Central bankers in government appointed committees

Journal of Public Economics 94 (2010) 363–379

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Journal of Public Economics j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / j p u b e

Central bankers in government appointed committees Marcela Eslava ⁎ Universidad de Los Andes, Department of Economics, Carrera 1 Este No 18 A - 70. Bloque W. Bogotá, Colombia

a r t i c l e

i n f o

Article history: Received 16 September 2008 Received in revised form 10 October 2009 Accepted 9 February 2010 Available online 12 February 2010 JEL classification: D71 E50 E58

a b s t r a c t I study the policy choices of members of a central bank committee, who are appointed by the government. Central bankers balance their desire to protect the Central Bank's reputation against their interest to be reappointed. Committees can be more successful than single central bankers at reducing inflation and insulating policy from government pressures. These gains are only achieved if the turnover rate of committee members is low and the committee is small. The former is associated with a low risk of being replaced for not supporting the government's preferred policy. The latter, meanwhile, implies high probability that a single vote affects policy, making any individual member more weary of potentially affecting the Central Bank's reputation through his vote. © 2010 Elsevier B.V. All rights reserved.

Keywords: Central bank Committee decision-making Reputation Reappointment

1. Motivation The optimal institutional design of an independent Central Bank (CB) has been a subject of great interest to legislators, central bankers, and academics alike. One of the sticky points concerns how much independence the CB should be given. The importance of CB independence to reduce inflation and alleviate ties between monetary policy and the electoral cycle is widely recognized. However, there is also ample agreement that the CB should not be completely isolated from the government. This is in part due to the need for coordination between monetary and fiscal policy, but also because the participation of publicly-elected officials in all policy decisions is a core principle of democratic systems. The dilemma is how to achieve the positive consequences of CB independence without completely isolating monetary policy from government influence. In practice, such a dilemma is frequently addressed by placing the design of monetary policy in the hands of a committee of central bankers, some of whose members are appointed by the government in office. The idea is that, while its power to make appointments creates a channel for the government to participate in the design of monetary policy, the larger set of decision-making mechanisms and appointment rules available for a committee can be taken advantage of to minimize the inflation bias and the political business cycles that result from government participation. Arrangements that stagger the

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terms of central bankers and limit the fraction of members changed during any given presidential period are applications of this idea.1 Several theoretical contributions in the literature give support to the choice of a committee-governed CB. One strand of this literature shows that the mechanisms for aggregating the policy preferences of members of a CB committee can be optimally designed to reduce inflation bias (Bullard and Waller, 2004; Faust, 1996; Dal Bó, 2006). Inflation bias is reduced if the representation of anti-inflation groups in the committee is disproportionately large relative to their representation in society. Such over-representation can be achieved either by setting rules that directly assign a majority of seats to these groups (Faust, 1996), or via appropriate supermajority rules (Bullard and Waller, 2004; Dal Bó, 2006). Another group of models shows that a committee design can reduce output cycles derived from swings in monetary policy following government changes (Lohman, 1997; Waller, 1989 and 2000). These are models where each government 1 Monetary policy in Colombia, for instance, is chosen by a committee of seven members serving four-year staggered terms. The original design was such that at each point in time the governing president would have appointed three of the seven members (including the Minister of Finance). The remaining members would be “inherited” from the previous government. The Bank's Website explains the underlying rationale as follows: “This system guarantees continuity in Bank policy while safeguarding it from the influences of political change, thus ensuring planning more in view of the long-term and garnering greater credibility with the public” (El Nuevo ordenamiento del Banco y su Junta Directiva, in http://www.banrep.gov.co). On another front, Waller (1989) documents how several historians of the U.S. Federal Reserve “argue that the board structure of the Federal Reserve Board of Governors was, in fact, chosen specifically to minimize the influence of partisan politics on the setting of monetary policy” (pp. 422–423).

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can change some members of the committee during its term, and it appoints central bankers who belong to its own party. Output cycles in these models arise from inflation surprises, in a context where inflation expectations are set before government elections, and where partisan appointments to the CB imply that the outcome of the election affects the preferences of the CB. The basic result is that a well designed committee can reduce inflation and output cycles because it generates more stable CB preferences than a single central banker arrangement. Cycles are reduced by arrangements that increase the stability of CB preferences, such as those that lengthen the terms of central bankers (Waller, 1989) and those that decentralize appointments (Lohman, 1997). They are also reduced when the government's incentives to make partisan appointments are diminished, as is the case when government nominations to the Central Bank must be confirmed by the congress (Waller, 2000). Appropriate committee designs can make governments value more the confirmation of their nominees by reducing the turnover rate of individual central bankers. In short, policy-making and appointment mechanisms for central bank committees can be designed to aggregate the given policy positions of potential or actual members optimally. Although this argument points to what are certainly important advantages of committee designs, it fails to recognize that central bankers embedded in such a setting may in fact not behave as they would in the absence of government appointments, or in a single central banker framework. Put another way, the policy positions of individual members are not given. This paper brings into the picture the potential effects of collective decision-making and government appointments on the individual choices of central bankers. Accounting for these dimensions makes clear that, under some circumstances, a committee structure may not help isolate monetary policy from the electoral cycle, or reduce inflation bias. In particular, the size of the committee and the fraction of members over whose appointment the government has influence are key determinants of the effects of collective decisionmaking and government appointments. The introduction of government power over appointments can affect the incentives faced by central bankers in ways that are particularly acute when informational asymmetries give rise to reputation concerns. As the previous literature has pointed out, central bankers may want to build a reputation for the CB as being tough on inflation (Backus and Driffill, 1985; Barro, 1986). Yet, this is not the only source of reputation-building incentives when appointments to the CB are a privilege of the government. In fact, a central banker may care about the government's perception of him (i.e., how close his preferences are to those of the government, and how loyal to the government he is), as such perception will likely affect his chances of being reappointed (Fratianni et al., 1997). Hence, a central banker may want to establish a reputation of being close to the government.2 Moreover, how these reputation-building incentives play into a central banker's decisions depends on whether policy is chosen by an individual or a committee. More specifically, reappointment probabilities are affected by government perceptions of individual central bankers, while inflation expectations are affected by the public's perception of the CB as a whole. Under a committee, but not in a single policy-maker environment, there is a potential dichotomy between individual members' reputation and the collective reputation of the CB. I argue below that the collective reputation of the CB is affected by actual policy, rather than by the vote of each individual central banker. Since the vote of a central banker may not be reflected in actual policy, each member knows that his vote will only have a limited effect on the collective reputation of the CB. Hence the

2 As an example, Ersenkal et al. (1985), find that the monetary base in the U.S. expands faster in the months preceding the decision as to the reappointment of the Federal Reserve Board chairman. Fratianni, Von Hagen, and Waller (1997) develop a framework in which a (single) central banker has an incentive to accommodate the wishes of the government to secure reappointment.

incentive to show closeness to the government is more likely to override the incentive to build a reputation of toughness for the CB in a committee. The conclusion that a committee design limits the influence of the government on monetary policy, therefore, does not hold in all contexts. Motivated by these considerations, I study how the reputationbuilding incentives faced by central bankers are affected by the government's appointment powers and by collective decisionmaking. I frame my analysis in a two-period model along the lines of Barro and Gordon (1983), where reputation considerations are taken into account (as in Backus and Driffill, 1985). I introduce committee decision-making and the possibility of the endogenous reappointment of CB members. In this setting I address the tension between the CB's collective reputation and members' individual reputations, as discussed above. The model in this paper shows that, compared to a single central banker setting, one based on a committee design has the potential to both increase the importance a central banker gives to building the CB's reputation as being tough on inflation, and reduce the importance given to reappointment considerations. The basic reason is that the greater stability of CB preferences under a committee, relative to a single central banker, implies higher importance of current inflation in determining future expectations. The realization of this potential advantage of committees, however, depends crucially on the specific features of the committee design, in particular its size and the fraction of its members the government can replace from one period to the next. An increase in the fraction of members replaced each period reduces the relevance of past inflation for future expectations of inflation, thus making central bankers less wary of raising current inflation. It also increases the risk of a central banker being removed from office if it is believed that his type is different from that of the government, thus generating more incentives to follow the government's preferred policy. As a result, a higher turnover rate leads to both higher inflation bias and more incentives to please the government by choosing its preferred policy. On the other hand, other things being equal, an increase in the size of the committee reduces the ability of any individual member to affect actual policy, thus increasing the importance central bankers give to influencing reappointment decisions relative to choosing their preferred policy. Small size committees, therefore, are better suited to reducing the influence of the government (for a given turnover rate). I also study some applications of the model. First, I extend the dynamics to illustrate the emergence of political cycles driven by the power of the government over CB appointments. I also show that extending the period of a central banker increases the importance he gives to protecting the reputation of the CB, but also potentially increases the importance he gives to reappointment. The realization of this latter potential depends on the specific rules governing reappointment; rules such that being reappointed today increases the likelihood that a member will be reappointed again in the future lead to greater influence of the government on policy. Finally, I study the possibility that central bankers may abstain from voting. This possibility arises naturally in the model since a members' vote affects his chances of being reappointed (perhaps negatively) with greater likelihood than it affects policy. I show that the risk of political cycles is increased by the possibility of abstaining from voting, especially in large committees and in those with high turnover rates. Riboni and Ruge-Murcia (2008) also have a model where members of a CB committee vote strategically over policy options, concerned by the potential dynamic implications of their votes. The focus of their paper is on the implications of having one member that holds disparate power, a question I do not address here. The effect of collective decision-making on the incentives faced by central bankers has also been addressed by Sibert (2003) and Mihov and Sibert (2006), in a framework similar to the one I use below. With respect to those studies, the current paper adds an analysis of the role of

