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Allocation efficiency in a competitive bribery game
Journal of Economic Behavior & Organization Vol. 42 (2000) 109–124
Allocation efficiency in a competitive bribery game Derek J. Clark a,b,∗ , Christian Riis a,c a
Department of Economics, NFH, University of Tromsø, N-9037 Tromsø, Norway Norwegian School of Management, BI Tromsø, P.O. Box 973, N-9260 Tromsø, Norway Department of Economics, Norwegian School of Management BI, P.O. Box 580, N-1301 Sandvika, Norway b
c
Received 16 June 1999; received in revised form 14 September 1999; accepted 10 November 1999
Abstract We consider the selection properties of a competitive bribery model in the presence of two types of asymmetry: unevenness between the competitors and unfairness in the contest rules. Only under very special conditions does the benchmark model yield allocation efficiency; in other cases, the effect on allocation efficiency of making the contest more unfair is ambiguous and parameter specific. We present conditions under which each result obtains. Our results indicate that it is socially optimal to run an unfair contest in order to redress the allocation inefficiency introduced when contestants are asymmetric. We show, however, that a selfish, income-maximizing bribee will discriminate in the opposite direction to that which society would prefer. ©2000 Elsevier Science B.V. All rights reserved. JEL classification: O17; C72 Keywords: Bribery; Allocation efficiency; Discrimination
1. Introduction In some countries, it is not uncommon to find government contracts awarded to firms which succeed in bribing a corrupt official. While such corruption has obvious costs, Lien (1986, 1990) has investigated a potential benefit of a competitive bribery procedure as a mechanism for ensuring ex post allocation efficiency; here allocation efficiency is narrowly defined as the lowest cost firm winning the contract. Two opposing models have been used to model this bribery procedure. Beck and Maher (1986) and Lien (1986) consider the case in which unsuccessful firms have their bribes refunded; if the highest bribe wins, then this procedure is a first-price sealed bid auction which, as is well known, can possess an ∗ Corresponding author. Fax: +47-77646021. E-mail addresses: [email protected] (D.J. Clark), [email protected] (C. Riis).
0167-2681/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 0 0 ) 0 0 0 7 7 - 9
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equilibrium exhibiting ex post allocation efficiency. In Lien (1986, 1990), an alternative set up is used in which all firms, successful or not, forfeit their bribes. Whilst, we prefer this latter type of model for analyzing corrupt activity, we show that Lien’s assumption that firms are ex ante identical is not innocuous. In this paper, we look at the role which ‘asymmetry’ plays in the competitive bribery procedure. 1 Following practice in the contest literature (see O’Keeffe et al., 1984), we can define two dimensions of asymmetry: (i) the game may be ‘uneven’ due to the fact that the contestants are expected to be different; (ii) an ‘unfair’ contest can arise when the players are treated differently. The game which we specify below, incorporates both of these elements and investigates the allocation properties which arise in equilibrium. Lien (1986) confines attention to an even and fair contest, whilst Lien (1990) considers an even, unfair contest. The unique equilibrium of the former is symmetric and hence allocation efficiency is ensured. In the latter paper, one firm is handicapped and must bribe proportionally more than the opponent in order to win; the equilibrium of this game is not symmetric which leads to the result that ‘the economy may suffer allocation inefficiencies from competitive bribery procedures, whenever there is some degree of discrimination’ (Lien, 1990; 157). In addition, Lien (1990) also shows that increasing the degree of discrimination makes matters worse (i.e. an increase in the probability that the most efficient firm is defeated). Intuitively, these results must obtain since firms are assumed to be ex ante identical and hence a symmetric equilibrium must result when there is no discrimination among the bribers. Thus, any move away from the symmetry of the no discrimination case must worsen the selection properties of the bribery procedure. When all bribes are forfeited, so that losers also bear the full cost of their bribe, one departs from the traditional first-price sealed bid auction procedure and hence, there is in general no mechanism to ensure allocation efficiency. We show that even a fair bribery procedure leads to inefficiency, except in the very rare case that firms are identical ex ante. Hence, changing the rules of the game may or may not enhance allocation efficiency, depending on the specific model parameters. Accordingly, increased discrimination in a competitive bribery procedure may well enhance its selection properties, contradicting Lien’s conclusion. In the literature, there are two views on corrupt activity. 2 Moralists argue that bribery has no positive effects, whilst reformists take the view that this mechanism may have some beneficial effects. Among these is allocation efficiency since the most efficient firm should be able to afford the largest bribe and win the contest. We demonstrate that the role of the bribee is critical in assessing the selection properties of the bribery procedure. It seems likely that the bribee is a selfish actor who desires to maximize his own income (in the form of bribe revenue). We show that this will lead to a direction of discrimination in exactly the opposite direction to that which society would prefer. Whilst, society would wish for the more efficient firm to be favored — and hence have a greater chance of winning — the bribee has an incentive to discriminate against this firm in order to even out the contest. This gives both firms an incentive to bribe more; the less efficient firm now has a real possibility 1 The model framework which we use is an all-pay auction under incomplete information; this possesses applications which go beyond bribery. In Clark and Riis (1996), we consider a firm which uses this type of mechanism to decide which worker to promote to the next level in its hierarchy. 2 See McMullan (1961), Leys (1965) and Heidenheimer (1970).
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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 vi . It is common knowledge, however, that vi 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 Di =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’, Bi ≥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 αB1 >B2 , where α>0 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 γ that the cost of bribing the official is represented by the function φBi , 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 x1 =αB1 and x2 =B2 as the effective bribe levels, the expected payoff of player i can be written as: 6 x γ 1 γ π1 (x1 ) = Pr(x1 > x2 )v1 − φ , π2 (x2 ) = Pr(x2 ≥ x1 )v2 − φx2 . (1) α 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 represent actual (rather than effective) bribe i levels, then α can be seen as depicting the relative moral cost of bribery between the two players; if α>(<)1 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).
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Assume that the equilibrium effective bribe functions, Gi (vi )=xi , are continuous and strictly increasing (except possibly at zero), with upper and lower supports x¯i and x i , and with inverse gi (xi )=vi . 7 Using the fact that the vi are distributed uniformly, we can write: x γ g2 (x1 ) − v 2 g1 (x2 ) − v 1 1 v1 − φ , π2 (x2 ) = v2 − φ(x2 )γ . (2) π1 (x1 ) = D2 α D1 The first-order conditions for an interior maximum of Eq. (2) are: g10 (x2 )v2 g20 (x1 )v1 φγ γ −1 − = 0, − φγ (x2 )γ −1 = 0. x 1 D2 αγ D1
(3)
Inserting vi =gi (xi ) into Eq. (3) yields a system of two differential equations: g20 (x)g1 (x) =
φD2 γ γ −1 x , αγ
g10 (x)g2 (x) = φD1 γ x γ −1 .
In Appendix A, it is shown that the unique solution to the system in Eq. (4) is: ! l λ v 1 g2 (x) = v¯2 Z λ/1+λ , where v1l = v 1 g1 (x) = v1l Z 1/1+λ , v¯1 λ 1/λ v1 v2 l iff v¯2 ≥ v 2 , otherwise v1 = v¯1 , v¯1 v¯2 D2 φD1 (1 + λ)x γ , λ≡ . Z ≡1+ 1+λ D1 α γ v¯1 v¯2 (v1 /v¯1 )
(4)
(5)
3. Allocation efficiency We are now in a position to look at allocation efficiency in the competitive bribery procedure. Recalling the definition of gi (xi )=vi , it can easily be calculated from Eq. (5) that: λ v1 v2 (>, =, <) . (6) x1 (>, =, <)x2 iff v¯1 v¯2 Eq. (6) depicts the role of the different parameters in determining the underlying selection properties of this asymmetric competitive bribery model. We can isolate several factors (or combinations of these) which influence selection: differences in the means and variances of the valuation distributions, the fairness of the competition, as well as the cost of bribery as depicted by γ . Notice that the shift parameter in the cost function, φ, does not affect the selection properties of the bribery procedure. Consider first the completely symmetric case in which v¯1 = v¯2 , v 1 = v 2 , α = 1; Eq. (6) indicates that the bribery game allocates efficiently since Firm i wins if and only if it has a higher valuation than Firm j. In other cases, allocation will not always be efficient. In the remainder of the paper we assume, with no loss of generality, that v¯1 ≥ v¯2 . 7
This assumption is fulfilled in equilibrium.
