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Working Paper 9601
DYNAMIC COMMITMENT AND IMPERFECT POLICY RULES
by Joseph G. Haubrich and Joseph A. Ritter
Joseph G. Haubrich is a consultant and economist at the
Federal Reserve Bank of Cleveland. Joseph A. Ritter is a
senior economist at the Federal Reserve Bank of St.
Louis. The authors wish to thank Rob Dittmar for
programming assistance and Avinash Dixit and seminar
participants at North Carolina State University, the
University of Iowa, the University of Pittsburgh, and the
Federal Reserve Bank of Atlanta for helpful comments.
Working papers of the Federal Reserve Bank of
Cleveland are preliminary materials circulated to
stimulate discussion and critical comment. The views
stated herein are those of the authors and are not
necessarily those of the Federal Reserve Bank of
Cleveland or of the Board of Governors of the Federal
Reserve System.
February 1996
Abstract
Examining the dynamics of commitment highlights some neglected features of time
inconsistency. We modify the rules-versus-discretion question in three ways: 1) A
government that does not commit today retains the option to do so tomorrow; 2) the
government's commitment capability is restricted to some class of simple rules; and
3) the government’s ability to make irrevocable commitments is restricted.
Three results stand out. First, the option to wait makes discretion relatively more
attractive. Second, the option to wait means that increased uncertainty makes
discretion more attractive. Third, because the commitment decision takes place in
"real time," policy choice displays hysteresis.
I. Introduction
In recent years, there has been a drift in economists' way of thinking about
policy rules versus discretion. Beginning with Kydland and Prescott (1977), a
theoretical presumption has developed in favor of rules, which allow outcomes
otherwise precluded by strategic behavior. This contrasts with the early monetarists,
who proposed simple rules because the monetary authority could not handle the
complexities of the actual economy. The theoretical case demonstrates how the
precommitment to fully state-contingent rules solves the time inconsistency problem
and is superior to discretion. In this paper we modify the rules-versus-discretion
question in three important ways: 1) A government that does not commit today retains
the option to do so tomorrow; 2) the government's commitment capability is limited to
a class of simple rules; and 3) the government’s ability to make irrevocable
commitments is restricted.
The first of these modifications raises the possibility that the government might
delay committing to a rule, with the outcome of the decision depending on the current
state of the world and, in our most general model, on history. In this sense it becomes
important that the decision be made in "real time." A broader implication, also
unrecognized in the previous literature, is that choosing discretion today has an option
value, since the government may still choose rules in the future. Previous work
considers a once-and-for-all choice between rules and discretion and does not allow a
1
future government to adopt rules. If the option to wait indeed has positive value--as
such options often do--it adds to the desirability of discretion.1
Complexity makes commitment more difficult. Conceiving, specifying, and
committing to all possible contingencies becomes prohibitively costly, if possible at
all. No one is surprised that the Federal Reserve Act of 1914 discusses neither Internet
cash nor monetary policy in the event of an oil crisis. This leads to our second
modification, that policymakers must choose not between discretion and optimal statecontingent rules, but between discretion and comparatively simple and imperfect rules
(as recently emphasized by Flood and Isard [1989] and Lohmann [1992]). Thus, it is
logically possible for policymakers to "regret" their commitment to a rule. Regret
makes questions of delay interesting, because a government would never delay
committing to a rule that was always better than discretion in every state.
Our third modification stems from the observation that governments cannot
make irrevocable commitments. Nevertheless, they do have a wide array of
commitment mechanisms, from campaign promises to constitutions. These
mechanisms involve a range of commitment costs (from zero to high) and, more
important, a range of reneging costs.
The remainder of this paper develops these themes in two main variations. In
section II, we explore what happens when governments can and cannot commit to
1
Our exploration of regret and the associated option value of waiting distinguishes this paper from
similar efforts, such as Cukierman and Meltzer (1986) and Flood and Isard (1989). Cukierman and
Meltzer discuss flexibility, but do not consider imperfect fixed rules and a fortiori miss the associated
option value. Admittedly, for some models (Barro and Gordon [1983] for example) the optimal rule is
simple, but generally the optimal state-contingent rules are rather complex. The option value of
waiting, of course, plays a key role in the analysis of irreversible investment (see Pindyck [1991] or
McDonald and Siegel [1986]).
2
fully state-contingent rules, using the standard simple model of monetary policy
traditional in the time-consistency literature. In section III we extend the model to
many periods and drop the assumption that commitment is a once-and-for-all decision.
We trace the consequences when only "simple" rules are feasible and when choosing
discretion today does not rule out choosing commitment in the future. Commitment,
however, once made, remains irrevocable. Some numerical examples explore the
significance of the results.
Section IV removes the rigid and unrealistic assumption that irrevocable
commitment is feasible. It provides a very general way of thinking about policy,
allowing costly commitment with costly reversal. In it we illustrate how decisions to
commit or renege depend on the commitment and reneging costs and on uncertainty in
the environment.
