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ECONOMIC REVIEW

FEDERAL RESERVE BANK
OF CLEVELAND

1
http://clevelandfed.org/research/review/

E C O N O M I C

R E V I E W

2001 Quarter 1
Vol. 37, No. 1

Sharing with a Risk-Neutral Agent

2

by Joseph G. Haubrich
In the standard solution to the principal–agent problem, a
risk-neutral agent bears all the risk. The author shows that, in
fact, multiple solutions exist, and often the risk-neutral agent is
not the sole bearer of risk. As risk aversion approaches zero,
the unique risk-averse solution converges to the risk-neutral
solution, wherein the agent bears the least amount of risk.
Even a small degree of risk aversion can result in agents bearing
significantly less risk than the standard solution suggests.

Monetary Policy and Self-Fulfilling
Expectations: The Danger of Forecasts
by Charles T. Carlstrom and Timothy S. Fuerst

9

What rule should a central bank interested in inflation stability
follow? Because monetary policy tends to work with lags, it is
tempting to use inflation forecasts to generate policy advice.This
article, however, suggests that the use of forecasts to drive policy
is potentially destabilizing. The problem with forecast-based policy is that the economy becomes vulnerable to what economists
term “sunspot” fluctuations.These welfare-reducing fluctuations
can be avoided by using a policy that puts greater weight on past,
realized inflation rates rather than forecasted, future rates.

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ISSN 0013-0281

2

Sharing with a RiskNeutral Agent
by Joseph G. Haubrich

Joseph G. Haubrich is an economist and consultant at the Federal
Reserve Bank of Cleveland.
He thanks Patricia Beeson and
Yukiko Hirao for helpful comments.

Introduction
In the classical principal–agent problem, a riskneutral agent bears all the risk. This particular
solution, while acknowledged as a special case, is
prominent in the minds of economists because the
more general risk-averse case does not easily yield
numerical results. For example, Jensen and Murphy
(1990) find a divergence between the risk actually
borne by chief executive officers and the risk-neutral
solution, which seems too large to be accounted for
by reasonable levels of risk aversion.
Although the standard risk-neutral solution is
correct, it is also misleading. Other solutions exist
wherein the agent does not bear all the risk, and
these may be considered more “natural,” since they
are limits of the risk-averse case.
Specifically, in Grossman and Hart’s (1983)
principal–agent problem, where there is a finite number of actions and states, many optimal sharing rules
exist; in only one does the agent bear all the risk.1
With a large enough stake in the project, the agent
will not shirk—and with a finite number of states and
actions, this stake need not be 100 percent.
Once agents have some risk aversion, the principal–
agent problem has a unique solution. For the two-state
case, limits can be computed as risk aversion approaches
zero. The risk-averse solutions do not converge to the

classic risk-neutral solution, however, but to the solution with the lowest risk for the agent. Because less risk
makes a risk-averse agent happier, he demands a lower
risk premium, in turn making the principal happier.
But exceptions occur. There are cases in which the
optimal action discretely shifts with an infinitesimal
increase in risk aversion. In this case, the sharing rule
(and thus the risk borne by the agent) differs substantially when the principal wants to induce distinctly
different actions.
By increasing the number of actions, the results
reduce to the standard continuous-action principal–
agent models (see Holmstrom [1979]). Under reasonable conditions, the set of risk-neutral solutions
shrinks to one.
This should introduce a note of caution to applications of the principal–agent model. The simple
risk-neutral solution is not a good approximation
of the optimal contract, even for arbitrarily low risk
aversion. It can be misleading to compare actual
contracts in which risk aversion is important—
executive compensation, for instance—with the
predictions of the risk-neutral principal–agent
model. Stated more positively, these results show
how principal–agent theory implies the relatively
flat sharing rules that are observed in practice.
■ 1 Although Grossman and Hart do not explicitly mention multiple
solutions in the risk-neutral case, they are careful in stating their theorems.
Hence, this result does not imply any error in their work.

3

–
Assumption (A2): {[U–G(a)]/K(a)} ≤ V (∞) for all a
in A.

I. Sharing Rules
The Model
First let us quickly review the assumptions, notation, and approach of the Grossman–Hart model.
For concreteness, assume the principal owns a firm,
but she delegates its management to the agent.
There is a finite number of outcomes (gross profit
states), q1 < q2 < … < qn. The principal, who is risk
neutral, cares only about the firm’s expected net
profit, defined as gross profit minus any payment to
the manager.
In managing the firm, the agent takes an action,
often thought of as effort, which the principal cannot
observe. The principal does observe the outcome,
however, and, like the agent, knows that different
actions determine the probability of the outcome
states. Both know πi (a), the probability of outcome qi
given action a. This probabilistic setting means the
agent might work hard but still have little output to
show for it. In choosing an action, the agent does not
know the ultimate result. Conversely, in seeing the outcome, the principal cannot deduce the agent’s action.
Actions belong to the finite set Α={a1, a2, a3 . . .
am}, making the principal’s expected benefit from an
action equal to
n

B(a) = ∑ πi(a)qi .
i=1

To avoid the problem of increasingly larger penalties
being imposed with progressively smaller probabilities (see Mirrlees [1976]), assume that πi (a) is
strictly greater than zero for all states and actions.
The agent likes income, but he dislikes effort.
His utility function, U(a,I), depends positively on
his income from the principal, I, and negatively on
his action, a. Grossman and Hart find it useful to
place the following restrictions on U(a,I):
Assumption (A1): U(a,I) has the form G(a) +
K(a)V(I), where V(I) is a real-valued, continuous,
strictly increasing, concave function with the
domain [I,∞] and lim = – ∞. G and K
←

I

I

are real-valued, continuous functions defined by
A, and K is strictly positive. For all a1, a2 in A
and I, J in (I,∞ ), [G(a1) + K(a1)V(I)] ≥ [G(a2) +
K(a2)V(I)] implies [G(a1) + K(a1)V(J)] ≥
[G(a2) + K(a2)V(J)].
–
The agent has a reservation utility U, that is, the
expected utility he can achieve working elsewhere.
–
Sometimes this is derived from an outside income I,
–
–
so that U = V(I). If the principal does not offer him
–
a contract worth at least U, the agent will take
another job. To make the model at all interesting,
some income level should induce the agent to work.
Grossman and Hart formalize this as

To see what happens when this assumption does
not hold, consider the negative exponential utility
–
–e–k(I–a) and U = 5. In this case, even infinite income
could not make the agent work.
If the principal could observe actions, it would
be straightforward to determine how much she pays
the agent for each action. Call this the first-best
cost, or CFB(a):
–
U [a,CFB(a)] = U, or
–
CFB(a) = h{[U–G(a)]/K(a)},
where h = V –1.
As Grossman and Hart put it, “CFB(a) is simply
the agent’s reservation price for picking action a. ”
Given this cost, the first-best optimal action maximizes the principal’s net benefit, B(a)–CFB(a).
Of course, the principal cannot observe the
agent’s actions, nor can she directly base pay on
effort. Instead, she chooses an incentive scheme,
Ii = {I1, I2, … In}, wherein payment Ii depends on
the observed final state, qi . Given this, the agent
will choose the action that maximizes his expected
utility. Knowing how the agent will react, the principal now can break her problem into two parts. For
each action, she calculates the least costly incentive
scheme that will induce the agent to choose that
course. This gives her the expected cost of motivating the agent to perform a particular action a,
n

C(a) = ∑πi (a)Ii. She then chooses the action with
i=1

the highest net benefit—that is, the one that maximizes B(a)–C(a).

Multiple Solutions
The possibility of multiple solutions arises from looking at the mathematics of the agent’s problem. With
risk neutrality, the concave programming problem
with a unique solution becomes a linear programming problem with multiple solutions. With a
risk-averse agent, the principal minimizes the agent’s
risk, subject to meeting the incentive constraints.
For a risk-neutral agent, only the incentives matter,
and any incentive-compatible risk configuration will
work. When the principal is not indifferent between
the two most desirable actions, multiple equilibria
can result. The larger the gap between the actions,
the more risk the agent can bear.
The traditional solution assigns all risk to the
agent, who delivers a fixed payment to the principal.

