This paper extends the model of Chortareas and Miller (2003) to the case of a continuum of central banker types. We derive two main results. First, whether the central banker candidate is too selfish or not enough, he has the same incentive to accept the contract and to breach. Second, a too selfish central banker is more costly for society than a benevolent central banker.monetary policy delegation, contract, selfishness

Eric Dehay

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 1 
  
In search of a selfish central banker? a note 
  
 
 
Eric Dehay  
University of Artois 
Abstract 
This paper extends the model of Chortareas and Miller (2003) to the case of a continuum of central banker types. We 
derive two main results. First, whether the central banker candidate is too selfish or not enough, he has the same 
incentive to accept the contract and to breach. Second, a too selfish central banker is more costly for society than a 
benevolent central banker.
 
Citation: Eric Dehay, (2011) ''In search of a selfish central banker? a note'', Economics Bulletin, Vol. 31 no.1 pp. 493-498. 
Submitted: Dec 17 2010.   Published: February 03, 2011. 
 
     1. Introduction
Chortareas and Miller (2003) [hereafter CM] consider a standard in￿ ation bias model in which monetary
policy is delegated through contracts but the central banker￿ s trade-o⁄ between social welfare and the
incentive scheme remains a private information. They show that it is convenient for a benevolent central
banker (CB) to pretend to be sel￿sh, accept the contract, and breach it. Consequently, a surprising positive
in￿ ation occurs. An important feature of their paper is the assumption that there are only two types of CB
candidates: the benevolent cares only about social welfare, whereas the sel￿sh one implements the contract.
This note extends the model of CM to the case of a continuum of CB types, that is, to the case where the
CB￿ s degree of sel￿shness can be too low (and even equal to zero as in CM) but also too high
1. Our results
are as follows. First, it is shown that a sel￿sh agent has an incentive to accept a contract designed for a
less sel￿sh one. Second, uncertainty about the CB￿ s sel￿shness may generate a negative in￿ ation surprise
and an output lower than the natural rate. Third, the social welfare loss resulting from the uncertainty is
asymmetric: a degree of sel￿shness that is too high is more costly than one that is too low. This note is
organized as follows. CM￿ s results are described in Section 2 as the benchmark case. Section 3 establishes the
results for a continuum of CB types. The conclusion summarizes the changes produced by the introduction
of this assumption.
2. Chortareas and Miller￿ s (2003) model
As in CM and following their notation, monetary policy delegation is represented by four equations.
y = yn + ￿(￿ ￿ ￿e) + " (2.1)
￿ = m + v ￿ ￿" (2.2)
LS = (y ￿ y￿)2 + ￿(￿ ￿ ￿)2 (2.3)
LBC = [(y ￿ y￿)2 + ￿(￿ ￿ ￿)2] ￿ ￿(to ￿ t￿) (2.4)
The ￿rst two equations employ the formulation adopted by Walsh (1995). (2.1) is the Lucas supply func-
tion where y, yn, ￿, and ￿e denote output, natural output, in￿ ation and in￿ ation expectations, respectively.
" is a shock with mean zero and variance ￿2
". The policy instrument is the rate of growth of the money
supply (m). The link between this instrument and in￿ ation is given by (2.2), where v is a control error (with
E(v) = 0, E(v2) = ￿2
v and E(v") = 0) and ￿" represents the direct impact of the shock on in￿ ation. (2.3)
is the society￿ s loss function where y￿ is the targeted output (y￿ = yn + z, with z > 0 the expansionary
bias) and ￿ is the preferred value for in￿ ation (zero by normalization). ￿, ￿, and ￿ are positive parameters.
