ECONOMICS OF PUBLIC GOOD PROVISION: AUDITING, OUTSOURCING AND
BRIBERY
GERVAN FEARON
Abstract. The paper investigates the choice of government to audit or outsource the provision of a public
good in the presence of a potential hidden bribe and information asymmetries.
An audit mechanism
is developed for characterizing the provision of the public good in the public sector given asymmetric
information about the cost of production between the bureau (i.e., producer) and the department (funding
source).
Outsourcing is represented by Nash bargaining between the government department and a
monopoly …rm over the price of the public good, including a hidden bribe. The study also relaxes the
assumption of a single-tier government underlying the Samuelson rule.
The key …ndings are as follows:
For small scale economies, bribery is predicted to be aimed at having an outsourcing contract persist even
when public sector provision of the public good dominates (based on welfare and unit costs measurements).
On the other hand, bribery in large scale economies is predicted to be aimed at motivating outsourcing when
the provision of the public good through public sector production is in the public interest. The …ndings also
predict that a reduction in the department’s corruptibility may result in increased prices as the …rm acts
to induce the department to maintain its choice of outsourcing over the public interest for auditing when
the …rm’s bargaining power is su¢ ciently low and the department’s preference towards bribes is relatively
high.
Date : May 18, 2006.
Key words and phrases. Public good, audit, outsource, bribery
.
I would like to thank Neil Bruce for facilitating my sabbatical term at the University of Washington where much of this
paper was written. I would also like to thank Fahad Khalil, Jacques Lawarree, and the other participants of the Economics
Seminar Series at the University of Washington for thier insightful comments. I am also grateful to Gregg Harbaugh for his
assistance with Mathematica.
1
2
GERVAN FEARON
Economics of Public Good Provision: Auditing, Outsourcing and Bribery
1. Introduction
This paper investigates the economic implications of bribes on the choice of government to audit
or outsource the provision of a public good.
This study is motivated by the rising role of the
private sector as an alternative to the provision of pubic goods and concerns about the economic
cost of corruption involved in government outsourcing ventures. Samuelson’s Rule characterizes the
optimal provision of a public goods under assumptions of a benevolent provider who knows all the
relevant information about the cost of production, the consumers’willingness-to-pay, and a uni…ed
or single-tier governmental institutional arrangement (Mortimont, 2005).
Fearon and Busch
(2005) have presented an optimal mechanism for meditating informational rents given information
asymmetries regarding the cost of production in a hierarchal bureaucracy. However, the analysis
was not particular to a public good such that there remains scope for a more formal relaxation of
the assumption of a single-tier governmental institution concerning the Samuelson Rule.
The economic implications of corruption has been notably highlighted by Shleifer and Vishny
(1993).
More recently, Dreher and Herzfeld (2005) have outlined the international e¤ort aimed
at addressing corruption, including the World Bank Anti-corruption Strategy of 1997, the OECD
Convention on Bribery of Foreign Public O¢ cial in International Business in 1997, and the United
Nations Anti-corruption Treaty in 2003.
For instance, the World Bank has withheld the transfer
of funding to a number of international projects and the Canadian 2006 federal election outcome
has been reportedly in‡uenced by concerns about corruption.
Types of corruption have been
categorized as petty corruption (e.g., theft of government funds, including kick-backs) and large
scale corruption (e.g., establishing a monopoly with direct or indirect interests) by Rauch (2001);
as grand corporation (bribes to a government o¢ cial to gain improved business deal or position
such as under conditions of privatization) by Glaeser (2001); and/or as grease money ( e.g., bribery
aimed at facilitating preferential access to government services) by Clarke and Xu (2004).
A
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
3
review of the transparency index suggests that corruption occurs at various scales of economic
activity.
Sheifer and Vishny (1994) suggest that the cost of corruption is directly related to the
number of layers of government bribed within a country and the secrecy implicitly necessary in these
activities.
Khalil and Lawarree (2005) using a principal-agent framework suggest that auditing
is only optimal if cheating occurs in equilibrium and cheating can be mitigated by increasing the
costs of collusion through the utilization of internal and exterior auditors.
The formal inclusion
of auditing, outsourcing and bribery within a single economic framework concerning the provision
of a public good remains a gap in the literature.
Rauch (2001) develops an hierarchal overlapping generation model consisting of a chief and
a large number of deputies with monitoring, corruption, and internal promotional opportunities.
The chief or head of the department is a Stakelberg leader and the deputies are followers. Internal
promotional opportunities are shown to assist in mitigating corruption by having the decision
making positions occupied by individuals with high integrity.
In comparison to the Stakelberg
model used by Rauch (2001), Fearon and Busch (2005) use an optimal auditing mechanism to
characterize the institutional arrangement within the public sector and show that the government
can reduce informational rents by switching between costly auditing of a public sector bureau and
outsourcing to …rms engaged in second price auction in the provision of a public sector good.
However, the Fearon and Busch (2005) framework has yet to be applied to the provision of a public
good.1
The research questions addressed in this study are as follows: First, how is the Samuelson’s
Rule modi…ed by the presence of asymmetric information and auditing within a hierarchical government framework? Second, under what conditions can a monopoly …rm be anticipated to o¤er
a government department a bribe aimed at in‡uencing the decision of outsourcing by government?
Third, how does the monopoly …rm bribing of a government department a¤ect cost, output and
the government’s choice whether or not to outsource the provision of the public good?
1
Fearon and Busch (2005) framework applies to a public sector good that may have private or mixed good characteristics
as oppsed to a public good.
A public good is considered to be non-rivalous, non-excludable and, for pure-public goods,
non-congestible.
4
GERVAN FEARON
These research questions are addressed through the development of a theoretical economic framework involving a government department, bureau and a monopoly …rm engaged in the provision of a
public good. The department is assumed to care about the public and private goods and, inversely
proportional to its integrity, the department also cares about a hidden bribe.
The department
chooses to …nance the public good produced in the public sector through an audit mechanism or
in the private sector by a monopoly …rm.
The key …ndings of the paper are as follows: First, a modi…ed Samuelson Rule is characterized
involving the marginal cost of auditing and the expected penalty from the bureau misreporting
its costs.
Bribery is found to distort the level of the public good provided and social welfare by
inducing the department to choose outsourcing over auditing in opposition to the public interest.
Second, a reduction in the department’s corruptibility may result in prices increasing as the …rm acts
to induce the department to maintain its choice of outsourcing over the public interest for auditing
when the …rm’s bargaining power is su¢ ciently low and the department’s preference towards bribes
is relatively high.
Correspondingly, an increase in the …rm’s bargaining power may support its
ability to resist the demands for bribes by the department or pro…t share allocated towards bribing
the department resulting in the price of the public good declining in response to increased bargaining
power of the …rm.
Third, a combination of reduction in the corruptibility, improvement in public
sector governance (e.g., penalty for the bureau misreporting costs), and a reduction in the bene…ts
from government-…rm collusive behavior is predicted to be necessary to combat corruption. These
results are consistent with the predictions of Khalil and Lawaree (2005).
The paper is organized into 3 sections. In section 1, the introduction is provided. In section 2,
the theoretical model is described, the equilibrium outcomes characterized, and comparative statics
as well as an application is provided. In section 3, the conclusions are presented.
2. The Model
Consider an economy consisting of N individuals with preferences over a public good, G 2 <+ ,
and a private good, xi 2 <N
+ , represented by ui (xi ; G) given i 2 f1; 2; ::; N g and per capita income
Ii . The government consists of a funding department and a bureau.
The department provides
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
5
a transfer to the bureau that possesses the technology to produce the public good and covers the
cost associated with auditing the bureau’s reported costs. The bureau is assumed to have private
information about the cost of producing the public good.
The department can choose to have the
public good produced in the public sector or outsourced to a private …rm.
The department is assumed to have a quasi-linear utility function, caring about G and xi in a
fashion consistent with a Benthamite social welfare function and caring about a bribe in quantity
B such that preferences are represented as
PN
i=1
ui (xi ; G) + B .
can be considered to be the
sensitivity of the department to a bribe (i.e., a metric for corruptibility) and, inversely, the level of
integrity of the department. When = 0, the department acts as a benevolent social planner in the
determination of the level of the public good and private goods in the economy. The price of each
xi is known and given by px > 0. The public good is produced either by a bureau or a …rm utilizing
a constant marginal cost technology. It is assumed that the bureau’s marginal cost c 2 [c; c]; c > 0;
is distributed according to a density f (c). The actual marginal cost is private information of the
bureau, while the department knows only the distribution of marginal cost.
As in Fearon and
Busch (2006), the bureau is an extended Niskanen bureau that cares about its discretionary budget
in excess of its production costs. Given the reported cost w, let t(w) denote the budget transferred
to the bureau by the department, G be the actual level of production of the public good, and
the bureau’s preferences be represented by v(w; G) = t(w)
cG. The bureau’s reservation payo¤ is
normalized at zero. The bureau chooses the cost of the public good to report, w, given its private
observation of c and transfer from the department, t(w).
The department possesses an audit technology that provides an unbiased signal, b
c, of the true
cost. The audit technology is costly resulting in the department facing a choice between the level
of the public good to provide and the audit level aimed at mitigating some of the informational
rents potentially earned by the bureau.
A principal-agent model based on the mechanism design
approach of Fearon and Busch (2005) is used to represent the department and bureau’s problem.
The department can choose to have the pubic good provided by an outside …rm (i.e., outsourcing)
as opposed to being produced by the bureau.
The department and …rm bargain over the price of
6
GERVAN FEARON
the public good given an anticipated demand schedule for the public good. The bargaining process
is represented by a variable-threat Nash bargaining game (Fearon, 2001).