In section V, we conclude by emphasizing three general results. First, the
option to wait, which we have restored to the policymaker's decision problem, makes
discretion relatively more attractive. Second, the option to wait means that increased
uncertainty makes discretion even more attractive. This is the “bad news principle” of
irreversible investment applied in a policy context. Third, by allowing the
commitment decision to take place in “real time,” we find that the policy choice
process displays hysteresis; which policy is in force at a given time depends on
history, not just the prevailing state.
II. Optimal Rules, Simple Rules, and Discretion
3
Most of the debate about rules versus discretion has taken place in the arena of
monetary economics. We continue this tradition, and in this section set out our model.
Though slightly specialized to highlight the main points, it derives from a fairly
general framework based on Flood and Isard (1989). We first use the model to
explore the distinctions between monetary policy under optimal rules, simple rules,
and discretion.
A. Basic Specification
The growth of base money, bt, relative to a velocity shock, vt (ignored
hereafter), determines the inflation rate, t:
(1)
t
= bt + vt .
Output depends on unexpected inflation, with the Federal Reserve focusing on the
deviation of output from a natural level. Because of distortions (for instance,
unemployment insurance or imperfectly clearing labor markets, depending on your
preferred ideology), that natural level may not be socially optimal.
Policymakers wish to minimize a social-loss function that reflects both output
deviations and inflation:
(2)
Lt = (bt − Et −1 bt − K + ut ) 2 + abt 2 .
The term bt − Et −1bt measures the unexpected base growth (or unexpected inflation),
K measures distortion (or the divergence between the natural level of output and the
socially optimal level), and ut, i.i.d with Eut = 0 , measures the productivity shock.
The parameter a measures the relative weight given inflation--as opposed to output-deviations.
4
The first step in finding the optimal policy is to minimize the loss function, Lt,
under both rules and discretion.
B. Discretion
From the first-order conditions for minimizing (2), we find
(3)
bt = (
1
)( Et −1bt + K − ut ) .
1+ a
This implies
(4)
Et −1
t
= Et −1bt =
K
a
As in Barro and Gordon (1983), the distortion term K determines the inflationary bias
of discretion. Actual base growth under discretion is
(5)
btD = −
ut
K
+ .
1+ a a
From this, we can calculate both the expected and realized social loss using equation
(2).
(6)
1+ a
Realized Loss: L =
a
(7)
Expected Loss: E t −1 LDt =
D
t
2
a
− K + 1 + a ut and
1+ a 2
a
K +
a
1+ a
2
u
.
The first term of equation (7) is the loss from the inflation bias of discretion, while the
second is the loss caused by output variance, some of which shows up in the inflation
rate via monetary policy.
5
C. Rules
Suppose the money supply, bt, cannot respond to ut, thus restricting the
policymaker to rules that are not state contingent. Then money only causes inflation;
it cannot reduce output variance. The best such rule sets bt = 0 in all periods. This is
the optimal rule without state contingency. If it were feasible, a better rule (which we
derive below) would let the base react to productivity shocks, but would avoid the
inflationary bias of pure discretion.
For the simple rule setting bt = 0 for all t, we can substitute into the loss
function.
(8)
Realized Loss: LRt = (ut − K ) 2 and
(9)
Expected Loss: Et −1 LRt = K 2 +
2
u
.
Equations (8) and (9) show that the rule has a lower inflation bias than does discretion,
but a higher output variance.
Discretion is better than the simple rule when LD − LR < 0 . Substitution from
equations (6) and (8) shows that this is the case when
(10)
u 2 > 1 +
1 2
K .
a
Notice that discretion is preferable in extreme times (that is, for large uts), when
the costs of shocks are especially high.
As inflation costs (a) increase, discretion is preferred in more and more states.
This may seem counterintuitive, but in fact it makes sense. Consider, for example, the
case of ut = 0. For the simple rule setting bt = 0, the loss due to inflation is 0. For
6
2
K
K2
discretion, the corresponding loss is a =
. As a increases, this cost
a
a
decreases. Because discretion weighs the inflationary costs of intervention, higher
inflation costs reduce the inflation bias of discretion. In the limit, with inflation
infinitely costly, discretion involves zero inflation.
Similarly, as K increases, discretion is preferred in fewer states. As the
distortion worsens, the inflation bias rises and it becomes worthwhile to sacrifice
discretion in favor of a rule. The relative return to the rule increases because the
higher distortion increases the inflation bias.
If the government can commit to a state-contingent rule, it can replicate
discretion's offset to productivity shocks while simultaneously eliminating the
inflationary bias. When feasible, this rule would let the monetary base react to
productivity shocks but avoid the inflationary bias of pure discretion. In our simple
model, it is possible to find this optimal rule. Its form illustrates several points about
the relationships among optimal rules, simple rules, and discretion.
To find the optimal state-contingent rule, we minimize the expected loss
function from equation (2):
2
n
min EL = ∑i =1 bi − ∑ g j b j − K + ui + abi2 gi ,
j =1
n
where gi denotes the probability of state i, bi denotes money growth in state i, and n
denotes the number of states.2 We end up with:
2
Nothing essential depends on using a discrete probability distribution. A continuous distribution
leads to identical results, but necessitates needlessly cumbersome notation.