M. Eslava / Journal of Public Economics 94 (2010) 363–379

government appointments; the effect of group policy-making differs when government appointments are brought into the picture, as discussed above. Moreover, Mihov and Sibert's models assume that the public obtains and uses information about individual votes in the CB. In this paper, meanwhile, the idea that individual votes are observed only imperfectly is pivotal to the concept of collective reputation. My contention is that people are more likely to form expectations based on actual policy even if individual voting records are available, as observing policy has a much lower cost than investigating individual votes.3 In this sense, my idea of collective reputation is closer to that proposed by Tirole (1996), wherein the reputation of an organization differs from that of each member, as the output of the latter is observed only imperfectly. My results contribute to the debate on the optimal institutional design of the CB. First, I qualify the conditions under which a committee structure might be preferable to a single central banker, in terms of making the incentives for central bankers consistent with minimal inflation bias and minimal variability of policy over the electoral cycle. Second, the model sheds light on considerations that should play an important role in defining the optimal size of the CB committee and the optimal turnover rate of its members. The optimal size of and turnover rate for a CB committee have been at the center

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of the policy debate,4 but the literature providing the arguments that should guide this discussion is in its first stages. Waller (1989, 2000) and Waller and Walsh (1996) provide an important line of argumentation, through models with purely partisan central bankers; in that context, they that show that lower turnover rates help to insulate policy from political influence, though at the cost of potentially locking bad policies into place. Sibert (2006), in turn, has reviewed experiments on group policy-making and used those experiments to make inferences about the optimal size of a monetary policy committee. This paper complements the existing models by making the policy positions of individual central bankers endogenous to the institutional design of the CB, and complements experimental evidence with a theoretical analysis of the incentives behind the behavior of committee members', in the specific context of monetary policy-making. The paper is divided into six sections, inclusive of this introduction. Section 2 discusses the general setting of the model, and solves the general problem of a central banker. Section 3 studies how that solution differs between the case of a single central banker and that of a committee. Section 4 introduces government appointments. Section 5 discusses some applications and related simple extensions of the model. Finally, Section 6 concludes.

2. Model: General setting My basic framework follows Barro and Gordon's model (1983), which is a standard framework for studying the institutional design of central banks. My analysis also relies on previous models of reputation-building by monetary authorities (Backus and Driffill, 1985; and Barro, 1986). I begin by studying the choice of monetary policy without imposing a specific institutional structure, and derive a general solution to the problem of a central banker. Even without further institutional detail, my general approach differs from the traditional setting in that, as is the case with many CB's, a central banker's term can be extended, that is, a central banker can be reappointed. I will later derive from the general solution obtained in this section more specialized ones; those analyze the form the solution takes under boards vs. single central bankers, and under government appointments vs. exogenous appointments. 2.1. The preferences of central bankers Consider a two-period economy with a Central Bank (CB) that rules over monetary policy. The CB is assumed to be independent from the government in that the latter does not participate directly in the choice of monetary policy. However, under some of the institutional arrangements I consider, the government is given the role of designating the central bankers. I assume that the CB has perfect control over the inflation rate, π, which summarizes monetary policy in the model. I also adopt the standard assumption that the preferences of central bankers reflect a trade-off between the costs imposed by inflation variability and the potential benefits of generating inflationary surprises (such as boosting employment in the short run). I assume that central bankers come in two types, which I call hawks and doves. Hawks give more importance to fighting inflation than do doves. In order to incorporate reputation-building considerations into the model, I assume that a central banker's type is his private information. A central banker's choice variable is his vote over π, which is equivalent to actual policy in the case of a single central banker, but not necessarily so under a committee arrangement. For simplicity, I follow Sibert (2003) and Mihov and Sibert (2006) in making the extreme assumption that a hawk central banker is mechanistic, always voting for zero inflation. The preferences of a dove central banker, identified as i, are captured by the following loss function: h i 2 d t−1 2 e Li = E ∑ βi πt −cðπt −πt Þ−bToit ; t=1

ð1Þ

where the superindex d refers to a dove, πt is the inflation rate for period t, πet is the public's expectation of inflation for period t, and βi ∈ [0, 1] is i's discount factor. The discrete variable oit is 1 if i is in office during period t, and 0 otherwise. The parameters c and b (where c N 0 and b ≥ 0) capture the value given by a dove policymaker to generating inflation surprises and being in office (relative to driving inflation to its 0 target level),

3 Sibert's paper does analyze the case in which voting records are not published. My results differ from hers even in this case, however, given my consideration of the reappointment process. 4 There are several recent examples of public debate around CB committee rules. In the US, right before President Obama was due to appoint the chairman of the Fed, incumbent chairman Bernanke organized a town hall meeting that was characterized by analysts as a reappointment campaign (Buiter, 2009; Pethokoukis, 2009). As a result, some questioned appointment rules for Fed directors (Buiter, 2009). In Colombia, the turnover rules of the CB have been in the spotlight as a result of a Constitutional amendment authorizing the president to run for reelection. CB rules were originally designed with a single-period president in mind; after the amendment, a reelected president can appoint all of the CB board. Another example of recent debate about CB committee rules is Alesina et al's (2005) piece recommending changes to the size of the Colombian CB board and the length of its members' terms.

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Fig. 1. Timing of events.

respectively.5 Central bankers face uncertainty stemming from two sources. First, if monetary policy is determined by a committee, each central banker faces uncertainty concerning what types his fellow committee members are. The other source of uncertainty for a central banker comes from the possibility that he will not be reappointed for period 2. I will consider different scenarios for how reappointment is decided. To simplify the notation, in what follows I drop the time subindexes, and indicate period 2 variables with a ′ mark. Focusing on a dove i who is in the CB in period one, and defining ri such that ri = 1 if i is reappointed for period 2 and 0 otherwise, i′s loss function can be rewritten as: d

2

e

2

e

Li = Efπ − cðπ − π Þ + βi ½ðπ′Þ − cðπ′ − π ′Þg − b½1 + βi Pr ðri = 1Þ:

ð2Þ

Loss function (2) makes clear that, when choosing what policy to support in period 1, a dove central banker takes into account the potential effect of his choice on three different factors: 1) current inflation, and therefore the central banker's current welfare; 2) next period expected inflation; and 3) his chances of being reappointed. Simply put, a dove central banker has a preference for generating an inflation surprise in period 1, but may be discouraged from doing so because high inflation damages the reputation of the CB, increasing future expectations of inflation. A dove central banker may also avoid high inflation (or stick to it) if it damages his chances of being reappointed (or benefits them). Reappointment is valuable both because i derives “ego rents”, b, from being in office in period 2, and because by being reappointed i has the chance to tilt period 2 policy toward his preferred inflation rate. The focus of this paper is on how different institutional arrangements affect the relative size and importance of the three incentives mentioned above. I allow the discount factor βi to vary across individual bankers, so that different dove central bankers may care more or less about the future implications of their choices; βi is characterized by a cumulative density function F(βi). Assuming that βi may vary from central banker to central banker is an alternative to assuming that dove central bankers may play mixed strategies. Without any of these assumptions, as will become clear later, equilibria where a member in a committee knows that his individual vote will not affect policy become prevalent in several of the different institutional contexts studied below. These equilibria rule out doves' concerns for the central bank's reputation, and have limited interest, since policy makers disregard possible consequences of their actions for the economy as a whole. The strategy in this paper is to consider a more general setting that can both generate equilibria where the probability that a single vote changes policy is zero, and equilibria where an individual central banker has some ability to affect policy. While allowing players to play mixed strategies is the standard way of addressing the issue of ineffective votes in committees (e.g. Sibert, 2003), the varying-βi assumption allows testing the robustness of results to alternative configurations of the forces underlying the choice of mixed strategies (and, as a result, testing different sets of plausible equilibrium mixed strategies). 2.2. Timing and information The timing of events, summarized in Fig. 1, is as follows. There are two periods, t = 1, and t = 2. Each period t is divided into three “subperiods”, t−, t, t+. At t− , central bankers are appointed (e.g., by the government). They can be newcomers, or, for t = 2, central bankers that were in office at t = 1 and were reappointed. At t, the public forms inflation expectations for the current period.6 At t+, the CB chooses π, which is observed by all players. As for the information structure, in any given period there is uncertainty about the central banker's type (central bankers' types, if the CB is a committee); that type is known only to himself. In period 1, only the unconditional probability that a central banker is a dove (which I denote as ϕ) is known to the public. Given the timing described above, however, in period 2 all players know what policy was chosen in period 1. They also know if a central banker is an incumbent reappointed at the end of period 1, or a newcomer. This information is used by the public to update beliefs about the types of central bankers in period 2 and, ultimately, to form πe′. 2.3. Solving the problem of a central banker The equilibrium vote of a central banker in each period maximizes his expected discounted utility, given the public's beliefs about the types of central bankers in both periods. These beliefs, based in period 1 on the unconditional probability distribution over types of central bankers, are

5 This loss function, linear in inflation surprises, ignores potential output stabilization incentives. This simplification is helpful in the context of this paper because the simultaneous introduction of government appointments, committee decision-making, and possible reappointments makes the problem quite difficult to solve by expanding considerably the set of possible states the central banker must consider. Some interesting consequences of CB collective decision-making for the ability of the monetary authority to stabilize output are analyzed by Dal Bó (2000), Waller (1989, 2000), and Lohman (1997). 6 An implicit assumption is that the length of a central banker's term and the length of nominal are equal. Although relaxing this assumption could have interesting implications for the variability of monetary policy within a central banker's term, I keep it throughout the paper for the sake of simplicity.

