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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 801, April 2004, Revised Version: late 2004--Screen Reader Version*

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 Policy

Susan Athey, Andrew Atkeson, and Patrick J. Kehoe¹

International 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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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$\$greater than this cutoff state$,$ the monetary authority chooses a constant inflation rate. Under no discretion, the monetary authority chooses some constant inflation rate regardless of its information$.$

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 Economy

A   The Model

Here 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 $t=0,1,\ldots.$

At the beginning of each period, agents choose individual action $z_{t}$ from some compact set. We interpret $z$ as (the growth rate of) an individual's nominal wage and let $x_{t}$ denote the (growth of the) average nominal wage. Next, the monetary authority observes the current realization of its private information about the state of the economy. This private information $\theta_{t}$ is an i.i.d., mean 0 random variable with support $\theta \in\lbrack\underline{\theta},\bar{\theta}]$, with a strictly positive density $p(\theta)$ and a distribution function $P(\theta)$. Given this private information $\theta_{t}$, referred to as the state, the monetary authority chooses money growth $\mu_{t}\$in some large compact set $[\underline{\mu},\bar{\mu}].$

The monetary authority maximizes a social welfare function $R(x_{t},\mu _{t},\theta_{t})$ that depends on the average nominal wage growth $x_{t},$ the monetary growth rate $\mu_{t}\$, and a privately observed state $\theta_{t}$. We interpret $\theta_{t}$ to be private information of the monetary authority regarding the impact of a monetary stimulus on social welfare in the current period. Throughout, we assume that $R$ is strictly concave in $\mu$ and twice continuously differentiable.

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:

$$R(x_{t},\mu_{t},\theta_{t})=-\frac{1}{2}\left[ (U+x_{t}-\mu_{t})^{2}+(\mu _{t}-\alpha\theta_{t})^{2}\right] .$$ (1)

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 $\theta$. Each period, inflation $\pi_{t}$ is equal to the money growth rate $\mu_{t}\$ chosen by the monetary authority. Unemployment is determined by a Phillips curve. The unemployment rate is given by

$$u_{t}=U+x_{t}-\mu_{t},$$ (2)

where $U$ is a positive constant, which we interpret as the natural rate of unemployment. In (1), $\alpha$ is a weight on the private information. Social welfare in period $t$ is a function of $u_{t}$ and $\pi_{t}$ and the state $\theta_{t}.$ Our benchmark example is derived from a quadratic objective function of the form

$$-\frac{u_{t}^{2}}{2}-\frac{(\pi_{t}-\alpha\theta_{t})^{2}}{2},$$ (3)

which is similar to that used by Kydland and Prescott (1977) and Barro and Gordon (1983). Using (2) and $\pi_{t}=\mu_{t}$ in (3), we obtain (1). Here the monetary authority's private information is about the social cost of inflation, but we develop our model for general specifications of the social welfare function $R(x_{t},\mu _{t},\theta_{t})$ which subsume (1) as a special case. Notice that in our general formulation, we allow the current payoff to vary with expected inflation, through $x_{t}$; with actual inflation, through $\mu_{t}\$; and with the state $\theta_{t}$. This formulation thus subsumes many other versions of the Kydland-Prescott and Barro-Gordon models in the literature.⁴

Throughout, a policy for the monetary authority in any given period, denoted $\mu(\cdot),$ specifies the money growth rate $\mu(\theta)$ for each level of the state $\theta.$ For any $x,$ we define the static best response to be the policy $\mu^{\ast}(\theta;x)\;$that solves $R_{\mu}$( $x,\mu(\theta),\theta$) $=0.$ We assume that if $x=\int \mu(\theta)p(\theta)~d\theta,$ then

$$\int R_{x} x,\mu(\theta),\theta p(\theta)~d\theta<0.$$ (4)

B   Two Ramsey Benchmarks

Before 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 $\mu^{R}(\cdot),$ yields the highest payoff that can be achieved in an economy with full information. The gap between that Ramsey payoff and the payoff in the economy with private information measures the welfare loss due to private information.

