Incorporating Rigidity and Commitment in the Timing Structure of Macroeconomic Games

1

Jan Libich2 La Trobe University, Dep. of Economics and Finance, and CAMA, ANU Petr Stehlik3 University of West Bohemia, Department of Mathematics Abstract This paper proposes a novel framework that generalizes the timing structure of macroeconomic (as well as other) games. Building on alternating move games and models of bounded rationality and rational inattention, the players’actions may be rigid, ie optimally chosen to be infrequent. This rigidity makes the game more dynamic/asynchronous and acts as a commitment mechanism. Therefore, it can enhance cooperation and often eliminate ine¢ cient equilibrium outcomes. We apply the framework to the Kydland-Prescott-Barro-Gordon monetary policy game and derive the conditions - the su¢ cient degree of policy commitment - under which the in‡uential time-inconsistency problem disappears. Interestingly, (i) this can happen even in a …nite game (possibly as short as two periods) and (ii) without reputation building. Furthermore, (iii) the required degree of policy commitment may be under some circumstances arbitrarily low and under others in…nitely high. Finally, (iv) this degree is a function of the structure of the economy and the policymaker’s preferences - among other it is increasing in wage rigidity, the policymaker’s temptation, and his impatience. The latter results suggests that policy commitment may substitute for patience in achieving time-consistency and credibility. The analysis hence provides an explanation for the observed trend towards explicit in‡ation targeting (commitment) and central bank independence (patience) as well as the fact that central banks initially lacking independence (eg those in New Zealand and UK) have more explicitly committed to targeting low in‡ation (than eg those in Germany and the US). We present empirical evidence for all the predictions of our analysis. Keywords: asynchronous moves, dynamic games, commitment, rigidity, time inconsistency, in‡ation targeting, central bank independence JEL classi…cation: C70, C72, E42, E61 ‘Some decisions by economic agents are reconsidered daily or hourly, while others are reviewed at intervals of a year or longer... It would be desirable in principle to allow for di¤erences among variables in frequencies of change and even to make these frequencies endogenous. But at present, models of such realism seem beyond the power of our analytical tools’. Tobin (1982) (quoted in Reis (2006)). 1 We thank Damien Eldridge, Andrew Hughes Hallett, Jonathan Kearns, Glenn Otto, Je¤ Sheen and the participants of the 23rd Australasian Economic Theory Workshop, 10th Australasian Macroeconomics Workshop, 34th Australian Conference of Economists, and seminars at the University of New South Wales, La Trobe University, University of Bayreuth, and University of Queensland. The second author gratefully acknowledges the support by the Ministry of Education, Youth and Sports of the Czech Republic, Research Plan No. MSM 4977751301. The usual disclaimer applies. 2 Corresponding author: Dr Jan Libich, La Trobe University, School of Business, Melbourne, Victoria, 3086, Australia. Phone: (+61) 3 94792754, Fax: (+61) 3 94791654, Email: [email protected]. 3 University of West Bohemia, Univerzitni 22, Plzen, 30614, Czech Republic. Phone: (+420) 777 846 682, Fax: (+420) 377 632 602, Email: [email protected].

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1. Introduction Some economic decisions are more frequent (less rigid) than others. The macroeconomic theory has long taken notice of various rigidities; in attempt to explain some observed phenomena. Empirical research followed and provided convincing micro-level evidence of the rigidity of, among other, price and wage setting.4 The presented paper takes rigidity a level up and incorporates it into the timing structure of macroeconomic games. The motivation is to bridge a gap between the micro-founded models of the economy used in macroeconomics, in which rigidities play a central role, and the rigidity-free solution concepts applied to these very models. This refers to both repeated games and the rational expectations solution – in both it is commonly assumed that players move simultaneously and do so each period. Since both the ‘simultaneity’and the ‘‡exibility’assumptions have been questioned,5 we provide an alternative framework to consider situations in which players’choices may be infrequent. This is consistent with the concepts of ‘economically rational expectations’ (Feige and Pearce (1976) and ‘rational inattention’ (Sims (2003), Reis (2006)) - in which information sets are a result of a cost/bene…t analysis by the agents. It will become apparent that as rigidity ties the hands of the players it makes the environment more dynamic and asynchronous and takes the role of commitment. This implies that it can help enhance cooperation between players in settings in which ine¢ cient outcomes may otherwise result in equilibrium. Our general setup can be summarized by one parameter ; 0

i t

1;

which denotes the probability that player i cannot move in period t. This nests (i) standard repeated games (in which it = 0, 8i; t), (ii) alternating move games of Maskin and Tirole (1988) t+1 t and Laguno¤ and Matsui (1997) (in which, 8t 2 N, it = ( 1)2 +1 and jt = ( 1) 2 +1 ), as well as (iii) the popular probabilistic speci…cation of rigidity by Calvo (1983) (in which it = i ,8i; t 2 N): In this paper we concentrate on (iv) the deterministic discrete rigidity setting of Taylor (1979), in which 0 if 8i and t = 1 + (n 1)ri where n; r 2 N; i t = 1 otherwise. This intuitive case (see Figure 1 for an example) does not only best …t Tobin’s (1982) observation quoted above but is also representative of the intuition and richness of asynchronous decision making. Naturally, we de…ne player i’s rigidity/commitment, ri 2 N, to be the number of periods for which the respective action cannot be altered.6 Let us spell out some of the advantages of our framework: Generality. The framework can be applied to any model in discrete time, continuous time as well as time scales (the latter being a recent generalized mathematical environment which 4 For recent surveys of empirical evidence see Apel, Friberg and Hallsten (2005) and Bewley (2002) respectively. For the seminal theoretical contributions see eg Fischer (1977), Taylor (1980), Calvo (1983), or Mankiw and Reis (2002). 5 In terms of simultaneity, Laguno¤ and Matsui (1997) argue that ‘[w]hile the synchronized move is not an unreasonable model of repetition in certain settings, it is not clear why it should necessarily be the benchmark setting. . . ’. In terms of incorporating some in‡exibility, in addition to the above see a growing body of literature examines some sort of inertia/stickiness/rigidity in updating/forming expectations (see eg Ball (2000), Mankiw and Reis (2002), Carroll (2003), Carroll and Slacalek (2006), Morris and Shin (2006)). 6 The terms rigidity and commitment can be used interchangeably in our framework. While a game theorist will …nd it natural to think of commitment (since the interest lies in the e¤ect on the game), a macroeconomist may want to use the term that better re‡ects the particular underlying circumstances. We will follow the latter practice in this paper.

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

i j k

i

ri

i j

rj

rk

3

i j

k

Figure 1. Timeline of an asynchronous game with deterministic rigidity/commitment in discrete time: an example with three players and ri = 3; rj = 4; rk = 6: nests both discrete and continuous time as special cases, see eg Bohner and Peterson (2001)).7 Further, unlike a standard repeated game our framework enables us to examine: (i) concurrent rigidity/commitment of more than one player (ii) various degrees of such rigidity/commitment (iii) endogenous determination of rigidity/commitment as players optimal choices. Some of these features (one at a time) have already been examined in games.8 This existing work provides a strong justi…cation and motivation for our general approach; for example Cho and Matsui (2005) argue that: ‘[a]lthough the alternating move games capture the essence of asynchronous decision making, we need to investigate a more general form of such processes. . . ’. It will be apparent than some of the results on the e¤ect of standard commitment will be re…ned and in fact partly quali…ed in the rigid world. Familiarity. The framework adopts all the main assumptions of a standard repeated game, eg it starts with a simultaneous move, rigidity/commitment is constant throughout each game, and all past periods’actions are observable (ie games of perfect monitoring). Realism. Players’rigidity and commitment introduce some asynchronicity in the game and make the game more dynamic. This combination of perfect and imperfect information is arguably a good description of many repeated real world interactions. Further, the framework does not rely on the in…nite horizon - a unique equilibrium (the e¢ cient one) can commonly be obtained in a …nite game even without reputational considerations. Finally, the framework captures Tobin’s observation quoted above about varying frequency of agents’actions and its endogeneity. Simplicity. While allowing for the above extensions some general results will be proven that demonstrate the tractability of our framework. In our deterministic setting the game can be solved by subgame perfection. Furthermore, the solution is often as simple as that of a one shot game since the most important ‘action’will occur in the initial simultaneous move (Theorem 2). To demonstrate the framework we use one of the most in‡uential macroeconomic games due to Kydland and Prescott (1977) and Barro and Gordon (1983) (referred to as BG). It is shown that the famous time inconsistency result may no longer obtain in the rigid world. Speci…cally, we derive the su¢ cient conditions for the e¢ cient ‘Ramsey’outcome of credibly low in‡ation - that is not a Nash equilibrium in the standard ‘rigidity-free’game - uniquely obtains in equilibrium 7 For detailed treatment of the framework in continuous and time scales calculi see Libich and Stehlik (2007) - a summary will be provided in Section 6 below. 8 Feature (i) is investigated in the alternating move games of Maskin and Tirole (1988), Laguno¤ and Matsui (1997) and Cho and Matsui (2005). In terms of (ii) Wen (2002) examines a …rst simple step in this direction. Finally, Bhaskar’s (2002) work in which leadership is endogenously determined o¤ers an avenue to incorporate (iii).

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of the rigid game (on the equilibrium path of all subgame perfect Nash equilibria, Theorem 3). It is further shown (Propositions 1 and 2) that the required degree of commitment is a function of the characteristics of the economy (eg wage rigidity) and the players’ preferences (eg the policymaker’s in‡ation aversion and discount factor). Interestingly, there exist circumstances under which this degree is arbitrarily low (Theorem 1) and in…nitely high (Theorem 3). Our …rst policy result is that both monetary policy commitment and patience/conservatism reduce in‡ation and its variability (under some - but not all - circumstances, Proposition 3). We discuss how the policymaker’s commitment has been achieved in the real world context drawing a link to the observed trend towards explicit in‡ation targeting and transparency. Since the in‡ation target (IT) is transparently incorporated in the central banking legislation, it cannot be frequently reconsidered and the choice of the long-run in‡ation level is therefore rigid.9 Similarly, policy patience/conservatism have come under the heading of central bank (goal) independence whereby independent central bankers have been granted a longer term in o¢ ce which arguably makes them more patient and in‡ation averse. Our second policy result then follows: explicit commitment to an IT can substitute for central bank goal-independence in ensuring the credibility of low in‡ation (Proposition 2). This substitutability o¤ers an explanation for the fact that ITs have been made more explicit in countries that had lacked central bank goal-independence in the late 1980s such as New Zealand, Canada, UK, and Australia rather than those with an independent central bank such as the US, Germany and Switzerland.10 The rest of the paper is structured as follows. Section 2 presents two versions of the BG game, one general and one speci…c. Section 3 sets up the scene of the deterministic setup that is the focus of this paper. Section 4 reports our results, …rst under fully patient players and then under impatience. Section 5 brings empirical evidence for our results, also reconciling some contradictory …ndings of the existing literature. Section 6 discusses their robustness and a number of extensions. Section 7 summarizes and concludes. 2. The Monetary Policy Game In the BG game there are two players, the policymaker g and the public p whose instruments are in‡ation, ; and wage in‡ation, w; respectively.11 The economy is described by a simple Lucas surprise-supply relationship (1)

yt = (

t

wt );

where t 2 N; > 0; and y denotes the output gap. The players’discount factors are g and p whereby we will call those with = 1 as (fully) patient and those with < 1 as impatient. The players’one period utility functions are the following (2) (3)

ugt =

(

upt =

~ )2 + yt ;

t

(

t

wt )2 ;

where ~ is the optimal in‡ation level (explicit or implicit target), and > 0 describes the policymaker’s relative weight between its objectives (of stable in‡ation and high output). The intuition is standard, the public cares about correctly expecting the in‡ation rate in order to set 9 The stochastic extension of this paper, Libich (2006), shows that this does not reduce the policymaker’s short run ‡exibility to stabilize shocks and output which we discuss in Section 6. 10 For the lively in‡ation targeting debate in regards to the Fed see Bernanke (2003), Goodfriend (2003), Kohn (2003), McCallum (2003), Friedman (2004), Mishkin (2004). 11 Unlike in Cho and Matsui’s (2005) alternating move game we do not need to assume that the public is composed of in…nitessimal agents. Moreover, our framework allows to examine heterogeneous agents within the public which we discuss in Section 6.

