1 Introduction

The literature on global games (inter alia Morris and Shin 2002b; Phelps 1983; Woodford 2002) has given rise to the importance of heterogenous information in decision making. Based on Keynes’ (1936) beauty context example, the argument goes, when agents need to form a view about the underlying state, they engage in trying to second-guess others’ actions. This occurs because the utility they derive individually depends not only on identifying the state correctly, but also on closing the distance between their own prediction and that of others. In the context of monetary policy, this methodology is used by Amato and Shin (2003) and Amato et al. (2003) to discuss how increasing Central bank transparency (in the form of making a public signal more precise) can, under certain circumstances, prove detrimental to social welfare. Indeed this is demonstrated in the seminal paper by Morris and Shin (2002a) (MS hereafter) in which they show that this arises because in their effort to second-guess others, private agents give disproportionate more emphasis on public information that is common knowledge, than is justified by its noise.Footnote 1 Svensson (2006) then argues that, true though this is, the parameterisation that is required for this to hold is very specific and actually implausible. If one was to assume plausible values he argues, then the benefits of increasing transparency are rescued and indeed dominate the range of outcomes (Blinder and Wyplosz 2005, raise similar concerns).

In this paper we would like to take this discussion further by introducing two innovations: first, we allow explicitly for a central bank (CB) to be one, of many, contributors to what constitutes the public signal. This is because, as commonly agreed in the literature, the Central bank has an important role in coordinating private sector expectations.Footnote 2 Monetary policy is now a game of two players, the central bank and the private sector, the latter being composed of an infinite number of private agents that are modelled in the MS fashion. This allows us to examine how increasing the precision of a signal (either private or public) affects the interaction between these two optimising players. Second, we then allow for the fact that increasing the precision of that signal does not come without costs. This is an important addition to the literature both because it is actually more plausible but also because it shows how the incentives of the two players may be distorted as a result, by introducing an element of free riding in their optimal actions.

The paper is organised as follows. Section 2 outlines the static game between the central bank and the private sector (PS). The central bank is a contributor to the public signal (but not necessarily the sole one) and the private sector then uses what is common knowledge in combination with its own information to decide on their action. Section 3 then shows how players’ welfare changes as the level of precision of the two signals increases. The underlying assumption here is that more information comes at no cost. Section 4 then lifts this assumption and introduces costs explicitly to show how the impact of greater information generates free-riding incentives. Section 5 summarises our results and concludes.

2 Monetary policy as an information game

The game between the two players based on the model presented in Demertzis and Viegi (2006) is summarised as follows. The central bank has a standard loss function in which it chooses the rate of inflation x to minimise the distance of inflation from the target x T and the output gap, y

$$ \min_{x}L_{CB} \vert \xi=\frac{1}{2}E\left[ \left( x-x^{T}\right) ^{2}+y^{2}\right] \label{CBLosses} $$
(1)

subject to a standard Lucas supply function, y = x − x e + ξ where ξ is a supply shock with zero mean and constant variance, \(\sigma_{\xi}^{2}\). Note that any central bank will have an objective x T irrespective of whether it has communicated it to the public clearly, or even at all. Optimisation of (1) implies that

$$ x \vert _{\xi}=\frac{x^{T}}{2}+\frac{x^{e}}{2}-\frac{\xi}{2} \label{inflation outcome} $$
(2)

where x is now the ex post inflation outcome conditional on the shock ξ and x e is private sector expectations about the relevant rate of inflation. Representation (2) is of a structural form in the sense that expectations are not solved for Svensson (2003) argues in favour of such a representation in order to indicate that factors like judgement that contribute to the way expectations are formed but cannot always be modelled, are an important contributor to monetary policy. Note that Eq. 2 is specific to the underlying Lucas supply function assumed but demonstrates that the outcome will be a function of both the policy the Central Bank pursues as well as what the private sector anticipates. We can re-group the terms in Eq. 2 and re-write it in a general  form:

$$ x \vert _{\xi}=\left( 1-r\right) \left( x^{T}-\xi\right) +rx^{e} \label{general form} $$
(3)

where r represents the strength with which expectations affect the final out come.Footnote 3 In a typical commitment game, where the central bank communicates its target x T and commits to it, expectations formed by all individuals collectively are equal to the CB’s objectives, x e = x T and the ex post outcome is

$$\left. {x^{*}} \right|_{\xi } = x^{T} - {\left( {1 - r} \right)}\xi $$
(4)
$$E{\left( {x^{*}} \right)} = x^{T} $$
(5)