It is assumed that the
negotiation between the department and …rm is subsequent to an un-modeled request-for-proposal
(RFP) aimed at identifying the short-list of potential …rms that possess the necessary technology
and are willing to engage in the bargaining process for the provision of the good.
It is assumed
that the RFP process results in only a single …rm being identi…ed due to considerations such as:
con…dentiality, patent protection and/or strategic technology regulation (e.g., vaccine production,
space exploration, and military or security application).
The monopoly …rm is assumed to maximize pro…t with the marginal cost of production given
by cf 2 [c; c] and density f (cf ). cf is independent of the bureau’s costs c.2 During the bargaining
process, the …rm can choose to o¤er the department a bribe, B
the department and …rm.
0 which is private information to
The bribe is aimed at in‡uencing the department’s decision in favour
of outsourcing.
The department’s production choice has two alternatives for the provision of the public good is
denoted by a.
It can either design an auditing mechanism for the bureau (denoted by a = A) or
the department can outsource the provision of the public good (denoted by a = C ). The bureau is
assumed not to o¤er the department a bribe which implies that B = 0 when a = A. Furthermore,
it is assumed that government regulation prohibits the department from making transfers to the
bureau without oversight.3
The government’s problem concerning the provision of the public good is represented as a sequential game.
The game is as follows: At stage I, the department chooses to have the public
good provided through an audit mechanism (a = A) or by outsourcing (a = C ).
If the department
chooses a = A, then the department designs a Fearon-Busch audit mechanism for the provision of
the public good.
If a = C , then the department and the monopoly …rm Nash bargain over the
2 An alternative cost distribution is presented in Garvie and Ware (1996) investigation of the regulation of a mixed (public
and private) market oligopoly with private, but correlated, information about production costs.
3 The oversight assumption re‡ects contemporary constructs regarding public sector governance and the political fallout
from the allocation of public funds in the absence of monitoring/auditing. Clearly, this assumption could be relaxed without
materially altering the results.
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
7
price for the provision of the public good, denoted by pf , and a hidden bribe to the department in
quantity B
0.
2.1. Public Good Provision. The institutional arrangement involves auditing the bureau or
outsourcing the provision of the public good (i.e., a 2 fA; Cg). The subgame perfect equilibrium
choices and expected payo¤s for each of these strategies is presented below.
2.1.1. Auditing subgame. For auditing (a = A), the supervisor possesses an auditing technology
that generates a signal of the true costs. This signal c^ is distributed on [c; c] and is conditional on
the actual value of the bureau’s cost realization by the density h(^cjc). The signal is assumed to
be unbiased such that
Rc
c
c^h(^
cjc)d^
c = c. It is also assumed that the conditional distribution satis…es
the monotone likelihood ratio property, or …rst order stochastic dominance. Let H(^cjc) denote the
(cumulative) distribution function and it is assumed that H(^cjc) < H(^cjc0 ) 8^c if c > c0 . Let q denote
the probability of an audit occurring with costs de…ned by m(q) and the …xed cost of auditing is
m(0) = m0 , where: m0 > 0.
As in Fearon and Busch (2006), the following is assumed:
Assumption A1: Audit costs m(q) satisfy: (i) m(0) > 0; (ii) m0 (q) > 0; m00 (q) > 0 for q 2 (0; 1);
(iii) m0 (0) = 0; and (iv) limq!1 m(q) = limq!1 m0 (q) = 1:
Auditing is treated as a principal-agent problem and addressed in the usual mechanism design
approach (Fearon and Busch, 2006). The department is the principal and receives a report from the
bureau, denoted by w. Based on these reports, the principal chooses an audit contract consisting
of a transfer from the department to the bureau, t(w), the level of good provided by the bureau,
G(w), and an auditing intensity q(w) 2 [0; 1] and the levels of the private good, xi (w), as functions
of the bureau’s reported cost (w).
Auditing returns a signal represented by b
c.
The bureau is
penalized in the case where the audit returns a cost signal below reported cost. It is assumed that
the penalty function is exogenous, proportional to the misreporting of costs, and does not reward
the bureau. It follows that the bureau is subject to a penalty of max[0; p(w
c^)] if assigned, where
c^ is the cost signal, and p is an exogenous parameter consistent with Fearon and Busch (2006).
The penalty is assumed to be paid into the consolidated revenue fund of the government and is
8
GERVAN FEARON
not available to the department for altering the level of the public good or being reimbursed to
individual consumers.
The bureau’s expected payo¤ from stating a cost of w when its true cost is c is
v(w; c)
=
(t(w)
= t(w)
c G(w))
c G(w)
q(w)Ec^[max[0; p(w
Z w
q(w)p
H(^
cjc)d^
c
c^)]jc]
c
Both v(w; c) and @v(w; c)=@w are continuous at w = c. By the revelation principle, attention can
be restricted to truth-telling mechanisms (see appendix A). The participation constraint for the
bureau is given by
(1)
v(c; c) = t(c)
c G(c)
q(c)p
Z
c
H(^
cjc) d^
c
c
0:
Global incentive compatibility requires that
(2)
v(c; c)
v(w; c) 8w; c 2 [c; c]:
A local incentive compatibility approach is used, as shown in appendix A (Jewitt, 1988). This
con…nes attention to the …rst derivative of the bureau’s value function v(c), denoted by v(c)
_ , which
is given by
(3)
v(c)
_
=
G(c)
q(c)p
Z
c
c
@H(^
cjc)
d^
c:
@c
Speci…cally, the bureau’s utility v( ) is considered to be the state variable, and the control variables
are t(w), G(w), q(w), and xi (w). Truth-telling together with (2) implies:
(4)
t(c) = v(c) + c G(c) + q(c)p
Z
c
c
H(^
cjc)d^
c:
The department has a balanced budget constraint given by
PN
Given truth telling and substituting t(c), the constraint becomes
G(c)c
q(c)p
Rc
c
H(^
cjc)dc^
i=1
PN
px xi + t(w) + m(q)
i=1
[Ii
px xi ]
m(q(c))
PN
i=1 Ii .
v(c)
0. Hence, the department maximizes its expected payo¤ by choosing a
contract (G(c; p; I); xi (c; p; I); q(c; p; I)) to o¤er to the bureau for any reported cost c given the bureau’s
participation constraint (1) and the incentive compatibility constraint (3). Optimal control theory
is used to solve this problem de…ned by the control variables (G(c; p; I); xi (c; p; I); q(c; p; I)) and state
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
variable fv(c)g. The department’s problem is:
max
(5)
fG( );xi ( );q( )g
s.t.
( N "Z
X
i=1
(i) v(c)
_
=
G(c)
#)
c
ui (xi (s); G(s))f (s)ds
c
q(c)p
Z
c
c
N
X
(ii)
[Ii
px xi (c)]
@H(^
cjc)
d^
c
@c
v(c)
G(c)c
q(c)p
Z
c
c
i=1
(iv) v(c)
9
H(^
cjc)d^
c
m(q(c))
0
0
(iv) v(c) = 0
As appendix A shows, the optimal contract (G(c; p; I); xi (c; p; I); q(c; p; I)) o¤ered by the department to the bureau is characterized by:
1
0
Rc
0
cjc)d^
c
c @ m (q( )) + p c @H(^
A 1
= +
R c @H(^cjc)
@ui (xi ( ); G( ))=@x( ) px
px
p
d^
c
N
X
@ui (xi ( ); G( ))=@G( )
(6)
i=1
0
(7)
m (q( )) =
N
X
(8)
[Ii
p
c
cF (c)
cf (c) F (c)
px xi ] m(q(c; p; I))
v(c; p; I)
Z
c
c
@c
@H(^
cjc)
d^
c
@c
G(c; p; I)c
Z
c
H(^
cjc)d^
c
c
q(c; p; I)p
Z
c
c
i=1
H(^
cjc)d^
c =0
where: M RSi = f@ui (xi ( ); G( ))=@G( )g =@ui (xi ( ); G( ))=@x( )). Let Ii = I for simplicity in what
follows. The audit level characterized in equation 7 is identical to the level determined in Fearon
and Busch (2006).
Using the Fearon and Bush (2006) approach, G(c) is solved using equation (8) and substituted
into the dynamic transition equation (i) resulting in cv(c)
_
v(c) since @[v(c)=c]@c = cv(c)
_
v(c) (see
appendix A). It follows that
(9)
v(c; p; I) = cK(I)
c
Z
c
where:
at c = c.
K(I) =
Rc
1
c s2
nP
N
c
1
s2
i=1 [I
(N
X
[I
px xi ]
m(q(c; p; I)) + q(c; p; I)p
c
i=1
px xi ]
Z
m(q(c; p; I)) + q(c; p; I)p
Rc
c
c
H(^
cjc)d^
c
)
ds
o
H(^
cjc)d^
c ds as v(c; p; I) = 0
It can be noted that v(c) is the bureau’s payo¤ relating to the informational rents
10
GERVAN FEARON
earned through the exploration of the asymmetric information about its costs, c.
This infor-
mational rent is maximal when c = c, earning the bureau v(c; p; I) = cK(I) > 0 and at its minimum when c = c with v(c; p; I) = 0. The expected transfer from the department to the bureau
is Ec [t(c; p; I)] =
o
Rcn
Rc
cjc)d^
c dc which includes a truth telling
v(c; p; I) + c G(c; p; I) + q(c; p; I)p c H(^
c
compensation given by v(c; p; I); the direct expenditures on the public good denoted by c G(c; p; I);
and the expected penalty or budget loss that the bureau can be expected to pay from reporting
w > c given by q(c; p; I)p
Rc
c
H(^
cjc)dc^.4
The department’s budget constraints can therefore be written as
(10)
N
X
px xi (c; p; I) + cG(c; p; I) =
i=1
N
X
I
v(c; p; I)
q(c; p; I)p
Z
c
i=1
c
H(^
cjc)d^
c
m(q(c; p; I))
It is clear that the public good could be …nanced by a lump-sum tax de…ned by Ec [T (c; p; I)]=N ,
where: Ec [T (c; p; I)] = Ec [t(c; p; I)] + Ec [m(q(c; p; I))].
