Patrick Guillaumont

Phu Nguyen-Van

Thi Kim

Cuong Pham
email: pham_tkc@parisnanterre.fr

Laurent Wagner

Economix

Equal opportunity and poverty reduction: how should aid be allocated?

Keywords: Development Aid, Efficiency, Equity, Growth Deviation, Poverty Reduction, Vulnerability JEL Classification: D63, F35, I30, O19

This paper analyses a model of aid allocation that aims to equalize the opportunity between recipient countries to reach a common poverty reduction goal. We propose a fair and efficient aid allocation based on a multi-criteria principle. The model considers structural handicaps in recipient countries in terms of lack of human capital and economic vulnerability, their initial poverty, and the natural gap between the growth rate required to reach a development goal and the observed one. We show that our proposed aid allocation favors poor and vulnerable countries with our multi-criteria principle. It substantially differs from the observed allocation. Analyses also shed light on the impact of the donors' aversion to the low natural growth gap in recipient countries on the optimal aid allocation and the marginal efficiency of aid.

Introduction

Developing countries, particularly the most vulnerable and low-income countries, need foreign aid, which may help them build up favorable conditions for a generation of economic growth and attainment of sustainable development. One of the most relevant questions concerns the ai allocation and factors determining an efficient allocation. Several donors, such as World Bank's International Development Association, Asian Development Bank, the European Development Fund, etc., employ performance-based allocation systems, prioritizing the countries in need and those with absorption capacity.

Several empirical analyses on aid allocation [START_REF] Easterly | Are aid agencies improving[END_REF][START_REF] Easterly | Where does the money go? Best and worst pratices in foreign aid[END_REF], Knack et al., 2011, etc.) emphasize two main characteristics in recipient countries: their need for assistance (low GDP per capita, low IDH, high child mortality, or post-conflict country), and their ability to use aid effectively (high government effectiveness index, high institutional performance, etc.). Other analyses underline the link between aid allocation and donors' preferences represented by colonial ties and language homogeneity between aid organizations and recipients (Berthelemy and Tichit, 2004;Maiden and Brockway, 2018). Donors' interests (strategic allies, commercial allies ) are potential criteria influencing donors' decisions. For instance, [START_REF] Couharde | The role of oil in the allocation of foreign aid: The case of the G7 donors[END_REF] underline the role of oil on bilateral aid policy, while in Nunnenkamp and Öhler (2011), donor strategic interests are reflected by the recipient's relative importance as a donor's export market and by the degree of recipient UN voting coincidence with a donor.

From a normative point of view, many issues are under debate regarding the optimal choice for aid allocation (Collier andDollar (2001, 2002), [START_REF] Wood | Looking ahead optimally in allocating aid[END_REF], [START_REF] Llavador | An equal-opportunity approach to the allocation of international aid[END_REF], [START_REF] Cogneau | Who deserves aid? Equality of opportunity, international aid, and poverty reduction[END_REF], [START_REF] Carter | Aid allocation rules[END_REF], etc.) For instance, Collier andDollar (2001, 2002) adopt a utilitarian vision by maximizing a social welfare function, the sum of aid-recipient countries' utilities. A country's utility is measured in terms of the number of poor, which is a decreasing aid function. Following Collier andDollar (2001, 2002), an optimal allocation of aid should maximize the reduction of the poor in recipient countries. According to the effectiveness principle, the optimal aid allocation proposed by Collier and Dollar is determined by the initial deprivation of recipient countries, institutional quality, and policy quality. In other words, compared to the observed allocation of aid, the Collier and Dollar's allocation gives more aid to the poorest countries implementing the highest policy quality (high CPIA). In the same vein, [START_REF] Wood | Looking ahead optimally in allocating aid[END_REF] includes an intertemporal aspect in his analysis and considers initial poverty and future poverty in aid donors' objective function.

This approach of Collier and Dollar is criticized for its lack of consideration for fairness. In this sense, [START_REF] Llavador | An equal-opportunity approach to the allocation of international aid[END_REF] propose an alternative way of calculating the optimal allocation of aid based on the Rawlsian principle. They analyze how aid can be distributed to equalize recipient countries' growth opportunities. Therefore, aid donors should give an allocation that compensates countries for bad initial circumstances so that the final differences in outcomes between countries will be only attributed to differences in their efforts, not to their initial circumstances.1 According to [START_REF] Cogneau | Who deserves aid? Equality of opportunity, international aid, and poverty reduction[END_REF], the optimal aid allocation of [START_REF] Llavador | An equal-opportunity approach to the allocation of international aid[END_REF] is paradoxically in favor of countries with high macroeconomic performances. They propose another method including equal opportunity but separating effort and circumstances of recipient countries. Their aid allocation shares poverty risk more fairly among the world's population, and their results show that donors should give more aid to the poorest countries than the observed aid allocation.

Recently, [START_REF] Carter | Aid allocation rules[END_REF] proposed an aid allocation rule that maximizes recipient countries' welfare rather than economic growth and considers recipient countries' absorptive capacity (measured by the World Bank's Country Performance Rating). Donors target a range of development outcomes by putting more weight on aid-funded consumption and less weight on economic growth. The division of aid between consumption and investment maximizes households' utility in recipient countries. In this setting, the objective of maximizing welfare in recipient countries may lead to an optimal allocation giving more aid to countries that are least able to stimulate economic growth.

The debate on foreign aid allocation in developing countries is still lively today. In this vein, our paper aims to design an efficient and fair distribution of aid based on a multi-criteria principle and within a utilitarian framework where aid donors maximize the sum of recipient countries' utilities. Our goal is not to dichotomize fairness and efficiency as two different aid allocation criteria. On the contrary, we show that it is possible to determine a fair allocation, which is also efficient. The difference between our paper and Collier andDollar (2001,2002), [START_REF] Llavador | An equal-opportunity approach to the allocation of international aid[END_REF], [START_REF] Cogneau | Who deserves aid? Equality of opportunity, international aid, and poverty reduction[END_REF] stems from three major points.

First, aid policy should address uneven economic conditions between countries and compensate them with foreign aid. Therefore, as in [START_REF] Mcgillivray | Reforming performance based aid allocation pratice[END_REF] and [START_REF] Guillaumont | Performance Assessment, Vulnerability, Human Capital, and the Allocation of Aid Among Developing Countries[END_REF], our analysis considers structural growth handicaps in recipient countries. We introduce economic vulnerability and lack of human capital in the growth equation. Our study assumes that a country with low human capital and a high economic vulnerability may encounter difficulties in formulating high-quality economic and social policies, inducing, therefore, a low possibility of achieving its development goal. It is fair to consider the lack of human capital and economic vulnerability in recipient countries.

Second, unlike previous studies, we focus on structural factors, mainly structural economic vulnerability, as essential elements determining aid effectiveness. Indeed, according to [START_REF] Guillaumont | Aid and performance: A reassessment[END_REF], aid is growth-enhancing in countries with a high economic vulnerability. Accounting for economic vulnerability in recipient countries may be viewed as compatible with an aid effectiveness principle.

Third, aid policy could be fair. However, unlike [START_REF] Llavador | An equal-opportunity approach to the allocation of international aid[END_REF] and [START_REF] Cogneau | Who deserves aid? Equality of opportunity, international aid, and poverty reduction[END_REF], which adopt the Rawlsian principle, we propose an alternative way to model this fairness. We assume that donors are sensitive to the natural growth deviation or gap in recipient countries. Their goal is to equalize the opportunity between recipient countries to reach a common poverty reduction goal. To formulate this assumption, we posit the utility of country i depending on the gap between its current growth rate (depending on received aid and structural handicaps) and its expected (or targeted) growth rate. The latter represents the growth rate that a country i has to reach if it wants to achieve a specific development goal. Hence, an efficient and fair allocation of aid should reduce poverty in recipient countries by considering the particular conditions (such as structural handicaps and poverty rate) and the natural growth deviation or gap in these countries.

Our optimal aid allocation with poverty reduction objective is then based on a multicriteria principle that incorporates initial poverty, lack of human capital, economic vulnerability, and natural growth deviation in recipient countries. Our analysis is implemented in two steps. The first step corresponds to estimating a growth equation depending on factors such as lack of human capital, economic vulnerability, and received aid amount. The second step uses the computed growth rates in recipient countries, obtained from the first step, to perform a simulation exercise showing a substantial difference between the optimal and observed allocation. We also indicate the meaningful impact of the donors' aversion to low growth deviation or gap in recipient countries on the optimal aid allocation.

The rest of the paper is structured as follows. Section 2 presents the theoretical model. Section 3 reports estimation results concerning the economic growth equation. Section 4 discusses the efficient and fair allocation of aid, as well as the marginal efficiency of aid. Section 5 concludes. Other details are presented in the Appendix.

A model of efficient and fair allocation of aid

This section proposes a theoretical model to determine aid allocation, equalizing countries' opportunities to reach a development goal. We take as a retrospective example the Millennium Development Goal (MDG), proposing to reduce by half the poverty by 2015 compared to 1990. It is all about approaching the most possible to the MDG by giving each recipient country the same probability of lifting out of poverty by half. We need to define an expected growth rate required to reach each recipient country's millennium goal and eventually compensate for the difference between this one and the effective growth rate. This difference can be used as an argument, among others, to justify the aid allocated. This principle involves considering both effectiveness and fairness in designing the optimal aid allocation. For this purpose, we assume that the utility of recipient country i, U i , which corresponds to the number of poor that can be reduced by economic growth, is defined by

U i [g i (A i )] = -η i h i N i αu(g i ) + (1 -α)v g i g * i (1)
where h i is a measure of poverty (such as the percentage of country i's population living below 2 dollars (in PPP) per day), η i = ∂h i ∂y i y i h i is the elasticity of poverty reduction with respect to per capita income y i , assumed to be a negative constant in Collier andDollar (2001, 2002), and N i is the population size. Country i's utility is a function of its growth rate of per capita income, u(g i ) with u ′ > 0 and u ′′ ≤ 0, and a function of the ratio between the observed growth rate and the growth rate expected to achieve the MDG, v(g i /g * i ) with v ′ > 0 and v ′′ ≤ 0. Parameter α ∈ [0, 1] represents the weight associated to the growth objective (α called as growthweight), and 1 -α is the weight relative to 'natural growth deviation' from the expected (or targeted) growth rate g * i (1 -α called as growth gap-weight). When α = 1 and u(g i ) = g i we exactly recover the utility function of Collier andDollar (2001, 2002) where U i = -η i h i N i g i . Thus, this expression gives the number of the poor that can be reduced by economic growth. However, this utility function, which is linear in the growth rate, implies that the marginal utility of growth U g (also interpreted as the marginal reduction of poverty) is constant. In this respect, the specification in (1) appears more general as it allows for a non-constant marginal reduction of poverty. More precisely, we assume that each country's utility is increasing and concave with respect to the economic growth rate g i . Its first and second derivatives are given respectively by

U g = ∂U i ∂g i = -η i h i N i αu ′ (g) + (1 -α)v ′ g i g * i 1 g * i > 0, (2)
U gg = ∂ 2 U ∂g 2 i = -η i h i N i (αu ′′ (g) + (1 -α)v ′′ g i g * i 1 g * 2 i ≤ 0. (3)
We observe that marginal utility of growth, U g , is decreasing with the ratio g i /g * i . In other words, the marginal effect of growth is stronger in countries where the natural growth deviation is lower (i.e., lower g i /g * i ). This decreasing effect is more important if the curvature of v is higher (or the relative risk aversion of v is higher).