M. Eslava / Journal of Public Economics 94 (2010) 363–379

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updated at the end of the first period following Bayesian updating. In this section, I analyze the problem of a central banker for a given set of beliefs by the public; left for Section 4 is the consideration of expectations formation and equilibrium. I denote dove i's vote as vi. Solving backwards, in period 2 i votes for the rate that minimizes his within-period loss function. This is i's preferred inflation rate, which I denote as πd in the rest of the paper. πd is given by: d

vi = π =

c : 2

ð3Þ

In period 1, meanwhile, i also considers the implications of his vote on the perceptions of others regarding his type and the “type of the CB”. Those perceptions in turn affect both future expected inflation and, potentially, i's chances of being reappointed. Since hawks always vote for π = 0, voting for any rate different from π = 0 will reveal i as a dove to those observing his vote. Choosing any vi ≠ 0 could also potentially lead to π ≠ 0, revealing to the public that the committee is dominated by doves. As a result, dove i will consider only two possible votes for period 1, vi = 0 and vi = πd, where the latter is his preferred vote among those revealing his type.7 The inflation rate implemented in a given period is the one that receives the most votes, and thus could be either π = 0 or π = π d. I eliminate the possibility of a tie by assuming that a CB committee has an odd number of members (which in fact is the case in many countries). It is useful at this point to introduce some additional notation. I use Prp(π′ = πd|π) to designate the probability assigned by the public to π′ = πd , given the realization of π. I use this notation to differentiate the probability assigned by the public to a given outcome from that assigned by the central banker, who knows his own type and can therefore use it as additional information. I also use ΔPr(x) ≡ Pr(x|vi = πd) − Pr(x|vi = 0) to denote the impact of i's vote on given outcome x, which could be π or ri). I also allow x = π′, in which case ΔPr(x) is defined as a function of v′i rather than vi; where v′i is the vote in period 2 of i or his replacement if he is not reappointed. In equilibrium, dove i chooses vi = 0 if and only if L(vi = 0) ≤ L(vi = πd). Using Eq. (2) and the notation introduced above, this condition can be written as in the following result: 1. Result 1 In period 2 a dove central banker i chooses v′i = πd. In period 1 i chooses vi = 0 if and only if the following condition holds: d

βi ΔPrðπ = π Þ

h i h i p d d d p d E Pr ðπ′ = π jπ = π Þjvi = π −E Pr ðπ′ = π jπ = 0Þjvi = 0 0:5 d

≥ ΔPrðπ = π Þ:

  4b d −βi ΔPrðri = 1Þ 2 + ð1−ϕÞΔPrðπ′ = π Þ c ð4Þ

If condition (4) fails to hold, i chooses vi = πd. Proof. See part A of Appendix.

□

The top row of condition (4) reflects the benefits of voting for π = 0 in terms of the CB's reputation, derived from the fact that low current inflation would reduce future expectations of inflation. Since i′s vote affects the CB's reputation only to the extent that it potentially affects the policy adopted, there are two components to this reputation incentive. The first is i's ability to affect policy, which I call the representation factor; it is reflected in ΔPr(π = πd). The second component is what I call the reputation factor, which is captured by E[Prp(π′ = πd|π = πd)|vi = πd] − E[Prp(π′ = πd|π = 0)|vi = 0]. The loss of reputation expected by i is conditioned here not only on π but also on vi, as under some institutional arrangements considered below whether i is reappointed or not matters for inflation expectations, and reappointment in turn may depend on vi. The middle row captures the benefits of vi = 0 in terms of reappointment. The sign this term has depends on how reappointment is determined. If appointments lie in the government's hands, this term likely implies an incentive to vote for the h government's preferredi rate (that 4b is, ΔPr(ri = 1) is negative if the government is hawk and positive if it is dove). The term in square brackets, 2 + ð1−ϕÞΔPrðπ′ = πd Þ , gives the c value of reappointment; it is given not only by the opportunistic reward of being in office, b, but also by i's desire to shape policy according to his preferences. Central bankers' incentives to increase their chances of being reappointed are frequently dismissed with the argument that central bankers are not opportunistic. The expression above makes the point that even an altruistic central banker (say b = 0) wants to remain in office, in order to see his beliefs about what is optimal for social welfare taken into account in the design of future policies. Finally, the term to the right of the inequality sign captures the fact that vi = 0 is costly for a dove from the point of view of period 1 since, in the absence of any consideration about the future, he would prefer π = πd to π = 0. As was the case with the reputation incentive, this current loss is only incurred if i's vote is reflected in actual policy; hence, representation also plays a role in this term. 2.4. Expectations and equilibrium choices When forming expectations of inflation for period 2, individuals know that central bankers will correctly represent their types during that period. Therefore, Prp(π′ = πd|π) is given by the probability the public assigns to the majority of period 2 central bankers (or the single central banker) being doves. This probability in turn depends on people's beliefs about the types of period 2 central bankers. For a newly appointed central banker, the public assigns a probability ϕ that he is dove, as past policy provides no information on that particular central banker. For central bankers reappointed from period 1, the public updates its beliefs using Bayesian updating. If i is reappointed for period 2, the public will assign: Prði is a dove jπÞ =

Prðπji is a doveÞ Prði is a doveÞ : PrðπÞ

ð5Þ

7 The assumption that a hawk always votes for zero inflation thus reduces the space of possible policies to a binary one, simplifying the problem. The greater simplicity comes at the cost of introducing asymmetries between the incentives faced by the two types, and precluding signaling by hawks. This paper, however, focuses on how the institutional environment shapes the incentives of those central bankers who are subject to reputation considerations. The qualitative conclusions on this specific question should not be affected by the asymmetric treatment of the two types.

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Letting w = Pr(vi = πd|i is a dove), we get:   d Prðπ ji is a doveÞ = w * Pr π jvi = π + ð1 − wÞ Prðπ jvi = 0Þ:

ð6Þ

In an equilibrium, w reflects the optimal strategy of doves, as presented in result 3. Note that w is the probability that condition (4) does not hold, given by 8 < Prðβi b ΨÞ = FðΨÞ w= and : 1

9 if 1 N Ψ N 0 = otherwise

;

;

ð7Þ

where Ψ=

1

h i h i E Pr p ðπ′ = πd jπ = πd Þj vi = πd −E Pr p ðπ′ = πd jπ = 0Þ jvi = 0 0:5

−

h i: 4b d ΔPrðri = 1Þ 2 + ð1−ϕÞΔPrðπ′ = π Þ

ð8Þ

c

ΔPrðπ = πd Þ

w summarizes the solution to the dove central banker's problem. It represents the votes of dove central bankers, where votes are optimal given expected inflation in each period, and expectations of inflation are rational given the optimal strategies of central bankers. The former condition is ensured by Eq. (8), and the latter by Eqs. (5) and (6). Note also that any given central banker faces an inference problem similar to that of the public, in that he cannot observe his colleagues' types. Like the public, he also uses w to evaluate the probability that any other dove central banker will choose vi = πd. This probability enters the problem of a central banker through ΔPr(π = πd). The discussion above presents a general characterization of the equilibrium choices faced by central bankers. I now analyze what these conditions imply in terms of the optimal institutional design of the CB. I present alternative institutional settings and study the choices of central bankers under each. My strategy is to study how institutional design affects the different incentives faced by central bankers (summarized by the three key concepts of reputation, reappointment, and representation, discussed above), and ultimately how they affect w. When discussing the convenience of one or another institutional design, I implicitly assume that CB institutions should seek to minimize both inflation bias and the government's influence over monetary policy. These two goals are frequent motivations for placing monetary policy in the hands of an independent CB; I simply take them as given in the discussion.8

3. The institutional dimension I: collective decision-making As a first approximation, I isolate the analysis of collective decision-making from the consideration of government appointments. I do so by assuming throughout this section that a central banker's reappointment is exogenously given and independent of his vote in the first period, such that ΔPr(ri = 1) = 0. For any ΔPr(π = πd) N 0, the condition under which a central banker votes for π = 0 can now be written as βi

h i h i p ′ d d d p ′ d E Pr ðπ = π jπ = π Þjvi = π −E Pr ðπ = π jπ = 0Þjvi = 0 0:5

N 1:

ð9Þ

The ΔPr(π = πd) term does not enter condition (9) because, without government appointments, the votes of central bankers only matter if they affect policy. As a result, any ΔPr(π = πd) N 0 suffices for policy considerations to be the only factors determining a central banker's choice. Given condition (9), vi = 0 is chosen in equilibrium if and only if the gain in the committee's collective reputation derived from low inflation is sufficient to overcome the current loss π = 0 imposes on a dove central banker. With this in mind, I analyze how reputation considerations are affected by the choice of a committee-based CB, as well as by some specific characteristics of the committee. A key argument in favor of committee arrangements, both in the informal tradition and in the literature reviewed above, is that they can be optimally designed to render CB preferences that are more stable than would be possible under a single central banker. The idea is simply that, by staggering the members' terms, a committee can be designed to outlive the maximum length of any single policymaker's term. Greater stability in the CB's preferences has benefits in this model, since it increases the importance given by the public to past policy as a signal of future policy, making it more costly for central bankers to choose high inflation. To retain the feature of greater stability under committees, I assume that i ´s term ends at the end of t, and compare the incentives he faces under a committee arrangement where he is outlived by colleagues versus those faced if policy depends solely on his choices. The key insight I obtain from this comparison is that staggering the terms of the committee members does not guarantee that inflation bias is reduced by a committee design, despite the fact that staggered terms do renders more stable CB preferences. Success in reducing inflation with respect to a single central banker depends on specific characteristics of the committee– the number of members and the fraction of members replaced at the end of period 1. 3.1. The single central banker A single central banker whose term ends in t = 1 chooses π = πd. In this model this is not imposed by assuming that i doesn't care about what happens once he is out of office. Rather, though his preferences do not change for period 2, he knows his choice of π does not have dynamic 8 My assumptions about the preferences of central bankers are in fact consistent with social losses that increase with inflation and inflation variability, since the preferences of central bankers are likely similar to those of other individuals in society. Note that political pressures constitute a source of undesirable variability of monetary policy, not related to shocks to economic fundamentals. There is also an underlying rationale for CB independence in the model: while reputation concerns may restrain the inflationary temptations of central bankers, I assume below that the government is immune to such concerns.