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Fig. 1. Erroneous selection; λ>1.
The locus of valuation combinations (v1 , v2 ) which gives x1 =x2 and is defined by v1 = v¯1
v2 v¯2
1/λ (7)
is graphed as 0e in Fig. 1, illustrating the case where λ>1; above and to the left of this line 8 , we have x2 >x1 (2 wins) below and to the right x1 >x2 (1 wins), and exactly on 0e, x1 =x2 (by assumption 2 wins). Only when the function in Eq. (7) coincides exactly with a 45◦ line from the origin in the relevant range (from v i to v¯i ), do we have efficiency in allocation. When this is not the case, the firm with the lower valuation may, on occasion, be awarded the contract. Notice that whenever v¯1 6= v¯2 , there are no settings of the other parameters which ensure complete efficiency in allocation in this model. 9 Area ‘abced’ in Fig. 1 represents the combinations of v1 and v2 which yield erroneous selection (i.e. that 2 is chosen when v1 >v2 ). When α is small, implying that the contest is heavily biased in favor of Firm 2, this area is large. Firm 1, being at a disadvantage, gives small bribes or, if v1 is sufficiently small, does not bribe at all. Hence he often loses to less efficient opponents. However, Firm 2, realizing that the opponent behaves in this way, acts less aggressively since it is likely that it wins the contract with a lower bribe (than if Firm 1 were at less of a disadvantage). Consequently, only if v1 turns out to be very large does Firm 1 actually win the game. As α is increased and Firm 1 is at less of a disadvantage, the area ‘abced’ shrinks (while 0e maintains the same start and end points) and the probability of wrong selection falls. Raising α sufficiently will eventually lead to a case in which 1>λ; 0h in Fig. 2 represents the graph of Eq. (7) for this situation. As before, above (below) 0h, we have x2 >(<)x1 . Two types of erroneous selection are now possible: (i) Firm 1 is chosen when v2 >v1 , represented in Fig. 2 by area ‘jkn’; (ii) Firm 2 is chosen when v1 >v2 which is the area ‘nrh’ in Fig. 2. Define v ∗ = [v¯2 /v¯1λ ]1/1−λ as the point at which 0h crosses the 45◦ line. If v 1 and v 2 are 0e is a straight line when λ=1. In a different setting, Clark and Riis (1996) investigate a prize structure which would lead to allocation efficiency in this case. 8 9
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Fig. 2. Erroneous selection; λ>2.
below v∗ , then both types of error are made. 10 Otherwise, only a type (ii) error occurs. For high values of v2 , a type (ii) error may occur; Firm 1 tends to underestimate the opponent, expects it to give a small bribe, and consequently loses if v2 is large. For low values of v2 , a type (i) error can occur, so that Firm 1 wins more often than allocation efficiency would dictate; Firm 2 is at a disadvantage relative to Firm 1, since α is large, and in addition has a low valuation, hence it bribes very little or not at all. Raising α increases the probability of a type (i) but simultaneously reduces that of a type (ii) error. When α is sufficiently large, then further increases in this parameter will raise the probability of choosing the least efficient firm, as the increase in the chance of a type (i) error outweighs the fall in the probability that a type (ii) error will be made. Let us now consider the result of Lien (1990) that making an unfair contest (α6=1) more unfair will increase the probability of choosing a less efficient contractor. We have two cases to consider: λ>1 and 1>λ. 3.1. λ>1 Take first the case represented by Fig. 1 (λ>1), where we have already noted that increasing α reduces the probability of wrong selection. The point is, however, that whether this increase in α represents a move to a more or a less fair contest is parameter specific. If D1 ≥D2 , then λ>1 implies 1>α and hence an increase in α makes the contest more fair and simultaneously reduces the probability of wrong selection. This is consistent with Lien (1990) and extends his result to a wider range of possibilities. 11 However, if D2 >D1 , it is permissible within Fig. 1 (where λ>1) to have α>1. Here an increase in α makes the contest less fair (i.e. more biased towards Firm 1) and reduces the probability of erroneous selection. This is the opposite of Lien’s result. 3.2. 1>λ Similarly, Lien’s result and its exact opposite can be generated when Fig. 2 (1>λ) is relevant. Assuming for illustrational simplicity that v 1 = v 2 = 0, and writing v¯ ≡ v¯2 /v¯1 , 10 Since the derivative ∂v /∂v from Eq. (7) tends to infinity as v tends to zero in this case, 0h will cross the 45◦ 2 1 1 line. 11 Lien’s assumption of ex ante symmetry obtains if D =D . 1 2
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the probability that the higher cost firm actually wins the contract in this case is θ =η+τ where, η=
1 − λ 1+λ/1−λ v¯ 2(1 + λ)
τ=
λ v¯ − + η. 1+λ 2
Here, the total probability that a mistake is made which is expressed by θ, is the sum of the probabilities of making a type (i) error, denoted by η, and a type (ii) mistake, τ . The comparative static effect of a change in α is given by −λγ 1+λ ∂θ 1+λ/1−λ = ln v ¯ . (8) 1 + 2v −1 + ∂α 1−λ α(1 + λ)2 Since 1 ≥ v, ¯ the term in square brackets in Eq. (8) is negative, making the sign of the whole expression ambiguous. It is straightforward to determine that ∂θ /∂α>0 as v¯ → 1, whilst ∂θ/∂α<0 as v¯ → 0. As we approach the situation in which the firms are approximately identical ex ante, increasing α makes mistaken selection more likely. Again this is Lien’s result. However, when the firms are expected to be very dissimilar (v¯ low), the opposite result obtains in which the probability of erroneous selection is decreasing in α. Intuitively, Firm 1 is expected to be much superior to Firm 2 in this case, so reducing the probability of mistakenly choosing the higher cost firm implies making the contest more biased toward Firm 1. When v¯ takes intermediate values, the effect of changing α on θ is not monotonic. 3.3. Efficiency loss due to erroneous selection Naturally, one is concerned with the efficiency loss which making a selection error entails, and not the probability of such an occurrence. 12 Normalizing v¯1 = 1, retaining the assumption that v¯i = 0 (i=1, 2), and writing v ∗ = v¯ 1/1−λ if 1>λ, and v∗ =0, otherwise (where v∗ is the intersection point between the 45◦ line and the curve at which x1 =x2 given in Eq. (7)), the expected efficiency loss in the competitive bribery procedure can be calculated as Z v¯ Z (v2 /v) Z v ∗ Z v2 ¯ 1/λ (v − v ) (v2 − v1 ) 1 2 dv1 dv2 + dv1 dv2 = 1/λ ∗ v ¯ v ¯ 0 (v2 /v) ¯ v v2 Z v¯ Z v2 vλ ¯ v¯ 2 λ (v2 − v1 ) dv1 dv2 = − + . = v¯ 2(2 + λ) 2λ + 1 6 ¯ 1/λ 0 (v2 /v) 12 As in Lien (1990), we focus in this section solely on losses due to erroneous selection. In our model, however, losses can occur in the transfer of bribes since their value may be different to giver and receiver. This deadweight loss complicates the calculations, whilst adding little to the analysis; it is, therefore, omitted in what follows. An expression for the expected deadweight loss of the bribery procedure is straightforward to calculate, and is available from the authors on request.
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Fig. 3. Efficiency loss due to erroneous selection.