4

The agent, then, receives

subject to
n

Ii = qi – [B(a*) – CFB(a*)].

∑ πi(a*) [G(a*) + Ii] ≥ U,

(IR)

The agent bears all the risk for shortfalls in q, and
the principal gets her expected benefit.

i=1
n

∑ π i(a*) [G(a*) + Ii] ≥

(IC)

i=1
n

∑ πi(a) [G(a*) + Ii] for a:a≠a*

i=1

qi – Ii = B(a*) – CFB(a*).

(FEAS)

Now, suppose the agent bears less risk and takes
only a fraction of the shortfall in q. Income in state i
becomes
(1) Ii = τ qi – t [B(a*) – CFB(a*)],

Ii ≤ ∞ for all i.

We now must determine the value of τ in equation (1) that will satisfy these conditions. This
means choosing τ to satisfy
n

n

i=1

i=1

∑ πi(a*)Ii = ∑ πi(a*) {τ qi – t[B(a*) – CFB(a*)]}

where t is a constant and τ measures the fraction
of risk borne by the agent. Proposition 1 gives
sufficient conditions for τ being less than one:

= CFB(a*),
resulting in

Proposition 1: Assume (A1)–(A2) and a riskneutral agent. If

∩ (a)≤1[τ (a), 1]
τ ∈ a:a≠a*and
τ

By construction, values between 0 and 1 satisfy
the individual-rationality constraint. Some values of
τ also satisfy the incentive-compatibility constraint,
as I will now show. Substituting equation (2) into
equation (1), the incentive scheme becomes

where

(β ) (
{α β [

τ (a) = 1
–

( + I)

1
B(a) – B(a*)

)

1 – 1 + G(a*) – G(a) ,
K(a) K(a*) K(a*) K(a)

]

}

then an optimal contract exists that pays the agent
Ii = τ qi – t [B(a*) – CFB(a*)] for some value of t.
This somewhat complicated condition guarantees
there is a “gap” or “jump” between the principal’s
payoff in different states.
The proof is straightforward and revealing. To
emphasize the underlying logic, I have made two
simplifying assumptions about utility, both of which
are easily generalized. First, I have specialized the
risk-neutral income utility to V(I) = I, rather than to
V(I) = α + β I. Second, I have used the additively
separable form of utility, setting U(a,I) equal to
G(a) + V(I) or, here, to G(a) + I.
Proof: For the optimal action, the principal calculates the least costly method of getting the agent to
choose action a*. The incentive scheme must minimize the principal’s expected payment to the agent
while still inducing him to act. This is a programming problem, including individual rationality (IR),
incentive compatibility (IC), and feasibility constraints (FEAS).
(P1)

B(a*) – CFB (a*)
(2) t = τ
.
B(a*) – CFB (a*)

(3) Ii = τ qi – τ B(a*) + CFB(a*).
This makes the incentive-compatibility constraint
n

(4) G(a*) + ∑ πi(a*)[τ qi – τ B(a*) – CFB(a*)] ≥
i=1
n

G(a*) + ∑ πi(a) [τqi – τB(a*) – CFB(a*),
i=1

which simplifies to
(5) G(a*)–G(a) ≥ τ [B(a)–B(a*)].
Whether a risk-neutral agent bears all the risk
depends on whether there is a gap between
G(a*)–G(a) and B(a)–B(a*).2 But this gap is not
solely a matter of chance: The principal chooses a*
to maximize B(a)–C(a) or, in the risk-neutral case,
B(a)–CFB(a). Because a* is the optimal action, it
satisfies B(a*)–CFB(a*) ≥ B(a)–CFB(a). Rearranging
and using the definition of CFB, we have
(6) G(a*) – G(a) ≥ B(a) – B(a*) ∀a ∈ A.

n

MIN ∑ π i(a*),
i=1

■

2 Haubrich (1994) provides several numerical examples of problems
of this type, showing that solutions do exist and the theorem is not vacuous.

5

If the inequality in equation (6) is strict, τ can
be less than one, meaning the agent does not assume
all the risk. There are three cases to consider, depending on the sign of each side of equation (6).
(i) Both G(a*)–G(a) and B(a)–B(a*) are positive.
In this case, a has the larger gross payoff but is
more costly to implement than a*. Clearly, if
equation (6) holds, any τ in the relevant range
of [0,1] will satisfy equation (5).
(ii) If G(a*)–G(a) is positive and B(a)–B(a*) is
negative, any τ works. In this case, the less
costly action, a*, also has the better payoff.
(iii) Both G(a*)–G(a) and B(a)–B(a*) are negative. In this case, a* is more costly but has a
better payoff. We usually think of this as the
“normal” case. With negative numbers,
division reverses signs, so equation (5)
implies that τ , the fraction of risk borne by
the agent, can fall anywhere in the interval

τ ∈ G(a*) – G(a) ,1 .
B(a) – B(a*)
With a more general utility function, this becomes
the condition stated in the proposition:

[

]

(β ) (
{α β [

(7) τ (a) = 1

1
B(a) – B(a*)

)
]

1 – 1 + G(a*) – G(a) .
K(a*) K(a) K(a*) K(a)
Even equation (7) understates the full range of
incentive schemes wherein the principal bears risk.
With more than two states, the sharing rule need
not be linear, and a single-parameter τ will not capture all possible deviations from the classic case. In
general, the solution set will be the convex hull of
extreme points, a multidimensional “flat” or “face”
of the constraint set for the linear programming
problem (P1).
–

( + I)

}

II. Convergence
Solutions in which the principal assumes some risk
are more than curiosities. As risk aversion approaches
zero, the risk borne by the agent converges to a
number less than one. The traditional solution offers
a poor approximation of this, even near zero.
The convergence results are for the two-state
case—the sole case with closed-form solutions for the
risk-averse problem. Answering convergence questions usually requires strong assumptions. For
instance, Grossman and Hart assume only two states,
or negative exponential utility. Without strong
restrictions, odd things can occur in the model: The
individual-rationality constraint may not bind, higher
profits may mean less money for the agent, or the
agent may get more money for less effort.

Limiting Cases
With only two states, the single-parameter τ fully
describes how much risk the agent bears. Usually,
the risk-averse solutions converge to the solution with
the smallest value (rather than the classic solution of
τ =1). Some exceptions exist because the optimal
action can switch at zero, which in turn causes a
discrete jump in the risk burden.
To explore convergence, we must first make sure
the utility functions do, in fact, converge. If we index
the income utility function by risk aversion γ , (γ, I ),
we embody this convergence as a new assumption.
Assumption (A3): As γ approaches 0, V(γ,I)
converges uniformly to α + β I, ( α,β ≠ 0), on the
interval [–qn, qn].
Although it is natural, this assumption does
restrict utility functions. For example, the negative
exponential function –e–γ(I–a) converges to zero,
a constant function that is inadmissible by assumption (A1).
The statement of proposition 2 requires a little
groundwork. First, the proof uses the closed-form
solution for the two-state case found by Grossman
and Hart:
–
–
(8) v1= π (a )[U–G(a )] – π (a )[U – G(a )]/K(a )
2 j
k
2 k
j
j

π 1(ak ) – π1(aj )
–
–
(9) v2= π (a )[U–G(a )] –π (a )[U – G(a )]/K(a )
2 j
k
2 k
j
j

π 2(ak ) – π 2(aj )
The derivation of these formulas depends crucially
on Grossman and Hart’s proposition 6, which proves
the agent is indifferent between the optimal action a*
and some less costly action. The existence of two possibilities makes convergence problematic. As risk
aversion falls, either the optimal action or the less
costly action may change. A change in the optimal
action matters for the convergence result, but it is not
clear whether a change in the less costly action makes
a difference. I have produced neither a proof nor a
counterexample for this case. Thus, the statement of
proposition 2 reflects these two possibilities.
The unique profit share for a given utility function and risk aversion is defined as τ (V,γ ), and the
the minimum in equation (7) is defined as τmin.
Proposition 2: Given assumptions (A1)–(A3),
if the optimal action and the indifferent alternative
action do not change for risk aversion in the neighborhood of zero, then
lim τ (V, γ ) =τmin.
γ →0

