Discretion Rather than Rules: Equilibrium Determinacy
and Forward Guidance with Inconsistent Optimal Plans
Jeffrey R. Campbell∗
Jacob P. Weber†
February 21, 2017
Abstract
We resolve the standard New Keynesian model’s local equilibrium indeterminacy
with a restriction that the central bank is subject to quasi-commitment (Schaumburg
and Tambalotti, 2007). Specifically, if the central banker succumbs to time inconsistency frequently enough, then there exists a unique, bounded equilibrium even under
a fixed interest rate path. The resulting framework features nontrivial, Odyssean forward guidance without a forward guidance puzzle. Optimal policy promises price level
targeting, but fails to deliver it.
∗
†
Federal Reserve Bank of Chicago and CentER, Tilburg University
Federal Reserve Bank of Chicago
We thank Gadi Barlevy, Marco Bassetto, Gauti Eggertsson, Simon Gilchrist, Alejandro Justiniano, and
Stephanie Schmitt-Grohe for insightul comments and discussion. Any errors that remain are our own. The
views expressed herein are those of the authors. They do not necessarily represent the views of the Federal
Reserve Bank of Chicago, the Federal Reserve System, or its Board of Governors.
JEL Codes: E12, E52.
Keywords: Quasi-Commitment, Sunspot Equilibria, Ramsey Problem, Monetary Policy.
1
Introduction
Consider the standard New Keynesian economy as presented in Giannoni and Woodford
(2005), wherein a central banker seeks to minimize deviations of output and inflation from
their steady-state values while satisfying an intertemporal-substitution (IS) curve and a
forward-looking Phillips curve (PC). If a transitory cost-push shock to the Phillips curve
creates inflationary pressure today, a credible promise of future policy tightening lowers
inflation expectations and reduces the shock’s effect on current inflation. However, this
strategy is time inconsistent; the central banker will be tempted to renege on her promise
when the time comes to make good on it (Kydland and Prescott, 1977).
Bodenstein et al. (2010) argue that, in the real world, central banks operate under an
“intermediate degree of credibility” between perfect commitment and complete discretion.
A quasi-commitment framework (Schaumburg and Tambalotti, 2007; Debortoli and Nunes,
2007), wherein the central banker faces a constant probability of reoptimizing and thereby
“succumbing” to time inconsistency, allows for exploration of these intermediate cases. We
show that replacing full commitment with quasi-commitment can
• eliminate local equilibrium multiplicity (Section 2.1),
• eliminate the “forward guidance puzzle” of Del Negro et al. (2012) (Section 4.1), and
• separate central bank promises (of price level targeting) from agents’ rational expectations (a permanent change in the price level) (Section 4.2).
Furthermore, this quasi-commitment modification delivers these results parsimoniously, requiring only one additional parameter representing the central banker’s credibility.
Quasi-commitment also has important ramifications for the implementation of the Ramsey policy. In it, the planner commonly solves for the best feasible allocation of inflation
and output subject to the sequence of PCs and then backs out the interest rates that satisfy
the IS curves. However, due to the familiar indeterminacy in the standard model, simply
announcing this interest rate path does not pin down the planner’s desired allocation as a
unique equilibrium. Svensson and Woodford (2005) ensure agents coordinate on the desired
allocation by promising to follow an active Taylor rule and so generate explosive outcomes off
the equilibrium path (Cochrane, 2011). We interpret such a promise as an extra tool which
appears when we write down the Ramsey problem in terms of instruments, rather than
allocations. We call this or any other unmodeled communication that coordinates agents’
expectations on a particular equilibria an “Open Mouth Operation” (OMO) and view it as
an undesirable artifact of the model’s indeterminacy. Quasi-commitment eliminates this tool
by eliminating local equilibrium indeterminacy.
1
Quasi-commitment obtains these results by modifying the IS curve. In the standard IS
curve the elasticity of output today with respect to expected output tomorrow is one. Under
quasi-commitment this elasticity becomes less than one, which we refer to as introducing
“Euler equation discounting” (McKay et al., 2016; Campbell et al., 2016). In Gabaix (2016),
the introduction of behavioral agents with myopia also achieves this discounting, with similar results for equilibrium determinacy and optimal policy. The main difference between
a quasi-commitment framework and such a behavioral model within the context of forward
guidance lies in how agents process central bank announcements. In the behavioral model,
agents discount central bank announcements because of the cognitive load associated with
processing information about long dated interest rate promises. With quasi-commitment,
agents discount such announcements due to their awareness that those promises might not
be fulfilled. Since we characterize forward guidance without deviating from an assumption
of complete rationality, and Gabaix (2016) does so without deviating from perfect commitment, these models are complements; a researcher may wish to use a convex combination of
behavioralism and quasi-commitment to gain determinacy without quantitative over-reliance
on either assumption.
Going forward, we lay the foundations for our analysis by reviewing the Ramsey planning
formulation of the central bank’s problem in Section 2. Therein, we present the planning
problem with perfect commitment and illustrate how the indeterminacy present in this formulation of the problem makes an OMO available. Then, we introduce quasi-commitment
by assuming that the central banker returns inflation and the output gap to zero with some
fixed probability in each time period (Schaumburg and Tambalotti, 2007). We show how
this can remove indeterminacy from the model and, in so doing, remove the OMO. In Section 3, we explicitly model the game associated with quasi-commitment, wherein an initial
central banker chooses a path of interest rates while aware that she may be succeeded by
different central bankers who will reoptimize, facing the same problem. We show that if the
the probability of reoptimization is high enough, then the resulting unique Markov perfect
equilibrium coincides with the Ramsey planning allocation under quasi-commitment. Section 4 considers the implications of quasi-commitment for the forward guidance puzzle and
price level targeting, and Section 5 concludes.