We also assume that the targeted growth rate g * i is independent of aid. It can be determined as follows. Adopting the MDG with the number of the poor reduced by haft between 1990 and 2015 means that the cumulated objective of poverty variation is -50% in 25 years. Let x denote the annual reduction rate of poverty, hence

h i N i (1 -0.5) = h i N i (1 -x) 25 . ( 4
)
We can easily find that x is equal to 0.0273. As g * i η i = -x, the targeted growth rate for country i, g * i , depends on the objective of annual poverty reduction rate (x i ) and the elasticity of poverty reduction with respect to per capita income (η i ),

g * i = -x/η i . (5)
Therefore, if η i = -2 as in Collier andDollar (2001, 2002), we obtain g * i = 0.0273/2 = 0.01365.

As the growth rate g i depends on the amount of aid A i , let us consider the following optimization program, which consists in choosing an aid allocation maximizing the sum of utilities of n countries under constraints on the total amount of aid, on the one hand, and the positiveness of aid, on the other hand,

max {A i } n i=1 n i U i (g i (A i )) (P1) s.t. n i A i y i N i = Ā (6) A i ≥ 0, ∀i (7)
where y i is per capita income, A i the amount of aid (measured as a percentage of total GDP of i), and Ā the total amount of aid available for allocating among all recipient countries. The constraint in equation ( 6) indicates the sum of aid given to all recipient countries is equal to the total amount of available aid.

The Lagrangian of problem (P1) can be written as follows.

L(A) = n i U i (g i (A i )) + λ Ā - n i A i y i N i + n i=1 µ i A i , (8)
where λ and µ i , i = 1, ..., n correspond respectively to the Lagrange multipliers of the constraint on the total available amount of aid and the positiveness of aid. The vector of candidates  ≡ ( Â1 , ..., Ân ), and multipliers λ, and μi must satisfy the following first order conditions (FOC), ∀i = 1, ..., n:

∂L Â ∂ Âi = U g ( Âi )g A ( Âi ) -λy i N i + µ i = 0 (9)
Âi ≥ 0, μi ≥ 0, μi Âi = 0 (10)

n i Âi y i N i = Ā (11)
where g A is the marginal effect of aid on the growth rate and U g the marginal effect of growth on country i's utility, given by equation (2). Condition ( 10) is about the complementarity between Âi and μi , i.e. μi = 0 if Âi ≥ 0, and μi > 0 if Âi = 0. Hence, for country i such that Âi ≥ 0, equation ( 9) gives U g (A i )g A (A i ) = λy i N i because μi = 0. Combining this with condition (11) will gives the values for Âi > 0 and λ. Finally, given Âi > 0 and λ, μj can be recovered from equation [START_REF] Carter | Dynamic aid allocation[END_REF], which only applies to country j such that Âj = 0, i.e. μj = λy j N j -U g (0)g A (0). This sketch about the solution of these FOCs appears simple. However, its implementation is cumbersome because it needs some combinatory calculations. Fortunately, certain softwares can help us to solve this problem. 2 Given the discussion above, we can state the following proposition:

Proposition 1 Considering the optimization program (P1) where each country's utility is increasing and concave with economic growth (i.e., conditions (2) and ( 3)), the optimal (efficient and fair) allocation of aid { Âi } n i=1 must verify the following conditions:

(i) Âi = 0 if U g ( Âi )g A ( Âi ) = λy i N i -μi and μi > 0, (ii) Âi > 0 if U g ( Âi )g A ( Âi ) = λy i N i and μi = 0, (iii) n i Âi y i N i = Ā,
where μi ≥ 0 is the multipliers associated with the positiveness of aid and λ is the multiplier (or shadow value) associated to the total aid.

It should be noted that the multiplier λ can be viewed as the marginal efficiency of aid as in Collier andDollar (2001, 2002). We observe from point (ii) of Proposition 1 that for any country i which receives a strictly positive amount of aid (i.e. Âi > 0), we have

λ = U g ( Âi )g A ( Âi ) y i N i . ( 12
)
In terms of our specification, this expression is equivalent to

λ = -η i h i N i (αu ′ + (1 -α)v ′ /g * i )g A ( Âi ) y i N i . ( 13
)
2 For the simulation of the optimal aid allocation (based on estimation results of the growth equation and a parametrization of the theoretical model) in the next section, we use Matlab (function fmincon) and R (package Rsolnp) to find the solution of the optimization problem. The implementation of these different packages generally give the same results.

In the case of the Collier andDollar' (2001, 2002) model, i.e. α = 1 and u(g i ) = g i , this multiplier becomes

λCD = -η i h i N i g A ( Âi ) y i N i . ( 14
)
Collier [START_REF] Collier | Can the world cut poverty in half? How policy reform and effective aid can meet intrenational development goals[END_REF][START_REF] Collier | Aid allocation and poverty reduction[END_REF] precisely defined this quantity as the number of the poor reduced by an increase of the total amount of aid by one unit.

From equation ( 13), we can compute for our model the equivalent number of the poor in country i that can be reduced by an increase of the total amount of aid by one unit (here in millions of dollars) as λi = λ

αu ′ + (1 -α)v ′ /g * i . ( 15
)
We observe that while λ is constant (i.e., there is a unique solution for this, see also Proposition 1) λi is different between countries receiving a positive amount of aid (because of different values of αu ′ + (1 -α)v ′ /g * i ). The only case for which the marginal efficiency of aid is the same for every recipient countries corresponds to the Collier-Dollar utility function, i.e., α = 1 and u(g i ) = g i implying λi = λ = λCD , ∀i.

We now turn to sufficient conditions for the optimization program's solution (P1). As the constraint is linear, the sufficient conditions for { Âi } n i=1 being the solution of the optimization problem (P1) are that the objective function is concave with { Âi } n i=1 . In other words, the second-order matrix

      ∂U 2 ∂ 2 A 1 ∂U 2 ∂A 1 A 2 . . . ∂U 2 ∂A 1 ∂An ∂U 2 ∂A 2 ∂A 1 ∂U 2 ∂A 2 2 . . . ∂U 2 ∂A 2 ∂An . . . . . . . . . . . . ∂U 2 ∂An∂A 1 ∂U 2 ∂An∂A 2 . . . ∂U 2 ∂A 2 n      
must be negative semi-definite. We observe that this matrix is diagonal, and its diagonal elements are given by

∂ 2 U Â ∂ Âi 2 = U gg ( Âi )g 2 A ( Âi ) + U g ( Âi )g AA ( Âi ) (16)
Hence, the negative semi-definiteness of this matrix corresponds to the negativity of ( 16) or equivalently g AA ( Âi )

g 2 A ( Âi ) ≤ - U gg ( Âi ) U g ( Âi ) , ∀i. (17)
The right-hand side term (-U gg /U g ), which is always positive, represents the utility functio's curvaturen with respect to the growth rate. It is also known as the coefficient of absolute risk aversion at g. The left-hand side term (g AA /g 2 A ) corresponds to the curvature of the growth rate with respect to aid. This inequality merely means that the absolute risk aversion of the objective function at g should be sufficiently high to warranty that { Âi } n i=1 is the solution of the maximization problem. Hence, we can state the following proposition on the sufficient condition:

Proposition 2 If the absolute risk aversion of the utility function with respect to the growth rate is sufficiently high as given by equation ( 17) (i.e. g AA ( Âi )/g 2 A ( Âi ) ≤ -U gg ( Âi )/U g ( Âi ), ∀i) the allocation of aid { Âi } n i=1 defined in Proposition 1 corresponds to the maximum of the optimization program (P1).

We remark that Proposition 2 automatically holds if g AA ( Â) ≤ 0. It also holds even with some positive g AA as long as condition (17) is satisfied.

3 Estimation of the growth equation

Foreign aid and economic growth

There is a consensus that aid rarely has a direct effect on the growth rate. Its impact on the recipients' economic growth, if significant, is often conditional on performance factors in recipient countries such as institutional quality, political stability, macroeconomic environment (Boone, 1996;[START_REF] Burnside | Aid, policies and growth[END_REF]Easterly, 2003;Moyo, 2009, Mosley, 2015, Temple and Van de Sijpe, 2017, Maruta et al., 2019, etc.). Collier andDollar (2001, 2002) underline that the growth effect of aid positively depends on recipient policy performance measured by CPIA. Such an analysis implies that countries with high performance in economic policies and institutions will receive more aid than others. Such studies focusing on the performance ignore the initial disadvantages of recipient countries and the stabilizing impact of aid. Indeed, aid may lead to higher growth rates in countries with high structural vulnerability [START_REF] Guillaumont | Aid and performance: A reassessment[END_REF]Chauvet andGuillaumont, 2004, 2009;[START_REF] Guillaumont | Aid and Remittances: Their Stabilizing Impact Compared[END_REF]Guillaumont andWagner, 2012, Chauvet et al., 2019).

From a development perspective, it is necessary to make aid fairer, without losing the efectiveness goal, by considering structural handicaps to the growth of recipient countries, particularly their economic vulnerability and lack of human capital. Indeed, as underlined in [START_REF] Guillaumont | Performance Assessment, Vulnerability, Human Capital, and the Allocation of Aid Among Developing Countries[END_REF], a country with a low human capital level is likely to have a low-performance score despite its great efforts. It may encounter difficulties to formulate a high quality of economic and social policies to achieve the objective of poverty reduction during their development process. Therefore, allocating more aid to countries suffering from low human capital and high economic vulnerability might be an excellent way to compensate for their initial disadvantages.

In the same line as [START_REF] Guillaumont | Aid and performance: A reassessment[END_REF] and [START_REF] Guillaumont | Performance Assessment, Vulnerability, Human Capital, and the Allocation of Aid Among Developing Countries[END_REF], we assume that the growth rate depends on structural handicaps represented by the degree of economic vulnerability and the lack of human capital. More precisely, the growth rate of country i in period t is a function of the degree of economic vulnerability (V it ), lack of human capital (H it ), level of aid relative to GDP (A it ), its squared term (A 2 it ) and the interaction of aid with economic vulnerability.

