Reconsidering DSGE models

Reconsidering DSGE models
with focus on New Keynesian models
William Heartspring
September 2, 2019 (ver 1.0)
William Heartspring Reconsidering DSGE September 2, 2019 (ver 1.0) 1 / 69

Table of Contents
1 Overview
2 Analysis 1: a transversality condition issue in an economy with the
representative consumer
3 Analysis 1: equilibrium independence and HANK model
4 Analysis 1: general impossibility result (GIR): equilibrium does not exist
due to time consistency issues
5 Analysis 2: Can central bank impose its monetary policy rule? Modern
Policy Ineﬀectiveness Proposition

Table of Contents
1 Overview

The general storyline
Analysis 1: the general impossibility result. Proves that there can be
no time-consistent sequential equilibrium for multiple-consumer DSGE
models, or in short, there cannot be an equilibrium for any
multiple-consumer DSGE model.
Analysis 2: the modern policy ineﬀectiveness proposition (MPIP).
This is a consequence of the general impossibility result, which
requires central bank bond (in New Keynesian terminology) Bt to be
a non-control variable of a consumer optimization problem in order to
save an equilibrium. Then interest rate-based monetary policy is
ineﬀective, unless augmented or its meaning changed as providing
lower bound for market interest rates.

Deterministic economy/rational expectation assumptions
For simplifying derivations without loss of generality, we assume that an
economy is deterministic. We also assume rational expectation.

Preview of Analysis 1
Discussion of an economy with one consumer. In case central bank
bonds Bt, as in typical New Keynesian models, exist, a transversality
condition mandates Bt = 0.
Discussion of an economy with multiple consumers: the general
impossibility result (GIR). Ratio of consumption of one observer to
another observer at the same time is required to be
history-dependent, but this is inconsistent with the concept of time
consistency required for an equilibrium of DSGE models.

Preview of Analysis 1: how GIR is proved
GIR is proven with a somewhat general form of budget constraints,
but we can demonstrate GIR without using a particular form of
budget constraints: if the assumption that feasible utility vectors
arising from feasible allocations form a convex set is satisfied, we can
invoke the Negishi procedure in Negishi (1960), which asserts that
linear combination of utility functions of consumers can be maximized
to get a potential equilibrium. With this, we can demonstrate GIR in
different or general contexts.
While each consumer optimization problem, given price vector for
each economic condition, guarantees time consistency as it is
equivalent to a recursive dynamic programming problem, it is the
question of finding equilibrium price vector amid interactions between
agents (consumers and firms, for example) that creates time
inconsistency issues.

Preview of Analysis 1: time consistency
That is, time consistency applies not only to each consumer
optimization problem but also to the combined result of
optimization problems agents solve, and it is the latter that
conventional understanding misses.
For DSGE models, assuming rational expectation and
deterministic economy, time consistency simply means that
present-time expectations agents have on outcomes at t = T
must equate to actual outcomes at t = T, which are fully
decided at t = T.
If economic structure, environment and states are identical at
t = 0 and t = T, then agents must decide identically.

Preview of Analysis 1 and Analysis 2
Combining the case of one consumer (transversality condition issue)
and multiple consumers (general impossibility result), we notice that
problems disappear if central bank bond Bt is dropped from control
variables of a consumer optimization problem.
We replace missing ﬁrst-order conditions to controlling Bt with
learning mechanisms.
We demonstrate that even if GIR issues do not exist, equilibrium
selection and determinacy issues are quite common, as can be seen in
the canonical New Keynesian model (IS-MP-PC) example. Thus
subjective expectations and learning mechanisms used to set them
cannot be ignored.

Preview of Analysis 2
If agents set expectations of economic variables without reference to
it and Bt, then whatever central bank does, monetary policy by purely
setting interest rate (rule) is ineﬀective. Modern Policy Ineﬀectiveness
Proposition (MPIP).
To allow for monetary policy purely based on setting interest rate on
central bank bonds, one must re-interpret central bank’s interest rates
as setting lower bounds for market interest rates. Market interest
rates do not have to equate to central bank interest rate because Bt
was dropped from the list of control variables in consumer
optimization problems.