6

To ease the notational burden and to emphasize
the logic, I present the proof for the additively separable case. The generalization to other utility functions is straightforward.
Proof: In the risk-neutral case, we know from equation (6) that

τ min=

G(ak ) – G(ai )
,
B(ai ) – B(ak )

which clearly depends on the optimal action ak and
a particular alternative ai. This implies an income
difference between states of
(10) I2 – I1 = G(aj ) – G(ak )/

π 1(aj ) – π1(ak ).
In the limit of the risk-averse case, optimal
incomes are given by the limits of equations (8)
and (9).
–
–
(11) I = v = π 2 (ai)[I – G(ak )] –π 2 [I – G(aj )]
1
1
π1 (ak ) – π1 (aj )
(12)

I2 = v2 =

–
–
π 2 (ai)[I – G(ak )] – π 1 [I – G(aj )]
.
π 2 (ak ) – π 2 (aj )

Because π1(aj ) + π 2(aj ) = 1, the probabilities can
be expressed in terms of π 1(•). Making this substitution and collecting terms yields
I1 = {[π (a ) – π (a )] I–+ [1 – π 1(a )] G(a ) –
1 k
1 j
k
j
[1 – p1(aj )]G(ak )} { π 1 (ak ) – π1 (aj )}–1
I2 = {[ π (a ) – π (a )] I–+ π (a )G(a )
1 j
1 k
1 k
j
– π1(ak )G(ak )} { π1 (aj ) – π1 (ak )}–1
Taking the difference and simplifying, we find
(13) I2 – I1 = G(aj ) – G(ak ) /

π1(aj ) – π1(ak ),
which matches equation (10).
The equality between equations (10) and
(13) depends on the constancy of both the optimal
action and the alternative action. I conjecture that
even if the alternative action switches in the neighborhood of zero, the equality (and thus the proposition) still holds.

Action Shifts
Proposition 2 does not hold when the optimal action
shifts at zero. Suppose one action is best at a risk aversion of zero and another at a risk aversion greater than
zero. As the action changes, so does the sharing rule.
The best way to illustrate this is a simple two-act
example. Here, the principal induces the better action
at zero risk aversion, but pays a flat fee and accepts
the lower action for risk aversion greater than zero.
We begin with B(a*)–C(a*)=B(a)–C(a), or indifference between the two actions, so that the switch
occurs at zero. This sets τ equal to one, meaning the
agent bears all the risk. We next want B(a2)–C(a2) <
B(a1)–C(a1), making the lower action preferred
for γ > 0. To do this, set V(I) = I – γI2. Then,
h(v) = [1 + (1 – 4 γv)] /2 γ. With h(v) in hand, we
can assess the second-best costs once we have calculated v1 and v2. The goal is to show that, in some
cases, ∂ C(a2)/∂γ > 0. If this is true, an increase in
leads the principal to prefer action a1, since the cost
of action a2 increases while the rest of the variables,
B(a2), B(a1), and C(a1), remain unchanged. (C [a1]
fixed payment independent of state. )
Simplifying v1 and v2 from equations (8) and
(9), we have:
– –2
v1 = I – γ I + [G(a ) – G(a ) + π (a )G(a ) –
1
2
1 1
2

π1(a2)G(a1)] [π1 (a2) – π1 (a1)]–1
– –2 π 1(a2)G(a1) – π1(a1)G(a2)
v2 = I – γ I +
π1 (a2) – π 1 (a1)
This represents a shift in the optimal action
induced by the principal. Both actions remain feasible. The last terms in each of these expressions are
constant with respect to γ, so we may rewrite them as
– –2
v1 = I – γ I + P
– –2
v2 = I –γ I + Q.
and solve for I1, I2, and C(a2):

1

– –2
2
I1 = 1 – 1 [1 – 4γ (I – gI + P)]
2γ 2γ

1
2
– –2
I2 = 1 – 1 [1 – 4γ (I – gI + P)] .
γ
γ
2
2

Notice that ∂I1/∂γ and ∂ I2/∂γ have the same sign,
matching ∂ C(a2)/∂γ. Explicitly calculating the first
of these derivatives, we have
1
1
–
∂ I1/ ∂γ = γ [1 – 4 γ (v1)]2 [v1 – I 2].

7

F I G U R E

Increasing the Number of Actions

1

Convergence of the Sharing Rule
τ (profit share)
1

lim τ > τ min

The lowest share of risk the agent can take, τ min, is
decreasing in the gap in the principal’s payoff
between the chosen and the indifferent act,
[G(ak ) – G(aj )]/[B(aj ) – B(ak )]. It seems intuitive
that as the number of actions increases, the gap
decreases and min moves toward one, its value in
the continuous-action case. But it is possible to
work the convergence so that exceptions occur. If B
and G are continuous functions, some condition on
the difference (such as lim  ak–ak+1 = 0) would
ensure the result.

lim τ = τ min

τ min

III. Conclusion
lim τ < τ min

δ (risk aversion)

0

SOURCE: Author’s calculations.

The first two terms are positive, while the last
–
–2
can be rewritten as [I – (1 +γ )I + P ]. As γ →0, the
– –2
–
last term approaches I – I + P. For values of I that
are not too large, that term is positive, and we have
the counterexample.
In this counterexample, the agent bears all the
risk if he is risk neutral, but assumes none if he is
even slightly risk averse. In other words, convergence fails in a spectacular way. But it may fail in
more prosaic fashions as well. The limit of the riskaverse case may be higher or lower than τ min.
Figure 1 schematically illustrates these possibilities.
Mathematically, convergence fails because of a
difference between min for the risk-neutral case and
for the limit of the risk-averse case. This difference is
(14) τ min – lim τ = [G(a1) – G(aj )] / [B(aj ) –
γ →0

B(a1)] – [G(ak ) – G(aj )] / [B(aj ) – B(ak )].
Indifference at zero risk aversion implies G(a1) +
B(a1) = G(ak ) + B(ak ), creating two distinct possibilities: Either ak or a1 can be the high-cost, high-benefit
action. If B(ak ) > B(a1), then G(ak ) < G(a1), and
vice versa. The sign of equation (14), then, can go
either way.
Despite the myriad possibilities, nonconvergence
remains a special case. To start with, the principal
must be indifferent between two different actions of
the risk-neutral agent, and she must strictly prefer one
action for arbitrarily small levels of risk aversion.

The traditional solution to the risk-neutral
principal–agent problem is misleading. With finite
states and finite actions, many solutions exist, and in
all but one of these the principal bears the risk. The
traditional solution cannot even claim to be the limiting case as risk aversion decreases: In fact, it is the
solution farthest away from the limit.
These results have two main consequences. First,
they caution us against using the traditional solution
as an approximation of the less tractable risk-averse
case. This may explain why Jensen and Murphy
(1990) found CEOs bearing a surprisingly low
amount of risk. It also explains, in part, why the
numerical calculations of CEO risk in Haubrich
(1994) were so small, even for very low levels of risk
aversion. Second, they illustrate the range, power,
and tractability of Grossman and Hart’s version of
the principal–agent model.
Nevertheless, the results presented here should
be taken as preliminary—brief observations of a rare
nocturnal animal. Proposition 1 provides sufficient,
but not necessary, conditions for multiple solutions
and does not characterize all possible solutions. The
convergence results require even stronger restrictions
and depend on the two-act case. Still, I believe the
scattered sightings reported here show a surprising—
and noteworthy—aspect of the principal–agent model.