Furthermore, G(c; p; I) = 0 if I
Ec [T (c; p; I)]
under the audit mechanism given the assumption that government regulation requires transfers
from the department to the bureau to be subject to oversight.
A modi…ed Samuelson’s Rule is characterized by equation 6, implying that the optimal combination of the public and private goods is determined by the sum of the marginal rate of substitution (M RSi ) being equated to imputed public good cost to private good price ratio.
The
imputed cost of the public good includes the direct production cost incurred by the bureau and
the marginal cost of auditing the bureau incurred adjusted by the expected penalty imposed
against the bureau for misreporting its cost given an audit intensity q(c).5 The imputed price
of the public good under the audit mechanism is represented by pb (c; p; I), where: pb (c; p; I) =
ih R
h
Rc
c
cjc)d^
c p c
c + m0 (q( )) + p c H(^
i
@H(^
cjc)
d^
c
@c
1
.
It can be observed from equation 6 that the level of the public good is inversely related to its
costs of production as intuition would suggest.
More particular, a rise in the marginal cost of
4 Equation 8 implies that q(:) is independent of income, I. It is also helpful to note that cf (c)
F (c) < 0 for some values
of c. It is assumed that c0 > c where c0 is de…ned by c0 = fcf (c) F (c) = 0g. The necessity of oversight assumption imply
no audit contract o¤ered were c 2 [c0 ; c].
5 The audit intensity is characterized by equation 7 and q(c; p), where: q(c; p) = 0 and @q(c; p)=@c > 0 and @q(c; p)=@p > 0.
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
11
auditing and a decrease in the penalty (p) implies a corresponding increase in the overall cost of
the public good and substitution away from this good.
For c = c, the audit intensity is zero as it can be easily shown that m0 (q(c; p; I)) = 0 as p
Rc
c
H(^
cjc)dc^ =
0 from equation 7 as well as v(c; p; I) > 0. It warrants noting that m(q(c; p; I)) = m(0) > 0. The
modi…ed Samuelson’s Rule in 6 simpli…es to the traditional Samuelson’s Rule, when c = c. The
traditional Samuelson’s Rule is therefore a nested result of the auditing subgame for c = c. Yet, the
…xed cost of auditing (i.e., m(0) > 0) and the information rents of the bureau (i.e., v(c; px ; I) > 0)
a¤ect the department’s budget constraint. The hierarchical structure of this economic framework
gives rise to allocations that are ex-ante e¢ cient yet can be improved upon were the department
able to directly and costlessly observe the bureau’s costs as under the traditional Samuelson’s Rule.
For c 2 (c; c], m0 (q(c; p; I)) > 0 causes a reduction in the level of the public good relative to when c = c,
assuming the public good is normal.6 It therefore follows that G(c; p; I) < G(c; p; I). If m0 (q(c; p; I))
is su¢ ciently large and/or p is su¢ ciently small, then there exists a potential for a corner solution
exhibited by G(c; p; I) = 0 and X =
PN
i=1 xi (c; p; I) =
hP
N
i=1
I
i
m(0) =px .
The inability to control
government administrative (audit) cost or the absence of a su¢ cient deterrent against cost misreporting may therefore result in the curtailment of public provision of the public good through an
audit mechanism.
Additionally, if the expected lump-sum tax per capita, fEc [T (c; p; I)]g=N , for
…nancing the public good is su¢ ciently large relative to per capita income, then the public good
will not be produced through the audit mechanism (i.e., q(c; p; I) = 0 for c 2 (c; c]) as it is assumed
that interdepartmental transfers without oversight is prohibited in the public sector).
Payo¤. The expected payo¤ of the department (i.e., expected indirect utility) from auditing
(i.e., q(c; p; I) > 0) is denoted by
(11)
W (c; p; I : A) =
N
X
Ec [Vi (pb (c; p; I); px ; I
T (c; p; I)=N : A)]
i=1
6 Gaube (2000) relaxes the assumption of public good being a normal good given a distortionary tax being used to …nance the
provision of public goods. Clearly, the use of a distortionary tax to …nance the public good a direct e¤ect (i.e., substitution and
income e¤ects) on the provision of the public good. An indirect e¤ect is also implied by the a¤ect associated with government
budget available to fund auditing and the minization of informational rents.
12
GERVAN FEARON
2.1.2. Outsourcing Subgame. For the outsourcing subgame(a = C), the …rm and the department
engage in Nash bargaining over the price (pf ) and the bribe (B ) at stage II. At stage III, the
department chooses the level of the public good given the negotiated price pf , represented by
the Marshallian demand G(pf ; px ; I). A bargaining outcome that improves upon the threat point
yields payo¤s: (i) a pro…t function M (pf ; cf ; B : C) = G(pf ; px ; I)(pf
an indirect utility function
PN
i=1 [V
cf )
B for the …rm, and (ii)
(pf ; px ; I : C)] + B for the department.
Assumption A2: It is assumed that @M (pf ; B)=@pf
0 at pf = c.7
A2 and the prohibition of public sector provision of the public good without oversight imply
that the department chooses outsourcing (a = C) and the …rm sets the price pf = c if auditing is
not conducted (i.e., q(c; p; I) = 0). Correspondingly, the department’s expected payo¤ is given by
PN
i=1
Ec [Vi (c; px ; I : C)] for income, I 2 [0; I0 ], and given T (c; p; I). I0 is de…ned by:
(
I0 = I :
N
X
Ec [Vi (pb (c; p; I); px ; I
T (c; p; I)=N : A)]
i=1
N
X
)
Ecf [Vi (c; px ; I : C)] = 0
i=1
The failure to reach a bargaining agreement results in the threat point payo¤s being realized.
The …rms’threat point pro…t is given by M0 = G(c; px ; I)(c cf ) if I
I0 and zero if otherwise. The
department’s threat point payo¤ is denoted by V0 = 0 for I 2 (0; I0 ] and V0 =
PN
i=1 [V
(pb (c; p; I); px ; I :
A)] for I > I0 . The bargaining power of the …rm relative to the department is denoted by
2 (0; 1).
The bargaining problem is therefore represented by a variable threat point Nash bargaining problem.
In what follows, the outsourcing subgame perfect equilibrium is determined through backwards
induction.
At stage III, the department chooses the allocation (xi (pf ; px ; I); G(pf ; px ; I))i=1;2;::;N maximizes:
PN
i=1
PN
i=1
u(xi ; G) + B subject to
M RSi = pf =px and
PN
i=1
PN
i=1
px xi + pf G
px xi + pf G =
PN
i=1
PN
i=1
I.
I.
The equilibrium is characterized by
The equilibrium allocation does not necessarily satisfy the Samuelson Rule as the negotiated price
of the public good, pf , is unlikely to equal the actual cost of production given the Nash bargaining
7 A2 is a simplifying assumption that could be relaxed without materially a¤ecting the results.
complicates the analysis without any signi…cant or transparent bene…ts.
The absence of A2
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
13
process involved in determining this price. If pf = cf , then the marginal rate of transformation is
equal to the price ratio and the Samuelson Rule holds.
At stage II, the …rm anticipates the department’s demand for the public good, G(pf ; px ; I), and
bargains with the department over the price (pf ) and hidden bribe (B ).8 The bargaining outcome
is de…ned by fpf ( ; ; cf ; I); B( ; ; cf ; I)g solving:
(12)
8
<
max [G(pf ; px ; I)(pf
fpf ;Bg :
cf )
B
M0 ]
"
N
X
[V (pf ; px ; I : C)] + B
V0
i=1
#1
9
=
;
As shown in appendix C, the subgame equilibrium outcome (pf ( ; ; cf ; I); B( ; ; cf ; I)) is characterized by
(13)
@G(pf ; px ; I)
(pf
@pf
(14)
B = (1
cf ) + G(pf ; px ; I) =
) [G(pf ; px ; I)(pf
cf )
1
(1
)
M0 ] +
N
X
@V (pf ; px ; I : C)]
@pf
i=1
[V0
!
V (pf ; px ; I : C)]
The subgame equilibrium bribe, B( ; ; cf ; I) involves a pro…t sharing (or pro…t extraction) and
a kick-back component re‡ective of the forms of corruption described by Rouch (2001), Glaeser
(2001), and Clarke and Xu (2004). The department shares in the …rm’s pro…ts from the provision
of the public good being outsourced in proportion to the bargaining power of the department
relative to the …rm (i.e., 1
).
On the other hand, the kick-back component represents a
compensation for the department choosing outsourcing (a = C ) when the public interest favors the
audit mechanism (i.e., W (c; p; I : A) >
PN
i=1
Ecf [Vi (pf ; px ; I : C)] for I
that pf ( ; ; cf ; I) > cf for pro…ts to support bribe B( ; ; cf ; I) > 0.
I0 ) Clearly, it is necessary
9
For I 2 (0; I0 ], A2 implies that pf ( ; ; cf ; I) = c and B( ; ; cf ; I) = 0 so the threat point is
M0 = G(c; px ; I)(c
cf ) . For I > I0 , pf ( ; ; cf ; I) and B( ; ; cf ; I) are characterized by equations 13
and 14. The subgame equilibrium expected payo¤ schedules are stated below.