In this paper however, we depart from the commitment set-up and allow agents to form expectations based on all the information that is available to them (therefore, inclusive of what the CB has announced). Every individual i will thus be forming an expectation of inflation \(x_{i}^{e}\), such that the collective outcome (for a continuum of agents) \(x^{e}=\int_{0}^{1}x_{j}dj\) is the expectation that is relevant to the inflation outcome. The timing of the game assumed has the central bank deciding what its objectives are first, private agents form expectations based on information available, shocks occur next and finally the CB reacts to the supply shock ξ.

2.1 The formation of expectations

We thus start by arguing that while the CB may be clear itself about what its objectives are, it is not always possible to assume that private individuals form expectations that are consistent with these objectives. It becomes important then to examine the information that is available to the private sector and how they use it to form expectations. Typically, every individual forms expectations based on two information sets, namely what is publicly available and therefore common to everyone, and what is available to them privately. Furthermore, every individual is aware of the fact that the ex post outcome of inflation x will be determined by Eq. 2, in other words will be affected partially by the policy the central bank pursues to attain its objectives, and partially by the average of expectations formed by the public.

However, as the individual is interested in predicting the ex post level of inflation correctly, she needs to interpret both components of Eq. 2 based on the information she has. Her objectives are captured by a standard expected dis-utility,Footnote 4

$$ \min_{x_{i}^{e}}u_{i}\left( \mathbf{x}^{e},x^{T}\right) \equiv\frac{1} {2}E_{i}(x_{i}^{e}-x)^{2} \label{PSutility1} $$
(6)

Note that subscript i in the expectations operator, indicates that the individual will be seeking to minimise her expected dis-utility, given her own perceptions. \(x_{i}^{e}\) is individual i’s expectation of what inflation will be and x is again the ex post inflation outcome from Eq. 3. We use x e to refer to the expectations profile over all agents. The objective of each individual i is thus to form expectations \(x_{i}^{e}\), as accurately as possible, which she will then use, for example, in wage negotiations. The individual decides her action \(x_{i}^{e}\) based on the first-order condition of Eq. 6. This is:

$$ \arg\min_{x_{i}^{e}}u_{i}\left( \mathbf{x}^{e},x^{T}\right) =E_{i}\left( x\right) $$

and from Eq. 3,

$$\begin{array}{*{20}c} {x^{e}_{i} = E_{i} {\left( x \right)}} \\ {x^{e}_{i} = E_{i} {\left[ {{\left( {1 - r} \right)}{\left( {x^{T} - \xi } \right)} + rx^{e} } \right]}} \\ {x^{e}_{i} = {\left( {1 - r} \right)}E_{i} {\left( {x^{T} - \xi } \right)} + rE_{i} {\left( {x^{e} } \right)}} \\ \end{array} $$
(7)

The optimal action for individual i is thus a function of three things: the objectives of the central bank and hence the policy it will pursue, the shock that will occur, and finally the average expectation formed by all individuals. Moreover, in forming expectations \(x_{i}^{e}\), individual i needs to evaluate these three things, captured here by the expectations operator, subscript i.Footnote 5 It follows that if \(x_{i}^{e}=x_{j}^{e}\) ∀ j, then \(x_{i}^{e}=x^{e}\) and individuals’ expectations are matched. However, although desirable, coordination between agents at any level of inflation is not sufficient; the optimal outcome occurs when agents coordinate at the objective pursued by the central bank. Coordination at any other expectation rate still leaves agents away from the level of inflation that the CB aims to achieve. Following Morris and Shin (2002a), we argue that information used by the agents is available in the form of a public signal common to all, and a private signal which is specific to each agent in the economy. These take the following form:

$${\text{Public signal: }}p = {\left( {x^{T} - \xi } \right)} + \eta $$
(8)
$${\text{Private signal: }}z_{i} = {\left( {x^{T} - \xi } \right)} + \varepsilon _{i} $$
(9)

