The B.E. Journal of Macroeconomics
Topics
Volume 11, Issue 1
2011
Article 3
The Cost Channel, Indeterminacy, and
Price-Level versus Inflation Stabilization
Sebastian Schmidt∗
∗
Goethe University Frankfurt, [email protected]
Recommended Citation
Sebastian Schmidt (2011) “The Cost Channel, Indeterminacy, and Price-Level versus Inflation
Stabilization,” The B.E. Journal of Macroeconomics: Vol. 11: Iss. 1 (Topics), Article 3.
Available at: http://www.bepress.com/bejm/vol11/iss1/art3
c
Copyright 2011
The Berkeley Electronic Press. All rights reserved.
The Cost Channel, Indeterminacy, and
Price-Level versus Inflation Stabilization∗
Sebastian Schmidt
Abstract
In a New Keynesian model with a cost channel of monetary transmission, discretionary inflation targeting may become susceptible to a multiplicity of equilibria and results in a larger
stabilization bias than in the standard model. Both observations can be attributed to a reduction
in the effectiveness of the traditional demand channel of monetary transmission. Based on this
insight, we evaluate the desirability of an alternative monetary policy regime, price-level targeting. We find that the latter is almost immune to equilibrium indeterminacy and remains successful
in stabilizing output and inflation. Due to the weak demand channel, the ability of the central
bank to manage private sector expectations takes center stage. This is accomplished by price-level
targeting through the introduction of history dependence, but not by inflation targeting.
KEYWORDS: price-level targeting, inflation targeting, New Keynesian model, cost channel, indeterminacy
∗
I am grateful to the editor, two anonymous referees, Elena Afanasyeva, Thomas Laubach, Volker
Wieland and Maik Wolters for useful comments.
Schmidt: The Cost Channel, Indeterminacy, and Price-Level Stabilization
1 Introduction
Recent empirical evidence in support of supply-side effects of monetary policy
transmission (e.g. Barth and Ramey, 2001; Chowdhury et al, 2006; Tillmann, 2008)
has led economists to introduce a cost channel into monetary business cycle models
to study implications for optimal monetary policy (e.g. Ravenna and Walsh, 2006;
Tillmann, 2009) and for equilibrium determinacy (e.g. Brueckner and Schabert,
2003; Gabriel et al, 2008; Surico, 2008; Aksoy et al, 2009; Llosa and Tuesta, 2009;
Christiano et al, 2010). A common finding of the latter is that inflation targeting
usually suffers from shrinking regions of determinacy. Due to the cost channel,
changes in the nominal interest rate have a direct effect on firms’ variable production costs, thereby weakening the effectiveness of the traditional demand channel
of monetary transmission. In modern macroeconomics it is widely recognized that
a reasonable monetary policy strategy should in any case prevent the existence of
such multiplicity of equilibria.1 A natural question to ask is therefore how the performance of alternatives to inflation targeting is altered by the presence of a cost
channel of monetary transmission.
In this paper, we investigate the stabilization properties of price-level targeting relative to inflation targeting in a New Keynesian model with a cost channel.
Our main results are twofold. First, determinacy regions under price-level targeting
remain nearly unaffected by the introduction of the cost channel whereas determinacy regions shrink under discretionary inflation targeting. Second, price-level
targeting continues to be potent in stabilizing inflation and the output gap, whereas
the stabilization trade-off deteriorates under inflation targeting.
In the standard New Keynesian model as analyzed by Clarida et al. (1999),
the equilibrium responses to asymmetric disturbances obtained under discretionary
optimization differ from those under optimal commitment policy. This so-called
stabilization bias arising under discretionary policy can be mitigated (and under
special conditions resolved) by delegating a price-level target to the central bank
(Vestin, 2006). For reasonable calibrations of this model, however, the gains from
price-level targeting relative to inflation targeting are rather small (e.g. Kryvtsov
et al, 2008). We find that explicit consideration of the cost channel suggests to
read these previous findings with caution, in that the difference in the stabilization
performance of the two monetary policy regimes turns out to be larger than in the
model without any supply-side effects of monetary policy.