8 Bargaining represents a discovery process that facilitates the …rm learning about the demand schedule of the department
and the department learning about the cost of production of the …rm.
Con…dentiality agreements are generally practical
realities of government-industry negotiations so often prohibit the release of propitiatory information such as the …rm’s costs.
Hence, the nature of the bargaining process my cultivate the establishment of trust and the maintenance of secrecy that Shleifer
and Vishny (1993) note are precurses and ingredients of bribery relationships.
9 The a¤ects of and
on the price and bribery levels are discussed in the comparative statics analysis to follow.
14
GERVAN FEARON
For the …rm:
(15)
M ( ; ; cf ; I : C) =
G(c; px ; I) [c cf ]
Ecf fG(pf ( ; ; cf ; I); px ; I) [pf ( ; ; cf ; I) cf ]g
for I < I0
for I I0
Ecf fB( ; ; cf ; I)g
For the department:
(16)
W ( ; ; cf ; I : C) =
(
PN
for I < I0
i=1 [V (c; px ; I : C)]
E
[V
(p
(
;
;
c
;
I);
p
;
I
:
C)]
+
E
[B(
;
;
c
;
f
f
x
cf
f I)] for I
i=1 cf
PN
I0
In what follows, it is helpful to refer to Figure 1.
Auditing
(a=A)
Department
Payoff
Outsourcing
(a=C with B>0)
Outsourcing
(a=C with B=0)
I0
I1
I2
Income level, I
Figure 1. Department payo¤ and Income Levels with High Outsourcing Costs
The equilibrium values for pf ( ; ; cf ; I) and B( ; ; cf ; I) in relationship to income level are now
considered.
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
15
For I > I0 and Ecf [pf ( ; ; cf ; I)] 2 [c; Ec [pb (c; p; I)]], the department’s expected payo¤ from outsourcing (a = C ) dominates the auditing mechanism (a = A) and is expressed by
W (c; p; I : A) so no bribe is o¤ered (i.e., B( ; ; cf ; I) = 0).
PN
i=1
Ecf [V ( : C)] >
For I > I0 and Ecf [pf ( ; ; cf ; I)] 2
(Ec [pb (c; p; I)]; c) and I being su¢ ciently large relative to Ec [Tc ; px ; I)]=N , it is clear that
PN
i=1
Ecf [V ( : C)] >
W (c; p; I : A) for some values of I > I0 so again the bribe is set at B( ; ; cf ; I) = 0. Let this level of
income be denoted by I1 as follows:
(
I1 = I :
N
X
Ecf [V (pf ( ; ; cf ; I); px ; I : C)]
i=1
)
W (c; p; I : A) = 0 given Ecf [pf ( )] 2 (Ec [pb ( )]; c)
It therefore follows that a bribe is only paid for values of I > I1 given Ecf [pf ( ; ; cf ; I)g 2
(Ec [pb (c; p; I)]; c).
The …rm’s expected pro…ts, Ecf [M ( ; ; cf ; I : C)], must be su¢ ciently large to
support Ecf [B( ; ; cf ; I)] > 0 and a payo¤ of the department given by
N
X
Ecf [V (pf ( ; ; cf ; I); px ; I1 : C)] + Ecf [B( ; ; cf ; I1 )] > W (c; p; I : A).
i=1
If so, a bribe o¤ered by the …rm (i.e., Ecf [B( ; ; cf ; I)] > 0) results in the department choosing
to outsource the provision of the public good at a higher income than would be supported in the
absence of a bribe. This level of income implied is denoted by I2 satisfying:
(
I2 = I :
N
X
Ecf [V (pf ( ; ; cf ; I); px ; I : C)] + Ecf [B( ; ; cf ; I)]
)
W (c; p; I : A) = 0 given Ecf [pf ( )] > Ec [pb ( )]
i=1
Clearly, B( ; ; cf ; I2 ) > 0 implies that the …rm’s pro…t must be su¢ cient to support the bribe
when I 2 (I1 ; I2 ).
On the other hand, no bribe is paid if pro…ts are insu¢ cient and I2 vanishes
from consideration.
The …rm with …xed cost is now considered.
Let the …rm …xed costs be greater than I0 and
consider Ecf [pf ( )] < Ec [pb ( )]. This implies that there exists an income level, say I3 , that leaves
the department indi¤erent between outsourcing and auditing.
This is shown in Figure 2.
If
a bribe is paid (i.e., B( ; ; cf ; I) > 0), then it follows that there is another income level, say I4 ,
satisfying I4 < I3 . Here a bribe is paid to the support outsourcing at I4 < I 3 even though the unit
16
GERVAN FEARON
Outsourcing
(a=C with B=0)
Outsourcing
(a=C with B>0)
Department
Payoff
Auditing
(a=A)
I0
I4
I3
Income level, I
Figure 2. Department payo¤ and Income Levels with Low Outsourcing Costs
cost from outsourcing is less the level achieved under the optimal audit mechanism. At high levels
of per capita income, unit cost may therefore be an insu¢ cient proxy for determining whether or
for determining whether or not there is public sector corruption.
2.2. Department Choice. At stage I, the department chooses a 2 fA; Cg.
For I < I0 , the
department does not audit the bureau such that the …rm sets its price at pf ( ; ; cf ; I) = c and
B( ; ; cf ; I) = 0. Hence, the department sets a = C for I < I0 . For I
I0 and Ecf [pf ( ; ; cf ; I)] 2
[c; Ec [pb (c; p)]], the department’s expected payo¤ from outsourcing dominates the auditing mech-
anism when the …rm has no …xed cost so the department chooses a = C without a bribe.
For
Ecf [pf ( ; ; cf ; I)] 2 (Ec [pb (c; p)]; c), the department chooses a = C without a bribe for I 2 (I0 ; I1 ).
For I > I1 , there is a potential for a bribe if it can be supported by the …rm’s pro…ts and it results
in the department choosing outsourcing when the public interest favors auditing.
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
17
De…nition 1. A bribery equilibrium is an allocation Ecf [B( ; ; cf ; I)] > 0 o¤ered by the …rm to the
P
department to support outsourcing (i.e., a = C) over auditing (i.e., a = A) when N
i=1 Ec [Vi (pf (:); px ; I) : C] <
PN
T (c; px ; I)=N ) : A].
i=1 Ecf [Vi (pb (:); px ; Ii
The speci…c conditions under which a bribe will be supported is conditional on combinations of
the bargaining power of the …rm, , and the sensitivity of the department to bribes (i.e., ).
the sets
D=
(
D
and
( ; ):
N
X
F
be de…ned by
N
X
Ecf [Vi (pf ( ; ; cf ; I); px ; I) : C] + Ecf [B( ; ; cf ; I)]
i=1
)
Ec [Vi (pb (c; p; I); px ; I
T (c; px ; I)=N ) : A]
i=1
F=
Proposition 1. If
Let
F\ D
6= ; and
( ; ) : Ecf [M ( ; ; cf ; I : C)]
PN
M0
Ecf [Vi (pf ( ; ; cf ; I); px ; I) : C] <
i=1
PN
i=1
Ec [Vi (pb (c; p; I); px ; I
T (c; px ; I)=N ) : A],
then there exists a combination ( ; ) supporting a bribery equilibrium.
The existence of a bribery equilibrium implies that the department’s choice regarding the provision of the public good deviates away from the public interest.10
The combination ( ; ) satisfying
a bribery equilibrium is:
N
X
Ec [Vi (pf ( ; ; cf ; I); px ; I) : C] + Ecf [B( ; ; cf ; I)]
i=1
N
X
Ec [Vi (pb (c; p; I); px ; I
T (c; px ; I)=N ) : A]
i=1
So,
N
X
Ecf [Vi (pf ( ; ; cf ; I); px ; I) : C] +
i=1
n
(1
) Ecf M ( ; ; cf ; I : C)
M0 +
N
X
[V0
o
V (pf ( ; ; cf ; I); px ; I : C)]
Ec [Vi (pb (c; p; I); px ; I
T (c; px ; I)=N ) : A]
i=1
The proposition therefore holds if
(17)
Ecf [M ( ; ; cf ; I : C)]
PN
i=1
Ec [Vi (pb (c; p; I); px ; I
T (c; px ; I)=N ) : A]
PN
i=1
Ecf [Vi (pf ( ; ; cf ; I); px ; I) : C]
10 The public interest is de…ned by the welfare acheived from the equilibrium choices determined by the benevolent social
planner (i.e., = 0).
18
GERVAN FEARON
given:
PN
i=1
Ec [Vi (pb (c; p; I); px ; I
by de…nition, Ecf [M ( ; ; cf ; I : C)]
T (c; px ; I)=N ) : A] >
0.
PN
i=1
Ecf [Vi (pf ( ; ; cf ; I); px ; I) : C] and,
Proposition 2. If the …rm o¤ers the department a bribe, Ecf [B( ; ; cf ; I)] > 0, then it necessarily
expands the range of income (i.e., budget level) over which outsourcing is chosen over the audit
mechanism.
The proposition suggests that the decision to o¤er a bribe by the …rm acts to prolong outsourcing
when public sector provision through an audit mechanism would otherwise be preferred or to
accelerate the move towards outsourcing when it would be otherwise preferred based on per unit
cost of the public good (see Figure 1 and 2).
This result therefore suggests that comparing private
and public sector unit cost of providing the public good may not be an adequate indicator of the
appropriateness to outsource nor does it necessarily indicate that bribery was not involved in the
decision making process of the government department.
3. Comparative Statics: Corruptibility and Bargaining Power
The a¤ect of corruptibility and bargaining power is material to the equilibrium choice of the
agents within the economy.
