Central banks and ambiguity

Central banks and ambiguity

International Review of Economics and Finance 17 (2008) 85 – 102 www.elsevier.com/locate/iref Central banks and ambiguity Willy Spanjers Department o...

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International Review of Economics and Finance 17 (2008) 85 – 102 www.elsevier.com/locate/iref

Central banks and ambiguity Willy Spanjers Department of Economics, Kingston University, Kingston-upon-Thames, Surrey KT1 2EE, United Kingdom Received 8 August 2003; received in revised form 24 January 2006; accepted 3 April 2006 Available online 9 August 2006

Abstract The purpose of this paper is to discuss the effects of ambiguity (or ‘non-calculable risk’) on the public's expectations about inflation and its impact on central bank policy. The effects of ambiguity are addressed in a textbook setting with a short run aggregate supply curve. Ambiguity about monetary policy can be characterised as a loss of central bank credibility. When the public is pessimistically inclined, its consequences are excessive inflation expectations and a national income below its natural rate. This result is obtained both in the context of ‘discretion’ and of ‘inflation targeting’, although the impact of ambiguity is less pronounced in the latter case. If the public is optimistic with respect to the monetary policy of the central bank, loss of credibility has no impact. © 2006 Elsevier Inc. All rights reserved. JEL classification: D81; E52; E58 Keywords: Ambiguity; Choquet Expected Utility; Central bank; Monetary Policy; Robust control

1. Introduction What does it take to establish confidence in a central bank? This question is of major importance for monetary policy makers. But answering it is tedious. Firstly, confidence is in the mind of the beholder. It is determined by cognitive processes that are difficult to reconcile with the established paradigms of economic decision making. Worse, it is not clear what the precise meaning of ‘confidence’ is in this setting. There are models that provide precise interpretations of ‘reputation’ in the context of incomplete information: they rely on the inability of the public to learn the ‘true’ preferences and objectives of the central bank. Depending on its ‘true’ preferences, the central bank may be tempted to deceive the public, by pretending to have different preferences. When considering a finite time horizon, the central bank eventually shows its true face. Meanwhile, the longer it keeps up appearances, the more likely it seems to the public that the pretended preferences are its true preferences and the better its reputation is.1 But this hardly is the mechanism that determines the confidence in present-day central banks. The actual decision processes in the leading central banks make it impossible to hide their true preferences for a sustained period of time. We E-mail address: [email protected] For such Bayesian reputation models see e.g. Kydland and Prescott (1977), Barro and Gordon (1983a,b), Backus and Drifill (1985) and Barro (1986). 1

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must therefore consider different ways to conceptualise the meaning of ‘confidence’ in central banks. This paper follows the approach to understanding confidence by studying situations in which it is absent, as taken in Spanjers (1998/2005). We consider the presence of ambiguity to be the main characteristic of situations in which there is a lack of confidence. This leads us to analyse the impact of ambiguity on the choice and effectiveness of monetary policy. We distinguish between ‘calculable’ and ‘non-calculable’ risk and refer to the latter as ambiguity.2 Knight (1921) considers entrepreneurs to be specialists in dealing with situations of ambiguity, i.e. situations for which there is no relevant experience to guide decision making. Keynes (1937) considers ambiguity to be one of the most characteristic features of decision making under uncertainty:3 […][A]t any given time facts and expectations were assumed to be given in a definite and calculable form; and risks, of which, though admitted, not much notice was taken, were supposed to be capable of an exact actuarial computation. The calculus of probability […] was supposed to be capable of reducing uncertainty to the same calculable status as that of certainty itself […]. Actually, however, we have, as a rule, only the vaguest idea of any but the most direct consequences of our acts. Sometimes we are not much concerned with their remoter consequences, even though time and change may make much of them. But sometimes we are intensely concerned with them, more so, occasionally, than with the immediate consequences. […] […] By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty […]. The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention, or the position of private wealth owners in the social system in 1970. About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know. Keynes continues:4 Now a practical theory of the future […] has certain marked characteristics. In particular, being based on so flimsy a foundation, it is subject to sudden and violent changes. The practice of calmness and immobility, of certainty and security, suddenly breaks down. New fears and hopes will, without warning, take charge of human conduct. The forces of disillusion may suddenly impose a new conventional basis of valuation. All these pretty, polite techniques, made for a well-panelled board room and a nicely regulated market, are liable to collapse. At all times vague panic fears and equally vague and unreasoned hopes are not really lulled, and lie but a little way below the surface. Modern examples of ambiguity include global warming, the BSE-crisis, bird-flu, the Gulf War, the South–East Asian crisis, New Economy technologies and the impact of 9/11. Risk may fail to be calculable for two basic reasons. Firstly, it may not be possible to assign a unique (subjective) probability distribution to different scenarios for the future. Secondly, it may be difficult to associate a unique outcome to each scenario. In either case there is ambiguity. The next question is how decisions are made in the face of ambiguity. As already indicated in the quotation from Keynes, decisions will depend on the decision-maker's attitude with respect to it. Optimists will hope for the best, pessimists will fear the worst. If one insists on referring to subjective probabilities, it represents a situation in which the decision-maker's probability assessment depends in a specific way on his choice of action.5 The first piece of evidence that decisions under ambiguity may fail to be compatible with the subjective expected utility approach was provided by the famous thought-experiment in Ellsberg (1961). Schmeidler (1982/89) and Gilboa (1987) provide an axiomatic foundation for decision making under ambiguity that can match that of subjective expected utility theory as provided by Savage (1954) and Anscombe and Aumann (1963). After this breakthrough, economists started to modify their standard analytical tools to deal with ambiguity.6

2

This distinction goes back to Knight (1921). He uses a different terminology but refers to the same concepts. Keynes (1937, pp. 113–114). 4 Keynes (1937, pp. 114–115). 5 See Spanjers (1999, Chapter 7). 6 Further developments include the unification of the approaches of Schmeidler (1982/89) and Gilboa (1987) (Sarin & Wakker, 1992), applications to portfolio choice (Dow & Werlang, 1992), and game theory (Dow & Werlang, 1994), and the ambiguity attitude (Ghirardato, Maccheroni & Marinacci, 2004). 3