Intuitively, with the demand channel of monetary transmission being restrained, the strength of the expectations channel of monetary transmission takes
1 Indeed,
some economists have argued that the failure to do so may have been an important
source of the Great Inflation experienced by the US economy in the 1970s, see e.g. Clarida et al.
(2000).
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The B.E. Journal of Macroeconomics, Vol. 11 [2011], Iss. 1 (Topics), Art. 3
center stage. Under price-level targeting, the lagged price level becomes an endogenous state variable. This makes monetary policy history dependent, allowing
the central bank to influence private sector expectations. Specifically, price-level
stationarity implies that an unexpected surge in contemporaneous inflation fosters
the central bank to engineer a below-trend inflation rate in the following periods,
so that the price level returns to its target path. As we show, this ability to induce
reversals in the price level has desirable impacts on macroeconomic stability and
the performance of monetary policy.
The remainder of the paper is organized as follows. Section 2 describes
a simple microfounded sticky-price model of the business cycle augmented with
a cost channel and introduces the general monetary policy framework. Section 3
determines optimal discretionary policy under inflation targeting and price-level
targeting, and contrasts their implications for equilibrium determinacy. Section 4
compares the stabilization performance of the two policy regimes. Finally, section
5 concludes.
2 The Model
The economy is represented by a prototypical microfounded New Keynesian model
of the business cycle augmented with a cost channel. Most of the modeling devices
are standard. Households consume a basket of differentiated goods and supply labor to firms in a perfectly competitive market. Due to monopolistic competition, the
goods-producing firms face a downward-sloping demand curve and it is assumed
that prices are set in a staggered fashion à la Calvo. We depart from the standard
model by assuming that part of the firms have to pay their wage bill prior to the
sales receipts which compels them to borrow the required funds from a financial intermediary. As a consequence, a cost channel of monetary transmission arises, that
is, changes in the interest rate have a direct effect on marginal costs and therefore
on firms’ pricing decisions.
2.1 Structural equations
Following Ravenna and Walsh (2006) and Christiano et al. (2010), private sector
behavior is summarized by two log-linearized equations
πt = κ̃ xt + β Et πt+1 + δ κ it + ut
1
xt = Et xt+1 − (it − Et πt+1 ) .
σ
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Schmidt: The Cost Channel, Indeterminacy, and Price-Level Stabilization
All variables are expressed in percentage deviations from the non-stochastic steady
state. Equation (1) is a generalization of the standard New Keynesian Phillips curve
(NKPC). Inflation, πt , depends on expected future inflation, the output gap, xt , and
the one-period nominal interest rate, it . The appearance of the latter in the NKPC
is specific to the introduction of the cost channel into the model. The output gap
represents the gap between the level of output under sticky prices and output under
flexible prices.2 As shown by Ravenna and Walsh (2006), the latter is no longer
independent of monetary policy once we introduce the cost channel. In order to
obtain a well-defined measure of the gap, we adopt the assumption of an interestrate peg in the flexible-price economy from Ravenna and Walsh. The cost-push
shock, ut , follows an AR(1) process with a mean-zero, constant-variance innovation
εt
ut = ρ ut−1 + εt ,
(3)
where ρ ∈ [0, 1). The coefficients are functions of the structural parameters of
θβ)
the model. In particular, κ̃ = κ (σ + η ) and κ = (1−θ )(1−
, where σ > 0 is the
θ
inverse of the intertemporal elasticity of substitution, η > 0 denotes the inverse
of the elasticity of labor supply, β ∈ (0, 1) is the subjective discount factor and
θ ∈ (0, 1) denotes the Calvo parameter representing the probability that a firm is
not allowed to reoptimize its price in a given period. The cost-channel coefficient
δ deserves particular attention. Empirical estimates of the parameter vary quite
substantially depending on the estimation method and error bands can be large.3
Some estimates of δ also exceed the value of one (Chowdhury et al, 2006; Ravenna
and Walsh, 2006), pointing to the presence of financial frictions. One way to allow
for values of δ > 1 in the model is to add financial intermediation costs which
increase in the amount of borrowing, see Tillmann (2009). In this case, δ is a
function of two objects, the fraction of firms constrained by the cost channel and
η
the magnitude of the financial friction. For the remainder, we assume δ ∈ [0, σ +
σ ),
η
where the upper bound on δ , σ +
σ > 1, ensures that the demand channel of monetary
transmission outweighs the supply-side channel. Equation (2) is an intertemporal IS
curve, revealing that the output gap depends on its own expected future realization
and the ex-ante real interest rate. Finally, for the purpose of this paper it is useful to
explicitly define the price level
pt ≡ pt−1 + πt .