As before,
represents the corruptibility of the department and the
relative bargaining power of the …rm vis-a-vis the department is represented by .
Comparative
statics are conducted using 13 and 14. The detail computations are shown in appendix D. For
this analysis, it is useful to de…ne two sets, namely,
U
and
L.
U
represents the set of
( ; ) involving the …rm having a relatively low bargaining power and the department having a
relatively high corruptibility (i.e., high sensitivity to bribery).
b( ) = 1
hP
N @ 2 V (pf ;px ;I:C)
i=1
@pf @pf
ih
@ 2 M (pf ;cf ; )
@pf @pf
i
1
( )
1
.
L
U
= f( ; ) :
< b ( )g, where:
is the set of ( ; ) re‡ecting the …rm
having relatively high bargaining power and the department having low corruptibility. Let
f( ; ) :
> b ( )g.
L
=
Given the department chooses a = C and a bribery equilibrium exists (i.e., B( ; ; cf ; I) > 0), the
a¤ect of corruptibility on the equilibrium price and bribery is as follows:
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
19
(:)
(1
) @M
dpf
@pf
=
d
jAj
dB
= f (1
d
)M (pf ; cf ; B) + Bg +
h
where: j A j=
( ; )2
L
.
If ( ; ) 2
u,
(1
then
)
dpf
d
n
(1
@ 2 M (pf ;cf ;B)
@pf @pf
> 0 and
dB
d
+
(:)
) @M
@pf
on
(1
PN
)
jAj
@ 2 V (pf ;px ;I:C)
i=1
@pf @pf
> 0.11
@M (pf ;cf ;B)
@pf
i
PN
@V (pf ;px ;I:C)
i=1
@pf
so j A j> 0 if ( ; ) 2
u
o
and j A j> 0 if
These results that the negotiated price of the public
good and the level of the bribe decline as the corruptibility declines. Intuitively, the importance of
the bribe to the department reduces the proportion of price levels required to …nance the bribery
of the department.
If ( ; ) 2
L,
then
dpf
d
< 0 and
dB
d
< 0 meaning that a reduction in the
corruptibility of the department results in a greater demand for a bribe to support the department
choosing outsourcing over auditing and, correspondingly, prices must be increased to support these
bribes.
The World Bank initiative for addressing corruption, includes a signi…cant e¤ort aimed at
improving integrity (i.e., reducing corruptibility) and good governance within government.
The
results above suggest that e¤orts to improve public sector integrity will likely not eliminate corruption without complementary e¤orts to address the …nancial capacity of …rms to pay bribes (e.g.,
monopolistic power or industrial concentration in the provision of public goods to government) and
the ability of governments to extract rents from …rms. Hence, these results reinforce the …ndings
of Khalil and Lawarree (2005) that e¤orts must also be made to increase the costs or penalty of
the collusion implicit in the conduct of corruption.
The a¤ect of bargaining power on the equilibrium price and bribe is as follows:
dpf
=
d
2 @M (:)
@pf
jAj
11 It can be shown that dB=d > 0 if ( ; ) 2
However, it is also clear that a bribery equilibrium requires that
u .
B > (1
)M (pf ; cf ; ) as the bribe includes the department’s share of pro…t, (1
)M (pf ; cf ; ), and compensation for the
payo¤ discrepancy from choosing a = C over a = A. Additionally, it is clear that were to be zero then bribe would be zero
so it reasonably follows that dB
> 0 for ( ; ) 2 u : Consistent with the conclusion formulated from the comparative statics
d
results.
20
GERVAN FEARON
dB
=
d
(
"
N
X
M (pf ; cf ; B) + V0
!#)
V (pf ; px ; I : C)
i=1
+
h
2 @M (:)
@pf
in
(1
)
PN
@M (pf ;cf ;B)
@pf
i=1
jAj
The a¤ect of bargaining power on the negotiated price of the public good is negative,
when ( ; ) 2
u,
dpf
d
@V (pf ;px ;I:C)
@pf
< 0,
meaning a …rm with relatively low bargaining power facilitates lower prices as its
bargaining power increases since the …rm is under less coercion to increase price in support of high
bribes for the department.
On the other hand, the a¤ect of the …rm’s bargaining power on the
level of the bribe is ambiguous even though it seems to be negative for ( ; ) 2
u.
The results suggest that the improvements in the integrity (i.e., reduction in corruptibility) of
the department can be anticipated to reduce prices and the level of the bribe when the integrity of
the department is relatively high and the …rm’s bargaining power is relatively low (i.e., ( ; ) 2
However, when integrity is relatively low (i.e., ( ; ) 2
L ),
u ).
an increase in integrity of the department
results in the …rm increasing the price levels to support higher bribes aimed at maintaining the
department’s decision to outsource when the public interest favors auditing. Furthermore, a …rm
with relatively small bargaining power will be more able to resist the department’s coercion for
higher prices in support of bribes given corruption (i.e., a bribery equilibrium).
The a¤ect of corruptibility and bargaining power on the equilibrium choice of the department is
implied by equation 17. Clearly, the set of ( ; ) 2
F\ D
6= ; also satis…es equation 17. Additional
examination of the a¤ects of corruptibility and bargaining power on the equilibrium outcome using
generalized functional forms is unlikely to yield more meaningful results than already derived. For
this reason, an example is presented in the application below using an assumed utility function for
the department.
3.1. Application. In what follows, the department’s payo¤ schedule is assumed to be
B . It can be shown that G(pf ; px ; I) = fN I=pf g and xi (pf ; px ; I) = (1
1
(1
)
1
N1
@pf ( ; ; cf ; I)=@
( ; )2
u.
px [ (1
)] 1
1
PN
i=1
x1i
G +
)fI=px g so pf ( ; ; cf ; I) =
1
cf1
(as shown in appendix E). Clearly, @pf ( ; ; cf ; I)=@ > 0 and
< 0 which implies the Cobb-Douglas based utility sets a restriction denoted by
Public good output, G(pf ; px ; I), is therefore decreasing in the level of corruptibility
o
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
(i.e., ) and the bargaining power of the department (1
).
21
In this respect, the department’s
bargaining power can be considered to be the level of coercion that can be levied against the …rm
to facilitate the extraction of bribes that is transmitted through to consumers by a high underlying
price of the public good.
Appendix E shows the …rm’s expected margin (i.e., revenues less production costs excluding the
bribe) for I
I0 can be written as12 :
8
<
c( ; ; c ; I : C)] = N I 1 (1
Ecf [M
f
:
(18)
)
f [1
1
The department’s expected payo¤ is W ( ; ; cf ; I : C) =
where:
]g 1
i=1
n
Ecf [V (pf ( ; ; cf ; I); px ; I : C)] = (1
)
1
N1
N1
px 1
Zc
(cf ) 1
f (cf )dcf
c
PN
(19)
N
X
1
px 1
i=1
;
Ecf [V (pf ( ; ; cf ; I); px ; I : C)]+ Ecf [B( ; ; cf ; I)],
8
o <Zc
:
9
=
(cf ) 1
c
f (cf )dcf
9
=
;
[ (1
)] 1
I:
The department’s expected payo¤ is linear in income (I ) as depicted in Figures 1 - 3.
A bribery equilibrium is de…ned by
PN
F
\
D
6= ; and
PN
i=1
Ecf [Vi (pf ( ; ; cf ; I); px ; I) : C] <
T (c; px ; I)=N ) : A].
Using equations 18 and 19, the boundary of F
n
o
c( ; ; c ; I : C)] = 0 and the boundary of D can be
can be characterized by gF ( ) =
: Ecf [M
f
i=1
Ec [Vi (pb (c; p; I); px ; I
characterized by gD ( ) =
: W ( ; ; cf ; I : C)
W (c; p; I : A)= 0 .
Figure 3 shows the relationship
between gF ( ) and gD ( ) in ( ; )-space based on the properties implied by gbF ( ) and gbD ( ) and
the numeric analysis shown in Appendix F13 .
12 The …rm’s expected pro…t (i.e., expected margin less the bribe) is:
c( ; ; cf ; )]
c( ; ; cf ; )]
Ecf [M ( ; ; cf ; )] = Ecf [M
E [B( ; ; cf ; I)] so Ecf [M ( ; ; cf ; )] =
E cf [ M
h
i cf
PN
( = ) V0
i=1 Ecf V (pf ( ; ; cf ; I); px ; I : C) .
(1
)=
13 It may helpful to write E [M
c( ; ; cf ; I : C)] = N If1 d1 f [1
]g =(1 ) . Let g
bF ( ) = 1 f1= gd1
which
cf
c( ; ; cf ; I : C)] = 0 when g
implies that Ecf [M
bF ( ) = . It is clear that @b
gF ( )=@ > 0, @ 2 g
bF ( )=@ @ < 0 and g
bF ( )
approaches 1 as approaches in…nity. The …rm’s expected margin is positive when g
bF ( ) < , meaning the bargaining power
of the …rm is su¢ ciently large to facilitate a positive expected margin for the …rm that is ah precurse to paying
a bribe.
i
PN
It is helpful to note that
:
C)] + E cf B( ; cf ; I)
V0 implies
i=1 Ecf [V (pf ( ; ; cf ; I); px ; I
PN
V0 .
As appendix E shows, g
bD :
! [0; 1] as de…ned by
i=1 Ecf [V (pf ( ; ; cf ; I); px ; I : C)] + M ( ; ; cf ; I)
o
n P
1
N
g
bD ( ) =
: i=1 Ecf [V (pf ( ; ; cf ; I); px ; I : C)] V0 = 0 . Speci…cally, g
bD ( ) = 1 f1= gfV0 =d2 g
and g
bD ( ) >
PN
implies that
E
[V
(p
(
;
;
c
;
I);
p
;
I
:
C)]
+
M
(
;
;
c
;
I)
>
V
.