We can illustrate the efficiency effects within a (v, ¯ λ) diagram (recall that v¯ ≡ v¯2 /v¯1 ); consider Fig. 3 which depicts iso-social-cost curves which are concentric to the global minimum at (1, 1). The further away from this point the iso-social-cost curve lies, the larger the efficiency loss associated with the competitive bribery procedure. Since we have defined λ = v/α ¯ γ , points to the right (left) of the 45◦ line reflect α<(>)1, whilst along the 45◦ line, we have α=1. Given v, ¯ the λ which minimizes the social cost of erroneous selection is unique and determined by 1 v¯ ∂ = − = 0. ∂λ (2 + λ)2 (2λ + 1)2
(9)
¯ in Fig. 3 and always lies above the 45◦ The locus of points satisfying Eq. (9) is denoted λm (v) m line. To the right of λ (v), ¯ is monotonically increasing in λ, and thus, the efficiency loss ¯ the efficiency loss is monotonically is monotonically decreasing in α. To the left of λm (v), decreasing in λ (increasing in α). ¯ Three areas are of interest in this figure, represented by Q (the area bounded by λm (v) ¯ and the 45◦ line) and S (below and to the right of the 45◦ and v¯ = 1), R (bounded by λm (v) line). The comparative static effect on the efficiency loss in selection of changes in α and γ are given in Table 1. Notice from Eq. (1) that the firms’ cost functions are identical up to a multiplicative term 1/α γ which indicates the cost of bribery to Firm 1 relative to Firm 2; when 1/α γ >1, Firm 1 has a larger marginal (and average) cost relative to Firm 2 for an identical bribe level. The intuition behind the results in Table 1 can be gleaned by looking at this factor, which we Table 1 The comparative static effect on the efficiency loss in selection of changes in α and γ
Q R S
∂/∂α
∂/∂γ
+ − −
+ − +
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shall denote by f. The assumptions which underlie in Fig. 3 imply that λ = vf ¯ . Notice that in areas Q and R, we have that α>1, whilst 1>α in S. Suppose we fix v¯ = V in Fig. 3, and start from a point where λ is low in Q, and hence f is small, implying that Firm 1 has a very low marginal cost of bribery relative to Firm 2. This leads to Firm 2 bidding little (if at all) and losing often to inferior Firm 1 types (a type (i) error occurs with a large probability). Knowing this, Firm 1 also bribes a small amount, but enough to beat almost all Firm 2 types. 13 Suppose now that we move horizontally from this point from left to right in Fig. 3 for fixed V. This movement can be accomplished by reducing α and/or reducing γ . Firm 2 is now at less of a cost disadvantage, and consequently, bids more and loses less often to inferior Firm 1 types. At the same time, it is also possible for a very efficient Firm 2 to beat inferior Firm 1 types, due to the fact that Firm 1 may tend to ‘slack off’ because it has an initial advantage in its lower relative cost. These two effects pull in opposite directions, but the net effect in area Q is that the efficiency loss decreases as α and γ fall. Reducing α and/or γ sufficiently to reach the point w in Fig. 3 gives the smallest welfare loss for v¯ = V due to mistaken selection in the competitive bribery procedure. By definition, any horizontal move away from w leads to an iso-social cost curve at a lower level of welfare. Thus, a further reduction in α and/or γ (and a move to the right from w into R) increases the efficiency loss; in Fig. 2, area ‘jkn’ shrinks by less than the enlargement in area ‘nrh’. As the relative marginal cost to Firm 1 becomes larger and larger, it offers successively smaller bribes and loses more often to inferior Firm 2 types. Simultaneously, Firm 1 beats more efficient opponents less and less often (until f becomes so large that this sort of error does not occur). Moving further to the right along v¯ = V into area S increases the efficiency loss even more. (Note that in area S, since 1>α, a horizontal move from left to right is achieved by reducing α and/or increasing γ ). One interpretation of increasing γ is as a toughening of the stance against large bribes. As indicated in Table 1, this reduces the expected efficiency loss in area R. However, in areas Q and S, the efficiency loss increases following this tougher stance; any gains due to a reduced level of bribery activity must thus be weighed up against the losses due to erroneous selection. The cases examined by Lien (1990) all lie along the horizontal line v¯ = 1 in Fig. 3, from which it is clear that any move away from the point of a fair contest (α=1 implying λ=1) must reduce social welfare; in other words, making an unfair contest more fair will improve welfare. 14 The delineations in Fig. 3 can be used to show the conditions under which this result holds (and when it does not) in our more general framework. In area Q, it is the case that α>1 and that the efficiency loss is increasing in this parameter; this is consistent with Lien’s result — making an unfair contest even more unfair reduces welfare. In area S, we have that increasing α reduces the efficiency loss and improves welfare; 1>α in this region so increasing this parameter implies making the contest more fair and Lien’s result again obtains. In area R, however, Lien’s result does not hold. In this region, an increase in α — implying a less fair contest — increases welfare. Notice that when the firms are not constrained to have the same expected efficiency (as depicted by v), it is socially optimal to 13 14
Recall that, in Fig. 2, the chance of a type (ii) error is low when λ takes low values. Recall again our (and Lien’s) narrow definition of social welfare as pertaining to efficiency in selection only.
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run an unfair contest (i.e. set α such that λ=λm (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 1+λ/γ 1/γ 1/γ 1+λ/γ λ v1 v2 v¯2 v¯2 , x2 = . x1 = v¯1 φ(1 + λ) v¯2 φ(1 + λ) Recalling that B1 =x1 /α, and B2 =x2 , the expected bribes can be calculated as follows: 1/γ Z v¯1 1+λ/γ v¯2 v1 1 1 dv1 , E (B1 ) = φ(1 + λ) α 0 v¯1 v¯1
v¯2 E (B2 ) = φ(1 + λ)
1/γ Z
v¯2
0
v2 v¯2
1+λ/λγ
1 dv2 v¯2
and hence, E(B1 ) + E(B2 ) =
v¯1 φ
1/γ
λ 1+λ
1/γ
(v/λ) ¯ 1/γ γ λ γ + . 1+λ+γ 1 + λ + γλ
(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). Recall that λ ≡ v/α ¯ γ . We have written Eq. (10) as a function of λ for analytical convenience; 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 1/λ∗ . 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 1/α ∗ where α ∗ 6=1. 2. Assume v¯ < 1. Then α ∗ <1.
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Proof. See Appendix B.
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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 1/α; 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, ¯ lies in the region where α>1. Hence, a planner would set α>1 and would discriminate λm (v), in favor of the expectedly more efficient firm. However, a selfish bribee discriminates in the opposite direction. 5. Summary We have focussed on the selection properties of a competitive bribery procedure when the contest may be unfair and/or the contestants are asymmetric. Previously, the literature has concentrated to a large degree on the case in which the unique equilibrium of the procedure is symmetric, thus, ensuring completely efficient allocation. Lien (1990) looks at a situation in which unfair contest rules are used to distinguish among ex ante identical firms and concludes that such unfairness leads to inefficient selection; furthermore, increasing the level of unfairness worsens allocation efficiency. In contrast to Lien’s model, we have allowed an unfair bribery procedure to be carried out between asymmetric players. Since there is no parameter setting in this case which guarantees allocation efficiency, the effect of making the contest unfair (or more unfair) cannot be determined unambiguously. We have presented cases in which Lien’s result holds, and when it does not. Furthermore, we have demonstrated that social efficiency (in the narrow sense of our definition) actually requires that the contest be ‘unfair’; this unfairness in the contest rules can be used to redress the adverse selection effects due to the fact that the players are asymmetric. In the setting of the amount of discrimination, we have considered the possibility that the bribee is a selfish, income-maximizing actor. Our results demonstrate that the contest which
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arises possesses a direction of discrimination which is the opposite to that which a social planner would set if he were interested in minimizing the efficiency loss from erroneous selection. Since bribery is a covert activity, this case would seem to be relevant from an empirical viewpoint, and contradicts the reformist argument that such a mechanism may lead to allocation efficiency.
Acknowledgements We would like to thank Tron Foss for helpful suggestions. The thoughtful comments of two anonymous referees are also gratefully acknowledged. Any errors are our own.
Appendix A This appendix derives Eq. (5) in the main text, and demonstrates uniqueness. The following lemma is useful in solving the model: Lemma 1. ¯ 1. G1 (v¯1 ) = G2 (v¯2 ) = x. 2. G1 (v1l ) = G2 (v2l ) = x = 0. 3. Either v1l = v 1 and/or v2l = v 2 . The proof of this is standard (see, for example, Clark and Riis, 1996) and is omitted here. Parts 1 and 2 of the lemma state that the firms’ equilibrium effective bribe functions have common upper and lower supports (x¯ and x), whilst Part 3 indicates that at most one player plays and atom at zero. Summing the two equations in Eq. (4) gives: D2 + D (A1) g20 (x)g1 (x) + g10 (x)g2 (x) = φγ x γ −1 1 αγ which has the general solution γ D2 + D1 + K g1 (x)g2 (x) = φx αγ where K is a constant of integration. From Lemma 1, we can determine that D2 + D K = v¯1 v¯2 − φ x¯ γ 1 . αγ From Eqs. (4), (A2) and (A3) we have that φD1 γ x γ −1 g1 (x) = 0. g10 (x) − v¯1 v¯2 − (x¯ γ − x γ )φ(D2 /α γ + D1 )
(A2)
(A3)
(A4)
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Solving the first-order differential equation in Eq. (A4) yields g1 (x) = v1l 1 +
φ(D1 + D2 /α γ )x γ v¯1 v¯2 − φ(D1 + D2 /α γ )x¯ γ
D1 /(D1 +(D2 /α γ ))
.