(4)
2 In
the presence of the cost channel, the steady state level of output differs from the one in the
standard model in that it depends on the level of the nominal interest rate in the steady state. Inflation
is zero in the steady state. Details of the derivations can be found in Ravenna and Walsh (2006) and
Christiano et al. (2010).
3 Furthermore, estimates generally vary over time (Tillmann, 2008) and across countries (Chowdhury et al, 2006).
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2.2 Society and policy objectives
Throughout, we distinguish between society’s preferences and those of the monetary authority. Society’s preferences are represented by the following objective
function
∞
2
1
2
Lt = Et { ∑ β j πt+
λ
x
+
(5)
t+ j },
j
2
j=0
where λ > 0 denotes the relative weight on output gap stabilization. For a specific
choice of λ , equation (5) represents an accurate linear-quadratic approximation to
household welfare, see Ravenna and Walsh (2006).4 In this paper, however, we
will consider alternative values for λ , possibly differing from the particular value
implied by welfare considerations.
Monetary policy is delegated to a central banker, whose objective function
is defined in terms of specific target variables
∞
1
j ′
LCB
t = Et { ∑ β yt+ j Qyt+ j }.
2
j=0
(6)
Here yt is the vector of target variables and Q is a diagonal positive semidefinite
matrix containing the central banker’s weights on the individual target variables.
Under discretion, the central banker minimizes (6) period-by-period subject to the
model equations.
2.3 Calibration
To illustrate results, we calibrate the model as summarized in Table 1. The period
length is one quarter. Most parameter values refer to the calibration in Ravenna
and Walsh (2006). In addition, the AR-coefficient of the cost-push shock is set to
ρ = 0.5 and the variance of the innovation, εt , is normalized to one.
4 In
principle, the adequate output gap measure in a loss function based on household welfare,
the deviation of output from the efficient level of output, differs from the output gap in equations (1)
and (2). This follows from the presence of monopolistic competition and from the dependence of
flexible-price steady state output on the steady state level of the nominal interest rate. To focus on
stabilization policy, we follow Ravenna and Walsh (2006) in assuming that steady state distortions
are eliminated by appropriate fiscal subsidies.
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Schmidt: The Cost Channel, Indeterminacy, and Price-Level Stabilization
Table 1: Calibration
Parameter
β
0.99
σ
1.50
η
1.00
κ
0.0858
ρ
0.50
3 Discretionary monetary policy and equilibrium determinacy
This section characterizes monetary policy under the discretionary inflation and
price-level targeting regime, respectively, and examines conditions for equilibrium
determinacy. Here, we focus on specific targeting rules, following the terminology
of Svensson (2003), i.e. optimality conditions implied by the monetary authority’s
objective function.
3.1 Inflation targeting
Under the inflation-targeting regime, yt = [πt , xt ]′ in (6) and q1,1 = 1, q2,2 = λ IT ,
where qk,m is the element from the kth row and mth column of matrix Q and λ IT > 0
denotes the central banker’s relative weight on output gap stabilization which may
differ from society’s λ . The first-order conditions of the minimization problem
result in the following (specific) targeting rule
πt = −
λ IT
xt .