Clearly,
g
b
(
)
has
a
form
similar
to
g
b
( ) as the
c
x
0
D
F
f
f
f
f
i=1
expected payo¤ of the department is weighted by the expected margin of the …rm.
22
GERVAN FEARON
Department’s
sensitivity to
bribe, θ
Department’s expected
payoff
Boundary for E[V(.)] =V0
as defined by gD(α).
θ0
Firm’s expected profit
Boundary for E[M(.)] = 0
as defined by gF(α).
α0
1
Firm’s Bargaining
Power, α
Figure 3. Department and Firm’s Indi¤erence Contours
The contour de…ned by gD ( ) represents the locus of points for which the department is indi¤erent
between outsourcing (upon accepting a bribe) and having the public good produced within the
public sector using an optimal audit mechanism. Outsourcing (upon accepting a bribe) dominates
auditing for values of corruptibility, , higher than the contour gD ( ).
On the other hand, the
contour de…ned by gF ( ) represents the expected pro…t (expected margin less the bribe) of the …rm
is zero.
The …rm therefore only makes a bribe for values of the department’s corruptibility higher
than the level de…ned by the contour gF ( ).
gD ( ) = gF ( ) =
0
and let
0
The intersection of the two contours is given by
= gF 1 ( 0 ) = gF 1 ( 0 ).
For simpli…cation, it is assumed that there is
only a single intersection.
A rise in the threat point payo¤ given from the optimal audit mechanism implies that the contour
de…ned by gD ( ) shifts upward and towards the left. Under these conditions, a higher corruptibility
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
23
for the department and a low bargaining power for the …rm would be needed to support even an
indi¤erence between outsourcing, with a bribe, and the optimal audit mechanism.
Proposition 3. If
>
0
and gD1 ( ) >
> gF 1 ( ), then a bribe o¤ered by the …rm is accepted by the
department choosing to outsource the public good even though it is not consistent with the interest
of the public. Otherwise, no bribe is o¤ered as it would not be accepted.
The proposition suggests that simply improving the integrity or reducing the corruptibility of
government o¢ cials (i.e., departments) does not necessarily curtail bribery.
Rather, the combi-
nation of increasing the department’s integrity and reducing the feasibility of …rms to o¤er bribes
may be needed.
It therefore suggests that public sector initiatives aimed at improving gover-
nance and integrity in government is only one component potentially needed to deter corruption.
This puts further weight to Khalil and Lawarree (2005) …ndings that corruption can be deterred
by increasing the cost or penalties associated with collusion. The proposition further stresses the
importance of the transparency of public accounts to deter corruption.
This transparency must
include information on service levels, quality and unit costs. The public must also have access to
data regarding program delivery, administration and auditing of public goods produced in-house
or through outsourcing.
4. Conclusion
The paper investigates the choice of government to audit or outsource the provision of a public
good in the presence of a potential hidden bribe and information asymmetries.
An audit mech-
anism is developed for characterizing the provision of the public good in the public sector given
asymmetric information about the cost of production between the bureau (i.e., producer) and the
department (funding source). Outsourcing is represented by Nash bargaining between the government department and a monopoly …rm over the price of the public good.
During the bargaining
process, the …rm may choose to o¤er the department a bribe aimed at in‡uencing the department’s
decision in favor of outsourcing when it is in opposition to the public interest (otherwise, a bribe is
not o¤ered provided). The study also relaxes the assumption of a single-tier institutional structure
24
GERVAN FEARON
within government that is considered to hold for the Samuelson Rule for the optimal provision of
the public good.
The key …ndings of the study are as follows:
First, bribery is predicted to distort the level of
the public good provided and social welfare by inducing the department to choose outsourcing over
auditing in opposition to the public interest.
Second, an increase in the department’s integrity
may result in increased prices as the …rm acts to induce the department to maintain its choice of
outsourcing over the public interest for auditing when the …rm’s bargaining power is su¢ ciently low
and the department’s preference towards bribes is relatively high.
Correspondingly, an increase
in the …rm’s bargaining power may support its ability to resist the demands for bribes by the
department resulting in the price of the public good declining in response to increased bargaining
power of the …rm.
Third, a combination of improvements in the integrity and public sector
governance (e.g., penalty for the bureau misreporting costs) and a reduction in the bene…ts from
government-…rm collusive behavior is predicted to be necessary to combat corruption.
These
results are consistent with the predictions of Khalil and Lawaree (2005). The results also suggest
that bribery for small and large scale economies in terms of per capita income is welfare reducing.
For small scale economies, bribery is predicted to be aimed at having an outsourcing contract
persist even when public sector provision of the public good dominates (based on welfare and unit
costs measurements). On the other hand, bribery in large scale economies is predicted to be aimed
at motivating outsourcing when the provision of the public good through public sector production
is in the public interest.
The study contributes to the literature by relaxing the assumption of a single-tier government
underlying the Samuelson rule and by characterizing a bribery equilibrium within a economic
framework that endogenize the institutional choice between outsourcing and an audit mechanism
for the public sector provision of a public good.
The scope for further study include empirical
estimation of the relationships predicted and application of numeric methods aimed at verifying
the conditions for the predictions to hold.
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
25
5. Appendix A
Bureau
The bureau’s expected utility from stating a cost of w when its true cost is c then is
v(w; c)
=
(t(w)
c G(w))
=
(t(w)
c G(w))
= t(w)
c G(w)
q(w)Ec^[max[0; p(w c^)]jc]
Z w
q(w)p
(w c^)h(^
cjc) d^
c
q(w)p
Z
(A1)
c
w
H(^
cjc)d^
c
c
The penalties both v(w; c) and @v(w; c)=@w are continuous at w = c. By the revelation principle,
attention can be restricted to truth-telling mechanisms. The participation constraint for the bureau
is given by
(A2)
v(c; c) = t(c)
c G(c)
q(c)p
Z
c
c
H(^
cjc) d^
c
0:
Global incentive compatibility requires that
(A3)
v(c; c)
v(w; c) 8w; c 2 [c; c]:
Know
(A4)
@v(w; c)
= t0 (w)
@w
0
c G (w)
0
q (w)p
Z
c
w
H(^
cjc)d^
c
q 0 (w)pH(^
cjc)
This restricts the derivative of the bureau’s value function v(c), denoted by v(c)
_ , and is given
by14
(A5)
v(c)
_
=
G(c)
q(c)p
Z
c
c
@H(^
cjc)
d^
c:
@c
Department
The department’s problem then is to maximize its expected payo¤ by choosing a contract
(G(w); x(w); q(w)) to o¤er to the bureau for any reported cost w.
14 v(c)
_
= @v(c; c)=@c after substituting @v(w; c)=@w = t0 (w)
@v(w; c)=@w = 0 with truthtelling so w = c.
c G0 (w)
q 0 (w)p
Rw
c
H(^
cjc)d^
c
q 0 (w)pH(^
cjc) and setting
26
GERVAN FEARON
The department faces a budget constraint given by
it can be shown that the budget constraint is:
N
X
(A6)
[Ii
px xi ] v(c)
c G(c)
q(c)p
Z
c
i=1
where: t(c) = v(c) + c G(c) + q(c)p
Rc
c
PN
i=1
px xi + t(c)
Ii .
After substituting t(c),
c
H(^
cjc) d^
ct(c)
m(q(c))
B
0
H(^
cjc)dc^.
The department o¤ers the bureau a contract fG(c; I); x(c; I); q(c; I)g solving the following optimal
control problem de…ned by the state variablev(c) and the contract consists of the control variables.
The department’s problem is:
max
fq( );xi ;G( )g
s.t.
15
( N "Z
X
i=1
(i) v(c)
_
=
G(c)
ui (xi (s); G(s))f (s)ds
c
q(c)p
Z
c
(ii)
N
X
[Ii
#)
c
px xi (c)]
v(c)
c
(A7)
@H(^
cjc)
d^
c
@c
G(c)c
q(c)p
Z
c
i=1
(iv) v(c)
c
H(^
cjc)d^
c
m(q(c))
B
0
0
(iv) v(c) = 0
Let (c) denote the (di¤erentiable) multiplier on the transition equation (i) and let (c) denote
the (di¤erentiable) multiplier on (ii). The …rst order condition for the optimal control problem
satis…es:16
15 See Kamien and Schwartz (1981) for an outline of the techniques used to solve such programming problems.
h
i
16 It is helpful to note that @ R h(x) f (z)dz =@x = f (h(x))h0 (x) f (g(x))g 0 (x).
g(x)
ECONOM ICS OF PUBLIC GOOD PROVISION:
N
X
[Ii
px xi ]
v(c)
i=1
v(c)
_
=
G(c)
q(c)p
G(c)c
Z
c
q(c)p
AUDITING, OUTSOURCING AND BRIBERY
Z
c
c
c
27
H(^
cjc)d^
c
m(q(c))
B
0 ? (c)
0
@H(^
cjc)
d^
c
@c
(A9)
v(c) = 0
0
(c) =
(A10)
(c)
(A10)
(c) = 0
N
X
i=1
(A11)
@ui (xi (c); G(c))
f (c)
@G(c)
@ui (xi (c); G(c))
f (c)
@xi (c)
Z c
@H(^
cjc)
(c)p
d^
c
@c
c
For (c)
0 and (c)
(c)
c (c) = 0
px (c) = 0
Z c
(c)p
H(^
cjc)d^
c
(A12)
(A13)
(c)m0 (q(c)) = 0
(A14)
c
0, it can be easily shown that fG(c; I); x(c; I); q(c; I)g satis…es
N
X
@ui (xi (c); G(c))=@G(c) c + (c)= (c)
=
@ui (xi (c); G(c))=@xi (c)
px
i=1
(A15)
(A16)
(A8)
N
X
i=1
px xi (c) + v(c) + G(c)c + q(c)p
Z
c
c
H(^
cjc)d^
c + m(q(c)) + B = N I i
For c = c, Sameulson’s Rule applies directly as (c) = 0 and m0 (q(c)) = 0.