(A5)
Similarly for Firm 2, g2 (x) =
v2l
φ(D1 + D2 /α γ )x γ 1+ v¯1 v¯2 − φ(D1 + D2 /α γ )x¯ γ
(D2 /α γ )/(D1 +(D2 /α γ ))
.
(A6)
From Eqs. (A5) and (A6), we can see that g1 (x) v1l
!1/D1 =
g2 (x)
!α γ /D2
v2l
and hence, v2l
v1l = g2 (x) g1 (x)
!D2 /D1 α γ .
(A7)
Furthermore, from the lemma and Eq. (A5), one can determine that " x¯ =
v¯1 v¯2 (1 − (v1l /v¯1 )(1+(D2 /D1 α φ(D1 + (D2 /α γ ))
γ ))
)
#1/γ .
(A8)
Combining Eqs. (A5), (A6), (A7) and (A8) gives Eq. (5) in the text. It remains to show that Eq. (5) is the unique solution of the model. By differentiating gi (x) and setting the result into the first-order conditions in Eq. (3), one finds: ! l v φλγ D1 v1 x γ −1 ∂π1 Z −1/1+λ − 1 (A9) = ∂x1 v1 D2 v1l Notice that if x=0, then Z=1, and that ∂Z−1/(1+λ) /∂x<0. Hence, when v1 < v1l , ∂π 1 /∂x1 <0 for all x1 >0; thus x1 =0 is the best response. When v1 ≥ v1l , then ∂π 1 /∂x1 (>, =, <)0 for Z −1/(1+λ) (>, =, <) v1l /v1 . Hence, there is a unique value of x1 which solves Firm 1’s maximization problem. The corresponding expression for Player 2 is ! v¯2 (v1l /v¯1 )λ φγ v2 x γ −1 ∂π2 −λ/1+λ = − Z (A10) ∂x2 v2 v¯2 (v1l /v¯1 )λ We have that ∂Z−λ/(1+λ) /∂x<0. Thus, when v2 ≤ v¯2 (v1l /v¯1 )λ , then ∂π 2 /∂x2 <0 for all x2 >0; thus x2 =0 is the best response. When v2 > v¯2 (v1l /v¯1 )λ , ∂π 1 /∂x1 (>, =, <)0 for Z −λ/(1+λ) (>, =, <) v¯2 (v1l /v¯1 )λ . This establishes the uniqueness of Firm 2’s best response.
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Appendix B Proof of proposition. (1) Differentiating Eq. (10) with respect to λ yields " 1 v¯1 1/γ λ(1/γ )−1 1 λγ ∂(E(B1 ) + E(B2 )) = − ∂λ γ φ 1+λ+γ 1+λ 1+λ+γ λ γ v¯ 1/γ − − 1 + λ + γλ 1 + λ 1 + λ + γλ If v¯ = 1, then Eq. (I) is zero at λ=1. We have that ∂ 2 (E(B1 ) + E(B2 )) = sign − [8γ + 7γ 2 + 5γ 3 − 4]. sign ∂λ2
(I)
(II)
λ=1
Hence, λ=1 is a local maximum (minimum) of Eq. (10) if γ > (<)γ˜ , where γ˜ ≈ 0.358 is the unique real root of the equation 8γ +7γ 2 +5γ 3 −4=0. For γ < γ˜ , this means that there are two global maxima which occur for λ<1 and λ>1 due to the symmetry property of Eq. (10) mentioned in the text. Consider γ ≥2. Then λ=1 is a local maximum of Eq. (10). Both terms in the square bracket in Eq. (I) are negative at this value of λ; furthermore the first term is strictly decreasing and the second strictly increasing in λ. Hence, Eq. (I) is strictly positive (negative) for λ<(>)1. Consequently, λ=1 is the global maximum for γ ≥2. Assume now that γ <2. Let λ1 be such that γ λ1 = 0. − 1 + λ1 1 + λ1 + λ 1 γ Then it is clearly the case that Eq. (I) is positive for λ≤λ1 . Furthermore, the symmetric nature of Eq. (10), as noted in the text, makes it sufficient to concentrate the search for turning points of this function (other than λ=1) to the interval [λ1 , 1]. Notice that for γ =1/2 and γ =1/3, Eq. (I) can easily be transformed to a polynomial of order 5 and 6, respectively. It can be shown that if γ =1/2, then Eq. (I) has no real roots in the interval (0, 1); if γ =1/3, then Eq. (I) has one real root in this interval. (The proof of this is available from the authors on request.) Hence, when γ =1/2, we have that λ=1 is a global maximum of Eq. (10) since Eq. (I) is positive in this case for all λ<1; recalling the definition of λ, this gives α ∗ =1 as the bribe-maximizing value of α for this case. When γ =1/3, Eq. (10) has one global maximum in the interval (0, 1). Let λ+ represent this root; then, by virtue of the symmetry in Eq. (10) indicated in the text, there is also a global maximum at 1/λ+ . Let us now consider other values of γ . We find that ∂ 2 (E(B1 ) + E(B2 )) ∂λ ∂γ 1 ln λ γ λ 1 − − 2 − = sign 1 + λ + γλ 1 + λ + γ 1+λ 1+λ+γ γ λ1/γ 1 − +λ(1 + λ) (1 + λ + γ λ)3 (1 + λ + γ )3
sign
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123
which is zero if λ=1, otherwise, positive for 1>λ>λ1 . Hence, λ=1 is the global maximum of Eq. (10) for all γ ≥1/2. It remains then to show that this is also the global maximum for γ ∈ (γ˜ , 1/2). Regrettably, we have fallen short of showing this formally. However, simply drawing out Eq. (10) shows that λ=1 is indeed the global maximum of Eq. (10) in this interval. (2) When v¯ < 1, the symmetry property of Eq. (10) described in the text no longer holds. Hence, we must also consider values of λ>1 in our search for a maximum. Assume first that γ <2. Define λ2 by γ λ2 1 =0 − 1 + λ2 1 + λ2 + γ and recall the definition of λ1 from Part (1). Notice that λ2 >1. When λ∈(0, λ1 ), Eq. (I) is strictly positive, and when λ∈(λ2 , ∞) Eq. (I), is strictly negative. Hence, the relevant range to look for a maximum in this case is the interval [λ1 , λ2 ]. The following can readily be seen to be the case: ∂ 2 (E(B1 ) + E(B2 )) >0 ∂λ ∂(−v) ¯ for λ∈[λ1 , λ2 ]. Hence, as v¯ is reduced below 1, the bribe-maximizing value of λ increases. Recalling that α = (v/λ) ¯ 1/γ , we see that this causes the bribe-maximizing value of α to fall below 1 as claimed. Now consider γ >2. The relevant range to look for the maximum of Eq. (10) is λ∈[λ3 , λ4 ], where λ3 γ 1 = 1 + λ3 1 + λ3 + γ γ λ4 = 1 + λ4 1 + λ4 + γ λ4 so that λ3 <1 and λ4 >1. We see that Eq. (I) is strictly positive for λ<λ3 and strictly negative for λ>λ4 . It can easily be seen that ∂ 2 (E(B1 ) + E(B2 )) <0 ∂λ ∂(−v) ¯ for λ∈[λ3 , λ4 ]. Hence, as v¯ is reduced below 1, the bribe-maximizing value of λ decreases (and is unchanged if γ =2). However, v/λ ¯ decreases and α ∗ again falls below 1. To see this, notice that ∂(E(B1 ) + E(B2 )) ∂λ (v=1,λ=1) ¯
=0<
∂(E(B1 ) + E(B2 )) ∂(E(B1 ) + E(B2 )) < ∂λ ∂λ (λ=v=k<1) ¯ (λ
for any v¯ = k < 1. Hence for v¯ < 1, a maximum of expected bribes occurs for λ > v, ¯ and ∗ hence, α <1 as claimed.