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Against this background, we examine the impact of ambiguity on monetary policy. The insight that ambiguity or incalculable risk plays an important role in monetary phenomena which include expectations is not new. Keynes, when discussing his General Theory in 1937, for instance writes:7 […] [P]artly on reasonable and partly on instinctive grounds, our desire to hold money as a store of wealth is a barometer of the degree of our distrust of our own calculations and conventions concerning the future. […] The possession of actual money lulls our disquietude; and the premium which we require to make us part with money is the measure of the degree of our disquietude. The significance of this characteristic of money has usually been overlooked; and in so far as it has been noticed, the essential nature of the phenomenon has been misdescribed. For what has attracted attention has been the quantity of the money which has been hoared, and importance has been attached to this because it has been supposed to have a direct proportionate effect on the price level […] [F]luctuations in the degree of confidence are capable of having quite a different effect, namely, in modifying not the amount that is actually hoarded, but the amount of the premium which has to be offered to induce people not to hoard. […] This, expressed in a very general way, is my theory of the interest rate. We analyse the monetary policy of a central bank in the context of a short run aggregate supply curve as in Mankiw (2003).8 In this framework, the public resents finding itself making false inflation predictions. The central bank dislikes inflation deviating from its optimal level or output being below its natural rate, but it likes output to exceed its natural rate. Our analysis distinguishes between strategic ambiguity on the one hand and state ambiguity of the central bank over the position of the short run aggregate supply curve on the other. Whereas the concept of strategic ambiguity derives from a game-theoretic setting, state ambiguity is closely related to the literature on robustness and on robust control. Friedman is claimed to have ‘expressed an enduring concern when he recommended that designers of macroeconomic policy rules acknowledge model uncertainty. His style of analysis revealed that he meant a kind of model uncertainty that could not be formalized in terms of objective or subjective probability distributions over models'.9 It seems that Friedman was concerned with ambiguity as described above. The literature on robustness springs from the observation that policy makers typically do not have complete confidence in their models. Therefore, rather than reaching for the optimal decision in one specific model, they would prefer to follow a policy that performs reasonably well over a range of plausible models. One problem with this approach is that the selection of the set of plausible benchmark models implicitly determines the objective function of the decision-maker. So ambiguity on structural parameters also implies ambiguity on the correct loss function.10 When applying robust control, the central bank bases its monetary policy on a distorted model that contains the worst conceivable development path, the so-called ‘worst case’, but than acts as if model uncertainty no longer exists.11 In some settings central banks, when facing parameter uncertainty, pursue a more aggressive policy than in its absence.12 In another interpretation of robust control, however, it is claimed that the central bank follows the same aggressive policy when it takes its approximation as the true model, but maximizes a target function that reflects an additional risk sensitivity.13 The model in the present paper is broadly in line with the latter interpretation of robust control. The decision-makers in the model consider standard objective functions that are extended with a term that reflects an additional sensitivity to uncertainty. Contrary to the above line of modelling, though, this additional term does not reflect an additional sensitivity to risk, but rather to ambiguity. Furthermore, adding this term is not ad hoc, but a natural result from applying the approach of Choquet Expected Utility to the problem at hand, for which we consider both a discretionary monetary policy and a policy of inflation targeting. To be fair, our model is not the first to apply the Choquet Expected Utility approach to monetary policy. Caglianrini and Heath (2000) notice that ambiguity can explain why interest rates typically move in steps, whereas one would 7

Keynes (1937, pp. 115–116). As indicated by Mankiw, the short run aggregate supply curve can be justified by the presence of sticky wages (Fischer, 1977), imperfect information (Friedman, 1968; Lucas, 1976), or by sticky prices (Rotemberg, 1982). 9 Hansen and Sargent (2003, p 582) as quoted in Wagner (2005b). 10 See Walsh (2005) as discussed in Wagner (2005b). 11 Wagner (2005a). See also Svensson (2000). 12 See Walsh (2003) as discussed is Wagner (2005b). 13 Hansen and Sargent (2005) as discussed in Wagner (2005b). 8

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expect them to move smoothly. Although they prefer the formalisation of ambiguity by Bewley (1986, 1987), their result seems a variation on Dow and Werlang (1992). Brock, Durlauf, and West (2004) consider a central bank with a Choquet expected objective function that represents a pessimistic decision-maker. The analysis of Chprits and Schipper (2003) is closer to the present paper, as they consider ‘double-sided intransparency’. They analyse the potential impact of ‘intransparency’, which they model as ambiguity, on the interaction between a central bank and a trade union, with a Philips-curve describing the trade-off between inflation and unemployment. The central bank follows an intransparent monetary policy, whereas the trade union follows an intransparent wage policy.14 The aim of our analysis, by contrast, is to provide a framework in which to discuss the role of confidence and credibility in contemporary central bank policy in a setting where both the central bank and the public face ambiguity.15 Another difference lies in the way ambiguity is treated. Whereas Chprits and Schipper confine their analysis to pessimism, we allow for an optimistic public as well. Furthermore, we introduce an innovation to the literature on ambiguous beliefs by expressly considering the role of perceived upside and downside risk. Finally, we discuss the real-life example of the European Central Bank to support the relevance of our argument. In the next section, we discuss decision making under ambiguity and the equilibrium concept we use in this paper. In Section 3 we derive the effect of strategic ambiguity about monetary policy on the inflation expectations of the public. We show that this strategic ambiguity results in excessive inflation expectations by a pessimistic public, which negatively affect output. We proceed by including the central bank in the analysis in Section 4. Both for a discretionary and an inflation targeting monetary policy, the effects of ambiguity point in the same direction, but inflation targeting dampens its impact. While the public is exposed to strategic ambiguity, the central bank faces ambiguity about the effectiveness of surprise inflation, which leads to interesting results. For a monetary policy with a flexible inflation target, ambiguity on the effectiveness of surprise inflation only influences monetary policy if the public has less than full confidence in the central bank. More surprisingly, when ambiguity does have an impact, it leads a cautious (i.e. pessimistic) central bank to loosen monetary policy, in order to increase inflation. In Section 5 we discuss the policy framework of the European Central Bank in the light of the model and results of this paper. Concluding remarks and suggestions for further research are provided in Section 6. 2. Decisions under ambiguity In this section we consider decision making under ambiguity using the approach of Choquet Expected Utility. Within this approach, we focus on a specific interpretation of the beliefs of the decision-makers. After considering individual decision-makers, we turn to their interaction, i.e. to equilibrium under ambiguity. 2.1. Decision under ambiguity Our approach to analysing the impact of ambiguity on decision-makers is a generalization of expected utility theory. Under expected utility, the preferences of a decision-maker can be represented by a utility function. This utility function assigns each random variable the expected value of the utility each outcome generates. More formally the representation is as follows. Consider a random variable x with probability distribution P, which assigns to each state of nature s an outcome x(s). The utility of each outcome x(s), when it occurs, is evaluated by the decision-maker according to his von Neumann– Morgenstern utility index u and leads to a utility level of u(x(s)). The preferences of the decision-maker can be represented by a utility function U, which is defined as the expected value of the utility index u over x with respect to P, i.e. U ðxÞ :¼ E p fuðxðsÞÞg:

ð1Þ

For decision making under ambiguity, this representation is generalized. Apart from a random variable x with an assessed probability distribution P and a von Neumann–Morgenstern utility index u, other variables determine the 14 Contrary to Chprits and Schipper (2003), Kissmer (2005), using a different kind of objective function, finds that increased transparency may reduce social welfare. 15 For a discussion of the New-Neoclassical-Synthesis and its historical roots, see Wagner (2005b).

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utility function U. These additional variables include the degree of optimism of the decision-maker, β ∈ [0, 1] and his level of confidence in his assessed probability distribution, γ ∈ [0, 1]. They also include the smallest value of the state of nature s and the largest value ¯s which the decision-maker considers to be plausible.16 We refer to s and ¯s as the ¯ ¯ downside risk and the upside risk of the state of nature, respectively. From these variables the utility function U that 17 represents the preferences of the decision-maker is obtained as     U ðxÞ :¼ gd Ep fuðxðsÞÞg þ ð1−gÞdbd max uðxðsÞÞ þ ð1−gÞd ð1−bÞd min uðxðsÞÞ : ð2Þ sa½ Ps ;¯s 

sa½ Ps ;¯s 

This utility function is the sum of three components. The first component reflects the expected utility of the random variable. Its weight equals the level of confidence in the assessed probability distribution, γ. The presence of ambiguity may cause the evaluation of the random variable to deviate from its expected utility, but it does not indicate whether this deviation is to the advantage or to the disadvantage of the decision-maker. The remaining two components quantify this deviation. Their combined weight equals the amount of ambiguity perceived by the decision-maker, (1 − γ). The degree to which the decision-maker is inclined to think that ambiguity leads to an advantageous outcome is represented by his degree of optimism β. This leads to the second term in the utility function, which contains the von Neumann–Morgenstern utility of the best plausible outcome. The weight of this term is the proportion β of the amount of ambiguity (1 − γ), i.e. (1 − γ) · β. Similarly, (1 − β) represents the degree to which the decision-maker is inclined to think that ambiguity leads to an outcome that is to his disadvantage. This is reflected in the final term of the utility function, which contains the von Neumann–Morgenstern utility value of the worst plausible outcome. This term is weighted by the remaining fraction (1 − β) of the amount of ambiguity (1 − γ), hence (1 − γ) · (1 − β).18 In this paper our main focus is on decision-makers who are pessimistically inclined, e.g. for whom β = 0. Furthermore, we restrict attention to probability distributions that concentrate all probability mass in a single state of nature t. Under these additional assumptions, the utility function of the decision-maker reduces to19   U ðxÞ :¼ gd uðxðtÞÞ þ ð1−gÞd min uðxðsÞÞ : ð3Þ sa½ Ps ;¯s 

This utility function weighs the utility of the outcome that would ‘normally’ occur by the level of confidence γ. Because the decision-maker is pessimistic, the remaining weight, (1 − γ), i.e. the level of ambiguity, is assigned to the worst plausible outcome. The associated reasoning of the decision-maker could be: ‘Although I am rather confident that state t will occur, I am not sure. As I do not really know what other state may occur instead of t, I had better taken into account the worst that may plausibly happen’. 2.2. Equilibrium under ambiguity When considering the interaction of decision-makers under ambiguity, it is useful to distinguish between state ambiguity and strategic ambiguity. State ambiguity refers to ambiguity about the environment in which the interaction takes place. It should be addressed in a similar way as randomness, the difference being the way the ‘expected’ pay-offs 16

For this particular restriction on the states of nature to have a meaningful interpretation, one must, of course, make certain assumptions about the random variable x. A discussion of why this approach is sensible and how it can be formulated in a more general way is beyond the scope of this paper. 17 We assume that the ambiguity faced by a decision-maker is represented by a belief function, which is then combined with his ambiguity attitude to obtain the capacity over which the Choquet integral is taken. 18 One may wonder if the weights associated to the ‘best case’ and ‘worst case’ can be interpreted as subjective likelihood estimates that the assessed probability distribution fails to apply and for which the ‘best case’ and the ‘worst case’ outcomes are obtained, respectively. Such interpretation is possible but implies that the decision-maker believes that the probabilities of the elementary states of nature in the model depend on his own actions. This requires the presence of an unmodelled actor who reacts on the decision-maker's actions, in a way that changes the probability distribution (e.g. Greek gods), see Kelsey and Milne (1997, 1999). This would imply that the model under consideration is a reduced form of an unspecified more appropriate model. As an alternative, one could assume a superstitious decision-maker. We prefer the interpretation provided in the paper to either of these alternatives. 19 Brock et al. (2004) suggest a loss function for the central bank that is based on this type of utility function. We extend their approach by allowing for ambiguity in the mind of the public as well. See also Epstein and Wang (1994) and Kuester and Wieland (2005).

