What Is the Aggregate Economic Rate of Return to Foreign Aid?

Abstract

In recent years, academic studies have been converging towards the view that foreign aid promotes aggregate economic growth. We employ a simulation approach to: (i) validate the coherence of empirical aid-growth studies published since 2008; and (ii) calculate plausible ranges for the rate of return to aid. Our results highlight the long run nature of aid-financed investments and the importance of channels other than accumulation of physical capital. We find the return to aid lies in ranges commonly accepted for public investments and there is little to justify the view that aid has had a significant pernicious effect on productivity.

Debate over the effectiveness of foreign aid as a tool to promote social and economic progress in developing countries has been sharp over many years. As noted early on by Mosley (1986), evaluations of foreign aid at the microeconomic and mesolevels have frequently found positive impacts. More recent studies in this vein support this view (e.g., Michaelowa 2004; Mishra and Newhouse 2009; Arndt et al. 2015). Controversies have been acute at the macroeconomic level. Even so, the large majority of up-to-date empirical studies in the economics literature have found positive impacts. More precisely, the full range of independent studies published since 2008 based on cross-country growth regressions report comparable results for the marginal effect of aid on growth. These studies suggest that receipt of foreign aid equal to 2.5 percent of GDP over a sustained period is expected to boost growth by approximately 0.25 percentage points on average. While some findings are statistically insignificant at conventional levels, in part reflecting noisy and sparse data, most are significant. Indeed, the broad magnitude and direction of these results are sufficiently similar to merit attention. This represents our point of departure.

In this study, we briefly review this new literature and go on to answer two specific questions: (i) are results from recent studies regarding the aggregate effect of aid on growth numerically coherent; and, if so (ii) what do they imply about the economic rate of return to aid? The first question is motivated by the notorious difficulty of pinning down causal effects in macroeconomic data. With respect to the assessment of foreign aid, these difficulties are compounded by the relatively low quality of data regarding income growth and foreign aid volumes in developing countries.

To answer the first question, we run numerical simulations of a dynamic neoclassical growth model, augmented with foreign aid. As deployed in various fields, from engineering (e.g., Lin and Liu 1998) to economics (e.g., Ashraf et al. 2009; Dalgaard and Erickson 2009), numerical simulations offer a transparent means to determine plausible magnitudes of empirical phenomena. Also, numerical simulations can help to think through the empirical implications of specific modelling challenges, such as the suitable time frame over which aid impacts on growth. The second question focuses on the comparative benefits of providing aid versus the financial costs of its provision. A positive long-run impact of aid on growth does not automatically imply that aid generates an acceptable return on investment when viewed over its life cycle. At the same time, if returns to aid are found to be high, this might suggest that there is scope to provide a larger share of development finance on nonconcessionary terms. Even though rates of return represent a standard criterion for evaluating investments at the project level, these issues have not been addressed in the recent aid literature.¹ Our simulation approach can be used directly to calculate a plausible domain for economic rates of return to aid at the macroeconomic level.

The remainder of this article is structured as follows: Section I briefly reviews the set of peer-reviewed studies of the macroeconomic impact of aid on growth published since 2008. Section II presents our model, how it is calibrated, and the outcome indicators. Section III derives insights from selected simulations that focus on macroeconomic effects from physical capital investment, human capital upgrading, and productivity impacts associated with aid. Section IV considers the distribution of outcomes for both individual and combined aid impacts, based on Monte Carlo simulations. Section V summarizes the main findings and reflects on their implications. Two supplementary appendices (A and B), referred to in the text, are available online at http://wber.oxfordjournals.org/.

To preview our simulation results, we find that the marginal effect of aid on growth is negative over a nonnegligible share of observations when assessed over a short time horizon (5 years). However, when the assessment window expands to 30 years, the macroeconomic effects of aid are consistently positive and highly comparable to findings from recent empirical studies. The average internal rate of return that corresponds to such results is around 11 percent. At the same time, we show that productivity and human capital accumulation effects are critical mechanisms through which aid can affect the macro-economy, especially over the long run. We conclude it is appropriate to view foreign aid as a long-term investment whose benefits cumulate slowly over long periods.

