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The Zero Bound on Interest Rates and Optimal Monetary Policy
Gauti Eggertsson and Michael Woodford
2003
Keywords: monetary policy
“To preview our results, we find that the zero bound does represent an important constraint on
what monetary stabilization policy can achieve, at least when certain kinds of real disturbances
are encountered in an environment of low inflation. We argue that the possibility of expansion of
the monetary base through central-bank purchases of a variety of types of assets does little if
anything to expand the set of feasible equilibrium paths for inflation and real activity that are
consistent with equilibrium under some (fully credible) policy commitment.”
“The key to dealing with this sort of situation in the least damaging way is to create the right
kind of expectations regarding the way in which monetary policy will be used subsequently, at a
time when the central bank again has room to maneuver. We use our intertemporal equilibrium
model to characterize the kind of expectations regarding future policy that it would be desirable
to create, and discuss a form of price-level targeting rule that — if credibly committed to by the
central bank — should bring about the constrained-optimal equilibrium.”
Liquidity trap definition: “when interest rates have fallen to a level below which they cannot be
driven by further monetary expansion”
I.
-
-
Is “Quantitative Easing” a Separate Policy Instrument?
QE: “expansion of the monetary base”
In this section, build a model that actually has money
“Our key result is an irrelevance proposition for open market operations in a variety of
types of assets that might be acquired by the central bank, under the assumption that the
open market operations do not change the expected future conduct of monetary or fiscal
policy”
“The point of our result is to show that the key to effective central-bank action to combat
a deflationary slump is the management of expectations,” not QE though
Model: MIU, flexible wages, Calvo-staggered prices, monopolistic competition in the goods
market
A. Households
Utility function: money-in-the-utility
-
Where Ct is a Dixit-Stiglitz aggregate of consumption over a continuum of goods with
elasticity of substitution theta
-
Dixit-Stiglitz price aggregator
(1)
-
Mt is monetary base
Ht(j) is quantity of labor supplied of type j (each industry employs industry-specific labor
with its own wage)
- u is concave, increasing in consumption, and increasing in real balances up to a satiation
point m̅ which is a function of consumption:
m̅ = m̅(Ct; ξt)
- “The existence of a satiation level is necessary in order for it to be possible for the zero
interest-rate bound ever to be reached; we regard Japan’s experience over the past several
years as having settled the theoretical debate over whether such a level of real balances
exists.” [unless JGBs have a liquidity premium]
- Don’t assume additive separability between consumption and real balances; thus money
can affect aggregate demand by affecting the marginal utility of consumption
- v is convex and increasing in labor
- ξt is the vector of exogenous disturbances
Budget constraint
- Assume complete financial markets, no limits on borrowing
- Budget constraint:
-
Qt,T is the “stochastic discount factor by which the financial markets value random
nominal income at date T in monetary units at date t”
δt is the opportunity cost of holding money, which is it/(1+it) where it is the nominal
interest rate on a riskless one period obligation purchased in period t
Wt is the nominal value of the household’s financial assets, including money, at the start
of period t
Πt(i) is the profits (revenue minus wage costs) in period t of the supplier of good i
wt(j) is the nominal wage earned by laborer type j in period t
Tth is net nominal tax liabilities of each household in period t
Optimization
- Euler equation:
(2)
-
FOC on money demand, assuming no corner solution of zero money demand:
-
If both consumption and money are normal goods, then from these can solve for unique
level of real balances L*
L* = L(Yt,it; ξt)
- And define L(Y, 0; ξt) = m̅(Y; ξt), the minimum level of real balances for which um = 0,
so that the function L is continuous at i = 0
- Equilibrium condition can be written as:
(3)
(4)
- Along with complementary slackness, i.e.:
1. it ≥ 0 and Mt/Pt = L(Yt,it; ξt)
OR
2. it = 0 and Mt/Pt ≥ L(Yt,it; ξt)
- Also required is a no-Ponzi condition and a transversality condition
-
(5, 6)
Where Dt is the (nominal) sum of total government liabilities, both money and debt
B. Firms
An infinite number of monopolistically competitive industries
- Production function in each industry is
yt(i) = Atf(ht(i))
- A is exogenous productivity common to all industries
- ht(i) is labor specific to industry i
- Supplier of good i sets price for that good which it will supply to meet demand each
period, hiring labor as necessary
- Can be shown that under profit maximization, taking labor supply and quantity demanded
as given from household optimization, profits each period are
-
ptj is the common price charged by other firms in industry j
ξtilde is exogenous shocks
If prices were fully flexible, pt(i) would be chosen each period to maximize this equation.