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wages at the market clearing real wage level (for a justi…cation based on Fischer-Gray contracts that is in line with our setting see Canzoneri (1985)). This is equivalent to ‘rational expectations’ in a rigidity-free repeated game.12 2.1. Long-run Perspective. Since our interest lies in the e¤ect of policy commitment we focus on long-run/average/trend outcomes of the game. To do so we have made the economy deterministic by disregarding shocks in (1) from which it follows that assuming out the policymaker’s aversion to output volatility in (2) is without loss of generality. It then follows that the policymaker’s instrument represents choosing average in‡ation or a certain level of a long-run IT.13 To obtain the equilibrium levels (denoted by star throughout) of the stage game we use (1)-(3) = wt : 2 The fact that t > ~ is the famous time inconsistency and in‡ation bias results of Kydland and Prescott (1977) and Barro and Gordon (1983).

(4)

t

=~+

2.2. Game Theoretic Representation. For game theoretic clarity we will restrict the players’ action set to two levels, low (L) and high (H). In its general form the game can be summarized by the payo¤ matrix in Figure 2 in which the payo¤s fa; b; c; d; q; v; x; zg are functions of the deep parameters of the model.

Public

Policymaker

L

H

L

a, q

b, v

H

c, x

d, z

Figure 2. General BG game payo¤s We will refer to (c a) as temptation, (a d) as the in‡ation cost, and (d b) as the disin‡ation cost. We follow Cho and Matsui (2005) who depict the most natural candidates for L and H the optimal level from (2) and the time-consistent level from (4) respectively 2 fL = ~ ; H = ~ +

g 3 w: 2 2.3. General and Speci…c BG Game. Throughout the paper we will use two versions of the game and refer to them as general and speci…c. The speci…c game will use the payo¤s derived from the model to numerically illustrate the results. Using (1)-(3) and dividing through by ( 2 )2 without loss of generality we obtain the payo¤s reported in Figure 3 and equation (5). (5)

c=1>a=0>d=

1>b=

2 and q = z = 0 > v = x =

1:

12 While in a rigidity-free environment the terms wage in‡ation and expected in‡ation can be used interchangeably, see eg Backus and Dri¢ ll (1985), it will become apparent that in the presence of wage rigidity these two di¤er. 13 Long-run IT means that the legislated horizon of the target is the business cycle or longer (inde…nite) - as is common in industrial countries, see Mishkin and Schmidt-Hebbel (2001). Since shocks have a zero mean they do not a¤ect the average/trend levels - this is shown in the stochastic extension of this paper, Libich (2006), which will be discussed in detail in section 6.

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

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Public

Policymaker

πL

πH

wL

wH

0, 0

-2, -1

1, -1

-1, 0

Figure 3. Speci…c BG game payo¤s In the general game payo¤s satisfy the following conditions (6)

c > a > d > b; q > v; q

z > x and c

a=d

b:

It is apparent that in our model the last term in (6), the equality of the size of temptation and disin‡ation cost, holds for all and :14 . 2.4. Stage Game With and Without Standard Commitment. We see from Figure 3 that the (H; H) outcome is the unique Nash equilibrium of the stage game, however, it is Pareto inferior to the non-Nash (L; L) outcome. Therefore, in a …nite horizon game without some form of reputation the e¢ cient (L; L) outcome is unachievable, ie the optimal low in‡ation target is time inconsistent and lacks credibility.15 The standard way to eliminate the time inconsistency and in‡ation bias is to impose the policymaker’s commitment - Stackelberg leadership. If the policymaker is the Stackelberg leader (…rst mover in the game) then (L; L) becomes the unique equilibrium outcome. It should be noted that this happens regardless of the exact payo¤s and discount factors. Our aim in the rest of the paper is to examine the robustness of these standard results in the rigid environment. Our analysis in Section 4 shows that allowing for various degrees of rigidity/commitment re…nes and partly quali…es the standard conclusions. 3. Deterministic Rigidity/Commitment 3.1. Assumptions. We adopt all the assumptions of a standard repeated game - a number of alternative speci…cations are discussed in Section 6. First, rigidity/commitment are discrete and constant throughout each game. Second, they are common knowledge. Third, all past periods’ moves can be observed. Fourth, the game starts with a simultaneous move with certainty which may be interpreted as re‡ecting some ‘initial’ uncertainty. Fifth, players are rational, have common knowledge of rationality and for expositional clarity they have complete information about the structure of the game and opponents’payo¤s. De…nition 1. An unrepeated asynchronous game with deterministic rigidity / commitment is an extensive game that starts with a simultaneous move of all players, continues with moves every ri periods, and …nishes after T periods, where T 2 N denotes the ‘least common multiple’ of ri ; 8i. An example of such a game in the form of a time line is presented in Figure 1 in which T (ri = 3; rj = 4; rk = 6) = 12. 14 We impose it in order to streamline our analysis without a¤ecting our main …ndings. The working version of the paper also includes the situations in which c a 6= d b; and they will be discussed in detail in Section 6. 15 By credibility of low in‡ation we will mean a situation of L accompanied by w L - whereas if it is accompanied by wH we will talk about ‘lack of credibility’.

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

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3.2. (Non)-Repetition. While this asynchronous game can be repeated we will restrict our attention to the unrepeated game (as depicted in Figures 1 and 4). This is possible because we will be deriving conditions under which the unique e¢ cient outcome uniquely obtains on the equilibrium path of the unrepeated game. If these conditions are satis…ed repeating the game and allowing for reputation building of some form would not a¤ect the derived equilibrium.16 The uniqueness implies that we can only focus on pure strategies without loss of generality. An additional advantage of out focus on the unrepeated game is the fact that the assumption of the public being composed of in…nitessimal agents (as in Cho and Matsui (2005)) is not required to avoid the Folk theorem. 3.3. Real World Interpretation. We will interpret rp as wage rigidity following Taylor (1979) and rg as the strength of the monetary policy long-run commitment. From the fact that represents setting a certain level of a long-run IT it follows that rg can be interpreted as the degree of the target’s explicitness. This is based on the assumption that the more explicitly the IT is stated in the central banking legislation the less frequently it can be altered (in the Taylor (1979) deterministic sense) or the less likely it is (in the Calvo (1983) probabilistic sense). Intuitively, legislating the target increases the (political, legal, and credibility) ‘cost’of renegotiating/altering the target. This assumption is supported by the fact that (i) there have only been very few occasions of a country changing its legislated IT (and these changes very only trivial) and, (ii) to our knowledge, no country has ever abandoned and explicit IT. As a real world example of deterministic rg the 1989 Reserve Bank of New Zealand Act states that the in‡ation target may only be changed in a Policy Target Agreement (PTA) between the Minister of Finance and the Governor and this can only be done on pre-speci…ed regular occasions (eg when a new Governor is appointed).17 It should further be noted that the absence of a legislated numerical target may not necessarily imply rg = 1; it has been argued that many countries pursue an in‡ation target implicitly (including the US, see e.g. Goodfriend (2003), or the Bundesbank and the Swiss National Bank in the 1980-90s, see Bernanke, et al. (1999)). In such cases we have rg > 1 but rg is smaller than under a fully-‡edged legislated IT. 3.4. Notation. We denote player i’s moves by ni and the number of his …nal move in the g p unrepeated game by N i . It then follows that N i = T (rri;r ) . Also, gnl and pln will denote a certain action l 2 fL; Hg in a certain node ni , eg pH 2 refers to the public’s high wage play in its second move. For notational parsimony we will introduce the notation for the case of interest, g g rg rp : Denote rrp 1 to be the players’ relative rigidity. Then b rrp c 2 N will be the integer g g value of relative rigidity (the ‡oor) and R = rrp b rrp c = [0; 1) denotes the fractional value of relative rigidity (the remainder).18 L Further, we denote b(:) to be the best response. For example, g1L 2 b(pL is 1 ) expresses that L L L the policymaker’s best response to the public’s initial w move and fg1 g = b(p1 ) expresses that it is the unique best response. Alternatively, to indicate a unique best response to the opponent’s current play (without specifying the exact node) the latter will be written as fg1L g = b(wL ): Let us also repeat here that a star denotes optimal play, ie p1 2 b(g1 ) expresses that the public’s optimal play in move 1 is the best response to the policymaker’s …rst move. Finally, threshold levels will be denoted by either upper or lower bar. 16 In this sense we can think of our analysis as the worst case scenario in which reputation cannot help in cooperation. 17 Since late 1990 the PTA was ‘renegotiated’ …ve times, ie roughly every three years. Only on two occasions the target level was changed: in 1996 from 0-2% to 0-3% and in 2002 to 1-3%. 18 It will be evident that R plays an important role since it determines the exact type of dynamics (asynchronicity) in the game.

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3.5. Recursive Scheme. Our proofs are based on the recursive scheme implied by our setup and number theory. Let us use kn to denote the number of periods between the ng -th move of the policymaker and the immediately following move of the public. This implies, for our case of interest rg rp ; that k1 = rp (see an example in Figure 4). From this it follows that the number

p1 g1

r

p

, k1

p2

k2

p3

g2

rg

k3

p4

p5

g3

Figure 4. An unrepeated asynchronous game with deterministic rigidity and commitment - an example of timing of moves with rg = 5 and rp = 3 (and kn and ni shown as a demonstration). of periods between the (ng + 1)-th move of the policymaker and the immediately preceding move of the public equals rp kn+1 . Using these we can summarize the recursive scheme of the game as follows: (7)

kn+1 =

kn Rrp kn + (1 R)rp

if if

kn Rrp ; kn < Rrp ;

Generally, kn is not a monotone sequence, see Figure 4. 3.6. History and Future. By convention, history in period t; ht ; is the sequence of actions selected prior to period t and future in period t is the sequence of current and future actions. It follows from our ‘observability’assumption that ht is common knowledge at t. Let us introduce the concept of ‘recent’history that will span from the last move of the opponent to the present period t.19 From (2) and (3) (in which the payo¤s in period t is only a function of the opponent’s action in t); the recursive mechanism in (7), and the …nite horizon of the unrepeated game it follows that anything prior to the recent history never a¤ects the players’play. Furthermore, we will see that even recent history has no a¤ect on some actions - we will refer to such actions as ‘history-independent’. 3.7. Strategies and Equilibria. A strategy for a certain player is a function that, 8ht ; t, assigns a probability distribution to the player’s action space. As common in macroeconomics, in this paper we will restrict our attention to pure strategies. A strategy of player i is then a vector that, 8ht , speci…es the player’s play 8ni . The asynchronous game will commonly have multiple Nash equilibria. To select among these we will use a standard equilibrium re…nement, subgame perfection, that eliminates non-credible threats. Subgame perfect Nash equilibrium (SPNE) is a strategy vector (one strategy for each player) that forms a Nash equilibrium after any history ht .20 19 For example, in terms of the policymaker’s ng > 1 move in period t it follows from using (7) that recent history starts in period t (r p kn ). 20 Note that the speci…cation of the players’ utility implies that all our SPNE will also be Markov perfect equilibria, for details see eg Maskin and Tirole (2001).

