This paper extends the results of Kobayashi (2003) and Ciccarone and Marchetti (2009) by considering the optimal choice of central bank conservativeness. It is shown that the government can choose a sufficiently populist but opaque central banker so that higher multiplicative uncertainty improves the social welfare only when the society is very conservative.Multiplicative uncertainty; optimal degree of conservativeness; Brainard conservatism; central bank transparency.

Meixing Dai

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 3 
  
Multiplicative uncertainty, central bank transparency and optimal degree of 
conservativeness 
  
 
 
Meixing Dai  
BETA, University of Strasbourg, France 
Abstract 
This paper extends the results of Kobayashi (2003) and Ciccarone and Marchetti (2009) by considering the optimal 
choice of central bank conservativeness. It is shown that the government can choose a sufficiently populist but opaque 
central banker so that higher multiplicative uncertainty improves the social welfare only when the society is very 
conservative.
Acknowledgments: I am grateful to an anonymous referee for helpful suggestions. 
Citation: Meixing Dai, (2010) ''Multiplicative uncertainty, central bank transparency and optimal degree of conservativeness'', Economics 
Bulletin, Vol. 30 no.3 pp. 1720-1726. 
Submitted: Apr 26 2010.   Published: July 06, 2010. 
 
     1. Introduction 
In a model without inflationary bias, Kobayashi (2003) has shown that an increase in Brainard-
type multiplicative uncertainty in the transmission mechanism of monetary policy improves 
social welfare if the degree of central bank opacity is high enough, i.e., if the weights attached to 
its objectives are sufficiently uncertain for the public, and vice versa. This is because the 
influence of preference uncertainty on inflation and output could be reduced through the 
Brainard’s (1967) conservatism principle. Ciccarone and Marchetti (2009) argue that this result 
holds only in a specific case where society as well as the central bank on average are 
“conservative”, i.e., in the social welfare loss, the weight attached to output is lower than unity. 
They show that there is a statistical bound on opacity, preventing an increase in multiplicative 
uncertainty from lowering enough the variability of output and hence the social loss.  
This paper assumes that the government, which represents the society, delegates the 
monetary policy decisions to an independent central bank with optimal expected degree of 
conservativeness. We re-examine in this framework how is plausible the possibility that an 
increase in multiplicative uncertainty improves the social welfare.  
 
2. The model without optimal delegation 
 
Kobayashi (2003) assumes that the monetary policy is delegated to a central banker who is 
randomly selected from society. The central bank minimizes the loss function: 
2 2 ) ( ) 1 ( y L
CB α λ π α − + + = ,    0 > λ ,      (1) 
where  y  is output in log terms, π  is inflation rate, and λ  denotes the relative weight assigned 
by the central bank on average to the output objective. A higher λ  corresponds to a lower 
expected degree of central bank conservativeness in the sense of Rogoff (1985).  ] , 1 [ λ α − ∈  is a 
random variable and hence its value is unobservable for the private sector ex ante, with expected 
value  0 ) ( = α E  and variance 
2
α σ  , which represents the degree of opacity about central bank 
preferences.  
The social welfare loss is given by: 
2 2 y L
S λ π + = .         ( 2 )  
Since the weights attached by the society to objectives equal the expected ones by the central 
banker, i.e.  1 ) 1 ( = +α E  and  λ α λ = − ) ( E , we have  ] [ ] [
S CB L E L E = .  
Aggregate supply is given by a conventional Lucas’s supply function with the natural rate of 
output normalized to zero: 
ε π π − − =
e y .          ( 3 )  
where 
e π  is expected inflation rate and ε  a supply shock with zero mean and variance 
2
ε σ .  
The central bank sets the money growth rate (m ) to control inflation, subject to 
multiplicative uncertainty in the transmission mechanism of monetary policy:  
m v) 1 ( + = π ,             ( 4 )  
where  v is a random variable with mean zero and variance 
2
v σ . All shocks are assumed to be 
mutually uncorrelated, so that  0 ] [ ] [ ] [ = = = ε α αε v E v E E .  
The timing of the game is as follows. 1) The private sector forms rational expectations. 2) 
Supply shock ε  is observed. 3) The policymaker sets m. 4) Shock v occurs, and then inflation  
and output are realized. The game is solved by backward induction. The minimization of Eq. (1) 
subject to Eqs. (3) and (4) yields the reaction function 
) 1 )( 1 (
) ( ) (
2
v
e
m
σ λ
ε α λ π α λ
+ +
− + − = . Substituting it into Eq. 
(4) gives 
) 1 )( 1 (
] ) ( ) )[( 1 (
2
v
e v
σ λ
ε α λ π α λ π
+ +
− + − + = . Rational expectations imply  0 =
e π . Thus, the equilibrium 
solutions for m , π  and  y  are given by 
) 1 )( 1 (
) (
2
v
m
σ λ
ε α λ
+ +
−
= ,          ( 5 )  
) 1 )( 1 (
) )( 1 (
2
v
v
σ λ
ε α λ
π
+ +
− +
= ,            ( 6 )  
ε
σ λ
σ λ α λ
) 1 )( 1 (
) 1 )( 1 ( ) )( 1 (
2
2
v
v v
y
+ +
+ + − − +
= .      ( 7 )  
The expected social loss is hence evaluated as: 
 