2
Ramsey Planning
Here we demonstrate how the well-known indeterminacy present in the standard model
makes an extra instrument available to the central banker in her corresponding Ramsey
problem. We then show how quasi-commitment can remove this instrument. Specifically,
2
there exists one and only one equilbrium with bounded inflation when the central banker
operates sufficiently far from perfect commitment even with a fixed nominal interest rate
path.
2.1
The Standard Model
We begin in a standard New Keynesian economy.1 The Phillips curve is
πt = k Yet + βπt+1 + mt
(1)
with cost-push shock m0 6= 0 and mt = 0 for all t > 0, and the IS curve is
1
Yet = − (it − πt+1 − i\ ) + Yet+1 .
σ
(2)
Here it denotes the nominal interest rate, i\ denotes the natural rate of interest, πt denotes
inflation and Yet denotes the output gap. The parameters satisfy σ, k ∈ (0, ∞) and β ∈ (0, 1).
We ignore the possiblity of an effective lower bound on nominal interest rates. The central
banker has a quadratic loss function of the following form, which can be derived from the
representative household’s utility function (Woodford, 2003) or from the central banker’s
legislative environment (Evans, 2011):
∞
X
t=0
β
t
1 2 λ e2
π + Yt .
2 t
2
(3)
A central banker solving the Ramsey problem chooses paths for it , πt , and Yet to minimize
her loss (3) while satisfying the sequences of constraints given by (1) and (2). The standard
approach solves for the paths of πt and Yet constrained only by the sequence of PCs, and then
backs out the necessary values for it using the sequence of IS curves.
Figure 1 presents an example solution with m0 = 1, or a one percent shock to the Phillips
curve in the initial time period (all of our numerical examples use the same value of β = 0.99).
If the central banker did nothing, so that it = i\ for all t, and agents’ inflation expectations
for π1 , π2 , ... = 0 were unchanged, then inflation today would have to increase by one percent.
The central banker can improve on this outcome by raising it above i\ , thereby reducing both
inflation and output. Because the loss function is quadratic, the central banker recieves a
first order gain from reducing inflation, at the cost of a second order loss from reducing
output. The optimal setting of i0 equates the marginal benefit of reducing inflation with the
1
See Galı́ (2008) for a derivation of these now-standard equations.
3
marginal cost of diminishing output. Given the parameter values used in Figure 1 and no
control over agent’s expectations of the future, this policy would yield π0 = .8 and Ye0 = −.8.
Since the central banker controls π1 , she can further improve outcomes by promising a
deflationary recession. Mechanically, the promised deflation in period one partially offsets
the cost-push shock, allowing the central banker to achieve both lower inflation today and
a smaller output gap (e.g. Campbell, 2013). This reduces inflation and increases output
today, yielding a first order gain, at the expense of a second order loss from the deflationary
recession in the future. Intuitively, such a policy improves outcomes by “spreading the
pain” of a transitory shock across multiple future periods. The dark blue line plots the first
few periods of the optimal plan for inflation, while the red line plots the same for output.
Although it is feasible to close these gaps at any time after t = 0, the planner chooses to close
them only asymptotically. The light blue line plots the price level, which is the accumulation
of inflation. Optimal policy eventually undoes all the inflation that was allowed to occur
in the initial time period. This is the familiar price level targeting result of Giannoni and
Woodford (2005).
The dashed black line plots the interest rate consistent with the IS curve given the optimal
choices of πt and Yet . When the central banker has no control over future expectations of
πt , Yet , the optimal initial interest rate jumps to i\ +1.6 percent. The Ramsey planner chooses
a more modest initial response of i\ + 0.24 percent, but she keeps the interest rate above i\
after the cost push shock has dissipated.
In the conventional interpretation of the exercise presented in Figure 1, the central banker
promises to keep interest rates above the natural rate after the cost-push shock has passed,
thereby creating deflationary expectations. However, this explanation fails to accurately
describe other, similar forward guidance experiments. To see this, consider Figure 2. This
plots the Ramsey solution given the same values for k and λ but a different value for σ. Since
the Phillips curve and loss function are unchanged, the chosen values for πt and Yet equal
those in Figure 1: and they are given by the dark blue and red lines, respectively. With the
particular IS curve chosen, the interest rate required by this plan is a constant: it = i\ . This is
puzzling because we have a deflationary recession without contractionary interest rate policy.
The solution lies in the New Keynesian model’s familiar indeterminacy result. The central
banker’s chosen allocation for t ≥ 1 coincides with one of the many equilibria consistent with
it = i\ . Indeed, this is the case whenever the parameters satisfy σk/λ = 1. Implementing this
example’s Ramsey allocation requires no contractionary interest rate policy, but coordinating
agents’ expectations with an Open Mouth Operation is essential.
We presented the“knife edge” case in Figure 2 to illustrate the use of a tool that is always
available and used by the central banker. To demonstrate the existence of this instrument
4
0.8
it ! i\
Yet
:t
Price Level
0.6
Percentage Points
0.4
0.2
0
-0.2
<=2
k = 0.25
6 = 0.25
-0.4
-0.6
-0.8
0
1
2
3
Elapsed Time Since Shock
4
Figure 1: The Standard Solution to the Ramsey Problem: m0 = 1
5
5
1.5
it ! i\
Yet
:t
Price Level
Percentage Points
1
0.5
0
<=1
k = 0.25
6 = 0.25
-0.5
-1
0
1
2
3
Elapsed Time Since Shock
4
Figure 2: Solution to the Ramsey Problem: m0 = 1 and
6
5
σk
λ
=1
analytically, consider the problem of the planner in terms of choosing the settings of the
instruments instead of allocations. We plug the PC (1) into the IS curve (2). The result is
a single, second order difference equation with forcing function xt :
k
πt − 1 + β +
πt+1 + βπt+2 = xt .