The level of G(c; I)
and x(c; I) at c = c is de…ned also by the department’s budget constraint as follows:
PN
i=1
px xi (c) +
v(c) + G(c)c + m0 = N Ii , meaning the transfer payments to the bureau, v(c), must still be covered
by the department even when the bureau reports a cost c = c.
It is now useful to solve for (c) by using A14 as follows:
28
GERVAN FEARON
(c)p
Z
c
c
(c)p
Z
@H(^
cjc)
d^
c
@c
c
(c)p
c
c
@H(^
cjc)
d^
c
@c
c
Z
0
(c) p
(c)m0 (q(c))
H(^
cjc)d^
c
Z
c
= 0
(A17)
c
H(^
cjc)d^
c + m0 (q(c))
(c)
p
(c)
(c)
p
(c)
Z
c
0
@H(^
cjc)
d^
c = p
@c
c
Z
=
c
@H(^
cjc)
d^
c =
@c
c
(c)
(c)
=
Z
c
c
p
p
Z
Rc
c
H(^
cjc)d^
c + m0 (q(c))
c
c
H(^
cjc)d^
c
m0 (q(c))
H(^
cjc)d^
c + m0 (q(c))
R c @H(^cjc)
p c @c d^
c
Using A14 and A10:
N
X
@ui (xi (c); G(c))
f (c)
@G(c)
i=1
(c)
c (c)
=
(c) + c 0 (c)
=
Known @[c (c)]=@c = (c) + c 0 (c) so (c) = [1=c]
"N Z
1 X c
(c) =
c i=1 c
(A19)
By B6, (c) =
0
(c)
(c)
(A18)
N
X
@ui (xi (c); G(c))
f (c)
@G(c)
i=1
(c) + c 0 (c) f (c)dc which implies:
c
@ui (xi (c); G(c))
@G(c)
f (c)dc
#
(c) so
(c)
(c)
=
=
(A20)
0
1
c
PN
1
c
=
c
For m0(q(c)):
Rc
0
i=1
PN
i=1
hP
N Rc
i=1 c
@ui (xi (c);G(c))
@G(c)
c
hP
f (c)
N Rc
i=1 c
@ui (xi (c);G(c))
@G(c)
@ui (xi (c);G(c))
@G(c)
1
c2
i=1 c
@ui (xi (c);G(c))
@G(c)
f (c)
f (c)dc
PN R c
PN R c
i=1 c
i
@ui (xi (c);G(c))
@G(c)
f (c)dc
i
@ui (xi (c);G(c))
@G(c)
f (c)dc
f (c)dc
ECONOM ICS OF PUBLIC GOOD PROVISION:
m0 (q(c))
(c)
p
(c)
=
=
p
Z
c
c
=
p
Z
Z
c
@H(^
cjc)
d^
c p
@c
c
8
@H(^
cjc) <
d^
c
: c PN
@c
Z
i=1
c
c
AUDITING, OUTSOURCING AND BRIBERY
29
c
H(^
cjc)d^
c
c
c
(A21)
hP
N Rc
i=1 c
@ui (xi (c);G(c))
@G(c)
Z c
@H(^
cjc)
cF (c)
d^
c
@c
cf (c) F (c)
p
c
@ui (xi (c);G(c))
@G(c)
f (c)
PN R c
i=1 c
f (c)dc
i
9
=
f (c)dc ;
@ui (xi (c);G(c))
@G(c)
H(^
cjc)d^
c
This equation can be expressed as m0 (q(c)) =
0
Rc
(p; q(c)) so m(q(c)) =
c
0
p
Z
c
c
H(^
cjc)d^
c
(p; q(c))dc and let
m(q(c)) = m0 + '(p; c). At c = c, it is known that m(q(c)) = m0 as q(c) = 0 and clearly for p = 0 it
would not be rational to monitor so implying @'(p; c)=@p > 0 and @'(p; c)=@c > 0: This result is
Known
N
X
@ui (xi (c); G(c))=@G(c)
i=1
=
@ui (xi (c); G(c))=@x(c)
N
X
@ui (xi (c); G(c))=@G(c)
i=1
=
@ui (xi (c); G(c))=@x(c)
c + (c)= (c)
px
0
1
Rc
m0 (q(c)) + p c H(^
cjc)d^
c
c
A 1
+@
R
c
@H(^
c
jc)
px
px
p c @c d^
c
(A22)
6. Appendix B
But from (A9) we can solve for x(c) and substitute into (A8). This yields
(B1)
N
X
[I i px xi (c; px ; I i )]
m(q(c; p))
v(c)
cG(c)m
q(c; p)p
0
c
i=1
(B2)
v(c)
_
=
G(c)
q(c)p
Z
c
c
Z
c
H(^
cjc)d^
c
=0
@H(^
cjc)
d^
c
@c
Solving for G(c) and substituting yields
(B3)
v(c)c
_
v(c) =
N
X
i=1
[I i px xi (c; px ; I i )] + q(c; p)p
Z
c
c
H(^
cjc)d^
c
m(q(c; p))
Using the Fearon and Busch (2005) method, it is known that @[v(c)=c]@c = [v(c)c
_
v(c)]=c2 so
30
GERVAN FEARON
(B4)
v(c) = cK(I i )
c
Z
c
c
1
s2
(N
X
[Ii
px xi (c; px ; Ii )] + q(c; p)p
Z
c
i=1
)
c
H(^
cjc)d^
c
m(q(c; p)) ds
where:
(B5)
K(I i ) =
Z
c
c
1
s2
(N
X
[Ii
px xi (c; px ; Ii )] + q(c; p)p
Z
c
H(^
cjc)d^
c
c
i=1
)
m(q(c; p)) ds
as v(c) = 0 and, therefore, v(c) reaches a maximum at c = c.
Example using a Cobb-Douglas utility function
In what follows, it will be useful analysis for a more closed form to assume a simplifying utility
function such as a Cobb-Douglas form.
The expected payo¤ of the department is given by a
Cobb-Douglas utility function (noting B = 0 for a = A) can be therefore expressed as:
(B6)
8 P
n
(1
N
1
N
<
X
(1
)
(c) px
i=1
n
Ec Vi (pb (c; p); px ; I i : A) =
PN
1
:
(1
)
pb (c; p)
i=1
i=1 Ec
)
o
N [Ii ]
px
(1
)
N [Ii
for I I0
o
T (c; px ; I)=N ] for I > I0
The expected payo¤ of the department with the Cobb-Douglas utility function is linear in per
capita income and inversely related to price as expected.
7. Appendix C
D. Firm-Supervisor Bargaining
The …rm-supervisor bargaining problem is de…ned by the choice of (pg ; B ) to solve the following:
(C1)
where:
8
"N
#1 9
<
=
X
max [G(pf ; px ; I)(pf cf ) B M0 ]
V (pf ; px ; I : C) + B V0
;
fpg ;Bg :
i=1
o
nP
PN
N
and M0 = G(c; px ; I)(c
V0 = max
i=1 V (pb (c; p); px ; I : A)
i=1 V (c; px ; I : E);
cf ) for
I < I1 and zero otherwise.
The equilibrium outcome fpf ( ; ; cf ; I); B( ; ; cf ; I)g for the Nash bargaining problem is characterized by
For pf ( ; cf ; I):
ECONOM ICS OF PUBLIC GOOD PROVISION:
[G(pf ; px ; I)(pf
+(1
cf )
B
1
M0 ]
) [G(pf ; px ; I)(pf
"
N
X
31
#1
@G(pf ; px ; I)
(pf cf ) + G(pf ; px ; I)
@pf
"N
# " P
#
N
X
@ i=1 V (pf ; px ; I : C)
M0 ]
V (pf ; px ; I : C) + B V0
@pf
i=1
V (pf ; px ; I : C) + B
i=1
cf )
AUDITING, OUTSOURCING AND BRIBERY
B
V0
For B( ; cf ; I):
[G(pf ; px ; I)(pf
+(1
cf )
) [G(pf ; px ; I)(pf
B
cf )
1
M0 ]
B
( 1)
"
M0 ]
"
N
X
N
X
V (pf ; px ; I : C) + B
V0
i=1
V (pf ; px ; I : C) + B
V0
i=1
#
#1
(C3)
=0
Using C2 and C3, it follows that
@G(pf ; px ; I)
(pf
@pf
cf ) + G(pf ; px ; I)
G(pf ; px ; I)(pf cf ) B M0
PN
V0
i=1 V (pf ; px ; I : C) + B
=
=
N
X
@V (pf ; px ; I : C)
@pf
i=1
1
(1
)
!"
#
G(pf ; px ; I)(pf cf ) B M0
(C4)
PN
V0
i=1 V (pf ; px ; I : C) + B
Hence,
@G(pf ; px ; I)
(pf
@pf
cf ) + G(pf ; px ; I)
B
1
=
=
(1
(1
)
N
X
@V (pf ; px ; I : C)
@pf
i=1
) [G(pf ; px ; I)(pf
cf )
!
