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The optimal sequencing of carrots
Journal of Development Economics 93 (2010) 1–6
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Journal of Development Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / d eve c
The optimal sequencing of carrots Jennifer L. Steele School of Economic Sciences, Washington State University, USA
a r t i c l e
i n f o
Article history: Received 26 August 2008 Received in revised form 1 June 2009 Accepted 10 June 2009 JEL classification: F35 H40 Keywords: Development Aid Public projects Corruption
a b s t r a c t When aid organizations contract with local agents aid funds have the potential to be diverted to purposes other than the intended project. A multi-stage game is presented where the benefit from the project is cumulative, with the application of funds in each stage increasing both the agent's and the organization's benefit from the project. As the agent's utility of diversion increases, the allocation in each stage decreases and the project takes more stages to complete. When contracting with agents with high utilities of diversion the optimal contract involves bloated projects and a side payment to the agent upon completion. If the organization's commitment to the contract is not credible both the agent's and the organization's benefit is reduced. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Corruption is estimated to account for between $20 and $40 billion dollars of development assistance annually (World Bank, 1997). In India, a study of 217 locally procured contracts in the health industry showed that 88% contained one or more indicators of fraudulent and corrupt practices (World Bank, 2007). Burnside and Dollar (2000, 2004) examine the factors affecting aid effectiveness, and find that aid has a positive effect on growth when good governance and strong institutions are present. They posit that giving aid to countries with good governance, provides incentives for countries with poor governance to improve their governance. Their results also point towards the need for aid organizations to structure the agreements to take into account both the nature of the project, and the circumstances in the country. Corruption and the design of optimal contracts are increasingly discussed in the development community. As debt forgiveness and renegotiation are becoming common and countries are becoming overly burdened with debt, many in the aid community are turning to grants as a more efficient option. Bulow and Rogoff (2005) argue that multilateral development banks should switch from administering loans to administering grants, reasoning that the transparency of grants could improve efficiency and decrease renegotiation. In IDA-14, the agreement for IDA replenishment from 2006 to 2008, it was decided that 30% of total assistance will be in the form of grants
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(Weiss, 2005). This comes amid attempts by donor countries to make aid more dependent on outcomes. In many instances conditionality is discounted as implausible and is rarely adhered to in practice. Both the World Bank (1992) and Mosley et al. (1995) assert that funds are generally released ex post regardless of compliance. When rampant corruption in the Reproductive and Child Health I project in India became apparent World Bank president Wolfowitz suspended the project but reinstated funding less than a year later (WSJ, 2007). In an analysis of conditionality in Africa Kanbur (2000) finds that one of the necessary reforms for aid effectiveness is for donors to stand firm on aid conditionalities and not release funds unless the conditions are met. With respect to public good projects, many papers have multiple parties contributing funds (Admati and Perry, 1991, Marx and Matthews, 2000). Marx and Matthews (2000) look at equilibrium funding levels for public good projects when a number of agents are contributing. They find that successful completion is dependent on the number of periods, period length and the similarity of the agent's benefit functions from the project. Breaking the project up into periods and shorter period length can lead to successful completion. These papers concentrate on how to alleviate the public goods problem, without including an option for diversion. Unlike traditional corporate finance models of international lending (Atkeson (1991), Marcet and Marimon (1992)) in the model presented here there is no repayment, with the contract structured as a grant as opposed to a loan. Requiring repayment would lower the agent's utility, and therefore his incentives. Aid organizations that wish to fund large scale projects in developing countries must often contract with local officials to
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J.L. Steele / Journal of Development Economics 93 (2010) 1–6
undertake them. In Section 2 the basic environment is presented where the benefit from the project is cumulative with both the agent and aid organization (principal) sharing in the benefit from the project. When contracting with an agent a lack of enforcement mechanisms can make it difficult to induce the agent to apply the funds toward the project. Breaking the allocation into multiple stages allows the principal to make future allocations contingent on the agent's history. This induces the agent to apply the funds to the project by making future allocations conditional on doing so, while funding the project one step at a time. In Mozambique Durao and Pavignani (1997) observe increased donor demands over time. In this model the cumulative nature of the benefit function allows for increasing allocations as the agent's opportunity cost of diversion increases with the size of the project. In Section 2.3 the design of the optimal contract reveals that any additional benefit in the final stage increases the agent's incentives to apply the funds to the project in all previous stages. This leads to two important results; the first is that the project is overinvested in, the second is that it may be optimal to allow the agent to divert a portion of the funds. The optimal contract explicitly takes into account the level of governance and enforcement mechanisms. In Section 3 the manner in which an aid organization can increase their benefit from a contract are outlined. Transparency International (TI, 2006) suggests that debarment or blacklisting of corrupt agents is an effective sanction. In this model the inclusion of reputation costs allows debarment to be included in the optimal contract as a cost of diversion, increasing the organization's benefit from the project. In the benchmark model conditionality is assumed to be credible in order to discuss what could be achieved with full commitment on the part of the principal. Conditionality is represented by the principal's ability to condition future allocations on application of current funds to a defined project. Building stronger conditionality into contracts would discourage corruption and allow for shorter contracts, increasing the benefit to all parties. When conditionality is not credible, the threat of reversion to a payoff-reducing contract supports a renegotiation contract. In Malueg (1988) he finds that an insurance company's threat of reversion to a higher premium upon deviation supports the best full-information contract. In Section 4 the conditionality assumption is relaxed and the effects on the optimal contract and welfare are outlined.
2. The model Aid organizations often contract with local agents who derive positive utility from diverting funds. This section develops an optimal contract between an aid organization and a local government agency. The aid organization (principal) may allocate funds to the local agent over many stages, and may give the agent a side payment in the final stage if the agent diverts none of the funds to other uses. The principal's optimal contract is constructed to maximize her benefit from a defined project. The principal is funding the project and realizes the cost of capital, normalized to be the amount allocated in stage t, pt. Given allocation pt the agent may then choose to divert the funds or to apply them to the project. Both the agent and principal get the same benefit, b(pt) from the project in stage t if the agent does not divert funds.1 Pt represents the sum of allocations given in stages s ≤ t that were applied to the project.
1 Since the project is the same for the principal and agent, the pattern of benefit from the project is assumed to be the same for each. However, the scale may be different, in which case the agent's utility of diversion can be scaled to equate his benefit with that of the principal.
