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«Bribes For Faster Delivery Amal Sanyal Lincoln University, New Zealand and Instituto de Analisis Economico, UAB, Barcelona Abstract: The paper models ...»

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We first construct a one-off game between the supervisor and the supplier in an optimal contract designed by the principal. The supervisor is contracted to provide a report {ρ} on the level of corruption with evidence, where ρ can take two values, Y* or 0. If the evidence is verified to be correct, the supplier is fined an amount fρ, while the supervisor gets a reward Pρ, Pf. The value of f is set such that πfρ≥ R(ρ) = R(Y*), π being the probability that the supervisor gets to learn the correct state of the market. All parameters are common knowledge.

If the supervisor finds out the true value of Y*(0), a bribe negotiation sets in between the supplier and the supervisor. In this negotiation, the reservation level payoff for the supervisor is Pρ, while that of the supplier is fρ. However since P f, there is no bribe rate β0 such that (f-β)ρ and (β-P)ρ are both positive, implying that the bargaining set is empty. Hence no bargain takes place, and the supplier has to pay the fine. But πfρ≥ R(ρ) = R(Y*), so the supplier will not take any bribes at all.

We now show that this contract does not sustain bribe-free equilibria over repeated encounters. Suppose the above bribe free equilibrium is enforced in every play in a repeated game. Then there is no bribe, but nor is there any reward for the supervisor. Intuitively, it appears that the supervisor would be better off accepting a feasible bribe from the supplier and ignoring the reward. More formally, let δ denote the common time discount rate for the supplier and the supervisor. Note that if the supervisor reports the defaulting supplier in the first period, then his earning stream is (Pρ, 0, 0, 0,......) with present value Pρ.

On the other hand if he takes a bribe β, β P, from the supplier each period, the earning stream is (βρ, βρ, βρ, βρ,....) with present value βρ/(1-δ). For any given P, there exist values of β such that P β P(1-δ).

Now consider the following game repeated indefinitely. The supplier first moves with two possible actions: ‘take bribe’ and ‘do not take’. Then the supervisor moves in. If the supplier has played ‘do not take’ then the supervisor has only one strategy, ‘report truthfully’. Otherwise the supervisor plays either ‘collude’ or ‘report truthfully’. The strategy ‘collude’ denotes accepting some bribe rate β P from the

supplier, and reporting 0. Now consider the following strategy profiles:

supplier: 'take bribe' in the first play of the game. Subsequently play 'take bribe' if the supervisor played 'collude' in the last play; 'do not take' otherwise.

supervisor: 'collude' if the agent has taken bribe, 'report truthfully' otherwise.

For all β such that f β P(1- δ ), this strategy profile is a Nash equilibrium for the repeated game. The pair 'never take bribe' for the agent and 'always report' for the supervisor constitutes another Nash

file:///A|/Workshop.htm (5 de 10) [15/12/2000 13:18:11] q1 = q0 + (n-y)

equilibrium. But the supervisor's profile in this latter equilibrium is dominated by his strategy in the earlier one. Assuming that the supplier knows it, this profile can be deleted, and the only equilibrium is the profile described above.

Given these observations there are two theoretically possible deterrent mechanisms. The first is to ensure that no β may exist such that f β P(1- δ ), by setting P ≥ f/(1- δ ). If the reward is set at such a high level, the equilibrium of the game is 'never take bribe', 'always report'. Bribes are not taken and the supervisor, too, never gets a reward. However, in this case, given the environment in which our problem is set, it is likely that the supervisor would try to alter the game by pre-committing to collude.

The second possibility is to employ supervisors for a sufficiently small number of periods so that the single-period reward for reporting, Pρ, exceeds the present value of the stream of feasible bribes significantly. In this case, too, the supervisor can pre-commit to collude, but because of the large payoff from defection, it is more difficult to make the pre-commitment appear credible. Note however that if the cost of hiring and training a supervisor is large, this solution will be costly for the principal.

The difficulty of designing a deterring fine-reward scheme highlights a difficult aspect of supply queue corruption. In a potentially collusive situation, the principal needs to base incentives on the positive contribution of the collusive group (Tirole, 1992). That requires the principal to have a way of assessing this contribution independently of the supervisor. For example, in the case of auditor-taxpayer collusion, the contribution can be assessed by the increase in tax revenue. In the case of work avoidance, it is assessed by increase in production. In these cases the reward schemes are based on those independently verifiable quantities. The supply queue problem eludes this solution because the principal has no way of assessing the relevant contribution independently of the supervisor. The contribution that the principal is looking for is a reduction in the practice of bribes. But while the existence of bribes can be established by 'hard evidence' (Tirole, 1986), its absence or reduction can not. The principal has to rely on the word of the supervisor as evidence of the absence or waning of bribe taking.





To appreciate the problem reconsider setting P f. We noted that it does not work because in equilibrium the supervisor gets no reward and hence develops an incentive to encourage bribe taking in repeated games. This suggests that a sustainable bribe-free equilibrium would need a contract that rewards the supervisor according to the degree of absence of corruption rather than corruption reported. But a contract of this kind can not be effectively used because the principal has no way of assessing the absence of corruption independently of the supervisor’s report. For example, suppose the contract makes the reward an increasing function of the degree to which bribe taking is reduced. This would ensure that rewards continue if the report is {0}. But in that case the supervisor's best strategy would be to allow the agent to take bribes against a side payment and also collect rewards by reporting {0}.

4. Introducing Competition This section explores the effects of introducing competition into the market. We first develop a model with two suppliers and then generalize it to many.

Assume that two suppliers 1 and 2 are hired, and each is to sell to one half of the market and produce that many sales receipts. Suppose both suppliers try to sell with bribes to a part of the market and set off two cut-off points Y1 and Y2 respectively.