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References Beck, P.J., Maher, M.W., 1986. A comparison of bribery and bidding in thin markets Economics Letters 20, 1–5. Clark, D.J., Riis, C., 1996. Rank-Order Tournaments and Selection, Memorandum No. 27. Department of Economics, University of Oslo, Norway, August. Heidenheimer, A., 1970. Political Corruption: Readings in Comparative Analysis. Holt, Rinchart & Winston, New York. Leys, C., 1965. What is the problem about corruption? Journal of Modern African Studies 3, 215–230. Lien, D., 1986. A note on competitive bribery games Economics Letters 22, 337–341. Lien, D., 1990. Corruption and allocation efficiency Journal of Development Economics 33, 153–164. McMullan, M., 1961. A theory of corruption Sociological Review 9, 181–200. O’Keeffe, M., Kip Viscusi, W., Zeckhauser, R.J., 1984. Economic contests: comparative reward schemes Journal of Labour Economics 2 (1), 27–56. Rose-Ackerman, S., 1975. The economics of corruption Journal of Public Economics 4, 187–203.
Allocation efficiency in a competitive bribery game Derek J. Clark a,b,∗ , Christian Riis a,c a
Department of Economics, NFH, University of Tromsø, N-9037 Tromsø, Norway Norwegian School of Management, BI Tromsø, P.O. Box 973, N-9260 Tromsø, Norway Department of Economics, Norwegian School of Management BI, P.O. Box 580, N-1301 Sandvika, Norway b
c
Received 16 June 1999; received in revised form 14 September 1999; accepted 10 November 1999
Abstract We consider the selection properties of a competitive bribery model in the presence of two types of asymmetry: unevenness between the competitors and unfairness in the contest rules. Only under very special conditions does the benchmark model yield allocation efficiency; in other cases, the effect on allocation efficiency of making the contest more unfair is ambiguous and parameter specific. We present conditions under which each result obtains. Our results indicate that it is socially optimal to run an unfair contest in order to redress the allocation inefficiency introduced when contestants are asymmetric. We show, however, that a selfish, income-maximizing bribee will discriminate in the opposite direction to that which society would prefer. ©2000 Elsevier Science B.V. All rights reserved. JEL classification: O17; C72 Keywords: Bribery; Allocation efficiency; Discrimination
1. Introduction In some countries, it is not uncommon to find government contracts awarded to firms which succeed in bribing a corrupt official. While such corruption has obvious costs, Lien (1986, 1990) has investigated a potential benefit of a competitive bribery procedure as a mechanism for ensuring ex post allocation efficiency; here allocation efficiency is narrowly defined as the lowest cost firm winning the contract. Two opposing models have been used to model this bribery procedure. Beck and Maher (1986) and Lien (1986) consider the case in which unsuccessful firms have their bribes refunded; if the highest bribe wins, then this procedure is a first-price sealed bid auction which, as is well known, can possess an ∗ Corresponding author. Fax: +47-77646021. E-mail addresses: [email protected] (D.J. Clark), [email protected] (C. Riis).
0167-2681/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 0 0 ) 0 0 0 7 7 - 9
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equilibrium exhibiting ex post allocation efficiency. In Lien (1986, 1990), an alternative set up is used in which all firms, successful or not, forfeit their bribes. Whilst, we prefer this latter type of model for analyzing corrupt activity, we show that Lien’s assumption that firms are ex ante identical is not innocuous. In this paper, we look at the role which ‘asymmetry’ plays in the competitive bribery procedure. 1 Following practice in the contest literature (see O’Keeffe et al., 1984), we can define two dimensions of asymmetry: (i) the game may be ‘uneven’ due to the fact that the contestants are expected to be different; (ii) an ‘unfair’ contest can arise when the players are treated differently. The game which we specify below, incorporates both of these elements and investigates the allocation properties which arise in equilibrium. Lien (1986) confines attention to an even and fair contest, whilst Lien (1990) considers an even, unfair contest. The unique equilibrium of the former is symmetric and hence allocation efficiency is ensured. In the latter paper, one firm is handicapped and must bribe proportionally more than the opponent in order to win; the equilibrium of this game is not symmetric which leads to the result that ‘the economy may suffer allocation inefficiencies from competitive bribery procedures, whenever there is some degree of discrimination’ (Lien, 1990; 157). In addition, Lien (1990) also shows that increasing the degree of discrimination makes matters worse (i.e. an increase in the probability that the most efficient firm is defeated). Intuitively, these results must obtain since firms are assumed to be ex ante identical and hence a symmetric equilibrium must result when there is no discrimination among the bribers. Thus, any move away from the symmetry of the no discrimination case must worsen the selection properties of the bribery procedure. When all bribes are forfeited, so that losers also bear the full cost of their bribe, one departs from the traditional first-price sealed bid auction procedure and hence, there is in general no mechanism to ensure allocation efficiency. We show that even a fair bribery procedure leads to inefficiency, except in the very rare case that firms are identical ex ante. Hence, changing the rules of the game may or may not enhance allocation efficiency, depending on the specific model parameters. Accordingly, increased discrimination in a competitive bribery procedure may well enhance its selection properties, contradicting Lien’s conclusion. In the literature, there are two views on corrupt activity. 2 Moralists argue that bribery has no positive effects, whilst reformists take the view that this mechanism may have some beneficial effects. Among these is allocation efficiency since the most efficient firm should be able to afford the largest bribe and win the contest. We demonstrate that the role of the bribee is critical in assessing the selection properties of the bribery procedure. It seems likely that the bribee is a selfish actor who desires to maximize his own income (in the form of bribe revenue). We show that this will lead to a direction of discrimination in exactly the opposite direction to that which society would prefer. Whilst, society would wish for the more efficient firm to be favored — and hence have a greater chance of winning — the bribee has an incentive to discriminate against this firm in order to even out the contest. This gives both firms an incentive to bribe more; the less efficient firm now has a real possibility 1 The model framework which we use is an all-pay auction under incomplete information; this possesses applications which go beyond bribery. In Clark and Riis (1996), we consider a firm which uses this type of mechanism to decide which worker to promote to the next level in its hierarchy. 2 See McMullan (1961), Leys (1965) and Heidenheimer (1970).
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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 vi . It is common knowledge, however, that vi 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 Di =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’, Bi ≥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 αB1 >B2 , where α>0 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 γ that the cost of bribing the official is represented by the function φBi , 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 x1 =αB1 and x2 =B2 as the effective bribe levels, the expected payoff of player i can be written as: 6 x γ 1 γ π1 (x1 ) = Pr(x1 > x2 )v1 − φ , π2 (x2 ) = Pr(x2 ≥ x1 )v2 − φx2 . (1) α 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 represent actual (rather than effective) bribe i levels, then α can be seen as depicting the relative moral cost of bribery between the two players; if α>(<)1 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).
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Assume that the equilibrium effective bribe functions, Gi (vi )=xi , are continuous and strictly increasing (except possibly at zero), with upper and lower supports x¯i and x i , and with inverse gi (xi )=vi . 7 Using the fact that the vi are distributed uniformly, we can write: x γ g2 (x1 ) − v 2 g1 (x2 ) − v 1 1 v1 − φ , π2 (x2 ) = v2 − φ(x2 )γ . (2) π1 (x1 ) = D2 α D1 The first-order conditions for an interior maximum of Eq. (2) are: g10 (x2 )v2 g20 (x1 )v1 φγ γ −1 − = 0, − φγ (x2 )γ −1 = 0. x 1 D2 αγ D1
(3)
Inserting vi =gi (xi ) into Eq. (3) yields a system of two differential equations: g20 (x)g1 (x) =
φD2 γ γ −1 x , αγ
g10 (x)g2 (x) = φD1 γ x γ −1 .
In Appendix A, it is shown that the unique solution to the system in Eq. (4) is: ! l λ v 1 g2 (x) = v¯2 Z λ/1+λ , where v1l = v 1 g1 (x) = v1l Z 1/1+λ , v¯1 λ 1/λ v1 v2 l iff v¯2 ≥ v 2 , otherwise v1 = v¯1 , v¯1 v¯2 D2 φD1 (1 + λ)x γ , λ≡ . Z ≡1+ 1+λ D1 α γ v¯1 v¯2 (v1 /v¯1 )
(4)
(5)
3. Allocation efficiency We are now in a position to look at allocation efficiency in the competitive bribery procedure. Recalling the definition of gi (xi )=vi , it can easily be calculated from Eq. (5) that: λ v1 v2 (>, =, <) . (6) x1 (>, =, <)x2 iff v¯1 v¯2 Eq. (6) depicts the role of the different parameters in determining the underlying selection properties of this asymmetric competitive bribery model. We can isolate several factors (or combinations of these) which influence selection: differences in the means and variances of the valuation distributions, the fairness of the competition, as well as the cost of bribery as depicted by γ . Notice that the shift parameter in the cost function, φ, does not affect the selection properties of the bribery procedure. Consider first the completely symmetric case in which v¯1 = v¯2 , v 1 = v 2 , α = 1; Eq. (6) indicates that the bribery game allocates efficiently since Firm i wins if and only if it has a higher valuation than Firm j. In other cases, allocation will not always be efficient. In the remainder of the paper we assume, with no loss of generality, that v¯1 ≥ v¯2 . 7
This assumption is fulfilled in equilibrium.