Table of Contents
1 Overview

Nature of Bt in a representative consumer economy
In the canonical cashless New Keynesian model, budget constraint and
accounting (in)equality enforce for realizable outcomes:
PtCt + QtBt ≤ WtNt + Ft + Bt−1 Consumer budget constraint
PtCt = WtNt + Ft Aggregate ﬁrm accounting identity
Ct ≤ Yt Consumption resource constraint
thus
QtBt ≤ Bt−1
where Qt = 1/(1 + it) is the price of bond Bt that unit quantity pays unit
money at t + 1.
Assume there is only one representative consumer, as in the model. If
agents are truly optimizing, then the ﬁrst budget inequality becomes
equality and thus: QtBt = Bt−1, but then this is inconsistent with optimal
behavior of the representative consumer, unless Bt−1 = 0. Also, if
B−1 = 0, then Bt = 0 forever. Already signs of a problem.

Model explanation here: Ct is consumption, Pt is price level, Wt is
nominal wage, Nt is amount of labor supplied, Ft are proﬁts of ﬁrms that
are entirely paid out to the representative consumer.
Why is PtCt = WtNt + Ft and Ct ≤ Yt? It does seem weird, given that in
disequilibrium, Ct can be higher than what is produced Yt. But when
consumers actually face disequilibrium, consumers cannot consume more
than what is produced, though they may consume less than what is
produced. Even in an disequilibrium that agents would wish to consume
more, they just end up consuming all of what are produced.
Note that QtBt = Bt−1 is not purely an equilibrium phenomenon. It holds
in all disequilibria, after factoring in physical resource constraints.

Why is QtBt = Bt−1 with positive B−1 non-optimal for the representative
household? Well, this means that all money that one obtained through
bonds are spent toward buying bonds again! Bonds do not act as an agent
of consumption smoothing! Thus if consumers are optimizing, B−1 = 0.
Can understand in terms of a transversality condition as well.
Does agent heterogeneity change the view? We show that it does not.

Table of Contents
1 Overview

Nature of Bt in a HANK model
HANK = heterogeneous-agent New Keynesian
But we do not need to specify the entire model - we only need the budget
constraint of each agent j and some properties of equilibria.
PtCj,t + QtBj,t ≤ WtNj,t + Fj,t + Bj,t−1
An equilibrium path of Cj,t, Pt, Qt, Wt, Nj,t, Fj,t is assumed to be
independent of initial Bj,−1. This does hold in a canonical HANK model.
Suppose that for initial value of Bj,−1, the transversality condition (TVC)
is satisﬁed.

Nature of Bt and transversality condition (TVC)
lim
T→∞
T
t=0
1
1 + it
Bj,T = 0
Suppose that Ba,j,t, including Ba,j,−1, is the equilibrium path of Bj,t
satisfying the above TVC. Let us change value of Bj,−1 such that
Bj,−1 = Ba,j,−1 + k.
Since equilibrium path of other variables is assumed to be unaﬀected by
Bj,−1, the budget constraint implies that:
Bj,T = Ba,j,T + k
T
t=0
(1 + it)
But this means that Bj,t does not satisfy the TVC.

Table of Contents
1 Overview

General impossibility result
One may ask whether introducing quantity of Bt into an interest rate rule
would allow TVC to be satisﬁed for any value of B−1.
We demonstrate that the problem is not of incorporating Bt into an
interest rate rule: even with heterogeneity, there is no time-consistent
equilibrium.
The model deals with a deterministic economy, despite being dynamic, for
simpliﬁcation purposes. Despite this, the result here applies without loss of
generality for stochastic cases as well.

A multiple-consumer dynamic model. Consumer j = 1, 2, .. utility function
Uj that consumer j maximizes by controlling {Cj,t, Bj,t, ..}:
Uj =
∞
t=0
(βj )t
uj (Cj,t, ..)
with Uj (C, ..) being a time-invariant utility function. Caution: U and u
diﬀerent. Let the budget constraint of each consumer be:
PtCj,t + (1 + it)−1
Bj,t + .. ≤ Bj,t−1 + ..
where .. terms do not contain any of Cj,t and Bj,t. Then we get:
u1(C1,t)
u2(C2,t)
=
(1 − µ)
µ
β2
β1
t
where u (C) = ∂u
∂C at particular value C.

