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Board of Governors of the Federal Reserve System This HTML version of this discussion paper is a revised and updated version of the same paper available as a PDF file at http://www.federalreserve.gov/pubs/ifdp/2004/801/ifdp801.pdf. The Optimal Degree of Discretion in Monetary PolicyInternational Finance Discussion Papers numbers 797-807 were presented on November 14-15, 2003 at the second conference sponsored by the International Research Forum on Monetary Policy sponsored by the European Central Bank, the Federal Reserve Board, the Center for German and European Studies at Georgetown University, and the Center for Financial Studies at the Goethe University in Frankfurt. NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. The views in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or any other person associated with the Federal Reserve System. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/. Abstract: How much discretion should the monetary authority have in setting its policy? This question is analyzed in an economy with an agreed-upon social welfare function that depends on the economy's randomly fluctuating state. The monetary authority has private information about that state. Well-designed rules trade off society's desire to give the monetary authority discretion to react to its private information against society's need to prevent that authority from giving in to the temptation to stimulate the economy with unexpected inflation, the time inconsistency problem. Although this dynamic mechanism design problem seems complex, its solution is simple: legislate an inflation cap. The optimal degree of monetary policy discretion turns out to shrink as the severity of the time inconsistency problem increases relative to the importance of private information. In an economy with a severe time inconsistency problem and unimportant private information, the optimal degree of discretion is none. Keywords: Rules vs. discretion, time inconsistency, optimal monetary policy, inflation targets, inflation caps JEL Classification: E5, E6, E52, E58, E61 Suppose that society can credibly impose on the monetary authority rules governing the conduct of monetary policy. How much discretion should be left to the monetary authority in setting its policy? The conventional wisdom from policymakers is that optimal outcomes can be achieved only if some discretion is left in the hands of the monetary authority. But starting with Kydland and Prescott (1977), most of the academic literature has contradicted that view. In summarizing this literature, Taylor (1983) and Canzoneri (1985) argue that when the monetary authority does not have private information about the state of the economy, the debate is settled: there should be no discretion; the best outcomes can be achieved by rules that specify the action of the monetary authority as a function of observables. The unsettled question in this debate is Canzoneri's: What about when the monetary authority does have private information? What, then, is the optimal degree of monetary policy discretion? To answer this question, we use a model of monetary policy similar to that of Kydland and Prescott (1977) and Barro and Gordon (1983). In our legislative approach to monetary policy, we suppose that society designs the optimal rules governing the conduct of monetary policy by the monetary authority. The model includes an agreed-upon social welfare function that depends on the random state of the economy. We begin with the assumption that the monetary authority observes the state and individual agents do not. In the context of our model, we say that the monetary authority has discretion if its policy is allowed to vary with its private information.2 The assumption of private information creates a tension between discretion and time inconsistency.3 Tight constraints on discretion mitigate the time inconsistency problem in which the monetary authority is tempted to claim repeatedly that the current state of the economy justifies a monetary stimulus to output. However, tight constraints leave little room for the monetary authority to fine tune its policy to its private information. Loose constraints allow the monetary authority to do that fine tuning, but they also allow more room for the monetary authority to stimulate the economy with surprise inflation. We find the constraints on monetary policy that, in the presence of private information, optimally resolve this tension between discretion and time inconsistency. Formally, we cast this problem as a dynamic mechanism design problem. Canzoneri (1985) conjectures that because of the dynamic nature of the problem, the resulting optimal mechanism with regard to monetary policy is likely to be quite complex. We find that, in fact, it is quite simple. For a broad class of economies, the optimal mechanism is static and can be implemented by setting an inflation cap, an upper limit on the permitted inflation rate. More formally, our model can be described as follows. Each period, the monetary authority observes one of a continuum of possible privately observed states of the economy. These states are i.i.d. over time. In terms of current payoffs, the monetary authority prefers to choose higher inflation when higher values of this state are realized and lower inflation when lower values are realized. Here a mechanism specifies what monetary policy is chosen each period as a function of the history of the monetary authority's reports of its private information. We say that a mechanism is static if policies depend only on the current report by the monetary authority and dynamic if policies depend also on the history of past reports. Our main technical result is that, as long as a monotone hazard condition is satisfied, the optimal mechanism is static. We also give examples in which this monotone hazard condition fails, and the optimal mechanism is dynamic. We then show that our result on the optimality of a static