Instead:
Calvo pricing
- Let 0<α<1 be the fraction of industries with prices that remain unchanged each period
- In any industry that revises its prices in period t, “the new price pt* will be the same”
- Price is implicitly defined by FOC from maximizing discounted sum of expected profits:
(7)
-
Stochastic discount factor is [from UMP]:
-
Price aggregator law of motion becomes with Calvo pricing:
(8)
(9)
C. Monetary authority
Monetary policy
- Assume interest-rate targeting rule similar to Taylor rule
-
-
-
(10)
Assume ϕ is nonnegative (i.e. doesn’t violate ZLB constraint)
This rule implies the central bank sets the monetary base to hit the prescribed interest rate
[thought interest rate targeting made money supply indeterminate?????????????????]
However, if the rule prescribes a zero interest rate, then this only gives a lower bound on
the money supply
If it is zero, suppose the monetary base rule is
(11)
Where ψ satisfies:
o ψ(Pt/Pt-1, Yt) ≥ 1
o ψ(Pt/Pt-1, Yt) = 1 if ϕ(Pt/Pt-1) = it > 0
QE can be understood as the choice of a function φ that is greater than one sometimes
What assets does the central bank buy when it varies the monetary base?
k types of securities
At end of period t, vector of nominal values of central bank assets is Mtωtm where ωtm is a
vector of “central bank portfolio shares”
These shares are determined by policy rule
(12)
Where the vector-valued function ωm has the property that its components sum to one
Since it depends on the same arguments as ϕ, assets bought could change at ZLB
Payoffs to securities in each state of the world are given by (state-contingent) vectors at
and bt and matrix Ft
-
A vector of asset holdings zt-1 gives at end of period t-1 to owner a quantity at’zt-1 of
money, a quantity bt’zt-1 of the consumption good, and a vector Ftzt-1 of securities that
may be traded in period t
This lets us deal with many types of assets:
“For example, security j in period t-1 is a one-period riskless nominal bond if bt(j) and
Ft(. , j) are zero in all states, while at(j) > 0 is the same in all states. Security j is instead a
one-period real (or indexed) bond if at(j) and Ft(. , j) are zero, while bt(j) > 0 is the same
in all states. It is a two-period riskless nominal pure discount bond if instead at(j) and bt(j)
are zero, Ft(i; j) = 0 for all i ≠ k, Ft(k, j) > 0 is the same in all states, and security k in
period t is a one-period riskless nominal bond.”