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consequences. This is so because the public is aware of the change of CB head, and thus assigns no value to period 1 policy as a signal about the policy that will be in place in period 2. 3.2. Reputation-building incentives in a committee setting Suppose now that monetary policy is decided by a committee, the members of which vote over possible policy choices. I assume again that dove member i of this committee knows that his term ends at the end of t = 1. As we shall see below, in the committee case i's vote affects the collective reputation of the committee so long as it has some effect on π, and even if i is no longer in office. Dove i may thus be led to vote for low inflation to protect the CB reputation. In this sense, a committee may strengthen a dove's reputation incentives to choose low inflation. To solve i's problem under a committee arrangement formally, I assume there are n = 2z + 1 members in the committee. Policies are decided by a simple majority rule, such that the inflation rate that obtains z + 1 or more votes is adopted.9 I also assume that at the end of period 1 exactly m members are removed from office, while the remaining n–m are reappointed.10 The probability the public assigns to π′ = πd is given by the probability that z + 1 or more members of the t = 2 committee are doves: p

d

n

d

Pr ðπ′ = π jπ = π Þ =

∑

x=z + 1

 h    i n d x d n−x Prðk is dovejπ= π Þ 1−Prðk is dovejπ =π Þ x

ð10Þ

  m of period 2 central bankers helped choose π. In other words, This probability depends on period 1inflation,  inasmuch as a fraction 1− m n

π contains information about a fraction 1− d

Prðk is dovejπ = π Þ = Prðk is dovejπ = 0Þ =

n

of the committee serving in t = 2. Formally:

m m þ ϕ + ð1− Þϕ ; n n

ð11Þ

m m − ϕ + ð1− Þϕ : n n

where ϕ+ and ϕ− refer to beliefs about a period 1 central banker i who was reappointed, with ϕ+ ≡ Pr(i is a dove|π = πd) and ϕ− ≡ Pr(i is a dove|π = 0). In turn, from Eqs. (5) and (6) it follows that that ϕ+ and ϕ− are given by:

Prði is dovejπÞ =

h i d wTPrðπ jvi = π Þ + ð1−wÞPrðπ jvi = 0Þ ϕ PrðπÞ

ð12Þ

where w ≡ Pr(vi = πd|i is a dove) is characterized by Eq. (7). Note that, since individual votes in period 1 are not known by the public, the individual identities of the reappointed members are not important; rather, what matters is the number of members reappointed, together with π. Using this fact and Eqs. (10) and (11), we can derive the reputation loss associated with choosing high inflation in period 1: h i h i E Pr p ðπ′ = πd jπ = πd Þjvi = πd −E Prp ðπ′ = πd jπ = 0Þjvi = 0 = Pr p ðπ′ = πd jπ = πd Þ−Prp ðπ′ = πd jπ = 0Þ ! x  n−x      n n mϕ + ðn−mÞϕþ mϕ + ðn−mÞϕþ mϕ + ðn−mÞϕ− x mϕ + ðn−mÞϕ− n−x = ∑ 1− − 1− n n n n x x=z + 1

ð13Þ

where the transition from the first to the second line follows from the fact that individual votes are not observed (see the discussion above). This reputation loss in general decreases both with the fraction of members replaced from one period to the next and with the size of the committee (for a given fraction of members replaced). Below I illustrate these results numerically and discuss the intuition; algebraic details can be found in part B of the Appendix.11 m An increase in the fraction of members replaced, , reduces the amount of information contained in period 1 policy that is relevant for n m expectations formation in period 2. The reputation loss expected from choosing high inflation thus decreases with . On the other hand, for a given n fraction of members replaced, the effect of the size of the committee is dominated by a representation effect.12 Representation is lower in larger committees, as the impact of any individual member's vote on policy choices diminishes with the size of the committee. The public thus learns less from policy about individual members' types when the committee is large. Put differently, policy reveals the votes of the majority, where the fraction of votes constituting a majority is decreasing with the size of the committee. The larger the committee, the less policy reveals about the “median member”. m Fig. 2 illustrates the effects of n and on the CB reputation loss associated with high π. What is being depicted is the reputation-building n incentive given by h i Pr p ðπ′ = πd jπ = πd Þ−Prp ðπ′ = πd jπ = 0Þ 0:5

;

for different levels of the ex-ante probability of being a dove, ϕ. The Figure assumes that w = 0.9 (which is consistent with plausible equilibrium values, as shown below) and c = 1, and depicts each (n, m) combination as a different line. Comparing the cases of (n = 3, m = 1) and (n = 9, m = 3), note that increasing the size of the committee while keeping the fraction of members to be replaced constant reduces the reputation 9 10 11 12

I do not address here the effect of different voting rules, something the literature has begun to look at (e.g. Dal Bó, 2006; and Riboni and Ruge-Murcia, 2010). This assumption closely matches the arrangement characterizing the Colombian CB. Though the appendix uses an approximation to Eq. (13) to show some algebraic results, the discussion, figures and tables in the main text use Eq. (13) exactly. The representation factor is ΔPr(π = πd). In the committee case it equals the probability i assigns to exactly z other members voting for πd.

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Fig. 2. Reputation-building incentive in a committee with an exogenous reappointment probability.

Table 1 Equilibrium probability of a dove choosing high inflation ω in committee with an exogenous reappointment probability. m Uniform Distribution

n

3 5 7 9 11 13 15 17

1

2

1 0.9461 0.9166 0.9022 0.8939 0.8887 0.8851 0.8826

1 1 1 1 0.9815 0.9608 0.9462 0.9354

Normal Distribution 3 1 1 1 1 1 1 0.9921

4

1

2

3

4

1 1 1 1 1 1 1

1 0.9672 0.9435 0.9310 0.9236 0.9187 0.9154 0.9131

1 1 1 1 0.9894 0.9777 0.9675 0.9592

1 1 1 1 1 1 0.9937

1 1 1 1 1 1 1

n is the number of members, m is the number of members replaced from one period to the next. ϕ = 0.5.The interval for the distribution of β is (0.8,1). When the distribution is normal, parameters are chosen so that 99% of the β fall in that interval.

incentive to choose low inflation, as discussed above. On the other hand, comparing (n = 9, m = 1) and (n = 9, m = 3) shows that the incentive to choose low inflation is also reduced if the fraction of members to be replaced grows while the size of the committee remains constant. These effects hold for any value of ϕ. As further illustration of these effects, Table 1 shows equilibrium values for the probability that a dove will vote for high inflation, w, for different combinations of n and m. This equilibrium probability is calculated using Eq. (7), while allowing for β to follow different distributions. In the left panel of Table 1, β is assumed to follow a uniform distribution over the [0.8, 1] interval, while in the right panel of Table 1 a normal distribution is assumed, with a mean of 0.9 and a variance such that 99% of observations fall in the [0.8, 1] interval.13 Holding n constant at any given value, a larger fraction of members to be replaced implies less incentive to vote for low inflation (that is, w takes a larger value). A similar m effect is seen when increasing the size of the committee while holding constant. Take, for instance, the highlighted cells of Table 1, which show n m 1 committees of different sizes, while holding constant at (approximately) . The equilibrium probability of voting for low inflation falls (that is, n 14 5 w grows) as the size of the committee increases. Note, however, that the overall effect of changes in n is ambiguous, since n has a direct positive m effect on w and an indirect negative effect through , if m stays constant. n In summary, a committee can imply more reputation-building incentives for a central banker than a monolithic arrangement, thus increasing the chances that low inflation is achieved. In particular, a dove who holds his last term in office during period 1 can choose low inflation for that period in a committee setting but not in a single central banker scenario. However, this is only possible if both the size of the committee and the fractions of members replaced from one period to the next are sufficiently small. Note that the assumption that the public observes π, but not the individual vi, is central to the result that a committee design has the potential to reduce inflation bias. In some of the literature (e.g., Sibert, 2003), a similar feature appears in the assumption that individual votes are not disclosed, or are disclosed only with a lag. These assumptions accurately reflect the disclosure policies of some CBs, but I take them as also capturing a more general phenomenon: the fact that the public likely bases expectations on past policies rather than individual votes, even if those votes are disclosed in the CB's public statements. On one side, while information about monetary policy choices makes the headlines, the positions of individual central bankers receive less attention, often being restricted to specialized media. This creates costs that may prevent the public from getting information about individual votes at the CB. Moreover, tracking the behavior of individual members adds one more layer to

13 If we assume a year-long period (that is, that nominal contracts are set for a year) these assumptions are consistent with an annual discount rate of between 0 and 25%. The patterns discussed are robust to choosing intervals of [0, 1] and [0.5, 1] for the βi. 14 This matrix of equilibrium probabilities is useful for analyzing the patterns that w follows as n and m change. The levels of w, however, should not be taken literally, as the extreme assumptions made about the preferences of conservative central bankers affect the overall incentives to choose low inflation (though not, I argue, the way in which those incentives are affected by n and m).