mechanism implies that the optimal policy has one of two forms:
either it has bounded discretion or it has no discretion. Under
bounded discretion, there is a cutoff
state: for any state less than this, the monetary authority chooses
its static best response, which is an
inflation rate that increases with the state, and for any
state We then show that we can implement the optimal policy as a repeated static equilibrium of a game in which the monetary authority chooses its policy subject to an inflation cap and in which individual agents' expectations of future inflation do not vary with the monetary authority's policy choice. In general, the inflation cap would vary with observable states, but to keep the model simple, we abstract from observable states, and the inflation cap is a single number. Depending on the realization of the private information, sometimes the cap will bind, and sometimes it will not. These results imply that the optimal constraints on discretion take the form of an inflation cap: the monetary authority is allowed to choose any inflation rate below this cap, but cannot choose one above it. We say that a given inflation cap implies less discretion than another cap if it is more likely to bind. We show that the optimal degree of discretion for the monetary authority is smaller in an economy the more severe the time inconsistency problem is and the less important private information is. It is immediate that we can equivalently implement the optimal policy by choosing a range of acceptable inflation rates. The optimal range will decrease as the time inconsistency problem becomes more severe relative to the importance of private information. Here the rationale for discretion clearly depends in a critical way on the monetary authority having some private information that the other agents in the economy do not have. Of course, if the amount of such private information is thought to be very small in actual economies, relative to time inconsistency problems, then our work argues that in such economies the logical case for a sizable amount of discretion is weak, and the monetary authority should follow a rather tightly specified rule. One interpretation of our work is that we solve for the optimal inflation targets. As such, our work is related to the burgeoning literature on inflation targeting. (See the work of Cukierman and Meltzer (1986), Bernanke and Woodford (1997), and Faust and Svensson (2001), among many others.) In terms of the practical application of inflation targets, Bernanke and Mishkin (1997) discuss how inflation targets often take the form of ranges or limits on acceptable inflation rates similar to the ranges we derive. Indeed, our work here provides one theoretical rationale for the type of constrained discretion advocated by Bernanke and Mishkin. Here we have assumed that the monetary authority maximizes the welfare of society. As such, the monetary authority is viewed as the conduit through which society exercises its will. An alternative approach is to view the monetary authority as an individual or an organization motivated by concerns other than that of society's well-being. If, for example, the monetary authority is motivated in part by its own wages, then, as Walsh (1995) has shown, the full-information, full-commitment solution can be implemented. Hence, with such a setup, monetary policy has no binding incentive problems to begin with. As Persson and Tabellini (1993) note, there many reasons such contracts are either difficult or impossible to implement, and the main issue for research following this approach is why such contracts are, at best, rarely used. Our work is related to several other literatures. One is some work on private information in monetary policy games. See, for example, that of Backus and Driffill (1985); Ireland (2000); Sleet (2001); Da Costa and Werning (2002); Angeletos, Hellwig, and Pavan (2003); Sleet and Yeltekin (2003); and Stokey (2003). The most closely related of these is the work of Sleet (2001), who considers a dynamic general equilibrium model in which the monetary authority sees a noisy signal about future productivity before it sets the money growth rate. Sleet finds that, depending on parameters, the optimal mechanism may be static, as we find here, or it may be dynamic. Our work is also related to a large literature on dynamic contracting. Our result on the optimality of a static mechanism is quite different from the typical result in this literature, that static mechanisms are not optimal. (See, for example, Green (1987), Atkeson and Lucas (1992), and Kocherlakota (1996).) We discuss the relation between our work and these literatures in more detail after we present our results. At a technical level, we draw heavily on the literature on recursive approaches to dynamic games. We use the technique of Abreu, Pearce, and Stacchetti (1990), which has been applied to monetary policy games by Chang (1998) and is related to the policy games studied by Phelan and Stacchetti (2001), Albanesi and Sleet (2002), and Albanesi, Chari, and Christiano (2003). The mechanism design problem that we study is related, at an abstract level, to some work on supporting collusive outcomes in cartels by Athey, Bagwell, and Sanchirico (2004), work on risk-sharing with nonpecuniary penalties for default by Rampini (forthcoming), and work on the tradeoff between flexibility and commitment in savings plans for consumers with hyperbolic discounting by Amador, Werning, and Angeletos (2004). However, our paper is both substantively and technically quite different from those. We discuss the details of the relation after we present our results. 