σ
(4)
Where xt ≡ − σk it − i\ + mt − mt+1 . For simplicity, from this point on consider xt to be
the relevant policy instrument. The full set of solutions to (4) for a particular path of xt is
given by a linear combination of two homogenous solutions and a particular solution. The
rates of decay of the homogenous solutions are each governed by one of the roots (ϕ, ψ) of
the characteristic polynomial:
k
1− 1+β+
q + βq 2 = 0.
σ
which are ϕ ∈ (0, 1) and ψ ∈ ( β1 , ∞). Since πt is governed by a second order difference
equation (with forcing function xt ) we generally need two restrictions to pin down a solution.
We can and do obtain one by assuming limt→∞ πt = 0. We describe this hereafter as “long
run inflation expectations are well anchored.” As noted by Cochrane (2011), this is not a
requirement for equilibrium or any agent’s optimality; rather it is a side condition. However,
infinitely many nonexplosive possible solutions for {πt } are consistent with (4). Imposing a
second restriction that π0 be fixed at some value and solving the system for t > 0 yields
πt = ϕt π0 −
t−1
X
ϕl+1
∞
X
ψ −j xt+j−l−1 .
j=0
l=0
From this exercise, it is clear that in solving the Ramsey problem for optimal allocations
and assuming that the central banker can pick an entire path for πt , we implicitly assume
that she can choose both the path of interest rates, which governs xt , and the level of π0 .
The interest rates alone do not pin down π0 ; the central banker accomplishes this with an
open mouth operation.
2.2
The Model with Quasi-Commitment
Perfect commitment might not be the most relevant case to consider for policymakers. The
benchmark from which we should deviate in considering forward guidance is probably far
from perfect commitment and much closer to complete discretion. As Donald Kohn noted,
“no central bank to date has adopted an Odyssean commitment strategy,” and Laurence
7
Meyer, who interviewed 100 market participants, reported that 98 of them understood
forward-looking statements of the Federal Reserve to be forecasts, rather than commitments
(Romer and Wolfers, 2012).
To relax the assumption of perfect commitment and allow for the consideration of “intermediate cases” (Bodenstein et al., 2010), consider a model after Schaumburg and Tambalotti
(2007) which allows for quasi-commitment. Here, a central banker promises paths of inflation
and output to minimize her loss facing the same transitory cost-push shock, and the same
PC and IS curve constraints. However, she succumbs to time inconsistency and will return
the economy to the optimal solution under discretion, πt = Yet = 0, with some constant
exogenous probability α in each time period. Thus after taking expectations, the Phillips
curve and the loss function are the same up to a new “discount factor” β(1 − α).
To solve the resulting Ramsey problem, the central banker minimizes
∞
X
t t
(1 − α) β
t=0
1 2 λ e2
π + Yt
2 t
2
subject to the constraints
πt = k Yet + β(1 − α)πt+1 + mt
where m0 6= 0 and mt = 0 for all t > 0. The solutions to this problem are the optimal
promises the central banker should make. Solving this problem involves exactly the same
procedure as with full commitment. However, quasi-commitment qualitatively changes the
IS curve used to back out the promised interest rates:
1
Yet = − (it − (1 − α)πt+1 − i\ ) + (1 − α)Yet+1
σ
The two forward-looking terms are the expectations of inflation and the output gap. These
are equal to the promises of the central banker discounted by the probability that she follows
through on those promises.
If α is high enough and we assume limt→∞ πt = 0, we will have determinacy in this model
and a path of interest rates alone will pin down a unique equilibrium.2 To see this, again
solve the model in terms of instruments instead of allocations. Plug the PC into the IS
curve to obtain a second order difference equation in terms of πt with the forcing function
2
Note this assumption has different empirical content than the standard assumption of well-anchored
inflation expectations. Here, we assume the central bankers promises converge to zero.
8
xt ≡ − σk (it − i\ ) + mt − (1 − α)mt+1 ,
k
πt − (1 − α) 1 + β +
πt+1 + β(1 − α)2 πt+2 = xt .
σ
(5)
Note that we can reduce (5) to to (4) by setting α = 0. In this case, we have equilibrium
multiplicity. Trivially, we can create a system with a unique solution by setting α = 1. This
leaves the central banker with no ability to commit to future policy actions.
Does an intermediate case exist where the central bank can commit to future policy with
some credibility, but the system still has a unique solution given any non-explosive {xt }? To
answer this in the affirmative, write down the expression for the roots of the characteristic
polynomial of (5)
1+β+
ϕ, ψ =
k
σ
q
2
± 1 + β + σk − 4β
2β(1 − α)
.
(6)
By construction, setting α = 0 recovers the roots of the characteristic polynomial of (4).
One of these is in the unit interval, and one of these always exceeds one. The expression in
(6) clearly shows we can get both roots to be as large as we want by choosing α to be close
to one. In this case, we will have determinacy given only our assumption that limt→∞ πt = 0.
Specifically, so long as α satisfies
α > α∗ ≡
q
2
− 1−β+
+ 1 + β + σk − 4β
k
σ
2β
(7)
we will have determinacy in the model.
To put numbers on this, consider the quarterly calibration of Schaumburg and Tambalotti
1
≈ 2/3, this implies that α∗ ≈ 0.32.
(2007): given their values of k = 0.1, β = 0.99 and σ = 1.5
To attach some intuition to this number, note that we can interpret α−1 as the expected
number of periods before we are brought back to steady state, or the expected horizon over
which the planner’s promises will be kept. At this α∗ , this expected duration is about three
quarters. However, Schaumburg and Tambalotti (2007) use a relatively high estimate of the
Phillips curve’s slope. Galı́ and Gertler (1999) estimate k = 0.023. Using this estimate for
the slope, but holding the other parameters fixed, yields α∗ ≈ 0.17, with a corresponding
expected duration of about seven quarters.3 Figure 3 provides values of α∗ for each ratio
k/σ ∈ [0, 1/2], which covers most of the possible ratios of empirical interest. By construction,
3
Galı́ and Gertler (1999) have an accompanying quarterly value of β = 0.942, but this makes practically
no difference to our results. See the estimated equation at the top of their page 207.