Assumption 2.1. b(·) is concave, weakly increasing and limp → ∞b′(P)= 0. Further, for some PN 0 b(P)NP. In Assumption 2.1 the principal's marginal benefit from the project is assumed to be weakly positive for all funding levels. Even a partially completed project has value. However, at some point additional resources have little effect on the benefit from the project, and the marginal benefit approaches zero. Assumption 2.1 also requires that the project is worth undertaking if there were no constraints, otherwise the principal would not optimally undertake the project. The agent has a positive marginal utility of diversion, x, but also suffers a reputation cost, Z N 0 if he diverts funds. The reputation cost parameter, Z, allows both the enforceability of contracts and the reputation cost to be built into the model. The agent and principal both discount the future at the same intertemporal discount rate δ ε (0,1). 2.1. Optimal contract Ex ante the principal outlines a contract comprised of a series of funding levels {pt}∞ t = 1. She commits to giving the agent pt in stage t, provided the agent applies the funds to the project in every stage k b t.2 If the agent applies the funds to the project in every stage k b t he receives allocation pt in stage t. If the agent chooses to divert funds in stage t the allocation in all future stages is zero and he forfeits any future benefit from the project. The principal continues to receive the benefit from all allocations applied to the project prior to t. Therefore if the agent deviates in t, he receives a payoff of xpt − Z from deviation, while the principal continues to receive the benefit from the project, b(Pt − 1), in all future stages. The optimal contract for the principal is the sequence of allocations that maximizes the following optimization problem: ∞
t
max ∑ δ ½bðPt Þ−pt
ð1Þ
fpt g∞ t=1 t=1
s.t. ∞
n
t
∑ δ bðPn Þ−δ ½xpt −Z≥0
for all t N 0
ð2aÞ
Pt = ∑ ps
for all t N 0:
ð2bÞ
n=t
s≤t
Eq. (2a) is the incentive compatibility constraint (ICC) for the agent. As long as it is met the agent will weakly prefer applying the funds to the project to diversion. If the constraint holds for all t, then we only need to compare diverting in t to always applying the funds to the project, rather than to diverting in some stage k ≠ t. 2.2. Side payments In certain situations the principal may want to give the agent side payments in order to induce the agent not to divert funds. Marx and Matthews (2000) incorporate a jump exogenously built into the benefit function, representing completion of the project, beyond which the marginal benefit of additional funding is zero. Like side payments, the jump provides additional incentive to the agent to complete the project, and increases cooperation. In the model presented here the side payment is costly for the principal, and
2 Ex ante commitment is optimal, and the principal is better off when commitment is credible, so it is implicitly assumed that the principal's cost of deviating is sufficiently high to prevent the principal from ever deviating. This assumption is relaxed in Section 4.
J.L. Steele / Journal of Development Economics 93 (2010) 1–6
therefore is only provided when the agent has a sufficiently high marginal utility of diversion. The agent's marginal utility from these side payments is x, the same as his marginal utility from diversion. In each period t the principal now gives the agent an allotment to apply towards the project, pt and a side payment, st. If the agent applies the allotment to the project and pockets the side payment he receives b(Pt) + xst. If he diverts the project funds he receives x(pt + st) − Z. The principal receives b(Pt) if the agent applies the allotment to the project, and b(Pt − 1) otherwise. In this case, the principal solves the following optimization problem: max ∞
∞
t
∑ δ ðbðPt Þ−pt −st Þ
ð3Þ
fpt ;st gt = 1 t = 1
s.t. ∞
n
t
∑ δ ðbðPn Þ + xsn Þ−δ ½xðpt + st Þ−Z≥0
for all t
ð4aÞ
pt ; st ≥0
for all t:
ð4bÞ
n=t
Eq. (4a) is similar to the incentive compatibility constraint seen earlier, but now allows for a positive side payment. If a principal chooses to give the agent a side payment she will delay the side payment until the last stage with a positive project allocation (pj). The following is a sketch of the proof: 1) The principal prefers to fund the project as early as possible. In each stage her benefit is an increasing function of the cumulative amount of funding. Therefore the principal would prefer to allocate an additional dollar today instead of tomorrow. 2) The agent's ICC must then bind in any stage t when the allocation in t + 1 is positive. 3) A side payment of st in stage t has the same cost as a side payment of δ− kst in stage k N t. It also has the same effect on the agent's ICC (i.e. the agent is indifferent between receiving st in stage t and δ− kst in stage k). However, the side payment in stage k also increases the possible project allocation in stage t. By delaying any side payment until the last stage with a positive allocation the principal is able to maximize the incentives for the agent to apply the funds to the project in all previous stages. Thus a contract with side payments can simply be written as the basic contract, with one side payment of sj in the last stage that includes a positive allocation.
3
Given that the ICC binds in every stage t b j and that there are no side payments in any stage t b j, the FOC for pt becomes ∞
s ′
t
∑ δ b ðPn Þ−δ
ð5aÞ
n=t
∞
n ′
t
t−1
∞
s≤t
n=t
n ′
+ λt ½ ∑ δ b ðPn Þ−δ x + ∑ λs ½ ∑ δ b ðPn Þ n=t
+
∞
∑
s=t + 1
∞
n ′
λs ½ ∑ δ b ðPn Þ≤0: n=s
The first line represents the direct marginal benefit to the principal in terms of the marginal benefit from the project, and the marginal cost of the allocation. The second and third lines represent the effects on the agent's incentives. The first term is the effect on the agent's ICC in stage t, the additional benefit from the project less the additional utility of diversion. The second term is the effect on all previous stages. An additional dollar in stage t makes the agent less likely to divert funds in any stage s b t because the payoff from future allocations increases. The final term is the effect on all future stages. An additional dollar in stage t increases the opportunity cost of diversion in all future stages. The allocation in each stage t b j is the allocation which solves
½
∞
n−t
∑ δ
n=t
bðPn Þ + Z
1x = p : t
ð6Þ
The allocation in each stage decreases as the agent's marginal utility of diversion increases, and as a result j will increase, taking more stages to complete the project. With a cumulative benefit function the opportunity cost of diverting funds is greater in later stages, allowing for larger allocations as the project progresses. If the agent has a sufficiently large marginal utility of diversion relative to his utility from the project, it may be optimal for the principal to give the agent a side payment in the last stage with a positive allocation. The principal would allow the agent to divert a specified portion of the allocation in the last stage, provided he applied all previous allocations to the project. Therefore the optimal contract involves an increasing sequence of allocations, with the possibility of a side payment in the final stage. The side payment will only be optimal if the agent has a sufficiently high marginal utility of diversion, an x greater than some cutoff point, xc. This cutoff point can be found by looking at the FOC for sj which becomes binding when c
x =
δj j−1
:
ð7Þ
t
∑ λt δ
t =1
2.3. Characterizing the optimal contract There must be some finite stage j, after which all allocations are zero. The intuition for this is twofold. First, the principal's benefit from the project is cumulative, so she strictly prefers funding the project earlier rather than later. Second, from Assumption 2.1 there comes a point where the marginal benefit from the project becomes sufficiently small that the principal no longer wishes to allocate any funds for it (the marginal cost outweighs the marginal benefit). The principal will allocate funds up until the agent's ICC binds in each stage, until some point where the marginal benefit from funding the project equals the marginal cost without the agent's ICC binding (stage j). Proposition 2.2. The principal's optimal contract will consist of a sequence of allocations, {pt}jt = 1 such that the agent is indifferent between applying the allocation to the project and to diversion in all stages t b j, plus a non-negative side payment sj in the final stage.
Definition 2.3. The socially optimal funding level, P°, is the funding level at which an efficient level of funding is invested. The socially optimal funding level is where: ′
b ðP-Þ = 1: 1−δ
ð8Þ
Any funding level beyond P° involves investment where the marginal benefit from the project is less than the marginal cost, and is therefore wasteful. The socially optimal contract is the contract where the allocation in the first stage is the socially optimal funding level, p1 = P°, and all subsequent allocations are zero. This is the best the principal can do, and is the contract she would offer if it did not induce diversion.