M0 ] +
(C5)
"
V0
N
X
V (pf ; px ; I : C)
i=1
8. APPENDIX D
Comparative Statics:
Note M0 = 0 for I > I0 and @M ( ; ; cf ; )=@pf = f@G(pf ; px ; I)=@pf g (pf
(D1)
(1
)
@M (:)
d
@pf
@M (:)
d + (1
@pf
)
cf ) + G(pf ; px ; I) so:
N
X
@ 2 M (:)
@ 2 V (pf ; px ; I : C)
dpf +
dpf = 0
@pf @pf
@pf @pf
i=1
!#
(C2)
=
0
32
GERVAN FEARON
M ( ; ; cf ; ) d + (1
"
@M ( ; ; cf ; )
)
dpf + V0
@pf
)M ( ; ; cf ; )d + (1
N
X
@V (pf ; px ; I : C)
dpf
@pf
i=1
Bd
N
X
!#
V (pf ; px ; I : C)
i=1
d(D2)
dB = 0
For dpf =fd g and dB=fd g:
(D3)
"
(1
)
(1
)
h
j A j=
PN
i=1
(1
@ 2 V (pf ;px ;I:C)
@pf @pf
@ 2 M ( ; ;cf ; )
@pf @pf
@M ( ; ;cf ; )
@pf
)
+
PN
@ 2 V (pf ;px ;I:C)
i=1
f @pf
PN @V @p
(pf ;px ;I:C)
i=1
@pf
@ 2 M ( ; ;cf ; )
@pf @pf
< 0.
+
PN
@ 2 V (pf ;px ;I:C)
i=1
@pf @pf
This implies that (1
#
0
) <
i
dpf
d
dB
d
=
"
(1
(:)
(1
) @M
@pf
)M ( ; ; cf ; ) + B
, where: j A j> 0 if (1
erwise, j A j< 0.
hP
N @ 2 V (pf ;px ;I:C)
i=1
@pf @pf
ih
#
@ 2 M ( ; ;cf ; )
@pf @pf
)
@ 2 M ( ; ;cf ; )
@pf @pf
i
1
+
. Oth-
(:)
(1
) @M
dpf
@pf
=
d
jAj
(D4)
dB
d
=
f (1
+
n
(1
n
)M ( ; ; cf ; ) + Bg (1
(:)
) @M
@pf
on
(1
)
)
@ 2 M ( ; ;cf ; )
@pf @pf
jAj
@M ( ; ;cf ; )
@pf
jAj
PN
i=1
+
PN
@ 2 V (pf ;px ;I:C)
i=1
@pf @pf
@V (pf ;px ;I:C)
@pf
o
o
(D5)
For dpf =fd g and dB=fd g:
(D6)
"
(1
(1
(D7)
)
)
@ 2 M ( ; ;cf ; )
@pf @pf
@M ( ; ;cf ; )
@pf
+
PN
@ 2 V (pf ;px ;I:C)
i=1
f @pf
PN @V @p
(pf ;px ;I:C)
i=1
@pf
0
dpf
=
d
#
dpf
d
dB
d
2 @M (:)
@pf
jAj
2
=4
@M (:)
@pf
h
M ( ; ; cf ; ) d + V 0
PN
i=1 V (pf ; px ; I : C)
3
i 5
ECONOM ICS OF PUBLIC GOOD PROVISION:
dB
d
h
=
(1
)
h
(
=
@ 2 M ( ; ;cf ; )
@pf @pf
in
2 @M (:)
(1
@pf
+
2 @M (:)
@pf
in
(1
PN
i=1
jAj
"
)
@ 2 V (pf ;px ;I:C)
i=1
@pf @pf
@M ( ; ;cf ; )
@pf
)
M ( ; ; cf ; ) + V 0
h
PN
N
X
AUDITING, OUTSOURCING AND BRIBERY
in
h
M ( ; ; cf ; ) d + V 0
jAj
!#)
V (pf ; px ; I : C)
i=1
PN
@M ( ; ;cf ; )
@pf
i=1
jAj
PN
i=1 V (pf ; px ; I : C)
o
@V (pf ;px ;I:C)
@pf
33
+
o
@V (pf ;px ;I:C)
@pf
9. APPENDIX E
Outsourcing Example using a Cobb-Douglas Function
PN
Assuming a Cobb-Douglas utility function (i.e., W (x; G; B) =
i=1
x1i
G + B , where: x =
(x1 ; x2 ; :::; xN )), it can be shown that G(pf ; px ; I) = fN I=pf g and xi (pf ; px ; I) = (1
@G(pf ; px ; I)
(pf cf ) + G(pf ; px ; I) =
@pf
(E1)
NI
p2f
!
)fI=px g so
cf
and
N
X
@V (pf ; px ; I : C)
=
@pf
i=1
(E2)
N
X
(1
1
N px (1
)
)
1
I
i=1
Using C4, E1 and E2, it can be therefore shown that pf ( ; ; cf ; I) =
)] 1
1
pf
1
(1
)
1
N1
px [ (1
1
cf1
. Clearly, @pf ( ; ; cf ; I)=@ > 0 and @pf ( ; ; cf ; I)=@ < 0.
The …rm’s payo¤ is:
(E3) M ( ; ; cf ; ) =
For I
(
NI
c
(c
cf )
[G(pf ; px ; I)(pf
cf )
M0 ]
I0 :
c( ; ; c ; )]
Ecf [M
f
= G(pf ; px ; I)(pf cf )
8
<
=
N I 1 (1
) 1 f [1
:
h
for I < I0
i
V
(p
;
p
;
I
:
C)
for
I I0
f
x
i=1
PN
V0
]g
1
1
N
1
px
1
Zc
c
(cf ) 1
f (cf )dcf
9
=
;
(E4)
io
(D8)
+
34
GERVAN FEARON
h
NI 1
c( ; ; cf ; )] =
It is helpful to write Ecf [M
(1
if 1
)
d1 f1= g
.
Let gF ( ) = 1
d1 f1= g
@ 2 gF ( )=@ @ < 0 with gF ( ) approaching 1 as
d1 f [1
(1
)
]g 1
1
i
c( ; ; cf ; )]
and Ecf [M
and it is clear that @gF ( )=@
0
> 0 and
approaches in…nity.
The department’s expected payo¤ is:
N
X
Ecf [V (pf ( ; ; cf ; I); px ; I : C)] + B( ; cf ; I)
i=1
=
N
X
Ecf [V (pf ( ; ; cf ; I); px ; I : C)] +
i=1
=
(1
)
N
X
(
(1
(E5)
) [M ( ; ; cf ; )
Ecf [V (pf ( ; ; cf ; I); px ; I : C)] + (1
M0 ] +
n
PN
where:
i=1 Ecf [V (pf ( ; ; cf ; I); px ; I : C)] = (1
C)]+ (1
For
N
X
i=1
i=1
PN
i=1
PN
i=1
V0 so (1
)
PN
i=1
px 1 I
N1
1
V0
!#)
V (pf ; px ; I : C)
i=1
8
o <Zc
:
(cf ) 1
f (cf )dcf
c
V0 implies (1
)
Ecf [V (pf ( ; ; cf ; I); px ; I : C)]+ (1
PN
i=1
9
=
;
[ (1
)] 1
I
Ecf [V (pf ( ; ; cf ; I); px ; I :
)M ( ; ; cf ; )
V0 .
Ecf [V (pf ( ; ; cf ; I); px ; I : C)] + M ( ; ; cf ; )
Ecf [V (pf ( ; ; cf ; I); px ; I : C)] + M ( ; ; cf ; ):
n
Ecf [V (pf ( ; ; cf ; I); px ; I : C)] + M ( ; ; cf ; ) = (1
+
It follows that we can write
C)]
)
Ecf [V (pf ( ; ; cf ; I); px ; I : C)]+ B( ; cf ; I)
)M ( ; ; cf ; )+ V0
)V0 and
(1
PN
N
X
)M ( ; ; cf ; ) + V0
i=1
It follows that
"
V0 implies 1
(1
d1 f1= g
PN
i=1
)
8
<
NI 1
:
)
1
(1
N1
)
1
px 1 I
f [1
Ecf [V (pf ( ; ; cf ; I); px ; I : C)] = d2 f [1
. Let gbD ( ) = 1 d1 f1= g(1
and @ 2 gF ( )=@ @ < 0 with gF ( ) approaching 1 as
)
8
o <Zc
:
f (cf )dcf
(cf ) 1
c
]g 1
1
px 1
N1
Zc
9
=
;
[ (1
(cf ) 1
c
]g 1
.
PN
i=1
and it is clear that @gF ( )=@ > 0
approaches in…nity.
Using Mathematica, numeric analysis was conducted for the Cobb-Douglas based utility function.
= 0:2; Ecf [cf ] = 25; px = 1; N = 100; and I = 20000.
f (cf )dcf
I (E6)
9
=
;
Ecf [V (pf ( ; ; cf ; I); px ; I :
10. APPENDIX F
Parameter values include setting
)] 1
ECONOM ICS OF PUBLIC GOOD PROVISION:
AUDITING, OUTSOURCING AND BRIBERY
1HWC
0.5
0.4
0.3
0.2
0.1
2
E
bribe
with
utility
Expected
.5
Figure 4. Department’s expected payo¤ including bribe
1HNM
0.5
0.4
0.3
0.2
0.1
2
E
bribe
less
Profits
Expected
.5
Figure 5. Firm’s expected pro…t adjusted for bribe
35
36
GERVAN FEARON
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Current address : York University, Toronto, Ontario, Canada.
This paper was primarily written while on sabbatical in the Department of Economics at the University of
Washington, Seattle, Washington, USA.
E-mail address : gfearon@yorku.ca, gfearon@u.washington.edu