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Fig. 1. Erroneous selection; λ>1.
The locus of valuation combinations (v1 , v2 ) which gives x1 =x2 and is defined by v1 = v¯1
v2 v¯2
1/λ (7)
is graphed as 0e in Fig. 1, illustrating the case where λ>1; above and to the left of this line 8 , we have x2 >x1 (2 wins) below and to the right x1 >x2 (1 wins), and exactly on 0e, x1 =x2 (by assumption 2 wins). Only when the function in Eq. (7) coincides exactly with a 45◦ line from the origin in the relevant range (from v i to v¯i ), do we have efficiency in allocation. When this is not the case, the firm with the lower valuation may, on occasion, be awarded the contract. Notice that whenever v¯1 6= v¯2 , there are no settings of the other parameters which ensure complete efficiency in allocation in this model. 9 Area ‘abced’ in Fig. 1 represents the combinations of v1 and v2 which yield erroneous selection (i.e. that 2 is chosen when v1 >v2 ). When α is small, implying that the contest is heavily biased in favor of Firm 2, this area is large. Firm 1, being at a disadvantage, gives small bribes or, if v1 is sufficiently small, does not bribe at all. Hence he often loses to less efficient opponents. However, Firm 2, realizing that the opponent behaves in this way, acts less aggressively since it is likely that it wins the contract with a lower bribe (than if Firm 1 were at less of a disadvantage). Consequently, only if v1 turns out to be very large does Firm 1 actually win the game. As α is increased and Firm 1 is at less of a disadvantage, the area ‘abced’ shrinks (while 0e maintains the same start and end points) and the probability of wrong selection falls. Raising α sufficiently will eventually lead to a case in which 1>λ; 0h in Fig. 2 represents the graph of Eq. (7) for this situation. As before, above (below) 0h, we have x2 >(<)x1 . Two types of erroneous selection are now possible: (i) Firm 1 is chosen when v2 >v1 , represented in Fig. 2 by area ‘jkn’; (ii) Firm 2 is chosen when v1 >v2 which is the area ‘nrh’ in Fig. 2. Define v ∗ = [v¯2 /v¯1λ ]1/1−λ as the point at which 0h crosses the 45◦ line. If v 1 and v 2 are 0e is a straight line when λ=1. In a different setting, Clark and Riis (1996) investigate a prize structure which would lead to allocation efficiency in this case. 8 9
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Fig. 2. Erroneous selection; λ>2.
below v∗ , then both types of error are made. 10 Otherwise, only a type (ii) error occurs. For high values of v2 , a type (ii) error may occur; Firm 1 tends to underestimate the opponent, expects it to give a small bribe, and consequently loses if v2 is large. For low values of v2 , a type (i) error can occur, so that Firm 1 wins more often than allocation efficiency would dictate; Firm 2 is at a disadvantage relative to Firm 1, since α is large, and in addition has a low valuation, hence it bribes very little or not at all. Raising α increases the probability of a type (i) but simultaneously reduces that of a type (ii) error. When α is sufficiently large, then further increases in this parameter will raise the probability of choosing the least efficient firm, as the increase in the chance of a type (i) error outweighs the fall in the probability that a type (ii) error will be made. Let us now consider the result of Lien (1990) that making an unfair contest (α6=1) more unfair will increase the probability of choosing a less efficient contractor. We have two cases to consider: λ>1 and 1>λ. 3.1. λ>1 Take first the case represented by Fig. 1 (λ>1), where we have already noted that increasing α reduces the probability of wrong selection. The point is, however, that whether this increase in α represents a move to a more or a less fair contest is parameter specific. If D1 ≥D2 , then λ>1 implies 1>α and hence an increase in α makes the contest more fair and simultaneously reduces the probability of wrong selection. This is consistent with Lien (1990) and extends his result to a wider range of possibilities. 11 However, if D2 >D1 , it is permissible within Fig. 1 (where λ>1) to have α>1. Here an increase in α makes the contest less fair (i.e. more biased towards Firm 1) and reduces the probability of erroneous selection. This is the opposite of Lien’s result. 3.2. 1>λ Similarly, Lien’s result and its exact opposite can be generated when Fig. 2 (1>λ) is relevant. Assuming for illustrational simplicity that v 1 = v 2 = 0, and writing v¯ ≡ v¯2 /v¯1 , 10 Since the derivative ∂v /∂v from Eq. (7) tends to infinity as v tends to zero in this case, 0h will cross the 45◦ 2 1 1 line. 11 Lien’s assumption of ex ante symmetry obtains if D =D . 1 2
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the probability that the higher cost firm actually wins the contract in this case is θ =η+τ where, η=
1 − λ 1+λ/1−λ v¯ 2(1 + λ)
τ=
λ v¯ − + η. 1+λ 2
Here, the total probability that a mistake is made which is expressed by θ, is the sum of the probabilities of making a type (i) error, denoted by η, and a type (ii) mistake, τ . The comparative static effect of a change in α is given by −λγ 1+λ ∂θ 1+λ/1−λ = ln v ¯ . (8) 1 + 2v −1 + ∂α 1−λ α(1 + λ)2 Since 1 ≥ v, ¯ the term in square brackets in Eq. (8) is negative, making the sign of the whole expression ambiguous. It is straightforward to determine that ∂θ /∂α>0 as v¯ → 1, whilst ∂θ/∂α<0 as v¯ → 0. As we approach the situation in which the firms are approximately identical ex ante, increasing α makes mistaken selection more likely. Again this is Lien’s result. However, when the firms are expected to be very dissimilar (v¯ low), the opposite result obtains in which the probability of erroneous selection is decreasing in α. Intuitively, Firm 1 is expected to be much superior to Firm 2 in this case, so reducing the probability of mistakenly choosing the higher cost firm implies making the contest more biased toward Firm 1. When v¯ takes intermediate values, the effect of changing α on θ is not monotonic. 3.3. Efficiency loss due to erroneous selection Naturally, one is concerned with the efficiency loss which making a selection error entails, and not the probability of such an occurrence. 12 Normalizing v¯1 = 1, retaining the assumption that v¯i = 0 (i=1, 2), and writing v ∗ = v¯ 1/1−λ if 1>λ, and v∗ =0, otherwise (where v∗ is the intersection point between the 45◦ line and the curve at which x1 =x2 given in Eq. (7)), the expected efficiency loss in the competitive bribery procedure can be calculated as Z v¯ Z (v2 /v) Z v ∗ Z v2 ¯ 1/λ (v − v ) (v2 − v1 ) 1 2 dv1 dv2 + dv1 dv2 = 1/λ ∗ v ¯ v ¯ 0 (v2 /v) ¯ v v2 Z v¯ Z v2 vλ ¯ v¯ 2 λ (v2 − v1 ) dv1 dv2 = − + . = v¯ 2(2 + λ) 2λ + 1 6 ¯ 1/λ 0 (v2 /v) 12 As in Lien (1990), we focus in this section solely on losses due to erroneous selection. In our model, however, losses can occur in the transfer of bribes since their value may be different to giver and receiver. This deadweight loss complicates the calculations, whilst adding little to the analysis; it is, therefore, omitted in what follows. An expression for the expected deadweight loss of the bribery procedure is straightforward to calculate, and is available from the authors on request.
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Fig. 3. Efficiency loss due to erroneous selection.