1 The EconomyA The ModelHere we describe our simple model of monetary policy. The
economy has a monetary authority and a continuum of individual
agents. The time horizon is infinite, with periods indexed by
At the beginning of each period, agents choose individual action
The monetary authority maximizes a social welfare function
A leading interpretation of the private information in our economy follows that of Sleet and Yeltekin (2003) and Sleet (2004). Individual agents in the economy have either heterogeneous preferences or heterogeneous information regarding the optimal inflation rate, and the monetary authority sees an aggregate of that information which the private agents do not see. (Informally, we imagine this private information takes resources to acquire, so that while agents in the economy feasibly can acquire the information, the costs involved in doing so outweigh the benefits.) When we pose our optimal policy problem as a mechanism design problem, we are presuming that the mechanism designer is a separate agent with no independent information of its own. We interpret the society's objective as a weighted average of the preferences of the heterogeneous agents. As a benchmark example, we use this function:
We interpret (1) as the reduced form that results from a monetary authority which maximizes a social welfare function that depends on unemployment, inflation, and the monetary authority's private information ![]() ![]() ![]()
where ![]() ![]() ![]() ![]() ![]() ![]()
which is similar to that used by Kydland and Prescott (1977) and Barro and Gordon (1983). Using (2) and ![]() ![]() ![]() ![]() ![]() Throughout, a policy for the
monetary authority in any given period, denoted
B Two Ramsey BenchmarksBefore we analyze the economy in which the monetary authority has private information, we consider two alternative economies. The optimal policies in these economies are useful as benchmarks for the optimal policy in the private information economy. One benchmark, the Ramsey policy,
denoted
The other benchmark, the expected Ramsey
policy, denoted For the Ramsey policy benchmark, consider an economy with full
information with the following timing scheme. Before the state
![]() ![]() ![]() ![]() ![]() For the other benchmark, consider an economy in which the
monetary authority is restricted to choosing money growth
subject to ![]() ![]() For our example (1), the Ramsey policy
obviously yields strictly higher welfare than does the expected
Ramsey policy. More generally, when
C The Dynamic Mechanism Design ProblemTo analyze the problem of finding the optimal degree of discretion, we use the tools of dynamic mechanism design. Without loss of generality, we formulate the problem as a direct revelation game. In this problem, society specifies a monetary policy, the money growth rate as a function of the history of the monetary authority's reports of its private information about the state of the economy. Given the specified monetary policy, the monetary authority chooses a strategy for reporting its private information. Individual agents choose their wages as functions of the history of reports of the monetary authority. A monetary policy in this environment is a sequence of
functions In each period, each agent chooses the action Each agent chooses nominal wage growth equal to expected
inflation. For each history
where we have used the fact that agents expect the monetary authority to report truthfully, so that ![]() ![]() The optimal monetary policy maximizes the discounted sum of social welfare:
where the future histories ![]() ![]() ![]() A perfect Bayesian equilibrium of
this revelation game is a monetary policy, a reporting strategy, a
strategy for wage-setting by agents
Note that this definition of a perfect Bayesian equilibrium includes no notion of optimality for society. Instead, it simply requires that in response to a given monetary policy, private agents respond optimally and truth-telling for the monetary authority is incentive-compatible. The set of perfect Bayesian equilibria outcomes is the set of incentive-compatible outcomes that are implementable by some monetary policy. The mechanism design problem is to choose a monetary policy, a reporting strategy, and a strategy for average wages, the outcomes of which maximize social welfare (7) subject to the constraint that these strategies are incentive-compatible. D A Recursive FormulationHere we formulate the problem of characterizing the solution to
this mechanism design problem recursively. The repeated nature of
the model implies that the set of incentive-compatible payoffs that
can be obtained from any period In our environment, this recursive method is as follows.
