D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124 111
of beating its rival due to the favorable discrimination, whilst the more efficient firm must bribe more to counteract this effect. This result has strong implications for the efficiency of
a competitive bribery procedure. The model and its solution are presented in Section 2. Section 3 examines allocation
efficiency and extends the generality of previous results in the literature. In Section 4, we consider the case in which the amount of discrimination is endogenously selected by an
income-maximizing bribee. Section 5 briefly summarizes the findings.
2. The model
Two firms compete for a government contract by bribing a corrupt official. The value to firm i=1, 2 of winning is private information and is represented by v
i
. It is common knowledge, however, that v
i
is drawn from a uniform probability distribution with supports [v
i
, ¯ v
i
]; notice that the supports of each player’s distribution are allowed to be different. Define D
i
= ¯ v
i
− v
i
. We adopt the interpretation that the higher the value of winning, the more cost-effective the firm is in carrying out the contract. Each firm gives an irretrievable
‘bribe’, B
i
≥ 0, to the corrupt official in the hope of being designated the winner. The rule
adopted by the official is that Firm 1 wins, iff αB
1
B
2
, where α0 is common knowledge; if α6=1 then the game is unfair.
3
Lien 1990 attributes this unfairness to the fact that the official may be on friendly terms with one of the firms at the outset. Another reasonable
interpretation is that one firm may possess an incumbent advantage. We further assume that the cost of bribing the official is represented by the function φB
γ i
, where φ, γ 0.
4
It is conceivable that the authorities may indirectly be able to affect this cost function by introducing rules or controls which make the purchase of influence more expensive. An
increase in γ can be regarded as adopting a stronger stance against large bribes than small, while raising φ may reflect an attempt to shift the nature of bribery from cash to in-kind
transfers.
5
By varying these two parameters, the marginal cost of bribery can be changed independent of its average cost.
Defining x
1
=αB
1
and x
2
=B
2
as the effective bribe levels, the expected payoff of player i
can be written as:
6
π
1
x
1
= Prx
1
x
2
v
1
− φ x
1
α
γ
, π
2
x
2
= Prx
2
≥ x
1
v
2
− φx
γ 2
. 1
3
Note that Firm 2 wins in the event of a tie. Since the equilibrium bribe functions are continuous, this assumption is innocuous, and is made for notational simplicity.
4
Using this exponential cost function makes for a wider range of possible interpretations of the model. Instead of thinking as bribes as monetary in nature, firms may more generally have to exert some kind of sunk effort in
order to win the contract. The cost function for this effort need not be linear.
5
A referee suggested this interpretation.
6
From 1, an alternative interpretation of α becomes apparent. If x
i
represent actual rather than effective bribe levels, then α can be seen as depicting the relative moral cost of bribery between the two players; if α1 then
the moral cost of bribery activity is lower higher for Firm 1 relative to Firm 2. Thus, unfairness and asymmetry are different sides of the same coin in this respect. In the first interpretation, α measures the relative benefit to
the firms in terms of an increased effective bribe level of increasing the level of the bribe, whereas the latter interpretation focuses explicitly on the cost of such an increase. On the moral costs associated with bribery, see
Rose-Ackerman 1975.
112 D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124
Assume that the equilibrium effective bribe functions, G
i
v
i
=x
i
, are continuous and strictly increasing except possibly at zero, with upper and lower supports ¯
x
i
and x
i
, and with inverse g
i
x
i
=v
i
.
7
Using the fact that the v
i
are distributed uniformly, we can write: π
1
x
1
= g
2
x
1
− v
2
D
2
v
1
− φ x
1
α
γ
, π
2
x
2
= g
1
x
2
− v
1
D
1
v
2
− φx
2 γ
. 2
The first-order conditions for an interior maximum of Eq. 2 are: g
′ 2
x
1
v
1
D
2
− φγ
α
γ
x
γ − 1
1
= 0,
g
′ 1
x
2
v
2
D
1
− φγ x
2 γ −
1
= 0.
3 Inserting v
i
=g
i
x
i
into Eq. 3 yields a system of two differential equations: g
′ 2
xg
1
x = φD
2
γ α
γ
x
γ − 1
, g
′ 1
xg
2
x = φD
1
γ x
γ − 1
. 4
In Appendix A, it is shown that the unique solution to the system in Eq. 4 is: g
1
x = v
l 1
Z
11+λ
, g
2
x = ¯ v
2
v
l 1
¯ v
1 λ
Z
λ 1+λ
, where v
l 1
= v
1
iff ¯
v
2
v
1
¯ v
1 λ
≥ v
2
, otherwise
v
l 1
= ¯ v
1
v
2
¯ v
2 1λ
, Z ≡
1 + φD
1
1 + λx
γ
¯ v
1
¯ v
2
v
1
¯ v
1 1+λ
, λ ≡
D
2
D
1
α
γ
. 5
3. Allocation efficiency