We can illustrate the efficiency effects within a (v, ¯ λ) diagram (recall that v¯ ≡ v¯2 /v¯1 ); consider Fig. 3 which depicts iso-social-cost curves which are concentric to the global minimum at (1, 1). The further away from this point the iso-social-cost curve lies, the larger the efficiency loss associated with the competitive bribery procedure. Since we have defined λ = v/α ¯ γ , points to the right (left) of the 45◦ line reflect α<(>)1, whilst along the 45◦ line, we have α=1. Given v, ¯ the λ which minimizes the social cost of erroneous selection is unique and determined by 1 v¯ ∂ = − = 0. ∂λ (2 + λ)2 (2λ + 1)2
(9)
¯ in Fig. 3 and always lies above the 45◦ The locus of points satisfying Eq. (9) is denoted λm (v) m line. To the right of λ (v), ¯ is monotonically increasing in λ, and thus, the efficiency loss ¯ the efficiency loss is monotonically is monotonically decreasing in α. To the left of λm (v), decreasing in λ (increasing in α). ¯ Three areas are of interest in this figure, represented by Q (the area bounded by λm (v) ¯ and the 45◦ line) and S (below and to the right of the 45◦ and v¯ = 1), R (bounded by λm (v) line). The comparative static effect on the efficiency loss in selection of changes in α and γ are given in Table 1. Notice from Eq. (1) that the firms’ cost functions are identical up to a multiplicative term 1/α γ which indicates the cost of bribery to Firm 1 relative to Firm 2; when 1/α γ >1, Firm 1 has a larger marginal (and average) cost relative to Firm 2 for an identical bribe level. The intuition behind the results in Table 1 can be gleaned by looking at this factor, which we Table 1 The comparative static effect on the efficiency loss in selection of changes in α and γ
Q R S
∂/∂α
∂/∂γ
+ − −
+ − +
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shall denote by f. The assumptions which underlie in Fig. 3 imply that λ = vf ¯ . Notice that in areas Q and R, we have that α>1, whilst 1>α in S. Suppose we fix v¯ = V in Fig. 3, and start from a point where λ is low in Q, and hence f is small, implying that Firm 1 has a very low marginal cost of bribery relative to Firm 2. This leads to Firm 2 bidding little (if at all) and losing often to inferior Firm 1 types (a type (i) error occurs with a large probability). Knowing this, Firm 1 also bribes a small amount, but enough to beat almost all Firm 2 types. 13 Suppose now that we move horizontally from this point from left to right in Fig. 3 for fixed V. This movement can be accomplished by reducing α and/or reducing γ . Firm 2 is now at less of a cost disadvantage, and consequently, bids more and loses less often to inferior Firm 1 types. At the same time, it is also possible for a very efficient Firm 2 to beat inferior Firm 1 types, due to the fact that Firm 1 may tend to ‘slack off’ because it has an initial advantage in its lower relative cost. These two effects pull in opposite directions, but the net effect in area Q is that the efficiency loss decreases as α and γ fall. Reducing α and/or γ sufficiently to reach the point w in Fig. 3 gives the smallest welfare loss for v¯ = V due to mistaken selection in the competitive bribery procedure. By definition, any horizontal move away from w leads to an iso-social cost curve at a lower level of welfare. Thus, a further reduction in α and/or γ (and a move to the right from w into R) increases the efficiency loss; in Fig. 2, area ‘jkn’ shrinks by less than the enlargement in area ‘nrh’. As the relative marginal cost to Firm 1 becomes larger and larger, it offers successively smaller bribes and loses more often to inferior Firm 2 types. Simultaneously, Firm 1 beats more efficient opponents less and less often (until f becomes so large that this sort of error does not occur). Moving further to the right along v¯ = V into area S increases the efficiency loss even more. (Note that in area S, since 1>α, a horizontal move from left to right is achieved by reducing α and/or increasing γ ). One interpretation of increasing γ is as a toughening of the stance against large bribes. As indicated in Table 1, this reduces the expected efficiency loss in area R. However, in areas Q and S, the efficiency loss increases following this tougher stance; any gains due to a reduced level of bribery activity must thus be weighed up against the losses due to erroneous selection. The cases examined by Lien (1990) all lie along the horizontal line v¯ = 1 in Fig. 3, from which it is clear that any move away from the point of a fair contest (α=1 implying λ=1) must reduce social welfare; in other words, making an unfair contest more fair will improve welfare. 14 The delineations in Fig. 3 can be used to show the conditions under which this result holds (and when it does not) in our more general framework. In area Q, it is the case that α>1 and that the efficiency loss is increasing in this parameter; this is consistent with Lien’s result — making an unfair contest even more unfair reduces welfare. In area S, we have that increasing α reduces the efficiency loss and improves welfare; 1>α in this region so increasing this parameter implies making the contest more fair and Lien’s result again obtains. In area R, however, Lien’s result does not hold. In this region, an increase in α — implying a less fair contest — increases welfare. Notice that when the firms are not constrained to have the same expected efficiency (as depicted by v), it is socially optimal to 13 14
Recall that, in Fig. 2, the chance of a type (ii) error is low when λ takes low values. Recall again our (and Lien’s) narrow definition of social welfare as pertaining to efficiency in selection only.
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run an unfair contest (i.e. set α such that λ=λm (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 1+λ/γ 1/γ 1/γ 1+λ/γ λ v1 v2 v¯2 v¯2 , x2 = . x1 = v¯1 φ(1 + λ) v¯2 φ(1 + λ) Recalling that B1 =x1 /α, and B2 =x2 , the expected bribes can be calculated as follows: 1/γ Z v¯1 1+λ/γ v¯2 v1 1 1 dv1 , E (B1 ) = φ(1 + λ) α 0 v¯1 v¯1
v¯2 E (B2 ) = φ(1 + λ)
1/γ Z
v¯2
0
v2 v¯2
1+λ/λγ
1 dv2 v¯2
and hence, E(B1 ) + E(B2 ) =
v¯1 φ
1/γ
λ 1+λ
1/γ
(v/λ) ¯ 1/γ γ λ γ + . 1+λ+γ 1 + λ + γλ
(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). Recall that λ ≡ v/α ¯ γ . We have written Eq. (10) as a function of λ for analytical convenience; 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 1/λ∗ . 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 1/α ∗ where α ∗ 6=1. 2. Assume v¯ < 1. Then α ∗ <1.
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Proof. See Appendix B.
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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 1/α; 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, ¯ lies in the region where α>1. Hence, a planner would set α>1 and would discriminate λm (v), in favor of the expectedly more efficient firm. However, a selfish bribee discriminates in the opposite direction. 5. Summary We have focussed on the selection properties of a competitive bribery procedure when the contest may be unfair and/or the contestants are asymmetric. Previously, the literature has concentrated to a large degree on the case in which the unique equilibrium of the procedure is symmetric, thus, ensuring completely efficient allocation. Lien (1990) looks at a situation in which unfair contest rules are used to distinguish among ex ante identical firms and concludes that such unfairness leads to inefficient selection; furthermore, increasing the level of unfairness worsens allocation efficiency. In contrast to Lien’s model, we have allowed an unfair bribery procedure to be carried out between asymmetric players. Since there is no parameter setting in this case which guarantees allocation efficiency, the effect of making the contest unfair (or more unfair) cannot be determined unambiguously. We have presented cases in which Lien’s result holds, and when it does not. Furthermore, we have demonstrated that social efficiency (in the narrow sense of our definition) actually requires that the contest be ‘unfair’; this unfairness in the contest rules can be used to redress the adverse selection effects due to the fact that the players are asymmetric. In the setting of the amount of discrimination, we have considered the possibility that the bribee is a selfish, income-maximizing actor. Our results demonstrate that the contest which
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arises possesses a direction of discrimination which is the opposite to that which a social planner would set if he were interested in minimizing the efficiency loss from erroneous selection. Since bribery is a covert activity, this case would seem to be relevant from an empirical viewpoint, and contradicts the reformist argument that such a mechanism may lead to allocation efficiency.
Acknowledgements We would like to thank Tron Foss for helpful suggestions. The thoughtful comments of two anonymous referees are also gratefully acknowledged. Any errors are our own.
Appendix A This appendix derives Eq. (5) in the main text, and demonstrates uniqueness. The following lemma is useful in solving the model: Lemma 1. ¯ 1. G1 (v¯1 ) = G2 (v¯2 ) = x. 2. G1 (v1l ) = G2 (v2l ) = x = 0. 3. Either v1l = v 1 and/or v2l = v 2 . The proof of this is standard (see, for example, Clark and Riis, 1996) and is omitted here. Parts 1 and 2 of the lemma state that the firms’ equilibrium effective bribe functions have common upper and lower supports (x¯ and x), whilst Part 3 indicates that at most one player plays and atom at zero. Summing the two equations in Eq. (4) gives: D2 + D (A1) g20 (x)g1 (x) + g10 (x)g2 (x) = φγ x γ −1 1 αγ which has the general solution γ D2 + D1 + K g1 (x)g2 (x) = φx αγ where K is a constant of integration. From Lemma 1, we can determine that D2 + D K = v¯1 v¯2 − φ x¯ γ 1 . αγ From Eqs. (4), (A2) and (A3) we have that φD1 γ x γ −1 g1 (x) = 0. g10 (x) − v¯1 v¯2 − (x¯ γ − x γ )φ(D2 /α γ + D1 )
(A2)
(A3)
(A4)
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Solving the first-order differential equation in Eq. (A4) yields g1 (x) = v1l 1 +
φ(D1 + D2 /α γ )x γ v¯1 v¯2 − φ(D1 + D2 /α γ )x¯ γ
D1 /(D1 +(D2 /α γ ))
.