The growth equation is estimated using a panel data framework (see, e.g., [START_REF] Islam | Growth empirics: A panel data approach[END_REF][START_REF] Caselli | Reopening the convergence debate: A new look at cross-country growth empirics[END_REF][START_REF] Durlauf | Growth econometrics[END_REF]. Following the literature [START_REF] Islam | Growth empirics: A panel data approach[END_REF][START_REF] Caselli | Reopening the convergence debate: A new look at cross-country growth empirics[END_REF], Durlauf et al. 2005, etc.), the growth equation can be expressed in a panel data framework as

ln y it = ρ ln y i,t-τ + β V V it + β AV A it V it + β H H it + +β A A it + β AA A 2 it + µ i + ν t + ε it (18)
where ln y it is the log real GDP per capita in international prices PPP 2005. The set of covariates includes the ratio of aid to GDP (A), its squared term (A 2 ), economic vulnerability (V ), lack of human capital (H), interaction of aid with V . The terms µ i represents the country fixed effects. The term ν t represents the time effects, which are treated as time dummies. 3

Data

The data employed to assess equation ( 18) are collected from the World Development Indicators of the World Bank, the Human Assets Index (HAI), and the FERDI Economic Vulnerability Index. 4 The WDI database provides information on aid, GDP, population, etc. Aid and GDP are expressed in real dollars, international prices PPP 2005, and measured in millions of dollars. These series are employed to compute GDP per capita (y) and the share of aid in GDP (A). We use the series on HAI, a composite index of education and health (see [START_REF] Closset | Human Assets Index Retrospective series: 2013 update[END_REF], to compute H as a measure of lack of human capital.

For this purpose, we simply calculate H = (100 -HAI)/100. The economic vulnerability index (V ), one of several criteria retained by the United Nations Committee for Development Policy in identifying the least developed countries, was proposed by the FERDI (see [START_REF] Cariolle | The economic vulnerability index: 2010 updated[END_REF], for more details). Following [START_REF] Cariolle | The economic vulnerability index: 2010 updated[END_REF], the vulnerability index encompasses the main determinants of structural vulnerability that can harm economic growth and poverty reduction in developing countries. The principal components entering then in the definition of economic vulnerability index are (1) population, (2) share of agriculture, forestry, and fisheries in GDP, (3) exports concentration, (4) remoteness from main world markets, adjusted for landlockness, (5) instability of exports receipts, (6) instability of agricultural production, and (7) homelessness due to natural disasters.

Values of lack of human capital H and economic vulnerability V are between 0 and 1. Our data are an unbalanced panel data sample covering 92 countries and the 1983-2011 period. As in the existing literature, we use data with some time span intervals to avoid business cycle effects. We adopt the 4-year interval (τ = 4) as [START_REF] Collier | Aid allocation and poverty reduction[END_REF]. More precisely, following [START_REF] Caselli | Reopening the convergence debate: A new look at cross-country growth empirics[END_REF] and [START_REF] Durlauf | Growth econometrics[END_REF], among others, y it (and then y i,t-τ ) correspond to GDP per capita observed in 1983, 1987, 1991, 1995, 1999, 2003, 2007, and 2011. Variables A, V , and H are defined as the averages over the 4 years preceding t, i.e., 1983-1986, 1987-1990[START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF]-1994[START_REF] Islam | Growth empirics: A panel data approach[END_REF][START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF], 1999[START_REF] Bourguignon | The growth elasticity of poverty reduction: explaining heterogeneity across countries and time series[END_REF], 2003[START_REF] Ram | Growth elasticity of poverty: Alternative estimates and a note of caution[END_REF][START_REF] Easterly | Are aid agencies improving[END_REF]-2010. There are eight waves of 4-year interval: 1983-1986, 1987-1990[START_REF] Easterly | Are aid agencies improving[END_REF]-2010. . Hence, as our model deals with a relatively small time dimension (T =8), the issue of nonstationarity is not crucial here. Furthermore, following the recommendation by Bond et al. (2010), the treatment of time effects as time dummies (one for each 4-year period) in our regressions allows capturing the common trend between variables and ensures no cross-country correlation in the model residuals. Definition and descriptive statistics on variables are summarized in Table 1.

Table 1 here 3 We also include a variable representing the quality of policies, Polity 4 (as in the existing literature), and its interaction with aid. However, the system-GMM estimation results using this variable show that coefficient of Polity 4 and the interaction term are insignificant. At the same time , the sample size is much reduced.

4 Data and estimation codes are available on request from the authors.

Econometric specification and estimation results

Equation ( 18)is a dynamic panel data model that can be estimated using the system-GMM method of [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. We note that this model considers two sets of regressors: (i) strictly exogenous regressors (including time dummies) and (ii) predetermined regressors (including ln y i,t-τ , A, V , and H). Outliers are excluded from the estimations. The estimation results by using the system GMM are reported in Table 2. 5 Results of the within fixed effects estimator are also reported for comparison purposes.6 Specification tests (Arellano-Bond tests for AR(1) and AR(2) in the regression residuals, Sargan and Hansen overidentifying restrictions tests) and tests for exogeneity of instruments are generally verified.

Table 2 here

We observe that results given by system GMM and within FE estimators are comparable. More precisely, the lagged term of GDP per capita is highly significant and positively affects the current GDP per capita level, which proves the usefulness of the dynamic setting. The linear term of aid has no effect, whereas its squared term has a negative effect. While economic vulnerability negatively affects income, its interaction term with aid has a significant and positive impact. This finding means that aid and economic vulnerability are complementary in the growth equation. In other words, when a country's vulnerability is high, aid is more efficient in terms of growth. The same result is found in [START_REF] Guillaumont | Aid and performance: A reassessment[END_REF]. Finally, lack of human capital is a handicap for growth as its coefficient is statistically significant and negative.

As the average annual growth rate of country i between periods t -τ and t is given by (1/τ )(ln y it -ln y i,t-τ ), it can therefore be computed from the estimated coefficients of equation ( 18) as follows:

g it ≡ (1/τ )(ln y it -ln y i,t-τ ) = ((ρ -1)/τ ) ln y i,t-τ + W ′ it β/τ, (19)
where W includes all right-hand side variables of equation ( 18), except y i,t-τ . This is the growth rate we will use in the subsequent simulations of efficient and fair aid allocation. We remark that the negative coefficient associated to A 2 in the growth equation (see Table 2) confirms the sufficient conditions of our optimization problem (i.e. g AA ≤ 0).

Simulation of the optimal aid allocation

To assess the theoretical allocation, we need an analytically tractable model. For simplicity's sake, we specify u(g i ) = g i and v(g i /g * i ) = (g i /g * i ) γ with γ > 0. We observe that γ > 0 measures the curvature of v with respect to ratio g i /g * i . This ratio, named as natural growth deviation, may be interpreted as natural growth deficit in case of g i < g * i or natural growth excess in case of g i > g * i . In this formulation, 1 -γ represents the relative 'risk' aversion to growth deviation. It should be noted that we do not exclude the case where g i > g * i as it is quite possible that g * i is low enough. Actually, many countries in our 2008 sample encounter this situation. A reduction in γ, represented by an increase of the function' curvature, means that aid donors are more sensitive to natural growth deviation. In other words, a reduction in γ means that donors are more averse to a low ratio g i /g * i . This implies that aid donors give more weight to countries with a ratio g i /g * i lower than other countries, all else being equal.

We think that the case γ > 1 should be excluded as it is not fair to favour the countries with a high ratio g i /g * i to the detriment of the countries with a low ratio

g i /g * i . With γ ∈ (0, 1), function v (g i /g * i )
is then increasing and concave with respect to g i /g * i . Figure 1 describes the behavior of function v following different values of γ.

0 ✲ g i /g * i ✻ v γ = 1 1 1 γ = 0.3 γ = 0.7 Figure 1: Behavior of v(g i /g * i ) = (g i /g * i )
γ with different values of γ (0 < γ < 1). A lower value of γ corresponds to a higher aversion to low natural growth deviation.

We use the estimation results of the growth equation above to simulate the amount of aid resulting from our theoretical model. We also compare these simulations with the observed data in 2008, which covers more than 90 countries. 7 The growth-aid relation for the 2008 data, based on the previously estimated coefficients, is presented in Figure 2. This relation has an increasing pattern.

Our simulations cover both the optimal allocations of aid obtained without the weighting scheme in the donors' objective (i.e., α = 1) and with the weighting scheme taking into account natural growth deviation (i.e., α < 1). Recall that both allocations, either with α = 1 or α < 1, rely on the same set of estimates obtained from the growth equation presented above (which focuses on the structural handicaps to growth in recipient countries). We consider the following assumptions for two models: (i) the headcount poverty rate is based on two types of the poverty line, 2 dollars/day and 1.25 dollars/day, which correspond to two sets of countries, 93 and 91 respectively, (ii) for comparison with [START_REF] Collier | Aid allocation and poverty reduction[END_REF], we assume that the growth elasticity of poverty reduction η is the same for all countries and η = -2. Complementary simulations are also done with elasticity varying with five groups of countries. Detailed simulations of optimal allocations are given in Appendix G (with constant elasticity) and F (with elasticity at the regional level).

The model without natural growth deviation corresponds to α = 1 (in this case, the value of γ does not affect the results). This situation corresponds to the donors' objective function depending only on economic growth obtained in recipient countries, as in [START_REF] Collier | Aid allocation and poverty reduction[END_REF]. However, differently from [START_REF] Collier | Aid allocation and poverty reduction[END_REF] focusing on policy factors, we estimate growth function focusing on aid recipients' structural handicaps (lack of human capital and economic vulnerability). As mentioned previously, aid may be growth-enhancing in countries suffering structural handicaps. The aid effectiveness principle should conduce the donors to give aid to such countries in high need. In this sense, the effectiveness principle is not necessarily incompatible with fairness. Regarding the model with natural growth deviation (i.e. α < 1), we consider two parameterizations: (i) α = 0.7, γ = 0.7 and (ii) α = 0.7, γ = 0.3. Recall that case (ii) with γ = 0.3 means that donors are more sensitive to natural growth deviation compared to the case (i) with γ = 0.7. Moreover, our model requires the value of the targeted growth rate g * related to η as shown previously in Section 2. With the assumption that η = -2 ∀i, we get g * = 0.01365, ∀i. Our results are manifold. 1. Our first result concerns the number of countries receiving development aid. While in 2008, the observed allocation of aid proposed a positive amount of aid (in GDP share) to all countries considered in our exercises (93 countries with the 2$/day poverty line and 91 countries with the 1.25$/day poverty line), our optimal allocation shortlists only around one-third of them.