Both η and ε i are normally distributed with zero mean and variance \(\sigma_{\eta}^{2}\) and \(\sigma_{\varepsilon}^{2}\) respectively. Furthermore, the two error terms are independent of x and of each other, such that \(E\left( \varepsilon_{i}\varepsilon_{j}\right) =0\) for i ≠ j. Contrary to MS then, the clarity of public information is not under the full control of the CB but is affected by a combination of the CB’s information strategy, general market information available and noise. Based on these two types of signals, MS show that action for agent i, applying Eqs. 8 and 9 on Eq. 7 then gives,

$$\begin{array}{*{20}c} {x^{e}_{i} = \frac{{\alpha p + \beta {\left( {1 - r} \right)}z_{i} }}{{\alpha + \beta {\left( {1 - r} \right)}}}} \\ { = x^{T} - \xi + \frac{{\alpha \eta + \beta {\left( {1 - r} \right)}\varepsilon _{i} }}{{\alpha + \beta {\left( {1 - r} \right)}}}} \\ \end{array} $$
(10)

where \(\alpha=\frac{1}{\sigma_{\eta}^{2}}\) and \(\beta=\frac{1}{\sigma _{\varepsilon}^{2}}\) represent the precision for the two information sets, respectively. We call this the MS action. It follows that expectations across all agents are then equal, to

$$x^{e} = {\int_0^1 {x_{j} d_{j} = x^{T} - \xi + \frac{{\alpha \eta }}{{\alpha + \beta {\left( {1 - r} \right)}}}} }$$
(11)

Equation 11 shows that the average expectation across all agents will be distorted by the (lack of) precision of the two signals as well as the value for the ‘beauty term’ r. For simplification purposes, we drop ξ as it does not change the qualitative nature of our results. The timing of the game then implies that the CB provides a public signal based on which the PS then forms expectations. In turn, this implies that expectations are predetermined at the time the central bank is asked to take action. In other words, the central bank needs to implement the policy rule it had announced in its signal, subject to Eq. 11,

$$\begin{array}{*{20}c} {L_{{CB}} = E{\left[ {{\left( {x - x^{T} } \right)}^{2} + {\left( {x - x^{e} } \right)}^{2} } \right]}} \\ {{\text{st}}{\text{. }}x^{e} = x^{T} + \frac{{\alpha \eta }}{{\alpha + \beta {\left( {1 - r} \right)}}}} \\ \end{array} $$
(12)

Note that average expectations across all possible public signals are

$$ E\left( x^{e}\right) =\int_{-\infty}^{+\infty}\left( x^{T}+\frac{\alpha \eta}{\alpha+\beta\left( 1-r\right) }\right) d\eta=x^{T} \label{Average expecations signal} $$
(13)

As \(E\left( x^{\ast}\right) =E\left( x^{e}\right) =x^{T}\) then Eq. 4 is indeed the optimal reaction consistent with Rational Expectations. We calculate next the losses incurred ex post by the two players. CB losses are found by substituting Eqs. 4 and 11 into Eq. 1:Footnote 6

$$\begin{array}{*{20}c} {L_{{CB}} = E{\left[ {{\left( {x^{*} - x^{T} } \right)}^{2} + {\left( {x^{*} - x^{e} } \right)}^{2} } \right]}} \\ { = \frac{\alpha }{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{2} }}} \\ \end{array} $$
(14)

Similarly, from Morris and Shin (2002a) we know that losses for the private sector collectively,Footnote 7 are:

$$\begin{array}{*{20}c} {L_{{PS}} = {\int_0^1 {{\left( {x^{e}_{i} - E{\left( {x^{*}} \right)}} \right)}^{2} di} }} \\ { = \frac{{\alpha + \beta {\left( {1 - r} \right)}^{2} }}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{2} }}} \\ \end{array} $$
(15)

It is straightforward to show that the slope of losses for both the central bank as well as society is not unequivocally signed in terms of changes in the precision of the public signal but depends on the parameterisation of r. We examine next how the incentives of the central bank and the private sector may be affected, if transparency (i.e. the precision of the two signals) was subject to choice.

3 Information and welfare

3.1 The role of strategic motive r

We first examine how the losses for the two players are affected by term r, the private sector’s strategic motive to be as close to the average expectations as possible.

Proposition 1

As private agents’ strategic motive increases, (dr > 0), both social as well as CB welfare deteriorate.