The other benchmark, the expected Ramsey policy, denoted $\mu^{ER},$ yields the highest payoff that can be achieved when the policy is restricted to not depend on private information. In our environment, there is no publicly observed shock to the economy; hence, this policy is a constant. The expected Ramsey policy is a useful benchmark because it is the best policy that can be achieved by a rule which specifies policies as a function only of observables. This policy is analogous to the strict targeting rule discussed by Canzoneri (1985).

For the Ramsey policy benchmark, consider an economy with full information with the following timing scheme. Before the state $\theta$ is realized, the monetary authority commits to a schedule for money growth rates $\mu(\cdot)$. Next, individual agents choose their nominal wages $z$ with associated average nominal wages $x.$ Then the state $\theta$ is realized, and the money growth rate $\mu(\theta)$ is implemented. The optimal allocations and policies in this economy solve the Ramsey problem:

$$\max_{x,\mu(\cdot)}\int R$$($$x,\mu(\theta),\theta$$   ) $$p(\theta)~d\theta$$

subject to $x=\int\mu(\theta)p(\theta)~d\theta.$ For our example (1), the Ramsey policy is $\mu^{R}(\theta)=\alpha\theta/2.$ Note that the Ramsey policy has the monetary authority choosing a money growth rate which is increasing in its private information. Thus, with full information, it is optimal to have the monetary authority fine tune its policy to the state of the economy. This feature of the environment leads to a tension in the economy with private information between allowing the monetary authority discretion for fine tuning and experiencing the resulting time inconsistency problem.

For the other benchmark, consider an economy in which the monetary authority is restricted to choosing money growth $\mu$ that does not vary with its private information. The equilibrium allocations and policies in the economy with these constraints solve the expected Ramsey problem:

$$\max_{x,\mu}\int R(x,\mu,\theta)p(\theta)~d\theta$$ (5)

subject to $x=\mu.$ For our example (1), the expected Ramsey policy is $\mu^{ER}=0.$

For our example (1), the Ramsey policy obviously yields strictly higher welfare than does the expected Ramsey policy. More generally, when $R_{\mu\theta}(x,\mu,\theta)>0,$ the Ramsey policy $\mu^{R}(\cdot)$ is strictly increasing in $\theta$ and yields strictly higher welfare than does the expected Ramsey policy.

C   The Dynamic Mechanism Design Problem

To 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
$\left\{ \mu_{t}(h_{t},\hat{\theta}_{t})\vert\mbox{ all }h_{t}\mbox{, }\hat{\theta}_{t}\right\} _{t=0}^{\infty}$, where $\mu_{t}(h_{t},\hat{\theta }_{t})$ specifies the money growth rate that will be chosen in period $t$ following the history $h_{t}=(\hat{\theta}_{0},\hat{\theta}_{1},\ldots ,\hat{\theta}_{t-1})$ of past reports together with the current report $\hat{\theta}_{t}.$ The monetary authority chooses a reporting strategy $\{m_{t}(h_{t},\theta_{t})\vert$ all $h_{t}$, $\theta_{t}\}_{t=0}^{\infty}$ in period $0,$ where $\theta_{t}$ is the current realization of private information and $m_{t}(h_{t},\theta_{t})$ $\in\lbrack\underline{\theta} ,\bar{\theta}]$ is the reported private information in $t.$ As is standard, we restrict attention to public strategies, those that depend only on public histories and the current private information, not on the history of private information.⁵ Also, from the Revelation Principle, we need only restrict attention to truth-telling equilibria, in which $m_{t}(h_{t},\theta_{t})=\theta_{t}$ for all $h_{t}$ and $\theta_{t}.$

In each period, each agent chooses the action $z_{t}$ as a function of the history of reports $h_{t}.$ Since agents are competitive, the history need not include either agents' individual past actions or the aggregate of their past actions.⁶

Each agent chooses nominal wage growth equal to expected inflation. For each history $h_{t},$ with monetary policy $\mu_{t}(h_{t},\cdot)$ given, agents set $z_{t}(h_{t})$ equal to expected inflation:

$$z_{t}(h_{t})=\int\mu_{t}(h_{t},\theta)p(\theta)~d\theta,$$ (6)

where we have used the fact that agents expect the monetary authority to report truthfully, so that $m_{t}(h_{t},\theta_{t})=\theta_{t}$. Aggregate wages are defined by $x_{t}(h_{t})=z_{t}(h_{t}).$

The optimal monetary policy maximizes the discounted sum of social welfare:

$$(1-\beta)\sum_{t=0}^{\infty}\int\beta^{t}R x_{t}(h_{t} ),\mu_{t}(h_{t},\theta_{t}),\theta_{t} p(\theta_{t} )~d\theta_{t},$$ (7)

where the future histories $h_{t}$ are recursively generated from the choice of monetary policy $\mu_{t}(\cdot,\cdot)$ in the natural way, starting from the null history. The term $1-\beta$ normalizes the discounted payoffs to be in the same units as the per-period payoffs.