Consider an operator on sets of the following form. Let
We say that the actions
and the incentive constraints
are satisfied for all ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() The payoff corresponding to
Define the operator ![]() ![]()
As demonstrated by Abreu, Pearce, and Stacchetti (1990), the set of incentive-compatible payoffs is the largest set ![]()
For any given candidate set of incentive-compatible payoffs
subject to the constraint that ![]() ![]() ![]() ![]() ![]() The best payoff problem is a mechanism design problem of
choosing an incentive-compatible allocation
Moreover, to prove our main result, we also need focus only on
the best payoff problem, which gives the highest payoff that can be
obtained from period 0 onward. For
completeness, however, notice that given some
exist by the definition of ![]() ![]() ![]() 2 Characterizing the Optimal MechanismNow we solve the best payoff problem and use the solution to characterize the optimal mechanism. Our main result here is that under two simple conditions, a single-crossing condition and a monotone hazard condition, the optimal mechanism is static. To highlight the importance of the monotone hazard condition for this result, we discuss in an appendix three examples which show that if the monotone hazard condition is violated, the optimal mechanism is dynamic. A PreliminariesWe begin with some definitions. In our recursive formulation, we
say that a mechanism is static if the
continuation value
Our characterization of the solution to the best payoff problem
does not depend on the exact value of We assume that the preferences are differentiable and satisfy a standard single-crossing assumption, that
This implies that higher types of monetary authority have a stronger preference for current inflation. Standard arguments can be used to show that the static best response ![]() ![]() Under the single-crossing assumption (A1), a standard lemma lets
us replace the global incentive constraints (10)
with some local versions of them. We say that an allocation is
locally incentive-compatible if it
satisfies three conditions:
wherever ![]() ![]() ![]()
Standard arguments give the following result: under the single-crossing assumption (A1), the allocation ( ![]() Given any incentive-compatible allocation, we define the
utility of the allocation at
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
while integrating ![]() ![]() ![]()
With integration by parts, it is easy to show that for interval endpoints ![]()
Using (18) and (20), we can write the value of the objective function ![]()
![]() Next we make some joint assumptions on the probability
distribution and the social welfare function. Assume that, for any
action profile
B Showing That the Optimal Mechanism Is StaticHere we show that the optimal mechanism is static by proving this proposition: Proposition 1: Under assumptions (A1) and A2), the optimal mechanism is static. The approach we take in proving Proposition 1 is different from the standard approach used by Fudenberg and Tirole (1991, Chapter 7.3) for solving a mathematically related principal-agent problem. To motivate our approach, we first show why the standard approach does not work for our problem. We discuss the forces that lead to the failure of the standard approach here because these forces suggest a variational argument we use to prove Proposition 1. The best payoff problem can be written as follows: Choose
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() The standard approach to solving either version of this problem
is to guess that the analog of constraints
for the first version of the best payoff problem and
for the second version, where ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() We also cannot use the ironing approach designed to deal with
cases in which the monotonicity constraint Before proving Proposition 1, we sketch our basic argument. Our
discussion of the first-order conditions of the relaxed problem
(22) and (23)
suggests that given any strictly increasing
Our objective is to show that the optimal continuation value
We next show that It is convenient in the proof of Proposition 1 to use a
definition of increasing on an interval
which covers the cases we will deal with in Lemmas 2 and 3. This
definition subsumes the case of Lemma 2 in which
In words, on this interval, the function ![]() ![]() ![]() ![]() ![]() ![]() ![]() Consider now some dynamic mechanism (
This policy ![]() ![]() ![]() ![]() ![]() ![]() We let (
for ![]() ![]() ![]() ![]() ![]() The delicate part of the variation is to construct the
continuation value
In the up variation, we determine the continuation values by
substituting
In the down variation, we use (19) in a similar way to get that ![]()
By construction, these variations are incentive-compatible. In the following lemma, we show that, if either variation is feasible, it improves welfare. LEMMA 1: Assume (A1)
and (A2), and
let (
PROOF: To see that the up variation improves welfare, use (21) to write the value of the objective function under this variation as
To evaluate the effect on welfare of a marginal change of this type, take the derivative of ![]() ![]()
which, with the form of ![]()
If we divide (31) by the positive constant ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() The down variation also improves welfare. The value of the objective function under this variation is ![]()
by arguments similar to those given before. Q.E.D. To gain some intuition for how these variations improve welfare,