4 4 4 4 4 4 4 48 4 4 4 4 4 4 4 47 6 4 48 4 47 6
2 2
) 1 ( ) 1 (
)] 1 ( ) 1 ( ) 1 ( 2 ) [(
) 1 ( ) 1 (
) (
] [ 2 2
2 2 2 2 2
2 2
2 2 2
y
v
v
v
S L E
σ
ε α
σ
ε α
σ λ
σ σ λ λ λ σ λ
λ
σ λ
σ σ λ
π
+ +
+ + + + − +
+
+ +
+
= .   (8) 
According to Eq. (8), an increase in 
2
v σ  reduces the variance of inflation, 
2
π σ , while it has 
ambiguous effect on the variance of output, 
2
y σ . Deriving  ] [
S L E  with respect to 
2
v σ  yields: 
2
2 2
2 2
2 ) 1 )( 1 (
] [
ε
α σ
σ λ
σ λ
σ v v
S L E
+ +
−
=
∂
∂ .        ( 9 )  
The conclusion obtained by Kobayashi is that if 
2 2
α σ λ <  , an increase in 
2
v σ  reduces social loss. 
For sufficiently high degree of opacity, the effect of an increase in multiplicative uncertainty on 
the variance of inflation will be greater than that on the variance of output. Brainard’s 
conservatism principle implies that the central bank varies its instrument less under multiplicative 
uncertainty than it would under multiplicative certainty (see Eq. (5)), improving hence the social 
welfare. 
Knowing that the probability distribution of α  ensuring the highest variance is the one that 
assigns positive probability values only to the extrema of α  (−1 and λ ) and zero elsewhere, 
Ciccarone and Marchetti (2009) have shown that the maximal value for 
2
α σ  is constrained, i.e. 
λ σα =
2
max , .
1 This implies that,  1 > ∀λ , i.e. the society is strictly “populist” (and the central bank 
is “populist” on average), the condition 
2 2
α σ λ <  will not be verified since 
2 2 λ λ σα < ≤ ,. In this 
case, increases in multiplicative uncertainty always reduce the expected social welfare. In the 
opposite case where  1 < λ , i.e. the society is strictly “conservative” (and the central bank is 
“conservative” on average), higher multiplicative uncertainty could reduce or increase expected 
social welfare according to the sign of the term  ) (
2 2
α σ λ −  in Eq. (9).  
In the following, we examine how the optimal choice of the expected degree of central bank 
conservativeness affects the effects of multiplicative uncertainty on the social welfare, in 
particular when  1 < λ .  
                                                 