7
(11)
bi = −
1
ut .
1+ a
Substituting into the loss function, we have
LFt =
a
a
u2 − 2
Ku + K 2 and
1+ a
1+ a
(12)
Realized Loss:
(13)
Expected Loss: Et −1 LFt =
a
1+ a
2
+ K2.
To understand the implications of fully state-contingent rules, it is important to
look at several relationships. Since the optimal rule ties the policymaker's hands, it
allows actions that would otherwise fall victim to time inconsistency. This makes the
optimal rule better than discretion. Second, by construction, the optimal rule
dominates a restricted or simple rule in expected value. In the current example, the
optimal rule turns out to be linear. Our simple rule is trivially linear: a constant
bt = 0 .
A tangential but important point is that while the optimal rule dominates both
discretion and the simple rule on average, it does not do so in all states. In some states
the other policies do better, as a comparison of equations (6), (8), and (12) shows. For
example, in states where 0 < ut < 2K, the simple rule does better because the optimal
rule’s response to ut is not worth the (small) amount of inflation that ensues. A rule
that attempts to exploit this inefficient response, however, changes expectations in a
way that on average hurts more than it helps. To illustrate, suppose we attempt to
revise the optimal rule by setting bt = 0 whenever 0 < ut < 2K. This lowers expected
inflation but increases the loss in states where state contingency is useful. The gain in
states where 0 < ut < 2K is offset by the loss in other states, even though policy is
8
unchanged in those other states. The response of individual behavior (in this case
expectations) distinguishes an equilibrium problem from a simple control problem.
III. Waiting to Commit
The approach we take only begins to differ from the standard approach in a
more dynamic setting. First we turn to a simple numerical example that helps to
clarify the notions of regret, delay, and option value.
A. A Two-Period Example
The simplification begins with the productivity shock, assuming ut is i.i.d. and
equals -x, 0, or +x with probabilities g1, g2, and g3.
Now, suppose Alan Greenspan wakes up and finds that today, u1 = 0. If he
says, "I commit," then the two-period social loss function is
(14) VR (0) = K 2 + g1 ( x 2 + 2 Kx + K 2 ) + g3 ( x 2 − 2 Kx + K 2 ) + g2 K 2 .
The first term, K2, measures the loss today, while the following three terms measure
the next period's expected loss.
If the Chairman chooses discretion today, the loss function becomes more
complicated because he may commit tomorrow, depending on the state:
(15)
VD (0) =
1+ a 2
K + g1
a
a x 2 + 2 Kx + 1 + a K 2
1+ a
a
1+ a 2
a 2
K + g2 K 2 .
+ g3
x − 2 Kx +
1+ a
a
9
Again, the first term is the loss today. The terms containing g1 and g3 are the
expected loss when the government chooses discretion next period, as it does when
u = ± x . The last term, g2K2, is the expected loss when the government commits to
rules tomorrow, as it does when u = 0.
Choosing rules or discretion comes down to comparing equation (14) with (15)
and then choosing the strategy with the smaller expected loss.
(16)
VR (0) − VD (0) =
−1 2
1 2 1 2
K + (g1 + g3 )
x − K .
a
a
1 + a
Removing the option value, that is, forcing the government to make an irrevocable
choice between rules and discretion, leads to a different expression because in the
value of discretion forever, VDF(0), the g2K2 term in (15) is replaced with g2
1+ a 2
K .
a
This makes the difference between rules and discretion forever:
(17)
VR (0) − VDF (0) =
-1 2
1 2 1 2
1
K + (g1 + g3 )
x − K − g2 K 2
a
a
a
1 + a
= VR (0) − VD (0) − g2
1 2
K .
a
For a range of cases, VR (0) − VD (0) > 0 and VR (0) − VDF (0) < 0 , so that correctly
valuing the option leads to choosing discretion, while ignoring it leads to choosing
rules. Taking account of the real-time aspect of decision making and properly valuing
the waiting option can reverse the policy decision. As an example, let
a = 1, x2 = 6K2, and g1 = g3 = 1/4.
10
Then V (0) − VD (0) = − K 2 +
R
VR (0) − VDF (0) = − K 2 +
1
(3K 2 − K 2 ) = 0, and
2
1
1
1
(3K 2 − K 2 ) − K 2 = − K 2 < 0.
2
4
4
This example shows both that the option can reverse the normal presumption of the
superiority of rules and that the option value may be sizable. Using equation (14), the
loss from adopting rules is 5K2, making the option difference 1/4K2, or 5 percent of
the total value.
This example also illustrates the bad-news principle. It shows why
policymakers may be correct in focusing on the problems of committing at an
inappropriate time. In (16), a mean-preserving spread in the distribution, say a
decrease in g2 and corresponding increase in g1 and g3, increases the relative
attractiveness of discretion. Furthermore, it does so in a particular way. What counts
in (16) is the effect in states 1 and 3, where we choose discretion instead of rules. The
benefit arising in state 2 does not appear. What matters is the loss in states with big
shocks (states 1 and 3) that are bad for rules.