κ̃ − δ κσ
(7)
Note, that for δ = 0, (7) reduces to the optimality condition of the standard New
η
Keynesian model as analyzed by Clarida et al. (1999). The assumption δ < σ +
σ
implies that inflation above target always requires the monetary authority to contract
demand below potential.
To examine the implications of the cost channel for equilibrium determinacy, combine (1), (2) and targeting rule (7), to get
πt =
λ IT (β + δ κ ) − (κ̃ − δ κσ ) δ κσ
λ IT + (κ̃ − δ κσ )2
Published by The Berkeley Electronic Press, 2011
Et πt+1 +
λ IT
λ IT + (κ̃ − δ κσ )2
ut .
(8)
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Equilibrium determinacy requires the expected-inflation coefficient in (8) to be less
than one in absolute value.5
Proposition 1 The model composed of (1), (2), (3) and targeting rule (7) exhibits
a unique rational expectations equilibrium {πt , xt , it }, if and only if either of the
following two cases is satisfied.
Case I:
β +δκ > 1
(9)
λ IT >
(10)
λ IT
(11)
κ̃ − δ κσ
(2δ κσ − κ̃ )
1+β +δκ
κ̃ − δ κσ
<
κ̃ .
β +δκ −1
Case II: Equation (10) and the inequality opposite to (9) hold.
Proof See Appendix A.1.
When the cost channel is weak, such that the inequality opposite to (9) holds, (10)
constitutes the sole condition for determinacy on λ IT . This condition is binding iff
δ > σ2+ση . Otherwise a determinate rational expectations equilibrium exists for all
λ IT > 0.6 Clearly, this result always applies to the standard model without the cost
channel, δ = 0. If, on the other hand, the cost-channel coefficient δ is large enough
such that (9) holds, (10) and (11) are the relevant conditions for determinacy. In
this case, the set of parameter values for λ IT is constraint from above and, for
δ > σ2+ση , also from below. Table 2 provides the determinacy conditions under
the baseline calibration for alternative values of δ . The second column contains
the results for inflation targeting. We observe that a cost-channel coefficient of
δ = 0.5 or δ = 0.75 does not impose particular tight constraints on the choice of
λ IT as the upper bounds are rather high. Increasing δ further changes the picture.
For instance, if δ = 1, values of λ IT below 0.0018 or above 0.2428 will induce
indeterminacy. Values of δ above one reduce the upper bound of the determinacy
region even further, while the lower bound first increases and then falls. In general,
Table 2 reveals that in a cost-channel economy neither the choice of an inflation
nutter nor of a central banker putting too much weight on output-gap stabilization
seems to be desirable if one wants to ensure a unique bounded rational expectations
equilibrium.
5 Llosa and Tuesta (2009) also consider determinacy conditions for inflation-targeting rules in the
presence of the cost channel but restrict the analysis to the case when δ = 1.
6 Throughout the paper, we refer to local determinacy of equilibrium.
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Schmidt: The Cost Channel, Indeterminacy, and Price-Level Stabilization
Table 2: Determinacy conditions for the baseline calibration
δ
Inflation targeting
Price-level targeting
IT
0
0<λ
0 < λ PT
0.5
0 < λ IT < 0.9789
0 < λ PT
IT
0 < λ < 0.4656
0 < λ PT
0.75
0.0018 < λ IT < 0.2428
0.0002 < λ PT
1
IT
0.0003 < λ PT
1.25 0.0027 < λ < 0.1183
0.0017 < λ IT < 0.0388
0.0001 < λ PT
1.5
3.2 Price-level targeting
Under the price-level-targeting regime, yt = [pt , xt ]′ in (6) and q1,1 = 1, q2,2 = λ PT ,
where λ PT > 0 denotes the central banker’s relative weight on output gap stabilization. The value function has the form
1
V (pt−1 , ut ) = min{ pt2 + λ PT xt2 + β Et V (pt , ut+1 )},
(12)
2
where the minimization is subject to
pt =
β +δκ
κ̃ − δ κσ
1
pt−1 +
Et p (pt , ut+1 ) +
xt
1+β +δκ
1+β +δκ
1+β +δκ
1
δ κσ
Et x (pt , ut+1 ) +
ut ,
+
1+β +δκ
1+β +δκ
(13)
and the law of motion of the cost-push shock (3).7 The functions p (pt , ut+1 ) and
x (pt , ut+1 ) denote the price level and the output gap that the central bank expects
to be realized in period t + 1 in equilibrium, contingent on the realization of the
exogenous state variable ut+1 . Specifically, we restrict attention to the minimumstate-variable (MSV) solution
p (pt−1 , ut ) = apt−1 + but
x (pt−1 , ut ) = cpt−1 + dut .