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By comparison, a contract where the project is funded in more than one stage will result in overinvestment. The total funding level of the project will exceed the socially optimal funding level. Agents with high marginal utilities of diversion not only slow down completion of the project by requiring multiple stages, but they also force the principal to overinvest in the project, creating bloated projects. To see this we look at the FOC for stage j, the last stage with positive funding, j−1 b′ ðPj Þ b′ ðPj Þ −1 + ∑ λn ≤0: 1−δ 1−δ n=1
stages weakly decreases. The increased allocation in each stage along with earlier completion increases the principal's net benefit from the project. With sufficient enforcement mechanisms the principal can obtain the socially optimal contract, with all funding occurring in the first stage. The socially optimal contract gives the principal the highest possible net benefit from the project. For a given marginal utility of diversion, x̂, the reputation cost at which the socially optimal contract is enforceable is
ð9Þ Z = x̂ P-−
In this case, if the total funding level were the same as the socially optimal funding level, then the incentive effects would have to be non-positive for the FOC to hold. However, an additional dollar in the final stage has positive incentive effects on all previous stages, so the summation of the incentive effects must be positive. The direct marginal benefit from the project must then be smaller than under the socially optimal contract, and the total funding level is therefore greater. The principal will overinvest in the project because doing so allows her to allocate more in earlier stages due to the incentive not to divert funds additional funding in the final stage creates. It is a tradeoff for the principal between additional funding in early stages and overinvestment in the last stage. For agents with sufficiently low marginal utilities of diversion, the socially optimal contract can be obtained. x° is the marginal utility of diversion at which the agent is indifferent between diversion and applying funds to the project given the socially optimal contract. It is defined as follows: x- =
bðP-Þ 1 +Z : 1−δ P-
ð10Þ
The marginal utility of diversion at which the agent is indifferent is increasing with his reputation cost and the discount rate. If x b x° the socially optimal contract can be successfully implemented. If x N x° the optimal contract will consist of more than one stage with positive allocations. Therefore agents can be separated into three groups. Those with sufficiently low marginal utilities of diversion, x ≤ x°, such that the socially optimal contract can be undertaken. Those with moderate marginal utilities of diversion, x ε (x°, xc] where the optimal contract will take more than one stage, and involve overinvestment. The final group has sufficiently high marginal utilities, x N xc such that not only does the contract involve multiple stages and overinvestment, but it is also optimal for the principal to give the agent a side payment upon completion. 3. Application to an aid environment As the principal's ability to punish the agent increases, the cost of diverting funds, Z, also increases. If there is a possibility of a future relationship between the principal and agent, making future projects conditional on completion of the current project can increase the cost of diversion for the agent. As Z increases the amount that the principal can allocate in each stage increases as well. The agent's ICC for the first stage is ∞
n−t
xp1 − ∑ δ n=1
bðPn Þ = Z:
ð11Þ
Due to the concavity of the benefit function, an increase in Z will allow the principal to increase the allocation in the first stage and all subsequent stages. This allows the principal to complete the project weakly earlier, as the allocation in each stage increases the number of
bðP-Þ : 1−δ
ð12Þ
An aid organization's benefit is maximized with the socially optimal contract, so increasing Z whether with the credible threat of punitive legal action after diversion, or the promise of future projects upon completion will increase their benefit from the project. Another way to increase efficiency, and shorten the length of time required to complete a project is by making the stages shorter. By increasing δ honesty becomes more valuable to the agent as the discounted value of future allocations is now greater. In practice there is likely a minimum stage length, dictated by the time required to monitor the agent and ensure compliance, and the length of time it takes for the defined portion of the project to be completed. From Eq. (7), as δ decreases, the marginal utility of diversion above which the agent gets a side payment increases. Diversion in stage t gives the agent a high payoff only in stage t. He must compare that to applying the funds to the project, and receiving the future benefit in all stages k ≥ t. As a result, with shorter stages the agent values his future utility more, is less likely to divert, and the marginal utility of diversion above which the agent would receive a side payment increases. 4. Non-credible threats and broken promises The optimal contract described in Section 2 depends on two key assumptions that will be relaxed here. The first is that the principal can credibly commit to cutting off all allocations upon diversion. The second is that the principal can commit to overinvesting in the project. Relaxing either assumption reduces both the principal's and agent's welfare. 4.1. Renegotiation If the agent does not believe that the principal will cease all allocations upon diversion, the optimal contract is no longer enforceable. In the optimal contract without renegotiation the agent's ICC is dependent upon the principal ceasing all allocations upon diversion. Given an optimal contract without renegotiation where x N x° the agent will always divert funds if the principal's threats are not credible. In this case, the most the principal can credibly threaten is to revert to some ‘threat’ contract upon diversion. By threatening reversion, she cannot achieve the optimal contract found in Section 2, but can achieve a ‘renegotiation’ contract that pareto dominates having no recourse upon diversion. In order to find the optimal contract with renegotiation we first define the threat contract and ensure that it is credible, then define the renegotiation contract that can be enforced by threatening reversion. If the agent diverts funds in stage k, the principal will revert to the threat contract for the remainder of the project. The threat contract is conditional on the total investment in the project prior to diversion, and the stage in which the agent diverts funds. If the agent diverts funds in stage k, in any stage t ≥ k + 1 he must choose between
J.L. Steele / Journal of Development Economics 93 (2010) 1–6
applying funds to the project, and diverting them, putting off investment in the project until the next stage. ∞
s−t
∑ δ
s=t
∞
s−t
bðPs;k Þ≥xpt;k + δ∑ δ s=t
bðPs;k Þ ∀t≥k + 1;
ð13Þ
where pt,k is the allocation in stage t following diversion in k, and Ps,k is the sum of all allocations applied towards the project up to and including stage s N k. In the threat contract, the most the principal can allocate in any stage without inducing diversion is the amount where the agent is indifferent between applying the funds to the project and diverting them. Any contract which gives the agent a lower payoff necessarily involves lower allocations, and is not credible as the principal can revert to the threat contract upon diversion and do strictly better. Once again the ICC must bind in every stage up to the final stage with a positive allocation, j. Following diversion in stage k, the allocation in any stage t ε (k, j) is ∞ 1 s−t ð1−δÞ∑ δ bðPs;k Þ = pt;k : x s=t
ð14Þ
4.2. No commitment When the principal is unable to commit to a stream of allocations, she is necessarily worse off whenever the project requires more than one stage.3 The optimal contract outlined earlier commits to a level above the socially optimal funding level and is therefore not credible if the principal cannot commit to overinvestment. Likewise, a commitment to any amount below the socially optimal amount is both not credible, and welfare reducing. Given the optimal contract the agent will choose to divert, knowing that the principal's promise to overinvest is not credible. In order to find the optimal contract without being able to overinvest, we take the total funding level as given and find the number of stages it takes to get there with the ICC binding in every stage t b j. The total funding level, number of stages and the allocation in each stage are the unique solution to the following set of equations: b′ ðPj Þ =1 1−δ
ð19aÞ
In stage j, the final stage with a positive allocation, the allocation is defined by the principal's FOC:
pt =
b′ ðPj;k Þ −1 + 1−δ
pj = Pj −pj−1 :
j−1
∑
n=k + 1
λn ð1−δÞ
b′ ðPj;k Þ = 0: 1−δ
ð15Þ
Comparing this equation to Eq. (9) we see that due to the increased incentive additional allocations provided for diversion, the total funding level in the threat contract is strictly less than in the optimal contract without renegotiation. The renegotiation contract can now be defined as the contract giving the principal the highest possible payoff, enforced by threatening reversion to the threat contract. The agent's benefit from diversion now includes the net present value of the threat contract received following diversion. In order to make the strongest case for the renegotiation contract and explore the worst-case scenario the reputation cost has been set to Z = 0. The agent's ICC in any stage t can be written as follows: ∞
s−t
∑ δ
s=t
r
r
bðPs Þ≥xpt +
∞
∑
s=t + 1
s−t
δ
bðPs;t Þ;
ð16Þ
where prt is the allocation in the renegotiation contract, when the agent has not diverted funds in any stage k b t, and Prt = ∑ts = 1prt. In the renegotiation contract, the allocation in all stages t b j is
½
∞ 1 ∞ s−t r s−t r ∑ δ bðPs Þ− ∑ δ bðPs;t Þ = pt : x s=t s=t + 1
ð17Þ
Upon diversion the agent not only gains the payoff from diversion, but also retains the present value of the threat contract. Without renegotiation the agent only gets the payoff from diversion, with no future contract. Therefore the allocation in each stage t b j of the contract without renegotiation is strictly greater than in the renegotiation contract,
½
½
∞ 1 ∞ s−t 1 ∞ s−t r s−t ∑ δ bðPs Þ N ∑ δ bðPs Þ− ∑ δ bðPs;t Þ : x s=t x s=t s=t + 1
ð18Þ
In the contract with renegotiation each funding level takes longer to reach than in the contract without renegotiation and as a result both the agent's and the principal's benefits are reduced.