8

References
Grossman, Sanford J., and Oliver D. Hart. “An
Analysis of the Principal–Agent Problem,”
Econometrica, vol. 51, no. 1 (January 1983),
pp. 7–45.
Haubrich, Joseph G. “Risk Aversion, Performance
Pay, and the Principal–Agent Problem,” Journal of
Political Economy, vol. 102, no. 2 (April 1994),
pp. 258–76.
Holmstrom, Bengt. “Moral Hazard and Observability,” Bell Journal of Economics, vol. 10, no. 1
(Spring 1979), pp. 74–91.
Jensen, Michael C., and Kevin J. Murphy. “Performance Pay and Top-Management Incentives,”
Journal of Political Economy, vol. 98, no. 2
(April 1990), pp. 225–64.
Mirrlees, James A. “The Optimal Structure of
Incentives and Authority within an Organization,”
Bell Journal of Economics, vol. 7, no. 1 (Spring
1976), pp. 105–31.

9

Monetary Policy and SelfFulfilling Expectations:
The Danger of Forecasts
by Charles T. Carlstrom and Timothy S. Fuerst
Charles T. Carlstrom is a senior
economic advisor at the Federal
Reserve Bank of Cleveland;
Timothy S. Fuerst is a professor
of economics at Bowling Green
State University.

Introduction
Economists have long argued that the best, surest
way for a central bank to do its job is to adopt some
sort of rule and stick to it. They reason that a central
bank’s short-term objectives may be inconsistent
with its long-term goals; a rule should prevent the
bank from undermining long-range goals for the
sake of short-term results.1
Using a rule also makes a central bank’s actions
more transparent. Consider the case of a central
bank whose long-term goal is inflation stability. If
the bank doesn’t specify its long- and short-term
objectives, the public is apt to misinterpret the
bank’s actions, making inflation stability difficult to
achieve. For example, to enhance the market’s functioning over the business cycle, a central bank may
make a temporary change, which the public may
confuse with a change in the bank’s long-term
objective. If no explicit reasons for the bank’s actions
are given, public expectations about future inflation
have no moorings.
But what kind of rule is best for a central bank
that wants to achieve stable inflation? One of the
earliest, most famous proposals was Milton Friedman’s constant-money-growth rule. He argued that
the monetary authority should ignore short-run considerations altogether, because attempts to stabilize
inflation—or even output—would ultimately make

matters worse. Long and variable amounts of time
pass before changes in the money supply affect prices,
so monetary authorities cannot be sure when and to
what degree their policies take effect. This, ironically,
means that stabilization policies would potentially be
destabilizing. Milton Friedman concluded that the
monetary authority should just commit to expanding
the money supply by a constant amount every year.
The chronic, widespread instability in money
demand that has been apparent to many economists
since at least the mid-1970s weakened this position,
and the unexpected shift in money demand in the
early 1990s signaled its demise. The relationship
between money and prices seems less predictable
now than it once did. Most policymakers now recognize that a constant-growth rule will not prevent
inflation from varying substantially over short and
long periods. A growing number of central banks
recently have moved toward the idea that they should
target the inflation rate directly. For example,
Canada, the United Kingdom, Sweden, and New
Zealand have all adopted explicit inflation targets.
Inflation targeting, however, is an objective, and
the best way to achieve it remains controversial. That
is, would it be better to respond proactively to stop
inflation before it increases or to respond only after
■ 1 This refers to the advantages of using rules because of the timeinconsistency problem. See, for example, Kydland and Prescott (1977) and
Barro and Gordon (1983).

10

realized inflation has crept up? While there is no
universal agreement on the best policy rule to stabilize
inflation, central banks with inflation targets have
found it necessary to base policy changes on inflation
projections. In New Zealand’s case, these projections
are set two to three years ahead. In the United Kingdom, they have a current target of 2.5 percent and
forecast inflation two years ahead in setting policy. In
fact, the central bank of New Zealand states that its
“inflation projections relative to the inflation target
range are the critical input in the quarter by quarter
formulation of monetary policy.”2
If monetary policy is to stabilize the inflation
rate over any but the longest time horizon, then the
monetary authority must look ahead and respond
to what inflation is expected to be. This is crucial,
given the long lags between monetary policy and
price changes. Without such forward-looking,
pre-emptive behavior, the monetary authority is
repeatedly responding to past inflation shocks,
many of which are temporary, with no bearing on
future inflation. The result may be unnecessarily
wide price swings.3
Although the United States has no official price
level target, it is clearly committed to not letting inflation accelerate. Consequently, we rely on forecasts.
As Alan Greenspan commented: “Implicit in any
monetary policy action or inaction is an expectation
of how the future will unfold, that is, a forecast.”4
Inflation targets are meant to reduce uncertainty
and tie down expectations. This paper argues, however, that far from pinning down expectations, such
a policy leaves expectations completely free.5 The
danger in basing monetary policy on forecasts is that
it creates a situation in which policy depends on
expected inflation and expected inflation, in turn,
depends on policy.6 As a result, nothing pins down
either one. The “anchor” that inflation targeting is
supposed to provide may well be illusory, leaving
monetary policy (and consequently real output)
without any moorings.
The consistent use of inflation forecasts, which is
necessary with strict inflation targeting, leaves the
system vulnerable to self-fulfilling inflation expectations. The pernicious event that triggers these selffulfilling cycles is known as a sunspot. We argue that
instead of using inflation forecasts in conducting
monetary policy, thereby creating the potential for
sunspot-induced volatility, the monetary authority
should respond aggressively to past inflation.
Although inflation can never be truly stable if only
past inflation is responded to, only in this way will
monetary policy truly provide the anchor that pins
down inflation expectations.

I. Sunspots and Lack of Coordination
The possibility of sunspot-induced, self-fulfilling
expectations can arise if monetary policy depends on
what the public is expected to do, and the public, in
turn, bases its behavior on policy actions. This can
lead to the well-known problem of “infinite regress,”
in which the public’s behavior and monetary policy
affect each other in turn, and there is nothing objective on which to “pin down” either. Outcomes are
determined by each side’s beliefs about what the
other is expected to do.
A simple noneconomic example illustrates this
possibility. Suppose Chuck’s decision about whether
to attend a party depends on whether he expects
Tim to go; Tim’s decision, in turn, depends on
whether he expects Chuck to be there. Now suppose
that Tim believes Chuck will go to the party only if
it rains in Tahiti. Tim’s belief will be self-fulfilling:
If it rains in Tahiti, Tim will go to the party (because
he expects Chuck to), and so will Chuck (because
Tim is going).
Economists refer to this as “sunspot” behavior.
An event is called a sunspot if it affects some economic variable (such as inflation) only because the
public believes it does. A sunspot is therefore purely
extraneous information (for example, rain in Tahiti)
that affects behavior because it leads to a circle of
self-fulfilling expectations. If the public expects
prices to be higher tomorrow, it acts on this belief,
setting in motion a series of forces that actually
cause prices to rise.
Perhaps the most famous economic example
of self-fulfilling expectations is that of the bank runs
during the Great Depression. Because of the firstcome-first-served rule, it was in depositors’ best
interest to withdraw their money whenever they
thought the bank might be in financial jeopardy.
But if everyone thought the bank was in financial
trouble, the ensuing run on the bank would itself
cause this trouble. The reason is that much of a
bank’s portfolio is tied up in assets that cannot be
easily liquidated, so that a bank run—or even the
■

2 See Huxford and Reddell (1996).

■ 3 To quote the Bank of Canada: “There are lags of a year to 18
months or more between monetary policy changes and their effects on
inflation and the economy. A chain of events is set in motion that affects
consumer spending, sales, production, employment, and other economic
indicators. This means that monetary policy must always be forwardlooking.” See <http:\\www.bankofcanada.ca/en/backgrounds/bgp1.htm>.
■

4 Greenspan (1994).

■ 5 Because this is a rational-expectations model, we use “forecast”
and “expected inflation” interchangeably.
■ 6 Sherwin (1997) writes that because of New Zealand’s inflation
targeting, “the [central bank] is more likely to be in a position of validating
market moves, rather than driving them overtly.”