(1 − µ)/µ can be understood as the ratio of lagrange multiplier of
consumers attached to the budget constraint: λj=1,t=0/λj=2,t=0. We
maintain notation µ as to allow a convenient connection to the Negishi
approach to GIR to be described - we describe the Negishi approach so
that we can extend GIR to economies where Bt does not exist.
Cj,t refers to consumption of agent j at time t, Bj,t refers to central bank
bonds agent j holds that are redeemed at the next period with central
bank setting interest rate it at time t.

u1(C1,t)
u2(C2,t)
=
(1 − µ)
µ
β2
β1
t
It may be immediately clear or unclear why the above result is
time-inconsistent. Thus let us provide an example of uj (C) = C−σj ,
meaning u is a CRRA utility function.
Since we have a deterministic economy, assuming consumer structure does
not change, we can use present-time consumption equilibrium outcomes to
trace back consumption distribution in the past if we just know aggregate
total consumption in the past, without requiring other non-consumer
speciﬁcations of economy in the past.

If σj = σ for every consumer j, then we even do not need to know
aggregate total consumption in the past to figure out distribution of past
consumption.
Past is a hologram that depicts future in a different way! In fact, we would
only need full economic structure at the last future period (or future
infinity) and consumer profiles to reconstruct distribution of equilibrium
consumption back to past infinity. (Aggregate consumption, however,
remains unknown.)
Why do we get such nonsense? In a traditional Arrow-Debreu general
equilibrium (ADM) model, this is not nonsense, because contracts of
future consumption delivery are all written at t = 0, so the result works as
a demand-side equilibrium condition. But in DSGE, we have consumers
only forming expectations of future consumption - they do not actually
decide future consumption. This is what causes nonsense, despite sharing
the result.

In ADM, all future consumption delivery contracts would be made at
t = 0, so it does not make sense to talk of agents forming a market
equilibrium separately at t = T, with state variables resulting from past
decisions. (There are no state variables in ADM, other than those at
t = 0.)
In DSGE, agents at t = 0 do not decide consumption at t = T - rather,
they form expectation of consumption at t = T. In a deterministic
economy, this expectation would have to equal to actual consumption at
t = T by time consistency. Thus it does make to sense to talk of an
equilibrium decision at t = T with state variables given separately from an
equilibrium decision at t = 0.

The general impossibility result thus is not that expected future
determining present decisions is problematic - it is that we do not need
anything other than consumer proﬁles to determine consumption
distribution in the past, once an equilibrium outcome is known today,
solved for known present-time state variables. The required distribution
condition is so binding that there is no way an equilibrium would have
formed in the past unless by sheer coincidence.
And of course, today is “new past” of future, future becomes “new today”!
Thus this is a problem of determining present-time decisions as well.

General impossibility result: resistance is futile
One may argue that we may resolve this issue if there are inﬁnitely many
sequential equilibria for one economy, and since µ is indeterminate, agents
pick µ that is history-consistent and thus time-consistent - future respects
past. But this is not a standard equilibrium selection mechanism - we will
see this in Analysis 2.
Though eventually, we will indeed use the “inﬁnitely many sequential
equilibria” idea as the starting point of Analysis 2 and as the way to evade
GIR.
Alternatively, one may argue that it is our use of a sequential equilibrium
that is problematic. What if a consumer chooses an optimal
time-consistent plan, instead of solving for an optimal sequential plan
which involves directly controlling present and future control variables to
maximize today’s time-discounted utility?

Unfortunately, we know that each consumer optimization problem solving
for an optimal sequential plan is equivalent to a recursive dynamic
programming problem under very weak and mild assumptions. (See SLP
for more information.)
A recursive dynamic programming problem involves the Bellman equation,
and this guarantees that time consistency is satisfied for each consumer
optimization problem. But did we not say that time consistency is not
there?
But a consumer solving its optimization problem is different from solving
for an entire model. A consumer has to be provided price vector for each
economic condition to start solving for a consumer optimization problem.
And solving for a model involves finding an equilibrium price vector. It is
time consistency for the whole model that is dissatisfied.