A perfect Bayesian equilibrium of this revelation game is a monetary policy, a reporting strategy, a strategy for wage-setting by agents $\left\{ z_{t}(\cdot)\right\} _{t=0}^{\infty},$ and average wages $\left\{ x_{t}\left( \cdot\right) \right\} _{t=0}^{\infty}$ such that (6) is satisfied in every period following every history $h_{t},$ average wages equal individual wages in that $x_{t}(h_{t})=z_{t}(h_{t})$, and the monetary policy is incentive-compatible in the standard sense that, in every period, following every history $h_{t}$ and realization of the private information $\theta_{t},$ the monetary authority prefers to report $m_{t}(h_{t},\theta_{t})=\theta_{t}$ rather than any other value $\hat{\theta}\in\lbrack\underline{\theta} ,\bar{\theta}].$ Note that since average wages $x_{t}(h_{t})$ always equal wages of individual agents $z_{t}(h_{t}),$ we need only record average wages from now on.

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 Formulation

Here 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 $t$ on is the same that can be obtained from period $0.$ Thus, the payoff from any incentive-compatible outcome for the repeated game can be broken down into payoffs from current actions for the players and continuation payoffs that are themselves drawn from the set of incentive-compatible payoffs. Following this logic, Abreu, Pearce, and Stacchetti (1990) show that the set of incentive-compatible payoffs can be found using a recursive method that we exploit here.

In our environment, this recursive method is as follows. Consider an operator on sets of the following form. Let $W$ be some compact subset of the real line, and let $\bar{w}$ be the largest element of $W$. The set $W$ may be interpreted as a candidate set of incentive-compatible levels of social welfare. In our recursive formulation, the current actions are average wages $x$ and a report $\hat{\theta}=m(\theta)$ for every realized value of the state $\theta.$ For each possible report $\hat{\theta},$ there is a corresponding continuation payoff $w(\hat{\theta})$ that represents the discounted utility for the monetary authority from the next period on. Clearly, these continuation payoffs cannot vary directly with the privately observed state $\theta.$

We say that the actions $x$ and $\mu(\cdot)$ and the continuation payoff $w(\cdot)$ are enforceable by $W$ if

$$w(\hat{\theta})\in W \hat{\theta}\in\lbrack\underline{\theta },\bar{\theta}],$$ (8)

$$x=\int\mu(\theta)p(\theta)~d\theta,$$ (9)

and the incentive constraints

$$(1-\beta)R x,\mu(\theta),\theta +\beta w(\theta)\geq(1-\beta)R x,\mu(\hat{\theta}),\theta +\beta w(\hat{\theta})$$ (10)

are satisfied for all $\theta$ and all $\hat{\theta}$, where $\mu(\theta )\in\lbrack\underline{\mu},\bar{\mu}].$ Constraint (8) requires that each continuation payoff $w(\hat{\theta})$ be drawn from the candidate set of incentive-compatible payoffs $W,$ while constraint (9) requires that average wages equal expected inflation. Constraint (10) requires that for each privately observed state $\theta,$ the monetary authority prefer to report the truth $\theta$ rather than any other message $\hat{\theta}.$ That is, the monetary authority prefers the money growth rate $\mu(\theta)$ and the continuation value $w(\theta)$ rather than a money growth rate $\mu (\hat{\theta})$ and its corresponding continuation value $w(\hat{\theta}).\;$

The payoff corresponding to $x,\mu(\cdot),$ and $w(\cdot)$ is

$$V x,\mu(\cdot),w(\cdot) =\int (1-\beta)R x,\mu(\theta),\theta +\beta w(\theta){\large ]}p(\theta)~d\theta.$$ (11)