The timing of events is as follows. The government chooses the monetary regime, the private sector forms
rational expectations, random shocks occur and the authority sets monetary policy.
If the discretionary solution applies, the equilibrium is found by minimizing (2.3) subject to (2.1) and
(2.2). This solution is characterized by the following expressions for, respectively, the reaction function,
expected in￿ ation, in￿ ation and output (with ￿ = 1=(￿2 + ￿)):
md = ￿￿z + ￿￿2E [m]d ￿ v + (￿ ￿ ￿￿)" (2.5)
E [￿]d = E [m]d =
￿z
￿
(2.6)
1Related papers are Ciccarone and Marchetti (2010) and Campoy, Candel-SÆnchez and Negrete (2009). Ciccarone and
Marchetti (2010) extend the model of CM to the common agency context but retain the assumption that there exist only two
types of CB. Campoy, Candel-SÆnchez and Negrete (2009) show that with sel￿shness uncertainty the CB￿ s choice in a menu of
contracts is a credible signal that reveals its true type to the private sector.
12
￿d =
￿z
￿
￿ ￿￿" and yd = yn + ￿￿" (2.7)
If monetary policy delegation is implemented using a linear in￿ ation contract, the CB￿ s loss function
(2.4) incorporates the social loss and a personal incentive scheme (to is the ￿xed reward and t the optimal
penalization rate). ￿ > 0 represents the degree of sel￿shness of the CB, that is, the extent to which he
responds to the incentive scheme rather than takes care of social loss. The monetary policy is found by
minimizing (2.4) subject to (2.1) and (2.2). The government chooses t so as to minimize (2.3) subject to the
policy outcomes. This yields:
mc = ￿￿z ￿
1
2
￿t￿ + ￿￿2E [m]c ￿ v + (￿ ￿ ￿￿)" (2.8)
t￿ =
2z￿
￿
(2.9)
E [￿]c = E [m]c = 0 (2.10)
￿c = ￿￿￿" and yc = yn + ￿￿" (2.11)
It is straightforward that the e⁄ectiveness of the contract depends on the degree of sel￿shness of the CB.
The problem is that it is not perfectly known a priori. CM assume that the government o⁄ers a contract
for a sel￿sh CB (￿ = ￿ > 0) and that a benevolent candidate (￿ = 0) masquerades as a sel￿sh one to become
the CB. Then, ￿ is the supposed degree of sel￿shness, whereas ￿ is the actual one. ￿ is the fraction of the
private sector believing that ￿ = ￿, that is, that the CB will respect the contract. 1 ￿ ￿ (with 0 < ￿ < 1) is
the fraction knowing that ￿ = 0, that is, that the CB is benevolent and will breach. The fraction ￿ sets its
expectations according to the contract equilibrium, whereas the fraction 1￿￿ sets them at the discretionary
one. Using (2.6) and (2.10), the private sector aggregated expectations are then given by:
E [￿]b = E [m]b = (1 ￿ ￿)
z￿
￿
(2.12)
Combining these expectations (2.12) and the reaction function of the benevolent CB (that reduces to 2.3)
gives the actual rate of money growth. The corresponding in￿ ation and output are:
￿b = ￿
￿
￿z
￿
￿￿
￿2(1 ￿ ￿) + ￿
￿
￿ ￿￿" and yb = yn + ￿(￿2￿z + ￿") (2.13)
CM￿ s main propositions are then as follows. First, a benevolent candidate has an incentive to accept a
contract designed for a sel￿sh CB and to breach it. Plugging (2.13) into (2.3), we obtain his expected loss
in this case (E [LCB]￿=0). Plugging (2.11) into (2.3), we obtain his expected loss if a truly sel￿sh CB was
appointed (E [LCB]￿>0). The benevolent candidate￿ s expected loss is always lower under breaching since the
following condition is always ful￿lled.
E [LCB]￿>0 ￿ E [LCB]￿=0 =
1
￿
2￿
￿
￿2(1 ￿ ￿) + ￿
￿2
￿2z2 > 0 (2.14)
Second, the benevolent CB who accepts and then breaches the contract produces an in￿ ation surprise 2.
Indeed, comparing (2.12) with in￿ ation in (2.13) gives:
￿b ￿ E [￿]b = ￿z￿￿ > 0: (2.15)