If the price vector given for a consumer optimization at t = 0 does not
change for other time periods, then consumers will not recognize time
inconsistency.
It is the fact that interactions with other parts of economy cannot
maintain time-invariance of the price vector that causes time inconsistency!

General impossibility result: the Negishi approach
One may argue instead that if central bank adopts diﬀerent monetary
policy such that the form of consumer budget constraints change, then the
general impossibility result (GIR) may not apply. Maybe, GIR is an artifact
of New Keynesian (Neo-Wicksellian) monetary policy!
This is why the derivation of GIR by the Negishi approach becomes
valuable: it demonstrates that GIR is the general result. The only way to
avoid this result is if there is no way to re-deﬁne “feasible” allocations
such that feasible utility vectors, with each vector’s ele-
ments being time-discounted utility Uj of agent j, cannot form a convex set.

Negishi (1960): solve the social planner problem that maximizes utility of
µU1 + (1 − µ)U2 with 0 ≤ µ ≤ 1 to ﬁnd a candidate equilibrium in case
there are two consumers.
But we require “feasible” utility vectors (U1, U2, ..) to form a convex set
with candidate equilibrium utility vectors to lie on the boundary of the set
in order for the Negishi approach to work.
“Feasible” is usually described in terms of resource constraints, and in
such a case, we require an allocation result to be Pareto-optimal. But we
may change the deﬁnition of what “feasible” is so that feasible utility
vectors do form a convex set, with equilibrium lying on the boundary of
the set. This change is proposed as to accommodate economies with
monopolistic competition.

But we now face the same time inconsistency issue when we solve the
social planner problem:
u1(C1,t)
u2(C2,t)
=
(1 − µ)
µ
β2
β1
t
when there is a resource constraint of:
C1,t + C2,t ≤ ....
where .... does not contain any of C1,t and C2,t. Thus in case we can
change the deﬁnition of a feasible allocation/utility vector suitably, we get
GIR without necessitating an asset like Bt that “connects” consumers.

One may try to escape the general impossibility result by changing utility
function uj to be time-variant, and time preference discount factor βj to
depend on time. This escape route misunderstands what the time
inconsistency issue in the general impossibility result is. We still get for the
two consumers deciding at t = 0:
u1,t=0(C1,t=0)
u2,t=0(C2,t=0)
=
(1 − µ)
µ
u1,t=1(C1,t=1)
u2,t=1(C2,t=1)
=
(1 − µ)
µ
β2
β1
where βj is time preference discount factor for utility arising from
consumption at t = 1 when deciding at t = 0. And this is the problem.

Consider CRRA utility such that ut(C) = C−σt . Then we realize that
computing consumption at t = 1 allows us to compute consumption ratio
at t = 0 without referencing other non-consumer details of economy at
t = 0.
It is only by sheer coincidence that this can be made consistent - that is
the problem.
It is possible to evade GIR by making a consumer have some form of
inertia. One straightforward example is consumers respecting expectations
formed in the past when there is multiplicity of possible equilibria. But
DSGE, in spirit, is forward-looking. Thus this evasion strategy is
non-standard.

General impossibility result: conclusion
We thus should not expect existence of a time-consistent sequential
equilibrium, and in general, DSGE models are in trouble. For the canonical
New Keynesian model with a representative consumer, the illusion of
having solved for an equilibrium has been possible because one has been
violating the TVC for Bt, or an interest rate rule does not make sense
because Bt = 0 in any circumstance, leaving central bank bond market
eﬀectively non-existent and interest rate policy ineﬀective.
Why did everyone not detect this? One reason: DSGE models in general
have no closed-form solutions. So one must use approximations and
numerical methods to solve for an equilibrium. Any sign of inconsistency
in a model simulation can easily be misread as an approximation error.

Toward Analysis 2
We can easily note that in the original form of GIR, it is Bj,t and it that are
causing the problem - common it for Bj,t “connects” together consumers
when they were not previously, and this causes time consistency problems.
Also, when there is only one consumer, it was also Bt and the TVC
attached to Bt that were causing problems.
Thus, we evade GIR and TVC issues if we drop Bt as a control variable of
a consumer optimization problem. But how can this be possible? This is
the starting point of Analysis 2.