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are calculated.20 In the presence of only state ambiguity, there is no reason to refrain from using the usual gametheoretic equilibrium concepts, since the sole impact of ambiguity is to change the objective functions of the players. Interacting decision-makers may also face a different type of ambiguity. It does not refer to the environment in which decisions are made, but rather to the choice of strategies by the others. We refer to such ambiguity as strategic ambiguity. When considering strategic ambiguity, there may be good reasons to apply equilibrium concepts that are specifically developed to deal with ambiguity about the strategy choices of other players. Such concepts have been introduced in Dow and Werlang (1994), Marinacci (2000) and Kelsey and Spanjers (1997/2004). These equilibrium concepts are appropriate for some situations of strategic ambiguity, but not for all. They look for combinations of ambiguous beliefs and strategies that are mutually consistent. Usually, this leads to a multitude of equilibrium beliefs and equilibrium strategies, where the model offers no obvious way to select any specific equilibrium. For our present purposes, these equilibrium concepts are of little use because of their indeterminacy. We want to consider strategic ambiguity, but on the other hand, we wish to find a unique equilibrium. This goal can be achieved by allowing for ambiguity with respect to the strategy choices of the players, but by assuming that the implied beliefs have a particular shape. This seems justified as in most applications ambiguity takes a specific form: beliefs are not just arbitrarily picked from a large range of possibilities. We believe that ambiguous beliefs are the outcome of a cognitive process, for now unspecified. This process not only determines the degree of familiarity with a given situation, and thus the ‘amount’ of ambiguity that is experienced, but also puts bounds on what are considered plausible values of the ambiguous variables. We identify these boundaries with the more familiar concepts of upside risk and downside risk, referring here to the maximum and minimum values of the state of nature that may plausibly be obtained. In this context, learning about an ambiguous variable consists of both finding more reliable values for the upside and the downside risk and of reducing the amount of ambiguity faced. One practical way of promoting this process of learning is by first observing how the ambiguous variable behaves when it is not interfered with. Once this experience is gathered, deliberate experiments may be used to learn more about its behaviour in extreme situations. In the absence of more solid theories as to how (strategic) ambiguity arises and develops, we consider beliefs that describe the level of ambiguity as an exogenously given characteristic of each individual player. In doing so, the mathematical treatment of strategic ambiguity becomes identical with that of state ambiguity where strategies by the other players represent the states of nature. The conceptual difference, of course, remains. The consequence of this is that, even though we consider games in which players face both state ambiguity and strategic ambiguity, standard Nash equilibrium and backward induction analysis can still be applied. The only difference is that players have somewhat unusual objective functions.21 In the remainder of the paper, Nash equilibrium in pure strategies is considered in the case of discretionary monetary policy. For this situation, the best response functions of the public and the central bank are derived and equilibrium is obtained for the strategies that are mutual best responses. To obtain the equilibrium for a policy of inflation targeting, backward induction is applied. Here the central bank announces its optimal strategy, assuming that the public reacts according to its best response function. In the next section we derive the best response function of the public before we analyse the equilibria for the different monetary policies in Section 4. 3. The public In this simple model, we first focus on the public's expectations of the rate of inflation. The public's expectations determine a range of economic decisions by consumers, firms and the government. These decisions influence the deviation of the level of output from its natural rate. For the purpose of simplification, we assume that the actual inflation rate is determined by the monetary policy of the central bank. This is not to say there are no other determinants of the rate of inflation. Rather, we do not consider them. We also abstract from interactions amongst the public that may influence the level of output. For example, we disregard the possibility of multiple equilibrium levels of expenditure and output and the coordination problems that result from this. 20 21

Here the expected utility is obtained by applying the Choquet integral to the appropriate representation of the beliefs. See Rothe (1996/98, 1996/99) for more general considerations on applying backward induction in ambiguous extended games.

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We assume the representative member of the public faces the decision problem to make the best possible prediction πe of the actual inflation rate π chosen by the central bank. In particular, the following quadratic objective function is maximized: max −ðp−pe Þ2 : e

ð4Þ

p

As is well-known, for any actual rate of inflation π chosen by the central bank this results in pe ðpÞ :¼ p

ð5Þ

as the best response of the public. 3.1. Strategic ambiguity faced by the public The public may, however, face strategic ambiguity regarding the decision of the central bank. Such beliefs can be interpreted as stating that the public expects the central bank to choose a particular inflation rate, but it is not quite sure if the central bank may not, in some unpredictable way, deviate from this action. We refer to this as strategic ambiguity, as it relates to the central bank's choice of strategy rather than to the exogenous economic environment, i.e. to the state of nature. The more confidence the public has in the central bank, the less emphasis it puts on potential unpredictable deviations. The confidence of the public that the central bank acts as anticipated is denoted by the level of confidence γ ∈ [0, 1], where (1 − γ) denotes the amount of strategic ambiguity. Assuming that the public is pessimistically inclined, its decision problem becomes   e 2 e 2 max g½−ðp−p Þ  þ ð1− gÞ min −ðp˜−p Þ : ð6Þ e p a½p ;p P ¯

p˜ a½p ;p P ¯

Here the maximal plausible downside risk and the maximal plausible upside risk with respect to the actual rate of inflation enter into the considerations of the public as psychological points of reference. The downside risk of inflation is represented by the minimal plausible inflation rate π; the upside risk by π¯. These estimates of upside and downside ¯ risk depend both on the particular situation under consideration and on the history from which it arises. We assume that the upside risk of the actual inflation rate exceeds the downside risk, for the given expected rate of inflation πe , which seems a plausible assumption to make. From this, we obtain as the optimal or best response inflation expectation of the public22,23: pe ðp; gÞ :¼ gp þ ð1−gÞ¯p :

ð7Þ

Proposition 1. For a pessimistic public, a decrease in the level of confidence in the central bank increases the expected rate of inflation. 3.2. Short run aggregate supply We use a simple short run aggregate supply curve to relate the actual and expected inflation rates to the level of output: yðp; pe Þ :¼ y n þ aðp−pe Þ;

ð8Þ

1 P When the downside risk with respect to expected inflation exceeds the upside risk, i.e. pe N ðp þ pÞ, we obtain pe ðp; gÞ :¼ gp þ ð1−gÞ Pp . This 2 P 1 P 1 p case occurs whenever pN 2g p þ ð1−2gÞP . 23 For an optimistic public, for each πe the best plausible outcome associated to the upside risk, is obtained by solving the term associated with e 2 (1 − γ)β, i.e. maxp˜ a½Pp ;P ˜−pe Þ2 . The resulting the decision problem for the public, since β = 1, is maxpe a½Pp ;P p  g½−ðp−p Þ  þ ð1−gÞd1d 0. So the p − ðp e best response expected inf lation for an optimistic public follows as π (π,γ): = π. 22

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where a denotes the effectiveness of surprise inflation. It should be noted, however, that the effectiveness of surprise inflation also represents the impact of incorrect inflation expectations by the public on the level of output. For any given choice of the actual inflation rate π by the central bank, the inflation expectation πe (π, γ) leads to a level of output y ¼ y n þ að1−gÞðp−¯ p Þb y n :