Define the operator $T$ that maps a set of payoffs $W$ into a new set of payoffs as

$$T(W) = v \vert x_{v},\mu_{v} (\cdot),w_{v}(\cdot) W,$$ (12)

$$v=V x_{v},\mu_{v} (\cdot),w_{v}(\cdot) .$$

As demonstrated by Abreu, Pearce, and Stacchetti (1990), the set of incentive-compatible payoffs is the largest set $W$ that is a fixed point of this operator:

$$W^{\ast}=T(W^{\ast}).$$ (13)

For any given candidate set of incentive-compatible payoffs $W,$ we are interested in finding the largest payoff that is enforceable by $W,$ or the largest element $\bar{v}\in$ $T(W).$ We find this payoff by solving the following problem, termed the best payoff problem:

$$\bar{v}=\max_{x,\mu(\theta),w(\theta)}\int (1-\beta)R x,\mu(\theta),\theta +\beta w(\theta ) p(\theta)~d\theta$$ (14)

subject to the constraint that $x,$ $\mu(\cdot)$, and $w(\cdot)$ are enforceable by $W$, in that they satisfy (8)-(10). Throughout, we assume that $\mu(\cdot)$ is a piecewise, continuously differentiable function.

The best payoff problem is a mechanism design problem of choosing an incentive-compatible allocation $x,\mu(\cdot),w(\cdot)$ which maximizes utility. Following the language of mechanism design, we now refer to $\theta$ as the type of the monetary authority, which changes every period. When we solve this problem with $W=W^{\ast},$ (13) implies that the resulting payoff is the highest incentive-compatible payoff. We will prove our main result in Proposition 1 for any $W.$ Hence, we will not have to explicitly solve the fixed-point problem of finding $W^{\ast}.$

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 $w_{0}(\theta)$ from the best payoff problem, a period $1$ policy and continuation value, $\mu_{w_{0}(\theta)}(\cdot)$ and $w_{w_{0}(\theta)}(\cdot),$ that satisfy

$$w_{0}(\theta)=\int\left[ (1-\beta)R\mbox{{\large (}}x_{w_{0}(\the... ...{w_{0}(\theta)}(z),z\mbox{{\large )}}+\beta w_{w_{0}(\theta)}(z)\right] p(z)~dz$$ (15)

exist by the definition of $T.$ Equation ( $\ref{promise})$ and its analog for other periods are sometimes referred to as a promise-keeping constraint. In our approach, we do not need to mention this constraint since it is built into the definition of the operator $T.$

2   Characterizing the Optimal Mechanism

Now 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   Preliminaries

We begin with some definitions. In our recursive formulation, we say that a mechanism is static if the continuation value $w(\theta)=\bar{w}$ for (almost) all $\theta.$ We say that a mechanism is dynamic if $w(\theta)<\bar{w}$ for some set of $\theta$ which is realized with strictly positive probability.

Our characterization of the solution to the best payoff problem does not depend on the exact value of $\beta.$ Hence, to simplify the notation, we suppress explicit dependence on $\beta$ and think of the term $\beta$ as being subsumed in the $w$ function and $1-\beta$ as being subsumed in the $R$ function.

We assume that the preferences are differentiable and satisfy a standard single-crossing assumption, that

$$R_{\mu\theta}(x,\mu,\theta)>0.$$ (A1)

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 $\mu^{\ast}(\theta;x)$ is strictly increasing in $\theta.$

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: $\mu(\cdot)$ is nondecreasing in $\theta$;

$$R_{\mu} x,\mu(\theta),\theta \frac{d\mu (\theta)}{d\theta}+\frac{dw(\theta)}{d\theta}=0$$ (16)

wherever $d\mu(\theta)/d\theta$ and $dw(\theta)/d\theta$ exist; and for any point $\theta_{i}$ at which these derivatives do not exist,

$$\lim_{\theta\nearrow\theta_{i}}R x,\mu(\theta),\theta _{i} +w(\theta)=\lim_{\theta\searrow\theta_{i}} R x,\mu(\theta),\theta _{i} +w(\theta ).$$ (17)

Standard arguments give the following result: under the single-crossing assumption (A1), the allocation ( $x,\mu(\cdot),w(\cdot)$) satisfies the incentive constraints (10) if and only if the allocation is locally incentive-compatible. (See, for example, Fudenberg and Tirole's 1991 text.)