11
B. Many Periods
Adequately capturing irreversibility requires a number of adjustments to the
model. First, it clearly needs more than one period. Second, to better focus on the
problems of regret, it is also helpful to revise the within-period time structure. In what
follows, we let the government observe the shock before the public does and before it
chooses to commit. The new time line, which leaves equations (1)-(12) intact, is as
follows:
Government sees ut → Government decides whether to commit, announces →
Economy revises expectations Et-1bit → Government chooses bt →; Economy sees ut;
production.
The contrived aspect here concerns observing the shock. After seeing today's
shock, the government chooses rules or discretion, but the public does not see ut until
much later. Some variant of this assumption appears in much of the literature. In
Cukierman and Meltzer (1986), for instance, the government has information on a
state variable that the public observes one period later. In Canzoneri (1985), the
government observes (perhaps noisily) a random disturbance that the public cannot.
In general, this new timing sequence will change the public's behavior. Seeing
what action the government takes provides information about the unseen shock to the
economy. In our specific model, however, the quadratic loss function and the
symmetry of the shocks mean that the public cannot extract useful information from
12
the government's decision to commit or not. People can infer the size, but not the sign,
of the shock, so that E(ut | government choice) = 0 and E(bt | government choice) = 0.3
Once the government chooses a simple rule, it must stick with that decision
forever, in effect setting bt = 0 permanently. By contrast, choosing discretion today
does not prevent choosing rules tomorrow.
In this setting, irreversibility introduces an option value whose worth is nonnegative.4 With a simple, non-state-contingent rule, regret exists. For example, the
government might regret committing to zero inflation and wish for discretion. This
point does not depend merely on the rule's extreme simplicity. The analysis holds
even with a more sophisticated state-contingent rule, as long as there are some states in
which discretion is preferred. As mentioned before, in some states the government
would even regret committing to the optimal state-contingent rule.
With many periods, policy choice comes down to comparing possible courses
of action. This is most naturally done using dynamic programming (see Ross [1983]).
For any policy (that is, for any set of bt choices by the government, denoted ), we have
a value function
3
This symmetry breaks down if we compare the optimal state-contingent rule with discretion. This
happens because the states where discretion is preferred is not symmetrical around zero: the
government’s decision with regard to commitment would give information to the public, who would
then update their expectations. The increased complexity adds to the signal extraction problem,
without any corresponding gain in economic content. This is one reason why we do not pursue the
comparison in this paper. The other, more important reason is that we consider simple, non-optimal
rules more realistic.
4
Our definition emphasizes the option as an option to commit. An alternative, complementary,
approach emphasizes the option as an option to undo a commitment (See Bernanke, 1983). The
difference in perspective explains why the bad-news principle and the option value look different, even
though both are aspects of the same phenomenon. Is it best to compare the optimal policy with
discretion forever (as we do) or with rules? The answer depends on what is most convenient for the
problem at hand.
13
∞
Vp (ut ) = E ∑
s
L(bt+s ,ut+s ).
s =1
Here the factor
discounts the future. To rule out reputational equilibria, we restrict
ourselves to nonrandomized policies and to those that depend only on today's shock
and whether or not the government has committed in the past. The government begins
this period by observing ut. If it chooses to commit to zero inflation (the optimal
simple rule), the loss is
(18)
a)
VR (ut ) = ( − K + ut ) 2 + E VR (ut+1 ) , from which we arrive at
b)
VR (ut ) = (ut − K) 2 +
1−
(K 2 +
2
u
),
where VR (u) denotes the value function for rules. The first term measures today's loss,
and the second gives the expected value of the problem tomorrow. Choosing
discretion forever yields a loss of
1 + a
a
ut + EVDF (ut +1 )
(19) a) VDF (ut ) =
−K +
a
1+ a
2
1 + a
a
b) VDF (ut ) =
ut +
−K +
a
1+ a
1−
2
1 + a K 2 + a
a
1+ a
2
u
.
The standard time-consistency literature, making the choice before any shocks
are observed, asks whether rules are better than discretion by comparing the expected
values of (18) and (19). The general case is more complicated because opting for
discretion today leaves the door open for choosing rules tomorrow. The loss to
choosing discretion today is
1 + a
a
(20) VD (ut ) =
ut + EVD (ut +1 ).
−K +
a
1+ a
2
14
Two different representations of EVD (ut ) turn out to be useful. Without any
simplifying, we can express this term as
∞
(21) EVD (ut ) = ∑
s =1
s
2
∑ g j (u j − K) +
j ∈URt+s
2
1 + a
1
,
−
+
g
K
u
∑ j a
j
1 + a
j ∉URt+s
where gj is the probability of state j and URt+s contains the period t+s states in which
the government Uses a Rule. Here, URt+s depends on history; that is, commitment to a
rule implies commitment in all future states.
Simplifying this expression takes a little work. First, note that the set of states
in which the government chooses to Commit to Rules, CR, does not vary with time.