we begin by emphasizing a critical insight: changing the inflation
for any given type not only has direct effects on the welfare of
that type, but also has indirect effects on the welfare of other
types through the incentive constraints. For example, making a
given type better off not only helps that type, but also makes that
type less tempted to mimic higher types. Thus, the continuation
values of those higher types can then be increased, if that is
feasible, as in the up variation. In that variation, the term
Using these ideas, let us now focus on the up variation, and
consider the effects of increasing More formally, let us derive expressions for the impact of the
flattening of the policy on the current payoffs ![]() ![]() ![]() ![]() ![]() ![]() ![]()
Hence, the impact on the utility of type ![]()
Notice from (34) that any change in the policy for some particular type ![]() ![]() ![]()
in the integral (30) can be thought of as the sum of the change in welfare for all types ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() In the down variation, the intuition for the derivative
(32) is the same as that for (31), except that, in this variation, a change in the
inflation rate chosen by type The following lemma proves that if LEMMA 2: Under (A1)
and (A2), in the
optimal mechanism, the continuation value function
PROOF: Since by assumption
![]() ![]() ![]() ![]() ![]() ![]() ![]() To complete the proof, we show that either the up variation or
the down variation is always feasible. Under the up variation, (
Figure 2 is a graph of
Under the down variation, (
for ![]() ![]() ![]() ![]() To ensure that the continuation value satisfies feasibility, we
use the up variation when the term
Q.E.D. In the next lemma, we show that
LEMMA 3: Under (A1)
and (A2),
In Appendix A, we prove that Together Lemmas 2 and 3 The Optimal Degree of DiscretionSo far we have demonstrated that the optimal mechanism is static. Now we describe three key implications of an optimal static mechanism for monetary policy: The optimal policy has either bounded discretion or no discretion; the optimal policy can be implemented by society setting an upper limit, or cap, on the inflation rate that the monetary authority is allowed to choose; and the optimal degree of discretion is decreasing the more severe is the time inconsistency problem and the less important is private information. A Characterizing the Optimal PolicyIn the optimal static mechanism, the monetary policy
subject to the constraints that ![]() ![]() ![]() ![]() ![]() ![]() We say that a monetary policy
where ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() We now show that the optimal policy has either bounded
discretion or no discretion. Here, as before, we can replace the
global incentive constraint in (38) with the
local incentive constraints, with the restriction that
where ![]() ![]() ![]() ![]() ![]() ![]() In the following proposition, we show that if the optimal policy
is not the expected Ramsey policy, then it must be of the form (
Proposition 2: Under assumptions(A1) and (A2), the optimal policy
Proof: We have argued that if the optimal policy is constant, then it must be an expected Ramsey policy, which has
no discretion. If the optimal policy is not constant, then it must
be of the form (
![]() ![]() ![]() ![]() ![]() ![]() ![]() B Implementing Optimal Policy with an Inflation Cap or a Range of Inflation RatesWe have characterized the solution to a dynamic mechanism design
problem. We now imagine implementing the resulting outcome with an
inflation cap, a highest allowable
level of inflation
The intuition for this result--that a policy with either bounded
discretion or no discretion can be implemented by setting an upper
limit on permissible inflation rates--is simple. In our
environment, the only potentially beneficial deviations from either
type of policy are ones that raise inflation. Under bounded
discretion, the types in
Clearly, we can also implement the optimal policy with a range
of inflation rates denoted
C Linking Discretion With Time Inconsistency and Private InformationSo far we have shown that the optimal policy has either bounded discretion or no discretion and discussed how to implement such a policy. Here we link the optimal degree of discretion to the severity of the time inconsistency problem and the importance of private information. We show that the optimal degree of discretion shrinks as the time inconsistency problem becomes more severe and private information becomes less important. The literature using general equilibrium models to study optimal
monetary policies suggests a qualitative way to measure the
severity of the time inconsistency problem. In most of this
literature, the time inconsistency problem is extremely severe, in
that the static Nash equilibrium is always at the highest feasible
inflation rate
In our reduced-form model, we can mimic the general equilibrium
models with the more severe problems by choosing a payoff function
We can mimic the general equilibrium models with less severe
problems by choosing a payoff function We summarize this discussion in a proposition: PROPOSITION 3: Assume
(A1) and (A2). Two
cases follow: (i) if the static best response satisfies
PROOF: Under (A1) and (A2), the optimal mechanism is static. To prove (i), note that in any equilibrium with bounded discretion,
Under ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() We prove (ii) by contradiction.