ð9Þ

Proposition 2. For a pessimistically inclined public, strategic ambiguity over the inflation rate π depresses output below its natural rate. 4. The central bank The next question we address is how a lack of confidence in the central bank affects its behaviour. We consider two cases: the case in which the monetary policy of the central bank adheres to an inflation target and the case in which the central bank chooses the inflation rate as it pleases, i.e. at its discretion. We first consider discretionary monetary policy. Thereafter the situation in which the central bank is committed to its announced inflation rate, inflation targeting, is analysed. 4.1. Discretion When the central bank is not committed to any type of target, it may choose its monetary policy to fit its own objectives. At the same time that the central bank determines its monetary policy, the public determines its inflation expectations. In the absence of ambiguity, the resulting interaction can be described by a Nash equilibrium in pure strategies in the appropriate normal form game.24 4.1.1. No ambiguity for the central bank We assume the decision problem of the central bank is:25 max bðyðp; pe Þ−y n Þ−ðp−pn Þ2 : p

ð10Þ

Thus, there is a certain level of inflation πn that the central bank considers most appropriate for the economy as a whole. Furthermore, the central bank would prefer a level of output above its natural rate yn and dislikes it being less. Taking the short run aggregate supply curve into account, the decision problem becomes max baðp−pe Þ−ðp−pn Þ2 : p

ð11Þ

For given inflation expectations πe of the public, the optimal choice of inflation rate by the central bank, i.e. its best response, is 1 pðpe Þ ¼ pn þ ba: 2

ð12Þ

24 Alternatively, one may assume that the central bank decides on the actual level of inflation after the public formed its inflation expectations. The public, of course, takes this into account in determining its inflation expectations. In the absence of ambiguity and for the objective function of the public considered in this paper, both approaches lead to the same outcome. The reason for this coincidence of outcomes is that the best possible indifference curve of the public, which is obtained as πe = π, coincides with its best response function. In the presence of ambiguity, as for different specifications of the public's objective function, these competing approaches lead to different outcomes. 25 Straightforward calculations show that the (main) qualitative results of this paper remain unchanged when the objective function of the central bank takes the more familiar basic form maxπ − b · (y(π,πe) − yn)2 − π2.

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1 P Fig. 1. Pessimism: discretion, γ b 1. Please note that for expositional purposes the location of the line pe ¼ ðp þ pÞ is disproportionately to the right 2 P in all figures.

The resulting Nash equilibrium between the private sector and the central bank is characterized by the intersection of the best response curves, which yields: •

1 pD ¼ pn þ ba; 2



  1 peD ¼ g pn þ ba þ ð1− gÞ¯ p N p; 2



 1 yD ¼ y þ að1−gÞ p − ¯ p þ ba b yn : 2



n

n

This result is illustrated in Fig. 1.26 4.1.2. State ambiguity of the central bank Next we address the question of how ambiguity faced by the central bank affects these results. For the central bank we focus on ambiguity concerning the position of the short run aggregate supply curve. We refer to this type of ambiguity as state ambiguity, as it concerns the exogenous economic environment, i.e. the state of nature, rather than the choice of strategy by the public. We consider both ambiguity over the natural rate of output and ambiguity with respect to the effectiveness of surprise inflation. Ambiguity with respect to the natural rate of output is a very real problem for central banks. However, as can be seen from the above, the natural rate of output does not occur in the reduced form of the decision problem of the central bank. Therefore, ambiguity over the natural rate of output has no impact on the central bank's policy. It should be noted that different specifications of the short run aggregate supply curve may well lead to the natural rate entering the decision problem of the central bank. In such cases, ambiguity with respect to the natural rate may influence monetary policy. The effectiveness of surprise inflation does enter into the decision problem of the central bank. Let δ denote the level of confidence of the central bank in its estimate a of the effect of surprise inflation. The belief of the central bank should be interpreted as stating: ‘We think a correctly represents the effectiveness of surprise inflation, but we are not

26

As a point of reference, the corresponding result for a central bank with the objective maxπ − b(y(π, πe) − yn)2 − (π − πn)2 is depicted in Fig. 2.

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Fig. 2. Pessimism: discretion, quadratic term.

sure. However, we have no clear indication what alternative value of a might apply'. If the central bank is pessimistic with respect to this variable, its decision problem becomes: 



n 2

n 2

max d½baðp−p Þ−ðp−p Þ  þ ð1−dÞ min ba˜ðp−p Þ−ðp−p Þ : e

e

a˜ a½ Pa ; a¯ 

p

ð13Þ

The upside risk and the downside risk regarding the effectiveness of surprise inflation now come into play. In the above expression a denotes the lowest value of a which the central bank still considers plausible, ā denotes the highest ¯ such value. Provided that πe ≥ π, as derived in the previous section, the minimum is obtained for ā and the decision problem reduces to27 max½da þ ð1−dÞ¯a bðp−pe Þ−ðp−pn Þ2 : p

ð14Þ

The optimal choice of inflation rate by the central bank for given expectations πe of the public and level of confidence δ is 1 pðpe ; dÞ :¼ pn þ b½da þ ð1−dÞ¯a: 2

ð15Þ

From this it follows that •

p ¼ pD þ DD ðdÞ



pe ¼ peD þ gdDD ðdÞ



y ¼ yD −adð1−gÞdDD ðdÞ 1 P where DD ðdÞ :¼ bða −aÞð1−dÞ denotes the increase in actual inflation due to the state ambiguity faced by the 2 central bank.

27

For πe b π the minimum is obtained for a and the expression changes accordingly. ¯

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Fig. 3. Pessimism: discretion, γ b 1, δ b 1.