Given any incentive-compatible allocation, we define the utility of the allocation at $\theta$ to be

$$U(\theta)=R$$($$x,\mu(\theta),\theta$$)$$+w(\theta ).$$

Local incentive-compatibility implies that $U(\cdot)$ is continuous and differentiable almost everywhere, with derivative $U^{\prime}(\theta )=R_{\theta}$( $x,\mu(\theta),\theta$). Integrating $U^{\prime}(\cdot)$ from $\underline{\theta}$ up to $\theta$ gives that

$$U(\theta)=U(\underline{\theta})+\int_{\underline{\theta}}^{\theta}R_{\theta } x,\mu(z),z dz$$ (18)

while integrating $U^{\prime}(\cdot)$ from $\bar{\theta}$ down to $\theta$ gives that

$$U(\theta)=U(\bar{\theta})-\int_{\theta}^{\bar{\theta}}R_{\theta } x,\mu(z),z dz.$$ (19)

With integration by parts, it is easy to show that for interval endpoints $\theta_{1}<\theta_{2},$

$$\int_{\theta_{1}}^{\theta_{2}}U(\theta)p(\theta)~d\theta=P(\theta... ...theta_{2})-P(\theta_{1})U(\theta_{1})-\int_{\theta_{1}}^{\theta_{2} }R_{\theta} x,\mu(\theta),\theta P(\theta)~d\theta.$$ (20)

Using (18) and (20), we can write the value of the objective function $\int_{\underline{\theta}}^{\bar{\theta}}U(\theta)p(\theta)~d\theta$ as

$$U(\underline{\theta})+\int_{\underline{\theta}}^{\bar{\theta}}\frac {1-P(\theta)}{p(\theta)}R_{\theta} x,\mu(\theta),\theta p(\theta)~d\theta$$ (21)

$$U(\bar{\theta})-\int_{\underline{\theta}}^{\bar{\theta}}\frac{P(\... ...heta}\mbox{{\large (}}x,\mu(\theta),\theta\mbox{{\large )} }p(\theta)~d\theta.$$

Next we make some joint assumptions on the probability distribution and the social welfare function. Assume that, for any action profile $x,\mu(\cdot)$ with $\mu(\cdot)$ nondecreasing,

$$\frac{1-P(\theta)}{p(\theta)}R_{\theta\mu} x,\mu(\theta),\theta \theta,$$ (A2a)

$$\frac{P(\theta)}{p(\theta)}R_{\theta\mu} x,\mu(\theta),\theta \theta.$$ (A2b)

We refer to assumptions (A2a) and (A2b) together as (A2) and, in a slight abuse of terminology, call them the monotone hazard condition. In our benchmark example (1), $R_{\theta\mu}$( $x,\mu(\theta),\theta$) $=1$, so that (A2) reduces to the standard monotone hazard condition familiar from the mechanism design literature, that $[1-P(\theta)]/p(\theta)$ be strictly decreasing and $P(\theta)/p(\theta)$ be strictly increasing.

B   Showing That the Optimal Mechanism Is Static

Here 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 $\mu(\theta)$ to maximize social welfare

$$U(\underline{\theta})+\int_{\underline{\theta}}^{\bar{\theta}}\frac {1-P(\theta)}{p(\theta)}R_{\theta}$$ ($$x,\mu(\theta),\theta$$   )$$p(\theta)~d\theta$$

subject to the constraints that $(i)$ $x=\int \mu(\theta)p(\theta)~d\theta,$ $(ii)$ $\mu(\theta)$ is nondecreasing, and $(iii)$ the continuation values defined by

$$w(\theta)\equiv U(\underline{\theta})+\int_{\underline{\theta}}^{... ...z\mbox{{\large )~}}dz-R\mbox{{\large (} }x,\mu(\theta),\theta\mbox{{\large )}}$$

satisfy $w(\theta)\leq\bar{w}$ for all $\theta.$ Alternatively, we can write the best payoff problem as choosing $\mu(\theta)$ to maximize