9
0.5
0.4
,$
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
k=<
Figure 3: α∗ versus k/σ for β = 0.99
α∗ = 0 if k = 0. Thus, even if credibility is nearly perfect, the model has a unique equilibrium
if the Phillips curve is very flat. Even if we use a high estimate for k such as the 0.1
from Schaumburg and Tambalotti (2007), and a low σ of 0.2 (which corresponds to an
intertemporal elasticity of substitution of 5), this would yield an α∗ ≈ 1/2. We conclude that
quasi-commitment provides a quantitatively promising approach to characterizing optimal
monetary policy without open mouth operations.
3
Quasi-Commitment as a Game
Under quasi-commitment, we demonstrated that an interest rate path determines unique
equilibrium paths for inflation and output for an initial central banker given a low enough
level of commitment and an assumption that re-optimization involves implementing the
discretion solution. We here check whether or not this result is robust to a specification
wherein future central bankers make decisions and re-optimize.
To that end, we specify the monetary policy environment as a game played by a continuum
of infinitely lived households, a continuum of infinitely lived firms, and a series of central
bankers with stochastic terms in office. In the first period of her term, a central banker
promises an infinite path of current and future interest rates. These rates will prevail so long
10
as she remains in office. However, she faces a constant probability of her term ending each
period. When this happens, she will be replaced by a new central banker who will choose
her own infinite path of current and future interest rates. The sequence of shocks mt is the
trivial, deterministic sequence m0 = 1 and mt = 0 ∀ t ≥ 1.
In the following sections we first delineate the primitives of the game by listing all players,
each player’s action space, and each player’s payoffs; second, define equilibrium; and third,
characterize the set of Markov perfect equilibria.
3.1
Primitives
The game’s list of players includes an initial central banker, a sequence of successor central
bankers with names in the positive integers, and unit measures of identical and infinitely-lived
households and firms.
The initial central banker, named 0, begins play by choosing and publicly announcing a
sequence of interest rates that will prevail for as long as she is in office. We denote her choice
for time t with i0t , and we gather these choices together into i0 ≡ {i00 , i01 , i02 , . . .}. After setting
i0 , central banker 0 makes no further choices. Her term ends stochastically with constant
hazard α. If it ends at date t, central banker t replaces her and makes her own onceand-for-all public choice of current and future interest rates, it ≡ {itt , itt+1 , itt+2 , . . .}. This
successor and all subsequent successors face the same constant probability of replacement,
α. Households and firms inhabit the standard New Keynesian economy examined above, and
they take their actions after any interest rate choices by a new central banker. Considering
households’ and firms’ actions with the level of detail present in Galı́ (2008) is unnecessarily
tedious for our purposes, because central bankers care only about the aggregates πt , and Yet .
The payoffs of the central bankers are relatively straightforward. All central bankers seek
to minimize the same expected loss:
E0
∞
X
t=0
β
t
1 2 λ e2
π + Yt .
2 t
2
Here, the expecation operator is nontrivial only because central bankers have stochastic
lifetimes. Households derive utility from consumption and leisure, and firms seek to maximize
their stock market values. Instead of providing detailed descriptions of these objectives, we
impose private sector optimality by requiring that πt and Yet satisfy the sequences of PCs
and IS curves.
11
3.2
Equilibrium Definition
Each period of calendar time encompasses two nodes of this game’s extensive form: after
period t’s hazard of central banker replacement has passed, the actions of a new central
banker occur at the first node. Private sector agents make their choices at the second. We
use h1t and h2t to denote the histories of play preceeding those two period t nodes. The first
history h1t includes πl , Yel for all l < t, the dates of all past replacements up to and including
the current time period, given by set Nt , and the past promises made on those dates, in with
n ∈ Nt and n < t. The second history h2t is identical except when t ∈ Nt , in which case h2t
also includes it . We use Ht1 and Ht2 to label the space of all such possible histories at time
t, and denote a generic history with ht ∈ Ht ≡ Ht1 ∪ Ht2 .
A strategy for central banker t, st , is a mapping of each h1t ∈ Ht1 into an infinite sequence
of interest rates, it . The private sector’s strategy is a pair of mappings π, Ye from any given
date t and history h2t ∈ Ht2 into values for current inflation and output. A strategy profile
is a collection of strategies {(π, Ye ), s0 , s1 , . . .}, one for the private sector and one for each
central banker. A strategy profile forms a Nash equilibrium if:
• each central banker’s strategy minimizes their common loss function given the private
sector’s strategy and all the other central banker’s strategies, and
• the private sector’s strategy satisfies
π(h2t , t) = k Ye (h2t , t) + Et βπ(h2t+1 , t + 1) + mt ,
h
i
1 j(h2t )
2
2
\
2
e
e
Y (ht , t) = −
i
− Et π(ht+1 , t + 1) − i + Et Y (ht+1 , t + 1) .
σ t
(8)
(9)
In (9), j(ht ) with ht ∈ Ht denotes the name of the central banker in office at the node with
history ht .
A period-t subgame is the game created by restricting all players’ actions to be consistent with some history ht . A subgame perfect equilibrium is a Nash equilibrium such
that for all t ≥ 0 and for all ht ∈ Ht the strategy profile induced by conditioning on history
ht forms a Nash equilibrium for that history’s corresponding subgame. A Markov strategy
is a strategy that depends nontrivially only on payoff-relevant variables. The only payoffrelevant variable for central banker t is mt . Because this follows a deterministic sequence, a
Markov strategy for central banker t is a single interest rate sequence, it . A Markov strategy for the private sector is a pair of mappings from mt and the remaining interest rates
j(h2 ) j(h2 )
in the current central banker’s promised sequence, {it t , it+1t , . . .}, into the real numbers.
