Incentives for Conditional Aid Effectiveness

Francisco Candel-Sánchez

doi:10.1515/jgd-2012-0018

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Incentives for Conditional Aid Effectiveness

Francisco Candel-Sánchez

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Aus der Zeitschrift Journal of Globalization and Development Band 5 Heft 1

Abstract

Can sanctions against foreign aid donors enhance the credibility of conditional aid policies? If such policies suffer from time inconsistency, the answer is positive. This paper proposes a mechanism to overcome the lack of credibility of conditional aid donations to developing countries. A scheme of policy-dependent transfers to the donor country is shown to achieve an optimal commitment outcome by improving the credibility of conditional aid programs. The scheme is devised to cover situations in which the cost of structural reforms is information privately owned by the recipient government.

Keywords: conditional aid policy; credibility; mechanism design

Corresponding author: Francisco Candel-Sánchez, Dept. de Fundamentos del Análisis Económico, Universidad de Murcia, 30100 Murcia, Spain, e-mail: fcandel@um.es

Acknowledgments

This paper has benefited from the comments and suggestions of Ravi Kanbur, Ashley Piggins and Stéphane Straub. I also thank two anonymous reviewers and the editors of this journal for very helpful comments. Financial support from the Spanish Ministry of Economy and Competitiveness through MEC/FEDER grant ECO2010-21624 is gratefully acknowledged.

Appendix

Proof of Proposition 1: First of all, we prove that, faced with y^opt, the recipient’s best response is z^opt. This conclusion follows whenever the inequality (k–1)y^opt>θ holds. Assumptions 1 and 2 imply that θ<k−1k(C¯−12γk)=(k−1)yopt. On the other hand, y^opt is the value for y that minimizes L(y, z^opt). Next we show that the pair (y^opt, z^opt) is the unique SPE of the game. To see this, consider any alternative strategy y₀≠y^opt If (k–1)y₀>θ then y₀ cannot belong to an equilibrium. The reason is that faced with y₀ the recipient would set z=1, but the value that minimizes L(y, z^opt) is y^opt≠y₀. If (k–1)y₀=θ any z₀ would be a best response. Consider z₀<1. The pair (θk−1, z0) cannot be an equilibrium, since aid level y^opt would induce reforms z^opt and the strategy pair (y^opt, z^opt) gives a higher payoff to the donor. For z₀=1, the donor would be better off by selecting y^opt, so (y₀, 1) is neither an equilibrium. Finally, if (k–1)y₀<θ the recipient sets up z=0. Let us call y0∗ the value for y₀ that minimizes the donor’s loss given that z=0. Observe first that any pair (y₀, 0) with y0≠y0∗ can never be an equilibrium since by definition y0∗ yields a better payoff than y₀. We have that (y0∗, 0) is neither an equilibrium. The reason is that the donor’s loss associated with the pair (y0∗, 0) is given by L0=C¯−14γ while the loss under (y^opt, z^opt) is given by Lopt=1k(C¯−14γk), and L^opt<L₀ since γ≥12C¯k−1k is implied by Assumption 2. Therefore, (y0∗, 0) is not a SPE of the game. This completes the proof.

Proof of Proposition 2:

Let p(y)=prob(θ≤ky). The donor’s problem is:

min{y}p(y)L(y, 1)+[1−p(y)]L(0, 0).

Let y^C be the solution of the above problem. We write the first order condition as:

p′(yC)[L(yC, 1)−L(0, 0)]+p(yC)L′(yC, 1)=0.

From this condition we obtain:

L′(yC, 1)=−p′(yC)p(yC)[L(yC, 1)−L(0, 0)]

The loss function is such that L(y^C, 1)<L(0, 0). Then, from the above FOC we deduce that L′(y^C, 1)>0. On the other hand, since y^opt minimizes function L(y, 1) we have L′(y^opt, 1)=0. The inequality L′(y^C, 1)>L′(y^opt, 1)=0 holds, and then y^C>y^opt.

Proof of Proposition 3: Faced with z^* in Eq. (5) the donor disburses

y∗=y(z∗)=(2γθk−1)1/2C¯−θk−1.

The minimal loss associated with any given z is:

l(z)=L(z, y(z))=A(z)[C¯−14γA(z)].

It is easy to see that l′(z)<0. As z^*<z^opt=1, we have that l(z^*)>l(z^opt). On the other hand, consumption level in the equilibrium pair (y^*, z^*) is given by

C∗=C¯−(θ2γ(k−1))1/2.

We find that C∗<Copt⇔k−1k2<2θγ. The latter inequality is true by Assumption 1. This finishes the proof.

Proof of Proposition 4: We solve the game by backwards induction. In the last stage of the game, the donor selects y to minimize L(y, z)+t(z, λ, θ˜)y. The first order condition of this problem yields the following reaction function:

(7)

y(z, λ, θ˜)=A(z)[C¯−1+t(z, λ, θ˜)2γA(z)]. (7)

There are several statements in the main text that are now proven here. Observe first that y(z) in Eq. (4) is above y(z, λ, θ˜) for all z and for all λ. This proves the claim that, for given reforms level z, the donor disburses more aid in the absence of a sanction scheme. We are also interested in the dependence on the reports λ and θ˜of the reaction function above.