(14)
(15)
The first-order condition of the optimization problem results in the following targeting rule
κ̃ − δ κσ
β λ PT
PT
pt + λ xt =
Et xt+1 ,
(16)
ω
ω
7 The
former equation results from combining (1) and (2) rewritten in terms of the price level.
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where ω = 1 + β (1 − a) + δ κ (1 − a − σ c). The details of the derivation are deferred to Appendix A.2. Note, that ω depends on the parameters a and c of the
MSV solution. In the case of price-level targeting, there is no simple closed-form
solution for these parameters and we employ a bisection method to obtain a, from
which one can then determine the other parameters of the solution functions.
Substituting (4) into (1) and (2), and using targeting rule (16) the model can
be written as
Azt = BEt zt+1 +Cut
ut = ρ ut−1 + εt ,
where zt = [pt , xt , pt−1 ]′ and


1
0
0


0
λ PT
A =  κ̃ −ωδ κσ
,
1
κ̃ −δ κσ
1
− 1+β +δ κ − 1+β +δ κ


B=
(17)
0
0
β +δ κ
1+β +δ κ
0
βλ
ω
δ κσ
1+β +δ κ

1
0
 . (18)
0
The form of C is omitted since it is not required for what follows. Define Ω ≡ B−1 A.
Since we have two free endogenous variables, xt and pt , and one predetermined
endogenous variable, pt−1 , the dynamic system (17) has a unique bounded solution
if and only if Ω exhibits two eigenvalues outside the unit circle and one eigenvalue
inside the unit circle.
Proposition 2 The model composed of (1), (2), (3), (4) and targeting rule (16)
exhibits a unique rational expectations equilibrium {πt , xt , pt , it }, if and only if
λ PT >
2δ κσ − κ̃
1
(κ̃ − δ κσ )
.
2
(ω + β ) (1 + β + δ κ )
(19)
Proof See Appendix A.3.
The parameter ω is strictly positive.8 Hence, if δ ≤ σ2+ση , condition (19) is trivially
satisfied and any λ PT > 0 will induce a unique rational expectations equilibrium. If
δ > σ2+ση , condition (19) constitutes a positive lower bound on the set of desirable
λ PT . The third column of Table 2 provides the determinacy conditions under the
baseline calibration. Low and intermediate values of δ do not impose any constraint
on λ PT . In the case of δ = 1, constraint (19) becomes binding but the implied lower
8 It
holds lim a λ PT = 0 and lim a λ PT = min{1, β +1δ κ }. It follows max{1, β + δ κ } ≤
λ PT →0
λ PT →∞
δ κσ
PT .
ω ≤ 1 + β + δ κ + κ̃ −
δ κσ , where ω is decreasing in λ
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Schmidt: The Cost Channel, Indeterminacy, and Price-Level Stabilization
bound on λ PT is close to zero. The same holds true for the two considered values of
δ > 1. Note however, that λ PT and λ IT are distinct objects, so that a direct comparison of their absolute values contains only limited information. In order to assess
the tightness of the lower bounds under price-level and inflation targeting we may
employ the relationship between the variance of the price level and the variance of
ρa
inflation under price-level targeting.9 It holds Var (pt ) = 2(1−1+
ρ )(1−a) Var (πt ). Substituting this expression into the loss function of the price-level-targeting central
bank, it follows that the lower bound under price-level targeting, λ PT , is tighter
ρ )(1−a)
than the lower bound under inflation targeting, λ IT , iff λ PT 2(1−1+
> λ IT . Unρa
der the benchmark calibration, this is not the case. In fact, it turns out that the
transformed lower bound corresponds to a slightly smaller value than the associated λ PT . For instance, in the case of δ = 1.25 the transformed lower bound equals
0.0002 whereas λ PT = 0.0003.