A Markov perfect equilibrium (MPE) is a subgame perfect equilibrium in which all
12
strategies in the strategy profile are Markov, and each central banker has the same strategy
as every other central banker facing the same mt .4
3.3
MPE Characterization
We demonstrate that all MPEs of this game share an outcome for inflation, output and
interest rates. In it, the values for interest rates promised by the initial central banker, i0 ,
and values played by the private sector given each of those interest rates, πt and Yet , are
identical to those that solve the Ramsey planning problem with identical parameter values.
Additionally, we verify that in every MPE of this game inflation contingent upon replacement
by a new central banker (or re-optimization as in Section 2.2) does equal zero, as postulated
by Schaumburg and Tambalotti (2007).
We split this demonstration into two propositions: one for the case where α > α∗ , in
which a unique MPE exists, and one where α ≤ α∗ , where there are multiple MPEs with
the same equilibrium outcome. Within each proposition, the argument proceeds as follows:
first, we show that any subgame conditional on a history ht such that j(ht ) = t > 0 (so that
central banker 0 is no longer in office and mt = 0 always) has a unique equilibrium outcome
in every MPE. In this outcome, it = i\ and πt = Yet = 0 always. Second, we show that any
subgame conditional on a history ht with j(ht ) = 0 (the initial central banker’s interest rates
are in effect), also has a unique equilibrium outcome.
Proposition 1. If α > α∗ , then there exists a unique Markov perfect equilibrium for the
quasi-commitment game. In it, the values for interest rates promised by the initial central
banker, i0 , and values played by the private sector given each of those interest rates, πt and
Yet , are identical to those that solve the Ramsey planning problem with identical parameter
values; the value of inflation conditional on replacement is always zero.
Proof. First we propose a particular profile of Markov strategies for the subgame beginning
after some hτ with j(h1τ ) > 0, and verify that it forms an MPE. Let all central bankers with
non-trivial choices in the subgame (i.e. every central banker t with t ≥ τ ) have strategies
st (h1t ) = it = {i\ , i\ , . . .}. Since we condition on a history such that mt is the same for
all central bankers, the requirement that our guessed strategy profile be Markov demands
that their strategies be identical. These strategies satisfy that requirement. For the private
sector, our proposal employs the unique Markov private sector strategy satisfying (8) and
(9) described in Section 2.2, where we demonstrated that given expectations of Yet = πt = 0
conditional on central banker replacement and α > α∗ , there was a unique mapping from any
4
In the literature, this is sometimes referred to as a “Symmetric MPE.”
13
promised interest rate path into sequences for output and inflation. Note that this strategy
yields Yet = πt = 0 if the remaining interest rates in the current central banker’s sequence
are {i\ , i\ , . . .}. Here, also note that inflation conditional on central banker replacement is a
constant: zero.
We now check that our guess forms an MPE. First, note that all central bankers recieve the
smallest possible loss in this strategy profile: zero. Therefore, no central banker can improve
her payoff by choosing a different strategy. This strategy profile implies that πt = 0 whenever
a new central banker takes office. This is the only requirement for the private sector strategy
to satisfy (8) and (9). Therefore this strategy profile forms a Nash equilibrium. Moreover,
since our choice of subgame hτ was arbitrary, this strategy profile forms a subgame perfect
equilibrium.
To show that this MPE is unique, note that within any MPE of this subgame the inflation
rate in the first period of each central banker’s term must be identical: since mt = 0 ∀ t in
this subgame, each central banker must have the same strategy in any MPE and the private
sector must therefore play the same values of πt and Yet when replacement occurs. We label
the constant value for expected inflation conditional on central banker replacement π 0 . Given
this constant and the assumption that α > α∗ , the analysis in Section 2.2 shows that there
are unique mappings from a central banker’s promised interest rates into sequences for Yet
and πt that are consistent with (8) and (9) (although that analysis assumed that inflation
conditional on replacement was π 0 = 0, the result easily generalizes to any π 0 ). So given π 0 ,
there is exactly one private sector strategy that is a best response, and hence consistent with
MPE. To show that there exists a unique MPE for this game, we must verify that only one
value of π 0 is consistent with MPE.
Because the private sector’s best response strategy is pinned down by π 0 , (8) and (9),
any particular central banker t can calculate her best response it by first solving for the
allocations of inflation and output that minimize her loss given the sequence of PCs, and
then backing out the required sequence of interest rates using the IS curves. It is tedious
but straightforward to show that a particular central banker’s optimal inflation in the first
period of her term is
πt =
θβα
π0
1 + θβ(1 − α)
where θ ∈ (0, 1) is the smaller solution to
k2
0 = 1− 1+
+ β(1 − α) θ + β(1 − α)θ2 .
λ
14
(10)
Since all central bankers in the profile have the same strategy, this becomes
π0 =
θβα
π0.
1 + θβ(1 − α)
(11)
Since the coefficient on the right hand side is less than one, (11) requires that π 0 = 0. Thus
our proposed strategy profile forms the only MPE to the subgame with j(hτ ) > 0.
With this proven, demonstrating that the complete game has a unique MPE is simple.
Append the interest rates that decentralize the initial central banker’s optimal solution under
quasi-commitment (given in Section 2.2) to the current strategy profile. Then, given α > α∗ ,
the resulting strategy profile forms the unique MPE to the full quasi-commitment game.
We now consider the case of α ≤ α∗ , beginning by noting a useful approximation result.
First, we define an eventually-constant sequence:
Definition 1. Eventually-Constant Sequences. An eventually-constant sequence is any sequence indexed by t ≥ 0 where there exists some T > 0 such that the sequence is constant
for all t ≥ T . If the value of this constant is X, we say “the sequence is eventually-constant
with value X.”