5
j−1
δ j−t bðPj Þ bðPs Þ + Z + xð1−δÞ x
s−t
∑s = t δ
for all t b j
ð19bÞ
ð19cÞ
The allocation in each stage where the ICC binds, pt, is smaller than in the benchmark model because the principal is unable to commit to overinvestment. As a result the project takes longer to reach each funding level, P, and both the principal and agent are worse off. The agent is clearly worse off because his benefit in each stage is lower. Implicitly the principal is worse off, if this contract increased her benefit the optimal contract found earlier would not have been a maximum. 5. Conclusion When an aid agency wishes to fund a project through grants, contracting with a local agent can lead to diversion due to insufficient enforcement mechanisms. Grants are becoming an increasingly popular means of international aid, but I show that when dealing with agents who have high marginal utilities of diversion, care needs to be paid to the structure of the contract. We see that as countries differ in both governance and enforcement mechanisms (institutions), the optimal contract must adjust as well. Given two identical projects the benefit from the optimal contract in a country with good governance is strictly greater than that from a country with poor governance. Unlike traditional international lending models, where loans are repaid, and the benefit from the project undertaken is not shared, the need for incentives leads to overinvestment. Bloated projects can therefore be optimal, because the allocation in the final stage provides incentives for the agent not to divert funds in all previous stages. Compared to the socially optimal contract, where the project is fully funded in the first stage, when dealing with agents who have an incentive to divert funds, the project will take more stages and be larger. When the agent derives little benefit from the project relative to diversion it may be optimal for the contract to specify an allocation in the final stage that the agent is allowed to divert. By promising an allocation in the last stage that the agent may keep, the principal is able to give larger allocations in early stages. 3
When x b x°, the socially optimal contract is optimal with or without commitment.
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J.L. Steele / Journal of Development Economics 93 (2010) 1–6
This model suggests that aid agencies should break large allocations up into smaller ones, increasing in size over time, and concentrate on increasing the reputation cost for the agent. A larger reputation cost allows the project to be completed earlier, with a larger allocation in each stage. This can be achieved by making future projects conditional on successful completion of the current one. It can also be achieved by increasing the capacity to punish the agent. One way of increasing the threat of punishment is to locate and seize the proceeds of diversion. If the agent believes there's a positive probability of the proceeds being confiscated, diversion becomes more costly. Shortening the length of each stage increases both the aid agency's and the agent's benefit. However, the stage length may be constrained by the length of time it takes to complete the required portion of the project, or the cost or time required to monitor. When conditionality is not credible, two separate results are obtained. The first is that when the threat to cease all allocations upon diversion is not credible, the principal can offer a renegotiation contract supported by the threat of reversion to a worst-case threat contract. The inability to commit in this manner lowers both the agent's and the principal's benefit from the project, and reduces the set of types for whom the principal will offer a contract. The second result looks at the case where the principal cannot commit to overinvestment. With the optimal contract the principal would be better off reneging in stage j and offering some allocation strictly less than the contract amount. In this case the principal's promise of pj in the optimal contract is not credible, and the allocation supportable in any stage t b j is lowered. In either of these two cases, the benefit from the project is diminished for both the principal and the agent. The aid agency is better off if they stick to their guns, with respect to both their threats and their promises. Acknowledgments For their help and patience on an earlier version of this paper I would like to thank Max Stinchcombe and Tom Wiseman. I am grateful to Dilip Mookherjee, two anonymous referees and participants at various seminars for their comments and discussion. Appendix A. Renegotiation contract When the principal is unable to commit to ceasing all allocations upon diversion, two maximization problems are needed. The first step is to identify the maximization problem for the threat contract for any stage k ≤ j: Threat contract maximization problem: ∞
maxfpt g∞
t=k+1
s
∑
s=k + 1
δ ½bðPs;k Þ−ps;k
ð20Þ
s.t. ∞
s−t
∑ δ
s=t
∞
s−t
bðPs;k Þ−½xpt;k + δ∑ δ s=t
k−1
r
Pt;k = ∑ ps + s=1
t
∑
s=k + 1
ps;k
bðPs;k Þ≥0
for all t N k
ð21aÞ
for all t N k
ð21bÞ
The solution to threat contract maximization problem gives us a set of allocations, {pt,k}t∞= k + 1 for all stages k ≤ j as a function of the r allocations applied to the project prior to k, Pk − 1. Next, we use the threat contract in the renegotiation contract maximization problem: ∞
maxfprt g∞
t=1
s
r
r
∑ δ ½bðPs Þ−ps
ð22Þ
s=1
s.t. ∞
s−t
∑ δ
s=t
r
r
t
∞
r
bðPs Þ−½xpt +
∑
s=t + 1
r
Pt = ∑ ps
s−t
δ
bðPs;t Þ≥0
for all t
ð23aÞ
for all t
ð23bÞ
s=1
t−1
r
Ps;t = ∑ pn + n=1
s
∑
n=t + 1
pn;t
for all t ≤ j; s N t ð23cÞ
References Admati, A., Perry, M., 1991. Joint projects without commitment. Review of Economic Studies 58 (2), 259–276. Atkeson, A., 1991. International lending with moral hazard and risk of repudiation. Econometrica 59 (4), 1069–1089. Bulow, J., Rogoff, K., 2005. Grants versus Loans for Development Banks. American Economic Review 95 (2), 393–397. Burnside, Craig, Dollar, D., 2000. Aid, policies, and growth. American Economic Review 90, 847–868. September. Burnside, C., Dollar, D., 2004. Aid, policies, and growth: revisiting the evidence. In: PR Working Paper, vol. 3251. World Bank. March. Durao, U.R., Pavignani, E., 1997. Aid, Change and Second Thoughts: Coordinating External Resources to the Health Sector in Mozambique (Annotation). Ministry of Health, Maputo. Kanbur, R., 2000. Aid, conditionality and debt in Africa. Revised version published in Finn Tarp (ed) Foreign Aid and Development: Lessons Learnt and Directions for the Future. InRoutledge. Malueg, D.A., 1988. Repeated insurance contracts with differential learning. Review of Economic Studies 55 (1), 177–181. Marcet, A., Marimon, R., 1992. Communication, commitment and growth. Journal of Economic Theory 58 (2), 219–249. Marx, L., Matthews, S., 2000. Dynamic voluntary contribution to a public project. Review of Economic Studies 67 (2), 327–358. Mosley, P., Harrigan, J., Toye, J., 1995. Aid and Power: the World Bank and Policy Based Lending2nd edition. Routledge. TI, 2006. Curbing corruption in public procurement. Handbook. Transparency International. Weiss, M., 2005. The World Bank: the International Development Association's 14th Replenishment (2006–2008). Technical Report, CRS Report to Congress. World Bank, 1992. World Bank structural and sectoral adjustment operations: the second OED review. Operations Evaluation Department Report No. 10870. InWorld Bank, Washington, D.C. World Bank, 1997. Stolen Asset Recovery (StAR) Initiative: Challenges, Opportunities and Action Plan. World Bank, Washington, DC. World Bank, 2007. Detailed Implementation Review: India Health Sector. Volume II. World Bank, Washington, DC. WSJ, 2007. World Bank Corruption: Bribery in India, and a Test for Bob Zoellick, editorial, Wall Street Journal, September 4, 2007.