Table of Contents
1 Overview

Overview of Analysis 2
Because of the general impossibility result, Bt should not be a control
variable of a consumer optimization problem.
This creates equilibrium indeterminacy issues, which we address by
learning mechanisms. Equilibrium indeterminacy issues arise in many
DSGE models even without the modification, so we cannot avoid
discussing learning mechanisms anyway. An example of the canonical New
Keynesian model is used to show that even before dropping a control
variable, there are equilibrium selection issues.
The required modification to DSGE models implies the Modern Policy
Ineffectiveness Proposition, which asserts that monetary policy in form of
an interest rate policy is ineffective, given that agents have no a priori
reason why an interest rate policy has to be effective. We then provide a
escape route from this ineffectiveness proposition.

Equilibrium indeterminacy may not really be a problem
Dropping Bt as a control variable of a consumer optimization problem can
lead to equilibrium indeterminacy.
However in many models, how agents actually learn (along with their
subjective expectation) is required to make sense of them. That is, we
cannot only study rational expectation equilibria. This is especially the
case when more than one rational expectation equilibria (REE) exist in a
model, which requires actual learning mechanisms to provide us which
REE is relevant.
Despite lack of rational expectation equilibrium determinacy, it is possible
that agents come to learn one equilibrium. We will approach this question
from the example of the canonical New Keynesian model.

The canonical New Keynesian model with modiﬁcation
Because of the general impossibility result and the TVC issue, we need to
modify the consumer optimization problem in the canonical New
Keynesian model:
max
{Ct ,Nt }
∞
t=0
βt (Ct)1−σ − 1
1 − σ
−
(Nt)1+ϕ
1 + ϕ
instead of
max
{Ct ,Nt ,Bt }
∞
t=0
βt (Ct)1−σ − 1
1 − σ
−
(Nt)1+ϕ
1 + ϕ
In both cases, they are subject to tbe budget constraint of:
PtCt + (1 + it)−1
Bt ≤ WtNt + Ft + Bt−1

Some brief explanations of variables: Ct refers to consumption of the
representative consumer, Nt refers to labor of the representative agent
utilized, Bt refers to quantity of central bank bonds, Wt refers to wage, Pt
refers to price level, Ft refers to profits of firm fully redistributed to the
representative consumer at each period because the consumer is a
shareholder of each firm.
The modification is that Bt is dropped from control variables of the
optimization problem. This invalidates the use of the IS equation in the
modified model.

The canonical New Keynesian model
The typical derived three equations of the canonical New Keynesian go,
after log-linearization: IS equation (representative consumer):
˜yt = Et [˜yt+1] − σ−1
(it − Et [πt+1] − rn
t )
PC equation (ﬁrms):
πt = βEt [πt+1] + κ˜yt
MP equation (monetary policy, Taylor rule):
it = ρ + φππt + φy ˜yt
The three log-linearized equilibrium conditions of the canonical
three-equation (IS-MP-PC) New Keynesian model. The model is very
basic, but still captures the core aspects of New Keynesian analysis. ˜y
refers to output gap, it nominal interest rate, π inﬂation rate, rn
t natural
rate of interest.

The modified model cannot have the IS equation, and the PC equation
needs to be changed as well. However, we may be able to save the IS
equation in a different form.
The IS equation in the modified model refers to how the consumer came
to learn relationship between current consumption and expected future
consumption.
˜yt = Et [˜yt+1] − σ−1
(im,t − Et [πt+1] − rn
t ) (IS-learning equation)
Notice that it of the IS equation is replaced with im,t in the IS-learning
equation.

IS-learning equation
im,t may not be equal to it - after all, one does not have to interpret im,t
as having anything to do with interest rate set on central bank bonds Bt.
Since we have not speciﬁed on how agents set im,t, the IS-learning
equation is largely a template to build upon. But we do not need to
specify an exact learning mechanism to make crucial observations.
This justiﬁes the use of the IS-learning equation in place of the IS
equation, but we then must ask whether central bank can enforce
im,t = it. The answer turns out to be no.