(This differs from URt in equation [21], where prior commitment does change the
action. URt answers the question, "At time t, in which state does the government use
rules?" CRt answers the question, "At time t, given that it can still choose, in which
states does the government commit to rules?") The time invariance of CR follows
from the simple form of equation (20). Then, recursively using equation 18(b) yields
(22) EVD (ut ) =
∑
j ∈CR
+ ∑ gj
j ∉CR
g j (u j − K) 2 +
1−
2
∑ g j (u j − K) +
1−
j ∈CR
(K
2
(K 2 +
+
2
u
) +
2
u
2
1 + a
a
) + ∑ g j
u j
−K +
1+ a
j ∉CR a
∑
j ∉CR
2
1+ a
a
u j +....
g j ∑ gi
−K +
a
1 + a
j ∉CR
The first term is the expected loss if we enter a state in which we choose rules
and adhere to them forever. The second term represents the loss today from using
discretion today only. The third term gives the loss from choosing rules in the period
after discretion. The fourth term gives the loss from choosing discretion again, with
this pattern repeating recursively.
15
Equation (22) simplifies to
(23)
EVD (ut ) =
1
2
∑ g j (u j − K) +
1−
j ∈CR
1 − ∑ gj
j ∉CR
(K 2 +
2
u
)
1+ a
a
u j ) 2 .
(− K +
+ ∑ g j
1+ a
a
j ∉CR
Finding the value function puts us in a position to examine the central issues of
regret, option value, and delay. Of course, different parameters can make rules or
discretion the better choice, but of interest here is what is unique to our model. To this
end, we focus on parameter values for which an irrevocable choice between rules and
discretion would favor rules. We then show that the possibility of future commitment
can make discretion today preferable, noting the importance of regret in that decision.
This increase in the attractiveness of discretion induces the government to choose
discretion in more states, a policy shift perhaps best interpreted as a delay in
commitment.
To rule out the trivial cases, we need some regret, so that simple rules do not
dominate discretion in every state of the world. If the loss from rules is less than the
loss from discretion in every state, it makes no sense to delay commitment or to
choose discretion. To have any regret, it must be that for some (but not all) shocks u,
1+ a
1
(u − K) >
u . We also want rules to do better in expected value
−K +
a
1+ a
2
2
terms than discretion forever, or else discretion forever is the obvious trivial choice.
This requires K 2 +
2
u
<
1+ a 2
a
K +
a
1+ a
2
u
16
, or
(24)
K2 >
a
1+ a
2
u
.
The problem for the government at t = 0 is to decide between
VR (uo ) = (uo − K) 2 +
(25) Rules:
1−
(K
2
+
2
u
)
and
1+ a
1
(26) Discretion: VD (uo ) =
uo + EVD (u1 ),
−K +
a
1+ a
2
where EVD (u1 ) is given by equation (21) or, equivalently, (23). It is also important to
know how VD (u0 ) and VR (u0 ) compare with discretion forever, VDF (u0 ) (given in
equation [19]).
Since VR (u) − VD (u) = [VR (u) − VDF (u)] − [VD (u) − VDF (u)] , we can see that
VD (u) − VDF (u) gives the option value of discretion -- the value of the ability to
abandon discretion and commit to rules. Since this quantity is ordinarily positive, we
can have VR (u) − VD (u) > 0 , even when VR (u) − VDF (u) < 0 (the standard criterion).
Note that since discounted future losses are lower for discretion,
EVD (ut ) <
EVR (u) , (because at worst, the discretion regime could commit next
period and attain equality) the government sometimes chooses discretion in states
where the one-period return favors rules. This conceivably could create a paradox
whereby we delay choosing rules forever, even though we prefer pure rules to pure
discretion. Actually, we can use equation (23) to demonstrate that this never occurs in
the case of irrevocable investment. Suppose the government never commits, so that
17
CR = ∅. Then (23) reduces to
1
1−
1 + a K 2 + 1
a
1+ a
2
u
which, from (19b), we
know is EVDF (ut ) . We assume, however, that discretion forever is worse than rules.
Along with eliminating such an "infinity paradox," the above calculation has
another implication. The government commits with a fixed positive probability in
each period, so with probability 1, the government eventually commits (by the BorelCantelli lemma).5
C. Numerical Example
To add a small degree of realism, the next example employs the infinitehorizon model, using parameter values we believe to be at least of the right order of
magnitude. While it cannot be called a test, nor even a calibration exercise, we try to
use plausible values for the effect of unanticipated money and the distribution of
output shocks. In this scenario, the government chooses discretion in about half the
states.
First differences of log GDP look somewhat like a standard normal. We
therefore assume that u is drawn from a discrete distribution that approximates a
normal. (For details, see section IV.) We choose a K value of 1.0, indicating that
long-run output differs from the socially optimal rate by 1.0 percentage point.
Following Barro (1987,
p. 469), we make the assumption implicit in equation (2)
that a 1 percent rise in money above expectations increases output by 1 percentage
point.
5
For a very different view of commitment problems using similar stochastic commitment techniques,
see Roberds (1987).
18
Two more parameter choices will fully specify the problem. Give inflation
twice the weight of output in the social-loss function, choosing an a of two. Next, set
, the discount factor, to 0.95. We think of the policymaker as choosing between rules
and discretion once a year.