For this purpose we compute ∂y(z, λ, θ˜)∂λ=−A(z)(1−z)[(k−1)yopt−θ˜]<0, and ∂y(z, λ, θ˜)∂θ˜=−A(z)(1−z)(1−λ)<0. This proves that, ceteris paribus, the donor’s equilibrium disbursement is lower the higher are λ and/or θ˜. Moreover, aid disbursements are more (less) sensitive to changes in θ˜ the lower (higher) is λ, and vice versa, since ∂2y(z, λ, θ˜)∂λ∂θ=A(z)(1−z)>0. This feature of the mechanism turns out to be critical to induce truthtelling on the recipient. As we will see below, the donor reports λ high enough so as to avoid the recipient’s misrepresentation of θ˜.

In the second stage of the game, the recipient selects z to maximize Gθ(y(z, λ, θ˜), z). We compute ∂Gθ(.)∂z=λ[(k−1)yopt−θ˜]+θ˜−θ. Let z^ be the value for z that maximizes G_θ(.). If ∂Gθ(.)∂z>0, then z^=1, and if ∂Gθ(.)∂z<0 then z^=0.

An equilibrium of the subgame starting in Stage 2 is given by z^=1 and y(1, λ, θ˜) if λ[(k−1)yopt−θ˜]+θ˜−θ>0. The equilibrium actions are z^=0 and y(0, λ, θ˜) when λ[(k−1)yopt−θ˜]+θ˜−θ<0, and z^∈[0, 1] and y(z, λ, θ˜) when

λ[(k−1)yopt−θ˜]+θ˜−θ=0. To compute the SPE of the game, we assume that both the donor and the recipient are aware of the outcomes derived from all possible equilibria of the subgame starting in Stage 2, and take them into account when announcing simultaneously λ and θ˜ in the first stage of the game.

Let us now analyze the (Nash) equilibrium actions in the announcement stage of the game. Let (λ∗, θ˜∗) be a pair of NE actions. Then,

λ∗∈argmin{λ}L(y(z^, λ, θ˜∗), z^)+t(z^, λ, θ˜∗)y(z^, λ, θ˜∗)

and θ˜∗∈argmax{θ˜}Gθ(y(z^, λ∗, θ˜), z^). Notice that if z^=1, then y(1, λ, θ˜)=yoptfor all λ and θ˜. Next, we show that the highest possible payoff for the donor is obtained when z^=zopt=1 and y(1, λ, θ˜)=yopt. For this purpose, observe first that L(y^opt, z^opt) is a minimum. It remains to show that t(z, λ, θ˜)>0 for all λ, θ˜,z≠z^opt and that t(zopt, λ, θ˜)=0 for all λ, θ˜. The latter is immediate. To prove the former we rewrite the inequality t(z, λ, θ˜)>0 as

2γA(z)(1−z){λ[(k−1)yopt−θ˜]+θ˜−12γk−1kA(z)}>0.

Provided that 1k<A(z)≤1 for z<1, Assumptions 1 and 2 imply that (k−1)yopt−θ˜>0 and Assumption 1 implies θ˜>12γk−1kA(z). Hence, t(z, λ, θ˜)>0 holds for all λ, θ˜ and z≠1. If the penalty rate is positive (except for z=1) and L(y^opt, z^opt) is the minimum social loss, the donor announces a value for λ with the goal of inducing z=1. Reporting λ^* such that λ∗[(k−1)yopt−θ˜]+θ˜−θ>0 for all θ˜ (including θ˜∗) dominates any other report. The donor’s announcement must be high enough (so as to induce z^=1 for any possible θ˜) since otherwise, for certain θ˜ it could be the case that λ[(k−1)yopt−θ˜]+θ˜−θ<0 and hence z^=0. Any equilibrium report from the donor must be such that the recipient can never be interested in performing z=0. In the absence of information about θ, the latter possibility is avoided by setting λ∗≥λ(θ˜)=θH−θ˜(k−1)yopt−θ˜. Function λ(θ˜) is a reaction function to any conjecture about θ˜. Now, given that λ∗∈[λ(θ˜), 1], the recipient can never get advantage from misrepresenting θ. It turns out that reporting truthfully (i.e. using the strategy s∗(θ)=θ for all θ) is optimal for the recipient. Then, for the pair (λ∗, θ˜∗) to be part of a SPE of the game induced by the mechanism, it is sufficient that λ∗≥λ(θ˜)and θ˜∗=s∗(θ)=θ for all θ.

Finally, since (k−1)yopt>θ for all θ, in equilibrium we have ∂Gθ(.)∂z=λ∗[(k−1)yopt−θ]>0. Therefore, the optimal reforms level is z^opt=1. Provided that the recipient performs z^opt=1 in Stage 2, the donor disburses y(1, λ^*, θ)=y^opt in Stage 3. This completes the proof.

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conditional aid policy; credibility; mechanism design