IS-learning equation
Agents may not even believe that it is a relevant factor in forming an
expectation, especially in case they believe monetary policy is neutral. But
this point holds regardless of monetary neutrality, given that a monetary
policy rule, or an interest rate rule, has interest rate determined by
economic variables. Thus, as far as an interest rate rule does not change,
agents may not feel a need to reference it in forming expectations.
From now on, we assume that agents do not initially reference it in
forming expectations.
The question is, what if central bank changes monetary policy? Would
agents be forced revise their expectations? The answer is no.

Rationale for dropping Bt as a control variable
But what is economic justiﬁcation for dropping Bt as a control variable?
Consumers may consider Bt as a vehicle to implement their plans and
expectations. That is, they consider Bt after they have chosen other
variables. This is not entirely unreasonable - the Walras law for general
equilibrium models states that if all markets other than one are in
equilibrium, then all markets must be in equilibrium as well.

Rationale for dropping Bt as a control variable
Furthermore, it is unreasonable to argue that agents cannot form some
deﬁnite economic outcomes when central bank bond markets do not exist,
regardless of existence of economic frictions such as price stickiness.
We assert that agents can form some equilibrium based on learning
mechanisms even when central bank bond markets are missing. Thus,
monetary policy as an interest rate policy works, if it does work, to change
equilibrium for better - purpose is not about allowing agents to form an
equilibrium.

Modern Policy Ineffectiveness Proposition
We thus now arrive at the Modern Policy Ineffectiveness Proposition
(MPIP). If we have Bt dropped as a control variable and consumers have
learning mechanisms that do not directly utilize it and Bt, then there is no
way that a pure interest rate policy can be effective. This is true by how
an optimization problem of a consumer was modified.

Overview of temporary digressions
We now make slight digressions, as stated in the beginning of Analysis 2:
we discuss the equilibrium indeterminacy and selection issues of the
canonical New Keynesian model, which serves as an example on why
learning is critical in understanding a DSGE model.
Thus until we return to the Modern Policy Ineﬀective Proposition (MPIP),
we will ignore the issue of the IS-learning equation being diﬀerent from the
IS equation. Thus until MPIP is discussed again, im,t = it.

New-Keynesian monetary policy
In the canonical New Keynesian model, monetary policy operates by
setting it, assuming present time is t = 0.
Monetary policy aﬀects economy only with expected nominal interest rate
path. A Taylor rule gives us central bank’s desired monetary policy.
The MP equation becomes a Taylor rule, when given appropriate
coeﬃcient restrictions for ensuring equilibrium uniqueness via the
Blanchard-Kahn condition.

Interest rate path does not provide equilibrium uniqueness
Suppose that expectations of future inﬂation rate are already known, and
let expected interest rate path be known as well. Does this allow us to
determine how agents would decide consumption/output?
The answer is yes. Thus the consumer optimization problem, which
assumes that price path and interest rate path are already known, is
guaranteed some solution.
But: unfortunately, from the combination of IS-PC equations and given
interest rate path, we cannot still determine the expected equilibrium path.

In the traditional general equilibrium understanding of the consumer
optimization problem, since central bank bond decisions only affect the
representative consumer, the MP equation has no role other than just
providing expected nominal interest rate path. The consumer and firm
optimization problems, in this understanding, must not directly factor in
the MP equation - they only must factor in expected nominal interest rate
path.
Reminder: DSGE is a dynamic sequential modification of Arrow-Debreu
general equilibrium models that have agents deciding based on a given
price vector, which includes interest rates.
In the traditional general equilibrium understanding thus, the MP equation
can only reflect the desired relationship between economic variables that
central bank aims.

That the MP equation can only reﬂect the desire central bank has would
not matter if one can determine a unique equilibrium using given nominal
interest rate path and IS-PC equations. The problem is that we cannot.
However, in the contemporary understanding of DSGE models, there is
emphasis on learning, and we will explore how learning may change the
above view.