4 Stabilization performance
Abstracting from equilibrium determinacy, the question remains to what extent
the presence of the cost channel alters the stabilization bias resulting from discretionary optimization. A convenient way to depict the central banker’s stabilization
trade-off is to reformulate society’s objective function in terms of the variance of
the output gap and inflation. In particular, for β → 1, the society’s loss function
evaluated under the unconditional expectations operator, E, becomes equivalent to
Var (πt ) + λ Var (xt ), a weighted average of the unconditional inflation variance and
output gap variance. Figure 1 plots variance frontiers for δ = 0 (no cost channel)
and δ = 1, characterizing the combinations of output-gap variance and inflation
variance attainable under price-level and inflation targeting, respectively. Different
points on the frontiers refer to different values of the preference parameter λ on
the interval [0.1, 1.2]. The central bank’s weight on output gap variation in its loss
function, λ j , j ∈ {IT, PT }, is determined numerically in order to minimize society’s
loss function for a given value of λ . Variance combinations to the south-west of a
particular frontier are unattainable in the respective regime whereas combinations
to the north-east are inefficient. Beginning with the standard model without the cost
channel, the figure shows that price-level targeting leads to an improved trade-off
between inflation and output-gap stabilization. This continues to hold true when we
move to the model with the cost channel (δ = 1). However, the gap between the
frontiers representing the two policy regimes widens as the stabilization trade-off
9 Appendix A.2 contains the expressions for the variances of the price level, inflation and the
output gap under price-level targeting.
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Figure 1: Variance frontiers
16
Price−level targeting (δ =0)
Inflation targeting (δ =0)
Price−level targeting (δ =1)
Inflation targeting (δ =1)
14
12
Var(x)
10
8
6
4
2
0
0
1
2
3
Var(π)
4
5
Notes: Variance frontiers are shown for the baseline calibration. Society’s preference parameter λ
is varied on the interval of [0.1, 1.2] and the central banker’s preference parameter λ j , j ∈ {IT, PT }
is determined numerically in order to minimize society’s loss function for a given value of λ .
under inflation targeting deteriorates. In the case of purely transitory exogenous
shocks, we can establish this result also analytically.
Proposition 3 For ρ = 0, the trade-off faced by discretionary inflation targeting
between inflation and output gap variability is always less favorable for δ = 1 than
for δ = 0.
Proof See Appendix A.4.
In contrast, under price-level targeting the frontier shifts only slightly, and leads
to an improved trade-off in the case of a sufficiently high preference parameter λ .
Intuitively, in the presence of the cost channel a change in the one-period nominal
interest rate has opposite effects on inflation via the demand-side and the supplyside channel of monetary transmission. Following the demand-side channel, a (sufficiently large) rise in the nominal rate raises the real interest rate. The induced
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Schmidt: The Cost Channel, Indeterminacy, and Price-Level Stabilization
reduction in aggregate demand leads to a fall in labor demand. Thus, equilibrium
wages decrease, thereby lowering firms’ marginal costs and attenuating potential
inflationary pressures. However, due to the supply-side channel, an increase in
the nominal rate raises firms’ borrowing costs, implying a direct positive effect on
marginal costs and thus on firms’ pricing decisions. The performance of pricelevel targeting is less affected by the introduction of the cost channel because it
employs the dependence of next period’s policy actions on current economic conditions. Price-level stationarity implies that any contemporaneous above-trend inflation episode has to be followed by enough below-trend inflation in order to bring
the price level back to its target path. But lower expected future inflation dampens current inflation, which reduces the trade-off for the monetary authority in the
contemporaneous period.