1 See also Ciccarone et al. (2007).   
 
3. Optimal degree of conservativeness 
 
We consider a policy game where monetary policy is delegated to an independent central 
bank. The government will optimally choose the expected degree of central bank 
conservativeness by minimizing the social loss function: 
2 2 y L
S δ π + = ,     0 < δ ,       ( 1 0 )  
where δ  is the weight assigned by the society to the output objective and it is different from λ , 
i.e., the expected weight assigned by the central bank to the same objective. Higher values of δ  
and ) ( α λ −  signify that the society and the central bank are more liberal or populist. The 
government is assumed, as the private sector, to be imperfectly informed about  ) ( α λ −  and 
considers the latter as a random variable with mean λ  and variance 
2
α σ .  
The central banker is assumed to be nominated by the government before the private sector 
forms inflation expectations. The timing of other events defined before is kept unchanged. Using 
backward induction, the optimal value of λ , i.e. 
* λ , is obtained by minimizing the social loss 
function (10) subject to Eqs. (6)-(7). The government’s minimization problem is written, using 
Eqs. (6)-(7) in Eq. (10), as:  
2
2 2
2 2 2 2
2 2
2 2 2
) 1 ( ) 1 (
) 1 ( ) 1 ( ) 1 ( 2 ) (
) 1 ( ) 1 (
) (
] [ min ε
α ε α
λ
σ
σ λ
σ λ λ λ σ λ
δ
σ λ
σ σ λ
v
v
v
S L E
+ +
+ + + + − +
+
+ +
+
= .     (11) 
The first-order condition is: 
  0
) 1 ( ) 1 (
) 1 ( 2
) 1 ( ) 1 (
) ( 2
0
/
2 3
2 2
/
2 3
2 2
2 2
=
+ +
+
× −
+ +
−
⇒ =
∂
∂
∂ ∂ ∂ ∂ 4 48 4 47 6 4 48 4 47 6
λ σ
ε α
λ σ
ε α
σ λ
σ σ
δ
σ λ
σ σ λ
λ
π y
v v
S L
.          (12) 
It follows that
2  
2 * ) 1 ( α σ δ δ λ + + = .             ( 1 3 )  
 
Proposition 1: The optimal expected weight attached by the central bank to the output objective 
is greater than that of the society, i.e.,  δ λ >
* . It is always greater than the degree of central bank 
opacity, i.e. 
2 *
α σ λ > , and increases with the latter.  
 
Proof: It follows directly from Eq. (13).    ■ 
 
The higher are δ  and λ , the less conservative (more liberal) become the society and the 
central bank (on average). Therefore, the condition  δ λ >
*  implies that the society is more 
conservative (less liberal) than the central bank on average. When the central bank is fully 
transparent, 
* λ  equals δ  in the absence of inflation bias. For 0
2 > α σ , the government has an 
incentive to nominate a relatively more liberal (less conservative) central banker to reduce the 
                                                 
2 The second-order condition is verified since  0
2
) 1 ( ) 1 (
) 1 )( 1 ( 2
2 4 *
2
* 2
2
> =
+ +
+ +
= ∂
∂
ε σ λ
σ δ
λ λ λ
σ
α
v
S L . 
  
negative effects of opacity on the social welfare, so that 
* λ  must be higher than δ  to neutralise 
the joint effects of inflation shocks and opacity. When λ  equals 
2
α σ , an increase in λ  reduces 
both 
2
π σ  and 
2
y σ  (see Eq. (12)), implying that 
* λ  is always greater than 
2
α σ .  
The result 
2 *
α σ λ >  is similar to the statistical higher bound on opacity found by Ciccarone 
and Marchetti (2009), i.e.,  λ σα =
2
max , . Meanwhile, their significations are quite different. The 
inequality 
2 *
α σ λ ≥  implies that the optimal expected weight attached by the central bank to the 
output objective must never be smaller than the degree of opacity, while the statistical bound on 
opacity  λ σα =
2
max ,  is a technical condition implied by probability considerations.  
 