Tables 3 and4 here Tables 5 and6 here Tables 3 and4 report simulation results following 5 different regions, while Tables 5 and6 following four income groups.The optimal choice is based on a multi-criteria principle, including initial poverty, natural growth deviation, and structural handicaps to the growth of recipient countries (lack of human capital and vulnerability).

Values reported in these tables represent the average amount for each region (Tables 3-4) or each income group (Tables 56). We remark that compared to the observed allocation, on average, our optimal allocation proposes to increase aid for Sub-Saharan Africa (Tables 3 and4) and low-income countries (Tables 5 and6).

2. We observe that initial poverty in recipient countries is one of the key elements which justifies a large aid flow. Tables 7 and8 report the observed aid allocation and the optimal aid allocation for ten countries with the highest poverty rates (based on 1.25$/day and 2$/day poverty lines, respectively). Compared to the observed allocation, our optimal allocations with (α = 1) or without natural growth deviation (α < 1) favor these developing countries and propose higher aid amounts to them. Note that all countries (except Haiti) that should receive a more significant aid amount belong to Sub-Saharan Africa.

Tables 7 and8 here 3. The third result relates to the difference between the allocation without natural growth deviation in Collier and Dollar (α = 1) and the allocation with natural growth deviation (α < 1). The latter allocation covers more countries than the former. More precisely, with the 2$/day poverty line (Table 3 or 5), the solution without growth deviation-weight (i.e., α = 1) only covers 33 countries. With the weighting scheme in the donors' objective (i.e., α = 0.7), the number of countries receiving a positive aid increases to 38 and 39 for γ = 0.7 and γ = 0.3, respectively.

A similar result can be found when the 1.25$/day poverty line is considered (Table 4 or 6).

4. The fourth finding is linked to the previous one. As a result of the increase in the number of countries receiving aid, for most of them, the optimal allocation with the weighting scheme (i.e., α = 0.7) may propose an aid amount lower than the case without (i.e., α = 1). Nevertheless, we can identify countries that take advantage of our multi-criteria principle compared to the case without this weighting scheme.

Tables 9 and10 here Table 9 identifies 12 "winning" countries (Angola, Bangladesh, Cambodia, Congo Republic, Ethiopia, Ghana, Guinea, Mauritania, Papua New Guinea, Senegal, Sudan and Tanzania) which would receive a higher aid amount in the case of α = 0.7, γ = 0.7 compared to the case of α = 1 (considering all criterion: lack of human capital, economic vulnerability, poverty rate and natural growth deviation). Yemen is added to this list when α = 0.7, γ = 0.3. Moreover, given α = 0.7, we observe that the difference in aid between two cases γ = 0.7 and γ = 0.3 is relatively important for countries with a low natural growth deviation (i.e., a low ratio g i /g * i ). Angola is a striking example. With only g i /g * i equal to 1.27, this country receives an aid flow of 0.0131% of GDP when γ = 0.7. If donors are more sensitive to low natural deviation, i.e., γ = 0.3, then the aid flow increases to 0.0495% of GDP. 8 When using the 1.25$ poverty line, the figures corresponding to γ = 0.7 and γ = 0.3 are respectively eight "wining" countries (Angola, Bangladesh, Congo Republic, Ethiopia, Guinea, Lao PDR, Papua New Guinea, and Tanzania) and 10 "winning" countries (the previous 8 countries plus Mauritania and Senegal).

Our analysis stresses that the optimal aid allocation is biased in favour of countries with a low ratio g i /g * i (given other factors such as economic vulnerability, lack of human capital, and poverty rate) when donors are more sensitive to the natural growth deviation in recipient countries.

5. We also compute the marginal efficiency of aid, i.e., the number of the poor reduced by increasing the total aid amount of 1 million dollars. It is shown that the optimal allocation with the weighting scheme in donors' objective (i.e., α = 0.7) gives higher marginal efficiency than the optimal allocation without this weighting scheme (i.e., α = 1). Considering the 2$/day poverty line, the reduction correpsonds to 131.78 people per million dollars when α = 1. Regarding the optimal allocation when α = 0.7, this quantity (denoted as λi , see also Section 2) varies among countries that receive a positive amount of aid. The average values of these λi for the whole sample are 140.96 for γ = 0.7 and 151.89 people per million dollars for γ = 0.3 (Table 11). Figures 3a-b give a full picture of the distribution of λi for the whole sample.

For the case with the 1.25$ poverty line, the number of people lifting from poverty in the optimal allocation when α = 1 is 88.85. When α = 0.7, the mean values of these λi are respectively 96.42 and 102.59 people for γ = 0.7 and γ = 0.3 (Table 12)and Figures 3c-d). 9

Tables 11 and12 here Figure 3 here

We observe that aid is more efficient in terms of poverty reduction in recipient countries with a high economic vulnerability. Indeed, the number of people lifted from poverty per million dollars of aid is higher than the average number for these countries. 10

8 Other countries widely benefit from the donors' sensitivity with respect to natural growth deviation are Congo Republic (g i /g * i = 2.51), Senegal (g i /g * i = 2.69), Yemen (g i /g * i = 2.53). 9 We remark that Angola is the country where aid is the lowest efficient among countries receiving a positive amount of aid. The numbers of people lifted from poverty (regarding the 2$ poverty line) are 77 and 70 for γ = 0.7 and γ = 0.3 respectively. If the poverty line is 1.25$/day, these numbers are respectively 58 people and 53 people for γ = 0.7 and γ = 0.3. In contrast, Burundi is the country where aid is the most efficient. Indeed, for the 2$ poverty line, there are 167 people (in the situation of γ = 0.7) and 205 people (γ = 0.3) that can escape poverty. When using the 1.25$ poverty line, these figures are respectively 112 people and 137 people per million dollars.

10 For the case of 2$ dollars poverty line, λ = 0.7, γ = 0.3, this number is 193.83 for Timor-Leste, 203.44 for Malawi, 184.5 for Zambia, 192.14 for Guinea-Bissau, 205.17 for Burundi, 199.01 for Comoros, 179.85 for Chad, 177.88 for the Gambia, which are considerably higher than the average level (151.89).

6. Finally, a reduction in the value of γ, equivalent to an increase in the donors' aversion to low natural deviation (1 -γ), significantly impacts the optimal aid allocation and raises the average marginal efficiency of aid. In other words, by giving more priority to countries with a low ratio of g i /g * i , aid donors choose an efficient and fair allocation that can help lift more people from poverty. For the 2$ poverty line, the average number of people increases from 140.96 to 151.89, and for the 1.25$ poverty line, this number is from 96.42 to 102.59 (Tables 11 and12).

Our model adopts the same growth elasticity of poverty reduction for all countries in our sample, η = -2. This choice is mainly motivated by the purpose of comparison with Collier andDollar (2001,2002). We remark that the values of this elasticity may vary with the poverty-related measures (poverty headcount, poverty gap, poverty line) and the changes in income measures (GDP per capita or survey mean income) (Ram, 2006, Amini andDan Bianco, 2016) and differs among countries [START_REF] Bourguignon | The growth elasticity of poverty reduction: explaining heterogeneity across countries and time series[END_REF][START_REF] Bourguignon | Aid effectiveness: Revisiting the trade-off between needs and governance[END_REF]. This elasticity also depends on macroeconomic activity (expansion, recession), as [START_REF] Aiyemo | Recessions and the vulnerable[END_REF] shows. This author reported that it is -2.5% during positive economic activity while it is -2.2% during recessions. These values are not far from the assumption of η = -2. However, they reported estimates for the GEB neither at the regional level nor at the country level. [START_REF] Ram | Growth elasticity of poverty: Alternative estimates and a note of caution[END_REF] is the only study that provided the growth elasticity at the regional levels, for the period 2002-2015. This elasticity is estimated with a value equal to -0.9 for East Asia and Pacific, -1.35 for Europe and Central Asia, -0.95 for Latin America and Caribbean, -1.6 for Middle East and North Africa, -0.81 for South Asia, and -0.72 for Sub-Saharan Africa.

In line with this literature and for comparison purposes, we also run simulations using Ram (2006)'s regional values. However, it should be noted that these calculations were based on the 1$ poverty line. Subsequently, [START_REF] Lenagala | Growth elasticity of poverty: estimates from new data[END_REF] and Ram (2011[START_REF] Ram | Income elasticity of poverty in developing countries: updated estimated from new data[END_REF] provided some updates and estimations based on other poverty lines (1.25$, 2$, and 2.5$ ) but they did not cover all the regions above. More precisely, there were no updates for countries from Europe and Central Asia, Latin America and Caribbean, Middle East and North Africa, East Asia (except China), South Asia (except India) whereas the estimate for Sub-Saharan Africa based on the 2$ poverty rate is very close, -0.73 [START_REF] Ram | Income elasticity of poverty in developing countries: updated estimated from new data[END_REF]. As the period studied by [START_REF] Ram | Growth elasticity of poverty: Alternative estimates and a note of caution[END_REF] is close to our study, we think it is reasonable to use Ram?s (2006) estimates in our simulations based on both the 1.25 and 2 poverty lines.

Our simulations on the optimal allocation of aid using the five different values of η are presented in Appendix F. Comparing these results with that given in Appendix G using the same elasticity for all regions, we can observe that the results mentioned above remain. Mainly, Table 13 and 14 indicate that countries with the highest poverty rate always take advantage of our principle, compared to the current allocation. Optimal allocations of aid in case of elasticity of poverty reduction at the regional levels give an aid flow slightly lower than allocations using η = -2. This difference is obtained because all five values of η are lower than 2 (absolute value).

Conclusion

In this study, we propose a model of aid allocation which aims to equalize the opportunity between recipient countries to reach a given common goal of poverty reduction. Based on a multi-criteria principle, our optimal allocation is fair and efficient. The modeling accounts for the natural deviation or gap between the growth rate required to reach a specific target and the actual growth rate observed in the recipient countries. To maximize poverty reduction in all recipient countries, we consider their structural handicaps to growth in terms of lack of human capital, economic vulnerability, initial poverty, and natural growth deviation.

This paper takes the Millennium Development Goal (MDG) as a retrospective example, proposing to reduce poverty by half by 2015 compared to 1990. It is all about approaching the MDG by giving to each recipient country the same probability of lifting itself out of poverty by half. We calculate the required growth rate from this goal, which is considered as given since it does not depend on foreign aid.