Contents lists available at ScienceDirect
Journal of Development Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / d eve c
The optimal sequencing of carrots Jennifer L. Steele School of Economic Sciences, Washington State University, USA
a r t i c l e
i n f o
Article history: Received 26 August 2008 Received in revised form 1 June 2009 Accepted 10 June 2009 JEL classification: F35 H40 Keywords: Development Aid Public projects Corruption
a b s t r a c t When aid organizations contract with local agents aid funds have the potential to be diverted to purposes other than the intended project. A multi-stage game is presented where the benefit from the project is cumulative, with the application of funds in each stage increasing both the agent's and the organization's benefit from the project. As the agent's utility of diversion increases, the allocation in each stage decreases and the project takes more stages to complete. When contracting with agents with high utilities of diversion the optimal contract involves bloated projects and a side payment to the agent upon completion. If the organization's commitment to the contract is not credible both the agent's and the organization's benefit is reduced. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Corruption is estimated to account for between $20 and $40 billion dollars of development assistance annually (World Bank, 1997). In India, a study of 217 locally procured contracts in the health industry showed that 88% contained one or more indicators of fraudulent and corrupt practices (World Bank, 2007). Burnside and Dollar (2000, 2004) examine the factors affecting aid effectiveness, and find that aid has a positive effect on growth when good governance and strong institutions are present. They posit that giving aid to countries with good governance, provides incentives for countries with poor governance to improve their governance. Their results also point towards the need for aid organizations to structure the agreements to take into account both the nature of the project, and the circumstances in the country. Corruption and the design of optimal contracts are increasingly discussed in the development community. As debt forgiveness and renegotiation are becoming common and countries are becoming overly burdened with debt, many in the aid community are turning to grants as a more efficient option. Bulow and Rogoff (2005) argue that multilateral development banks should switch from administering loans to administering grants, reasoning that the transparency of grants could improve efficiency and decrease renegotiation. In IDA-14, the agreement for IDA replenishment from 2006 to 2008, it was decided that 30% of total assistance will be in the form of grants
E-mail address: [email protected]. 0304-3878/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jdeveco.2009.06.003
(Weiss, 2005). This comes amid attempts by donor countries to make aid more dependent on outcomes. In many instances conditionality is discounted as implausible and is rarely adhered to in practice. Both the World Bank (1992) and Mosley et al. (1995) assert that funds are generally released ex post regardless of compliance. When rampant corruption in the Reproductive and Child Health I project in India became apparent World Bank president Wolfowitz suspended the project but reinstated funding less than a year later (WSJ, 2007). In an analysis of conditionality in Africa Kanbur (2000) finds that one of the necessary reforms for aid effectiveness is for donors to stand firm on aid conditionalities and not release funds unless the conditions are met. With respect to public good projects, many papers have multiple parties contributing funds (Admati and Perry, 1991, Marx and Matthews, 2000). Marx and Matthews (2000) look at equilibrium funding levels for public good projects when a number of agents are contributing. They find that successful completion is dependent on the number of periods, period length and the similarity of the agent's benefit functions from the project. Breaking the project up into periods and shorter period length can lead to successful completion. These papers concentrate on how to alleviate the public goods problem, without including an option for diversion. Unlike traditional corporate finance models of international lending (Atkeson (1991), Marcet and Marimon (1992)) in the model presented here there is no repayment, with the contract structured as a grant as opposed to a loan. Requiring repayment would lower the agent's utility, and therefore his incentives. Aid organizations that wish to fund large scale projects in developing countries must often contract with local officials to
2
J.L. Steele / Journal of Development Economics 93 (2010) 1–6
undertake them. In Section 2 the basic environment is presented where the benefit from the project is cumulative with both the agent and aid organization (principal) sharing in the benefit from the project. When contracting with an agent a lack of enforcement mechanisms can make it difficult to induce the agent to apply the funds toward the project. Breaking the allocation into multiple stages allows the principal to make future allocations contingent on the agent's history. This induces the agent to apply the funds to the project by making future allocations conditional on doing so, while funding the project one step at a time. In Mozambique Durao and Pavignani (1997) observe increased donor demands over time. In this model the cumulative nature of the benefit function allows for increasing allocations as the agent's opportunity cost of diversion increases with the size of the project. In Section 2.3 the design of the optimal contract reveals that any additional benefit in the final stage increases the agent's incentives to apply the funds to the project in all previous stages. This leads to two important results; the first is that the project is overinvested in, the second is that it may be optimal to allow the agent to divert a portion of the funds. The optimal contract explicitly takes into account the level of governance and enforcement mechanisms. In Section 3 the manner in which an aid organization can increase their benefit from a contract are outlined. Transparency International (TI, 2006) suggests that debarment or blacklisting of corrupt agents is an effective sanction. In this model the inclusion of reputation costs allows debarment to be included in the optimal contract as a cost of diversion, increasing the organization's benefit from the project. In the benchmark model conditionality is assumed to be credible in order to discuss what could be achieved with full commitment on the part of the principal. Conditionality is represented by the principal's ability to condition future allocations on application of current funds to a defined project. Building stronger conditionality into contracts would discourage corruption and allow for shorter contracts, increasing the benefit to all parties. When conditionality is not credible, the threat of reversion to a payoff-reducing contract supports a renegotiation contract. In Malueg (1988) he finds that an insurance company's threat of reversion to a higher premium upon deviation supports the best full-information contract. In Section 4 the conditionality assumption is relaxed and the effects on the optimal contract and welfare are outlined.
2. The model Aid organizations often contract with local agents who derive positive utility from diverting funds. This section develops an optimal contract between an aid organization and a local government agency. The aid organization (principal) may allocate funds to the local agent over many stages, and may give the agent a side payment in the final stage if the agent diverts none of the funds to other uses. The principal's optimal contract is constructed to maximize her benefit from a defined project. The principal is funding the project and realizes the cost of capital, normalized to be the amount allocated in stage t, pt. Given allocation pt the agent may then choose to divert the funds or to apply them to the project. Both the agent and principal get the same benefit, b(pt) from the project in stage t if the agent does not divert funds.1 Pt represents the sum of allocations given in stages s ≤ t that were applied to the project.
1 Since the project is the same for the principal and agent, the pattern of benefit from the project is assumed to be the same for each. However, the scale may be different, in which case the agent's utility of diversion can be scaled to equate his benefit with that of the principal.