$$U(\bar{\theta})-\int_{\underline{\theta}}^{\bar{\theta}}\frac{P(\... ...theta}\mbox{{\large (}}x,\mu(\theta),\theta\mbox{{\large )} }p(\theta)~d\theta$$

subject to the constraints $(i),$ $(ii),$ and $(iii),$ with the continuation values defined by

$$w(\theta)\equiv U(\bar{\theta})-\int_{\theta}^{\bar{\theta}}R_{\t... ...z\mbox{{\large )~}}dz-R\mbox{{\large (}} x,\mu(\theta),\theta\mbox{{\large )}}$$

satisfying $w(\theta)\leq\bar{w}$ for all $\theta.$

The standard approach to solving either version of this problem is to guess that the analog of constraints $(ii)$ and $(iii)$ do not bind, take the corresponding first-order conditions of either version to find the implied $\mu(\cdot),$ and then verify that constraints $(ii)$ and $(iii)$ are in fact satisfied at that choice of $\mu(\cdot).$ If we take that approach here, it fails. The first-order conditions with respect to $\mu(\theta)$ are

$$\frac{1-P(\theta)}{p(\theta)}R_{\theta\mu} x,\mu(\theta),\theta =\lambda$$ (22)

for the first version of the best payoff problem and

$$-\frac{P(\theta)}{p(\theta)}R_{\theta\mu} x,\mu(\theta),\theta =\lambda$$ (23)

for the second version, where $\lambda$ is the Lagrange multiplier on constraint $(i)$. The solution to these first-order conditions (22) and (23), from the relaxed problem in which we have dropped constraints $(ii)$ and $(iii),$ implies a decreasing $\mu(\cdot)$ schedule. To see why, note, for example, that the left side of equation (22) is the increment to social welfare from marginally increasing $\mu(\cdot)$ at some particular $\theta$ and adjusting the continuation values $w(\cdot)$ for $\theta^{\prime}\geq\theta$ to preserve incentive-compatibility, while the right side is the cost in terms of welfare from raising expected inflation $x.$ Under assumption (A2a), the benefits of raising $\mu(\cdot)$ are higher for low values of $\theta$ than for high values of $\theta$. Thus, in the relaxed problem, it is optimal to have a downward-sloping $\mu(\cdot)$ schedule$.$ Similar logic applies to (23). Clearly, then, the solution to the relaxed problem violates at least one of the dropped constraints $(ii)$ or $(iii)$, and hence, we cannot use this standard approach.

We also cannot use the ironing approach designed to deal with cases in which the monotonicity constraint $(ii)$ binds, because in our problem, the constraint that binds is constraint $(iii)$, which is not dealt with in that approach. Instead, in the proof of Proposition 1 that follows, we use a variational argument to show that constraint $(iii)$ binds for all $\theta$ at the solution to the best payoff problem. (We discuss below the reason our model differs from others in the literature.)

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 $\mu(\cdot)$schedule, a variation that flattens this schedule will improve welfare if it is feasible in the sense that the associated continuation value satisfies constraint $(iii).$ Our proof of Lemma 1 formalizes this logic.

Our objective is to show that the optimal continuation value $w(\cdot)$ is constant at $\bar{w}.$ We prove this by contradiction. We start with the observation that $w(\cdot)$ is piecewise-differentiable since $\mu(\cdot)$is piecewise-differentiable and (16) holds. We first show that $w(\cdot)$ must be a step function. If not, there is some interval over which $w^{\prime }(\theta)$ is nonzero, and hence, from local incentive-compatibility, $\mu(\cdot)$is strictly increasing. In Lemma 2, we show that a variation that flattens $\mu(\cdot)$ over that interval is feasible. From Lemma 1, we know it is welfare-improving.

We next show that $w(\cdot)$ must be continuous, and since it is a step function, it must be constant. We prove this by showing that if either $\mu(\cdot)$ or $w(\cdot)$are discontinuous at some point $\theta,$ then (17) implies that $\mu(\cdot)$ must be increasing in the sense that it jumps up at that point. In Lemma 3, we show that a variation that flattens $\mu(\cdot)$ in a neighborhood of that point is feasible, and again from Lemma 1, we know that it is welfare-improving.