Now consider the following useful result:
Lemma 2. Consider the solution to the minimization problem of a central banker under
quasi-commitment, where replacement in each period with some probability α will result in
allocations π 0 and Ye 0 for that period, which do not depend on the date of replacement. The
central banker chooses interest rates, inflation and output to minimize the loss function
∞
X
1 2 λ e2
t t
e
(1 − α) β
π + Yt
Loss {πt }, {Yt } ≡
2 t
2
t=0
subject to the constraints
πt = k Yet + β(1 − α)πt+1 + απ 0 + mt
1
Yet = − it − (1 − α)πt+1 − απ 0 − i\ + (1 − α)Yet+1 + αYe 0
σ
Let {πt? }, {Yet? }, and {i?t } denote the unique solution to this problem. For any > 0, there
exist eventually-constant sequences for the interest rate, inflation, and output, denoted {i0t },
{πt0 } and {Yet0 }, which satisfy the same constraints and yield a loss such that
Loss
{πt0 }, {Yet0 }
− Loss
15
{πt? }, {Yet? }
< .
The proof is provided in an online appendix. This Lemma is useful for the proof of a
unique equilibrium outcome for case of α ≤ α∗ because all Markov private sector strategies
that are best responses (i.e. satisfy (8) and (9)) must prescribe the same values for πt and
Yet given a particular eventually-constant interest rate path and values π 0 and Ye 0 :
Lemma 3. Consider the strategy of a particular central banker τ in the quasi-commitment
game where each future central banker t > τ has an identical strategy to each other central
banker t > τ and the private sector’s strategy is Markov, playing π 0 and Ye 0 given the i0 which
will be played in the first period of each future central banker’s term. For every eventuallyconstant interest rate sequence {it } promised by central banker τ , there exist unique sequences
{πt } and {Yet } that satisfy (8) and (9). These sequences {πt } and {Yet } are also eventuallyconstant.
Proof. If central banker τ promises constant interest rates for all t ≥ T , and identical strategies for all central bankers t > τ imply the same values for it conditional on replacement,
then after time T all payoff relevant variables for the private sector are the same. Thus the
private sector plays a constant inflation/output combination for all t ≥ T , which we label π 00
and Ye 00 . These values can be calculated from (8) and (9), using π 0 , Ye 0 , and the level of interest rates central banker τ promises after T . The fact that all these values are constant pins
down a unique solution to the second order system given by (8) and (9); solving backwards
from T yields the unique paths for {πt } and {Yet }.
Thus by Lemma 3, an eventually-constant strategy for a particular central banker “guarantees” a particular action from the private sector, given that all other bankers play identical
strategies to all other central bankers and that the private sector strategy is Markov and
a best response satisfying (8) and (9). Further, by Lemma 2 such an eventually-constant
strategy can always be found which will guarantee a loss in equilibrium which is within of
the minimum loss possible given the identical strategies played by all other central bankers,
and given that the private sector strategy is Markov and a best response. With this result,
we can now demonstrate
Proposition 2. If α ≤ α∗ , then there exists a unique equilibrium outcome in every Markov
perfect equilibrium for the quasi-commitment game. In it, the values for interest rates
promised by the initial central banker, i0 , and values played by the private sector given each of
those interest rates, πt and Yet , are identical to those that solve the Ramsey planning problem
with identical parameter values; the value of inflation conditional on replacement is always
zero.
16
Proof. First we propose a particular profile of Markov strategies for the subgame beginning
after some ht with j(h1t ) = t > 0. As in Proposition 1, let all central bankers with nontrivial choices in the subgame (i.e. every central banker t with t ≥ τ ) have strategies
st (h1t ) = it = {i\ , i\ , . . .}. Since we condition on a history such that mt is the same for all
central bankers, the requirement that our guessed strategy profile be Markov demands that
their strategies be identical. These strategies satisfy that requirement. For the private sector,
recall that a Markov strategy is a pair of mappings from mt and the remaining interest rates
j(h2 ) j(h2 )
in the current central banker’s promised sequence, {it t , it+1t , . . .}, into current values of
inflation and output, Yet and πt . We guess a strategy that is consistent with (8) and (9)
wherein Yet = πt = 0 if the remaining interest rates in the current central banker’s sequence
are {i\ , i\ , . . .}. Thus, we have inflation conditional upon central banker replacement π 0 = 0.
Note here that assuming (8) and (9) hold and that π 0 = 0 does not completely specify
a best-response strategy for the private sector, as it did in Proposition 1: because α ≤ α∗ ,
there exist infinitely many choices for inflation and output given other possible interest rate
promises it 6= {i\ , i\ , . . .} that are consistent with the assumption that the private sector
strategy is a best response, so long as it is not eventually-constant. These choices are
indexed by the non-explosive, homogenous solutions to the system given by (8) and (9) with
π 0 = 0. For now, consider one such Markov, best response strategy for the private sector
that plays Yet = πt = 0 given a promise of {i\ , i\ , . . .}, and one particular choice of πt and
Yet out of the many possible choices for every interest rate sequence that is not eventually
constant-consistent with (8) and (9) and π 0 = 0.
With a complete specification of the private sector’s strategy, we now have a complete
specification of our strategy profile, and proceed to verify that such a strategy profile is an
MPE. First, note that all central bankers recieve the smallest possible loss in such a strategy
profile: zero. Therefore, no central banker can improve her payoff by choosing a different
strategy. Such a strategy profile implies that π 0 = 0, which is the only requirement for the
private sector strategy to satisfy (8) and (9). Therefore such a strategy profile forms a Nash
equilibrium. Moreover, since our choice of subgame hτ was arbitrary, this strategy profile
forms a subgame perfect equilibrium.