Thus gross nominal return Rt(j) on the jth asset between period t-1 and t is
-
Where qt is the vector of nominal asset prices in period t trading
No arbitrage condition ensures that equilibrium prices must satisfy:
-
(14)
Nominal transfer to the Treasury each period, i.e. seigniorage, becomes: where Rt is the
vector of returns from (13)
-
-
(13)
(15)
D. Fiscal authority
Fiscal authority: followd rule for evolution of total government liabilities Dt
-
(16)
Dt is both the monetary base, the rule for which is specified above, and other types of
securities, which are a subset of the k securities purchased by the central bank
Let ωjtf indicate the share of government debt at the end of period t that is of type j
Let Bt = Dt – Mt be the total nominal value of non-monetary liabilities
Let Tth be the nominal value of the primary budget surplus (taxes minus transfers, since
no government consumption)
The flow government budget constraint then is:
-
Government determines the composition of its debt according to function:
-
Again where components of vector-valued ωtf sum to one
-
(17)
E. Equilibrium and the irrelevance proposition
Equilibrium: a collection of stochastic processes {pt*, Pt, Yt, it, qt, Mt, ωtm, Dt, ωtf}, “with each
endogenous variable specified as a function of the history of exogenous disturbances to that
date”, that satisfy the aggregate demand block of the model (2)-(6), aggregate supply conditions
(7) and (9), conditions (10)-(12) specifying monetary policy, and conditions (16)-(17) specifying
fiscal policy each period
Irrelevance proposition: “The set of paths for the variables{pt*, Pt, Yt, it, qt, Mt, ωtm, Dt, ωtf} that
are consistent with the existence of a rational-expectations equilibrium are independent of the
specification of the functions ψ in equation (1.11), ωm in equation (1.12), and ωf in equation
(1.17)”
- Recall ψ is the QE function, ωm is the composition of assets purchased, and ωf is the
composition of fiscal authority liabilities
- The set of restrictions on the set of processes can be written in a form that does not
involve the irrelevant variables Mt, ωm, or ωf and so does include the functions ψ, ωm, or
ωf
- Proof:
- Note first, for all m ≥ m̅(C;.), it is true that u(C, m;.) = u(C, m̅;.) because it’s a satiation
point
- Differentiating, we see that uc(C, m;.) does not depend on the exact value of m if it
exceeds the satiation point
- Thus at the ZLB we can replace uc(Yt, Mt/Pt;.) that appears several times in our
equilibrium relations by the following λ function, using the fact that Mt/Pt = L at all levels
of real balances at which uc depends on the level of real balances
-
i.e. rewrite uc as a function that doesn’t include Mt/Pt
We can similarly replace um(Yt, Mt/Pt;.)(Mt/Pt) that appears in (5) by
-
Since Mt/Pt = Lt when real balances are lower than satiation, and um = 0 when real
balances are higher
Finally we can express nominal profits as the following, after having substituted λ(.,.;.)
for the marginal utility of real income in the wage demand function that is used in
deriving the profit function
-
-
Thus we can rewrite the system of equations that are the equilibrium relations as follows,
none of which include ψ, ωm, ωf
-
Along with relations (9), (10), and (16) as before
Thus generalized QE (LSAP, Operation Twist) does not matter if policy rules do not change
- “If the commitments of policymakers regarding the rule by which interest rates will be set,
on the one hand, and the rule by which total private sector claims on the government will
be allowed to grow, on the other, are fully credible, then it is only the choice of those
commitments that matters.”
- Also no portfolio balance effect, despite distinguishing among different assets
o “Our conclusion differs from that of the literature on portfolio balance effects for
a different reason. The classic theoretical analysis of portfolio balance effects
assumes a representative investor with mean-variance preferences. This has the
implication that if the supply of assets that pay off disproportionately in certain
states of the world is increased (so that the extent to which the representative
investor ’ s portfolio pays off in those states must also increase), the relative
marginal valuation of income in those particular states is reduced, resulting in a
lower relative price for the assets that pay off in those states. But in our generalequilibrium asset pricing model, there is no such effect. The marginal utility to the
representative household of additional income in a given state of the world
depends on the household’s consumption in that state, not on the aggregate payoff
of its asset portfolio in that state. And changes in the composition of the securities
in the hands of the public do not change the state-contingent consumption of the
representative household — this depends on equilibrium output”
-
The irrelevance proposition is in the spirit of Wallace (1981)’s irrelevance proposition for
open market operations. Wallace’s analysis is usually useless, because money is only a
store of value in his model, so an equilibrium where short-term Treasuries have a higher
return than money is not possible, something that is very often observed. But at the ZLB,
the marginal liquidity services provided by money is zero, so Wallace’s analysis is
correct
What about the assumption of complete markets and no debt constraints?