Objectives of temporary digressions
There are two problems in determining a unique equilibrium:
Even for the solution of the combined IS-MP-PC equations, there are
inﬁnitely many equilibria.
Agents must believe in the MP equation in order for the MP equation
to be operative. Thus, how does central bank make agents believe in
the MP equation it announces?
We tackles these two issues as to possibly allow determination of a unique
equilibrium, evolving around how agents must decide and learn. We then
show why making agents believe in the right MP equation and
expectations that support an intended equilibrium may be impossible.

Backward induction and equilibrium determinacy in DSGE
models
Typically, DSGE models have inﬁnitely many equilibria. Thus, one must
choose one equilibrium out of those.
The logic of forward-looking DSGE models is that solution checking must
be possible by setting expectations of future outcomes in an optimally
consistent way and use these to determine present-time outcomes
recursively. This essentially is a backward induction strategy.
There are equilibria that cannot be obtained from the backward induction
strategy - we may consider them to be invalid equilibria.

models
For an inﬁnite-duration model, in order for this to work, we need to have a
stable future limit. This is essentially what one means by selecting a
locally-bounded equilibrium in DSGE models.
If parameters of a log-linearized model satisfy the Blanchard-Kahn
condition, then there exists one locally-bounded equilibrium, which ensures
equilibrium uniqueness/determinacy.
If a solution for the log-linearized model is not locally bounded, then there
exists no stable future limit for the solution such that backward induction
can be used.

models
Because of reliance on a log-linearized approximative model, it is possible
that an equilibrium eliminated can turn out to be a valid locally bounded
equilibrium on a diﬀerent steady-state log-linearization procedure.
If a full non-linear model only allows for one steady-state, then this is not
a problem.
If not, then in order to eliminate this issue, we require central bank to be
able to enforce a particular steady state. Example: 2% inﬂation target
(steady state), combined with the Taylor rule (MP equation).

Backward induction and equilibrium determinacy
But this does not give us how central bank may enforce the steady state
target, though easy to implement theoretically: if central bank detects
deviation from the target, then reacts in discretionary ways - such as
controlling money supply, or discretionary policy on nominal interest rate -
but discretion restricted for implementing a monetary policy rule.
When ZLB (zero lower bound) is there, we often do have two steady-state
equilibria. Thus the concern of how central bank enforces one particular
steady-state becomes a valid issue.

The explanation given is already in Michael Woodford’s [Interest and
Prices], but with details missing or unclear in the book. Missing details:
the backward induction foundation and the role steady state plays in this
foundation.
The argument/explanation is not without objections - for example, see
Cochrane (2017) argues that a locally bounded equilibrium is too sensitive
to future outcomes that are very far away from now.
Also, one may complain that this is like assuming away tendency of
equilibrium toward the long-run reference equilibrium, instead of explaining
tendency. After all, why do we have to solve an equilibrium with the
backward induction strategy? Other strategies are not naturally
forward-looking, but still do not necessarily contradict forward-looking
nature of models. Thus depends on how we view backward induction as
fundamental for forward-looking models.

In a way, the locally bounded equilibrium constraint takes a role that a
transversality condition used to take to secure a unique equilibrium. In the
modiﬁed canonical New Keynesian model a TVC no longer applies, so this
comes to matter in such a case.

ZLB and equilibrium determinacy
Zero lower bound (ZLB) complicates equilibrium determinacy matters.
The problem is that once agents reach the ZLB, then there is not much
central bank can do with conventional policy to escape it.
Again from Woodford’s
[Interest and Prices]. The
45-degree line represents
the Fisher equation:
it = r + Et[πt+1], with r
assumed to be constant.
The bold line represents
the simple Taylor rule of
it = ρ + φππt.

A locally-bounded equilibrium,
initially not considering ZLB, in
this setting must satisfy
Et[πt+1] = πt. If agents start
from any equilibrium that is not
at intersection of the Taylor rule
and the Fisher equation, then
inflation diverges away either
toward positive infinity or
negative infinity. The arrows
trace out how inflation changes
over time, starting from A.

But if ZLB is there (here ZLB is
zero nominal interest rate), then
if agents start from an
equilibrium that is below the
intended equilibrium inﬂation
rate, then we reach the ZLB
steady state. This equilibrium
path is indeed a locally-bounded
equilibrium path!
If central bank fails to respond
aggressively before reaching ZLB,
then conventional monetary
policy can do not much.