19
Figure 1 shows the results of this example using these parameters.6 The top
panel plots the difference between VR(u) and VD(u), or between the value of
committing to rules and adopting discretion in a given state. Since we use a loss
function, a positive value means discretion is better, and a negative value means rules
are better.
Notice that for any u shock between -1.02 and +1.02, the social loss from
discretion exceeds that from rules. Consequently, the monetary authority should
commit to rules. For larger shocks, the monetary authority should choose discretion.
For 30.
1 percent of the time, discretion is preferable to rules.
The bottom panel shows the importance of considering option value. If we
compare using rules forever with using discretion forever, we would choose rules in
every state. The possibility of future commitment and its associated option value
changes discretion from a dominated policy to one preferred in a majority of states.
Another perspective is the "delay probability," or the expected time until a
commitment is made. For example, if we interpret each decision as a yearly meeting
date, the probability that the policymaker will go five years without committing to
rules is (1-0.699)5 = 0.0025. The independent nature of the shocks in this example
means that even though commitment is chosen in fewer than half the states, the
probability of ending up in those states at least once increases rapidly. In other words,
we hit the "absorbing barrier" quickly.
6
The most straightforward way to produce figure 1 is to use equation (23). We instead used a more
general approach described in the next section.
20
There are really two vantage points on these numbers. One stresses the large
number of states where the government prefers discretion. The other stresses the short
horizon until commitment. A model with serial correlation would tend to reconcile the
two vantages, because it would keep the economy in states with discretion for a longer
time. Without having incorporated this into the formal model, though we do not hold
obdurately to this point.
This model, too, illustrates the bad-news principle. A mean-preserving spread
makes discretion preferable in more states. Increasing the variance of the distribution
by 10 percent reduces the commitment region to the range of -0.957 to +0.957, so that
rules are adopted only 64.6 percent of the time. The probability of delaying for five
years rises to 0.0055.
IV. Entering and Exiting Commitment
The obvious impossibility of inescapable commitment (recently emphasized by
McCallum, 1995) calls for a sophisticated approach to modeling commitment, not an
abandonment of the insights generated by the time-inconsistency literature. The model
so far has allowed only simple, inescapable commitment. We now generalize this,
allowing the policymaker to enter and exit commitment (or, more generally, any policy
regime) at a cost.
Mechanisms forcing governments to commit irrevocably are almost impossible
to imagine. It is not difficult, however, to think of mechanisms that make it costly for
a government to alter its policy. A constitutional amendment, for example, is difficult
to put into place and difficult to repeal. Ordinary legislation has lower costs at both
21
ends. Governments can, in effect, tie their hands loosely or tightly, but can always
escape if they have the will to bear the corresponding levels of pain.
It is important to understand that entry and exit do not destroy the possibility of
commitment. Once the policymaker commits in time period t , the rule is in effect for
that time period at least. Another decision is made at t + 1. Likewise for discretion. In
this discrete time framework, we don’t allow shifts in midstream, between FOMC
meetings, or when Congress is out of session. Thus, commitment can tie the hands
and reduce the possible choices of the policymaker long enough to influence the
public’s expectations.7
We maintain the traditional semantics of commitment and discretion, but we
wish to highlight a bias in tone that creeps into the discussion when commitment is not
irrevocable. This innovation forces us to words like “renege” and “weasel,” although
they have clear negative connotations that we consider unfortunate. We interpret the
results of this section as a model of optimal behavior and tolerate the terminology only
to fit our paper into the literature on rules and discretion. The terminology does have
one advantage, though: The emotional evocation reminds us of strategic aspects of the
problem that might otherwise get overlooked in the formalism.
Thinking about the problem as entering or exiting commitment deepens the
analogy to irreversible investment. Our extended model now resembles an extension
of the irreversible-investment model, namely, Dixit's [1989] model of firm entry and
exit. For many of these questions, the continuous-time approach set forth in Dixit and
7
This does not exclude the possibility that the commitment cost may be some sort of bond posted for
credibility. But the foregoing analysis does say that if another commitment technology exists that does
not require such a costly bond, it benefits the economy.
22
Pindyck (1994) generally proves more convenient. Rigorously formulating questions
of time inconsistency, however, brings up serious difficulties in the theory of
stochastic differential games. This is particularly true of the monetary-policy question,
because the unanticipated-money model does not easily generalize to continuous time.
Fortunately, the discrete-time approach, though less elegant, suffices for many
important problems. In this we follow Lambson [1992], who used it to model entryexit decisions.
A. Model Solution
We modify the model of section III by adding costs for entering and exiting
commitment. A policymaker committing to rules in period t pays a cost C. Once
committed, a policymaker may on renege -- or “weasel out” of -- rules and return to
discretion by paying cost W. The problem becomes finding the boundaries where the
policymaker switches between discretion and rules. The model produces four
boundaries: an upper and a lower boundary for moving from rules to discretion, and
an upper and a lower boundary for moving from discretion to rules. With i.i.d shocks,
the zero mean of the u shocks and the quadratic loss function conspire to produce a
rules region centered on zero. As before, small shocks imply that the government
chooses (or stays with) rules, while large shocks imply it chooses (or stays with)
discretion.