2CM also show that if the CB￿ s sel￿shness is private information but the CB is sel￿sh indeed (and not benevolent), then a
negative in￿ation surprise occurs.3
3. A continuum of central banker types
CM￿ s model highlights the costs of appointing a CB who assigns not enough importance to personal
reward compared to social welfare. However, what are the costs of appointing a CB who assigns too much
importance to his private welfare? We generalize the model by assuming that the actual degree of sel￿shness
of the CB candidate (￿) can be higher or lower than that selected to de￿ne the contract (￿). As in CM, the
government o⁄ers a contract for a CB with the degree of sel￿shness ￿ = ￿ > 0 (see 2.9). A candidate with
the degree ￿ = ￿ + ￿ masquerades as a ￿ one to become the CB. ￿ can be seen as a positive or negative
"sel￿shness-gap" (￿ 2 [￿￿ ￿;+1[). The CB candidate￿ s loss function is then:
LCB;￿ = [(y ￿ y￿)2 + ￿(￿ ￿ ￿)2] ￿ (￿ + ￿)(to ￿ t￿) (3.1)
This framework includes CM￿ s model as a particular case where ￿ = ￿￿. The equilibrium is found by
minimising (3.1) subject to (2.1) and (2.2). Plugging the penalization rate (2.9) into the ￿rst-order condition
gives the CB reaction function:
m￿ = ￿
￿
￿
￿￿z + ￿￿2E [m]￿ ￿ v + (￿ ￿ ￿￿)" (3.2)
￿ remains the fraction of the private sector believing that the CB has a degree of sel￿shness ￿ = ￿ > 0 and
will respect the contract. This fraction ￿ sets its expectations according to the contract equilibrium (see
2.10). 1 ￿ ￿ (with 0 < ￿ < 1) is the fraction knowing that the actual degree of sel￿shness is ￿ = ￿ + ￿ and
that the CB will breach. This fraction 1 ￿ ￿ sets its expectations according to the reaction function (3.2).
Aggregated expectations follow as:
E [￿]￿ = E [m]￿ = ￿(1 ￿ ￿)
￿z￿
￿￿
(3.3)
Inserting (3.3) back into (3.2) and using the resulting expression obtains us the realized in￿ ation and
output:
￿￿ =
￿
￿￿
￿
￿￿
￿z
￿
￿
￿
￿
￿2(1 ￿ ￿) + ￿
￿
￿ ￿￿" and y￿ = yn +
￿
￿￿
￿
￿
￿￿2￿z + ￿￿" (3.4)
The following propositions posit the ￿ndings of this note.
Proposition 1. Whether the CB candidate is too sel￿sh or not enough, he has the same incentive to accept
the contract and to breach.
Proof. We ￿rst calculate the CB￿ s expected loss when he breaches. Substituting (3.4) into (3.1) and taking
expectations, we get :
E[LCB;￿]￿+￿ = z2 +
￿￿2z2
￿￿
￿
2(￿ ￿ 1) +
￿
￿
￿
￿
￿2(￿
2 ￿ 1) ￿ ￿
￿
￿
￿ (￿ + ￿)to + ￿￿￿2
" (3.5)
If the CB does not breach, he brings about the optimal contract in￿ ation rate (see 2.11). Plugging the
in￿ ation rate into (2.11) and the private sector expectations (3.3) into (2.1), we get the output. Substituting
this outcome and the in￿ ation rate into (3.1) and taking expectations, the CB￿ s excepted loss when he abides
by the contract is given by:
E[LCB;￿]￿ = z2 +
￿￿2z2
￿
2￿
2 (￿ ￿ 1)[￿￿2(￿ ￿ 1) + 2￿￿] ￿ (￿ + ￿)to + ￿￿￿2
" (3.6)4
The CB breaches when E[LCB;￿]￿ > E[LCB;￿]￿+￿. Computing (3:6) ￿ (3:5) yields:
￿2￿2z2
￿
2￿
2 ￿
￿
￿2(1 ￿ ￿) + ￿
￿2
(3.7)
Whatever the sign of ￿, this expression is positive. As a consequence, a CB who is too sel￿sh has the
same incentive to cheat as one who is not enough sel￿sh. This result replicates and extends CM￿ s ￿nding
(Proposition 2). For example, remember that the benevolent CB in CM is the particular case where ￿ = ￿￿
in our model. A CB candidate with the symetric excessive sel￿shness (￿ = ￿) has the same incentive to
cheat as the benevolent one. ￿
Proposition 2. Unexpected in￿ation can be positive or negative and output can be higher or lower than the
natural rate depending on whether the CB assigns not enough or too much importance to personal reward.
Proof. Ignoring any supply shock and comparing (3.3) with the in￿ ation in (3.4) yields:
￿￿ ￿ E [￿]￿ = ￿
￿
￿
￿￿￿z (3.8)