Proposition 2: For given degree of opacity, if the society is sufficiently conservative, i.e. 
2
2 2
1 α
α α
σ
σ σ δ
+
− < , the government can choose a relatively more liberal central banker to ensure that an 
increase in multiplicative uncertainty improves the social welfare. Inversely, if  2
2 2
1 α
α α
σ
σ σ δ
+
− > , the 
government cannot avoid that higher multiplicative uncertainty worsens the social welfare. 
 
Proof: Substituting 
* λ  given by Eq. (13) in the social loss function (11) and deriving the 
resulting function with respect to 
2
v σ  yield: 
         .
) 1 )( 1 (
)] ) 1 ( [
 
) 1 ( ) 1 (
)] ( ) 1 ( 2 [
) 1 ( ) 1 (
) ( ] [ 2
2 2
2 2 2
/
2 2 2
2 2 2
/
2 2 2
2 2 2
2
2 * 2 2 2 2
ε
α
λ
α
σ σ
ε α
σ σ
ε α σ
σ λ
σ σ δ δ
σ λ
σ σ λ λ λ
δ
σ λ
σ σ λ
σ
π
v v v v
S
v y v
L E
+ +
− + +
=
+ +
+ − +
× +
+ +
+ −
=
∂
∂
∂ ∂ ∂ ∂ 4 48 4 47 6 4 4 4 48 4 4 4 47 6 4 48 4 47 6
  (14) 
Imposing  0 2
] [ <
∂
∂
v
S L E
σ  leads to  0 )] 1 ( [
2 2 2 2 < + + + − α α α σ δ σ σ . Solving the inequality for δ  gives:  
) 1 (
2
2 2
α
α α
σ
σ σ
δ
+
−
< .           ( 1 5 )  
The opposite result, i.e.  0 2
] [ >
∂
∂
v
S L E
σ , is obtained when the condition (15) is reversed.  ■ 
 
Proposition 2 implies that if the society is sufficiently conservative, i.e. δ  is lower enough, 
the government can nominate a relatively less conservative (more liberal) central banker so that 
0 2
] [ <
∂
∂
v
S L E
σ . In effect, an increase in opacity reinforces the negative effects of multiplicative 
uncertainty on the social welfare (see Eq. (14)), which can be counterbalanced by a decrease in 
the expected degree of central bank conservativeness (see Eq. (12)). If the society is very 
conservative, the government will be able to choose a central banker who is liberal enough 
compared to the society to more than neutralize the negative effects of higher multiplicative 
uncertainty and hence to improve the social welfare.    
Since Eq. (13) implies that 
2 *
α σ λ ≥ , then for  1
2 > α σ , we always have 
2 2 *
α σ λ > . This implies 
that, when the central bank is highly opaque and liberal, an increase in multiplicative uncertainty 
will always deteriorate the social welfare, i.e.  0 2
] [ >
∂
∂
v
S L E
σ , according to Eq. (14).  
 
Proposition 3: If the degree of conservativeness of the society is low enough so that δ  is greater 
than 2071 , 0 max ≈ δ , the government will never be able to choose a relatively more liberal central 
banker to ensure that an increase in multiplicative uncertainty improves the social welfare.  
 