Assumption 2.1. b(·) is concave, weakly increasing and limp → ∞b′(P)= 0. Further, for some PN 0 b(P)NP. In Assumption 2.1 the principal's marginal benefit from the project is assumed to be weakly positive for all funding levels. Even a partially completed project has value. However, at some point additional resources have little effect on the benefit from the project, and the marginal benefit approaches zero. Assumption 2.1 also requires that the project is worth undertaking if there were no constraints, otherwise the principal would not optimally undertake the project. The agent has a positive marginal utility of diversion, x, but also suffers a reputation cost, Z N 0 if he diverts funds. The reputation cost parameter, Z, allows both the enforceability of contracts and the reputation cost to be built into the model. The agent and principal both discount the future at the same intertemporal discount rate δ ε (0,1). 2.1. Optimal contract Ex ante the principal outlines a contract comprised of a series of funding levels {pt}∞ t = 1. She commits to giving the agent pt in stage t, provided the agent applies the funds to the project in every stage k b t.2 If the agent applies the funds to the project in every stage k b t he receives allocation pt in stage t. If the agent chooses to divert funds in stage t the allocation in all future stages is zero and he forfeits any future benefit from the project. The principal continues to receive the benefit from all allocations applied to the project prior to t. Therefore if the agent deviates in t, he receives a payoff of xpt − Z from deviation, while the principal continues to receive the benefit from the project, b(Pt − 1), in all future stages. The optimal contract for the principal is the sequence of allocations that maximizes the following optimization problem: ∞
t
max ∑ δ ½bðPt Þ−pt
ð1Þ
fpt g∞ t=1 t=1
s.t. ∞
n
t
∑ δ bðPn Þ−δ ½xpt −Z≥0
for all t N 0
ð2aÞ
Pt = ∑ ps
for all t N 0:
ð2bÞ
n=t
s≤t
Eq. (2a) is the incentive compatibility constraint (ICC) for the agent. As long as it is met the agent will weakly prefer applying the funds to the project to diversion. If the constraint holds for all t, then we only need to compare diverting in t to always applying the funds to the project, rather than to diverting in some stage k ≠ t. 2.2. Side payments In certain situations the principal may want to give the agent side payments in order to induce the agent not to divert funds. Marx and Matthews (2000) incorporate a jump exogenously built into the benefit function, representing completion of the project, beyond which the marginal benefit of additional funding is zero. Like side payments, the jump provides additional incentive to the agent to complete the project, and increases cooperation. In the model presented here the side payment is costly for the principal, and
2 Ex ante commitment is optimal, and the principal is better off when commitment is credible, so it is implicitly assumed that the principal's cost of deviating is sufficiently high to prevent the principal from ever deviating. This assumption is relaxed in Section 4.
J.L. Steele / Journal of Development Economics 93 (2010) 1–6
therefore is only provided when the agent has a sufficiently high marginal utility of diversion. The agent's marginal utility from these side payments is x, the same as his marginal utility from diversion. In each period t the principal now gives the agent an allotment to apply towards the project, pt and a side payment, st. If the agent applies the allotment to the project and pockets the side payment he receives b(Pt) + xst. If he diverts the project funds he receives x(pt + st) − Z. The principal receives b(Pt) if the agent applies the allotment to the project, and b(Pt − 1) otherwise. In this case, the principal solves the following optimization problem: max ∞
∞
t
∑ δ ðbðPt Þ−pt −st Þ
ð3Þ
fpt ;st gt = 1 t = 1
s.t. ∞
n
t
∑ δ ðbðPn Þ + xsn Þ−δ ½xðpt + st Þ−Z≥0
for all t
ð4aÞ
pt ; st ≥0
for all t:
ð4bÞ
n=t
Eq. (4a) is similar to the incentive compatibility constraint seen earlier, but now allows for a positive side payment. If a principal chooses to give the agent a side payment she will delay the side payment until the last stage with a positive project allocation (pj). The following is a sketch of the proof: 1) The principal prefers to fund the project as early as possible. In each stage her benefit is an increasing function of the cumulative amount of funding. Therefore the principal would prefer to allocate an additional dollar today instead of tomorrow. 2) The agent's ICC must then bind in any stage t when the allocation in t + 1 is positive. 3) A side payment of st in stage t has the same cost as a side payment of δ− kst in stage k N t. It also has the same effect on the agent's ICC (i.e. the agent is indifferent between receiving st in stage t and δ− kst in stage k). However, the side payment in stage k also increases the possible project allocation in stage t. By delaying any side payment until the last stage with a positive allocation the principal is able to maximize the incentives for the agent to apply the funds to the project in all previous stages. Thus a contract with side payments can simply be written as the basic contract, with one side payment of sj in the last stage that includes a positive allocation.
3
Given that the ICC binds in every stage t b j and that there are no side payments in any stage t b j, the FOC for pt becomes ∞
s ′
t
∑ δ b ðPn Þ−δ
ð5aÞ
n=t
∞
n ′
t
t−1
∞
s≤t
n=t
n ′
+ λt ½ ∑ δ b ðPn Þ−δ x + ∑ λs ½ ∑ δ b ðPn Þ n=t
+
∞
∑
s=t + 1
∞
n ′
λs ½ ∑ δ b ðPn Þ≤0: n=s
The first line represents the direct marginal benefit to the principal in terms of the marginal benefit from the project, and the marginal cost of the allocation. The second and third lines represent the effects on the agent's incentives. The first term is the effect on the agent's ICC in stage t, the additional benefit from the project less the additional utility of diversion. The second term is the effect on all previous stages. An additional dollar in stage t makes the agent less likely to divert funds in any stage s b t because the payoff from future allocations increases. The final term is the effect on all future stages. An additional dollar in stage t increases the opportunity cost of diversion in all future stages. The allocation in each stage t b j is the allocation which solves
½
∞
n−t
∑ δ
n=t
bðPn Þ + Z
1x = p : t
ð6Þ
The allocation in each stage decreases as the agent's marginal utility of diversion increases, and as a result j will increase, taking more stages to complete the project. With a cumulative benefit function the opportunity cost of diverting funds is greater in later stages, allowing for larger allocations as the project progresses. If the agent has a sufficiently large marginal utility of diversion relative to his utility from the project, it may be optimal for the principal to give the agent a side payment in the last stage with a positive allocation. The principal would allow the agent to divert a specified portion of the allocation in the last stage, provided he applied all previous allocations to the project. Therefore the optimal contract involves an increasing sequence of allocations, with the possibility of a side payment in the final stage. The side payment will only be optimal if the agent has a sufficiently high marginal utility of diversion, an x greater than some cutoff point, xc. This cutoff point can be found by looking at the FOC for sj which becomes binding when c
x =
δj j−1
:
ð7Þ
t
∑ λt δ
t =1
2.3. Characterizing the optimal contract There must be some finite stage j, after which all allocations are zero. The intuition for this is twofold. First, the principal's benefit from the project is cumulative, so she strictly prefers funding the project earlier rather than later. Second, from Assumption 2.1 there comes a point where the marginal benefit from the project becomes sufficiently small that the principal no longer wishes to allocate any funds for it (the marginal cost outweighs the marginal benefit). The principal will allocate funds up until the agent's ICC binds in each stage, until some point where the marginal benefit from funding the project equals the marginal cost without the agent's ICC binding (stage j). Proposition 2.2. The principal's optimal contract will consist of a sequence of allocations, {pt}jt = 1 such that the agent is indifferent between applying the allocation to the project and to diversion in all stages t b j, plus a non-negative side payment sj in the final stage.