11

rumor of one—would be a self-fulfilling prophecy.
Deposit insurance was instituted to eliminate this.
Self-fulfilling expectations usually occur when
multiple players’ actions depend on each other but
cannot be coordinated. There would be no problem
if Chuck and Tim could coordinate their decision
about whether to attend the party. Similarly, lack of
coordination was crucial in the bank-run problem.
The possibility of a disastrous run on an otherwise
healthy bank would have been eliminated had
depositors been able to coordinate their actions
before deciding whether to clean out their accounts.
Knowing that others were contemplating withdrawals only out of fear that everyone else would do
so would have removed depositors’ need to withdraw their funds.

II. Self-Fulfilling Expectations and
Monetary Policy
In monetary policy, the two agents that lead to selffulfilling prophecies are the monetary authority and
the public. Such prophecies have become more likely
because central banks around the world have found
it in their best interest to operate off interest rates
(in this country, the federal funds rate). Because the
interest rate is the chosen policy instrument, the
money supply (inflation’s primary determinant) is
no longer controlled directly by the monetary authority. It is supplied at whatever level is necessary to
achieve the interest rate target. Hence, the potential
problem with this approach is that changes in public
expectations will indirectly influence money growth,
which directly affect (and may even justify) these
expectations.
Although our focus is on the response to
expected inflation, the basic problem arises under
the more extreme assumption of a pure funds rate
peg, where monetary policy promises to keep the
nominal interest rate constant. Suppose prices today
increase. This lowers real money balances, putting
upward pressure on nominal interest rates. To keep
interest rates constant, the monetary authority must
increase the money supply to accommodate the
price increases. But at the end of this cycle, real
money balances (and hence interest rates) are back
where they started.7 In this example, the central
bank’s promise to keep interest rates constant
obliged it to increase money whenever prices rose.
Thus, prices and nominal money are not pinned
down. This indeterminacy is related to the classical
statement of monetary neutrality, in which one-time
money-supply changes have no real effect, because
all dollar prices would respond by the same proportion. With an interest rate peg, the nominal money
supply is free in each period because money is supplied at whatever level is needed to keep the peg.

Sunspot events can lead to these unanticipated
changes in the money supply and prices, but they
have no real effect because there is monetary neutrality. Using sunspots is like letting a roulette wheel
determine how many zeros should be added (or
subtracted) every period from money and prices.
This classical indeterminacy is referred to as a
purely nominal indeterminacy. It affects prices and
all nominal quantities but affects neither expected
inflation nor any real variable. But nominal indeterminacy, coupled with a nominal rigidity, leads to a
situation in which expected inflation (and thus real
economic variables) is not pinned down.
Suppose that prices are fixed today having been
set one period in advance. Now consider a sunspotinduced increase in expected inflation, which puts
upward pressure on nominal interest rates (as nominal rates include an expected inflation premium).
This obliges the monetary authority to increase the
money supply in order to keep nominal interest
rates constant. Since prices are fixed today, firms
respond to increased money by raising their prices
tomorrow. This completes the circle. An increase in
expected inflation causes prices to increase tomorrow. Furthermore, because prices were fixed today,
the increase in money today would have stimulated
real output. Thus not only is expected inflation not
anchored, but real output is also without moorings.8
A pure funds rate peg, therefore, would make
money supply and prices vulnerable to random
sunspot events. In principle, nothing governs the
size of these sunspots, so prices could be quite
volatile and the costs of sunspots quite large.
Of course, the Federal Reserve does not maintain
a pure funds rate peg. Instead, changes in inflation
(whether past or future) enter heavily into its policy
decisions. The question asked in this paper is whether
it is better to be proactive and use a forward-looking
rule, raising the funds rate when inflation is expected
to increase, or to use a backward-looking rule,
responding after prices start to rise. For simplicity, we
consider rules in which policy responds only to inflation and not to output.9 We argue that only with
an aggressive, primarily backward-looking rule is
indeterminacy not a problem. This timing difference
mitigates the coordination difficulty because the
monetary authority does not “move” until long after
the public does.

■ 7 This result is due to Sargent and Wallace (1975). For a general
equilibrium analysis, see Carlstrom and Fuerst (1995), which shows that
this nominal indeterminacy becomes real in a limited-participation model.
■

8 The formal details of this logic will be spelled out below.

■ 9 Responding to future output will be similar to responding to
future inflation, making self-fulfilling expectations more likely to occur.
Responding to past output, however, will make them less likely.

12

To show this, we develop a simple economic
model,10 first considering a purely flexible-price
economy and analyzing the conditions in which
nominal indeterminacy will arise, then demonstrating that this nominal indeterminacy will become real
when prices are sticky. We conduct the analysis in a
perfect-foresight environment because, as is well
known, a necessary and sufficient condition for
indeterminacy and sunspot fluctuations in a rationalexpectations environment with shocks is for there to
be indeterminacy in the corresponding-perfect
foresight model (without shocks).11

growth process that supports these rules is endogenous and can be backed out of the CIA constraint.
Two extreme forms of this rule are α = 1, in which
the monetary authority follows a pure backwardlooking interest rate rule, and α = 0, in which the
monetary authority follows a pure forward-looking
interest rate rule.
An equilibrium for this economy (see appendix 1)
consists of equation (1) and

III. A Flexible-Price Model

(3a)

Consider a model economy consisting of numerous
infinitely lived households with preferences over
consumption, ct , and disutility over hours worked,
Lt. For simplicity, we restrict per-period utility to
U(c,T–L) = ln(ct ) – Lt , where T = total time endowment. To reflect people’s preference for today rather
than tomorrow, utility is assumed to be discounted
over time at a constant rate β ≡ 1/(1+ ρ ) < 1.
Households maximize the infinite discounted value
of per-period utilities.
Firms produce the consumption good by using
labor supplied by the household according to the
simple linear production function, ct = f (Lt ) = Lt.
To buy the consumption good, households must
first acquire money, Mt. Thus, we assume the cashin-advance (CIA) constraint, Mt = Ptct.12 The
importance of this assumption is that cash that is
being held cannot be invested where it would earn a
nominal rate of return, it. The nominal interest rate
is therefore the opportunity cost of holding money,
while the benefit of holding money comes from the
consumption it provides.
Monetary policy is assumed to operate off an
interest rate objective, where interest rates increase
whenever some average of past and future inflation
increases.

(4) Mt = Pt f (Lt ).

^

(1) it = τ [α π^ t–1 + (1 – α ) π^ t+1],
where it = it – iss, π^t = πt – πss, iss =ρ + πss,
^

where it ( πt ≡

Pt
Pt–1

–1) denotes the nominal interest

rate (inflation rate) at time t, iss ( πss) is the steadystate or long-run nominal interest rate (inflation
rate). The “hats” thus denote deviations from steady
state, and ρ is the fixed, steady-state, real interest
rate. According to the policy rule in equation (1),
the nominal interest rate increases (decreases) from
its long-term trend whenever inflation is higher
(lower) than its long-term trend. Monetary policy is
said to be aggressive (passive) if τ >(<) 1. The money

(2a)

Uc (t)
U (t + 1)
U
, where Ux = ∂
= β Rt c
χ
∂
Pt
Pt+1
UL (t)
f (t )
= L
Uc (t)
Rt

Equation (2a) is the standard Fisher equation.
The left side shows the utility lost by forgoing $1
and hence 1/P fewer units of consumption today.
The right side shows how much (in terms of utility)
is gained tomorrow by investing that dollar bill and
earning R = 1+i dollars, which buys R/P units of
consumption and provides Uc(t+1) worth of enjoyment tomorrow. On the margin, these two must
be equal.
Equation (3a) is the marginal condition for labor.
The left side shows how much utility (measured in
terms of the consumption good) one loses from working one more hour, while the right side shows the
marginal productivity of labor, or how much more
consumption one gets by working one more hour.
Because this is a monetary economy, this labor
market expression differs from the textbook condition. Defining (1–tw) = 1/R, we see that the nominal
(gross) interest rate distorts the economy just as a
wage (or, equivalently, a consumption tax) of tw
would. The nominal interest rate acts like a consumption tax because households must acquire cash
before buying the consumption good (constraint 4);
this has an opportunity cost of R in terms of forgone
interest. This, in turn, is equivalent to a wage tax,
because labor income is used to purchase consumption. This monetary distortion has an important
effect on the existence of sunspot equilibria when
one assumes that the central bank conducts policy
according to a nominal interest rate rule like the celebrated Taylor rule (Taylor 1993).
■ 10 The model contained in this paper is extremely simple, although
the results are quite general. Carlstrom and Fuerst (1999) present technical
details in more general environments.
■

11 Farmer (1999) offers a useful discussion of these issues.

■ 12 A CIA constraint implies that the interest elasticity of money
demand is zero and that the velocity of money is unity. Similar results arise
in a more general money-in-the-utility-function framework containing
today’s money balances. See Carlstrom and Fuerst (1998, 1999).