Our proposed aid allocation substantially differs from the observed allocation. In particular, poor and vulnerable countries are "winners" as they should receive higher aid following our multi-criteria principle. Our analysis shows that it is possible to design a fair allocation of aid, which is also an efficient one since we observe that aid may be more efficient in poverty reduction in recipient countries with a high economic vulnerability. In addition, our results shed light on the impact of the weighting scheme in the donors' objective on the allocation of aid in favor of countries with a low natural growth deviation. We also show the meaningful impact of the donors' aversion to natural growth deviation on the optimal allocation of aid and its marginal efficiency. By giving more priority to countries with a low natural growth deviation or gap, aid donors can help to lift more people from poverty.

It is undeniable that performance (i.e., good governance, good macroeconomic policy in recipient countries, etc.) is the most frequent argument to explain aid allocation decided by several multilateral development banks. Our study proposes that the performance approach could be improved by considering the vulnerable initial situation in recipient countries to obtain the same initial opportunities for development. This fairness focus is not necessarily incompatible with the effectiveness principle one. Further extensions are needed to check the robustness of our results. The theoretical analysis can also be extended to include the dynamic aspect of aid allocation as official development aid may result from a dynamic interaction between donor and recipient countries. It is also promising to have other time horizons and targeted growth rates to achieve different development goals.

This paper examines the choice from the donors' point of view and discusses criteria and conditions for an optimal aid allocation, taking into account recipient characteristics and donors' preferences. However, the aid effectiveness relies not only on aid forms and donors' decisions but also on the recipient countries' circumstances and how aid is used. Several internal factors of a country determining its economic development are not included in the donors' aid allocation decisions. The normative approach adopted by our analysis should be completed by asking why the same amount of aid is efficient in one country and not in another one. Moreover, the question related to the conditions under which a recipient country can get out of poverty should be deeply analyzed in parallel with our normative approach. In brief, analyses on the effectiveness and allocation of aid are perfectly complementary. Appendix B: Optimal allocations of aid Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural growth deviation corresponds to α = 1. The last two columns correspond to the model with natural growth deviation, a lower value of γ means a higher donors' sensitivity with respect to growth deviation in recipient countries. Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita). The model of [START_REF] Collier | Aid allocation and poverty reduction[END_REF] corresponds to α = 1. The last two columns correspond to our model with α = 0.7, γ = 0.7, g * = -0.0273/η = 0.01365, ∀i.

Tables and Figures

Appendix C: "Winning" countries

(%GDP) (%GDP) (%GDP) (% GDP) (% GDP) (%GDP) (% GDP) α = 1 α = 0.7 α = 0.7 α = 1 α = 0.7 α = 0.7 γ = 0.7 γ = 0.3 γ = 0.7 γ = 0.
The last row reports the marginal efficiency of aid of the Collier and Dollar' model ( λCD ) and our model ( λ and the average of λi ). Notes: The model of [START_REF] Collier | Aid allocation and poverty reduction[END_REF] corresponds to α = 1. The last two columns correspond to our model with α = 0.7, γ = 0.7, g * = -0.0273/η i = 0.01365, ∀i.

The last row reports the marginal efficiency of aid of the Collier and Dollar' model ( λCD ) and our model ( λ and the average of λi ). Notes: The model of [START_REF] Collier | Aid allocation and poverty reduction[END_REF] corresponds to α = 1. The last two columns correspond to our model with α = 0.7, γ = 0.3, g * = -0.0273/η = 0.01365, ∀i.

The last row reports the marginal efficiency of aid of the Collier and Dollar' model ( λCD ) and our model ( λ and the average of λi ).

Appendix G: Detailed optimal allocations of aid with growth elasticity of poverty reduction at regional level

Figure 2 :

 2 Figure 2: The estimated relation between growth and aid for the 2008 data.

Figure 3 :

 3 Figure 3: Marginal efficiency of aid, λi . (a) poverty line = 2$/day, γ = 0.7, (b) poverty line = 2$/day, γ = 0.3, (c) poverty line = 1.25$/day, γ = 0.7, (d) poverty line = 1.25$/day, γ = 0.3. The horizontal lines correspond to the Collier-Dollar utility function, λCD = 131.78 for poverty line = 2$ per day and = 88.85 for poverty line = 1.25$/day.

  Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita). The model of[START_REF] Collier | Aid allocation and poverty reduction[END_REF] corresponds to α = 1. The last two columns correspond to our model with α = 0.7, γ = 0.3, g * = -0.0273/η = 0.01365, ∀i.The last row reports the marginal efficiency of aid of the Collier and Dollar' model ( λCD ) and our model ( λ and the average of λi ).

Notes:

  Simulations are run under the following assumption about the regional values for growth elasticity of poverty reduction (η i , cf. Ram, 2006): -0.90 (East Asia & Pacific countries), -0.95 (Latin America & Caribbean), -1.60 (Middle East & North Africa), -0.81 (South Asia & Central Asia), and -0.72 (Sub-Saharan Africa). The model of Collier and Dollar (2002) corresponds to α = 1. The last two columns correspond to our model with α = 0.7, γ = 0.7, g * i = 0.0273/η i . The last row reports the marginal efficiency of aid of the Collier and Dollar' model ( λCD ), our model ( λ and the average of λi ).

  Notes: Simulations are run under the following assumption about the regional values for growth elasticity of poverty reduction (η i , cf.[START_REF] Ram | Growth elasticity of poverty: Alternative estimates and a note of caution[END_REF]: -0.90 (East Asia & Pacific countries), -0.95 (Latin America & Caribbean), -1.60 (Middle East & North Africa), -0.81 (South Asia & Central Asia), and -0.72 (Sub-Saharan Africa). The model of Collier and Dollar (2002) corresponds to α = 1. The last two columns correspond to our model with α = 0.7, γ = 0.3, g * i = 0.0273/η i . The last row reports the marginal efficiency of aid of the Collier and Dollar' model ( λCD ), our model ( λ and the average of λi ).

  53 λ = 210.85 average { λi } = 40.03 Notes: Simulations are run under the following assumption about the regional values for growth elasticity of poverty reduction (η i , see Ram, 2006): -0.90 (East Asia & Pacific countries), -0.95 (Latin America & Caribbean), -1.60 (Middle East & North Africa), -0.81 (South Asia), and -0.72 (Sub-Saharan Africa). The model of Collier and Dollar (2002) corresponds to α = 1. The last two columns correspond to our model with α = 0.7, γ = 0.7, g * i = 0.0273/η i . The last row reports the marginal efficiency of aid of the Collier and Dollar' model ( λCD ), our model ( λ and the average of λi ).

Notes:

  Simulations are run under the following assumption about the regional values for growth elasticity of poverty reduction (η i , see[START_REF] Ram | Growth elasticity of poverty: Alternative estimates and a note of caution[END_REF]: -0.90 (East Asia & Pacific countries), -1.35 (Europe & Central Asia), -0.95 (Latin America & Caribbean), -1.60 (Middle East & North Africa), -0.81 (South Asia), and -0.72 (Sub-Saharan Africa). The model of Collier and Dollar (2002) corresponds to α = 1. The last two columns correspond to our model with α = 0.7, γ = 0.3, g * i = 0.0273/η i . The last row reports the marginal efficiency of aid of the Collier and Dollar' model ( λCD ), our model ( λ and the average of λi ).

Table 1 :

 1 Definition of variables and descriptive statistics

	Variable	Obs Mean Std. Dev.	Min.	Max.
	Growth rate of GDP per capita (g) 838	1.618	4.543	-28.546 27.271
	log GDP per capita (ln y)	971	7.969	0.958	4.6134 10.051
	Economic vulnerability index (V )	923	0.413	0.123	0.130	0.730
	Lack of human capital (H)	1007 0.428	0.243	0.012	0.965
	Aid (A, in GDP share)	958	0.091	0.125	0	1.451
	Poverty headcount, 1.25$/day	759	0.193	0.223	0	0.926
	Poverty headcount, 2$/day	759	0.325	0.287	0	0.985

Table 2 :

 2 Estimation results of the growth equation

		Within FE	System GMM
	Variable	Coefficient Std. Err. Coefficient Std. Err.
	ln y i,t-τ	0.683 * *	0.024	0.943 * *	0.020
	Aid	-0.219	0.392	-0.883	0.620
	Squared aid	-1.019 *	0.620	-1.324 * *	0.496
	Vulnerability	0.101	0.115	-0.206 * *	0.087
	Aid × Vulnerability	1.503 * *	0.740	2.455 *	1.263
	Lack of human capital	-0.527 * *	0.141	-0.259 * *	0.074
	Period 87-90	-0.085 * *	0.029	-0.032 *	0.018
	Period 91-94	-0.099 * *	0.025	-0.029	0.022
	Period 95-98	-0.087 * *	0.023	-0.020	0.016
	Period 99-02	-0.058 * *	0.020	0.003	0.014
	Period 03-06	-0.067 * *	0.016	-0.025 *	0.013
	Period 07-10	0.026 *	0.014	0.062 * *	0.010
	Intercept	2.740 * *	0.227	0.704 * *	0.195
	Number of observations		700		693
	Number of countries		109		109
	Existence of fixed effects, F (108, 572)		4.50 * *
	Hausman test, random vs. fixed effects, χ 2 (12)	151.12 * *
	Arellano-Bond test, AR(1)				-4.80 * *
	Arellano-Bond test, AR(2)				-1.44
	Sargan test of overid. restrictions, χ 2 (50)			130.63 * *
	Hansen test of overid. restrictions, χ 2 (50)			64.35 *
	Difference-in-Hansen tests of exogeneity
	Hansen test excluding group, χ 2 (44)				54.38
	Difference, χ 2 (6)				9.97

Notes. Dependent variable: ln y it . Results obtained with the within fixed effects estimator and the one-step system GMM estimator of

[START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]

using robust standard errors.

* and * * denote significance at the 10% and 5% levels, respectively. Strictly exogenous regressors include time dummies. Predetermined regressors are ln y i,t-τ , Aid, Vulnerability, Lack of human capital.

