full
indexation. The firm rebates profits back to households and therefore discounts dividends
by the household’s stochastic discount factor, Λt .
The profit maximization problem is
∞
max
{pj,t, ,yj,t+s }
p
Et ∑ Λt+s Πt−1,t+s−1
θj [
ζ
s=0
10
pj,t
yj,t+s − mct+s yj,t+s ]
pt+s
subject to
−
pt+s
) .
yj,t+s = yt+s (
pj,t
Note that this is the problem for the firm conditional on updating in period t. The firm sets
its price knowing that there is a probability that it will not be able to in future periods.
Substituting the constraint into the objective function and taking the first order condition
gives the symmetric pricing rule written recursively as
p#
t =
p X1,t
p − 1 X2,t
X1,t = Λt mct yt pt p + βθp (1 + πtp )−ζp p Et X1,t+1
X2,t = Λt yt pt p
−1
+ βθp (1 + πtp )ζp (1−p ) Et X2,t+1 .
If prices were completely flexible, the reset price would simply be a markup over nominal
marginal cost.
3.2
Households
There is a continuum of households indexed by h ∈ [0, 1]. Much like the function of final
goods firms, households sell their labor to a labor packer who bundles the different sorts of
labor into a final aggregate labor input according to the function
nt = ( ∫
1
0
w −1
w
nh,t dh)
w
w −1
where w is the elasticity of substitution between different types of workers. The labor
packer buys the differentiated labor at a nominal wage of Wh,t and sells the labor to the
intermediate good firms at a nominal wage rate of Wt . Profit maximization yields the labor
−
demand equation nh,t = nt (Wh,t /Wt ) w .
The aggregate wage index is given by
Wt = (∫
1
0
1−w
Wh,t
)
1
1−w
.
As in Erceg et al. (2000), there is perfect insurance across households and utility is
additively separable in consumption and leisure. Households choose capital, utilization,
investment, consumption, bonds, wages, and hours. Because of complete insurance, all
households choose the same value of every variable except for hours and wages. Therefore,
11
we drop the h subscript except where necessary. The household’s problem is to maximize
⎡
⎢
Et ∑ β ⎢
⎢
t=0
⎣
∞
t ⎢ (ct
1+η ⎤
nh,t
⎥
− bct−1 )1−σ − 1
⎥
−ψ
1−σ
1 + η ⎥⎥
⎦
subject to
ct + It +
Wh,t
Bt+1 − Bt
χ2
kt Bt
Πt
+ [χ1 (ut − 1) + (ut − 1)2 ]
≤ ii−1 + Rt ut kt +
Nh,t +
+ Tt
Pt
2
Zt Pt
Pt
Pt
2
τ It
kt+1 = Zt [1 − (
− 1) ] It + (1 − δ)kt
2 It−1
The first constraint says that the sum of real consumption, real investment, change in real
bond holdings and utilization cost cannot exceed the sum of interest income from bonds,
utilization adjusted income from capital, profits from owning the intermediate good firms and
lump sum transfers from the government. The second constraint is the capital accumulation
equation, which takes investment adjustment costs into account. The variable Zt is an
investment specific technology shock. When Zt increases, a given amount of final goods will
produce more capital. The parameter b ∈ [0, 1) governs the degree of habit persistence. The
first order conditions are
λt = (ct − bct−1 )−σ − bβ Et (ct+1 − bct )−σ
1
Rt = [χ1 + χ2 (ut − 1)]
Zt
λt+1
λt = β(1 + it ) Et
p
1 + πt+1
2
τ It
It
It
It+1
It+1 2
λt = µt Zt [1 − (
− 1) ] − µt Zt τ (
− 1)
+ β Et [µt+1 Zt+1 τ (
− 1) (
)]
2 It−1
It−1
It−1
It
It
χ2
λt+1
+ (1 − δ)µt+1 }
µt = β Et {Rt+1 ut+1 λt+1 − [χ1 (ut+1 − 1) + (ut+1 − 1)2 ]
2
Zt
The next step is solving for the optimal reset wage and hours conditional on being able to
adjust wages in period t. Like intermediate good firms, households can partially index their
w
nominal wages to inflation so that Wh,t+s = Πζt−1,t+s−1
Wh,t . In any given period, a household
cannot adjust their nominal wage with probability 1 − θw . The household maximization
problem is
⎡
⎤
∞
n1+η
(ct+s − bct+s−1 )1−σ − 1
h,t+s ⎥
s⎢
⎢
⎥
max Et ∑ (βθw ) ⎢
−ψ
{nh,t+s ,Wh,t }
1−σ
1 + η ⎥⎥
⎢
t=0
⎣
⎦
12
subject to
ζ
−w
w
Wh,t ⎞
⎛ Πt−1,t+s−1
nh,t+s = nt
Wt
⎠
⎝
ζw
ct + It +
Πt−1,t+s−1 Wh,t
Bt+1 − Bt
χ2
kt Bt
Πt
+ [χ1 (ut − 1) + (ut − 1)2 ]
≤ ii−1 + Rt ut kt +
Nh,t +
+ Tt
Pt
2
Zt P t
Pt
Pt
Substituting the first constraint into the second constraint and the objective function and
then taking the first order condition gives the optimal real reset wage
1+w η
(wt# )
w H1,t
w − 1 H2,t
=
H1,t = ψwt w
(1+η)
Nt1+η + θw β(1 + πtp )−ζw w (1+η) Et (1 + πt+1 )w (1+η) H1,t+1
H2,t = λt wtw Nt + θw β(1 + πtp )ζw (1−w ) Et (1 + πt+1 )w −1 H2,t+1
If households can reset their wages every period then the real wage is a constant markup
over the household’s marginal rate of substitution.
3.3
Shocks, Policy, and Market Clearing
The government consumes a time varying share of total output given by gt = ωt yt where
ωt = (1−ρg )ω +ρg ωt−1 +g,t . The government runs a balanced budget in every period implying
gt = Tt . The nominal interest rate follows the Taylor rule
it = ρi it−1 + (1 − ρi ) [i + φπ (πtp − π) + φy (yt /yt−1 − 1)] + i,t .
If φy = 0, this is a strict inflation targeting rule; if φπ = φy , this is a nominal GDP targeting
rule.
The processes for TFP and investment specific technology are given by
At = (1 − ρa ) + ρa At−1 + a,t
Zt = (1 − ρz ) + ρz Zt−1 + z,t
respectively. Integrating total output across all intermediate good firms give
syt =
1
where νt = ∫0 (pj,t /pt )
−p
At n1−α
k̂tα
t
νt
dj is the deadweight loss due to price dispersion. Finally, the market
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clearing condition is
yt = ct + It + gt + [χ1 (ut − 1) +
14
χ2
kt
(ut − 1)2 ] .
2
Zt
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