Definition 2.3. The socially optimal funding level, P°, is the funding level at which an efficient level of funding is invested. The socially optimal funding level is where: ′
b ðP-Þ = 1: 1−δ
ð8Þ
Any funding level beyond P° involves investment where the marginal benefit from the project is less than the marginal cost, and is therefore wasteful. The socially optimal contract is the contract where the allocation in the first stage is the socially optimal funding level, p1 = P°, and all subsequent allocations are zero. This is the best the principal can do, and is the contract she would offer if it did not induce diversion.
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J.L. Steele / Journal of Development Economics 93 (2010) 1–6
By comparison, a contract where the project is funded in more than one stage will result in overinvestment. The total funding level of the project will exceed the socially optimal funding level. Agents with high marginal utilities of diversion not only slow down completion of the project by requiring multiple stages, but they also force the principal to overinvest in the project, creating bloated projects. To see this we look at the FOC for stage j, the last stage with positive funding, j−1 b′ ðPj Þ b′ ðPj Þ −1 + ∑ λn ≤0: 1−δ 1−δ n=1
stages weakly decreases. The increased allocation in each stage along with earlier completion increases the principal's net benefit from the project. With sufficient enforcement mechanisms the principal can obtain the socially optimal contract, with all funding occurring in the first stage. The socially optimal contract gives the principal the highest possible net benefit from the project. For a given marginal utility of diversion, x̂, the reputation cost at which the socially optimal contract is enforceable is
ð9Þ Z = x̂ P-−
In this case, if the total funding level were the same as the socially optimal funding level, then the incentive effects would have to be non-positive for the FOC to hold. However, an additional dollar in the final stage has positive incentive effects on all previous stages, so the summation of the incentive effects must be positive. The direct marginal benefit from the project must then be smaller than under the socially optimal contract, and the total funding level is therefore greater. The principal will overinvest in the project because doing so allows her to allocate more in earlier stages due to the incentive not to divert funds additional funding in the final stage creates. It is a tradeoff for the principal between additional funding in early stages and overinvestment in the last stage. For agents with sufficiently low marginal utilities of diversion, the socially optimal contract can be obtained. x° is the marginal utility of diversion at which the agent is indifferent between diversion and applying funds to the project given the socially optimal contract. It is defined as follows: x- =
bðP-Þ 1 +Z : 1−δ P-
ð10Þ
The marginal utility of diversion at which the agent is indifferent is increasing with his reputation cost and the discount rate. If x b x° the socially optimal contract can be successfully implemented. If x N x° the optimal contract will consist of more than one stage with positive allocations. Therefore agents can be separated into three groups. Those with sufficiently low marginal utilities of diversion, x ≤ x°, such that the socially optimal contract can be undertaken. Those with moderate marginal utilities of diversion, x ε (x°, xc] where the optimal contract will take more than one stage, and involve overinvestment. The final group has sufficiently high marginal utilities, x N xc such that not only does the contract involve multiple stages and overinvestment, but it is also optimal for the principal to give the agent a side payment upon completion. 3. Application to an aid environment As the principal's ability to punish the agent increases, the cost of diverting funds, Z, also increases. If there is a possibility of a future relationship between the principal and agent, making future projects conditional on completion of the current project can increase the cost of diversion for the agent. As Z increases the amount that the principal can allocate in each stage increases as well. The agent's ICC for the first stage is ∞
n−t
xp1 − ∑ δ n=1
bðPn Þ = Z:
ð11Þ
Due to the concavity of the benefit function, an increase in Z will allow the principal to increase the allocation in the first stage and all subsequent stages. This allows the principal to complete the project weakly earlier, as the allocation in each stage increases the number of
bðP-Þ : 1−δ
ð12Þ
An aid organization's benefit is maximized with the socially optimal contract, so increasing Z whether with the credible threat of punitive legal action after diversion, or the promise of future projects upon completion will increase their benefit from the project. Another way to increase efficiency, and shorten the length of time required to complete a project is by making the stages shorter. By increasing δ honesty becomes more valuable to the agent as the discounted value of future allocations is now greater. In practice there is likely a minimum stage length, dictated by the time required to monitor the agent and ensure compliance, and the length of time it takes for the defined portion of the project to be completed. From Eq. (7), as δ decreases, the marginal utility of diversion above which the agent gets a side payment increases. Diversion in stage t gives the agent a high payoff only in stage t. He must compare that to applying the funds to the project, and receiving the future benefit in all stages k ≥ t. As a result, with shorter stages the agent values his future utility more, is less likely to divert, and the marginal utility of diversion above which the agent would receive a side payment increases. 4. Non-credible threats and broken promises The optimal contract described in Section 2 depends on two key assumptions that will be relaxed here. The first is that the principal can credibly commit to cutting off all allocations upon diversion. The second is that the principal can commit to overinvesting in the project. Relaxing either assumption reduces both the principal's and agent's welfare. 4.1. Renegotiation If the agent does not believe that the principal will cease all allocations upon diversion, the optimal contract is no longer enforceable. In the optimal contract without renegotiation the agent's ICC is dependent upon the principal ceasing all allocations upon diversion. Given an optimal contract without renegotiation where x N x° the agent will always divert funds if the principal's threats are not credible. In this case, the most the principal can credibly threaten is to revert to some ‘threat’ contract upon diversion. By threatening reversion, she cannot achieve the optimal contract found in Section 2, but can achieve a ‘renegotiation’ contract that pareto dominates having no recourse upon diversion. In order to find the optimal contract with renegotiation we first define the threat contract and ensure that it is credible, then define the renegotiation contract that can be enforced by threatening reversion. If the agent diverts funds in stage k, the principal will revert to the threat contract for the remainder of the project. The threat contract is conditional on the total investment in the project prior to diversion, and the stage in which the agent diverts funds. If the agent diverts funds in stage k, in any stage t ≥ k + 1 he must choose between
J.L. Steele / Journal of Development Economics 93 (2010) 1–6
applying funds to the project, and diverting them, putting off investment in the project until the next stage. ∞
s−t
∑ δ
s=t
∞
s−t
bðPs;k Þ≥xpt;k + δ∑ δ s=t
bðPs;k Þ ∀t≥k + 1;
ð13Þ
where pt,k is the allocation in stage t following diversion in k, and Ps,k is the sum of all allocations applied towards the project up to and including stage s N k. In the threat contract, the most the principal can allocate in any stage without inducing diversion is the amount where the agent is indifferent between applying the funds to the project and diverting them. Any contract which gives the agent a lower payoff necessarily involves lower allocations, and is not credible as the principal can revert to the threat contract upon diversion and do strictly better. Once again the ICC must bind in every stage up to the final stage with a positive allocation, j. Following diversion in stage k, the allocation in any stage t ε (k, j) is ∞ 1 s−t ð1−δÞ∑ δ bðPs;k Þ = pt;k : x s=t