Assume that
In Proposition In general, the choice of the optimal inflation cap depends on
the importance of private information relative to the severity of
the time inconsistency problem. In our benchmark example, the
parameter When the objective function satisfies (1), the
condition
For policies of the bounded discretion form (39), we think of
PROPOSITION 4: Assume
(1), (A1), and (A2a).
If
We prove this proposition in Appendix D. Figure 5 illustrates
the proposition for two economies with different degrees of
relative importance of private information and severity of time
inconsistency problems,
4 Comparison to the LiteratureOur result on the optimality of a static mechanism is quite different from what is typically found in dynamic contracting problems, that static mechanisms are not optimal. Using a recursive approach, we have shown how our dynamic mechanism design problem reduces to a simple quasi-linear mechanism design problem. Our result is thus also directly comparable to the large literature on mechanism design with broad applications, including those in industrial organization, public finance, and auctions. (See Fudenberg and Tirole's 1991 book for an introduction to mechanism design and its applications.) In this comparison, the continuation values in our framework correspond to the contractual compensation to the agent in the mechanism design literature. Our result that the optimal mechanism is static, so that the continuation values do not vary with type, stands in contrast to the standard result in the mechanism design literature that under the optimal contract, the compensation to the agent varies with the agent's type. In this sense, our result is also quite different from what is found in the mechanism design literature. The key feature of our model that distinguishes it from much of the dynamic incentive literature is the feasibility constraint
The implication of this constraint is that in our model the continuation values of one type cannot be traded off against other types as they can be in many other models. To highlight the importance of this constraint, we consider a highly stylized example in Appendix E that replaces the constraint ![]()
and show that the resulting optimal value of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() A large class of dynamic incentive models include a feature like (43); they might usefully be thought of as debt models. Early versions of these include the private debt models of Green (1987), Thomas and Worrall (1990), Atkeson (1991), and Atkeson and Lucas (1992, 1995) while later versions include the government debt models of Sleet and Yeltekin (2003) and Sleet (2004). All of these models share the feature that optimal contracts are dynamic because in each of these settings a low continuation for one type can be traded off against a high continuation value for another type. In this sense, the debt models share many of the features of models with constraints of the form (43) rather than those with constraints of the form (42). Having a constraint like (42) rather than (43) is important for our result that the optimal mechanism is static, but it is not sufficient, for at least two reasons. First, even in our model, we have given examples in which the optimal mechanism is dynamic when our monotone hazard condition is violated. Second, the information structure also matters. In our model, private agents receive no direct information about the state of the economy. If private agents receive a noisy signal about the state before the monetary authority takes its action, then our result goes through pretty much unchanged; the noisy signal is just a publicly observed variable upon which the inflation cap is conditioned. If, however, private agents receive a noisy signal about the information the monetary authority received after the monetary authority takes its action, then dynamic mechanisms in which continuation values vary with this signal may be optimal. Sleet (2001) considers such an information structure and shows that the optimality of the dynamic mechanism depends on the parameters governing the noise. He finds that when the public signal about the monetary authority's information is sufficiently noisy, having the monetary authority's action depend on its private information is not optimal; hence, the optimal mechanism is static. In contrast, when this public signal is sufficiently precise, the optimal mechanism is dynamic. The logic of why a dynamic mechanism is optimal is roughly similar to that in the literature of industrial organization which follows Green and Porter (1984) on optimal collusive agreements that are supported by periodic reversion to price wars, even though these price wars lower all firms' profits. Our work here is also related to some of