Suppose that economy is stuck at ZLB. Central bank wishes agents to
return to the intended equilibrium. What can it do in conventional interest
rate-based monetary policy?
Because of ZLB, it cannot lower nominal interest rate to raise inﬂation.
When real interest rate rt is assumed to be constant, this conventional
understanding of lowering nominal interest rate to raise inﬂation is
contradictory according to the Fisher equation.
After all, why not just raise nominal interest rate as to reach the intended
equilibrium? Assuming agents are forward-looking, past history does not
matter for determination of present and future outcomes. One can raise
nominal interest rate but still not violate the Taylor rule and the Fisher
equation and achieve the intended equilibrium.

New Keynesian models typically have sticky price such that real interest
rate rt is aﬀected by monetary policy as well. In such a case, to raise
inﬂation rate in short-run, one needs to lower nominal interest rate, which
is blocked by ZLB. Thus one seems to be stuck at ZLB.
If we forget about learning issues for now, then it is still possible, even in
sticky-price models, that central bank simply raises nominal interest rate
to implement the intended equilibrium, and still not violate the Taylor rule
and the Fisher equation. This is now often referred to as a Neo-Fisherite
position. Again, agents are assumed to be forward-looking, so past
expectation history does not matter here.
Question: Is the Neo-Fisherite position really tenable?

Learning
What we need to ask is this: even if agents try to cling onto ZLB
steady-state expectations of future outcomes, can central bank act to
enforce change of expectations but still maintain the Taylor rule?
We ask a more general question: can central bank change either the MP
equation (for example, its parameters/coeﬃcients) or expectations agents
have using an interest rate policy? We will see that the answer is no.
We now return to the general case of it = im,t and the modiﬁed canonical
New Keynesian model. We may rephrase the question as follows: can
central bank act by an interest rate policy to enforce it = im,t? The
answer is no, as we have already seen, unless agents already believe that
monetary policy matters for economic outcome.

What about monetary policy that is not interest
rate-based?
MPIP does not invalidate monetary policies that are not purely interest
rate-setting policies. In reality, central bank can control monetary
quantities by selling and purchasing government bonds, which also act as
collaterals. If monetary quantities matter in an economy - one convenient
modeling trick being the money-in-utility idea - then an interest rate rule
really is a convenient surrogate and a summary for monetary quantity
controls. Modeling this requires allowing for several types of central bank
bonds of diﬀerent maturity, and carefully modeling beneﬁts of money and
monetary assets.

But a turn...
This seems to spell an end to a purely interest rate-based monetary policy.
But there is a way to escape: since Bt is dropped from control variables of
consumer optimization problems, we now can allow for interest rate of
central bank bonds diverging from market interest rates.
If there is heterogeneity in information, then it is possible that agents
know diﬀerent market interest rates. Because interest rates on central
bank bonds are publicly announced, they can reasonably be assumed to be
known by everyone. From this view, interest rates on central bank bonds
form lower bounds of possible market interest rates.

Conclusion for Analysis 2
The modern policy ineffectiveness proposition (MPIP) arises from the
general impossibility result, which requires us to sacrifice central bank
bonds Bt as a control variable of a consumer optimization problem.
MPIP states that an interest rate policy is ineffective, unless learning
mechanisms and beliefs of agents already accept that interest rate policy
matters.
We can escape MPIP easily - one easy alternative, as said before, is to
adopt an alternative monetary policy. What MPIP then invalidates is the
concept of monetary policy in cashless economy.
But we can even save an interest rate policy, if we re-interpret interest
rates on central bank bonds as setting lower bounds on market interest
rates.

Conclusion
Even though ways to escape the general impossibility result (GIR) and the
modern policy ineﬀective proposition were provided, they still imply that
the conventional understanding of DSGE models is not tenable, and we
need to have a new consistent understanding of DSGE models.
There always have been something weird about DSGE models endorsing
the single rate of interest idea, when we know that in Arrow-Debreu
general equilibrium models (ADM models), there usually are multiple rates
of interest. We now know that intuitive suspicions against DSGE models
are indeed justiﬁed by theoretical analysis.

Reconsidering DSGE models

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