Table 3 :

 3 Observed and optimal allocations of aid for 5 regions (based on the 2$/day poverty line, 2008 data)

	Region	Observed aid Optimal aid Optimal aid Optimal aid
		(%GDP)	(%GDP)	(%GDP)	(% GDP)
			α = 1	α = 0.7	α = 0.7,
				γ = 0.7	γ = 0.3
	Sub-Saharan Africa	0.10	0.1588	0,151	0.1431
	Middle East and North Africa 0.0386	0.0279	0.0245	0.0259
	Latin America and Caribean	0.0233	0.0113	0.0011	0.0097
	East-Asia and Pacific	0.0283	0.0192	0.0198	0.0195
	South and Central Asia	0.0276	0.0195	0.0193	0.0186
	# recipients	93	33	38	39

Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural growth deviation corresponds to α = 1. The last two columns correspond to the model with natural growth deviation, a lower value of γ means a higher donors' sensitivity with respect to growth deviation in recipient countries. Our sample contains 93 countries: 41 countries in Sub-Saharan Africa, 11 countries in Middle East and North Africa, 24 countries in Latin America and Caribean, 9 countries in East-Asia and Pacific, and 8 countries in South and Central Asia

Table 4 :

 4 Observed and optimal allocations of aid for 5 regions (based on the 1.25$/day poverty line, 2008 data)

	Region	Observed aid Optimal aid Optimal aid Optimal aid
		(%GDP)	(%GDP)	(%GDP)	(% GDP)
			α = 1	α = 0.7	α = 0.7
				γ = 0.7	γ = 0.3
	Sub-Saharan Africa	0.10	0.1553	0,1469	0.1366
	Middle East and North Africa 0.0386	0.0226	0.0186	0.0153
	Latin America and Caribean	0.0233	0.0121	0.0115	0.0109
	East-Asia and Pacific	0.0332	0.0084	0.0116	0.0125
	South Asia	0.0312	0.0124	0.0132	0.0134
	# recipients	91	30	31	35

Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural growth deviation corresponds to α = 1. The last two columns correspond to the model with natural growth deviation, a lower value of γ means a higher donors' sensitivity with respect to growth deviation in recipient countries. Our sample contains 91 countries: 41 countries in Sub-Saharan Africa, 11 countries in Middle East and North Africa, 24 countries in Latin America and Caribean, 8 countries in East-Asia and Pacific, and 7 countries in South Asia

Table 5 :

 5 Observed and optimal allocations of aid for 4 income groups (based on the 2$/day poverty line, 2008 data)

	Group	Observed aid Optimal aid Optimal aid Optimal aid
		(%GDP)	(%GDP)	(%GDP)	(% GDP)
			α = 1	α = 0.7	α = 0.7
				γ = 0.7	γ = 0.3
	Low income	0.1289	0.2416	0.2186	0.2149
	Low middle income	0.0555	0.0462	0.0449	0.0439
	Upper middle income 0.0169	0	0	0
	High income	0.00425	0	0	0
	# recipients	93	33	38	39
	Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with
	respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural
	growth deviation corresponds to α = 1. The last two columns correspond to the model with
	natural growth deviation, a lower value of γ means a higher donors' sensitivity with respect
	to growth deviation in recipient countries. Our sample contains 93 countries: 24 low income
	countries, 35 low middle income countries, 30 upper middle income countries, and 4 high income
	countries.

Table 6 :

 6 Observed and optimal allocations of aid for 4 income groups (based on the 1.25$/day poverty line, 2008 data)

	Group	Observed aid Optimal aid Optimal aid Optimal aid
		(%GDP)	(%GDP)	(%GDP)	(% GDP)
			α = 1	α = 0.7	α = 0.7
				γ = 0.7	γ = 0.3
	Low income	0.1289	0.2351	0.223	0.2117
	Low middle income	0.0555	0.0406	0.0382	0.0366
	Upper middle income 0.0180	0	0	0
	High income	0.0043	0	0	0
	# recipients	91	30	31	35

Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural growth deviation corresponds to α = 1. The last two columns correspond to the model with natural growth deviation, a lower value of γ means a higher donors' sensitivity with respect to growth deviation in recipient countries. Our sample contains 91 countries: 24 low income countries, 35 low middle income countries, 28 upper middle income countries, and 4 high income countries.

Table 7 :

 7 Observed and optimal allocations of aid of 10 countries with highest poverty rate (based on the 1.25$/day poverty line, 2008 data)

	Country	Poverty Observed aid Optimal aid Optimal aid Optimal aid
		rate	(%GDP)	(%GDP)	(%GDP)	(% GDP)
				α = 1	α = 0.7	α = 0.7
					γ = 0.7	γ = 0.3
	Central African Rep. 0.6283	0.1312	0.3229	0.3128	0.3022
	Haiti	0.6358	0.1419	0.2909	0.2760	0.2611
	Zambia	0.6627	0.0843	0.3570	0.3298	0.3030
	Nigeria	0.6646	0.0066	0.1679	0.1591	0.1510
	Tanzania	0.6676	0.1138	0.1294	0.1303	0.1302
	Malawi	0.6734	0.2280	0.4103	0.3914	0.3716
	Rwanda	0.6766	0.1996	0.3339	0.3163	0.2984
	Madagascar	0.7162	0.0902	0.2902	0.2775	0.2644
	Burundi	0.8058	0.3214	0.4773	0.4660	0.4540
	Congo Dem. Rep.	0.8615	0.1707	0.3888	0.3843	0.3794

Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural growth deviation corresponds to α = 1. The last two columns correspond to the model with natural growth deviation.

Table 8 :

 8 Observed and optimal allocations of aid of 10 countries with highest poverty rate (based on the 2$/day poverty line, 2008 data)

	Country	Poverty Observed aid Optimal aid Optimal aid Optimal aid
		rate h	(%GDP)	(%GDP)	(%GDP)	(% GDP)
				α = 1	α = 0.7	α = 0.7
					γ = 0.7	γ = 0.3
	Central African Rep. 0.8017	0.1312	0.3108	0.2990	0.2861
	Zambia	0.8104	0.0843	0.3294	0.2978	0.2666
	Mozambique	0.8187	0.2155	0.3009	0.2883	0.2746
	Nigeria	0.8411	0.0066	0.1341	0.1283	0.1216
	Rwanda	0.8482	0.1996	0.3159	0.2954	0.2739
	Tanzania	0.8741	0.1138	0.1133	0.1149	0.1145
	Malawi	0.8755	0.2280	0.4000	0.3785	0.2550
	Madagascar	0.8943	0.0902	0.2735	0.2586	0.2427
	Burundi	0.9321	0.3214	0.4652	0.4507	0.4346
	Congo Dem. Rep.	0.9449	0.1707	0.3805	0.3743	0.3672

Table 9

 9

	: "Winning" countries when accounting for natural growth deviation (g i /g * i ). Op-
	timal allocations of aid. Headcount poverty rate based on the 2$/day poverty line, 2008
	data.
	Country	H	h	V	g i /g * i	Optimal aid Optimal aid Optimal aid
						(%GDP)	(% GDP)	(% GDP)
						α = 1	α = 0.7	α = 0.7
							γ = 0.7	γ = 0.3
	Angola	0.69 0.72 0.47 1.27	0	0.0131	0.0495
	Bangladesh	0.43 0.79 0.22 3.26	0.0336	0.0412	0.0452
	Cambodia	0.41 0.54 0.37 3.06	0	0.0156	0.0255
	Congo Republic	0.38 0.74 0.42 2.51	0	0.0001	0.0211
	Ethiopia	0.64 0.69 0.25 3.42	0.1140	0.1174	0.1188
	Ghana	0.32 0.47 0.33 3.61	0.0125	0.0165	0.0162
	Guinea	0.58 0.69 0.26 3.41	0.0960	0.1004	0.1023
	Mauritania	0.47 0.48 0.49 3.06	0	0.0335	0.0439
	Papua New Guinea 0.43 0.64 0.42 3.63	0.0681	0.0717	0.0710
	Senegal	0.54 0.50 0.35 2.69	0	0.0270	0.0436
	Sudan	0.45 0.45 0.48 3.29	0.0174	0.0383	0.0411
	Tanzania	0.52 0.87 0.27 3.64	0.1133	0.1149	0.1145
	Yemen	0.43 0.46 0.44 2.53	0	0	0.0121

Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural growth deviation corresponds to α = 1. The last two columns correspond to the model with natural growth deviation, a lower value of γ means a higher donors' sensitivity with respect to growth deviation in recipient countries. H: lack of human capital; h: poverty line; V : economic vulnerability index; g i /g * i : natural growth deviation

Table 10 :

 10 "Winning" countries when accounting for natural growth deviation (g i /g * i ). Optimal allocations of aid. Headcount poverty rate based on the 1.25$/day poverty line, 2008 data.

	Country	H	h	V	g i /g * i	Optimal aid Optimal aid Optimal aid
						(% GDP)	(% GDP)	(% GDP)
							α = 1	α = 0.7	α = 0.7
							γ = 0.7	γ = 0.3
	Angola	0.69 0.56 0.47 0.85		0	0.0274	0.0642
	Bangladesh	0.43 0.47 0.22 3.06		0.0101	0.0220	0.0296
	Congo Republic	0.38 0.53 0.42 2.39		0	0.0145	0.0332
	Ethiopia	0.64 0.32 0.25 3.10		0.0600	0.0698	0.0746
	Guinea	0.58 0.42 0.26 3.31		0.0820	0.0883	0.0925
	Lao PDR	0.35 0.34 0.44 3.30		0.0073	0.0238	0.0296
	Papua New Guinea 0.43 0.42 0.42 3.60		0.0634	0.0690	0.0706
	Tanzania	0.52 0.67 0.27 3.73		0.129	0.1303	0.1302
	Mauritania	0.47 0.23 0.49 2.13		0	0	0.0004
	Senegal	0.54 0.25 0.35 2.05		0	0	0.0022
	Appendix D: Marginal efficiency	λ

Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural growth deviation corresponds to α = 1. The last two columns correspond to the model with natural growth deviation, a lower value of γ means a higher donors' sensitivity with respect to growth deviation in recipient countries. H: lack of human capital; h: poverty line; V : economic vulnerability index; g i /g * i : natural growth deviation

Table 11 :

 11 Marginal Efficiency λ. Headcount poverty rate based on the 2$/day poverty line, 2008 data.

	Region		Average	λ
		α = 1 α = 0.7 α = 0.7
			γ = 0.7 γ = 0.3
	Sub-Saharan Africa	131.78 141.48 154.66
	Middle East and North Africa 131.78 162.98 150.1
	Latin America and Caribean	131.78 148.81 166.52
	East-Asia and Pacific	131.78 130.1	130.58
	South Asia and Central Asia	131.78 132.75 135.5
	Whole sample average	131.78 140.96 151.89
	# recipients	33	38	39

Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural growth deviation corresponds to α = 1. The last two columns correspond to the model with natural growth deviation, a lower value of γ means a higher donors' sensitivity with respect to growth deviation in recipient countries. The marginal efficiency value corresponds to the average value of 31 recipients over 41 countries in Sub-Saharan Africa; 1 recipient over 11 countries in Middle East & North Africa (and 2 recipients when γ = 0.3); one recipient over 24 countries in Latin America & Caribean, 3 recipient over 9 countries in East-Asia and Pacific; 2 recipient over 8 countries in South Asia and Central Asia.