the repeated game literature in industrial organization about supporting collusion in oligopolies. Athey and Bagwell (2001) and Athey, Bagwell, and Sanchirico (2004) solve for the best trigger strategy-type equilibria in games with hidden information about cost types. Athey and Bagwell (2001) show that, in general, the best equilibrium is dynamic (nonstationary). In this equilibrium, a firm which sets low prices gets a lower discounted value of profits from then on. Athey, Bagwell, and Sanchirico (2004) show that when strategies are restricted to be strongly symmetric, so that all firms receive the same continuation values even though they take observably different actions, a different result emerges. In particular, under some conditions, the best equilibrium is stationary and entails pooling of all cost types. When those conditions fail, and when firms are sufficiently patient, there may be a set of stationary and nonstationary equilibria that yield the same payoffs. (The latter result relies heavily on the Revenue Equivalence Theorem from auction theory.) 5 ConclusionWhat is the optimal degree of discretion in monetary policy? For economies in which private information is not important and time inconsistency problems are severe, the optimal degree of discretion is zero. For economies in which private information is important and time inconsistency problems are less severe, it is not zero, but bounded. More generally, the optimal degree of discretion is decreasing the more severe is the time inconsistency problem and the less important is private information. For all of these economies, the optimal policy can be implemented by legislating and enforcing a simple inflation cap. In our simple model, the optimal inflation cap is a single number because there is no publicly observed state. If the model were extended to have a publicly observed state, then the optimal policy would respond to this state, but not to the private information. To implement optimal policy, therefore, society would need to specify a rule for setting the inflation cap, where the cap would vary with public information. Equivalently, society could specify a rule for setting ranges for acceptable inflation, where these ranges would vary with public information. We interpret these rules as a type of inflation targeting that is broadly similar to the types actually practiced by a fair number of countries. (For a discussion of inflation targeting in practice, see Bernanke and Mishkin (1997).) To keep our theoretical model simple, we have abstracted from exotic events which are both unforeseeable and unquantifiable. Anyone interpreting the implications of our results for an actual society, therefore, should keep in mind that to handle such exotic events, the optimal policy rule would need to be adapted to deal with them, perhaps by the addition of some type of escape clauses. Appendix A: Proof of Lemma 3Here we prove Lemma 3, that under (A1) and (A2), the optimal
allocation
PROOF. In Lemma 2, we showed that in an optimal
allocation Consider now the first type of discontinuity, when
Under these assumptions,
Suppose that for the chosen interval
To show that the up variation is feasible inside the interval ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
Using ![]()
Consider first (46). By construction ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Consider next (47). Note that we can rewrite (44) as ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Now consider the second type of discontinuity, when
Appendix B: Optimal Policy without Monotone HazardsHere we give three examples in which our monotone hazard
condition (A2) is violated and in which the optimal mechanism is
dynamic. In the first two examples, we assume that the hazard
For the first two examples, assume that at the point
To interpret this inequality, note that the left side is the conditional mean of the function ![]() ![]() ![]() ![]() ![]() It is easy to show that a two-piece uniform distribution with
In the first example, the linear
example, we make the calculations trivial by assuming that
In the third example, the discrete example, ![]() ![]() All three of these examples satisfy the single-crossing property
(A1). In the first two examples,
The Linear ExampleAny solution to the mechanism design problem must have the two-piece form
This follows because the arguments used in Lemmas 1 and 2 can be applied separately to the intervals ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() The mechanism design problem then reduces to the linear problem
of choosing ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
where the inequality follows from ( ![]() ![]() ![]() ![]() The Benchmark ExampleNow assume that the policy
This variation is similar to the one in the linear example.