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the (as we shall see) already sophisticated calculations involved in forming inflation expectations. It seems relevant to question whether the general public, or even the players involved in wage setting, analyze every policy decision with such a high level of sophistication.15 Before proceeding to an analysis of government appointments, it is important to note that the discussion of committee decisions in this section has been based on Eq. (9), which holds for any ΔPr(π = πd) N 0 (see Eq. (4)). However, there is also another equilibrium where all central bankers vote for low inflation (i.e., w = 0); as a result, ΔPr(π = πd) = 0. In this case, representation is zero and no member has an incentive to deviate from voting for low inflation. More generally, if we were to assume that all central bankers respond to reputation (rather than treating hawks as mechanistic), any situation in which all types pool voting for the same rate is an equilibrium, simply because none of them influence policy individually. This is an extreme example of the reduction of what I have called representation in committees, as compared to single central banker settings. As an implication, central bankers can potentially be more responsive to considerations other than the welfare consequences of policies when acting within committees. A concern for being reappointed is one of these potential alternative considerations. This is the central theme of next section. 4. The institutional dimension II: government appointments Consider now the possibility that the appointments (and reappointments) of central bankers are not exogenously determined, but rather are government choices. The government appoints central bankers for period 1, decides whether a central banker is reappointed at the end of that period, and chooses who will replace a central banker that is not reappointed. The president has a seat (but not a vote) on the Central Bank committee; hence, he can observe the votes of the central bankers.16 As a result, the reappointment of an incumbent central banker is based on his vote for period 1 policy. As for the appointment of newcomers (at the beginning of period 1, or to replace a retired incumbent in period 2), there is no relevant information available to distinguish one potential central banker from another; thus, the government chooses randomly which candidates fill the vacant spots.17 My modeling of the government is extremely simple. I leave aside the political process that determines who the president is, and assume that the government's type is known to everybody. I also focus on partisan appointments by assuming that governments want to appoint members of their own type. Since members who vote for high inflation reveal themselves as doves, a dove government tries to reappoint these members. For the same reason, a hawk government tries to replace central bankers who vote for high inflation. This strategy on the part of the government is consistent with previous models where government appointments to the CB are partisan (e.g. Waller 1992; Havrilesky, 1995, chapter 9). How does the fact that the government chooses CB appointments affect the vote of a dovish central banker during the first period? Consider again the condition under which i chooses vi = 0, Eq. (4), which I write here in a slightly different form: i h i h i1 0 h p d 4b E Pr ðπ′ = πd jπ = πd Þjvi = πd −E Pr p ðπ′ = πd jπ = 0Þjvi = 0 ΔPrðri = 1Þ 2 + ð1−ϕÞΔPrðπ′ = π Þ c AN 1: ð14Þ − βi @ 0:5 ΔPrðπ = πd Þ The government's partisan reappointment strategy implies that voting for low inflation increases one's chances of being reappointed if the government is hawk, and decreases them if the government is dove. That is, ΔPr(ri = 1)is negative (positive) when the government is hawk (dove). The reappointment factor thus creates incentives to vote for the rate preferred by the government. Furthermore, notice from the second term that the value given to reappointment relative to other objectives is decreasing in ΔPr(π = πd). This reflects the fact that the probability of being reappointed depends on the central banker's vote, while his other objectives are only affected by the actual choice of policy. Thus, representation now plays the role of determining how much each central banker values reappointment: if each vote has a low impact on the actual choice of policy, the decisions of central bankers will be mainly driven by reappointment incentives. As a result, the influence of the government on monetary policy is greatest when representation is lowest. I now return to the analysis of specific institutional designs, with an emphasis on how they affect the balance between reappointment and reputation considerations. 4.1. Government appointments and the case of a single central banker Let us start with the case of single central banker. We focus again on a dove central banker i whose period expires at the end of t, though now giving the government the power to extend that period after i′s choice of π has been observed (at the end of the first period). Remember, first, that the single central banker has perfect power over the choice of policy, such that ΔPr(π = πd) = 1. If g = {h, d} represents the type of the government, then the general reappointment strategy discussed above implies that: ΔPrðr = 1 jg = dÞ = 1 and

ð15Þ

ΔPrðr = 1jg = hÞ = −1: The structure of reappointments affects the incentives faced by central bankers, not only because they value reappointment, but also because current inflation only affects the CB's reputation in the future if the central banker is reappointed. The government's reappointment strategy 15 My results imply that inflation bias is smaller if the public bases inflation expectations on past inflation rather than individual votes, because in the former case a central banker is aware that he can affect the CB's reputation even if he is not reappointed. This makes even central bankers who are in their last term in office wary of increasing inflation, which I would argue is a fair representation of the decisions of many central bankers. My model contrasts in this respect with A. Sibert's (2003) result that publishing the votes of central bankers reduces inflation bias, obtained in the context of a model where central bankers optimize solely over their horizon in office. 16 It is indeed frequent that the government (for instance the Finance Minister or a member of his staff) participates in the regular meetings of CB committees. However, more than trying to mimic an actual CB design, this assumption is meant to capture the fact that the official in charge of deciding which central bankers stay in office is likely to have more information than the public about the past choices of central bankers. Compared to the general public, this official faces both lower costs and larger incentives to acquire such information. 17 The assumption that new appointments are uninformed decisions reflects the focus of this paper on the effects of appointments on the decisions of incumbent central bankers. Given that interest, the focus is on reappointments as opposed to the appointment of newcomers. For an analysis of the effects of the partisan appointment of newcomers, though in the absence of reputation concerns, see A. Alesina's (1987) model, and the subsequent extensions to the case of committees (e.g., Waller, 1989; Lohman, 1997).

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implies that the central banker knows he will be reappointed if and only if he votes for the inflation rate preferred by the government. Moreover, if i is reappointed, the public assigns p

d

Pr ðπ′ = π jπÞ = Prði is a dovejπÞ = 1

if π = π

d

and p

d

Pr ðπ′ = π jπÞ = Pr ði is a dovejπÞ =

ð16Þ

ð1−wÞϕ 1−wϕ

if π = 0

If, on the other hand, i is not reappointed, the public assigns p

d

Pr ðπ′ = π Þ = ϕ:

ð17Þ

Using Eqs. (15) through (17), condition (14) under which a dove central banker chooses low inflation can be written as     1 ð1−wÞϕ 4b ϕ− + + ð1−ϕÞ N1 0:5 1−wϕ c2 and    1 4b ð1−ϕÞ− 2 + ð1−ϕÞ N1 βi 0:5 c

 βi

if g = h ð18Þ if g = d:

In each row of condition (18) the first term inside the square parenthesis represents the expected reputation loss from choosing high inflation. This varies with the type of government, inasmuch as i′s vote will affect his reappointment chances differently under a hawk or a dove government. The second term in the square parentheses represents reappointment considerations, which create incentives to vote for the government's preferred rate. From condition (18) there is greater likelihood of low inflation under hawk governments than under dove ones. To see this clearly, rewrite the ð1 + ϕwÞð1−ϕÞ 1−ϕw

4b 4b when g = h, and ð1−ϕÞ− 2 when g = d. The latter expression is unambiguously smaller. In fact, under c2 c 4b the specific functional forms chosen here, a dove central banker will never choose low inflation if the government is dove, since ð1−ϕÞ− 2 b1. From c

LHS of condition (18) as

+

a broader perspective, when appointments to the CB are the government's choice, the political cycle can be a source of variability for monetary policy. I turn now to the question of how collective decision-making affects this link between monetary policy and government preferences. 4.2. Collective policy-making and government appointments

Consider now government appointments in the committee case. I continue to assume that the committee has n members and that exactly m of them are replaced between period 1 and period 2. I also continue to assume that the public uses π but not individual votes to form expectations about π′. Under these two assumptions, it is the case that inflation expectations depend both on past inflation and on how many members from the previous period were reappointed (that is, m), but not on which specific members were reappointed. As in Section 2, the probability the m public assigns to high inflation in the second period is the probability that the majority of period 2 members is dove, where for a fraction (1− ) of n the members (those who remain from the previous period) the probability that each one is dove is conditioned on π. The effect of past inflation on future expectations thus continues to be captured by Eq. (13), which I reproduce here. Pr p ðπ′ = πd jπ = πd Þ−Prp ðπ′ = πd jπ = 0Þ ! x  n−x      n n mϕ + ðn−mÞϕþ mϕ + ðn−mÞϕþ mϕ + ðn−mÞϕ− x mϕ + ðn−mÞϕ− n−x = ∑ 1− − 1− n n n n x x=z + 1