(A5)
Similarly for Firm 2, g2 (x) =
v2l
φ(D1 + D2 /α γ )x γ 1+ v¯1 v¯2 − φ(D1 + D2 /α γ )x¯ γ
(D2 /α γ )/(D1 +(D2 /α γ ))
.
(A6)
From Eqs. (A5) and (A6), we can see that g1 (x) v1l
!1/D1 =
g2 (x)
!α γ /D2
v2l
and hence, v2l
v1l = g2 (x) g1 (x)
!D2 /D1 α γ .
(A7)
Furthermore, from the lemma and Eq. (A5), one can determine that " x¯ =
v¯1 v¯2 (1 − (v1l /v¯1 )(1+(D2 /D1 α φ(D1 + (D2 /α γ ))
γ ))
)
#1/γ .
(A8)
Combining Eqs. (A5), (A6), (A7) and (A8) gives Eq. (5) in the text. It remains to show that Eq. (5) is the unique solution of the model. By differentiating gi (x) and setting the result into the first-order conditions in Eq. (3), one finds: ! l v φλγ D1 v1 x γ −1 ∂π1 Z −1/1+λ − 1 (A9) = ∂x1 v1 D2 v1l Notice that if x=0, then Z=1, and that ∂Z−1/(1+λ) /∂x<0. Hence, when v1 < v1l , ∂π 1 /∂x1 <0 for all x1 >0; thus x1 =0 is the best response. When v1 ≥ v1l , then ∂π 1 /∂x1 (>, =, <)0 for Z −1/(1+λ) (>, =, <) v1l /v1 . Hence, there is a unique value of x1 which solves Firm 1’s maximization problem. The corresponding expression for Player 2 is ! v¯2 (v1l /v¯1 )λ φγ v2 x γ −1 ∂π2 −λ/1+λ = − Z (A10) ∂x2 v2 v¯2 (v1l /v¯1 )λ We have that ∂Z−λ/(1+λ) /∂x<0. Thus, when v2 ≤ v¯2 (v1l /v¯1 )λ , then ∂π 2 /∂x2 <0 for all x2 >0; thus x2 =0 is the best response. When v2 > v¯2 (v1l /v¯1 )λ , ∂π 1 /∂x1 (>, =, <)0 for Z −λ/(1+λ) (>, =, <) v¯2 (v1l /v¯1 )λ . This establishes the uniqueness of Firm 2’s best response.
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Appendix B Proof of proposition. (1) Differentiating Eq. (10) with respect to λ yields " 1 v¯1 1/γ λ(1/γ )−1 1 λγ ∂(E(B1 ) + E(B2 )) = − ∂λ γ φ 1+λ+γ 1+λ 1+λ+γ λ γ v¯ 1/γ − − 1 + λ + γλ 1 + λ 1 + λ + γλ If v¯ = 1, then Eq. (I) is zero at λ=1. We have that ∂ 2 (E(B1 ) + E(B2 )) = sign − [8γ + 7γ 2 + 5γ 3 − 4]. sign ∂λ2
(I)
(II)
λ=1
Hence, λ=1 is a local maximum (minimum) of Eq. (10) if γ > (<)γ˜ , where γ˜ ≈ 0.358 is the unique real root of the equation 8γ +7γ 2 +5γ 3 −4=0. For γ < γ˜ , this means that there are two global maxima which occur for λ<1 and λ>1 due to the symmetry property of Eq. (10) mentioned in the text. Consider γ ≥2. Then λ=1 is a local maximum of Eq. (10). Both terms in the square bracket in Eq. (I) are negative at this value of λ; furthermore the first term is strictly decreasing and the second strictly increasing in λ. Hence, Eq. (I) is strictly positive (negative) for λ<(>)1. Consequently, λ=1 is the global maximum for γ ≥2. Assume now that γ <2. Let λ1 be such that γ λ1 = 0. − 1 + λ1 1 + λ1 + λ 1 γ Then it is clearly the case that Eq. (I) is positive for λ≤λ1 . Furthermore, the symmetric nature of Eq. (10), as noted in the text, makes it sufficient to concentrate the search for turning points of this function (other than λ=1) to the interval [λ1 , 1]. Notice that for γ =1/2 and γ =1/3, Eq. (I) can easily be transformed to a polynomial of order 5 and 6, respectively. It can be shown that if γ =1/2, then Eq. (I) has no real roots in the interval (0, 1); if γ =1/3, then Eq. (I) has one real root in this interval. (The proof of this is available from the authors on request.) Hence, when γ =1/2, we have that λ=1 is a global maximum of Eq. (10) since Eq. (I) is positive in this case for all λ<1; recalling the definition of λ, this gives α ∗ =1 as the bribe-maximizing value of α for this case. When γ =1/3, Eq. (10) has one global maximum in the interval (0, 1). Let λ+ represent this root; then, by virtue of the symmetry in Eq. (10) indicated in the text, there is also a global maximum at 1/λ+ . Let us now consider other values of γ . We find that ∂ 2 (E(B1 ) + E(B2 )) ∂λ ∂γ 1 ln λ γ λ 1 − − 2 − = sign 1 + λ + γλ 1 + λ + γ 1+λ 1+λ+γ γ λ1/γ 1 − +λ(1 + λ) (1 + λ + γ λ)3 (1 + λ + γ )3
sign
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which is zero if λ=1, otherwise, positive for 1>λ>λ1 . Hence, λ=1 is the global maximum of Eq. (10) for all γ ≥1/2. It remains then to show that this is also the global maximum for γ ∈ (γ˜ , 1/2). Regrettably, we have fallen short of showing this formally. However, simply drawing out Eq. (10) shows that λ=1 is indeed the global maximum of Eq. (10) in this interval. (2) When v¯ < 1, the symmetry property of Eq. (10) described in the text no longer holds. Hence, we must also consider values of λ>1 in our search for a maximum. Assume first that γ <2. Define λ2 by γ λ2 1 =0 − 1 + λ2 1 + λ2 + γ and recall the definition of λ1 from Part (1). Notice that λ2 >1. When λ∈(0, λ1 ), Eq. (I) is strictly positive, and when λ∈(λ2 , ∞) Eq. (I), is strictly negative. Hence, the relevant range to look for a maximum in this case is the interval [λ1 , λ2 ]. The following can readily be seen to be the case: ∂ 2 (E(B1 ) + E(B2 )) >0 ∂λ ∂(−v) ¯ for λ∈[λ1 , λ2 ]. Hence, as v¯ is reduced below 1, the bribe-maximizing value of λ increases. Recalling that α = (v/λ) ¯ 1/γ , we see that this causes the bribe-maximizing value of α to fall below 1 as claimed. Now consider γ >2. The relevant range to look for the maximum of Eq. (10) is λ∈[λ3 , λ4 ], where λ3 γ 1 = 1 + λ3 1 + λ3 + γ γ λ4 = 1 + λ4 1 + λ4 + γ λ4 so that λ3 <1 and λ4 >1. We see that Eq. (I) is strictly positive for λ<λ3 and strictly negative for λ>λ4 . It can easily be seen that ∂ 2 (E(B1 ) + E(B2 )) <0 ∂λ ∂(−v) ¯ for λ∈[λ3 , λ4 ]. Hence, as v¯ is reduced below 1, the bribe-maximizing value of λ decreases (and is unchanged if γ =2). However, v/λ ¯ decreases and α ∗ again falls below 1. To see this, notice that ∂(E(B1 ) + E(B2 )) ∂λ (v=1,λ=1) ¯
=0<
∂(E(B1 ) + E(B2 )) ∂(E(B1 ) + E(B2 )) < ∂λ ∂λ (λ=v=k<1) ¯ (λ
for any v¯ = k < 1. Hence for v¯ < 1, a maximum of expected bribes occurs for λ > v, ¯ and ∗ hence, α <1 as claimed.
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