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Given the large number of nodes in the game reporting fully characterized SPNE would be cumbersome. We will therefore focus on the equilibrium path of the SPNE, ie actions that will actually get played.21 To simplify the language let us de…ne the following. De…nition 2. Any SPNE in which both players play L in all their moves on the equilibrium path, (iL n ) ; 8n; i; will be called a Ramsey SPNE. 4. Results Our aim is to revisit the standard results with and without commitment (reported in Section 2.4) allowing for the rigid environment and various degrees of commitment. This section shows that they are re…ned and partly quali…ed. First, we show that the e¢ cient outcome in which both players play L throughout can be achieved, in a …nite game, even without assuming the policymaker’s …rst move and without reputational considerations. Second, we derive the exact degree of commitment that ensures this and show that it is a function of various characteristics of the economy and the players’preferences. Third, and perhaps surprisingly, it is demonstrated that under some circumstances the required commitment is arbitrarily low and under others even an in…nitely strong commitment is insu¢ cient. The latter …nding in particular is in contrast to the intuition of the standard commitment concept whereby policy commitment guarantees (L; L) under all circumstances. For the sake of transparency - to better expose the solution and intuition of the rigid environment - we do the following: (i) We complement the results of the general BG game in which only (6) is required to hold with those of the speci…c BG game, in which (5) is satis…ed. (ii) We …rst report the necessary and su¢ cient conditions (that are a function of R) and then turn to the su¢ cient conditions (that hold 8R). (iii) We …rst focus on the game under patient players, p = g = 1; and only then extend the …ndings under players’impatience (to separate the e¤ects of public’s discounting, p < 1; and the policymaker’s discounting, g < 1; we examine them in turns). As the intuition of the rigid environment is independent of the players’discount factor most of the results will carry over. It will be shown that while the public’s impatience improves cooperation the policymaker’s impatience has the opposite e¤ect (which is in line with the results of Cho and Matsui (2005)). 4.1. Patient Players. Let us now report our …rst set of results. Proposition 1. Consider the general unrepeated asynchronous BG game in which (6) holds and assume patient players, p = g = 1. Any SPNE of the game is Ramsey if and only if 8 c d a b p p if R = 0; > > a dr = a dr > > > < (1+R)(c d) p d) p r = a b+R(c r if R 2 (0; R); (8) rg > rg (R) = a d a d > > > > > : c d (1 R)(a b) rp = a b Rrp if R 2 [R; 1); a d a d

where R = (9)

q v z x+q v .

In the speci…c BG game, in which (5) holds, this happens if and only if rg 2 rp

3 ;2 [ 2

5 ;1 : 2 p

g

r r 21 To demonstrate, for our example in Figure 4 with r p = 3; r g = 5 each SPNE consists of P P 2(s+f 1) = s=1 f =1

254 actions whereas on its equilibrium paths there are rp + r g = 8 actions.

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The necessary and su¢ cient threshold commitment level rg (R) in (8) is, 8R; increasing in wage rigidity, rp ; temptation, c a; and the disin‡ation cost, d b; and decreasing in the in‡ation cost, a d. Proof. To prove the claims it su¢ ces to show that under the stated circumstances L is the policymaker’s unique best play in all his nodes for all histories ht , ie every optimal move gn is ‘history independent’. As the public’s unique best response to L is L this will ensure wL throughout the equilibrium as well. We solve the game backwards and prove the statements by a mathematical induction argument with respect to the policymaker’s moves, focusing on the case rg > rp . First, we prove that on the equilibrium path L will be played in the policymaker’s last move ng = N g (the inductive basis). Then, supposing that it holds for some ng N g , we show that the same is true for g (n 1) as well (the inductive step). This will demonstrate that on the equilibrium path we 22 uniquely have iL For the details of the proof see Appendix A. n ; 8i; n. Intuitively, similarly to Barro and Gordon (1983) an output gain from in‡ation surprise is only temporary and is followed by the public’s ‘punishment’in terms of high expectations and wages. However, unlike in the Barro-Gordon simultaneously repeated game the public’s punishment in the rigid world is not arbitrary - it is the public’s optimal play and its length is uniquely determined by wage rigidity and policy commitment. For L to be time consistent and credible the punishment has to more then o¤set the temptation. This happens if the policymaker’s commitment rg is su¢ ciently strong relative to the public’s wage rigidity rp as well as several other important variables a¤ecting the payo¤s fa; b; c; d; q; v; x; zg: In our model these payo¤s are functions of the structure of the economy, ; and the policymaker’s preferences, : In the real world they arguably depend on various other factors such as Union power, the way agents form expectations, political economy factors (lobby groups, political cycles), institutional setting of monetary and …scal policy etc. This re…nes the conclusion made under the standard commitment concept in which a committed policymaker (acting as a Stackelberg leader) ensures (L; L) regardless of these factors. It is also illustrative to consider why some low commitment values (in the speci…c BG game g in the interval rrp 2 (0; 32 ) [ (2; 52 )) fail to deliver any Ramsey SPNE. It is because the relative length of the public’s punishment is insu¢ cient to discourage the policymaker from in‡ating. For example under rg = 4; rp = 3 there would be no punishment whatsoever (the play on L L the equilibrium path is g1H ; pH 1 ; pn=f2;3;4g ; gn=f2;3g ). It is shown in the next section that this ‘forgiving’behaviour (in which wages are reduced before the start of the disin‡ation) disappears if the public is su¢ ciently impatient or has adaptive expectations. 4.2. The Public’s Impatience. This section shows that the public’s discounting may weaken the above su¢ cient conditions for Ramsey SPNE and hence improve cooperation. The following result is a general …nding that Section 6 shows to apply to other classes of games as well - under some circumstances even a marginal amount of (relative) commitment is su¢ cient to uniquely achieve an e¢ cient outcome. Theorem 1. Consider the general unrepeated asynchronous BG game in which (6) holds and assume a patient policymaker, g = 1, a su¢ ciently impatient public, 0 p p < 1 where p is some upper bound, and a su¢ ciently high in‡ation cost, a d > a 2 b = c 2 d . Then for all rg 2 (1; 1); rp 22 It will become evident that for most parameter values satisfying (8) and (9) there will be a unique Ramsey SPNE but since our attention will be on the equilibrium path we will not examine the number of Ramsey SPNE (o¤-equilibrium path behaviour) in detail.

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any SPNE of the game is Ramsey. Proof. This claim that, for some parameter values any rg > rp is su¢ cient, is proven in Appendix B. Intuitively, an impatient public will disregard the future and always play wt 2 b( t ).23 Therefore, it will never reduce wages before the start of the disin‡ation, ie always punish in‡ating and make the disin‡ation costly. As this decreases the policymaker’s temptation to in‡ate, it reduces the degree of commitment su¢ cient to achieve time-consistency and credibility.2 4 We explicitly formulate this result since it shows that credibly low in‡ation can uniquely obtain in equilibrium in a game theoretic setting that ‘approaches’the BG repeated game - the required degree of relative commitment may be arbitrarily low. The next section however shows that this conclusion changes - sometimes radically so - if the policymaker is impatient. 4.3. The Policymaker’s Impatience. In this section we demonstrate that the policymaker’s discounting of the future worsens coordination. Despite this we can still derive two main general results. The …rst one extends the …nding of Lemma 1 under the policymaker’s impatience and greatly simpli…es the solution of the game. The second one shows whether and under what circumstances a su¢ cient degree of commitment to uniquely ensure the Ramsey SPNE to exists. These results are followed by several policy related …ndings that o¤er testable hypotheses. Theorem 2. Consider the general unrepeated asynchronous BG game in which (6) holds. Then for all fR; g ; p ; a; b; c; d; q; v; x; zg the su¢ cient conditions to ensure any SPNE to be Ramsey, namely 8ng ; fgnL g = b(wL ) and fgnL g = b(wH ); obtain at ng = 1 (the initial simultaneous move). Proof. See Appendix C. This property means that regardless of the exact dynamics, it su¢ ces to focus on the initial simultaneous move assuming that all further relevant conditions hold.25 If the strongest condition for ng = 1 is satis…ed we then know that a unique (and e¢ cient) equilibrium outcome obtains throughout. This property is very convenient as it signi…cantly reduces the number of steps required to solve the rigid game. Theorem 3. Consider the general unrepeated asynchronous BG game in which (6) holds. (i) If the policymaker is su¢ ciently patient, r r a d (5) rp 2 rp 1 = ; (10) g > g = c b 3 then there exists a su¢ cient threshold rg 2 N such that for all rg > rg and for all frp ; R; c; d; q; v; x; zg any SPNE of the game is Ramsey.26

p ; a; b;

23 Note that since in all but the initial move the players never move simultaneously this implies p n>1 2 b( t 1 ) which is observationally equivalent to backward looking expectations and will be discussed in Section 6. 24 The implication of this is worth noting. The theorem implies that ‘myopic’ behaviour by the public of the tit-for-tat variaty may be optimal in the rigid world (for both the public and the policymaker) serving as a credible threat. 25 Speci…cally, using k and k with (19), (24), and (26) implies that anything that happens in periods 1 2 t > r g + rp (1 R) can be ‘skipped’. 26 The claim implies that the necessary bound r g depends on g . But for the sake of expositional clarity we use r g instead of r g ( g ) (and similarly for all thresholds in the rest of this section). This also implies that the conditions r g > r g and g > g are not su¢ cient for the uniqueness of Ramsey SPNE individually - they have to hold jointly.