Proposition 3. In presence of ambiguity over the effectiveness of surprise inflation under a discretionary policy, all other things being equal, a cautious central bank chooses a higher level of inflation than in its absence. The intuition for this result is as follows. If the public expects a higher inflation rate than the rate chosen by the central bank, this depresses output below its natural rate. This is bad for the central bank, as it also cares about the level of output. In the presence of ambiguity, additional weight is placed on the worst state. For the central bank, the worst state is that the excessive inflation expectations have a large impact on the level of output. This occurs when the impact of deviations of actual inflation from expected inflation have a large impact on the level of output, i.e. when surprise inflation is most effective. To counteract this potentially harmful effect of excessive inflation expectations, the central bank is prepared to accept a larger deviation of the inflation rate from its preferred value πn. This causes the central bank to increase the rate of inflation. The result is illustrated in Fig. 3.28 4.2. Inflation targeting The major contemporary central banks in essence follow inflation targets. Therefore, one may argue, the most relevant case is not central bank discretion. One should rather focus on the question of how ambiguity affects the behaviour of a central bank with an inflation target. In order to consider the comparative static effects of ambiguity on central bank behaviour, we introduce some flexibility into the inflation targeting framework. We maintain what we consider to be the basic property of a monetary policy with an inflation target: the central bank implements whatever inflation rate it announces. We deviate from the usual approach, however, by allowing the central bank to announce any inflation rate it seems fit.29 The announced inflation rate may depend both on the strategic ambiguity of the public concerning the central bank's behaviour and on the state ambiguity faced by the central bank itself. 4.2.1. No ambiguity for the central bank As before, the objective of the central bank is max bðyðp; pe Þ−yn Þ−ðp−pn Þ2 : p

28

ð16Þ

The corresponding result for an optimistic public and a pessimist central bank is illustrated in Fig. 4. Allowing the central bank to change its inflation target is not the ad hoc assumption it may seem. The European Central Bank can effectively choose its inflation target between 0% and 2%. The Federal Reserve, which follows an ‘unofficial inflation target’, has even more room for manoeuvre. The inflation target of the Bank of England was, in December 2003, changed by the government from 2.5% 12-month increase in the RPIX to 2% 12-month increase in the (H)CPI. 29

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Fig. 4. Optimism: discretion, γ b 1, δ b 1.

Since the announcement of the central bank is, in principle, credible, the central bank can predict how its announcement influences the inflation expectations of the public. Anticipating that the inflation expectations of the public are pe ðp; gÞ ¼ gp þ ð1−gÞ¯ p;

ð17Þ

as derived in Section 3, the decision problem of the central bank becomes max bað1−gÞðp−¯ p Þ−ðp−pn Þ2 : p

ð18Þ

From this it follows that, due to strategic ambiguity, the central bank's optimal choice of the actual inflation rate π for a given level of confidence on the part of the public is 1 pðgÞ ¼ pn þ bað1−gÞ N pn : 2

ð19Þ

The equilibrium for an inflation targeting monetary policy is described by: •

1 pT ¼ pn þ bað1−gÞ ¼ pD þ DT ðgÞ 2



1 peT ¼ gp þ ð1−gÞ¯ p ¼ gpn þ ð1−gÞ¯ p þ bað1−gÞg ¼ peD þ gdDT ðgÞ 2



 1 yT ¼ y þ að1−gÞ p − ¯ p þ bað1−gÞ ¼ yD −ad ð1−gÞd DT ðgÞ 2



n

n

where DT ðgÞ :¼ − 12 bag denotes the (negative) increase in inflation due to inflation targeting instead of discretion. Therefore, even when the central bank is bound by the level of inflation it announces, a lack of confidence by the private sector leads to excessive inflation. There are, however, two noteworthy differences between the inflation targeting case and that of a discretionary monetary policy. Firstly, the level of confidence γ influences the central bank's choice of inflation rate. In the case of discretion, the inflation rate is independent of the strategic ambiguity faced by the public. Secondly, in the case of inflation targets, the deviation of the actual inflation rate π from the optimal rate πn is less than under ‘discretion’. Similarly, the level of output is closer to its natural rate. This result is illustrated in Fig. 5.

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Fig. 5. Pessimism: inflation target, γ b 1.

4.2.2. State ambiguity of the central bank How does state ambiguity with respect to the location of the short run aggregate supply curve affect monetary policy when the central bank targets inflation? As before, ambiguity about the natural rate of output has no impact on monetary policy. So what is the impact of ambiguity with respect to the effectiveness of surprise inflation? Again, the level of confidence of the central bank in the value of a is denoted by δ. The decision problem of a central bank that faces a public with a level of confidence γ is now given by   max d½bað1−gÞðp−¯p Þ−ðpn −pÞ2  þ ð1−dÞ min ba˜ ð1−gÞðp−¯p Þ−ðpn −pÞ2 ð20Þ a ; a¯  a˜ a½P

p

Since π − π¯ ≤ 0, the minimum in the second part of the expression is obtained for a¯; as in the case of discretion. Therefore, the decision problem reduces to max b½da þ ð1−dÞ¯ a ð1−gÞðp−¯ p Þ−ðpn −pÞ2 : p

ð21Þ

The optimal choice of inflation rate by the central bank, for given levels of confidence γ and δ, is 1 pðg; dÞ :¼ pn þ ½da þ ð1−dÞ¯a bð1−gÞ z pn : 2

ð22Þ

From this we obtain: •

p ¼ pT þ DT ðd; gÞ



pe ¼ peT þ gdDT ðd; gÞ



y ¼ yT −adð1−gÞdDT ðd; gÞ 1 P where DT ðd; gÞ :¼ bða −aÞð1−gÞð1−dÞ denotes the increase in inflation caused by the state ambiguity faced by the 2 central bank.

Proposition 4. Consider a central bank that targets inflation. Suppose that the central bank and the public are both pessimistically inclined. Provided that the private sector faces ambiguity, the ambiguity faced by the central bank increases inflation.

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Fig. 6. Pessimism: inflation target, γ b 1, δ b 1.

If the public has full confidence in the central bank, then the state ambiguity faced by the central bank neither affects inflation nor output. The result is illustrated in Fig. 6.30 The ambiguity faced by the central bank only matters when the private sector faces ambiguity about the central bank. If the private sector has full confidence in the announcements of the central bank, its expected rate of inflation equals the announced level and output remains at its natural rate. The state ambiguity faced by the central bank affects the output of the economy only to the extent that the expected rate of inflation differs from the inflation rate announced by the central bank. Comparing the level of output under discretion with that under inflation targeting, we find that inflation targeting leads to a higher level of output when yD −ad ð1−gÞd DD ðdÞ N yT −ad ð1−gÞd DT ðd; gÞ:

ð23Þ

From this we obtain DD ðdÞ N DT ðgÞ þ DT ðd; gÞ;

ð24Þ

which, after substitution, yields ð¯a−aÞð1−dÞ N −ag þ ð¯ a−aÞð1−gÞð1−dÞ:

ð25Þ

This leads to the following proposition. Proposition 5. Suppose that the central bank and the public are both pessimistically inclined. Then inflation targeting is better suited to deal with ambiguity than a discretionary monetary policy.