We now characterize the set of MPEs of this subgame, and show that they all must
satisfy the requirements of our guess, and have the same equilibrium outcome of it = i\ and
πt = Yet = 0 always. First, note within any MPE of this subgame the interest rates played by
central bankers must be identical (although not necessarily constant). Because the private
sector has a Markov strategy, the inflation rate in the first period of a central banker’s term
must be the same for all central bankers, which we again call π 0 . Given this constant, a
particular central banker’s optimal inflation in the first period of her term is given by (10).
17
We show (10) must hold and that π 0 = 0 in every MPE of this game using proof by
contradiction: assume some other MPE to this game exists where (10) is not satisfied for a
particular central banker. This central banker is receiving, in this equilibrium, a discretely
worse payoff than if (10) were satisfied; let this discrepancy be denoted by . By Lemma 2
and Lemma 3, an eventually-constant interest rate sequence exists which provides a higher
payoff to the central banker, so that whatever strategy the central banker is using is not
a best response. We have a contradiction, and therefore conclude that every MPE to this
game must satisfy (10). Since all strategies in the profile are Markov, and each central banker
faces the same problem, for a particular central banker’s strategy to be part of an MPE the
resulting equilibrium inflation must satisfy (11), requiring π 0 = 0.
Since we must have π 0 = 0 in every MPE of this subgame, it is simple to verify that
only strategy profiles like those in our guess can form MPEs. To do so, note that a constant
interest rate sequence (the only Markov strategy available for central bankers) and a best
response private sector strategy which satisfies (8), (9) and π 0 = 0 implies that the strategy
for central bankers must be st (h1t ) = it = {i\ , i\ , . . .}. Thus, only strategy profiles like those
in our guess consitute MPEs of this subgame.
Now consider subgames formed by conditioning on some ht where the initial central
banker’s interest rates are in effect (j(h1t ) = 0). We proceed as in Proposition 1, proposing
a strategy profile where the initial central banker promises the interest rates calculated in
Section 2.2 in response to the shock m0 , given π 0 = 0, and the private sector plays any
strategy where inflation and output are the solutions to the Ramsey problem as presented
in Section 2.2 conditional on the central banker’s strategy, and otherwise one of the many
strategies that satisfies (8), (9) and π 0 = 0 given other possible interest rate sequences.
This satisfies the definition of an MPE, since the payoff for the central banker is as high as
possible, and the private sector strategy satisfies (8) and (9) given π 0 = 0.
To prove all MPEs must have outcomes identical to the one in this proposal, we again
use proof by contradiction: assume an MPE exists which satisfies all the above assumptions
(the private sector strategy satisfies (8) and (9) given π 0 = 0, and the central banker plays
the unique interest rate path consistent with the Ramsey optimum in Section 2.2) except
the assumption that the private sector plays a strategy where inflation and output are the
solutions to the Ramsey problem given the central banker’s strategy. This other MPE has
a discretely worse payoff for the central banker than in our proposal. Therefore, the central
banker’s strategy in this other MPE cannot be a best response; by Lemma 2 and Lemma
3, whatever the discrete difference between the payoff in this other MPE, the central
banker can improve by choosing an appropriate eventually-constant interest rate sequence.
Because the central banker’s strategy is not a best response in this other MPE, we have a
18
contradiction. Therefore the assumption is false, and no MPE exists which does not satisfy
the assumptions in our proposal.
Combining the results for each subgame, we have have that in every MPE of the full game
the values for interest rates promised by the initial central banker, i0 , and values played by
the private sector given each of those interest rates, πt and Yet , are identical to those that
solve the Ramsey planning problem with identical parameter values.
Thus, regardless of α, every MPE of this game results in values for interest rates, inflation,
and output identical to those in the Ramsey planning problem in Section 2.2. However, note
that the proofs of these propositions are sensitive to the fact that all payoff-relevant state
variables are constant after some time T , as this allows us to use Lemma 2 and Lemma 3
in the proof of Proposition 2. If we considered a game with persistent or serially correlated
shocks mt , for example, a different approach would be necessary.
4
Optimal Monetary Policy with Quasi-Commitment
We demonstrate here two results: first, that the conventional forward guidance puzzle is
eliminated with quasi-commitment, and second that while optimal policy is characterized by
the central banker promising price level targeting, in expectation the price level permanently
deviates from its pre-shock level.
4.1
Eliminating the Forward Guidance Puzzle
We here confirm that quasi-commitment, like other forms of introducing discounting into
the IS curve, mitigates the forward guidance puzzle. The forward guidance puzzle is the
prediction of the standard model that forward guidance gets more powerful, and has a greater
impact on the present, the further in the future it is promised. The impact is unbounded
the farther ahead in time the promised action takes place (Del Negro et al., 2012).
First, we review the conventional forward guidance puzzle result. Consider the problem
under perfect commitment without shocks (mt = 0 ∀t). Let the interest rate track the
natural rate perfectly (it = i\ ) up to some time T , where it will be lowered five basis points.
Assuming that we are at the discretion solution after time T (πt = Yet = 0 for all t > T )
what happens to inflation?
As Figure 4 illustrates, inflation today is exploding backwards from T , so that inflation
is higher in the present than at time T . The level of inflation today increases as we push T
to infinity. This unintuitive result is the standard forward guidance puzzle.