ð19Þ

where ϕ+ = Pr(i is a dove|π = πd) and ϕ− = Pr(i is a dove|π = 0) ≡ ϕ−. The influence of the committee size and the turnover rate of its members on this expression were illustrated in Fig. 2. Remember from that discussion that the incentive to protect the CB reputation may lead dove i to vote for low inflation, if the committee and the turnover rate of its members are both small. It should be noted that, although expression (19) is identical to the corresponding equation in Section 2, the value of the reputation incentive will in the case with government appointments depend on the type of the government in period 1. Intuitively, the public takes into account the fact that, as will become clear below, a central banker's choice in period 1 is not independent of the type of the government if the central banker wants to increase his chances of being reappointed. Mechanically, this is captured by the fact that ϕ+ and ϕ− depend on w ≡ Pr(vi = πd|i is a dove) (see Eq. (12)), and w in turn changes with the type of the government (see below). As for the reappointment incentive, I now assume that the president chooses who are the m committee members to be replaced between period 1 and period 2. Given his preference to keep central bankers of his same type, the president starts by replacing members who voted for the policy he likes least. Let V− g be the number of members other than i who voted against the rate preferred by the government (for instance, those members who voted for πd if the government is hawk). If m or more members voted against the government's preference (i.e. if V− g ≥ m), the president randomly chooses m members from those who voted against his preferred inflation rate, and replaces them. If, on the contrary, the number of members that voted against his preference is less than the number of members to be replaced (V− g b m), the president replaces all those V− g members, and randomly chooses the remaining m − V− g bankers to be replaced from the pool of members that voted for his preferred rate.18 These assumptions are used to derive ΔP(r1 = 1), which is then plugged, together with Eq. (13), into Eq. (14) to obtain the condition under which vi = 0 in the committee case with government appointments. This derivation is shown in part C of the Appendix. Below, I discuss the results. The above assumptions about the government's reappointment strategy imply an incentive for i to vote for the inflation rate preferred by the government. Under these assumptions, ΔP(r1 = 1) is positive if the government is dove and negative if it is hawk. How strong this incentive is vis18 If i also votes against the rate preferred by the government, V− g b m implies that there are at most m votes against the government (including that of i, if he voted against the government). It is thus still the case under V− g b m that the president replaces all members that voted against his preferred rate (including i).

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Fig. 3. Reappointment incentive if i's vote affects policy.

Fig. 4. Reappointment incentive.

a-vis other considerations, particularly the incentive to protect the CB reputation and the current gain from high inflation, depends on the value given to reappointment and the relative influence of vi on i′s chances of being reappointed compared to its influence on π. Take first the reappointment value of voting for the government's preferred rate, given by the absolute value of   4b d ΔPrðri = 1Þ 2 + ð1−ϕÞΔPrðπ′ = π Þ : c The relationship between this value and the committee's size and turnover rate is illustrated in Fig. 3. The figure assumes b = 0; larger values of b would obviously imply greater incentives to affect reappointment (especially in the case of a single central banker, since ΔPr(ri = 1) is larger in this case) but do not change the qualitative patterns described below. As above, the figure also assumes that w = 0.9. Note first that this value of reappointment is much smaller for committees than for a single central banker (represented by the thin solid line). This is so because the government has perfect power over the permanence of a single central banker (not so for a committee member), and because reappointment gives a single central banker perfect power over period 2 policy, making the incentive to seek reappointment stronger for a single central banker than for a member of committee. For sufficiently low values of b, the value given to reappointment is also small with respect to the range of values for the effect of reputation-building shown, for instance, in Fig. 2. Obviously, this value is further reduced by rules that limit the number of members over whose appointment the government has power (as seen, for instance when comparing the lines for (n = 9, m = 1) and (n = 9, m = 3) in Fig. 3). The ability of collective decision-making to minimize the effect of reappointment considerations, however, is considerably reduced when representation is brought into the picture. A central banker's vote affects his chances of reappointment with a probability of 1 but only affects policy, and the reputation of the CB, with a probability of ΔPr(π = πd). This tension between individual and collective reputation does not arise in the single central banker case. As a result, the potential benefit from a committee design in terms of isolating the CB from government pressures is reduced. Fig. 4 illustrates the reappointment incentive after taking representation into account (that is, the value of reappointment from Fig. 3, but now divided by ΔPr(π = πd)). Note this is the second term in the LHS of condition (14). Relative to Fig. 3, the reappointment incentive for members of committees moves closer to that faced by a single central banker. Once again, the size of the committee and the rate of turnover of its members are critical in determining the extent to which committees reduce government pressures over monetary policy. In particular, a smaller n increases representation, reducing the relative importance given to reappointment (compare (n = 3, m = 1) and (n = 9, m = 3)).19 Moreover, the overall 19 The size effect can potentially change its sign due to the presence of an opposite, but second order, effect: the value i assigns to being reappointed decreases with n, because his ability to affect period 2 policy is reduced. This explains why the dashed line in Fig. 4 is not always above the thick solid one. However, for most of the relevant range of ϕ, the effect of n the reappointment incentive is increasing as discussed above.

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Fig. 5. Benefit of choosing low inflation in a committee.

reappointment incentive is smaller for lower turnover rates (compare (n = 9, m = 1) and (n = 9, m = 3)), reflecting the fact that the likelihood that i will be penalized for voting against the government's preference grows with the number of members the government can replace. Comparing the magnitudes of the reputation building and reappointment incentives depicted, respectively, in Figs. 2 and 4, it is clear that the reappointment incentive is not negligible. This is so even though Fig. 2 is built under the assumption that b = 0. This analysis thus shows that even ideologically driven central bankers want to keep their jobs and are therefore not immune to government pressures, contradicting the often heard argument that government appointments do not distort policy because most central bankers are not opportunistic. There is of course a fundamental difference between opportunistic and partisan central bankers, in that the latter are driven by a desire to choose the policy they believe is best. It is also the case that opportunistic central bankers (those with b N 0) have an additional reason for wanting to stay in office, and thus are confronted with stronger reappointment incentives. The balance between the different incentives faced by i – reputation, reappointment, and the current gain from high inflation – determines i′s choice of vi, as captured by condition (14). Fig. 5 shows this balance, depicting the term inside the parenthesis in the LHS of condition (14), under the same assumptions and conventions used for Figs. 3 and 4. For ease of exposition, I focus the discussion below on the case where the government is dove (grey lines); for this case, the two goals we have assumed for the institutional design of the CB – increasing reputationbuilding incentives and reducing reappointment considerations – act in the same direction in terms of what inflation rate i should vote for. Fig. 5 shows that, when the government is dove, the overall benefit of voting for low inflation, against the preferences of the government, is m m larger under a committee than with a single banker only under some assumptions for n and . An increase in the turnover rate, , reduces the n n relevance of past inflation for future expectations of inflation, thus diminishing the ability of reputation considerations to contain inflation bias. It also increases the risk that a central banker will be removed from office if he is believed to be of a different type than the government, thus strengthening incentives to follow the government's preferred policy. This is why, in the Figure, a (n = 9, m = 1) committee generates larger incentives to vote for low inflation than does a (n = 9, m = 3) committee. In fact, while the former design (with a smaller turnover rate), is always better than a single central banker in terms of reducing inflation bias, the same cannot be said for the (n = 9, m = 3) committee. On the other hand, an increase in the size of the committee, n, reduces the ability of any member to have an effect on actual policy. Central bankers thus m 1 become more concerned with influencing reappointment than with choosing policy. This is why, keeping = , the benefit of voting for low n 3 inflation is larger if n = 3 than if n = 9 for all values of ϕ. Perhaps the most dramatic expression that, against common wisdom and the previous literature, a committee structure ends up increasing the influence of the government if the committee is not properly designed, is the possibility of a pooling equilibrium around the inflation rate preferred by the government in a committee scenario. Note that under a hawk government there is at least one such pooling equilibrium, where each central banker votes for low inflation. This is so because a pooling equilibrium leads to zero representation; if all the other committee members are voting for π = 0, i′s vote will have no impact on policy. Given this, i′s vote will only affect his reappointment chances, and i will not want to deviate from voting for the government's preferred rate. Under a more general setting where hawk central bankers are not mechanistic, but also respond to reappointment incentives, there is a pooling equilibrium where all committee members vote for the rate preferred by the government, no matter what the government's type is. Table 2 shows equilibrium values of w. As in Table 1, it is assumed that ϕ = 0.5, and that β can follow either a uniform distribution between 0.8 and 1 or a normal distribution with 99% of observations concentrated between 0.8 and 1. Given the existence of a pooling equilibrium with w = 0 under a hawk government, for this type of government the table only shows values of w other than w = 0 (and only for combinations of n and m for which equilibria with w ≠ 0 exist). The key question here is whether certain combinations of n and m can make i deviate from voting for the rate preferred by m m the government. Table 2 indicates that the answer is affirmative only for small enough n and , and that the effect of seems to dominate.20 n

n

5. A discussion of applications and related extensions I have used the model presented above to address the question of whether committees are more or less fit to fight inflation and insulate policy from government pressures, and how this ability depends on the size of the committee and the rate of turnover of its members. The model can be extended in different ways to address more specific questions about the optimal design of the monetary authority, and to overcome some limitations imposed by the simplifying assumptions adopted above. I sketch here three simple extensions that illustrate uses for the model. 20 It is important to keep two facts in mind when reading Table 2. First, for the hawk government case there is no tension between reappointment and reputation: both generate m m incentives to vote for low inflation. Second, while the size effects I have discussed are for a given , is obviously not independent from n.

n n

M. Eslava / Journal of Public Economics 94 (2010) 363–379

375

Table 2 Equilibrium probability of a dove choosing high inflation ω (other than ω = 0). m Hawk Government

Dove Government

Uniform Distribution

n

Single

3 5 7 9 11 13 15 17

1

2

0.7070 . . . . . . . 0.7178

1 0.7757 . . . . . .

Normal Distribution 3 1 . . . . . .

4

1

2

1 1 . . . . .

0.7629 . . . . . . 0.7253 0.7356

1 0.8619 . . . . . .

Uniform Distribution 3 1 0.894 . . . . .