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 $d\mu(\theta)/d\theta>0$ for some interval and the case of Lemma 3 in which $\mu(\cdot)$ jumps up at $\tilde{\theta}.$ We say that $\mu(\cdot)$ is increasing on $(\theta_{1},\theta_{2})$ if $\mu(\cdot)$ is weakly increasing on this interval and there is some $\tilde{\theta}$ in this interval such that $\mu(\theta)<\tilde{\mu}$ for $\theta<\tilde{\theta}$ and $\mu(\theta)>\tilde{\mu}$ for $\theta>\tilde{\theta}$, where $\tilde{\mu}$ is the conditional mean of $\mu(\cdot)$ on this interval, namely,

$$\tilde{\mu}=\frac{\int_{\theta_{1}}^{\theta_{2}}\mu(\theta)p(\theta)~d\theta }{P(\theta_{2})-P(\theta_{1})}.$$ (24)

In words, on this interval, the function $\mu(\cdot)$ is weakly increasing and is strictly below its conditional mean $\tilde{\mu}$ up to $\tilde{\theta}$ and strictly above its conditional mean after $\tilde{\theta}.$⁷ Throughout, we will also say that the policy $\mu(\cdot)$ is flat at some particular point $\theta$ if the derivative $\mu ^{\prime}(\theta)$ exists and equals zero at that point.

Consider now some dynamic mechanism ( $x,\mu(\cdot),w(\cdot)$) in which the policy $\mu(\cdot)$ is increasing on some interval, say, $(\theta_{1},\theta_{2}).$ In our variation, we marginally move the function $\mu(\cdot)$ toward its conditional mean on this interval and adjust the continuation values to preserve incentive-compatibility. In particular, our variation moves our original policy $\mu(\cdot)$ marginally toward a policy $\tilde{\mu}(\cdot)$ defined by

$$\tilde{\mu}(\theta)=\left\{ \begin{array}[c]{c} \tilde{\mu}\mbox{... ...\in(\theta_{1},\theta_{2})\\ \mu(\theta)\mbox{ otherwise} \end{array} \right. .$$ (25)

This policy $\tilde{\mu}(\cdot)$ differs from the original policy $\mu(\cdot)$ only on the interval $(\theta_{1},\theta_{2}),\$and there the original policy $\mu(\cdot)$ is replaced by the conditional mean $\tilde{\mu}$ of the original policy over the interval. Clearly, the expected inflation under $\tilde{\mu}(\cdot)$is the same as the expected inflation under the original policy.

We let ( $x(a),\mu(\cdot;a),w(\cdot;a)$) and $U(\cdot;a)$ denote our variation and the associated utility. The policy $\mu(\cdot;a)$ in our variation is a convex combination of the policy $\tilde{\mu}(\cdot)$ and the original policy $\mu(\cdot)$ and is defined by

$$\mu(\theta;a)=a\tilde{\mu}(\theta)+(1-a)\mu(\theta)$$ (26)

for $a\in\left[ 0,1\right] .$ (For a graph of $\mu(\cdot;a)$, see Figure 1.) Clearly, the expected inflation in our variation $\tilde{x}(a)$ equals that of the original allocation $x$ for all $a\in\left[ 0,1\right] .$

The delicate part of the variation is to construct the continuation value $w(\cdot;a)$ so as to satisfy the feasibility constraint $w(\theta;a)\leq \bar{w}$ for all $\theta,$ in addition to incentive-compatibility. It turns out that we can ensure feasibility if we use one of two ways to adjust continuation values. In the up variation, we leave the continuation values unchanged below $\theta_{1}$ and pass up any changes induced by our variation in the policy to higher types by suitably adjusting the continuation values to maintain incentive-compatibility. In the down variation, we leave the continuation values unchanged above $\theta_{2}$ and pass down any changes induced by our variation in the policy to lower types by suitably adjusting the continuation values to maintain incentive-compatibility.