- “The absence of borrowing limits is also innocuous, at least in the case of a
representative-household model, because in equilibrium the representative household
must hold the entire net supply of financial claims on the government. As long as the
fiscal rule (equation 16) implies positive government liabilities at each date, any
borrowing limits that might be assumed can never bind in equilibrium. Borrowing limits
can matter more in the case of a model with heterogeneous households. But in this case
the effects of open-market operations should depend not merely on which sorts of assets
are purchased and which sorts of liabilities are issued to finance those purchases, but also
on how the central bank’s trading profits are eventually rebated to the private sector (that
is, with what delay and how distributed across the heterogeneous households), as a result
of the specification of fiscal policy. The effects will not be mechanical consequences of
the change in composition of assets in the hands of the public, but instead will result from
the fiscal transfers to which the transaction gives rise; it is unclear how quantitatively
significant such effects should be.”
The important assumption is that OMOs have no consequence for future interest rate policy or on
the evolution of total government debt liabilities Dt at present or in the future
- “In our view it is more important to note that our irrelevance proposition depends on an
assumption that interest rate policy is specified in a way that implies that these openmarket operations have no consequences for interest rate policy, either immediately
(which is trivial, because it would not be possible for them to lower current interest rates,
which is the only effect that would be desired), or at any subsequent date. We have also
specified fiscal policy in a way that implies that the contemplated open-market operations
have no effect on the path of total government liabilities {Dt} either, whether
immediately or at any later date. Although we think these definitions make sense, as a
way of isolating the pure effects of open-market purchases of assets by the central bank
from either interest rate policy on the one hand or fiscal policy on the other, those who
recommend monetary expansion by the central bank may intend for this to have
consequences of one or both of these other sorts.”
For example, a helicopter drop is not just a change in the function φ in the policy rule
- First, the expansion is supposed to be permanent
- In this case, the function ϕ that defines interest rate policy is also being changed in a way
that will be relevant at a future date when money supply no longer exceeds satiation
- Second, a helicopter drop implies a change in fiscal policy as well
- The operation increases the nominal value of total government liabilities, and this is often
presumed to be permanent as well
-
-
-
“First of all, it is typically supposed that the expansion of the money supply will be
permanent. If this is the case, then the function φ that defines interest rate policy is also
being changed, in a way that will become relevant at some future date, when the money
supply no longer exceeds the satiation level. Second, the assumption that the money
supply is increased through a helicopter drop rather than an open-market operation
implies a change in fiscal policy as well. Such an operation would increase the value of
nominal government liabilities, and it is generally at least tacitly assumed that this is a
permanent increase as well.”
“This explains the apparent difference between our result and that obtained by Auerbach
and Obstfeld (2003) in a similar model. These authors assume explicitly that an increase
in the money supply at a time when the zero bound binds carries with it the implication of
a permanently larger money supply, and that there exists a future date at which the zero
bound ceases to bind, so that the larger money supply will imply a different interest rate
policy at that later date. Clouse and others (2003) also stress that maintenance of the
larger money supply until a date at which the zero bound would not otherwise bind
represents one straightforward channel through which open-market operations while the
zero bound is binding could have a stimulative effect, although they discuss other
possible channels as well.”
Thus, helicopter drops are not subject to the irrelevance proposition
Thus future policy is important
- “And this question can be addressed without explicit consideration of the role of openmarket operations by the central bank of any kind. Hence we shall simplify our model —
abstracting from monetary frictions and the structure of government liabilities altogether
— and instead consider how it is desirable for interest-rate policy to be conducted, and
what kind of commitments about this policy it is desirable to make in advance”
II.
How Severe a Constraint is the Zero Bound?