Consider an alternative policy that lowers inflation for types at
or below
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
Since ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() It is straightforward, but somewhat tedious, to show that the
associated continuation values
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Note that if
![]() ![]() ![]() ![]() The Discrete ExampleNow let the types be
for ![]() ![]() ![]() ![]() ![]()
We now give an example in which the hazard (
![]() ![]() ![]() ![]() ![]() In SumIn each of the three examples, we have shown that welfare could be improved relative to a static policy by raising inflation for high types and lowering inflation for low types so as to keep expected inflation constant. In the first two examples, this improved welfare because there were sufficiently few high types relative to low types; we could raise inflation a lot for the types who valued it more and lower it only a little for the types who valued it less. In the third example, even though the distribution of types is uniform, the high types valued inflation so much more than the low types that raising inflation for the high types and lowering it for the low types still improved welfare. Appendix C: Implementation with an Inflation CapHere we prove that the equilibrium outcome in an economy with an inflation cap is the optimal outcome of the mechanism design problem. We show this result formally using a one-shot game in which we drop time subscripts. With an inflation cap of An equilibrium of this one-shot game consists of aggregate wages
To implement the best equilibrium in the dynamic game, we choose
Whenever bounded discretion is optimal, we choose the cap to be the money growth rate chosen by the cutoff type ![]()
where ![]() PROPOSITION 5: Assume
(A1), (A2), and that the inflation cap
PROOF: We establish this result in two steps. We
first show that the monetary authority will choose the upper bound
We next show that if bounded discretion is optimal in the
dynamic game, then in the associated static game with the inflation
cap, all types choose the bounded discretion policies. For all
types
Appendix D: Proof of Proposition 4Here we prove Proposition 4, which links monetary policy discretion to both time inconsistency and private information. PROOF: The optimal policy with bounded discretion
is found as the solution to the problem of choosing
![]() ![]() ![]() ![]() ![]()
Let ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
We can show that, under (A2a), this derivative is strictly decreasing in ![]() ![]() ![]() ![]() The fact that (58) is strictly decreasing in
Finally, to complete the proof of Proposition 4, we must show
that when
![]() ![]() Appendix E: The Role of Our feasibility Constraint![]() Here we develop a highly stylized example (about traffic congestion) that illustrates the importance of the feasibility constraint
in generating our result that the optimal policy is static. In the example, we replace this constraint with the constraint
and show that the resulting optimal mechanism differs radically from ours. To be concrete, consider a mechanism design problem of choosing
![]() ![]() ![]() ![]() ![]()
and ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() It is easy to see that here the optimal Note that the result that the first best is incentive-compatible
is special to this functional form in which payoffs are linear in
![]() ![]() ![]() ![]() ![]() ![]() ![]() How could we interpret our model and results in this road
congestion context? Suppose that using tolls is not feasible, and
the only way to ration road use is to make people wait to get on
the road. Let
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The authors thank the editor and the referees for very useful comments, Kathy Rolfe for excellent editorial assistance, and the NSF for generous financial assistance. The views expressed are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. Return to text 2. Our approach here is different from that in the early literature on rules vs. discretion, as is our notion of discretion. The early literature assumes that society has no mechanism for committing to rules governing monetary policy. As does Taylor (1983), we find the legislative approach more appealing for advanced economies. Return to text 3. For some potential empirical support for the idea that the Federal Reserve possesses some nontrivial private information, see the work of Romer and Romer (2000). As we discuss below, we interpret this private information in our economy along the lines of Sleet and Yeltekin (2003) and Sleet (2004). Return to text 4. Note that the inflation rate that
enters the period Also, for simplicity, our formulation abstracts from direct costs due to future inflation. One interpretation of this feature is that it captures what happens in the cashless limit of a sticky price model. Return to text 5. For a discussion of the large class of environments for which this restriction does not alter the set of equilibrium payoffs, see Fudenberg and Tirole's 1991 text. Return to text 6. For details of why this is true, see the work of Chari and Kehoe (1990). Return to text 7. Note that this definition of
increasing is stronger than the
definition of a function weakly
increasing on an interval because our definition rules out a
function that is constant over the interval. But our definition is
weaker than the definition of a function strictly increasing over an interval because ours
allows for subintervals over which
8. Note that the best policy with no
discretion, the expected Ramsey policy,
will not typically be a special case of a policy with bounded
discretion. Specifically, when
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