23
The optimal policy switches between the two quadratic loss functions with cost
C of committing to rules, that is, of moving from discretion to rules, and cost W (for
weaseling) of moving from rules to discretion.8
To solve the infinite-horizon model with switching costs, we use a discrete
state-space approach. The shock ut is a Markov chain with n states
i
. The
probability of transition to state j from state i is
gij =
(
∑
n
k =1
j
(
)
k
)
where
is the normal density function with mean 0 and variance
2
. This produces a
Markov chain that is similar to white noise with normal innovations. We set the range
of possible states to include 6
2
on each side of 0.
The policymaker is faced with a problem that has two state variables, ut and
current value--rules of discretion--of a policy variable. Thus the value function for this
problem is an n × 2 matrix where the columns correspond to rules and discretion.
Denote the columns by V R and V D . To solve the model, we choose an initial value
function and iterate on the following mappings:
V D ( i ) = min {LD ( i ) + E[V D ( )|
C, NC
i
], LR ( i ) + C + E[V R ( )|
8
i
]}
Allowing weaseling adds a component similar to the “escape clause” models of Flood and Isard
(1989) and Lohmann (1992), who consider a cost to renege. In one sense, we generalize those models
by allowing a positive cost of recommitment and allowing delay in recommitment. In another sense,
those models are more general in that they use more general state-contingent rules. Such rules can be
embedded in our dynamic framework. We consider simple rules in order to focus on the dynamics.
24
V R ( i ) = min {LR ( i ) + E[V R ( )|
W, NW
i
], LR ( i ) + W + E[V D ( )|
i
]} .
Since the distribution of u is discrete, the following rules determine the regime
switching points. Thesuch
upperthat
commitment boundary,
V R(
C
) + C ≤ V D(
C
C
) + C ≤ V D(
C
, is the largest
).
The lower
such that
commitment boundary,
V R(
C
C
, is the smallest
).
such that
The upper weasel boundary is the smallest
V D(
W
) + W ≤ V R(
W
).
such that
The lower weasel boundary is the largest
V D(
W
) + W ≤ V R(
W
).
To solve the model in the case of irrevocable commitment at zero cost, we set C to
zero and W to an extremely large number.
B. Regime Switching
The actual numeric solutions are less interesting than the comparative statics.
Starting from a baseline of: K = 1, a = 2,
= 1, C = W = 1,
= 0.95, figures 2 to 5 depict
the solutions as we vary parameters one at a time. The state space has 401 nodes
evenly distributed from −6
2
to +6
2
.
Figure 2 plots the commitment and weasel thresholds as the commitment cost
changes, keeping the weasel cost fixed at 1. Notice that for any particular commitment
25
cost, the Fed adopts rules for “small” shocks on either side of zero, as it did in the
discrete time model. For larger shocks the Fed adopts discretion. This is a natural
consequence of the quadratic loss function.
The probability of being outside the area where rules are better for the current
period does not change as C increases. Thus, as the cost of committing to rules
increases, the range over which the policymaker is willing to commit shrinks. It will
disappear altogether if C is high enough; the option to commit is worthless if its
exercise price is too high. Thus positive commitment cost destroys the result that
commitment will happen in finite time with probability one.
Another prominent feature is that the weasel boundary is further out than the
commit boundary. Were there no cost of switching between regimes, the boundaries
would be the same, at LD (u) = LR (u) , where the expected loss from continuing
discretion just matches the expected loss from using rules. Adding a commitment cost
drives a wedge between the two value functions, and requires that the policymaker
gain even more from rules. This means moving the boundary further into the area
where rules are better, i.e., closer to zero. Similarly, a cost to backing out of rules
(weaseling your way out) means shifting the boundary even further into the area where
discretion is preferred, i.e., away from zero.
Figure 2 shows that as the cost of commitment increases, the Fed is less likely
to commit. As the cost increases, the relative benefits of rules over discretion must
also increase, and so the commitment boundary shrinks towards zero. For high
enough cost, commitment never occurs.
26
Figure 3 highlights a key point not easily noticed in section III’s model;
namely, the importance of history. Because the weasel and commit boundaries differ,
in some states of the economy (levels of u) current policy depends on past policy. For
anything above the upper commit line and below the upper weasel line, a policymaker
committed to rules sticks with rules, and a policymaker using discretion sticks with
discretion. Quite apparently, then, it is incorrect to judge policy simply on the current
state of the economy, and particularly inappropriate to naively contrast current policy
with past policies at a similar state of the economy or stage of the business cycle. In a
word, our model predicts hysteresis in monetary policy.
Implicit in the hysteresis is something so obvious as to possibly escape notice:
Over time, the policymaker switches from rules to discretion, and from discretion to
rules. Regimes shift. Discretion, commitment, and weaseling out of commitment will
all occur. As an example, figure 3 shows one path for independent shocks, the
commitment and weasel boundaries, and shades the time spent committed to rules.
The figure makes it easy to see both the historical dependence and the switches
between rules and discretion. This shifting reemphasizes a point stressed by Flood
and Garber (1984) in their work on the gold standard: To evaluate a policy rule, the
entire dynamic policy sequence must be analyzed, including those periods where
discretion reigns.