5 Conclusion
This paper examines the implications of the cost channel of monetary transmission
for the desirability of two alternative monetary policy regimes, inflation targeting
and price-level targeting. Employing a standard New Keynesian model of the business cycle with a cost channel, we confirm that discretionary inflation targeting can
suffer from strongly shrinking determinacy regions. In contrast, price-level targeting remains almost insusceptible to equilibrium indeterminacy, independently of
the size of the cost channel. Furthermore, the stabilization trade-off under inflation
targeting deteriorates with increasing importance of the cost channel, whereas the
stabilization performance under price-level targeting is virtually unaffected. Accordingly, the quantitative difference in the performance of inflation and price-level
targeting increases, suggesting that, to the extent that the cost channel is empirically
relevant, comparison exercises based on the standard New Keynesian model underestimate the gains from price-level targeting. Both policy regimes differ in their
ability to make policy history dependent. This property of price-level targeting
turns out to be especially important once we take supply-side effects of monetary
policy transmission explicitly into account.
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Appendix A
Appendix A.1 Proof of Proposition 1
From (8), define γ ≡
λ IT (β +δ κ )−(κ̃ −δ κσ )δ κσ
.
λ IT +(κ̃ −δ κσ )2
Equilibrium determinacy requires
−1 < γ < 1. If β + δ κ ≤ 1, the condition γ < 1 is equivalent to
λ IT (1 − β − δ κ ) > − (κ̃ − δ κσ ) κ̃ ,
(20)
η
IT > 0 and can therefore be omitted.
which, given δ ∈ [0, σ +
σ ), is satisfied for all λ
If β + δ κ > 1, the condition γ < 1 is equivalent to
λ IT <
κ̃ − δ κσ
κ̃ ,
β +δκ −1
(21)
which represents condition (11) in Proposition 1.
The condition γ > −1 is equivalent to
λ IT >
κ̃ − δ κσ
(2δ κσ − κ̃ ) ,
1+β +δκ
(22)
which represents condition (10) in Proposition 1.
Appendix A.2 Price-level targeting under discretion
The value function has the form
1 2
pt + λ PT xt2 + β Et V (pt , ut+1 )}.
2
We restrict attention to the minimum-state-variable (MSV) solution
V (pt−1 , ut ) = min{
p (pt−1 , ut ) = apt−1 + but
x (pt−1 , ut ) = cpt−1 + dut .
(23)
(24)
(25)
Combining (1) and (2) rewritten in terms of the price-level with (24) and (25) written one period ahead leads to
1
κ̃ − δ κσ
1 + ((β + δ κ ) b + δ κ d) ρ
pt−1 +
xt +
ut ,
(26)
ω
ω
ω
where ω = 1 + β (1 − a) + δ κ (1 − a − σ c). Solving minimization problem (23)
with respect to xt subject to (26) results in the following first-order condition
pt =
pt
∂ pt
∂ V (pt , ut+1 ) ∂ pt
+ λ PT xt + β Et
= 0.
∂ xt
∂ pt
∂ xt
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Schmidt: The Cost Channel, Indeterminacy, and Price-Level Stabilization
It is
∂ V (pt−1 , ut )
∂ pt
∂ V (pt , ut+1 ) ∂ pt
= pt
+ β Et
∂ pt−1
∂ pt−1
∂ pt
∂ pt−1
PT
λ
= −
xt .
κ̃ − δ κσ
(28)
Writing (28) one period ahead and taking expectations, the resulting equation can
be substituted into (27) to arrive at
κ̃ − δ κσ
β λ PT
pt + λ PT xt =
Et xt+1 ,
ω
ω
(29)
which is targeting rule (16) representing optimal discretionary price-level targeting.