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(ii) If the policymaker is su¢ ciently impatient, r r b (5) rp 1 rp d = (11) ; g < g = a b 2 then for all fR; p ; a; b; c; d; q; v; x; zg even an in…nitely strong commitment, rg ! 1; does not deliver any Ramsey SPNE. Proof. See Appendix D. The e¤ect of impatience in the rigid framework is similar to the intuition of repeated games in which a punishment for defection is a¤ected by the discount factor of the deviating player. In both cases it is harder (or even impossible) to deter an impatient player from defecting. However, the second result quali…es the intuition of the standard commitment concept in which Stackelberg leadership of the policymaker uniquely ensures (L; L) for all g - in the rigid environment this is no longer the case. While claim (i) of Theorem 3 reports the su¢ cient bound g it does not provide a su¢ cient commitment level rg - it only shows its existence. This is because the claim is proven 8R and we have seen in Proposition 1 that the value of R determines the exact dynamics and hence the necessary and su¢ cient commitment, rg (R) (where obviously rg (R) rg ). Nevertheless, since Proposition 1 showed that the special case R = 0 is representative of the more asynchronous cases ((8) shows that the thresholds rg (R) for R = (0; 1) do not di¤er qualitatively from rg (0)), we will investigate the policymaker’s impatience under R = 0 and extend our conclusions for all R. Proposition 2. Consider the general unrepeated asynchronous BG game in which (6) holds and assume g > g (R) and R = 0. Then the necessary and su¢ cient threshold rg (0) is increasing in wage rigidity, rp ; temptation, c a; and the disin‡ation cost, d b; and decreasing in the in‡ation cost, a d and the policymaker’s discount factor g : The negative relationship between rg and g implies that the policymaker’s commitment and patience are substitutes in achieving the Ramsey SPNE. Proof. Appendix E shows that the necessary and su¢ cient commitment level is (12) ! ! rp rp (c d) (c a) (a b) (d b) (5) g g g r > rg (0) = log g = log g = log g (2 a d a d

rp g

1):

from which the implied necessary and su¢ cient patience threshold is r r p b a (5) rp rp d rp c (13) = = 0:5; g > g (0) = a b c d It it straightforward to see that both arguments of the logarithm in (12) are increasing in d and decreasing in a; the former is further decreasing in b and the latter increasing in c; which proves the claims on the e¤ect of fc a; d b; a dg: To prove the remaining claims on the negative relationship of rg vs rp and rg vs g further steps are required. For graphical demostration of these relationships see Figure 5 (in which the thresholds from (12) and (13) are plotted), for formal proofs see Appendix E. This implies that a less patient policymaker needs to commit more strongly (make its low in‡ation target more explicit) to ensure credibility. It can therefore be argued that (i) the intuition of the patient policymaker environment is unchanged; rg (R) under patience and impatience is a function of the same variables with the same signs (compare Propositions 1 and 2). Further,

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Figure 5. Dependence of rg (0) on rp and g (from (12) and (13) for the speci…c game (5)). The dotted asymptotes correspond to the necessary and su¢ cient thresholds g (0) for each particular rp . while rg does not exists 8 g it can still be concluded from the …rst claim of Theorem 3 that the results are fairly robust to the policymaker’s discounting.27 In recent years there has been a heated debate about the e¤ects of monetary policy commitment and explicit IT in particular (discussed in Section 5 below). The following proposition summarizes the adverse consequences of insu¢ cient monetary policy commitment, rg < rg (R); and provides testable hypotheses that contribute to this debate. Proposition 3. Consider the general unrepeated asynchronous BG game in which (6) holds and rg satis…es (14)

1

rg < rg (R):

(i) Then L is time-inconsistent (lacks credibility) and, compared to the case rg > rg (R), the average level of in‡ation is higher. (ii) If g > g (R), R = (0; 1); and, in addition to (14), rg also satis…es (15)

1

rg (R) < rg ;

where rg (R) is some lower bound, then and only then in‡ation variability is higher than under rg > rg (R). Proof. By de…nition, if rg > rg (R) then the optimal L level of in‡ation obtains uniquely on the equilibrium path, ie (long-run) in‡ation is always on target and hence its variability is zero. 27 For example, Figure 5 shows the following for the speci…c game in (5). If r p = 1 (which implies R = 0; 8r g ) then under g = 1 we have r g > r g (0) = 2 (from (8)), under g = 0:99 we have r g > rg (0) 2:01; under g = 0:8 we have rg > rg (0) 2:29, and under g = 0:51 we have r g > r g (0) 5:81 (all from (12)). Put di¤erently, if r p = 1 then for all g > 1 0:62 the value r g = 3 su¢ ces to uniquely ensure Ramsey SPNE.

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Conversely, if rg < rg (R) then the level H obtains on the equilibrium path for at least one ng . This increases average in‡ation. For this to also increase the variability of in‡ation additional conditions are required that will ensure that the level H does not obtain 8ng ; ie both L and H occur in equilibrium For details of the proofs see Appendix F. Two features are worth noting. First, the variability result obtains even in the absence of shocks, ie for trend/long-run in‡ation. This arises because the gains and costs of in‡ating vary in time with kn (not only due to shocks as in the BG model). Second, there exist circumstances under which claim (i) applies but claim (ii) does not. For example for all rg and rp such that R = 0 there exist(s) SPNE that has/have either H or L uniquely on the equilibrium path (but not their combination) so volatility may not be higher (for this reason (15) never includes the case rg = 1): 5. Empirical Evidence Our analysis has several testable implications. The level of in‡ation and its variability are shown to be weakly decreasing (and hence the policy’s credibility increasing) in the degree of the policymaker’s: (i) long-run commitment, rg ; (ii) patience, g ; and (iii) conservatism, ; (implied by (4) in the spirit of Rogo¤ (1985)). Further and interestingly, our model implies a negative relationship between patience/conservatism on one hand and commitment on the other due to their substitutability (Proposition 2). It should however be noted that all these results obtain weakly, ie only under some circumstances whereas under others there may be no relationship between these variables. This quali…cation is crucial in terms of empirical testing as it may guide the choice of the sample and control variables. We will …rst discuss suitable proxies of these variables, then examine the patience-commitment relationship, and then revisit their e¤ect on in‡ation and its variability. In doing so some con‡icting empirical …ndings of the literature will be reconciled based on our theoretical results. 5.1. Proxies. In terms of (i) the policymaker’s (long-run) commitment was interpreted above as the degree of explicitness of the IT - see Section 3.3 for a discussion. While there exist no index that would measure the target’s explicitness the closest proxies are arguable the pivotal features of the regime, namely the degrees of (goal) transparency and accountability (see eg Bernanke et al (1999)). In terms of (ii) and (iii) it can be argued that patience and conservatism are a function of several characteristics of monetary policy, most importantly the degree of central bank goalindependence (goal-CBI).28 First, goal-independent central bankers are commonly more conservative (tougher on in‡ation) in the spirit of Rogo¤ (1985). Second, they have a longer term in o¢ ce which is likely to translate into more patient behaviour (see eg Eggertsson and Le Borgne (2003)). In the past two decades the real world has seen a move in the direction of increasing goal-CBI and greater length of the banker’s term has come as one of the arrangements (see for example Waller and Walsh (1996)).2 9 28 It should be stressed that our paper makes prediction about the goal -CBI, not instrument-CBI (on this

distinction see Debelle and Fischer (1994)). This is because both and g relate to the parameters in the policymaker’s objective function. 29 On the length in o¢ ce for 93 countries see Mahadeva and Sterne (2000), Table 4.4. While the norm of 5-7 years is only marginally longer than the policymaker’s term, in the majority of cases (in industrial countries) the Governor gets reappointed which makes the expected term in o¢ ce signi…cantly longer. The U.S. o¤ers itself as a good example.

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nd

2

Sl o Po vak rtu ia A gal us G tri a re ec e Ita ly

1

Sw ed e EC n B

U SA M J e x ap a ic oT n ur ke y Sp A a Ch rg i en n ile tin D a en Ire m a Fr Lu land rk a Cz nc xe e ec N mb h et u R ep Ge herl rg r m an . an d s y

3

Sw itz H er l un an ga d ry Fi nl an d

N ew Ic el Ze an al d a

or e

Po la nd

4

K

ad a Ca n

A ccountability lIt

5

A us tra lia a N Be or lg w iu ay m En gl an d

5.2. Institutional Relationships: IT vs Goal-CBI. Using these proxies implies that there exists substitutability between explicit in‡ation targeting and goal-CBI in ensuring low in‡ation and high credibility. This novel prediction is supported by several studies that report a negative correlation between (goal) CBI and accountability, eg Briault, Haldane and King (1997), de Haan, Amtenbrink and Eij¢ nger (1999), and Sousa (2002) (see Figure 6 for an example). Note that virtually all top left hand corner countries are explicit in‡ation targeters.3 0

0 2

3

4

5 6 Inde pe nde nc e

7

8

Figure 6. Central bank accountability vs independence, Sousa (2002). We use the ‘…nal responsibility’component of accountability, see Appendix for details on the criteria, countries, and scores. The correlation coe¢ cient equals 0:78 (the t-value equals 6:94). Despite the arguable shortcomings of any such index this …nding seems robust as it has been obtained using di¤erently constructed indices for di¤erent countries and periods. If we plot Sousa (2002) …nal responsibility against the length of term in o¢ ce (which is one of the criteria in his CBI index) the picture remains roughly the same. Furthermore, in a comprehensive dataset of Fry et al. (2000) the length of term in o¢ ce is negatively correlated to accountability procedures (that apply when targets are missed or must be changed) in both industrial and transition countries. 30

It should be mentioned that this …nding does not seem to be a result of omitted variables: all the countries in the sample have comparable in‡ation levels and existing economic theory does not identify any other reasons/variables for this negative relationship. In fact, the conventional view that accountability should go hand in hand with independence to be consistent with democracy (for a widely cited example see King (1998)) implies that the correlation should be positive.

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Finally, Hughes Hallett and Libich (2006a) present evidence that transparency, too, is negatively correlated to goal-CBI. For example, it is shown that the correlation between transparency in Eij¢ nger and Geraats (2006) and goal-CBI in Briault, Haldane and King (1997) is 0:86 (with the t-value equaling 4:46).31 5.3. E¤ect on IT and CBI In‡ation. For the purposes of empirical testing it is important to note the exact nature of our results. The analysis implies that a more explicit long-run IT reduces the level of in‡ation and its the variability, but only if the initial level of explicitness is insu¢ cient to achieve the Ramsey SPNE, rg < rg (R) (see Proposition 3). Otherwise rg may have no long-run e¤ect.32 Therefore, the results are not equivalent to the claim that IT countries will have lower in‡ation and its volatility than non-IT countries. This is because the latter group’s implicit IT may still be su¢ ciently explicit, ie in the region of rg > rg (R). This is however what most of the literature has done and which received some criticism, see eg Gertler (2003). Our analysis implies a criterion to distinguish whether this is or isn’t the case - it suggests to examine the average level of in‡ation (say over the past ten years), . If > L (arguably the case of many transition and developing countries) then rg < rg (R) is implied and empirical analysis of such sample will …nd the explicitness of in‡ation targeting to be negatively correlated with both the level of in‡ation and its volatility. In contrast, if = L (arguably the case of most industrial countries) then rg > rg (R) is implied and our model predicts no correlation. Both predictions are supported; papers that only include industrial countries …nd weak and/or insigni…cant e¤ects of in‡ation targeting on in‡ation and its volatility, eg Ball and Sheridan (2003) and Willard (2006) whereas those with larger samples …nd strong and signi…cant e¤ects, eg Corbo, Landerretche and Schmidt-Hebbel (2001). Furthermore, in line with the predictions of our model, in‡ation has been found negatively correlated with accountability (Briault, Haldane and King (1997)) as well as with transparency (Chortareas, Stasavage and Sterne (2002) and Fry et al. (2000)). See also Debelle (1997) who …nds in‡ation targeting to increase the policy’s credibility. All these papers include either pre1980 in‡ation data or emerging/developing countries. In contrast, papers that only focus on industrial countries and use recent data often …nd no correlation, see eg Eij¢ nger and Geraats (2006). Similarly, goal-CBI was found to be associated with lower in‡ation, see Grilli, Masciandaro and Tabellini (1991), Cukierman, Webb and Neyapti (1992), Alesina and Summers (1993), Eij¢ nger, Schaling and Hoeberichts (1998). However, using more recent data in‡ation is uncorrelated to goal-CBI among industrial countries, see eg Fry et al. (2000). 6. Robustness and Extensions This section discusses some extensions and implies that our results are robust to a number of alternative speci…cations and assumptions. 6.1. Game Theoretic Issues. Generality. It follows from the above proofs that relaxing the general game constraint on the size of temptation, c a = d b (which will allow for any model 31 For welfare implications of these institutional features and a more detailed empirical analysis see Hughes Hallett and Libich (2006a,b). The papers also demonstrate that the Debelle and Fischer (1994) distinction between goal and instrument CBI is crucial. Instrument CBI has come as a part of in‡ation targeting (as one of the prerequisites of the regime, see eg Masson, Savastano and Sharma (1997), Blejer and et al. (2002)) its correlation with transparency and accountability in most indices is positive, see eg Chortareas, Stasavage and Sterne (2002). 32 Below it will be argued, based on Libich (2006), that in the presence of shocks (ie in te short run) there may be an additional, anchoring, e¤ect of an IT.