5. The European Central Bank Although the results of the paper are plausible, one may argue that some indication of its relevance for real life monetary policy is called for. Therefore, we have a closer look at the example of the European Central Bank, where the reasoning outlined in this paper is relevant. Further potentially interesting examples are mentioned as suggestions for further research in the concluding remarks. 30

The corresponding result for an optimistic public and a pessimistic central bank is depicted in Fig. 7.

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Fig. 7. Optimism: inflation target, γ b 1, δ b 1.

At the introduction of the euro, the public (and quite a number of academic monetary economists) faced strategic ambiguity with respect to the monetary policy of the European Central Bank, as its monetary policy was untested. In conducting its monetary policy, the European Central Bank seems to be aware of the impact ambiguity may have on results of its monetary policy. In a speech Issing (1999), a member of the executive board of the European Central Bank, observes: Academic economists hardly need any reminding that central bankers have to make decisions in a world of pervasive uncertainty. However, while the academic profession has made tremendous progress in analysing risk in well-defined stochastic economies, the ‘Knightian’ uncertainty that confronts central bankers is altogether of another dimension. Among the various forms of uncertainty that central bankers face, the uncertainty about how the policy instrument affects inflation and economic activity — the monetary transmission mechanism — and the uncertainty about the state of the economy — the data — appear to weigh particularly heavily. Regarding the ECB's inflation target, he continues: […][I]t was […] important that the ECB clearly defined what it means by price stability. […] The definition […] contains two very precise statements. The ECB does not consider inflation above 2% as price stability; and the ECB does not consider deflation as price stability. Being more precise in the form of a point target is likely to be counterproductive […] With respect to the desirability of flexibility, Issing (1999) notes: […][I]n an environment of high uncertainty about the economy, flexibility — some would call it discretion — is important […] this is why a central bank cannot afford to be strictly bound by any simple policy rule which may be optimal for a given structure of the economy, but breaks down as soon as the structure changes. […] […][T]he strategy chosen by the ECB is neither monetary targeting nor inflation targeting nor even a mixture of these two approaches well known to observers. It is a new strategy designed for a unique situation with which the ECB was confronted before the start of Stage Three. This seems to indicate that the ECB purposely retained the option to adjust its inflation target between bounds. The representation of inflation targeting with a changing target, as analysed in this paper, seems to be in line with the policy framework the ECB chose in the presence of a relatively high level of ambiguity.

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Thus, it not only seems that the thinking of the ECB, when it was facing a high level of ambiguity on the introduction of the euro, was closely in line with the mechanisms outlined in this paper. It also seems that the design of the monetary policy targets of the ECB is closely related to the inflation targeting with flexible targets, which Proposition 5 suggests is a suitable institutional framework for dealing with ambiguity this context. Strangely, though, the excessive inflation expectations that would be expected through the mechanism addressed in this paper do not seem to be present in the data on inflation expectations provided by the ECB.31 Perhaps, the questionnaire being targeted on experts, it reflects the expectations of the better informed who face less ambiguity. The general public, however, is likely to experience higher levels of ambiguity than the experts. 6. Concluding remarks This paper shows that ambiguity may affect monetary policy in a number of ways. When considering the strategic ambiguity faced by a pessimistically inclined public (the most plausible case), we find that a lack of confidence in the central bank is harmful. It leads to a level of output that is below its natural rate. This effect is larger when central banks can choose their monetary policy at their discretion compared to them adhering to an inflation target. Regarding the state ambiguity faced by central banks, we similarly find that the impact of state ambiguity on inflation is less if monetary policy is guided by credible announcements, rather than being left to the central banks' discretion. One may feel that, thus far, an important question regarding our model has not been addressed: Is the model in line with the usual assumption of rational expectations? In a straight forward interpretation this would mean to ask if the equilibrium values of the decision-makers' point expectations equal the expected values of the relevant variables in equilibrium. Is hardly surprising that this property fails to hold, as the ambiguity faced by the decision-makers is explicitly taken into account and ‘distorts’ the objective functions that determine the equilibrium beliefs. A more sophisticated version of rational expectations, which refers to perfect foresight regarding the outcomes in all states of nature, does apply. Due to the presence of ambiguity, however, this perfect foresight no longer translates into the expected value of the (degenerated) probability distribution that is applied in this paper. Finally, we note that ambiguity has no impact on the expected rate of inflation of an optimistic public. For further, more empirically oriented, research on the issues raised in this paper, we would like to make two suggestions. Firstly, to obtain broader insights on the effects of strategic ambiguity faced by the public, it would be useful to have a closer look at the effects monetary policy of in countries in Central and Eastern Europe. After disposing of their communist governments and starting on the way to becoming market economies, the public in the Central and Eastern European countries faced a higher level of ambiguity about their central banks and their monetary policies due to the lack of relevant past experience with these central banks. Tough monetary policy in the early days may have helped to establish confidence in some central banks. The prospect of membership of the EU may also have had a confidence enhancing effect, as it reduced the scope for governments and central banks to pursue unpredictable monetary policies. According to the analysis of this paper, this gain in confidence may lead to lower inflation rates and output levels closer to their natural rate. It would be interesting to see if empirical data supports the conclusions of this paper and, if so, to establish the order of magnitude of the effects of ambiguity. Secondly, one may want to have a closer look at the United States to examine the impact of optimism by the public regarding the central bank. In recent years, the attitude of the public regarding the Federal Reserve, with the glorification of Alan Greenspan, has become one of optimism. When the public is optimistic regarding the actions of the central bank, ambiguity has no impact on the expected rate of inflation. An optimistic public associates ambiguity with the best, rather than with the worst plausible outcome. For the public, the best outcome occurs when the actual rate of inflation equals the rate of inflation it expects. It would be interesting to see if empirical evidence supports this hypothesis and, if so, if there are indications about what may have caused the attitude of the public to shift from pessimism to optimism. It may also provide insights on how, in general, to best hand-over of the chair of the Federal Reserve without compromising the optimistic attitude of the public or creating strategic ambiguity over monetary policy.

31

at http://www.ecb.int/stats/prices/indic/forecast/html/index.en.html.