First suppose Y1≠ Y2. Without any loss of generality, let Y2 Y1. This implies that file:///A|/Workshop.htm (6 de 10) [15/12/2000 13:18:11] q1 = q0 + (n-y) 1 sells at a bribe to [Y1, Y2) and sells 1/2 –(Y2-Y1) units without bribes.

2 sells at a bribe to [Y2,1] and sells 1/2 – (1-Y2) units without bribes.

Average waiting time offered by the two suppliers for the non-bribe segment are:

In equilibrium the average waiting time offered to residual buyers must be equal for the two suppliers. If they are not, some consumers will keep shifting from [0,Y1) to [Y1, Y2) thus altering both Y1and Y2.

Therefore in equilibrium The two sides of equation (5) represent the share of the bribe paying buyers as also the average waiting time offered to them by the two suppliers. In equilibrium they are equal.

Given this equality, let db and dn denote the common delivery time offered by the two suppliers for bribe paying and non-bribe parts of the market, and let b1and b2 be their bribe rates. For the buyer at Y2 the

following holds:

Y2(Q-db)-p-b1 =Y2(Q-db)-p- b2, so that b1=b2.

Therefore the suppliers’ revenues given by b1(Y2-Y1) and b2(1-Y2) are also equal.

The equilibrium therefore implies that buyers in [Y1, 1] are equally shared by the two suppliers, serviced with identical average waiting and charged equal bribes.

Clearly this outcome is equivalent to setting Y1=Y2=Y, with each supplier selling at random to half the buyers in [Y, 1] with bribes and to half the buyers in [0,Y) without bribes.

Revenue maximizing equilibrium for both suppliers is given by the choice of Y1 (or Y2) that maximizes either b1(Y2-Y1) [or b2(1-Y2)]. Write the revenue function for 1 as Y1(dn-db)(Y2-Y1) =Y1[(Y2-Y1)2+Y1/2 –(Y2-Y1)](Y2-Y1). In view of (5), this simplifies to R=(1/8)Y1(1-Y1)(Y12+ 2Y1-1), with maximum at Y1= 0.7726. The equilibrium bribe rate = Y1(dn –db)=(1/4)Y1(Y1 +2Y1-1) equals 0.2205.

Compared to the one supplier case, the bribe rate is seen to fall significantly. But the proportion of the market that pays bribe is not significantly affected. The fall in the bribe rate, nearly a half, is pronounced because with two suppliers the average delay is halved in the bribe part and reduced by nearly half in the non-bribe part of the market. Since the bribe rate depends on the difference between the qualities, it falls significantly. This is entirely a scale effect resulting from introducing more resources. But competition

file:///A|/Workshop.htm (7 de 10) [15/12/2000 13:18:11] q1 = q0 + (n-y)

does not introduce any strategic element to undermine bribe taking as such.

The argument can be repeated with three suppliers to show that in equilibrium they will take bribes and equally share the bribe-paying segment. If y1y2 y3 are the cutoff points set by the three suppliers, then the requirement that in equilibrium the waiting time offered by each supplier to residual buyers must be equal, leads to 1-y3 = y3-y2 = y2-y1. This in turn would imply that the bribe rates and revenues of three suppliers are equal. Thus y1 separates bribe payers from residual buyers and will be set so as to maximize the identical revenue of the three suppliers.

In general suppose there are ν suppliers. Let Yν denote the separation point between bribe payers and residual buyers, so that each supplier sells with bribes to (1-Yν)/ν buyers, and without bribes to Yν/ν buyers. The average waiting times are [(1- Yν)/ν]2 +Yν/ν and (1-Yν)/ν for the non-bribe and bribe parts of the market. The revenue of individual suppliers is given by the sequence on ν Rν = (1/ν)Yν(1-Yν)[{(1- Yν)/ν}2 +Yν/ν- (1-Yν)/ν]. (6) The sequence of first order conditions for maximum can be written as 6Yν2 –6Yν +1 = (1/ν)(1-6Yν+9Yν2-4Yν3), (7) while the second order conditions are (2Yν-1) (-2Yν2+3Yν-1)/ν. (8) As ν→∞, (7) approaches the limiting equation 6Yν2 –6Yν +1 = 0, which solves as Yν* = 0.7886, where (8) is satisfied.

Thus bribes can not be eliminated by competition and at least 21per cent of the market remain under its

influence. We may summarize these observations in the following proposition:

Proposition 2: With increase in the number of suppliers bribes do not disappear and the share of the market under bribes asymptotically approaches a definite limit. But the bribe rate falls rapidly.

In view of proposition 2 that bribes are not eliminated by competition, arguably an alternative objective of the principal could be to reduce the average waiting time for residual buyers. Clearly as ν increases, the average waiting time for residual buyers [(1-Y*ν)2]/ν2 + Yν*/ν falls, and goes to zero in the limit.

However with increase in ν the principal introduces and pays for more delivery-related resources as wages. Therefore it is useful to get a measure of how waiting time improves in relation to additional resources spent.

If there were no bribes, the residual group of buyers would enjoy an average waiting time of Y*ν/ν with ν suppliers in the market. Let dn,ν denote their average waiting time when there is a bribe-taking equilibrium. The sequence [Cν] = [dn,ν/(Y*/ν) –1] = [(1-Y*2 )/νY*] provides an indication of how waiting time for residual buyers improves as more resources are spent. Table I shows the values of Yν* and Cν agaist ν.

Table I: Values of Y* and Cν against ν file:///A|/Workshop.htm (8 de 10) [15/12/2000 13:18:11] q1 = q0 + (n-y)

–  –  –

1 0.75 0.0833 2 0.7726 0.0335 3 0.7785 0.0210 4 0.7812 0.0153 5 0.7828 0.0120 6 0.7838 0.0099.........



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