19
Forward Guidance at T = 7 with Puzzle, , = 0
Percentage Points
0.15
:t
it ! i\
0.1
0.05
0
-0.05
0
1
3
4
5
6
7
8
9
8
9
Forward Guidance at T = 7 without Puzzle, , = 0.5
0.02
Percentage Points
2
0
-0.02
-0.04
-0.06
0
1
2
3
4
5
6
7
Figure 4: Effect of Forward Guidance at time T = 7 quarters ahead
We can prove this analytically: by choosing some α > α∗ , ensuring determinacy since
both roots (6) will be ϕ, ψ ∈ (1, ∞), we can solve for inflation in terms of xt . We will then
show the derivative of π0 with respect to xt is decreasing in time t. If α > α∗ , the solution
to the system given by the Phillips curve and IS curve under quasi-commitment is:
πt =
∞
X
l=0
−l
ϕ
∞
X
ψ −j xt+j+l .
j=0
Now write down the expression for π0 , and take the derivative with respect to xt :
∞
∞
dπ0
d X −l X −j
=
ϕ
ψ xj+l
dxt
dxt l=0
j=0
=
ϕψ(ψ −(t+1) − ϕ−(t+1) )
.
ϕ−ψ
Since the value of the derivative decreases with time, there is no forward guidance puzzle.
20
0.8
it ! i\
Yet
:t
Price Level
0.6
Percentage Points
0.4
0.2
0
-0.2
<=2
k = 0.25
-0.6
6 = 0.25
, = 0.3
Dashed lines are the -rst t periods of the in-nite horizon perfect commitment solution.
-0.4
-0.8
0
1
2
3
Elapsed Time Since Shock
4
5
Figure 5: Central bank’s promises under quasi-commitment.
4.2
Qualitative Optimal Policy: Price Level Targeting
In Figures 5 and 6, we illustrate the qualitative results for optimal monetary policy. The
dashed lines in each graph represent the solution to the same problem with perfect commitment (α = 0). Note that the optimal promises that the central banker makes are qualitatively
the same as before, since the only difference in solving the optimization problem is the discount factor: β for the perfect commitment central banker with α = 0 and (1 − α)β for the
quasi-commitment central banker with α ∈ (0, 1).
However, expectations differ sharply. Under perfect commitment, agents’ expectations
are identical to the central banker’s promises. Under quasi-commitment, they will be different
by a factor of (1 − α)t . Thus, under quasi-commitment, it remains optimal to promise price
level targeting, but not optimal to make agents believe in price level targeting. As indicated
in Figure 6, the price level does not return to its pre-shock level in expectation.
21
0.8
it ! i\
Yet
:t
Price Level
0.6
Percentage Points
0.4
0.2
0
-0.2
<=2
k = 0.25
-0.6
6 = 0.25
, = 0.3
Dashed lines are the -rst t periods of the in-nite horizon perfect commitment solution.
-0.4
-0.8
0
1
2
3
Elapsed Time Since Shock
4
Figure 6: Agent’s expectations under quasi-commitment.
22
5
5
Conclusion
In this paper we demonstrated the ability of a quasi-commitment framework (Schaumburg
and Tambalotti, 2007; Debortoli and Nunes, 2007) to parsimoniously eliminate both equilibrium multiplicity and the “forward guidance puzzle” of Del Negro et al. (2012). By
eliminating equilibrium multiplicity, we also removed the Open Mouth Operation from the
central banker’s toolbox.
References
Bodenstein, M., J. Hebden, R. Nunes, et al. (2010). Imperfect credibility and the zero lower
bound on the nominal interest rate. Board of Governors of the Federal Reserve System.
Campbell, J. (2013). Odyssean forward guidance in monetary policy: A primer. Forward
Guidance: Perspectives from Central Bankers, Scholars and Market Participants, Vox
eBook .
Campbell, J. R., J. D. Fisher, A. Justiniano, and L. Melosi (2016). Forward guidance and
macroeconomic outcomes since the financial crisis. In NBER Macroeconomics Annual
2016, Volume 31. University of Chicago Press.
Cochrane, J. H. (2011). Determinacy and identification with taylor rules. Journal of Political
Economy 119 (3), 565–615.
Debortoli, D. and R. C. Nunes (2007). Loose commitment. FRB International Finance
Discussion Paper (916).
Del Negro, M., M. Giannoni, and C. Patterson (2012, October). The Forward Guidance
Puzzle. FRBNY Staff Reports (574).
Evans, C. (2011). The Fed’s dual mandate: responsibilities and challenges facing US monetary policy. Speeech at European Economics and Financial Centre, London, 7 September
2011. Accessed online 12/12/16.
Gabaix, X. (2016). A behavioral New Keynesian model. Working paper accessed online
November 6th, 2016.
Galı́, J. (2008). Monetary Policy, Inflation, and the Business Cycle: An Introduction to the
New Keynesian Framework. Princeton, New Jersey, USA: Princeton University Press.
23
Galı́, J. and M. Gertler (1999). Inflation dynamics: A structural econometric analysis.
Journal of monetary Economics 44 (2), 195–222.
Giannoni, M. and M. Woodford (2005, March). Optimal Inflation-Targeting Rules. In
M. Woodford and B. Bernanke (Eds.), The Inflation Targeting Debate, pp. 93–172. University of Chicago Press.
Kydland, F. E. and E. C. Prescott (1977). Rules rather than discretion: The inconsistency
of optimal plans. The Journal of Political Economy, 473–491.
McKay, A., E. Nakamura, and J. Steinsson (2016). The power of forward guidance revisited.
The American Economic Review 106 (10), 3133–3158.
Romer, D. and J. Wolfers (2012). Comments and Discussion. Brookings Papers on Economic
Activity 1, 74–80.
Schaumburg, E. and A. Tambalotti (2007). An investigation of the gains from commitment
in monetary policy. Journal of Monetary Economics 54 (2), 302–324.
Svensson, L. and M. Woodford (2005, March). Implementing Optimal Policy through
Inflation-Forecast Targeting. In M. Woodford and B. Bernanke (Eds.), The Inflation
Targeting Debate, pp. 19–91. University of Chicago Press.
Woodford, M. (2003). Interest and Prices: Foundations of a Theory of Monetary Policy.
Princeton, New Jersey, USA: Princeton University Press.
24