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Acknowledgments Helpful comments and suggestions from Paul Auerbach, Subrata Ghatak, Friedrich Kissmer, Helmut Wagner, the participants in the research seminars of the Department of Economics of Kingston University and the University of Hagen, and the participants in the 3rd Annual Conference of the European Economics and Finance Society in Gdansk are gratefully acknowledged. Any remaining errors are my own. References Anscombe, F., & Aumann, R. (1963). A definition of subjective probability. Annals of Mathematical Statistics, 34, 199−205. Backus, D., & Drifill, J. (1985). Inflation and reputation. American Economic Review, 75, 530−538. Barro, R. (1986). Reputation in a model of monetary policy with incomplete information. Journal of Monetary Economics, 17, 3−20. Barro, R., & Gordon, D. (1983a). A positive theory of monetary policy in a natural rate mode. Journal of Political Economy, 91, 589−610. Barro, R., & Gordon, D. (1983b). Rules, discretion and reputation in a model of monetary policy. Journal of Monetary Economics, 12, 101−122. Bewley, T. (1986). Knightian decision theory: Part I. Cowles Foundation Discussion Paper, Vol. 807. New Haven: Yale University. Bewley, T. (1987). Knightian decision theory, Part II. Cowles Foundation Discussion Paper, Vol. 835. New Haven: Yale University. Brock, W., Durlauf, S., & West, K. (2004). Model uncertainty and policy evaluation. Some theory and empirics. NBER Working Paper, Vol. 10916. Cambridge: NBER. Caglianrini, A. and Heath, A. (2000). Monetary policy making in the presence of Knightian uncertainty, Mimeo, Economic Research Department (Reserve Bank of Australia, Sydney). Chprits, E. and Schipper, B. (2003). Knightian intransparency of central bank and labor union, Mimeo, Department of Economics (University of Bonn, Bonn). Dow, J., & Werlang, S. (1992). Uncertainty aversion, risk aversion and the optimal choice of portfolio. Econometrica, 60, 197−205. Dow, J., & Werlang, S. (1994). Nash equilibrium under Knightian uncertainty: Breaking down backward induction. Journal of Economic Theory, 64, 305−324. Ellsberg, D. (1961). Risk, ambiguity and the savage axioms. Quarterly Journal of Economics, 75, 643−669. Epstein, L., & Wang, T. (1994). Intertemporal asset pricing and Knightian uncertainty. Econometrica, 62, 283−322. Fischer, S. (1977). Long-term contracts, rational expectations and the optimal money supply rule. Journal of Political Economy, 85, 191−205. Friedman, M. (1968). The role of monetary policy. American Economic Review, 58, 1−17. Ghirardato, P., Maccheroni, F., & Marinacci, M. (2004). Differentiating between ambiguity and ambiguity attitude. Journal of Economic Theory, 118, 133−173. Gilboa, I. (1987). Expected utility with purely subjective non-additive probabilities. Journal of Mathematical Economics, 18, 65−88. Hansen, L., & Sargent, T. (2003). Robust control of forward-looking models. Journal of Monetary Economics, 50, 581−604. Hansen, L., & Sargent, T. (2005). Robust control, Manuscript. New York: New York University. Issing, O. (1999). The monetary policy of the ECB in a world of uncertainty. Speech held to the policy panel at the Conference on Monetary Policy Making under Uncertainty, Frankfurt am Main. Kelsey, D., & Milne, F. (1997). Induced preferences, dynamic consistency and Dutch books. Economica, 64, 471−481. Kelsey, D., & Milne, F. (1999). Induced preferences, non-additive beliefs, and multiple priors. International Economic Review, 40, 455−477. Kelsey, D., & Spanjers, W. (1997/2004). Ambiguity in partnerships, The Economic Journal, 114, 528–546. Keynes, J. (1937). The general theory of employment. From: Quarterly Journal of Economics, Collected Writings, Vol. 14 (pp.109−123). London: Macmillan. Kissmer, F. (2005). Uncertain degree of central bank independence: Transparency vs. opacity, Mimeo (University of Hagen, Hagen). Knight, F. (1921). Risk, uncertainty and profit. Boston: Houghton Miffin. Kuester, L., & Wieland, V. (2005). Insurance policies for monetary policy in the Euro area, CFS Working Paper Nr. 2005/13 (Center for Financial Studies, University of Frankfurt). Kydland, F., & Prescott, E. (1977). Rules rather than discretion: The inconsistency of optimal plans. Journal of Political Economy, 85, 473−492. Lucas, R. (1976). Some international evidence on output-inflation tradeoffs. American Economic Review, 63, 326−334. Marinacci, M. (2000). Ambiguous games. Games and Economic Behavior, 31, 191−219. Mankiw, G. (2003). Macroeconomics, 5th Edition. New York: Worth Publishers. Rotemberg, J. (1982). Monopolistic price adjustment and aggregate output. Review of Economic Studies, 40, 517−531. Rothe, J. (1996/98). Uncertainty aversion and backward induction, Mimeo (London School of Economics, London). Rothe, J. (1996/99). Uncertainty aversion and equilibrium in extended form games, Mimeo (London School of Economics, London). Savage, L. (1954). Foundation of statistics. New York: Wiley. Sarin, S., & Wakker, P. (1992). A simple axiomatisation of non-additive expected utility. Econometrica, 60, 1255−1272. Schmeidler, D. (1982/89). Subjective probability and expected utility without additivity, Econometrica, 57, 571–587. Spanjers, W. (1998/2005). Loss of confidence and currency crises, Discussion Paper 2005/02, School of Economics (Kingston University, Kingstonupon-Thames). Spanjers, W. (1999). Liquiditätsversorgung bei Knight'scher Unsicherheit: ein theoretischer Vergleich von Banken und Märkten, (Provision of liquidity under Knightian uncertainty: A theoretical comparison of banks and markets), Habilitation thesis (University of Saarland, Saarbrücken). Svensson, L. (2000). Robust control made simple, Mimeo (Princeton University, Princeton).

102

W. Spanjers / International Review of Economics and Finance 17 (2008) 85–102

Wagner, H. (2005a). Unsicherheit in der Zentralbankpolitik (Uncertainty in central bank policy), Mimeo (University of Hagen, Hagen). Wagner, H. (2005b). Information und Unsicherheit in der Geldtheorie (Information and uncertainty in monetary theory), Mimeo (University of Hagen, Hagen). Walsh, C. (2003). Implications of a changing economic structure for the strategy of monetary policy. Monetary policy and uncertainty: Adapting to a changing economy (Federal Reserve Bank of Kansas City's Jackson Hole Symposium, Wyoming). Walsh, C. (2005). Endogenous objectives and the evaluation of targeting rules for monetary policy. Journal of Monetary Economics, 52, 889−911.