Proof

We can demonstrate this by looking at the way losses vary with r.

$$\begin{array}{*{20}c} {L_{{CB}} = \frac{\alpha }{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{2} }}} \\ {\frac{{\partial L_{{CB}} }}{{\partial r}} = \frac{{2\alpha \beta }}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }} > 0} \\ \end{array} $$
(16)

□

Similarly for the private sector,

$$\begin{array}{*{20}c} {L_{{PS}} = \frac{{\alpha + \beta {\left( {1 - r} \right)}^{2} }}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{2} }}} \\ {\frac{{\partial L_{{PS}} }}{{\partial r}} = \frac{{2\alpha \beta r}}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }} > 0} \\ \end{array} $$
(17)

The intuition behind this result is that the externality of the ‘beauty contest’ term moves the action away from the otherwise optimum. A positive value of r induces private sector agents to reduce the weight attached to their private information by (1 − r)β. So, as r increases,Footnote 8 the weight that the private sector puts on the public signal is greater than what is justified by the relative precision of the two signals. Similarly, for the central bank, the increase in the private sector’s strategic motive increases the distance between its inflation objective and the expectations that the private sector forms. Two further comparisons are worth mentioning. First, irrespective of parameterisation, there will always exist a range of values for α for which the central bank incurs higher losses by increasing the precision of the public signal further (i.e. \(\frac{\partial L_{CB}}{\partial\alpha}>0\) for \(\alpha\in\left( 0,\tilde{\alpha}\right) \)). For the private sector this is only true for a very specific choice of parameter values, outwith which, increasing α is always beneficial to its welfare. Second, comparing the two deteriorations in losses, the CB in fact stands the most to lose because while it incurs the costs of this increased externality, it does not face any of the benefits associated with this extra term (incorporated in the individual’s utility function). It is straight-forward to see that by comparing Eq. 17 to Eq. 16. We turn next to how increasing the public and private signal precision affects the losses of the two players. Implicit in our approach is the idea that the exact level of precision is subject to choice for the two players accordingly.

3.2 Public signal

We first demonstrate how increases in the precision of the public signal (identified often with increases in transparency) affect the two players’ losses in the  game.

$$\frac{{\partial L_{{CB}} }}{{\partial \alpha }} = \frac{{ - \alpha + \beta {\left( {1 - r} \right)}}}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }}$$
(18)
$$\frac{{\partial L_{{PS}} }}{{\partial \alpha }} = - \frac{{\alpha + \beta {\left( {r - 1} \right)}{\left( {2r - 1} \right)}}}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }}$$
(19)

As argued by MS, increasing the level of precision of the public signal will not always be beneficial to the private sector (or indeed the central bank as we show here), as expressions (18) and (19) are not unequivocally signed. We summarise next the conditions required for the derivatives to sign one way. Starting with the central bank, for transparency to be detrimental to its losses, (i.e.,\(\frac{\partial L_{CB}}{\partial\alpha }>0\)), the necessary condition is that

$$\begin{array}{*{20}r} {\hfill {\beta {\left( {1 - r} \right)} > \alpha }} \\ {\hfill {{\left( {1 - r} \right)} > \frac{\alpha }{\beta }}} \\ \end{array} $$
(20)

The CB will always have a hump shaped function such that the optimal level is at α = 0; but once this is violated (for example, because the central bank is only one of many contributors to public information and not the exclusive one), then it is always optimal to try to increase α as much as possible such that losses return to zero. As, we will show in the next section, this result does not hold if increasing the public signal is not without cost. Turning next to the private sector, the necessary condition for social losses to be increasing in transparency, i.e. ,\(\frac{dL_{PS}}{d\alpha }>0\), (as discussed by MS and Svensson 2006) is that

$$\begin{array}{*{20}c} {{f{\left( r \right)} > \frac{\alpha }{\beta }}} \\ {{{\text{where }}f{\left( r \right)} \equiv {\left( {1 - r} \right)}{\left( {2r - 1} \right)}}} \\ \end{array} $$
(21)

For Eq. 21 to be true, a necessary condition is that parameter

$$r \in {\left( {\frac{1}{2},1} \right)}$$
(22)

If \(r\in\left[ 0,\frac{1}{2}\right] \) or r = 1 then \(\left( 1-r\right) \left( 2r-1\right) \leq0\) and Eq. 21 is violated. This implies that for transparency to be detrimental to welfare, agents must put a sufficiently high (although not exclusive) weight on the ‘beauty contest’ term. However, beyond that Svensson (2006) also mentions that f(r) has a maximum equal to \(\frac{1}{8}\) when \(r=\frac{3}{4}\). This implies that even if Eq. 22 holds, condition (21) maybe violated if

$$ \frac{\alpha}{\beta}\geq\frac{1}{8} $$
(23)

In other words if the public signal precision is greater than one eighth of the private signal precision, social welfare benefits from increases in transparency. Moreover, if r is on the other side of \(\frac{3}{4}\) but satisfying condition (22), then the likelihood of transparency being beneficial is even greater.