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featuring time inconsistency), does not alter the qualitative nature of our results.33 Similarly, Libich and Stehlik (2007) and Libich, Hughes Hallett and Stehlik (2007) examine other classes of games and …nd the intuition to carry over. Most importantly, the conditions can be derived under which in games with multiple Nash equilibria the e¢ cient outcome is uniquely selected by subgame perfection. For example, the Battle of Sexes can be summarized by (the …rst part of) our general constraints in (6) with two modi…cation, c < a and q < z; so the results of Theorems 1-3 still obtain. In these games additional insights are obtained that contribute to the literature on equilibrium selection. Endogenous ri . It should be noted that all ri ’s can be endogenized as players’optimal choices. This seems desirable - while rigidity has been found important most common macroeconomic models take it as given.34 Libich (2007) is a step in this direction - it formalizes the concept of ‘economically rational expectations’(Feige and Pearce (1976)) by incorporating various realistic costs into the players’objectives (that are some function of ri ) and letting them choose their ri ’s optimally (at the beginning of the game). In terms of the public it postulates a wage bargaining cost and a cost of updating expectations (processing information) about some stochastic process (shock). In terms of the policymaker a cost of explicit commitment is considered (such as implementation or accountability cost of an explicit IT). The paper shows that the public’s optimal rp is a decreasing function rg : This can be thought of as an ‘anchoring e¤ect’of explicit IT that has been found empirically (see eg Gurkaynak et at (2005)). Probabilistic ri . Deterministic rigidity/commitment of Taylor (1980) can be reinterpreted as a probabilistic one in the spirit of the Calvo (1983). In such case the average/expected length of time between each move is 1 1 i which is equivalent to our deterministic ri . In a companion paper Libich and Stehlik (2007) we examine this probabilistic version explicitly. We show that the intuition remains the same, ie under a su¢ ciently committed and patient policymaker, g > g and g > g , the Ramsey SPNE uniquely obtains. More Instruments. Each player can have a number of choice variables, each with a certain i 2 N, to be the number of periods for degree of rigidity/commitment. We can then de…ne rm which player i’s instrument m cannot be altered. The next sections discusses an example of this in Libich (2006) in which both the public and the policymaker have two instruments. More Players - eg Heterogeneous Public. The number of players can easily be increased - for example, Libich (2006) models heterogeneous public (that may not bargain wages collectively). The players’set is then I = fg; pj g where j 2 [1; J] denotes a certain Union (individual) PJ with wage rigidity of rjp and relative size Pj such that 1 Pj = 1. The paper shows that the necessary and su¢ cient condition of the speci…c BG game for the case R = 0; p = g = 1; equivalent to Proposition 1, (8), generalizes from rg (0) > 2rp to

(16)

g

r (0) > 2

J X

Pj rjp :

j=1

This demonstrates that the nature of the results remains unchanged.35

33 There is one exception, Theorem 2. Under c a 6= d b the su¢ cient condition may no longer be related to ng = 1; see (30) and (31). Therefore, the remaining results may alter quantitatively but their qualitative nature will remain intact. 34 Hahn (2006) is one of the notable exceptions in endogenizing price rigidity in the New Keynesian framework. Further, Bhaskar (2002) proposed an alternative way to endogenize the timing of games. 35 For example, with three equally sized Unions the condition becomes r g (0) > 5 (r p + r p + r p ): 2 3 6 1

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Continuous Action Space. Allowing the players’actions to be continuous has the natural consequence of increasing rg (R) compared to the above analysis. This is intuitive; if the policymaker can engage in arbitrarily small in‡ation surprises his temptation to do so increases, the public’s punishment decreases, and hence a stronger commitment is required. Continuous Time. Libich and Stehlik (2007) present analogous results for continuous time, t 2 R, which can incorporate not only the players heterogeneity but also the probabilistic models. Roughly speaking, if we denote by f : [0; rg ] ! [0; 1] a non-decreasing function which describes a (possibly probabilistic) distribution of the public’s (various Unions’) reactions, then the condition analogous to rg > 2rp is (17)

Zr

g

f (t)dt >

rg : 2

0

Time Scales. Both continuous and discrete models can be illustratively generalized using time scales (a recent mathematical tool see eg Bohner and Peterson (2001) for a comprehensive treatment). It enables us to neatly unify and extend all of the above mentioned setups and results. A time scale T is de…ned as a nonempty closed subset of the real numbers R. In the analysis, the so-called ‘jump operators’play a key role. The main contribution of this environment is the ability to consider non-constant (heterogeneous) rigidity/commitment. This generalization is arguably realistic and hence important in many settings in economics, econometrics, as well as other disciplines.36 Libich and Stehlik (2007) show that the condition analogous to rg > 2rp and (17) is Zr

g

f (t) t >

rg : 2

0

where the LHS is called ‘delta integral’such that 8 rg R g > > Zr < f (t)dt if 0 f (t) t = g rP 1 > > : f (t) if 0 t=0

T 2 R; T 2 Z:

This shows that time scale calculus, while nesting both continuous and discrete time as special cases, allows for even more ‡exible analysis of repeated dynamic interactions with heterogeneous time steps. 6.2. Macroeconomic Issues. Substitutability. The negative relationship between goal-CBI and the explicitness of the IT is derived in Hughes Hallett and Libich (2006b) through an entirely di¤erent avenue. The paper uses a standard simultaneously repeated game but explicitly incorporates these features in the macroeconomic model. Short Run Stabilization. As the paper takes a long-run view it is imperative to consider whether the …ndings are quali…ed in the presence of shocks. This is because some in‡ation targeting opponents (see eg Kohn (2003), Friedman (2004) or Greenspan (2003)) have expressed concerns that a legislated numerical IT may reduce the policymaker’s ‡exibility to react to shocks and stabilize output. 36 For an interesting application of time scales in economics see Biles, Atici and Lebedinsky (2005). The authors

model payments to an agent (eg capital income or dividents) arriving an unevenly spaced intervals.

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

19

Our companion paper Libich (2006) utilizes the rigid framework to investigate these concerns in detail. It uses a stochastic New Keynesian type environment and a standard quadratic objective function. Existence of shocks requires to incorporate an additional policy instrument, the interest rate, which is selected every rig periods (and which can be though of as short-run commitment). The paper shows that allowing for disturbances does not alter the conclusions of the presented paper if the IT is speci…ed as a long-run objective (achievable on average over the business cycle). This is because shocks have a zero mean, ie they do not a¤ect the average/trend level of in‡ation.37 Sticky Expectations. While the analysis examined rigidity in wage setting the insights also apply to the public’s (infrequent) adjustment of expectations. Libich (2006) explicitly models this in the rigid framework by incorporating a cost of updating/processing information and allowing for the possibility of sticky expectations. This goes in the spirit of the models of ‘rational inattention’ (see eg Sims (2003), Reis (2006)) and bounded rationality (see eg Gigerenzer and Selten (2002)). Adaptive Expectations. There is a large body of empirical research showing that backward looking expectations are important (see eg Fuhrer (1997)). Libich (2007) considers a simple case of adaptive/static expectations in the rigid framework and shows that the outcomes of such static behaviour are equivalent to those under su¢ ciently impatient public studied above, p < g - in both cases the public will disregard the policymaker’s future periods’play. This implies that all our …ndings carry over. 7. Summary and Conclusions This paper proposes a simple framework that generalizes the timing structure of games with emphasis on macroeconomic games. As most such real world games are arguably …nite, dynamic and most importantly, rigid, our framework combines these characteristics. We show that, similarly to reputation in repeated games, players’rigidity draws a link between successive periods and can therefore serve as a commitment device. This can enhance cooperation and eliminate ine¢ cient outcomes from the set possible equilibria. We apply the framework to the in‡uential Kydland-Prescott-Barro-Gordon game and show the conditions under which the in‡ation bias disappears. Speci…cally we derive the exact degree of policy commitment that makes low in‡ation time-consistent and credible. It is interesting to note that (i) this can happen in a …nite game (possibly as short as two periods) since the required levels of commitment may be rather low, (ii) reputation building is not necessary, and (iii) the policy commitment may substitute for central bank goal-independence (conservatism and/or patience) in achieving credibility. These results o¤er an explanation for the convergence to low in‡ation and high credibility in industrial countries over the past two decades - as a consequence of explicit in‡ation targeting and CBI. Note that this explanation is independent of the three standard remedies in the literature, namely (i) the Rogo¤ (1985) conservative central banker, (ii) the Barro and Gordon (1983) reputation building, and (iii) the Walsh (1995) optimal incentive contract (dismissal procedure). Furthermore, their substitutability helps explain why in‡ation targets have been made more explicit by countries with low degree of goal-CBI such as New Zealand, Canada, and the UK, than the relatively goal-independent central banks in the US, Germany, and Switzerland. We do 37 The paper in fact …nds the opposite, the policymaker’s ‡exibility under an explicit long-run IT is likely to

increase which reduces the volatility of both in‡ation and output in equilibrium. This is due to the ‘anchoring’ e¤ect discussed above which makes the interest rate instrument more e¤ective not only in stabilization of in‡ation but also of output. For arguments and results in the same spirit see Orphanides and Williams (2005), Bernanke (2003), Goodfriend (2003) and Mishkin (2004).

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

20

not only bring empirical support for all our …ndings but also reconcile some con‡icting results of the existing empirical literature. 8. References Backus, D., Dri¢ ll, J., 1985. Rational Exp ectations and Policy Credibility Following a Change in Regim e. Review of Econom ic Studies 52, 211-21 Ball, L., 2000. Near-Rationality and In‡ation in Two M onetary Regim es. Johns Hopkins - Departm ent of Econom ics Johns Hopkins - Departm ent of Econom ics Ball, L., Sheridan, N., 2003. Does In‡ation Targeting M atter? International M onetary Fund IM F Working Pap ers: 03/129 Barro, R.J., Gordon, D.B., 1983. Rules, Discretion and Reputation in a M o del of M onetary Policy. Journal of M onetary Econom ics 12, 101-21 Bernanke, B.S., Laubach, T., M ishkin, F.S., Posen, A., 1999. In‡ation targeting: Lessons from the international exp erience. Princeton University Press, Princeton. Bernanke, B.S., Wo odford, M ., 2005. The in‡ation-targeting debate. University of Chicago Press, NBER Studies in Business Cycles, vol. 32. Chicago and London. Bewley, T.F., 2002. Fairness, Reciprocity, and Wage Rigidity. p. 36 pages. Cowles Foundation Yale University Cowles Foundation Discussion Pap ers: 1383 Bhaskar, V., 2002. On Endogenously Staggered Prices. Review of Econom ic Studies 69, 97-116 Biles, D.C., Atici, F.M ., Leb edinsky, A., 2005.