13

It will be simpler to work in log deviations, so we
rewrite (2a) and (3a), plugging in the assumed functional forms, as
^

(2b) it = π^t+1 + (c^t+1 – c^t ) and
^

(3b) it = –c^t.
Equation (2b) states that the nominal interest
rate consists of an inflation component, π^t+1, and
^
a real component. The real interest rate, it –π^ t+1,
increases whenever consumption is expected to
rise over time. Because households prefer a constant
consumption stream to a variable one, whenever
tomorrow’s consumption is expected to be greater
(less) than today’s, households will tend to borrow
(save) more to smooth out their consumption, thus
exerting upward (downward) pressure on interest
rates.
Eliminating (3b) by substituting out consumption, we obtain13
^
(5a) it+1 = π^t+1 .
This expression, along with the monetary policy
rule (1), will determine whether self-fulfilling
prophecies are possible. We first analyze the case in
which monetary policy looks ahead and show that
doing so makes it vulnerable to sunspot-induced
fluctuations.

Forward-Looking Interest Rates ( α = 0)
Suppose the central bank conducts policy according
to the forward-looking rule, α = 0 in equation (1).
Substituting this into (5a) yields

()

(6) π^ t+2 = τ1 π^ t+1.
Two observations are in order. First, expression
(6) starts with πt+1 (expected inflation between
today and next period), so that the current price
level is always free for all values of τ , that is,
P
πt ≡ t – 1 is completely free (Pt–1 is predetermined
Pt–1
by history). This condition does not affect real
behavior at this stage and is exactly the analogue of
the nominal indeterminacy with pegged interest
rates discussed above. Second, the path of expected
inflation and thus real behavior is determinate if and
only if the mapping in (6) is explosive.14 Hence,
there is real determinacy if and only if τ < 1.
Where does this indeterminacy come from? By
definition, indeterminacy results when current consumption and the real interest rate move in opposite
directions. This is true because multiple stationary
paths are possible if the paths of the endogenous
variables are not explosive (that is, c^ t+1 <  c^t  ) .

This guarantees that the real interest rate (which
equals consumption growth) will be inversely related
to current consumption. In a labor-only economy,
this suggests that indeterminacy is possible if and
^
only if the real interest rate (it – π^ t+1) and the labor
^
market distortion, the nominal interest rate, (it )
move together. By definition, this occurs when τ > 1.

Backward-Looking Interest Rates ( α = 1)
Now suppose that the central bank conducts
policy according to the backward-looking rule.
Substituting into (5) the monetary policy rule, α = 1
in equation (1), we have
(7) π^t+1 = τ π^t.
The economy is determinate if and only if τ > 1.
An aggressive (τ > 1) backward-looking rule pins
down the entire inflation sequence, including the
current πt, so that there is both real and nominal
determinacy. That is, if the monetary authority
responds aggressively to past inflation, the initial
price level (and thus the initial money stock) is also
Pt
pinned down (recall that πt ≡
– 1 so that
Pt–1
pinning down πt also determines Pt ). This contrasts
sharply with both an interest rate peg and the
forward-looking rule, where there is always nominal
indeterminacy.
The intuition for nominal and real determinacy
is as follows: Suppose a 1 percent increase in the
current price level Pt (and hence πt). The backwardlooking rule implies that next period’s nominal rate
must rise τ percent. This increase in the future
nominal rate (consumption tax) leads to an increase
in current consumption, thereby decreasing today’s
real rate. The policy rule, however, implies that the
current nominal rate does not respond to πt. Hence,
the decline in the real rate must lead to an increase
in πt+1 (to offset the drop in the real rate). This
increase will be greater than the initial increase in π t
(see equation [7] with τ > 1). Continuing down this
path would be explosive, so it is not a possible
equilibrium path.
This confirms McCallum (1981), who argues
that the monetary authority could eliminate the
nominal indeterminacy associated with interest rate
rules by having a nominal anchor. He suggests that
this could be achieved by responding to a nominal
variable. Our result, however, also shows that merely
responding to a nominal variable, such as past
■ 13 The same equation arises in a general money-in-the-utilityfunction framework containing today’s real money balances (Carlstrom
and Fuerst [1999]).
■

14 See Farmer (1999).

14

inflation, is not enough. To ensure nominal determinacy, the monetary authority must respond
aggressively to past inflation (τ > 1).
In appendix 2, we consider the more realistic case
of a mixed rule, in which the monetary authority
reacts to both past and expected inflation (α ≠ 0).
We show that to avoid nominal indeterminacy in a
flexible-price economy (and thus real indeterminacy
in a sticky-price economy) the monetary authority
must react aggressively to inflation ( τ > 1) and most
of their response must be from past inflation
( α > 1/2).

IV. Sunspots and Money Growth
Do changes in inflation correspond to different levels of money? Yes, from the CIA constraint, different values of inflation (and hence current prices)
correspond to different levels of money. That is,
nominal indeterminacy is equivalent to having more
than one money growth process to support a given
interest rate objective.15 For example, adding an
independent, identically distributed (i.i.d.) shock to
the money growth process does not affect nominal
or real interest rates.
Despite the nature of this indeterminacy, the
monetary authority is not spinning a roulette wheel
to determine money growth. The key is that expected
inflation (and hence money growth) can depend on
sunspot events whenever the central bank operates off
interest rates so that money growth is endogenous. In
effect, sunspot events matter in this model economy
because the central bank allows them to matter by
passively varying the money supply to hit the interest
rate objective in (1).
A natural criticism of the previous analysis is that
it was conducted in a flexible-price monetary model
in which only anticipated inflation had real effects.
For example, many would find it peculiar that there
can be nominal indeterminacy without any effect on
real variables. It is hard to imagine that this extra
price volatility would be of no consequence. We
argue below that nominal indeterminacy is important because, in the presence of nominal rigidity, it
becomes real indeterminacy. Expected inflation and
real activity will now be free whenever the initial
price level is free.

V. A Sticky-Price Model
Because changes in money feed directly into prices,
a flexible-price model implies that i.i.d. shocks to
money would have no effect on real variables.16
From the CIA constraint, Mt= Ptct, it is obvious that
i.i.d. monetary shocks will have real effects in the
presence of sticky prices. If prices are fixed, then
changes in money must feed into corresponding

changes in consumption, as the rest of this section
will illustrate more formally. The end result,
however, is that nominal indeterminacy is important
because it becomes real with sticky prices.
In the simple type of nominal rigidity considered
in this section, prices are fixed one period in
advance. Because prices are sticky, we must move
away from perfect competition, where firms have no
pricing power. Imperfect competition implies that
prices will no longer be equal to marginal cost;
instead, there will be a markup, 1/ z, over marginal
cost.17
Real wages will no longer be constant and equal
to one (Wt /Pt = ƒ' (Lt ) = 1). Now, workers will be
paid less than their marginal productivity
RtUL (t) Wt
=
= zt ƒ' (Lt ) = z t < 1,
Uc (t)
Pt
where z t is marginal cost (see appendix 3 for details).
Notice that marginal cost, z t, acts like a wage tax of
(1– zt ). Written in log-deviation form, (3b) becomes
^
(3c) it = z^t – c^t , where zt= ln( zt ) – ln( zss ).