Table 12 :

 12 Marginal Efficiency λ. Headcount poverty rate based on the 1.25 $/day poverty line, 2008 data.

	Region

Notes: Simulations are run under the assumption η = -2 (elasticity of poverty reduction with respect to income per capita) and g * = 0.01365, ∀i. The model without consideration for natural growth deviation corresponds to α = 1. The last two columns correspond to the model with natural growth deviation, a lower value of γ means a higher donors' sensitivity with respect to growth deviation in recipient countries. The marginal efficiency value corresponds to the average value of 29 recipients over 41 countries in Sub-Saharan Africa; 1 recipient over 11 countries in Middle East & North Africa; 1 recipient over 24 countries in Latin America & Caribean, 2 recipient over 8 countries in East-Asia and Pacific; 2 recipient over 7 countries in South Asia.

Table 13 :

 13 Observed and optimal allocations of aid of 10 countries with highest poverty rate, based on the$1.25/day poverty line (2008 data) and growth elasticity of poverty reduction following regions.

	η = -2	Regional η

Notes: The model without consideration for natural growth deviation corresponds to α = 1. The columns with α = 0.7 correspond to the model with natural growth deviation.

Table 14 :

 14 Observed and optimal allocations of aid of 10 countries with highest poverty rate, based on the $2/day poverty line (2008 data) and growth elasticity of poverty reduction following regions.

	η = -2	Regional η

Table 15 :

 15 Actual and optimal allocations of aid, headcount poverty rate based on $2/day poverty line, α = 0.7, γ = 0.7, and η = -2

	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.7	α = 0.7, γ = 0.7
	Algeria	0.0019	0	0	0
	Angola	0.0052	0	0.0134	76.91
	Argentina	0.0004	0	0	0
	Bangladesh	0.0239	0.0336	0.0412	125.93
	Belize	0.0199	0	0	0
	Benin	0.0961	0.1575	0.1496	137.24
	Bhutan	0.0806	0	0	0
	Bolivia	0.0389	0	0	0
	Botswana	0.0562	0	0	0
	Brazil	0.0003	0	0	0
	Burkina Faso	0.1199	0.2199	0.2094	141.15
	Burundi	0.3214	0.4652	0.4507	166.55
	Cambodia	0.0755	0	0.0156	122.21
	Cameroon	0.0234	0	0	0
	Cape Verde	0.1464	0	0	0
	Central African Rep. 0.1312	0.3108	0.2990	149.62
	Chad	0.0631	0.3545	0.3216	156.39
	Chile	0.0006	0	0	0
	China	0.0003	0	0	0
	Colombia	0.0041	0	0	0
	Comoros	0.0785	0.3741	0.3361	165.38
	Congo, Dem. Rep.	0.1707	0.3805	0.3743	157.56
	Congo, Rep.	0.0553	0	0.0001	111.86
	Costa Rica	0.0023	0	0	0
	Cote d'Ivoire	0.0279	0	0	0
	Djibouti	0.1313	0	0	0
	Dominican Republic	0.0035	0	0	0
	Ecuador	0.0044	0	0	0
					continued on next page

Table 16 :

 16 Actual and optimal allocations of aid, headcount poverty rate based on $2/day poverty line, α = 0.7, γ = 0.3, and η = -2

	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.3	α = 0.7, γ = 0.3
	Algeria	0.0019	0	0	0
	Angola	0.0052	0	0.0495	70.28
	Argentina	0.0004	0	0	0
	Bangladesh	0.0239	0.0336	0.0452	122.89
	Belize	0.0199	0	0	0
	Benin	0.0961	0.1575	0.1410	143.20
	Bhutan	0.0806	0	0	0
	Bolivia	0.0389	0	0	0
	Botswana	0.0562	0	0	0
	Brazil	0.0003	0	0	0
	Burkina Faso	0.1199	0.2199	0.1981	151.18
	Burundi	0.3214	0.4652	0.4346	205.17
	Cambodia	0.0755	0	0.0255	118.51
	Cameroon	0.0234	0	0	0
	Cape Verde	0.1464	0	0	0
	Central African Rep. 0.1312	0.3108	0.2861	169.25
	Chad	0.0631	0.3545	0.2901	179.85
	Chile	0.0006	0	0	0
	China	0.0003	0	0	0
	Colombia	0.0041	0	0	0
	Comoros	0.0785	0.3741	0.2980	199.01
	Congo, Dem. Rep.	0.1707	0.3805	0.3672	186.69
	Congo, Rep.	0.0553	0	0.0211	105.87
	Costa Rica	0.0023	0	0	0
	Cote d'Ivoire	0.0279	0	0	0
	Djibouti	0.1313	0	0	0
	Dominican Republic	0.0035	0	0	0
	Ecuador	0.0044	0	0	0
					continued on next page

Table 17 :

 17 Actual and optimal allocations of aid, headcount poverty rate based on $1.25 per day poverty line, α = 0.7, γ = 0.7, and η = -2

	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.7	α = 0.7, γ = 0.7
	Algeria	0.0019	0	0	0
	Angola	0.0052	0	0.0274	57.94
	Argentina	0.0004	0	0	0
	Bangladesh	0.0239	0.0102	0.0220	83.45
	Belize	0.0199	0	0	0
	Benin	0.0961	0.1373	0.1317	91.19
	Bhutan	0.0806	0	0	0
	Bolivia	0.0389	0	0	0
	Botswana	0.0562	0	0	0
	Brazil	0.0003	0	0	0
					continued on next page

Table 18 :

 18 Actual and optimal allocations of aid, headcount poverty rate based on $1.25/day poverty line, α = 0.7, γ = 0.3, and η = -2

	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.3	α = 0.7, γ = 0.3
	Algeria	0.0019	0	0	0
	Angola	0.0052	0	0.0642	52.65
	Argentina	0.0004	0	0	0
	Bangladesh	0.0239	0.0102	0.0296	80.01
	Belize	0.0199	0	0	0
	Benin	0.0961	0.1373	0.1262	93.49
	Bhutan	0.0806	0	0	0
	Bolivia	0.0389	0	0	0
	Botswana	0.0562	0	0	0
	Brazil	0.0003	0	0	0
	Burkina Faso	0.1199	0.2058	0.1864	99.53
	Burundi	0.3214	0.4773	0.4540	137.14
	Cambodia	0.0755	0	0	0
	Cameroon	0.0234	0	0	0
	Cape Verde	0.1464	0	0	0
	Central African Rep. 0.1312	0.3229	0.3022	113.48
	Chad	0.0631	0.3423	0.2792	118.57
	Chile	0.0006	0	0	0
	China	0.0003	0	0	0
	Colombia	0.0041	0	0	0
	Comoros	0.0785	0.3830	0.3123	133.67
	Congo, Dem. Rep.	0.1707	0.3888	0.3794	124.50
	Congo, Rep.	0.0553	0	0.0332	73.93
	Costa Rica	0.0023	0	0	0
	Cote d'Ivoire	0.0279	0	0	0
	Djibouti	0.1313	0	0	0
	Dominican Republic	0.0035	0	0	0
	Ecuador	0.0044	0	0	0
					continued on next page

Table 19 :

 19 Actual and optimal allocations of aid, headcount poverty rate based on $2/day poverty line, α = 0.7, γ = 0.7, and regional elasticity of poverty reduction

	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.7	α = 0.7, γ = 0.7
	Algeria	0.0019	0	0
	Angola	0.0052	0	0
	Argentina	0.0004	0	0
	Bangladesh	0.0239	0.0240	0.0265	55.59
	Belize	0.0199	0	0
	Benin	0.0961	0.1189	0.0987	62.04
	Bhutan	0.0806	0	-0
	Bolivia	0.0389	0	0
	Botswana	0.0562	0	0
	Brazil	0.0003	0	0
	Burkina Faso	0.1199	0.1888	0.1686	63.92
	Burundi	0.3214	0.4546	0.4354	73.19
	Cambodia	0.0755	0	0.0321	52.23
	Cameroon	0.0234	0	0
	Cape Verde	0.1464	0	0
	Central African Republic 0.1312	0.2970	0.2745	67.30
	Chad	0.0631	0.3159	0.2752	68.74
	Chile	0.0006	0	0
	China	0.0003	0	0
	Colombia	0.0041	0	0
	Comoros	0.0785	0.3259	0.2964	72.16
	Congo, Dem. Rep.	0.1707	0.3722	0.3652	70.21
	Congo, Rep.	0.0553	0	0
	Costa Rica	0.0023	0	0
	Cote d'Ivoire	0.0279	0	0
					continued on next page

Table 19

 19

			-continued from previous page
	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.7	α = 0.7, γ = 0.7
	Mali	0.1144	0.2189	0.2022	65.20
	Mauritania	0.1251	0	0
	Mexico	0.0001	0	0
	Morocco	0.0166	0	0
	Mozambique	0.2155	0.2791	0.2623	67.30
	Namibia	0.0244	0	0
	Nepal	0.0549	0.1116	0.1042	59.73
	Nicaragua	0.1192	0	0
	Niger	0.1144	0.2682	0.2500	65.68
	Nigeria	0.0066	0.0872	0.0698	60.69
	Pakistan	0.0093	0	0
	Panama	0.0013	0	0
	Papua New Guinea	0.0381	0.0786	0.0940	54.47
	Paraguay	0.0080	0	0
	Peru	0.0038	0	0
	Philippines	0.0003	0	0
	Rwanda	0.1996	0.2939	0.2634	68.88
	Sao Tome and Principe	0.2559	0	0
	Senegal	0.0801	0	0
	Seychelles	0.0140	0	0
	Sierra Leone	0.1974	0.2688	0.2568	66.53
	South Africa	0.0042	0	0
	Sri Lanka	0.0184	0	0
	St. Lucia	0.0181	0	0
	Sudan	0.0483	0.2159	0.2768	42.07
	Suriname	0.0286	0	0
	Swaziland	0.0232	0	0
	Syrian Arab Republic	0.0030	0	0
	Tanzania	0.1138	0.0864	0.0754	60.52
					continued on next page

Table 19

 19

			-continued from previous page
	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.7	α = 0.7, γ = 0.7
	Timor-Leste	0.0844	0.3074	0.2961	63.45
	Togo	0.1048	0.2029	0.1861	66.34
	Tunisia	0.0088	0	0	0
	Turkey	0.0015	0	0	0
	Uganda	0.1159	0.1503	0.1266	64.27
	Uruguay	0.0011	0	0	0
	Venezuela	0.0002	0	0	0
	Vietnam	0.0290	0	0	0
	Yemen, Rep.	0.0172	0.1406	0.2180	40.38
	Zambia	0.0843	0.2961	0.2560	69.63
			λCD =57.10	λ=324.61	average { λi }=61.61

Table 20 :

 20 Actual and optimal allocations of aid, headcount poverty rate based on $2/day poverty line, α = 0.7, γ = 0.3, and regional elasticity of poverty reduction