Naturally, the same holds for the private sector, even if for some parameterisations it has to put up with increasing losses before it starts to benefit from an increasing α again.

Figure 1 demonstrates this point for r = 0.7, for which both the CB as well as the PS have to put up with a deterioration in their welfare for increases within a given range of public signal precision (see Appendix A for alternative parameterisation of r). Note that for α = 0, the central bank is at its optimal point whereas the PS is at a positive level of losses. As mentioned earlier, this maybe the optimal level for the central bank but it is a very unlikely outcome (Svensson 2006).

Fig. 1
figure 1

Losses against public signal precision (r = 0.7)

Full size image

Furthermore, it is also true that increases in transparency, do not affect the two players in the same way. It follows that:

Proposition 2

For any increase in the precision of the public signal, the private sector benefits more (or loses less) than the central bank.

Proof

It is straight forward to prove this by comparing Eq. 18 to Eq. 19 by taking the difference between the two

$$\begin{array}{*{20}c} {\frac{{\partial L_{{CB}} }}{{\partial \alpha }} - \frac{{\partial L_{{PS}} }}{{\partial \alpha }} = \frac{{ - \alpha + \beta {\left( {1 - r} \right)}}}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }} - \frac{{ - \alpha + \beta {\left( {1 - r} \right)}{\left( {2r - 1} \right)}}}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }}} \\ { = \frac{{2\beta {\left( {1 - r} \right)}^{2} }}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }} > 0} \\ \end{array} $$

Figure 2 also demonstrates this for a given parameterisation of r(= 0.6).□

Fig. 2
figure 2

Marginal losses against public signal precision (r = 0.6)

Full size image

Coupled with the fact that, for plausible parameterisations, increases in transparency are always beneficial to the private sector, this proposition implies that private agents are justified to lobby for increases in α from the central bank or even for any other source that contributes to public information.

3.3 Private signal

The effects of increases in the precision of private information are different in that both parties benefit from the increase in its precision demonstrated by

$$\frac{{\partial L_{{CB}} }}{{\partial \beta }} = \frac{{2\alpha {\left( {r - 1} \right)}}}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }} \leqslant 0$$
(24)
$$\frac{{\partial L_{{PS}} }}{{\partial \beta }} = \frac{{{\left( {r - 1} \right)}{\left[ {\beta {\left( {1 - r} \right)}^{2} + \alpha {\left( {1 + r} \right)}} \right]}}}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }} \leqslant 0$$
(25)

This can also be seen in Fig. 3 for a given parameterisation of r(= 0.7) but holds also in general.

Fig. 3
figure 3

Losses against private signal precision (r = 0.7)

Full size image

However, the difference between the two is not unequivocally signed and thus there are instances for which one of the two players benefits the most.

$$\begin{array}{*{20}c} {\frac{{\partial L_{{CB}} }}{{\partial \beta }} - \frac{{\partial L_{{PS}} }}{{\partial \beta }} = \frac{{2\alpha {\left( {r - 1} \right)}}}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }} - \frac{{{\left( {r - 1} \right)}{\left[ {\beta {\left( {1 - r} \right)}^{2} + \alpha {\left( {1 + r} \right)}} \right]}}}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }}} \\ { = \frac{{{\left[ {\alpha + \beta {\left( {r - 1} \right)}} \right]}{\left( {r - 1} \right)}^{2} }}{{{\left[ {\alpha + \beta {\left( {1 - r} \right)}} \right]}^{3} }}} \\ \end{array} $$
(26)

The important point of this section however, is that for plausible (and even conservative) parameter values both in terms of r as well as ratios of public to private signal precision, increasing the levels of precision of either of the players’s signal is beneficial to both. We examine next how (and whether) this may be distorted if one, or possibly both had to incur costs in order to increase the precision of their own signal.