Not surprisingly, as W increases, the weasel boundaries move out (see figure
4). It is somewhat more surprising that the commitment boundaries are insensitive to
W. This is because eventually crossing a weasel boundary is a low-probability event
and therefore has little impact on the decision to commit. The commitment boundaries
27
are completely flat only because the state space is not fine enough; the commitment
boundaries narrow slightly with a very fine state space.
Increasing the variance of the shocks
2
causes the policymaker to narrow the
commitment ranges because it lowers the probability that later periods’ shocks will fall
in the range where the rules loss is less than the discretion loss, and thus increases the
value of the waiting option. Two things can happen (depending on parameter values)
when the variance gets large: 1) Rules may suddenly become inferior to discretion
forever, meaning the commitment range cuts off and drops to zero or 2) the
commitment range gradually disappears. We show the latter case in figure 5. The
weasel boundaries tend to be relatively insensitive to changes in the variance, mostly
because reneging is a low- probability event.
At first, it seems that a ought to increase the likelihood of commitment, but, as
mentioned above, increasing a decreases the inflation bias. This effect dominates the
direct influence of increased desire to avoid inflation, so that commitment never
occurs if a is large enough. For some values of K and
2
, discretion forever may be
strictly preferred to rules.
C. Beyond Monetary Policy
We have noted, the points made here apply generally to questions of time
inconsistency, not just the particular class of Barro-Gordon models. Bordo and
Kydland (1992), for example, interpret the gold standard as a rule containing
contingencies in case of wars and financial panics. Even with such contingencies, they
recognize the possibility of regret, because a fully contingent rule would create “ a
lack of transparency and possible uncertainty among the public regarding the will to
28
obey the original plan” (p. 8). One advantage of a simple rule like Bordo and
Kydland's interpretation of the gold standard is that the contingencies---wars and
financial panics---are readily verified. This makes credible commitment easier. A
complicated rule may lose some of the benefits of commitment because it is more
costly to verify the government's compliance.
In our framework, the gold standard can have two slightly different
interpretations. It may be seen as an imperfectly state-contingent rule that has been
abandoned in favor of discretion since the advent of the Bretton Woods system World
War II. Alternatively, because the gold standard did not bind government’s hands in
times of war, these could be seen as times when the government abandoned the rule in
favor of discretion, returning to rules at a later time. Our own view is that the lack of
waretime constraints points more to abandonment of a standard, and thus to the sort of
entry and exit considerations we have analyzed in this paper.
More generally, the tractability of the quadratic loss model makes it a natural
approximation for many time inconsistency problems (along with many other
economic problems as well) Thus, additional examples like restraining the lender of
last resort from bailing out insolvent institutions (with regret in a true financial crisis),
granting patents for the exclusive use of new technology (with regret in cases such as
AZT), or allowing constitutions to bind future legislatures could illustrate of our main
point.
For many applications, a continuous-time approach is more powerful,
particularly when the dynamic game of policy choice takes a simple form. We explore
these issues more deeply in a companion paper (Ritter and Haubrich, 1995).
29
V. Conclusion
Sometimes the right answer is inherent in the right question. The standard
analysis of the choice between rules and discretion has not asked the right question.
This failure may underlie the frustration felt on both sides of the of issue, by both the
starry-eyed theorists and hard-nosed practitioners, who have mostly talked past each
other. The decision regarding rules versus discretion occurs in real time, not at some
mythical starting date. That means that, because opting for discretion today leaves
open the possibility of adopting rules later on, it is often the better choice. Previous
work, by ignoring this option, has ignored an important advantage of discretion.
Like other options, the option to wait increases in value as uncertainty
increases--and so the value of discretion increases as well. Policy, then, has a “badnews principle” because the ability to avoid regret leads us to wait: Only news about
increased regret matters for the policy choice. But while the option-value results may
explain delay and refusal to adopt simple monetary targets or tax reforms during
recessions or wars, they do not generally justify permanently abandoning such rules.
Eventually, when the time is right, the government should commit--at least for a while.
When commitment to rules is no longer an irrevocable choice made at the
beginning of time, optimal policy looks more dynamic. Periods of rules alternate with
periods of discretion, depending both on the state and the history of the economy .
Policy at a given point in the business cycle may look quite different from policy at a
similar point in an earlier cycle. Such seeming confusion nevertheless reflects a
coherent, optimal choice.
30
In principle, the notion of commitment as irreversible investment can be
applied to other areas like tariff agreements, deficit reduction, and tort reform. In this
sense, our work complements recent studies on the political economy of resistance to
reforms (Fernandez and Rodrick [1991]), as well as on the delay in their
implementation (Alesina and Drazen [1991]). Our approach emphasizes delay and
resistance as an optimal response to an uncertain future. It also suggests the possibility
of hysteresis resulting from
Our findings are by no means the last word on the rules-versus-discretion
debate. We hope that by clarifying some neglected issues--regret, future commitment,
and the bad-news principle--we will contribute to clearer insight and a more focused
dialogue.
31
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