4

1

2

1 1 0.9047 . . . .

1 1 1 1 0.9761 0.9581 0.9455 0.9362 1

1 1 1 1 1 1 1 1

3 1 1 1 1 1 1 1

Normal Distribution 4

1

2

3

4

1 1 1 1 1 1 1

1 1 1 1 0.9868 0.9757 0.9664 0.9589 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

n is the number of members, m is the number of members replaced from one period to the next. ϕ = 0.5. A "." indicates there is no equilibrium ω other than ω = 0. The interval for the distribution of β is (0.8,1). When the distribution is normal, parameters are chosen so that 99% of the β fall in that interval.

5.1. The cyclicality of monetary policy Policy swings and consequent output cycles have been a central concern of the literature on central banking institutions. One source of policy cycles that the literature has focused on is the electoral cycle. If governments affiliated with different parties have different policy preferences, changes in government may lead to policy swings; with electoral uncertainty these shifts further imply output cycles (Alesina, 1987). These changes are not granted by economic fundamentals and under usual assumptions about welfare they have negative effects.21 Waller (1989, 2000) and Lohman (1997) extend that model to the case of monetary committees, and study the effects of changing some features of the committees, with findings that I discussed in the introduction. By looking into the effects of committees on the policy positions of central bankers rather than assuming these positions are given, this paper points to an additional source of electoral cycles in monetary policy. The model presented above implies that even the same central banker may choose different policies under different governments, when his permanence in office is decided precisely by the government. Put in terms of the different incentives faced by central bankers in the exposition above, the reappointment incentive is a source of policy cyclicality. Although this result is implied by the comparison between the cases of dovish and hawkish governments in Section 4, the dynamics of the model can be enriched to better illustrate cycles. Consider, in particular, an overlapping generations version of the model. Central bankers live for n−m n+m two periods, but the horizon for the economy if infinite. In any given period the CB committee is made up of “old” and “young” central 2 2 bankers. Each government is in office for one period and there is alternation between dovish and hawkish governments. During his period in n−m office, the president decides which m young central bankers are not reappointed; those m members plus all old members are replaced in the 2 following period by young central bankers randomly chosen from the unconditional distribution of central bankers. All other assumptions remain as in the basic framework presented in the preceding sections. The main difference between the two versions of the model is that, with overlapping generations, at each point in time there are both old and young central bankers. The former vote for high inflation if and only if they are doves, while the latter vote for high inflation if they are doves with probability w, where w depends on the type of the government, and thus cycles between wg = d and wg = h. Being this the only difference with the two-period model, a young dove policy maker continues to behave as discussed in Section 4.2.22 From the discussion in that section, wg = d N wg = h. The distance between wg = d and wg = h implies electoral cycles in the votes of CB members, with the difference increasing in both m and n. As discussed above, larger committees increase the relative importance of reappointment vis-a-vis other considerations, while higher turnover rates imply a higher risk of not being reappointed for not supporting the policy preferred by the government. From this perspective, reducing political cycles in monetary policy requires putting policy in the hands of smaller committees with members that face a lower probability of being removed by the government. 5.2. Extending the horizon of central bankers The assumption that central bankers live for two periods has been helpful to simplify an already complicated framework. However, it has some restrictive implications. The overlapping generations version discussed above addresses some of those issues, but leaves others open. In particular, since in the second and final period every central banker correctly represents his type, there are abrupt differences between the choices of a dove central banker in the two periods of his life. Moreover, policy in the second period is, as a consequence, highly dependent on the unconditional distribution of types. The basic model can be modified to consider central bankers who face longer horizons. Besides addressing the issues just mentioned, the comparison between different horizons is useful to study an additional dimension of the institutional design: the maximum number of terms an individual can serve as member of the CB committee. Extending the horizon potentially affects both the incentive to be reappointed and the incentive to protect the central bank reputation. Consider, in particular, extending the basic model of Section 2 to three periods: t = 0, 1, 2. Each dove central banker in office in period 1 is identical to the period 1 central banker studied in Sections 2 through 4: he can potentially be reappointed to period 2, and faces the 21 In my model, inflation (i.e. policy) variability is indeed assumed to generate utility losses. More generally, output cycles associated with policy changes are also welfaredecreasing. 22 The mechanical difference between the two models is that if dove central banker i is not reappointed, his replacement is known vote for π′ = π with probability ϕwg in the overlapping generations model and with probability ϕ in the two period model. This affects the relative importance of reappointment vis-a-vis reputation: being reappointed is more valuable with overlapping generations because the likelihood that i′s replacement votes for i′s preferred policy is reduced. The main insights from Section 4.2 (that wg = d N wg = h, and the way in which this difference is affected by n and m), however, continue to hold.

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M. Eslava / Journal of Public Economics 94 (2010) 363–379

considerations analyzed above. In turn, a dove central banker in office in period 0 faces potential reappointment to period 1. The consequences for the reputation of the CB of his policy choices in t = 0 can potentially extend two periods into the future. The condition under which a dove in office in t = 1 votes for π = 0 continues to be condition (4), where I keep π and π′ to refer to inflation in t = 1 and t = 2, respectively. Introducing π0 to refer to inflation in t = 0, and vi0 for dove i′s vote in that period, the corresponding condition for choosing vi0 = 0 is given by: d

βi ΔPrðπ0 = π Þ

h i h i E Prp ðπ = πd Þ + βi Prp ðπ′ = πd Þjπ0 = πd ; vi0 = πd −E Prp ðπ′ = πd Þ + βi Pr p ðπ′ = πd Þ jπ0 = 0; vi0 = 0

0:5    4b 4b d d Δ Prðri = 1Þ d ≥ Δ Prðπ0 = π Þ: −βi ΔPrðri0 = 1Þ 2 + ð1−ϕÞwΔPrðπ = π Þ + βi 2 + ð1−ϕÞΔPrðπ′ = π Þ 0 ΔPrðri0 = 1Þ c c

ð20Þ

Following the notation in previous sections, I am using ΔPr(x0) ≡ Pr(x0|vi0 = πd) − Pr(x0|vi0 = 0). Also, to capture the possibility that i′s vote in period 0 affects his chances of being reappointed one period ahead (from t = 1 to t = 2), I have used Δ0Pr(ri = 1) = Pr(ri = 1|vi0 = πd) − Pr(ri = 1| vi0 = 0). Comparing expressions (4) and (20) shows that extending the horizon of a central banker has two effects. First, it extends reputation building incentives to consider the effects of vi0 on inflation expectations not only in the following period, but also in periods further into the future. Reputation building concerns are thus increased. On the other hand, the effects of vi0 on reappointment further into the future are also brought into the picture, implying that reappointment incentives are also potentially increased. Whether vi0 matters for reappointment not only in the current period but also further into the future, and how much it matters, depends on the specific rules concerning reappointment. Consider, for instance, rules that create some incumbency advantage: by being reappointed at the end of t = 0, i improves his chances of being reappointed again in t = 1. In this case the incentives to please the government in t = 0 are maximized. What this discussion suggests is that extending the horizon of a central banker reduces inflation bias (by strengthening reputation building incentives), potentially at the cost of increasing the likelihood of observing political cycles in policy. Reappointment rules, however, can be designed to minimize this cost. For instance, rules that preclude reappointment for more than two periods reduce reappointment incentives in period t = 0 (and, if central bankers care about general welfare even beyond their own terms in office, do not imply costs in terms of inflation bias). 5.3. Central bankers that abstain from voting Throughout the paper I have followed the literature in assuming that a CB committee member does not abstain from voting. Opening the door to the possibility of abstaining is an interesting potential extension. The likelihood that a central banker chooses to abstain should be greatest when his vote is unlikely to affect policy, while it has other implications, possibly negative for the central banker. This possibility arises naturally when government appointments are being considered and a central banker's vote affects his chances of being reappointed, as in Section 2. Going back to the basic setting of that section, suppose that in any period the central banker first chooses whether to vote or abstain, and then, if he chose the former, he decides whether to vote for π = 0 or π = πd. His decision follows a backward induction analysis: he solves for his optimal vote if he were to vote, and based on this optimal rule decides whether he should vote or not. Assume, further, that each member has little chances of affecting current policy (that is, ΔPr(π = πd) is low). As a consequence, i′s vote will mostly affect his chances of being reappointed. He is thus most likely to abstain if, in the case he were to vote, he would do so against the government preferred rate. Votes would thus be more likely cast by members who vote with the government, with the consequence that political cycles become more likely. This analysis suggests another reason why CB committees should be small and with low turnover rates. These two features make it less likely that central bankers have sufficiently low chances of affecting policy and sufficiently high chances of being removed by the government, that they decide to vote only if they will support the government's preferred policy.

6. Concluding remarks This paper studies how government appointments and committee design affect the structure of incentives faced by central bankers. This is in contrast with previous literature that either abstracts from the influence of the government, or assumes that a central banker always votes in line with the preferences of the government that appointed him. I also explore the idea that collective decisionmaking creates a dichotomy between the collective and individual reputations of board members, in a context where the former depends on past policy outcomes, and the latter on the members' individual decisions. Key insights are obtained from this approach. First, an incumbent central banker obtains benefits from voting for the policy stance preferred by the government in charge of evaluating his permanence in the committee. Second, in a committee there is a greater probability that the vote of a central banker will affect his reappointment chances than policy. his increases the importance central bankers give to reappointment. In fact, there are always equilibria in which all members vote in the first period for the rate preferred by the government; in such equilibrium each individual member knows with certainty that he will not be able to affect policy,

but that his vote will affect his chances of staying in office. Finally, the influence of the government on the choices of a central banker increases with the government's ability to remove members of the committee, given by the turnover rate. As a result of all of the above, the potential ability of committee designs to reduce government influence depends on the specific characteristics of the committee in terms of size and turnover rate. The influence of the government grows with the size of the committee (which reduces representation, thus making reappointment relatively more important), and with the turnover rate.