4 Introducing costs

The analysis so far assumes that there are no costs associated with acquiring information that contributes to the increase of the precision of the signal (either public or private). We consider this rather implausible as we believe that acquiring greater information is always costly. This argument is of course relevant when seen from the private sector’s point of view but is it also relevant for a Central Bank. It is argued that as the Central Bank is intended to pursue the public’s interest then, for any given resources, it should strive to acquire as much information as possible so that it takes the most informed decision. Naturally for any given amount of information it has, the Central Bank always retains a certain degree of discretion in terms of what level of transparency it operates at. The point that we wish to raise next however, is that if information is costly to acquire, then the Central Bank (and the private sector) is quicker to stop looking to increase it for itself. Moreover, this gives rise to a “free-ride” effect from both sides, in the sense that both players prefer the other to invest in increasing the information set available. We show this next.

We assume that costs are positive and unbounded and adopt a general Box–Cox transformation.

$$\begin{array}{*{20}c} {C_{{CB}} = c_{{CB}} \alpha ^{{ - \lambda }} } \\ {C_{{PS}} = c_{{PS}} \beta ^{{ - \lambda }} } \\ \end{array} $$
(27)

where \(c_{CB}^{{}}>0\), and \(c_{ {PS}}^{{}}>0\), λ is negative. For λ = − 1 the marginal cost is constant (linear), and λ = − 2 (quadratic) is the functional form applied by Svensson (2006), who assumes increasing marginal costs. The rationale behind Svensson’s choice is that information is increasingly costly to acquire (‘...producing better forecasts requires more and better -qualified staff’). Each of the two agents now faces the sum of welfare losses and the costs of increasing the level of precision of the signal.

$$ T=L+C $$

In what follows we assume that costs are linear, (λ = − 1), for analytical tractability, but we can generalise our results for all values of λ < 0 (see Appendix B).

4.1 Linear costs (λ = − 1)

We assume first, that it is only the CB that incurs costs in increasing the precision of the public signal. The private sector receives information some of which is common to everyone and some specific, but does not incur any costs. Furthermore, we argue that the CB is one, of many contributors to public information and therefore, it is reasonable to assume that α > 0. Faced therefore, with a certain positive level of precision, we examine next how the costs of increasing this information affects the CB’ s incentives to increase transparency.

Figure 4 shows that the existence of costs, clearly prevents the CB from increasing the level of precision of public information indefinitely. Furthermore, it is also clear, that if the level of precision of public information is to the left of point A,Footnote 9 then the CB will not try to increase it, as by doing so, it increases its losses. However, if public information is to the right of point A then the CB does have an incentive to increase the level of precision up to point B. Beyond that however, the costs of going further outweigh the benefits of doing so. It is worth noting that as the degree of non-linearity increases (λ↓) then both the total costs as well as marginal costs increase (i.e. \(C_{\lambda}^{\prime}<0,\) and \(C_\lambda ^{\prime \prime } > 0\)). This implies that the cut-off point B beyond which the central bank would not wish to increase the level of precision any longer is displaced leftwards (reduced, see Appendix B). Beneficial though this is to society’s welfare, the CB is faced with a local minimum which distorts its incentives to increase its contribution to public information. Naturally, if public information was to increase through some other source, then the CB would benefit in the same manner as described in the previous section.

Fig. 4
figure 4

Losses and total (linear costs against public signal) (r = 0.5)

Full size image

We assume next that both the central bank as well as the private sector incur (linear) costs when wishing to improve the quality of their signal. Both players are then faced with having to minimise the following:

$$\begin{array}{*{20}c} {T_{{CB}} = L_{{CB}} + C_{{CB}} } \\ {T_{{PS}} = L_{{PS}} + C_{{PS}} } \\ \end{array} $$

An optimising central bank is now faced with the following first-order condition:

$$ \frac{\partial T_{CB}}{\partial\alpha}=\frac{-\alpha+\beta\left( 1-r\right) }{\left[ \alpha+\beta\left( 1-r\right) \right] ^{3}}+c_{CB}^{{}}=0 \label{FOC CB1} $$
(28)

Ensuring that optimisation leads to minimisation of the function, requires that the second-order condition is met, i.e.

$$ \frac{\partial^{2}T_{CB}}{\partial\alpha^{2}}=\frac{2(\alpha-2\beta\left( 1-r\right) )}{\left[ \alpha+\beta\left( 1-r\right) \right] ^{4}}>0 \label{SOC CB} $$
(29)

For Eq. 29 to be true, the sufficient condition is that α > 2β(1 − r) holds. However, in equilibrium, as both conditions need to be satisfied, the following restriction needs also to apply.

$$ c_{CB}^{{}}>\frac{\beta\left( 1-r\right) }{\left[ \alpha+\beta\left( 1-r\right) \right] ^{3}} $$

The first-order condition is a third-order polynomial and solving it explicitly for an optimal level of precision of central bank information α does not provide much insight. However, we can apply the implicit function theorem to characterise the strategic substitutability between the two signals in equilibrium.