Proof: To determine the higher bound on δ  under which an increase in multiplicative 
uncertainty improves the social welfare under opacity, one maximizes the function 
2
2 2
1 sup
α
α α
σ
σ σ δ
+
− = . The first-order condition is: 
0
) 1 ( 2
2 1
2 2
2 2
2
sup =
+
− −
=
∂
∂
α
α α
α σ
σ σ
σ
δ
. 
Solving  0 2 1
2 2 = − − α α σ σ  leads to  1716 . 0
2 ≈ α σ . Hence, the maximal value for  sup δ  is 
2071 , 0 max ≈ δ .
3  ■ 
 
 
Figure 1: The thresholds of liberalism for the society and the central bank. 
                                                 
3 The second-order condition is verified since  0
2 2 2
2
2
2 2
sup
2
) 1 ( 2
) 1 (
1716 , 0
) ( < ≈
+
+ −
≈
∂
∂
α α
α
α
α σ σ
σ
σ
σ
δ
. 
Degree of opacity 
0  0.1  0.2  0.3 0.4 0.5 0.6 0.7 0.8 0.9  1 
0 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1 
Threshold of lambda  
Threshold of delta  
Figure 1 draws two curves representing respectively the threshold of liberalism of the society 
(i.e.  2
2 2
1 sup
α
α α
σ
σ σ δ
+
− = , bold lower line) and that of the central bank (i.e. 
2 2
sup sup
*
sup ) 1 ( α α σ σ δ δ λ = + + = , dashed higher line) under which an increase in multiplicative 
uncertainty improves the social welfare. The curve representing  sup δ  is concave, increasing in 
2
α σ  if  1716 , 0
2 < α σ  and vice versa. The curve corresponding to 
*
sup λ  is increasing in 
2
α σ . The 
difference between them, i.e.  2
2 2
1
) 1 (
sup
*
sup ) (
α
α α
σ
σ σ δ λ
+
+ = − , is increasing in 
2
α σ  at the rate 
0 2 2
2 2 2
2
sup
*
sup
) 1 ( 2
2 3 ) (
> =
+
+ +
∂
− ∂
α
α α α
α σ
σ σ σ
σ
δ λ
.  
Proposition 3 and Figure 1 suggest that for high or very low degrees of opacity, the society 
must be very conservative for being able to choose a relatively more liberal (less conservative) 
central bank to implement a monetary policy which allows an increase in multiplicative 
uncertainty to improve the social welfare.  
 
4. Conclusion 
 
In this paper, we have examined the effects of multiplicative uncertainty on the social 
welfare in a framework where the government can optimally choose the central banker in office 
under imperfect information about the latter’s preferences. It is shown that the optimal expected 
weight assigned by the central bank to the output objective (the inverse of which is a measure of 
central bank conservativeness) is always greater than that of the society as well as the degree of 
central bank opacity. In general, the society must be very conservative for allowing the choice of 
a relatively more liberal but opaque central bank to improve the social welfare when 
multiplicative uncertainty becomes higher.  
 
References: 
Brainard, W.C. (1967) “Uncertainty and the effectiveness of policy” American Economic Review 
57, 411–425. 
Ciccarone, G. and E. Marchetti (2009) “Revisiting the role of multiplicative uncertainty in a 
model without inflationary bias” Economics Letters 104, 37-39. 
Ciccarone, G., G. Di Bartolomeo and E. Marchetti (2007) “Unions, fiscal policy and central bank 
transparency” The Manchester School 75, 617-633.  
Kobayashi, Teruyoshi (2003) “Multiplicative uncertainty in a model without inflationary bias” 
Economics Letters 80, 317–321.  
Rogoff, Kenneth (1985) “The optimal degree of commitment to a monetary target” Quarterly 
Journal of Economics 100(4), 1169–1190. 

Multiplicative uncertainty in a model without inflationary bias”

Revisiting the role of multiplicative uncertainty in a model without inflationary bias”

The optimal degree of commitment to a monetary target”

Uncertainty and the effectiveness of policy”

Unions, fiscal policy and central bank transparency”

Multiplicative uncertainty, central bank transparency and optimal degree of conservativeness

http://www.accessecon.com/Pubs/EB/2010/Volume30/EB-10-V30-I3-P157.pdf

Multiplicative uncertainty, central bank transparency and optimal degree of conservativeness

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