In the up variation, we determine the continuation values by substituting $U(\theta;a)$
$=R$( $x,\mu(\theta;a),\theta$) $+$ $w(\theta;a)$ into (18) to get that $w(\theta;a)$ is defined by

$$w(\theta;a)=U(\underline{\theta})+\int_{\underline{\theta}}^{\theta}R_{\theta } x,\mu(z;a),z dz-R x,\mu(\theta;a),\theta .$$ (27)

In the down variation, we use (19) in a similar way to get that $w(\theta;a)$ is defined by

$$w(\theta;a)=U(\bar{\theta})-\int_{\theta}^{\bar{\theta}}R_{\theta } x,\mu(z;a),z dz-R x,\mu(\theta;a),\theta .$$ (28)

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 ( $x,\mu(\cdot),w(\cdot)$)$\;$be an allocation in which $\mu(\cdot)$ is increasing on some interval $(\theta_{1},\theta_{2}).$ Then the up variation and the down variation both improve welfare by increasing the objective function ( $\ref{obj}).$

PROOF: To see that the up variation improves welfare, use (21) to write the value of the objective function under this variation as

$$V(a)=U(\underline{\theta})+\int_{\underline{\theta}}^{\bar{\theta}} \frac{1-P(\theta)}{p(\theta)}R_{\theta} x,a\tilde{\mu} (\theta)+(1-a)\mu(\theta),\theta p(\theta)~d\theta.$$ (29)

To evaluate the effect on welfare of a marginal change of this type, take the derivative of $\tilde{V}(a)$ and evaluate it at $a=0$ to get

$$\frac{dV(0)}{da}=\int_{\underline{\theta}}^{\bar{\theta}}\frac{1-P(\theta )}{p(\theta)}R_{\theta\mu} x,\mu(\theta),\theta \left[ \tilde{\mu}(\theta)-\mu(\theta)\right] p(\theta)~d\theta$$ (30)

which, with the form of $\tilde{\mu}(\cdot),$ reduces to

$$\frac{dV(0)}{da}=\int_{\theta_{1}}^{\theta_{2}}\frac{1-P(\theta)}{p(\theta )}R_{\theta\mu} x,\mu(\theta),\theta \left[ \tilde{\mu}-\mu(\theta)\right] p(\theta)~d\theta.$$ (31)

If we divide (31) by the positive constant $P(\theta_{2})-P(\theta _{1})$, then we can interpret (31) to be the expectation of the product of two functions, namely, $f(\theta)$ defined as $[1-P(\theta)]R_{\theta\mu }(x,\mu(\theta),\theta)/p(\theta)$ and $g(\theta)$ defined as $\tilde{\mu} -\mu(\theta)$, where $p(\theta)/[P(\theta_{2})-P(\theta_{1})]$ is the density of $\theta,$over the interval $(\theta_{1},\theta_{2}).$ By assumption (A2a), we know that the function $f$ is strictly decreasing. Because the function $\mu(\theta)$ is increasing on the interval $(\theta_{1},\theta_{2})$, the function $g$ is decreasing on this interval in the sense that $g(\theta)$ is weakly decreasing and lies strictly below its conditional mean for $\theta<\tilde{\theta}$ and strictly above its conditional mean for $\theta>\tilde{\theta}.$ By the definition of a covariance, we know that $Efg=~$cov $(f,g)+(Ef)(Eg)$, where the expectation is taken with respect to the density $p(\theta)/[P(\theta_{2})-P(\theta_{1})].$ By the construction of $\tilde{\mu}$ in (24), we know that $Eg=0,$ so that $Efg=~$cov$(f,g)$, which is clearly positive because $f$ is strictly decreasing and $g$ is decreasing on the interval $(\theta_{1},\theta_{2})$. Thus, (31) is strictly positive, and the variation improves welfare.

The down variation also improves welfare. The value of the objective function under this variation is

$$V(a)=U(\bar{\theta})-\int_{\underline{\theta}}^{\bar{\theta}}\fra... ...tilde{\mu}(\theta)+(1-a)\mu (\theta),\theta\mbox{{\large )}}p(\theta)~d\theta.$$

Hence,

$$\frac{dV(0)}{da}=\int_{\theta_{1}}^{\theta_{2}}\frac{P(\theta)}{p(\theta )}R_{\theta\mu} x,\mu(\theta),\theta \left[ \mu(\theta)-\tilde{\mu}\right] p(\theta)~d\theta>0$$ (32)

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 $\frac{1-P(\theta)}{p(\theta)}$ measures the importance of higher types $1-P(\theta)$ relative to the rate at which changing $\mu(\theta)$ affects expected inflation as measured by $p(\theta).$ When continuation values are adjusted for types below a given type