When the CB places not enough weight on his private welfare (￿ < 0), ￿￿￿E [￿]￿ > 0: the actual in￿ ation
rate exceeds the expected one. In this case, y￿ > yn (see 3.4). When the CB is too sel￿sh (￿ > 0),
￿￿ ￿ E [￿]￿ < 0: the actual in￿ ation rate is lower than the expected one and then y￿ < yn. This result
replicates and extends CM￿ s ￿nding (Proposition 1). A CB who is too sel￿sh reduces in￿ ation expectations
(see 3.3). However, the expected in￿ ation is higher than their actual value because incomplete information
exists about the CB type (￿ > 0). ￿
Proposition 3. A CB candidate who is too sel￿sh relative to the contract o⁄ered by the government is more
costly for the society than one who is not enough sel￿sh.
Proof. Using (3.4) to substitute for in￿ ation and output in (2.3), we get the expected social loss when the
CB breaches the contract:
E[LS;￿]￿+￿ = z2 +
￿2￿2z2
￿￿
2 ￿[￿2(￿ ￿ 1)2 + ￿] + ￿￿￿2
" +
2￿
￿
￿￿￿2z2 (3.9)
Whatever the sign of ￿ 6= 0, the three ￿rst terms of (3.9) are the same ones. If ￿ < 0, the last term is
negative. If ￿ > 0, the last term is positive. Hence, the cost of appointing a wrong candidate is asymmetric:
for a given absolute value of the sel￿shness-gap, a CB who is too sel￿sh deteriorates social welfare more
sharply than one who is not enough 3. The intuition behind this result is as follows. As long as a fraction of
the private sector believes that the CB will respect the contract (￿ > 0), a negative (positive) sel￿shness-gap
produces a positive (negative) in￿ ation surprise and an output higher (lower) than the natural rate. Because
of the expansionary bias (z), the net cost of less in￿ ation but less output when the CB is too much sel￿sh
exceeds the net cost of more output but more in￿ ation when the CB is not enough sel￿sh. The higher the
degree of sel￿shness selected to de￿ne the contract (￿) or the higher the degree of conservatism (￿), the
smaller the asymmetry. ￿
3Tillmann (2008) is a related paper. He shows that a CB who is too conservative is more costly than a one who is too
liberal.5
4. Conclusion
The aim of this paper has been to extend the model of Chortareas and Miller (2003) and to explore the
e⁄ects of appointing a CB who assigns too much importance to his private welfare. Our results replicate
and extend CM￿ s ￿ndings that only take into account the case of a benevolent CB. We have shown that a
CB candidate who is too sel￿sh has the same incentive to breach the contract by which monetary policy is
delegated than a not enough sel￿sh one. Thus, when the CB￿ s sel￿shness is private information, in￿ ation
surprise can be negative or positive and output can be lower or higher than the natural rate. Finally, we ￿nd
that a CB who is not enough "self-interested" is cheaper for society than a CB who is too "self-interested".
References
[1] Campoy, J.C., F. Candel-SÆnchez and J.C. Negrete J.C. (2009) "Can Central Bank screening solve the in￿ationary bias?"
Working paper - Universidad de Murcia - (http://webs.um.es/juancris)
[2] Ciccarone, G. and E. Marchetti (2010) "Linear contracts, common agency and central bank preference uncertainty" Public
Choice, published online: 17 august
[3] Chortareas, G.E. and S.M. Miller (2003) "Central banker contracts, incomplete information, and monetary policy surprises:
In search of a sel￿sh central banker?" Public Choice 116: 271-295
[4] Tillmann, P. (2008) "The conservative central banker revisited: Too conservative is more costly than too liberal" European
Journal of Political Economy 24: 737-741
[5] Walsh, C. (1995) "Optimal contracts for central bankers" American Economic Review 85 (March): 150-67

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