Combining (26) and (29), it follows from (24) that
a =
b =
λ PT ω
(κ̃ − δ κσ )2 + λ PT ω 2 + β λ PT (1 − ω a)
λ PT ω + λ PT ρ [2β bω − β (1 + β bρ ) + δ κ (b + σ d) (ω − β ρ )]
(κ̃ − δ κσ )2 + λ PT ω 2 + β λ PT (1 − ω a)
(30)
. (31)
Using (26) and (24), it follows from (25) that
(1 − a) (1 − (β + δ κ ) a)
κ̃ − (1 − a) δ κσ
1 − b [ω − (β + δ κ ) ρ ]
d = −
.
κ̃ − (1 − ρ ) δ κσ
c = −
(32)
(33)
Employing the solution functions, the following expressions for the variance of the
price level, the inflation rate and the output gap can be derived:
(1 + ρ a) b2
Var (εt )
(34)
(1 − ρ a) (1 − a2 )
2 (1 − ρ ) b2
Var (εt )
Var (πt ) =
(35)
(1 − ρ a) (1 + a)
(1 + ρ a) b2 c2 + (1 − ρ a) 1 − a2 d 2 + 2ρ 1 − a2 bcd
Var (εt ) .
Var (xt ) =
(1 − ρ a) (1 − a2 )
(36)
Var (pt ) =
Published by The Berkeley Electronic Press, 2011
13
The B.E. Journal of Macroeconomics, Vol. 11 [2011], Iss. 1 (Topics), Art. 3
Appendix A.3 Proof of Proposition 2
The characteristic polynomial of Ω is given by P (µ ) = −µ 3 + A2 µ 2 − A1 µ + A0 ,
where
ω
(37)
A0 =
β (β + δ κ )
1
(κ̃ − δ κσ )2
1+β +δκ
+
ω
+
PT
β + δ κ β λ (β + δ κ ) β (β + δ κ )
1+β +δκ ω
δ κσ
=
+ −
(κ̃ − δ κσ ) .
β +δκ
β β λ (β + δ κ )
A1 =
(38)
A2
(39)
Furthermore,
κ̃ − δ κσ
P (1) = −κ̃
β λ PT (β + δ κ )
ω 1+β +δκ
κ̃ − δ κσ
P (−1) = 2 1 +
+
(κ̃ − 2δ κσ ) .
β
β +δκ
β λ PT (β + δ κ )
(40)
(41)
If P (1) < 0 and P (−1) > 0, continuity implies that the characteristic equation has
an odd number of roots inside the unit circle. Suppose, all three eigenvalues would
lie inside the unit circle, then |det (Ω) | = |A0 | < 1. But det (Ω) = A0 = β (β ω+δ κ ) > 1
which follows from ω ≥ max{1, β + δ κ }. Hence, if P (1) < 0 and P (−1) > 0, Ω
has two eigenvalues outside the unit circle and one eigenvalue inside the unit circle.
η
PT > 0 and that P (−1) > 0
Observing that, given δ ∈ [0, σ +
σ ), P (1) < 0 for all λ
is equivalent to condition (19) in Proposition 2 completes the proof.
Appendix A.4 Proof of Proposition 3
The solution functions for πt and xt under discretionary inflation targeting have the
form
πt = f IT δ , λ IT ut
xt = gIT δ , λ IT ut .
IT
κ̃ −δ κσ
IT δ , λ IT = −
For ρ = 0, we have f IT δ , λ IT = IT λ
.
2 and g
IT +(κ̃ −δ κσ )2
λ
κ̃
−
δ
κσ
)
λ
+(
Define λ̂ IT such that gIT 1, λ̂ IT = gIT 0, λ IT . This implies the following relationship between λ̂ IT and λ IT
η
λ IT + κ 2 σ η .
σ +η
IT
IT
IT
IT
<f
1, λ̂
for all λ IT .
It is then straightforward to verify that f
0, λ
λ̂ IT =
http://www.bepress.com/bejm/vol11/iss1/art3
14
Schmidt: The Cost Channel, Indeterminacy, and Price-Level Stabilization
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