Essentially, z ss is a measure of firms’ monopoly
power. The smaller becomes, the greater is the
monopoly power enjoyed by firms. Thus, in contrast to the perfectly competitive example in the previous section, where all firms earned zero profits
(zss =1), with imperfect competition firms will earn
profits ( zss < 1).
Solving for c^t and c^t+1, and substituting them
into (2b), yields
^

(5b) it+1 = π^t+1 + z^ t+1 – z^ t ,
where, as before, the “hats” denote log deviations.
Notice that when z^t = 0( z t = zss for all t), equation
(5b) collapses to the flexible-price economy of (5a).
That is, with flexible prices, zt = z ss, and imperfect
competition would have no bearing on the previous
analysis. But how is z^t determined when prices
are sticky?
When a firm sets its prices in advance, it agrees
to produce at the level needed to satisfy demand at
the fixed price (see the CIA constraint [4]). To
increase production, the firm must hire labor,
■ 15 Hence, sunspots can be avoided if the central bank specifies the
exact money growth process used to support the desired interest rate target
(see, for example, Coleman, Gilles, and Labadie [1996]) or tightly restricts
the money growth process (the well known “minimum-state vector solution” of McCallum [1983, 1999]). Both assumptions, however, amount to
moving to a money growth operating procedure.
■ 16 The fact that i.i.d. money shocks have no effect on real interest
rates is also at the heart of the nominal indeterminancy in the previous
section.
■

17 See appendix 3 or chapter 5 of Walsh (1998).

15

bidding up the real wage (and thus marginal cost).
From the labor equation, this implies that the real
wage (and thus marginal cost) will vary with the
level of production. Remember that zt arises because
of firms’ pricing power. Pricing in advance implies
that this pricing power is determined from the
^
prices currently being charged. Therefore zt is free,
except to the extent that the CIA constraint (4)
must be satisfied. Dividing the CIA constraint by
Mt–1, log linearizing, and solving for consumption
from (3c), we obtain
M
^
^
(8) zt = g^t + it where g^t = ln( M t ) – gss and gss = πss.
t–1

We have used the fact that the existing prices
(Pt ) and beginning-of-period money stock (Mt–1)
are predetermined (fixed).
It is important to notice that the firm’s marginal
cost, or markup, will be determined if and only if
both money growth, g^t, and nominal interest rates,
it , are determined. Since nominal indeterminacy
implies that money growth is not determined, this
suggests that real determinacy with sticky prices
requires that there be no nominal or real indeterminacy with flexible prices. Once money growth is
determined, marginal cost adjusts to ensure that the
CIA constraint (8) is satisfied.
A key insight in showing this is that since prices
can adjust after one period, zt+j = zss for all j ≥ 1;
but at time t, zt need not equal zss because prices are
predetermined.18 Equilibrium in this economy,
therefore, consists of two separate pieces. When
prices are sticky,
^
^
^
(5c) zt + (it+1– πt+1) = 0;

and when prices are flexible,
^

(5a) it+j – π^t+j = 0

for all j ≥ 2.

Is z t pinned down? Consider first the flexibleprice part (5a). If the flexible-price economy has
both real and nominal determinacy, then πt+1 is
uniquely determined. If this occurs, then sticky
prices (5c) imply that zt is determined. But if there
is nominal indeterminacy, so that πt+1 is not pinned
down, then zt must also be free. The money balances
that support this cycle can then be backed out of the
CIA constraint (8). Remarkably, the presence of
nominal indeterminacy in the flexible-price economy implies that expected inflation (and thus real
variables) are free in a sticky-price economy.
To understand this, consider a forward-looking
monetary policy rule. Suppose there is a sunspot
increase in expected inflation. The monetary policy
rule implies that today’s funds rate must increase in
response. To achieve this, the monetary authority

lowers today’s money growth, temporarily lowering
output (and hence consumption) and thus increasing
the real interest-rate. Given this monetary contraction, firms’ preset prices will be too high tomorrow.
Therefore, the monetary authority must increase
tomorrow’s money growth to keep the nominal rate
in a neutral position. This increases expected inflation
today, which means that the initial increase in
expected inflation was self fulfilling.
This analysis suggests that since real indeterminacy results whenever expected inflation is not
pinned down, a policy in which the monetary
authority targets and stabilizes inflation expectations
(τ = ∞ and α = 0) might be determinate. The
advantage of such a policy is that by stabilizing the
price level, the central bank stabilizes marginal cost,
so that the real economy behaves like a flexible-price
model with stabilized prices. The disadvantage is
that this policy is also subject to real indeterminacy,
as equation (5a) shows. An expected inflation target
pins down πt+1, and from (5a) determines it+1.
But it is completely free. With expected inflation
pegged, this freedom in the nominal rate corresponds to saying that the real rate (and thus real
behavior) is subject to sunspot fluctuations. Therefore, an expected inflation target will have real
indeterminacy whether prices are flexible or sticky.

VI. Empirical Relevance
At this point, the reader may ask whether this danger
is of more than academic interest. Doesn’t every central bank look at current economic conditions in
determining monetary policy? The answer is, of
course, yes, but this observation does not belie our
central point. The key issue is whether current conditions influence policy because of what they tell us
about the future or because of what they tell us about
the past. The sunspot problem arises when central
banks use past information to generate forecasts and
use these forecasts to drive monetary policy.
How robust is this analysis to more complicated
and realistic interest rate policies? Far from being a
razor-edge result, the problem that forward-looking
rules lead to indeterminacy is extremely robust to
the exact formulation of the policy rule and to the
modeling environment. Previous research suggested
the opposite, arguing that aggressive current and
forward-looking rules did not lead to indeterminacy
in more complicated sticky-price environments.

■ 18 As noted earlier, to determine whether a model economy is subject to sunspots fluctuations, it is sufficient to examine the perfect-foresight
version of the model for arbitrary initial conditions. The initial condition is
the predetermined price.

16

These models, however, ignored the transactions
role of money and did not have capital.19 Carlstrom
and Fuerst (2000) demonstrate that correcting
either of these omissions leads once again to the
results emphasized in this paper.
But if the use of forecasts is so dangerous, why are
inflation-targeting countries currently not experiencing any major problems? First, these countries have
small, open economies, so the effect from their own
monetary policy (and hence the impact of sunspots
on domestic activity) would be muted. Because of
the counterfactual comparison, it’s also impossible to
judge whether they are having problems. Yet even if
we accept that everything is currently working well
in these inflation-targeting countries, we are not dissuaded from our main point. We would like the
central bank to follow a monetary strategy that works
well under all circumstances. Inflation-forecast
targeting rules may work well some of the time,
perhaps even most of the time. But they clearly
do not work well under a wide range of conditions
and therefore should be avoided.
It is also important to note that the severity of
the problem today might not indicate its severity
tomorrow. In fact, the severity of the problem is
likely to increase over time because sunspots involve
a coordination problem. That is, the agents in the
economy must reach an implicit agreement on what
this sunspot event is and base their forecasts on that
agreement. This coordination takes time. Unfortunately, sunspots evolve because of the nature of
self-fulfilling expectations, particularly if the true
causes of inflation are not completely understood.
If some variable is thought to cause inflation, the
nature of self-fulfilling expectations is such that
incorrectly latching on to this variable will help
validate the belief that this variable causes inflation.
Suppose that either the public or the monetary
authority falsely believes that capacity utilization in
and of itself causes future inflation. Even if changes
in capacity utilization have no direct impact on
expected inflation, they nonetheless initiate a chain
of events that cause expected inflation to rise. Over
time, the belief in a direct causal connection
between the two will become entrenched because
inflation typically increases following high capacityutilization numbers. To borrow a phrase, over time
the public may learn to believe in sunspots.20 This
suggests that while sunspot behavior may not arise
immediately, it is probably only a matter of time
before it does. But there is no conceptual limit on
the size and frequency of these sunspots (or consequently on the volatility of inflation and output).