Central bank is still powerful at ZLB since it’s the expected path of short term rates, not the
current level alone, which is important; but ZLB is a genuine constraint
Log-linearize model above around zero inflation SS since this is optimal (not Friedman’s
deflation, since abstracting from transaction frictions)
- Under zero inflation, “it is easily seen” real rate of interest (and also nominal rate of
interest) is r̅ = β-1 – 1 > 0 [since interest rate will equal rate of time preference rho]
- Because log linearization only works for small disturbances, and 0 might be too far away,
we have to abstract and say that there is some interest paid on money im > 0 that cannot
be reduced, and is the lower bound
As Woodford has shown in Interest and Prices, the log linear approximated equilibrium relations
can be summarized by two equations each period:
- A forward looking IS relation
-
And a forward looking AS relation (NK Phillips Curve)
-
Where πt = log(Pt/Pt-1), i.e. inflation
xt is the “welfare-relevant” output gap
it is now the continuously compounded nominal interest rate, corresponding to log(1+it)
using the notation of the previous section
ut is exogenous “cost push disturbances” that shift the NKPC
rnt is exogenous variation in the Wicksellian natural rate that shifts the IS relation
(24) is log linearized equation (2)
(25) is log linearized equation (7)-(9) eliminating log(p*t/Pt)
We can omit the log linear equation for money demand since we don’t care about it 
“The other equilibrium requirements of the earlier discussion can be ignored in the case
that we are interested only in possible equilibria that remain forever near the zeroinflation steady state, because they are automatically satisfied in that case”
For the inflation target π* to be hit, the real interest rate must equal the Wicksellian rate
However, the ZLB prevents this from holding if 𝑟𝑡𝑛 < −𝜋 ∗
- E.g. if target inflation is zero, if natural rate is negative, it won’t be able to be hit
- Higher inflation target thus allows central bank to be less likely to have this problem
- But inflation still distortionary
Suppose for illustration the natural rate of interest is unexpectedly -2% in the first period then
reverts back to steady state r̅ =4% > 0 with constant probability 10% every period (expected
liquidity trap length of 10 periods)
-
With 2% inflation target, natural rate can be hit by nominal rate
This is why some call for a higher inflation target. But if inflation target doesn’t have to be fixed,
we can have even more optimal policy:
III.
The Optimal Policy Commitment
What is the optimal policy to satisfy (24) and (25)?
- Assume, for now that policy is fully credible
- Purely forward-looking policy (i.e. neglecting past conditions) would do no good
The government loss function can be derived from a second-order Taylor expansion of the utility
function of the representative household (see again Interest and Prices)
-
Again, this is abstracting from transaction frictions; if they exist, the loss function
includes an additional term proportional to (it – im)2, a la Friedman rule
However, because
Minimize loss function under the constraint (from ZLB and IS equation):
-
Giving the Lagrangian
-
The FOCs are
-
This cannot be solved algebraically, due to nonlinearities of (31); must use numerical
methods [fuck]
Optimal policy
- Is history dependent – note Lagrange multipliers are lagged (t-1)
- Depends on keeping interest rate lower than otherwise in the future
- Even promising inflation 10 years down line, i.e. not that soon, has an effect
- However, higher inflation in the future will have distortionary effects, and there is a
tradeoff; so even under optimal policy with commitment, there will be some negative
effect today
- Further, although there will be a higher price level in the future, the price level will
ultimately be stabilized
Suppose for illustration again natural rate negative in period zero then reverts to SS with fixed
probability. Under optimal policy: (again where lines indicate different paths of natural rate)
-
Optimal policy has you create a boom and inflation once the natural rate returns to SS
Price level increases, but eventually stabilizes
Contrast interest rate paths in comparison with a zero-inflation target:
Full comparison where natural rate returns to SS, stochastically, at period 15:
- Policy rate is kept at zero five periods longer
- As a consequence, deflation and output gap largely avoided
IV.
