118 D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124
run an unfair contest i.e. set α such that λ=λ
m
¯v in Fig. 3, not α=1, which would imply λ = ¯
v . Lien’s symmetric model is incapable of capturing this fact.
4. Endogenous discrimination
In the analysis so far, we have assumed that the degree of discrimination is exogenously determined. If the bribee is not regulated, then it seems reasonable to assume that he will set
a level of discrimination in accordance with his private goal of income bribe maximization. The question we ask in this section is which α is it that maximizes the expected sum of
bribes? It is this degree of discrimination which we would expect to arise endogenously from an unregulated bribery procedure.
To aid the calculation, assume again that v
1
= v
2
= 0. From Eq. 5, one can then
determine that x
1
= v
1
¯ v
1 1+λγ
¯ v
2
φ 1 + λ
1γ
, x
2
= v
2
¯ v
2 1+λγ λ
¯ v
2
φ 1 + λ
1γ
. Recalling that B
1
=x
1
α, and B
2
=x
2
, the expected bribes can be calculated as follows: E B
1
= ¯
v
2
φ 1 + λ
1γ
1 α
Z
¯ v
1
v
1
¯ v
1 1+λγ
1 ¯
v
1
dv
1
, E B
2
= ¯
v
2
φ 1 + λ
1γ
Z
¯ v
2
v
2
¯ v
2 1+λλγ
1 ¯
v
2
dv
2
and hence, EB
1
+ EB
2
= ¯v
1
φ
1γ
λ 1 + λ
1γ
γ 1 + λ + γ
+ ¯
vλ
1γ
γ λ 1 + λ + γ λ
. 10
If granted the freedom to choose, a completely selfish bribee would select the value of α
i.e. the degree of discrimination which maximizes the expression in Eq. 10. Re- call that λ ≡ ¯vα
γ
. We have written Eq. 10 as a function of λ for analytical conve- nience; to find the bribe-maximizing parameter setting, we optimize Eq. 10 with respect
to λ and then recover the implied value of α. When ¯v = 1, Eq. 10 is symmetric in the sense that if λ
∗
maximizes this expression, then so does 1λ
∗
. Hence in searching for a maximum of Eq. 10, for the case ¯v = 1, it is sufficient to focus on the interval
λ∈ [0, 1].
The following proposition indicates the setting of the discrimination parameter which maximizes the income of the bribee. Let α
∗
denote the bribe-maximizing value of α.
Proposition.
1. Assume ¯v = 1, and let ˜ γ
denote the unique real root of the equation 8γ +7γ
2
+5γ
3
−4=0 ˜
γ ≈ 0.358. If γ ≥ ˜
γ , then α
∗
=1. If γ ˜ γ
, then two solutions exist, α
∗
and 1α
∗
where α
∗
6=1. 2. Assume ¯v 1. Then α
∗
1.
D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124 119
Proof.
See Appendix B. The first case dealt with in the proposition concerns ex ante identical bribing firms. The
bribe-maximizing level of discrimination depends upon the parameter in the cost function γ . When this is sufficiently small below ˜
γ , the selfish bribee finds it profitable to discriminate
in favor of one of the competitors. The direction of discrimination does not matter to the bribee, since the expected level of bribes in Eq. 10 attains the same value for α and 1α;
naturally, these parameters give the same magnitude of discrimination, and since the bribing firms are expected to be identical they lead to a common ex ante bribe level. This case then
is characterized by a bribee who discriminates between expectedly identical firms; the intuition is as follows. When γ is small, the cost function possesses a quickly diminishing
marginal cost of bribing once a certain level is reached. Hence, additional bribes can be given which cost the bribing firm quite little. Introducing discrimination stimulates one firm
to increase its bribe, whilst that of the other falls by only a small amount relative to the situation without discrimination. When the bribing firms are ex ante identical and the cost
parameter γ is sufficiently large at least ˜
γ , then this effect is no longer present and the
bribee finds it profitable to run a fair contest. Case 2 in the proposition deals with asymmetric bribing firms, in which Firm 1 is expected
to be most efficient. Notice that the income-maximizing bribee discriminates against this firm in order to ‘even up’ the uneven contest and encourage Firm 2 to bribe. This is the
opposite direction of discrimination to that which a social planner would choose if he were
interested in minimizing the welfare loss as a consequence of erroneous selection. In Fig. 3, notice that the locus of points which depicts the minimal welfare loss from wrong selection,
λ
m
¯ v
, lies in the region where α1. Hence, a planner would set α1 and would discriminate in favor of the expectedly more efficient firm. However, a selfish bribee discriminates in the
opposite direction.
5. Summary