VII. Conclusion
A fundamental contribution of the last three
decades of economic research is that private-sector
expectations have an enormous influence on the
business cycle and on the effect of government
policy changes. This paper illustrates a natural
corollary: If monetary policy is based on expected
inflation, and expected inflation is influenced by
monetary policy, then there is real danger that a
forward-looking policy will worsen matters by
creating the possibility that extra uncertainty is
introduced into the economy.
To avoid indeterminacy, the monetary authority
must move aggressively against inflation. The key
is that it must react primarily to past inflation rather
than expected inflation. The basic problem with a
proactive agenda is that money growth is endogenous. A backward-looking interest-rate rule may
eliminate self-fulfilling expectations by committing
the central bank to moving future funds rates in
response to today’s price movements. This mitigates
the coordination problem because the monetary
authority does not move until long after the
public has.21
It would be a mistake, however, to conclude that
central banks should be completely backward looking. Basing policy on the future is of no consequence, or is actually desirable, if monetary policy is
firmly grounded in the past. Similarly, looking
entirely ahead is no problem in certain highly
volatile times. The difficulty arises only when the
monetary authority consistently bases the bulk of its
actions on the future.
To avoid this, the central bank should place the
most weight on past movements in the inflation rate
(or output). As long as this link between current
interest rates and past inflation is aggressive enough,
the central bank can eliminate the possibility of selffulfilling behavior. An immediate implication of this
■ 19 Clarida, Gali, and Gertler (2000) and Kerr and King (1996) are
good examples of this line of research. By “ignoring the transactions role
of money,” we mean that they assume that end-of-period money affected
the money-in-the-utility function. This is equivalent to saying that how
much money you conserve on transaction costs is determined by how
much money you leave the store with versus how much you had entering
it. See Carlstrom and Fuerst (1999) for a discussion. Benhabib, SchmittGrohe, and Uribe (2001) are also critical of the modeling structure in
Clarida, Gali, and Gertler (2000) and in Kerr and King 1996).
■ 20 See Woodford (1990). Carlstrom and Fuerst (2001) also show
that sunspots are learnable if the public has rational expectations over
anticipated inflation but the central bank must learn about the coefficients
of anticipated inflation. If both the central bank and the public are required
to learn, learnability is generally much more difficult to achieve.
■ 21 It doesn’t completely end the coordination problem because the
public’s movement is based on their expectation of the monetary authority’s
future action. This is why a backward-looking rule must also be sufficiently
aggressive to eliminate sunspots.

17

analysis is that inflation targeting over short horizons,
which necessarily involves forecasts, is a potentially
dangerous policy because it will always be susceptible
to sunspots.
Many may wish to reject this conclusion by arguing that theoretical models ignore some aspects of
reality. Surely, for example, no central bank uses a
rule as simple as the one posited above. But any
model must ignore some components of reality.
A good model incorporates the salient features of
reality and ignores the rest. An examination of the
operating procedures of many inflation-targeting
central banks leads us to conclude that one salient
characteristic of their policy rule is closely approximated by the forward-looking rule that we write
down.22 Theoretical modeling is particularly important here because only theory can shed light on the
effects of a monetary policy regime that has not
been used before. In the current context, theory has
a clear warning. Central banks may increase the
volatility of both inflation and real output in their
attempt to minimize such volatility through the use
of forecasts.

Appendix 1
The household’s maximization problem is given by
∞

t

Max ∑ β U(ct ,T–Lt ),
t=0

s.t.

Pt ct ≥ Mt–1 + Xt + Bt–1 Rt–1 – Bt

Mt = Mt-1 + Xt + Bt–1 Rt–1 – Bt – Pt ct+ Pt f (Lt ),
where Bt denotes bond holdings (in zero net supply),
and monetary injections, Xt, are assumed to be given
to households at the beginning of the period. (The
remaining notation is as in the text.) Notice that we
assume that the household makes the production
decision directly. This is without loss of generality.
In the sticky-price model (see appendix 3), it is
convenient to separate the firm from the household.
Household optimization is defined by the
binding cash constraint and the following Euler
equations:
(A1) Uc (t)/Pt = Rt β Uc (t+1)/Pt+1
(A2) Ul (t)/Pt = β fL(t)Uc (t+1)/Pt+1 .
Substituting (A1) into (A2), we have
(A3) Ul (t)/Uc (t) = fL(t)/Rt .
Given our functional forms
Rt
(A4) c1 = β
(1 + π
t

1
ct+1
)
t+1

( )

(A5) ct = 1 .
Rt
Taking logarithms and subtracting their longterm (steady-state) values yields (2b) and (3b),
where the approximations used are ln(Rt ) ≈ it and
ln(1+ π t ) ≈ π t.

Appendix 2
Plugging (5a) into (1) yields the following equation:
F = τ [α π^ t–1 + (1 – α ) π^t+1 ] – π^t.
The eigenvalues of this equation are given by

λ1 =

λ2=

■

[

1 – 1 – 4 ατ 2 (1 – α )

[

2τ (1 – α )

1 + 1 – 4 ατ 2 (1 – α )
2τ (1 – α )

1/2

]

and

1/2

]

22 We are not alone in this belief; see, for example, Taylor (1999).

18

For determinancy, both of these must lie outside the
unit circle, which occurs only if the eigenvalues are
complex. This leads us to the two following necessary and sufficient conditions for determinancy:
[1– 4 ατ 2 (1–α )] < 0
2
and 1 – [ 1 – 4 ατ (1 – α)] < 1.
2
4 τ (1 – α )2
Solving these two equations yields the conditions
discussed in the text, namely, that the monetary
authority must react aggressively to inflation (τ > 1)
and that the bulk of their response must be from
past inflation ( α >1/2).

Appendix 3
In this appendix, we consider a popular model of
monetary non-neutrality—a model with sticky prices.
We utilize the standard model of imperfect competition in the intermediate goods market,23 omitting
any discussion of household behavior because it is
symmetric with appendix 1. The sole exception is
that the firm now faces its own decision problem with
an objective of maximizing profits which are then
paid out to the representative household.
In this economy, final goods are produced in a
perfectly competitive industry that utilizes intermediate goods in production. The CES production
function is given by

sticky price is given by the solution to the following
maximization problem:

{

Pts = arg max Et–1 µ t+1PtYt

–η

( ) [ ( ) ]}
Pts
Pt

where Et–1 is the expectation conditional on time
t–1 information. The firm’s optimal preset price is
thus given by:
Pts =

[( )

η Et–1 ( µ t+1 Pt η + 1ztYt )
η – 1 E ( µ P ηY )
t–1
t+1 t t

]

In a model without preset prices, this equation
would hold at time t, and thus imply that zt = z ss.
As for production, the intermediate firm hires
labor from households utilizing the CRS production
function from before. Imperfect competition implies
that factor payments are distorted. With zt as marginal cost, we then have Wt /Pt= zt ƒ' (Lt ) = z t.
Coupling this condition with the household
optimization conditions yields the labor market
condition utilized in the text.

1

Yt = { ∫ [yt (i) (η –1)/ η ] di} η /( η –1)
0

where Yt denotes the final good and yt(i) denotes the
continuum of intermediate goods, each indexed
by i ∈ [0,1]. The implied demand for the intermediate good is thus given by
Pt (i) –η
Pt
where Pt(i) is the dollar price of good i, and Pt is the
final goods price.
Intermediate goods firm i is a monopolist producer of intermediate good i. (We henceforth omit
the firm-specific notation as all firms are symmetric.)
The intermediate goods firm is owned by the household and pays its profits out to the household at the
end of each period. Because of the cash-in-advance
constraint on household consumption, the firm discounts its profits using µt+1 ≡ β Uc(t+1)/Pt+1, the
marginal utility of $1 in time t+1. Therefore the
yt (i) = Yt

Pts
– zt ,
Pt

[ ]

■

23 See, for example, Chapter 5 of Walsh (1998).

19

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