ð14Þ
4.2. No commitment When the principal is unable to commit to a stream of allocations, she is necessarily worse off whenever the project requires more than one stage.3 The optimal contract outlined earlier commits to a level above the socially optimal funding level and is therefore not credible if the principal cannot commit to overinvestment. Likewise, a commitment to any amount below the socially optimal amount is both not credible, and welfare reducing. Given the optimal contract the agent will choose to divert, knowing that the principal's promise to overinvest is not credible. In order to find the optimal contract without being able to overinvest, we take the total funding level as given and find the number of stages it takes to get there with the ICC binding in every stage t b j. The total funding level, number of stages and the allocation in each stage are the unique solution to the following set of equations: b′ ðPj Þ =1 1−δ
ð19aÞ
In stage j, the final stage with a positive allocation, the allocation is defined by the principal's FOC:
pt =
b′ ðPj;k Þ −1 + 1−δ
pj = Pj −pj−1 :
j−1
∑
n=k + 1
λn ð1−δÞ
b′ ðPj;k Þ = 0: 1−δ
ð15Þ
Comparing this equation to Eq. (9) we see that due to the increased incentive additional allocations provided for diversion, the total funding level in the threat contract is strictly less than in the optimal contract without renegotiation. The renegotiation contract can now be defined as the contract giving the principal the highest possible payoff, enforced by threatening reversion to the threat contract. The agent's benefit from diversion now includes the net present value of the threat contract received following diversion. In order to make the strongest case for the renegotiation contract and explore the worst-case scenario the reputation cost has been set to Z = 0. The agent's ICC in any stage t can be written as follows: ∞
s−t
∑ δ
s=t
r
r
bðPs Þ≥xpt +
∞
∑
s=t + 1
s−t
δ
bðPs;t Þ;
ð16Þ
where prt is the allocation in the renegotiation contract, when the agent has not diverted funds in any stage k b t, and Prt = ∑ts = 1prt. In the renegotiation contract, the allocation in all stages t b j is
½
∞ 1 ∞ s−t r s−t r ∑ δ bðPs Þ− ∑ δ bðPs;t Þ = pt : x s=t s=t + 1
ð17Þ
Upon diversion the agent not only gains the payoff from diversion, but also retains the present value of the threat contract. Without renegotiation the agent only gets the payoff from diversion, with no future contract. Therefore the allocation in each stage t b j of the contract without renegotiation is strictly greater than in the renegotiation contract,
½
½
∞ 1 ∞ s−t 1 ∞ s−t r s−t ∑ δ bðPs Þ N ∑ δ bðPs Þ− ∑ δ bðPs;t Þ : x s=t x s=t s=t + 1
ð18Þ
In the contract with renegotiation each funding level takes longer to reach than in the contract without renegotiation and as a result both the agent's and the principal's benefits are reduced.
5
j−1
δ j−t bðPj Þ bðPs Þ + Z + xð1−δÞ x
s−t
∑s = t δ
for all t b j
ð19bÞ
ð19cÞ
The allocation in each stage where the ICC binds, pt, is smaller than in the benchmark model because the principal is unable to commit to overinvestment. As a result the project takes longer to reach each funding level, P, and both the principal and agent are worse off. The agent is clearly worse off because his benefit in each stage is lower. Implicitly the principal is worse off, if this contract increased her benefit the optimal contract found earlier would not have been a maximum. 5. Conclusion When an aid agency wishes to fund a project through grants, contracting with a local agent can lead to diversion due to insufficient enforcement mechanisms. Grants are becoming an increasingly popular means of international aid, but I show that when dealing with agents who have high marginal utilities of diversion, care needs to be paid to the structure of the contract. We see that as countries differ in both governance and enforcement mechanisms (institutions), the optimal contract must adjust as well. Given two identical projects the benefit from the optimal contract in a country with good governance is strictly greater than that from a country with poor governance. Unlike traditional international lending models, where loans are repaid, and the benefit from the project undertaken is not shared, the need for incentives leads to overinvestment. Bloated projects can therefore be optimal, because the allocation in the final stage provides incentives for the agent not to divert funds in all previous stages. Compared to the socially optimal contract, where the project is fully funded in the first stage, when dealing with agents who have an incentive to divert funds, the project will take more stages and be larger. When the agent derives little benefit from the project relative to diversion it may be optimal for the contract to specify an allocation in the final stage that the agent is allowed to divert. By promising an allocation in the last stage that the agent may keep, the principal is able to give larger allocations in early stages. 3
When x b x°, the socially optimal contract is optimal with or without commitment.
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J.L. Steele / Journal of Development Economics 93 (2010) 1–6
This model suggests that aid agencies should break large allocations up into smaller ones, increasing in size over time, and concentrate on increasing the reputation cost for the agent. A larger reputation cost allows the project to be completed earlier, with a larger allocation in each stage. This can be achieved by making future projects conditional on successful completion of the current one. It can also be achieved by increasing the capacity to punish the agent. One way of increasing the threat of punishment is to locate and seize the proceeds of diversion. If the agent believes there's a positive probability of the proceeds being confiscated, diversion becomes more costly. Shortening the length of each stage increases both the aid agency's and the agent's benefit. However, the stage length may be constrained by the length of time it takes to complete the required portion of the project, or the cost or time required to monitor. When conditionality is not credible, two separate results are obtained. The first is that when the threat to cease all allocations upon diversion is not credible, the principal can offer a renegotiation contract supported by the threat of reversion to a worst-case threat contract. The inability to commit in this manner lowers both the agent's and the principal's benefit from the project, and reduces the set of types for whom the principal will offer a contract. The second result looks at the case where the principal cannot commit to overinvestment. With the optimal contract the principal would be better off reneging in stage j and offering some allocation strictly less than the contract amount. In this case the principal's promise of pj in the optimal contract is not credible, and the allocation supportable in any stage t b j is lowered. In either of these two cases, the benefit from the project is diminished for both the principal and the agent. The aid agency is better off if they stick to their guns, with respect to both their threats and their promises. Acknowledgments For their help and patience on an earlier version of this paper I would like to thank Max Stinchcombe and Tom Wiseman. I am grateful to Dilip Mookherjee, two anonymous referees and participants at various seminars for their comments and discussion. Appendix A. Renegotiation contract When the principal is unable to commit to ceasing all allocations upon diversion, two maximization problems are needed. The first step is to identify the maximization problem for the threat contract for any stage k ≤ j: Threat contract maximization problem: ∞
maxfpt g∞
t=k+1
s
∑
s=k + 1
δ ½bðPs;k Þ−ps;k
ð20Þ
s.t. ∞
s−t
∑ δ
s=t
∞
s−t
bðPs;k Þ−½xpt;k + δ∑ δ s=t
k−1
r
Pt;k = ∑ ps + s=1
t
∑
s=k + 1
ps;k
bðPs;k Þ≥0
for all t N k
ð21aÞ
for all t N k
ð21bÞ
The solution to threat contract maximization problem gives us a set of allocations, {pt,k}t∞= k + 1 for all stages k ≤ j as a function of the r allocations applied to the project prior to k, Pk − 1. Next, we use the threat contract in the renegotiation contract maximization problem: ∞
maxfprt g∞
t=1
s
r
r
∑ δ ½bðPs Þ−ps
ð22Þ
s=1
s.t. ∞
s−t
∑ δ
s=t
r
r
t
∞
r
bðPs Þ−½xpt +
∑
s=t + 1
r
Pt = ∑ ps
s−t
δ
bðPs;t Þ≥0
for all t
ð23aÞ
for all t
ð23bÞ
s=1
t−1
r
Ps;t = ∑ pn + n=1
s
∑
n=t + 1
pn;t
for all t ≤ j; s N t ð23cÞ
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