Implementing optimal policy
Implementation of optimal policy
- Can be committed to via a policy rule that is a function of what has occurred before that
date
- Because describing policy as the (state-contingent) interest rate path leaves variable
indeterminate, not an appropriate policy rule
- Instead, describe rule as function of deviations of inflation, output gap from desired paths
- Only needs to observe price level and output gap; not equilibrium interest rate
Each period there is a predetermined price level target pt*; the central bank chooses the interest
rate it to achieve, if possible
-
And otherwise sets it = 0.
𝑝̃𝑡 is the output gap adjusted price level index defined as (see Interest and Prices)
-
If 𝜆 = 𝜅, then the natural rate of output follows a deterministic trend and this is NGDP
targeting; unlikely though
The target for the next period is then determined by
-
Where Δ𝑡 is the shortfall in period t
-
This achieves the optimal commitment solution
“If the price-level target is not reached, because of the zero bound, the central bank
increases its target for the next period. This, in turn, increases inflation expectations
further in the trap, which is exactly what is needed to reduce the real interest rate.”
The best way to make a rule credible is to conduct policy over time that demonstrates the
commitment
- “The best way of making a rule credible is for the central bank to conduct policy over
time in a way that demonstrates its commitment. Ideally, the central bank’s commitment
to the price-level targeting framework would be demonstrated before the zero bound
came to bind (at which time the central bank would have frequent opportunities to show
that the target did determine its behavior).”
- In normal times, when natural rate of interest is nonnegative, Δ𝑡 = 0
A simpler version of this rule approximates this and does not have any special proviso for the
liquidity trap:
-
“More easily communicated to the public”; fully optimal if ZLB never binding
Constant price level target: the target for the gap adjusted price level is fixed at all times
Comparison with optimal policy: (welfare loss relative to strict inflation target of zero)
If this was instead inflation targeting instead of level targeting (first differencing:
𝜆
𝜋𝑡 + 𝜅 (𝑥𝑡 − 𝑥{𝑡−1} ) = 0), when the ZLB binds this would be a disaster. Central bank would
deflate after leaving liquidity trap
- i.e., the level targeting (history dependence) aspect is key
V.
Other issues
What if ZLB binds forever?
- Consistent with all equilibrium conditions
- …EXCEPT transversality condition (6) (which recall is equivalent to the condition that
households hit their intertemporal budget constraint)
- To answer the question of whether this is possible requires a more complete specification
of the fiscal policy rule
- With a Ricardian policy rule a la Benhabib, Schmitt-Grohe, and Uribe (2001), you can
get multiple equilibria including a perpetual liquidity trap
- However with a fiscal policy rule that does not allow for rapid contraction in government
liabilities, e.g. a balanced budget rule, however, such a case is not possible
- Also e.g. a rule not to contract monetary base in combination with a commitment for a
nonnegative asymptotic present value of public debt
Other aspects of expectations management
- What policies might contribute to desirable expectations
-
-
Demonstrate resolve by following rule before crisis hits
Manipulating money supply “can be helpful, even though irrelevant to interest rate
control as a way of communicating to the private sector the central bank’s belief about
where the price level ought properly to be (and hence the quantity of base money that the
economy ought to need)”
i.e. QE can be an important communication device
Thus Svensson’s “fool-proof method” is just a communication and commitment device
The time inconsistency problem and output gap-adjusted price level targeting
- If the central bank is credibly able to commit to OGA-PLT, then ZLB not too serious
- However, if the central bank is not able to credibly commit to future actions, then central
bank each period will end up targeting strict inflation of zero
- Result is a prolonged contraction
- “Deflationary bias,” as opposed to inflationary bias of Kydland and Prescott
Preventing time inconsistency
- If central bank cares about distortions caused by taxes, then tax cuts financed by money
printing can be an effective commitment mechanism
- Purchases of real assets (e.g. real estate) can be thought of as another way of increasing
nominal government liabilities, which would affect inflation incentives similar to deficit
- Purchases of foreign exchange: seigniorage off of foreigners