Proposition 3

In equilibrium, the optimal value of public signal precision, as affected by the central bank, is decreasing in the precision of the private signal.

Proof

Let \(F:=\frac{\partial T_{CB}}{\partial\alpha}\); then the implicit function theorem implies that

$$ \frac{\partial\alpha^{\ast}}{\partial\beta}=-\frac{F_{\beta}}{F_{\alpha} }=\frac{(1-r)(2\alpha-\beta(1-r))}{2\beta(1-r)-\alpha} \label{IFT CB} $$
(30)

□

The restriction on the second-order condition (α > 2β(1 − r)), implies that it is straight forward to show that \(\frac{\partial\alpha^{\ast} }{\partial\beta}<0\). In other words, if the private sector reduces the level of precision of its own signal, then in order to maximise its utility, the CB is forced to increase its contribution to the public signal, in equilibrium.

Turning to the private sector, the FOC is now

$$ \frac{\partial T_{PS}}{\partial\beta}=\frac{-\left( 1-r\right) \left[ \beta\left( 1-r\right) ^{2}+\alpha\left( 1+r\right) \right] }{\left[ \alpha+\beta\left( 1-r\right) \right] ^{3}}+c_{ {PS}}^{{}}=0 \label{FOC PS} $$
(31)

It is straight forward to demonstrate that the second-order condition is always met for the private sector.

$$ \frac{\partial^{2}T_{PS}}{\partial\beta^{2}}=\frac{2(1-r)^{2}(\alpha +\beta\left( 1-r\right) +2\alpha r)}{\left[ \alpha+\beta\left( 1-r\right) \right] ^{4}}>0 $$
(32)

Proposition 4

In equilibrium, the optimal value of private signal precision, as dictated by the representative private agent, is decreasing in the precision of the public signal.

Proof

Let \(G:=\frac{\partial T_{PS}}{\partial\beta}\); then the implicit function theorem implies that

$$ \frac{\partial\beta^{\ast}}{\partial\alpha}=-\frac{G_{\alpha}}{G_{\beta} }=-\frac{\beta+\alpha(1+r)+\beta r(2r-3)}{(1-r)(\alpha+\beta(1-r)^{2}+2\alpha r)} \label{IFT PS} $$
(33)

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The restriction for the central bank’s second-order condition (α > 2β (1 − r)) helps show that \(\frac{\partial\beta^{\ast}}{\partial\alpha} <0\). Similarly to the central bank, if the precision of public information increases, it is optimal for the private sector to reduce the precision of its own information.

In summary, the result above shows that, provided the first-order conditions for both the central bank and the private sector and the second-order condition for the central bank are satisfied, public and private information are strategic substitutes in line with Morris and Shin’s contribution. However, as pointed out by Svensson with reference to this set-up, it is very unlikely that providing public information is detrimental to social welfare in the absence of costs. When introducing costs on the other hand, strategic substitutability is a very likely outcome.

5 Conclusions

We have presented a model where the CB, as a provider of public information and private sector agents, as providers of private information, interact strategically. First, we replicate the result found by MS that, in the presence of the so-called “beauty-contest externality” that arises when private sector agents try to second-guess other private sector agents’ expectations, more precise public information may be detrimental to welfare. As a next step, we attach costs to the provision of more precise public and private information. When these costs are taken into account, we find an optimal level for the precision of private information and a local optimum for the precision of public information. The global optimum is reached for zero precision of public information, confirming the result by MS. However, as pointed out by Svensson, since the CB is not the only provider of public information, it is implausible that the global optimum can be reached, and very well possible that the local optimum is the best strategy for the CB. We derive the exact conditions under which this is the  case.

Our second contribution is the strategic substitutability between the precision of public and private information that arises once we take the costs of providing information into account. We show that, in equilibrium, a marginal increase in the precision of the private signal induces the central bank to reduce the precision of the public signal. The reverse also holds: A marginal increase in the precision of the public signal induces the private sector to reduce the precision of its information.