An application of tim e scales to econom ics.

M athem atical and

Com puter M odelling Blejer, M .I., Ize, A., Leone, A.M ., Werlang, S., 2000. In‡ation Targeting in Practice: Strategic and op erational issues and application to em erging m arket countries. International M onetary Fund. Bohner, M ., Peterson, A., 2001. Dynam ic equations on tim e scales. Birkhauser Boston, 704 Inc., Boston, M A, 2001. Briault, C., Haldane, A., King, M ., 1997. Indep endence and Accountability. In: Kuro da Ie (ed.) Towards m ore e¤ective m onetary p olicy. M acm illan Press in asso ciation with Bank of Japan, London, pp. 299-326. Calvo, G.A., 1983. Staggered Prices in a Utility-M axim izing Fram ework. Journal of M onetary Econom ics 12, 383-98 Canzoneri, M .B., 1985. M onetary Policy Gam es and the Role of Private Inform ation. Am erican Econom ic Review 75, 1056-70 Carroll, C., 2003. M acroeconom ic Exp ectations of Households and Professional Forecasters. Quarterly Journal of Econom ics 118, 269-298 Carroll, C., Slacalek, J., 2006. Sticky Exp ectations and Consum ption Dynam ics. Cho, I., M atsui, A., 2005. Tim e Consistency in Alternating M ove Policy Gam es. Japanese Econom ic Review 56(3), 273-294 Chortareas, G., Stasavage, D., Sterne, G., 2002. Do es It Pay to Be Transparent? International Evidence from Central Bank Forecasts. Federal Reserve Bank of St. Louis Review 84, 99-117 Corb o, V., Landerretche, O., Schm idt-Hebb el, K., 2001.

Assessing In‡ation Targeting after a Decade of World

Exp erience. International Journal of Finance and Econom ics 6, 343-68 Cukierm an, A., Webb, S.B., Neyapti, B., 1992. M easuring the Indep endence of Central Banks and Its E¤ect on Policy Outcom es. World Bank Econom ic Review 6, 353-98 de Haan, J., Amtenbrink, F., Eij¢ nger, S.C.W ., 1999. Accountability of Central Banks: Asp ects and Quanti…cation. Banca Nazionale del Lavoro Quarterly Review 52, 169-93 Deb elle, G., 1997. In‡ation Targeting in Practice. International M onetary Fund IM F Working Pap ers: 97/35 Deb elle, G., Fischer, S., 1994. How indep endent should a central bank b e? In: Goals, Guidelines, and Constraints Facing M onetary Policym akers. FRB of Boston, Conference Series No. 38. Eggertsson, G.B., Le Borgne, E., 2003. A Political Agency Theory of Central Bank Indep endence. International M onetary Fund IM F Working Pap ers: 03/144 Eij¢ nger, S., Schaling, E., Hoeb erichts, M ., 1998. Central Bank Indep endence: A Sensitivity Analysis. Europ ean Journal of Political Economy 14, 73-88

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Eij¢ nger, S.C.W ., Geraats, P.M ., 2006. How Transparent Are Central Banks? Europ ean Journal of Political Economy, (forthcom ing) Feige, E.L., Pearce, D.K., 1976.

Econom ically Rational Exp ectations: Are Innovations in the Rate of In‡ation

Indep endent of Innovations in M easures of M onetary and Fiscal Policy? The Journal of Political Economy Vol. 84, No. 3. (Jun., 1976), 499-522. Fischer, S., 1977. Long-Term Contracts, Rational Exp ectations, and the Optim al M oney Supply Rule. Journal of Political Economy 85, 191-205 Friedm an, B.M ., 2004. W hy the Federal Reserve Should Not Adopt In‡ation Targeting. International Finance 7, 129-36 Fry, M ., Julius, D., M ahadeva, L., Roger, S., Sterne, G., 2000. Key Issues in the Choice of a M onetary Policy Fram ework. In: M ahadeva L & Sterne G (eds.) M onetary Fram eworks in a Global Context. Routledge, London. Fuhrer, J., 1997. The (Un)im p ortance of Forward Lo oking Behaviour in Price Sp eci…cations. Journal of M oney, Credit and Banking 29(3), 338-50 Gertler, M ., 2003. M onetary Policy and Uncertainty: Adapting to a Changing Economy, sym p osium sp onsored by the Federal Reserve Bank of Kansas City. Jackson Hole, W yom ing, August 28 - 30 Gigerenzer, G., Selten, R., 2002. Bounded Rationality. The M IT Press. Go o dfriend, M ., 2003. In‡ation Targeting in the United States? National Bureau of Econom ic Research Inc NBER Working Pap ers: 9981 Greenspan, A., 2003. M onetary Policy and Uncertainty: Adapting to a Changing Economy, sym p osium sp onsored by the Federal Reserve Bank of Kansas City. Jackson Hole, W yom ing, August 28 - 30, 2003 Gurkaynak, R.S., Sack, B., Swanson, E., 2005. The Sensitivity of Long-Term Interest Rates to Econom ic News: Evidence and Im plications for M acroeconom ic M o dels. Am erican Econom ic Review 95, 425-36 Haan, J.d., Amtenbrink, F., Eij¢ nger, S.C.W ., 1998. Accountability of central banks : asp ects and quanti…cation. Tilburg University Center for Econom ic Research Discussion Pap er: 54 Hahn, V., 2006. Endogenous Attentiveness of Price-Setters and Central Bank Conservatism . In: Heidelb erg University, m im eo. Heidelb erg University, m im eo Hughes Hallett, A., Libich, J., 2006a. Central bank indep endence, accountability, and transparency: dem o cratic com plem ents or strategic substitutes? Centre for Econom ic Policy Research, London, CEPR DP 5470 Hughes Hallett, A., Libich, J., 2006b. Explicit In‡ation Targets, Com munication, and Central Bank Indep endence: Friends or Foes? . m im eo, George M ason University King, M ., 1998. The in‡ation target …ve years on. London Scho ol Of Econom ics, 99 Kohn, D.L., 2003. Rem arks at the 28th Annual Policy Conference: In‡ation Targeting: Prosp ects and Problem s. Federal Reserve Bank of St. Louis, St. Louis, M issouri Kydland, F.E., Prescott, E.C., 1977. Rules Rather Than Discretion: The Inconsistency of Optim al Plans. Journal of Political Economy 85, 473-91 Laguno¤, R., M atsui, A., 1997. Asynchronous Choice in Rep eated Co ordination Gam es. Econom etrica 65, 1467-77 Libich, J., 2006. In‡exibility of in‡ation targeting revisited: m o delling the ‘anchoring’ e¤ect. Centre for Applied M acro econom ic Analysis, CAM A W P 2/2006 Libich, J., 2007. An explicit in‡ation target as a com m itm ent device. Journal of M acro econom ics, forthcom ing Libich, J., Stehlik, P., 2007. Rational Inattention in Gam es. m im eo, La Trob e University Libich, J., Hughes Hallett, A., Stehlik, P., 2006. M onetary and Fiscal Policy Interaction W ith Various Typ es and Degrees of Com m itm ent. m im eo, George M ason University M ankiw, N.G., Reis, R., 2002. Sticky Inform ation versus Sticky Prices: A Prop osal to Replace the New Keynesian Phillips Curve. Quarterly Journal of Econom ics 117, 1295-1328 M askin, E., Tirole, J., 1988. A Theory of Dynam ic Oligop oly, I: Overview and Quantity Com p etition with Large Fixed Costs. Econom etrica 56, 549-69 M askin, E., Tirole, J., 2001. M arkov Perfect Equilibrium : I. Observable Actions. Journal of Econom ic Theory 100, 191-219

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22

The Scop e for In‡ation Targeting in Developing Countries.

International M onetary Fund IM F Working Pap ers: 97/130 M cCallum , B.T., 2003. In‡ation Targeting for the United States. Shadow Op en M arket Com m ittee, available at http://www.som c.rochester.edu/M ay03/M cCallum .p df M ishkin, F.S., 2004. W hy the Federal Reserve Should Adopt In‡ation Targeting. International Finance 7, 117-27 M orris, S., Shin, H.S., 2006. Inertia of Forward-Lo oking Exp ectations. m im eo, Princeton University Orphanides, A., W illiam s, J.C., 2005. Im p erfect Knowledge, In‡ation Exp ectations, and M onetary Policy. Journal of Econom ic Dynam ics and Control Elsevier 29(11), 1807-8 Reis, R., 2006. Inattentive Consum ers. Journal of M onetary Econom ics 53 (8) Rogo¤, K., 1985. The Optim al Degree of Com m itm ent to an Interm ediate M onetary Target. Quarterly Journal of Econom ics 100, 1169-89 Sim s, C.A., 2003. Im plications of Rational Inattention. Journal of M onetary Econom ics 50 (3), 665-690 Sousa, P., 2002. Central bank indep endence and dem o cratic accountability. m im eo, Portucalense University Svensson, L.E.O., 1999. In‡ation Targeting as a M onetary Policy Rule. Journal of M onetary Econom ics 43, 607-54 Taylor, J.B., 1979. Staggered Wage Setting in a M acro M o del. Am erican Econom ic Review 69, 108-13 Taylor, J.B., 1980. Aggregate Dynam ics and Staggered Contracts. Journal of Political Economy 88, 1-23 Tobin, J., 1982. M oney and Finance in the M acro econom ic Pro cess. Journal of M oney, Credit and Banking 14 (2), 171-204. Waller, C.J., Walsh, C.E., 1996.

Central-Bank Indep endence, Econom ic Behavior, and Optim al Term Lengths.