Acknowledgments I thank Miguel Rueda for his superb research assistance. I also thank Allan Drazen, John Shea, Debbie Minehart, Antonio Merlo, and two anonymous referees for their many good comments. Useful discussions were also provided by participants at seminars at the University of Maryland, the 4th meeting of the Political Economy Network of LACEA, the 8th Annual Meeting of LACEA, and the Norges' Bank Workshop of Monetary Policy Committees. All remaining errors are mine.

M. Eslava / Journal of Public Economics 94 (2010) 363–379

377

Appendix A. Proof of result 3 Since his vote can only take the values of 0 and πd , the problem of dove central banker i in period 1 can be rewritten as23: Min fvi g " d

Prðπ = π jvi Þ

c2 − 4

!

#   d + Prðπ = 0jvi Þ½βi EðL′ jvi ; π = 0Þ; + βi E L′ jvi ; π = π

where EðL′ jvi ; πÞ =

−

! i c2 h Prðri = 1j vi ÞPrðπ′ = πd j ri = 1Þ + Prðri = 0 jvi ÞPrðπ′ = πd jri = 0Þ + 4

! i c2 h p E Pr ðπ′ = πd j πÞj vi −bPrðri = 1 jvi Þ: 2

I have abstracted from the constant terms, and made use of the fact that πe′ = πd * Prp(π′ = πd|π). I now use the specifics of each case (single central banker and a CB committee) to derive a unique expression. For the committee case, as discussed in Section 2, E[Prp(π′ = πd|π)|vi] = Prp(π′ = πd|π). This is because we assume that inflation expectations depend on past inflation, rather than individual votes, and because the number of members not reappointed is constant. Terms involving solely Prp(π′ = πd|π) are thus constants in the dove problem. Abstracting from these and other constants, we can re-write the dove problem for the committee case as Min fvi g c2 Prðπ = π jvi Þ − 4

!

d

d

! i c2 h p Pr ðπ′ = πd jπ = πd Þ−Prp ðπ′ = πd jπ = 0Þ 2

+ Prðπ = π jvi Þβi ! c2 ð1−ϕÞPrðri = 1jvi ÞΔPrðπ′ = πd Þ: −βi bPrðr1 = 1jvi Þ + βi − 4

The last term makes use of the fact that, if ri = 0, i's replacement votes for πd in the second period with a probability of ϕ. Since a dove chooses vi = 0 if L(vi = 0) ≤ L(vi = πd), result 3 follows, using Prp(π′ = πd|π) = E[Prp(π′ = πd|π)|vi] for any vi. For the single central banker case, meanwhile, the problem can be written as Min fvi g d

Prðπ = π jvi Þ −

c2 4

! + βi

! ! i c2 h p ′ c2 d i ′ d E Pr ðπ = π j π = v Þjvi −βi bPrðr1 = 1 jvi Þ + βi − ð1−ϕÞPrðri = 1jvi ÞΔPrðπ = π Þ: 2 4

The expression in result 3 follows, taking into account that ΔPr(π = πd) = 1 for the single central banker case. B. Reputation loss in committees as a function of n and m Taking a first order approximation to Eq. (13) around ϕ+ = ϕ and ϕ− = ϕ one obtains  n m þ ϕ −ϕ− ∑ Pr ðπ′ = π jπ = π Þ−Pr ðπ′ = π jπ = 0Þ≈ 1− n x=z + 1 p

d

d

p

d

"

n x

! x

n−x

ϕ ð1−ϕÞ

# x n−x − ; ϕ 1−ϕ



ð21Þ

  m m which is increasing in ðϕþ −ϕ− Þ 1− . The reputation loss is thus decreasing in if (ϕ+ − ϕ−) N 0. To see that, indeed, (ϕ+ − ϕ−) N 0 work from n n Eq. (5): 

þ

−

ϕ −ϕ

! Prðπ = πd ji is a doveÞ 1−Prðπ = πd ji is a doveÞ − ≡ Prði is a dovejπ = π Þ−Prði is a dove jπ = 0Þ = ϕ Prðπ = πd Þ 1−Prðπ = πd Þ ! ! d d d Prðπ = π ji is a doveÞ−Prðπ = π Þ Prðπ = π ji is a doveÞ−Prðπ = πd j i is a hawkÞ =ϕ  = ϕð1−ϕÞ  Prðπ = πd Þ 1−Prðπ = πd Þ Prðπ = πd Þ 1−Prðπ = πd Þ d

= ϕð1−ϕÞ

wΔPrðπ = πd Þ  Prðπ = πd Þ 1−Prðπ = πd Þ

where the fourth line uses the fact that Pr(π = πd) = ϕPr(π = πd|i is a dove) + (1 − ϕ)Pr(π = πd|i is a hawk) , and the fifth line uses Eq. (6). 23

Here, I make use of the fact that cπd =

c2 , 2

 2 c2 and −cπd + πd = − : 4

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M. Eslava / Journal of Public Economics 94 (2010) 363–379

Note that (ϕ+ − ϕ−) does not depend on m because it refers to the probability that a member known to have been reappointed is a dove. Also, it is directly related to ΔPr(π = πd). Moreover, note that d

ΔPrðπ = π Þ = Pr

n−1 =

  n−1 d of others vote for π 2

!

n−1 n−1 2 ð1−ϕwÞ 2

ðϕwÞ

n−1 2

ð22Þ

N0

and hence (ϕ+ − ϕ−) N 0. The size of the committee affects negatively the representation factor, ΔPr(π = πd) (see the text and Eq. (22)). It thus has a negative impact on + (ϕ − ϕ−). The final impact of size on Eq. (21) depends also on its effect on the third term of this Eq. (second line). This effect is negative in general, but can be positive for values of ϕ far from 0.5 (see Eslava, 2007, for a detailed discussion). The negative impact of n on representation dominates for the relevant range of w, and as a result Eq. (12) decreases with n. C. Condition to vote for π = 0 in the committee case with government appointments Assuming that the government is hawk, central banker i faces the following probabilities of being reappointed: d

Prðri = 1jvi = π ; g = h; V−g Þ = 1−

m if V−g ≥ m V−g + 1

and d Prðri = 1 jvi = π ; g = h; V−g Þ = 0 if V−g b m while Prðri = 1jvi = 0; g = h; V−g Þ = 1 if V−g ≥ m and Prðri = 1 jvi = 0; g = h; V−g Þ = 1−

m−V−g if V−g b m n−V−g

where V− g + 1 in the first of these expressions is the total number of votes against the rate preferred by the hawk government when i votes for πd and V− g other members vote for πd as well. The government's strategy and the implied reappointment probabilities when the government is dove are equivalent to the ones just described, only vi = 0 is replaced by vi = πd, and V− g is the number of members other than i who voted for π = 0. Let ρ be the probability assigned by i that any other member votes against the government's preferred rate (ϕw if the government is hawk and (1 − ϕw) otherwise). If the government is hawk, then the reappointment incentive is given by ΔPðr1 = 1; g = hÞ = "



m−1

∑

V−g = 0

ð23Þ

!  n−1 V−g n−1−V−g m−V−g −1 + ρ ð1−ρÞ V−g n−V−g



n−1

∑

V−g = m

!#  m n−1 V−g n−1−V−g −1 ρ ρ 1− x V−g + 1

ð24Þ

If the government is dove a similar expression applies, letting V− g equal the number of votes for π = 0. In general. the change in the probability of i being reappointed can be written as ΔPðr1 = 1Þ = " 

m−1



∑

V−g = 0

ð25Þ

!  n−m n−1 V−g n−1−V−g ρ ð1−ρÞ + V−g n−V−g

n−1

∑

V−g = m



!#  m n−1 V−g n−1−V−g ρ ρ ; x V−g + 1 

where the negative sign is for the case where the government is hawk, and where we have used Moreover, note that

d

ΔPrðπ′ = π Þ =

n−1 n−1 2

!

n−1 n−1 2 ð1−ϕÞ 2 :

ðϕÞ

m−V−g −1 n−V−g



 =−

n−m n−V−g

 :

ð26Þ

M. Eslava / Journal of Public Economics 94 (2010) 363–379

379

The condition for i choosing vi = 0 in the committee case with government appointments is obtained by plugging expressions (22), (26), (13), and (23) into condition (14) to obtain: 0

n

∑

n

!

þ

mϕ + ðn−mÞϕ n

x     þ n−x −  − mϕ + ðn−mÞϕ mϕ + ðn−mÞϕ x mϕ + ðn−mÞϕ n−x 1− − 1−

1

B C n n n B C x x=z + 1 B C B C 0:5 B C 2 0 1 3 B " C ! ! # B C n−1 n−1 n−1 m−1 n−1 n−1 n−1 C x n−1−x n−m x n−1−x m 4b βi B B ∑ + ∑ ρ ð1−ρÞ ρρ T4 2 + ð1−ϕÞ@ n−1 AðϕÞ 2 ð1−ϕÞ 2 5 CN 1: B C n−x x + 1 c x = m x x x = 0 B C 2 B C 0 1 B C n−1 n−1 n−1 B C @ A @ AðϕwÞ 2 ð1−ϕwÞ 2 n−1 2

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