	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.3	α = 0.7, γ = 0.3
	Algeria	0.0019	0	0
	Angola	0.0052	0	0
	Argentina	0.0004	0	0
	Bangladesh	0.0239	0.0240	0.0346	53.05
	Belize	0.0199	0	0
	Benin	0.0961	0.1189	0.1122	58.69
	Bhutan	0.0806	0	0
	Bolivia	0.0389	0	0
	Botswana	0.0562	0	0
	Brazil	0.0003	0	0
	Burkina Faso	0.1199	0.1888	0.1763	61.45
	Burundi	0.3214	0.4546	0.4312	76.83
	Cambodia	0.0755	0	0.0339	51.93
	Cameroon	0.0234	0	0
	Cape Verde	0.1464	0	0
	Central African Republic 0.1312	0.2970	0.2751	66.92
	Chad	0.0631	0.3159	0.2733	69.28
	Chile	0.0006	0	0
	China	0.0003	0	0
	Colombia	0.0041	0	0
	Comoros	0.0785	0.3259	0.2879	74.85
	Congo, Dem. Rep.	0.1707	0.3722	0.3641	71.85
	Congo, Rep.	0.0553	0	0
	Costa Rica	0.0023	0	0
	Cote d'Ivoire	0.0279	0	0
	Djibouti	0.1313	0.0786	0.1138	50.05
	Dominican Republic	0.0035	0	0
	Ecuador	0.0044	0	0
					continued on next page

Table 20

 20

			-continued from previous page
	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.3	α = 0.7, γ = 0.3
	Egypt, Arab Rep.	0.0106	0	0
	El Salvador	0.0111	0	0
	Ethiopia	0.1248	0.1671	0.1770	48.06
	Fiji	0.0129	0	0
	Gabon	0.0049	0	0
	Gambia, The	0.0948	0.2353	0.1481	67.61
	Ghana	0.0515	0	0
	Guatemala	0.0140	0	0
	Guinea	0.0983	0.0740	0.0743	54.52
	Guinea-Bissau	0.1605	0.3309	0.3060	73.00
	Guyana	0.0872	0	0
	Haiti	0.1419	0.2807	0.2666	63.89
	Honduras	0.0424	0	0
	India	0.0017	0	0
	Indonesia	0.0025	0	0
	Iran, Islamic Rep.	0.0003	0	0
	Iraq	0.1187	0	0
	Jamaica	0.0067	0	0
	Jordan	0.0326	0	0
	Kenya	0.0448	0	0
	Lao PDR	0.0955	0.1106	0.1049	59.23
	Lesotho	0.0667	0.0525	0.0893	63.35
	Madagascar	0.0902	0.2573	0.2297	67.06
	Malawi	0.2280	0.3828	0.3496	76.31
	Malaysia	0.0007	0	0
	Maldives	0.0301	0	0
	Mali	0.1144	0.2189	0.2066	63.45
	Mauritania	0.1251	0	0
	Mexico	0.0001	0	0
					continued on next page

Table 20

 20

			-continued from previous page
	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.3	α = 0.7, γ = 0.3
	Morocco	0.0166	0	0
	Mozambique	0.2155	0.2791	0.2630	66.93
	Namibia	0.0244	0	0
	Nepal	0.0549	0.1116	0.1032	60.06
	Nicaragua	0.1192	0	0
	Niger	0.1144	0.2682	0.2527	64.21
	Nigeria	0.0066	0.0872	0.0880	56.93
	Pakistan	0.0093	0	0
	Panama	0.0013	0	0
	Papua New Guinea	0.0381	0.0786	0.0882	55.56
	Paraguay	0.0080	0	0
	Peru	0.0038	0	0
	Philippines	0.0003	0	0
	Rwanda	0.1996	0.2939	0.2618	69.55
	Sao Tome and Principe	0.2559	0	0.0022	55.75
	Senegal	0.0801	0	0
	Seychelles	0.0140	0	0
	Sierra Leone	0.1974	0.2688	0.2587	65.65
	South Africa	0.0042	0	0
	Sri Lanka	0.0184	0	0
	St. Lucia	0.0181	0	0
	Sudan	0.0483	0.2159	0.2137	57.48
	Suriname	0.0286	0	0
	Swaziland	0.0232	0	0
	Syrian Arab Republic	0.0030	0	0
	Tanzania	0.1138	0.0864	0.0888	55.89
	Timor-Leste	0.0844	0.3074	0.2730	72.19
	Togo	0.1048	0.2029	0.1887	65.35
	Tunisia	0.0088	0	0
					continued on next page

Table 20

 20

			-continued from previous page
	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.3	α = 0.7, γ = 0.3
	Turkey	0.0015	0	0	0
	Uganda	0.1159	0.1503	0.1337	62.04
	Uruguay	0.0011	0	0	0
	Venezuela	0.0002	0	0	0
	Vietnam	0.0290	0	0	0
	Yemen, Rep.	0.0172	0.1406	0.1593	53.13
	Zambia	0.0843	0.2961	0.2524	70.74
			λCD =57.10	λ=126.61	average { λi }=62.94

Table 21 :

 21 Actual and optimal allocations of aid, headcount poverty rate based on $1.25/day poverty line, α = 0.7, γ = 0.7, and regional elasticity of poverty reduction

	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.7	α = 0.7, γ = 0.7
	Algeria	0.0019	0	0	0
	Angola	0.0052	0	0	0
	Argentina	0.0004	0	0	0
	Bangladesh	0.0239	0.0072	0.0111	35.82
	Belize	0.0199	0	0	0
	Benin	0.0961	0.1072	0.0846	39.95
	Bhutan	0.0806	0	0	0
	Bolivia	0.0389	0	0	0
					continued on next page

Table 21

 21

			-continued from previous page
	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.7	α = 0.7, γ = 0.7
	Botswana	0.0562	0	0	0
	Brazil	0.0003	0	0	0
	Burkina Faso	0.1199	0.1828	0.1585	41.34
	Burundi	0.3214	0.4712	0.4563	47.63
	Cambodia	0.0755	0	0	0
	Cameroon	0.0234	0	0	0
	Cape Verde	0.1464	0	0	0
	Central African Republic 0.1312	0.3123	0.2951	43.88
	Chad	0.0631	0.3155	0.2683	44.53
	Chile	0.0006	0	0	0
	China	0.0003	0	0	0
	Colombia	0.0041	0	0	0
	Comoros	0.0785	0.3631	0.3166	47.13
	Congo, Dem. Rep.	0.1707	0.3855	0.3787	45.64
	Congo, Rep.	0.0553	0	0	0
	Costa Rica	0.0023	0	0	0
	Cote d'Ivoire	0.0279	0	0	0
	Djibouti	0.1313	0	0.0653	23.84
	Dominican Republic	0.0035	0	0	0
	Ecuador	0.0044	0	0	0
	Egypt, Arab Rep.	0.0106	0	0	0
	El Salvador	0.0111	0	0	0
	Ethiopia	0.1248	0.1425	0.1720	24.20
	Fiji	0.0129	0	0	0
	Gabon	0.0049	0	0	0
	Gambia, The	0.0948	0.1861	0.1154	42.85
	Ghana	0.0515	0	0	0
	Guatemala	0.0140	0	0	0
	Guinea	0.0983	0.0592	0.0492	38.54
					continued on next page

Table 22 :

 22 Actual and optimal allocations of aid, headcount poverty rate based on $1.25/day poverty line, α = 0.7, γ = 0.3, and regional elasticity of poverty reduction

	Country	Actual aid Optimal aid	Optimal aid Marginal efficiency λi
		(%GDP)	(%GDP)	(%GDP)	(people/$ million)
			α = 1 α = 0.7, γ = 0.3	α = 0.7, γ = 0.3
	Algeria	0.0019	0	0	0
	Angola	0.0052	0	0.0051	22.00
	Argentina	0.0004	0	0	0
	Bangladesh	0.0239	0.0072	0.0224	33.74
	Belize	0.0199	0	0	0
	Benin	0.0961	0.1072	0.1018	37.35
	Bhutan	0.0806	0	0	0
	Bolivia	0.0389	0	0	0
	Botswana	0.0562	0	0	0
	Brazil	0.0003	0	0	0
	Burkina Faso	0.1199	0.1828	0.1688	39.30
	Burundi	0.3214	0.4712	0.4537	49.57
	Cambodia	0.0755	0	0	0
	Cameroon	0.0234	0	0	0
	Cape Verde	0.1464	0	0	0
	Central African Republic 0.1312	0.3123	0.2964	43.33
	Chad	0.0631	0.3155	0.2690	44.42
	Chile	0.0006	0	0	0
	China	0.0003	0	0	0
	Colombia	0.0041	0	0	0
	Comoros	0.0785	0.3631	0.3101	48.62
	Congo, Dem. Rep.	0.1707	0.3855	0.3783	46.26
	Congo, Rep.	0.0553	0	0	0
	Costa Rica	0.0023	0	0	0
	Cote d'Ivoire	0.0279	0	0	0
					continued on next page

[START_REF] Llavador | An equal-opportunity approach to the allocation of international aid[END_REF], the effort variable is defined by economic management, the weighted average of three macroeconomic markers: budget surplus relative to GDP, inflation, and trade openness. The initial circumstances or initial disadvantages of country i are defined as the component of the growth rate, which is not explained by effort or aid.

The difference-GMM method of[START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF] was also considered but proved less satisfactory than the system GMM through specification tests. As noted in[START_REF] Roodman | How to do xtabond2: An introduction to difference and system GMM in Stata[END_REF], when performing the system GMM, all strictly exogenous regressors are used as instruments in both transformed equation and levels equation. Predetermined regressors are also valid instruments for the levels equation since they are assumed to be uncorrelated with the contemporaneous error term. Moreover, we use all available lags of the predetermined variables in levels as instruments for the transformed equation and the contemporaneous first differences as instruments in the levels equation. Finally, following[START_REF] Roodman | How to do xtabond2: An introduction to difference and system GMM in Stata[END_REF], we specify one instrument for each variable and lag distance (rather than one for each time period, variable, and lag distance) to reduce the bias in small samples with an increased number of

instruments.6 We do not present the GLS random effects estimator here as it is dominated by the within fixed effects estimator according to the Hausman test.

We use the 2008 data because that contains a much higher number of countries than other years.

Appendix A: List of countries Sub-Saharan Africa: Angola, Botswana, Burkina Faso, Burundi, Cameroon, Cape Verde, Central African Rep., Chad, Comoros, Congo, Dem. Rep., Congo, Rep., Cote d'Ivoire, Ethiopia, Gabon, Gambia, Ghana, Guinea, Guinea-Bissau, Kenya, Lesotho, Madagascar, Malawi, Mali, Mauritania, Namibia, Niger, Nigeria, Rwanda, Sao