Am erican Econom ic Review 86, 1139-53 Walsh, C.E., 1995. Optim al Contracts for Central Bankers. Am erican Econom ic Review 85, 150-67 Wen, Q., 2002. Rep eated Gam es with Asynchronous M oves. Departm ent of Econom ics Vanderbilt University Working Pap ers: 0204 W illard, L., 2006. Does In‡ation targeting M atter: A Reassessm ent. CEPS W P 120, Princeton University Wu, T., 2004. Does in‡ation targeting reduce in‡ation? An analysis for the OECD industrial countries. Banco Central do Brazil Working Pap er 83

Appendix A. Proof of Proposition 1 gnL

Proof. For some to be the policymaker’s unique optimal play 8ht two conditions need to be satis…ed in that node n: Speci…cally, it must hold that fgnL g = b(wL ) and fgnL g = b(wH ), ie L is the unique best response to both current wL and wH : Induction Basis: ng = Ng under R = 0: This special case is illustrative of the intuition of the proof. Here we have, due to rg > rp , T (rg ; rp ) = rg and therefore N g = 1 (and N p = rg ). Solving backwards, we know that pn>1 2 b(g1 ) due to perfect information in np > 1. Further, from the public’s rationality and complete information it follows that p1 2 b(g1 ). Using this, the two required conditions for fg1L g = b(wL ) and fg1L g = b(wH ) are the following (18)

arg > crp + d(rg

(19)

brp + a(rg

rp );

rp ) > drg :

The left-hand side (LHS) and the right-hand side (RHS) will throughout report the policymaker’s payo¤s from playing, in a certain node, L and H respectively. Examine (18) for H example. It assumes pL 1 and says that if the policymaker in‡ates (plays g1 ) he manages to surprise the public and gets a boost in output, c. This however only lasts for rp periods after which the public would switch to wH and ‘punish’the policymaker with a d payo¤ for the rest of the unrepeated game (let us stress again that we only need to examine the unrepeated game for reasons explained in Section 3.2). Rearranging these yields (20)

rg > rg (0) =

c a

a d p r = d a

b p (5) p r = 2r ; d

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

23

where the two fractions only use the general payo¤ constraints in (6) (they are a result of (18) and (19) respectively) and the last term also uses the speci…c payo¤ from (5).38 This implies g g that all rrp = f3; 4; 5; : : :g uniquely deliver Ramsey SPNE and that in the case of rrp = 2 there exists both Ramsey and non-Ramsey SPNE. Induction Basis: ng = Ng under R = (0; 1]: From De…nition 1 it follows that the number g p of the policymaker’s moves in the unrepeated game is N g = T (rrg;r ) . The two conditions analogous to (18) and (19) are the following (21)

arg > crp R + d(rg

rp R) and

brp R + a(rg

c a

a d p Rr = d a

b p (5) Rr = 2Rrp ; d

rp R) > drg ;

and hence rg >

(22)

which is, due to R < 1, weaker than (20) for all constraints satisfying (6). This means that if L (20) holds a patient policymaker will …nd it optimal to play gN for all histories. g g Inductive Step: n + 1 ! n (if applicable, ie if 1 ng < N g ): We assume that the policymaker’s unique best play in the (ng + 1)-th step is L regardless of the public’s preceding play (ie that gn+1 is history-independent), and we attempt to prove that this implies the same assertion for the ng -th step. Intuitively, this means that if the policymaker in‡ates he …nds it optimal to immediately disin‡ate. Two scenarios are possible for each of the two conditions of interest, fgnL g = b(wL ) and fgnL g = b(wH ). The disin‡ation will either be costly - lacking credibility (due to excessive wages wH the payo¤ b occurs for at least one period) or costless (only accompanied by wL ). Whether the disin‡ation is costly or costless depends on the public’s play preceding the disin‡ation, which in turn depends on the recursive scheme in (7) and the public’s preferences. As discussed in Section 3.6 the public’s optimal play in any np is never a¤ected by play prior to the recent history. In particular, due to the …nite horizon the public plays the best response to either the immediately preceding or the immediately following move of the policymaker. Therefore, the costly disin‡ation conditions for all relevant nodes of the policymaker - analogous to (18) and (19) respectively - are the following (23)

arg + a[rp

(rp

kn+1 )] > ckn + d(rg

(24)

bkn + a(rg

kn ) + a[rp

(rp

kn ) + b[rp

(rp

kn+1 )];

kn+1 )] > drg + b[rp

(rp

kn+1 )];

where the last d terms express the total cost of disin‡ation. Similarly the costless disin‡ation conditions analogous to (18) and (19) respectively are (25) (26)

arg > ckn + d[rg bkn + a(rg

kn

kn ) > d[rg

(rp (rp

kn+1 )] + c(rp kn+1 )] + c(rp

kn+1 ); kn+1 );

where the last c components express the output gain from the public’s switch prior to the start of the disin‡ation (note again that under the general constraints (6) the two conditions in each set yield the same inequality). Which of these two sets of conditions is relevant to a certain ng (and hence what the value of R in (8) is) depends on the public’s payo¤s fq; v; x; zg, and importantly on kn+1 . Speci…cally, if (27)

z(rp

kn+1 ) + vkn+1

x(rp

kn+1 ) + qkn+1 ;

38 Note that under the general constraints from (6) the conditions implied by (18) and (19) are equivalent which will be apparent to be the case for all pairs of such conditions.

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

24

then (23) and (24) obtain otherwise (25) and (26) are the relevant conditions. Rearranging yields p z x (5) r 39 rp = : (28) kn+1 z x+q v 2 The following result will dramatically simplify the proof. Lemma 1. Consider the general unrepeated asynchronous BG game in which (6) holds and L assume fgn+1 g = b(wL ) = b(wH ) (ie (20) from the induction basis is satis…ed). Then out of the conditions for fgnL g = b(wL ) and fgnL g = b(wH ); those regarding the initial move ng = 1 are the strongest (in the weak sense) 8R. Proof. Due to the fact that kn attains its unique maximum value, rp ; at ng = 1; it su¢ ces to show that the strength of all the four conditions (ie the RHS) in (23), (24), (25) and (26) is non-decreasing in kn : These equations can be, respectively, rearranged into rg >

(c

d)kn a

(a d

b)kn+1

and rg >

a a

b (kn d

kn+1 );

d)(kn + rp kn+1 ) (a b)kn + (c d)(rp kn+1 ) and rg > : a d a d Recall from (7) that kn+1 is a function of kn : Since in all four conditions the RHS is decreasing in kn+1 we need to use kn+1 = kn Rrp from (7) to substitute away kn+1 : This yields (29)

rg >

(c

rg >

(30)

(c

d

a + b)kn + (a a d

b)Rrp

a a

and rg >

b p Rr ; d

d)rp (1 + R) (a b c + d)kn + (c d)rp (1 + R) and rg > : a d a d The fact that the RHS of all four inequalities is non-decreasing in kn for all values satisfying the general constraints in (6) completes the proof. (31)

rg >

(c

Continuing the proof of Proposition 1 and substituting kn+1 = k2 = rp (1 q v (5) 1 (32) R R= = ; z x+q v 2

R) into (28) yields

which features in (8) and (9). Substituting kn = k1 = rp into (30) and (31) yields, together with (20) and (6), equation (8). Using (5) with (8) then yields (9). Noting that rg (R) in (8) is increasing in c and d and decreasing in a and b implies the last claim and completes the proof of Proposition 1. Appendix B. Proof of Theorem 1 Proof. We start by noting that the value of p only a¤ects the necessary and su¢ cient conditions (8) of Proposition 1 through the value of R (ie when the conditions apply rather than what conditions apply). It then follows that the …nding of Lemma 1 still holds under the public’s impatience. Formally, the generalization of (27) under discounting is, using kn+1 = k2 = rp (1 R); the following r p kn+1

(33)

z

X t=1

r p kn+1

p

t 1 p

+v

r X

t=r p kn+1 +1

t 1 p

x

X t=1

p

t 1 p

+q

r X

t 1 p :

t=r p kn+1 +1

39 To illustrate using our game in Figure 4, if g H then the disin‡ation in the next move, g L ; is costly since 1 2 H L the public’s unique optimal play is pH 2 (the best response to g1 ): In contrast, if g2 then the disin‡ation in g3 is L costless since the public’s optimal play is p4 (the best response to g3 ; not g2 ):

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

25

It is now apparent that the public’s impatience reduces the value of R and hence may alter the su¢ cient condition for Ramsey SPNE. Instead of deriving analytically p from (33) we focus on the extreme case p = p = 0 which is a su¢ ciently low threshold for all frg ; rp ; q; v; x; zg satisfying (6). Under p = 0 (33) becomes z x from which it follows that R = 0: Therefore, (25) and (26) no longer apply and (23) and (24) become the relevant condition 8ng ; R = (0; 1], and for all fa; b; c; d; q; v; x; zg satisfying (6). Hence we need to show that any rg > rp satisfy the following two conditions from (8): (i) g a b rg a b 40 In terms under R = 0 it holds that rrp a d and (ii) 8R = (0; 1) it is true that r p a d R. rg g p of (ii) realize that any r > r have, from the de…nition of R; the property that rp 1 + R. This implies that claim (ii) can be rewritten as 1 + R aa db R. Divide both sides by R to obtain a b 1 1 R +1 a d . To see that this is satis…ed utilize two characteristics. First, R + 1 > 2 since R < 1. a b 1 a b Second, rearrange a 2d b into 2 a d . Combining these gives R + 1 > 2 a d which g completes the proof of (ii). In terms of (i) note that under R = 0 all rg > rp satisfy rrp 2. Using this jointly with 2 aa db completes the proof. Appendix C. Proof of Theorem 2 Proof. We need to prove an extension of Lemma 1 under g < 1 (note that the proof of Theorem 1 argued that p does not a¤ect the necessary and su¢ cient conditions except for the value of R which only a¤ects the su¢ cient condition). Under g < 1 (24) and (26) become41 (34)

b

t 1 g

+a

t=1

(35)

b

r g +kn+1

g

kn X

r X

t 1 g

+a

kn X

g

t 1 g

t=1

r

r X

+a

t 1 g

>d

t=r g +1

t=kn +1

t 1 g

g

r g +kn+1

g

X

r X

t 1 g

+b

t=1

g

r +kn+1

X

t 1 g

r X

+c

t 1 g :

t=r g r p +kn+1 +1

t=1

t=kn +1

t 1 g :

t=r g +1

p

>d

X

Focusing on (34) it can be rearranged into (d

b)

kn X

(a

b = (a

d) + (d (a

b)

g

r P

t 1 g

t 1 g

(a

X

b)

b) and split the …rst series to obtain t 1 g

(a

d)

< 0:

r g +kn+1

g

kn X

t 1 g

t=r g +1

t=kn +1

r X

t 1 g

(a

X

b)

t 1 g

< 0;

t=r g +1

t=1

t=1

add

r X

d)

t=1

Use a

r g +kn+1

g

t 1 g

to both sides and collect the terms

t=kn +1 g

(a

b)

r X

(a

d)

t=1

g

(a

b)

r X

t 1 g

< (a

b)

t=1

Adding up the series on the RHS we obtain r X

r g +kn+1

g

t 1 g

(a

t=1

d)

r X t=1

t 1 g

t 1 g :

t=kn +1

g

t 1 g

X

< (a

b)

kn g

1

r g +kn+1 kn g

1

:

g

40 We have used the property of (6) that c d = a b to collapse the two equalities in (8). 41 It again su¢ cies to consider these two conditions since under the general constraints (6) equations (23) and

(25) yield the same respective conditions.

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

Rrp from (7) to substitute away

Since the RHS is decreasing in kn+1 we need to use kn+1 = kn kn+1 : This yields g

(a

g

r X

b)

t 1 g

(a

d)

t=1

r X

t 1 g

< (a

b)

26

kn g

r g Rr p g

1 1

t=1

:

g

Note that the RHS is increasing in kn , ie the condition is the strongest at ng = 1 since kn has a unique maximum in k1 : The same can be analogously shown for (35) which we omit due to the space constraint. The realization that for R = 0 we have N g = 1 …nishes the proof. Appendix D. Proof of Theorem 3 Proof. Claim (i): Realize that (10) implies 0 < g < 1 for all assumed values. First, the inequality ja dj < jc bj (direct consequence of (6)) ensures that the argument of the square root is positive. Moreover, the inequalities a > d and c > b imply that this argument is less than one. It is apparent in (8) that the strongest possible necessary and su¢ cient condition (highest rg (R)) obtains under costless disin‡ation if the public’s payo¤s fq; v; x; zg are such that R ! 1: Furthermore, we have shown in Theorem 1 that the public’s impatience weakens the su¢ cient conditions. Therefore, it su¢ ces to focus on the analog of (26) under g < 1 = p , (35),42 assuming that kn = k1 = rp and kn+1 = k2 ! 0 (the latter leading to R ! 1): Substituting this into (35) yields (36)

(a

d)

g

p

r gX rp

t 1 g

> (d

b)

t=r p +1

r X

t 1 g

+ (c

r X

a)

t 1 g :

t=r g r p +1

t=1

Therefore, in this proof it su¢ ces to show that (10) implies (36). First, let us realize that (10) can be rewritten as c b+d a rp ; g > c b which can be rearranged into d

0<1 Since

2r p g

b+c a d

rp g

a1 rp g

:

> 0, we have 2r p g

0< Consequently, for each

g

1

b+c a d

rp g

a1 rp g

!

:

= (0; 1) there exists rg 2 N such that for all rg > rg ! rp d b+c a1 g rg 2r p 1 : g < g rp a d g

Multiplying both sides by (a

(a d)

rp g

d)

rp g

r g 2r p g

1

d)

rp g

r g

1 1

g

g 2r

> (d

b+c

we obtain

a)(1

rp g ):

> 0 and split the right-hand side

p

> (d g

2r p g

> 0 and dividing by

Moreover, we divide both sides by 1 (a

d

1 b) 1

rp g g

42 Again the condition implied by (25) will be equivalent to (35).

+ (c

1 a) 1

rp g g

:

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

27

Note that the three fractions are in fact partial sums of geometric series with quotient (37)

0 > (d

b)

rp X

t 1 g

(a

d)

r gX rp

t 1 g

+ (c

rp X

a)

t=r p +1

t=1

g

t 1 g :

t=1

Finally, we realize that p

g

r X

t 1 g

r X

>

t 1 g ;

t=r g r p +1

t=1

which, in connection with c a > 0 and equation (37), implies that (36) holds and completes the proof of claim (i). Claim (ii): Realize that (11) implies 0 < g < 1 for all assumed values since a > d and jd bj < ja bj (from (6)). It is apparent in (8) that the weakest possible necessary and su¢ cient condition (lowest rg (R)) obtains under costly disin‡ation if the public’s payo¤s fq; v; x; zg are such that R ! 0: Furthermore, we have shown in Theorem 1 that the public’s impatience weakens the su¢ cient conditions. Therefore, we can focus on the analog of (24) under 0 = p g < 1, (34),43 assuming that kn+1 = k2 ! kn = k1 = rp (the latter leading to R ! 0): Substituting this into (34) yields g

(38)

(a

r X

d)

t 1 g

+ (a

b)

t=r p +1

r gX +r p

p

t 1 g

> (d

b)

t=r g +1

r X

t 1 g :

t=1

Using the formula for a …nite sum of geometric series and rearranging yields rp g )[(a

(1

b)

rp g

(d

(5)

rp rp g )(2 g

b)] = (1

1) > 0:

The conditions under which this is satis…ed are reported in (11) (the fact that there may be no Ramsey SPNE implies that the inequality in (11) must be strict). Appendix E. Proof of Proposition 2 Proof. Under the policymaker’s impatience the conditions analogous to (18) and (19) become g

(39)

a

r X

p

t 1 g

>c

t=1

r X

g

t 1 g

+d

r X

p

t 1 g

and

b

t=r p +1

t=1

r X

g

g

t 1 g

+a

r X

t 1 g

>d

t=r p +1

t=1

r X

t 1 g :

t=1

which can be rewritten into 1 a 1

rg g g

1 >c 1

rp g

+d

rp

rg rp g

1

g

1

>

g

and

rp g

1 b 1

+a

rp

rg rp g

1

g

1

g

1 >d 1

rg g

:

g

Dividing by (1 g ), taking the logarithms, and rearranging yields (12) which can only be satis…ed if (13) holds. Note two properties which follow from (12). First, the arguments of the logarithms in (12) are positive if and only if (13) holds. Second, both the base and the argument of the logarithms in (12) are then (strictly) between 0 and 1 - to see this realize that 0<

c a

d d

rp g

c a

a a = d a

b d

rp g

d a

b c < d a

d d

c a

a a = d a

b d

d a

b = 1: d

This implies that rg (0) in (12) is increasing in rp and, after simple manipulations, that it is also increasing in c and d and decreasing in a and b. Proving that rg (0) is also decreasing in g 43 Again the condition implied by (23) will be equivalent to (34).

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

28

requires further calculations. Focus on the latter element of (12) and rewrite it into rg

p b r d g

ln( aa

=

ln

Our task is to show that rg is decreasing in

d b a d)

:

g

on the considered domain ! r b rp d ;1 : a b g

D :=

For the sake of clarity, we simplify the notation by de…ning :=

a a

b d ; ! := d a

b ; r := rp ; := d

g:

Therefore, we want to show that the function r

ln(

f( ) =

!)

ln

is decreasing in , or equivalently that f 0 ( ) < 0 on D. Obviously, r

f ( )= =

r

1

r

0

!

1

ln

r

ln(

!)

2

r

r

ln (

ln

r r

(

r

!) ln( !) ln2

!)

:

Since the denominator is always positive, it su¢ ces to show that r

r

ln

r

(

r

!) ln(

!) < 0;

or equivalently r

( ) := r

ln < (

r

!) ln(

r

!) =: ( )

on the considered domain D. Taking into account de…nitions of and ! we observe that (1) = 0 = (1). Therefore it su¢ ces to show that 0 ( ) > 0 ( ) for all 2 D. But this is satis…ed since: 0

r

2

r 1

x

( )>

r 1

ln + r x

0

( )

> r xr

r ln > ln( r rp g

(a

d)

rp g

>

r

1

ln(

r

r

!) + r xr

1

!) !

a b rp d b a d g a d rp > (a b) g (d b) >

0 > (d

b)(

rp g

1);

where the last inequality is trivially satis…ed since d > b and g < 1. Noting that the two elements of (12) equal for all values of the general game (ie the proof would be analogous for the former part) completes the proof.

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

29

Appendix F. Proof of Proposition 3 Proof. Claim (i). The fact that rg (R) rp 1; 8R; is implied by Theorems 1 and 3. Speci…cally, realize that if rg = rp (which is the standard repeated game) then H will be played uniquely in g p H equilibrium in all moves, (iH will be played in at least n ) ; 8i; n. Further, if r < r then 8R, one move on the equilibrium path (the easiest way to see this is to note that the policymaker has the …nal move which will uniquely be H ): Claim (ii). Under the circumstances of claim (ii) (if there exists rg satisfying (15)), both L and H obtain on the equilibrium path which increases in‡ation variability. Intuitively, due to rg < rg (R) the policymaker in‡ates in at least one ng but due to g > g (R) and rg > rg (R) he …nds it optimal to eventually disin‡ate, ie L uniquely obtains on the equilibrium path for at least one ng : For an example with rg > rp see Section 4.1 that describes the case rg = 4; rp = 3. Note however that (15) also includes cases of rg < rp : For example in the speci…c BG game with g p g = p = 1, under r = 4; r = 5 both players play H throughout on the equilibrium path with the exception of their second moves in which they both play uniquely L: Appendix G. Central Bank Independence Index (Sousa, 2002) Each of the following criteria is assigned up to one point, for more details see Sousa (2002). PERSONAL INDEPENDENCE 1. Appointment of the central bank board members 2. Mandate duration of more than half of the CB board members. 3. Fiscal policymaker’s participation at central bank meetings. POLITICAL INDEPENDENCE 4. Ultimate responsibility and authority on monetary policy decisions. 5. Price stability 6. Banking supervision 7. Monetary policy instruments ECONOMIC AND FINANCIAL INDEPENDENCE 8. Policymaker …nancing 9. Ownership of the central bank’s (equity) capital Appendix H. Central Bank Accountability Index (Sousa, 2002) Criteria and methodology adopted from De Haan et al. (1998). We only use the ‘…nal responsibility’ component that best proxies the policymaker’s LR commitment (explicitness of the IT). Each of the following criteria is assigned up to one point, for more details see Sousa (2002). FINAL RESPONSIBILITY 1. Is the central bank subject of monitoring by Parliament? 2. Has the policymaker (or Parliament) the right to give instruction? 3. Is there some kind of review in the override procedure? 4. Has CB possibility for an appeal in case of an instruction? 5. Can the CB law be changed by a simple majority in Parliament? 6. Is past performance a ground for dismissal of a central bank governor?

Incorp orating Rigidity and Com m itm ent in the Tim ing Structure of M acro econom ic Gam es

30

Appendix I. Evaluation Table Independence

Accountability Sousa (2002)*

Final

1.00 0.00 1.00 0.00 0.00 1.00 1.00 0.00 1.00 0.00 1.00 1.00 1.00 1.00 1.00 0.00 1.00 1.00 0.00 0.00 1.00 0.00 0.00 1.00 0.00 0.00 1.00 1.00 1.00 1.00 1.00 0.00 0.00

Total#

2.83 2.16 3.16 1.50 1.83 1.83 3.16 2.83 3.66 2.66 3.66 3.16 3.16 3.16 3.66 3.33 3.16 3.16 3.66 2.16 3.16 2.33 3.16 2.16 1.83 2.16 3.16 3.50 3.16 3.16 3.33 2.83 1.83

Financial

1.25 0.50 1.66 1.75 0.50 2.00 1.58 2.16 2.50 1.00 2.50 1.50 1.50 1.91 1.91 1.75 1.00 2.16 0.75 0.75 1.25 1.83 2.41 1.83 1.58 1.25 1.50 1.00 0.75 2.75 2.08 1.66 2.00

Economic

Political

1 Argentina 2 Australia 3 Austria 4 Belgium 5 Canada 6 Chile 7 Czech Republic 8 Denmark 9 EMU - ECB 10 England 11 Finland 12 France 13 Germany 14 Greece 15 Hungary 16 Iceland 17 Ireland 18 Italy 19 Japan 20 Korea 21 Luxemburg 22 Mexico 23 Netherlands 24 New Zealand 25 Norway 26 Poland 27 Portugal 28 Slovakia 29 Spain 30 Sweden 31 Switzerland 32 Turkey 33 USA

Personal

Country

/

Sousa (2002)*

Responsibility

Index

5.08 2.66 5.82 3.25 2.33 4.83 5.74 4.99 7.16 3.66 7.16 5.66 5.66 6.07 6.57 5.08 5.16 6.32 4.41 2.91 5.41 4.16 5.57 4.99 3.41 3.41 5.66 5.50 4.91 6.91 6.41 4.49 3.83

2 5 1 4 4 3 2 2 1 4 2 2 2 1 2 4 2 1 3 4 2 2 2 4 5 3 1 1 2 1 2 3 2

*Assessment is based on the situation in January 2002. # Excludes aspect 9 due to missing observations.