I. Recent Studies

This section highlights the principal findings of recently published empirical studies that focus on the aid-growth relationship.² A meaningful starting point is Rajan and Subramanian (2008, hereafter RS08), who introduced a pair of influential innovations. First, they signaled a movement away from a reliance on cross-country dynamic panel data methods. Rather, their preferred strategy involved a long-run cross-section regression in which both Aid/GDP and growth are taken as averages over relatively extended periods (up to 40 years). This responds to the insight that aid given at time t may only have a growth impact after $t+n$ years, and that this impact may yield benefits over an extended period. Second, to address the endogeneity of aid, RS08 deploy external instrumental variables rather than the internal instruments commonly deployed in dynamic panel estimators.

Table 1 summarizes core results from recent papers that address aid and growth. To the best of our knowledge, the table covers the full population of studies that meet the following criteria: they (i) refer to an average aggregate aid-growth relation for developing countries as a group; (ii) include data spanning at least 30 years; (iii) attempt to address the endogeneity of aid; and (iv) are published in a peer-reviewed economics journal over the period from November 2008 to July 2014.³ As the various results included in the table use alternative specifications, an attempt has been made to select estimates from comparable models. In some instances nonlinear specifications involving a squared aid term are included. For these we report the marginal effect of Aid/GDP on growth evaluated by fixing Aid/GDP at 2.5 percent.⁴ In the final column, we report the probability associated with a two-tailed test that the reported point estimate (beta) is not different from zero.

Table 1.

Summary of Recent Aid-Growth Studies

Studya	Reference	Period	Spec.b	Beta	Std. Error	≈ Prob.
RS08	Table 4, col. 1	1960–2000	Linear	0.06	0.06	0.30
RS08	Table 4, col. 2	1970–2000	Linear	0.10	0.07	0.17
MR10	Table 4, col. 1	1960–2000	Linear	0.08	0.03	0.01
AJT10	Table 6, col. 2	1960–2000	Linear	0.09	0.04	0.02
AJT10	Table 4, col. 4	1970–2000	Linear	0.13	0.05	0.01
CRBB12	Table 7, col. 6	1970–2005	Nonlinear	0.15	0.06	0.01
CRBB12	Table 7, col. 10	1970–2005	Nonlinear	0.31	0.17	0.07
CRBB12	Table 9, col. 9	1971–2005	Nonlinear	0.27	0.13	0.04
CRBB12	Table 9, col. 9	1971–2005	Nonlinear	0.42	0.20	0.04
KSV12	Table 2, col. 5	1970–2000	Linear	0.05	0.05	0.32
LM12	Table 3, col. 4	1960–2001	Linearc	0.85	0.43	0.05
NDHKM12	Table 1, col. 4	1960–2006	Linear	−0.02	0.01	0.14
B13	Table 3, col. 1	1960–2000	Linear	0.12	0.04	0.00
B13	Table 3, col. 1	1970–2000	Linear	0.18	0.07	0.01
HM13	Table 2, row 1	1971–2003	Lineard	−0.01	0.00	0.00
AJT15	Table 1, col. 2	1970–2007	Lineare	0.30	0.18	0.09
Mean effect	Unweighted			0.19	0.06	0.00
	Weighted			0.12	0.04	0.00

^aRS08 is Rajan and Subramanian (2008); MR10 is Minoiu and Reddy (2010); AJT10 is Arndt et al. (2010); CRBB12 is Clemens et al. (2012); KSV12 is Kalyvitis et al. (2012); NDHKM12 is Nowak-Lehmann et al. (2012); LM12 is Lessmann and Markwardt (2012); B13 is Brückner (2013), HM13 is Herzer and Morrissey (2013); and AJT15 is Arndt et al. (2015).

^bFor nonlinear specifications (which involve a squared aid term) the marginal effect of Aid/GDP on growth is evaluated by fixing Aid/GDP at 2.5 percent; standard errors are also approximate for these cases.

^cFor comparability, this result is the marginal effect due to aid assuming no decentralization of government spending; however, the authors find that if the degree of decentralization exceeds 7 percent, the marginal effect of aid on growth is no longer statistically significant.

^dThis estimate controls for investment and is derived as an average from country-specific regressions.

^eBeta coefficient and standard errors are adjusted to raw values from standardized values as reported in the study.

Standard errors for the unweighted and weighted mean effects are derived from the set of beta coefficients reported in the table (see text). Probability is based on the normal distribution.

Source: Authors' collation from citations listed in Google Scholar.

In all but two cases, the beta coefficients in table 1 are positive.⁵ Three quarters are approximately significant at the 10 percent level, and just two are insignificant at the 20 percent level. The simple average of the point estimates for the average marginal effect of Aid/GDP on growth is 0.19; weighting by the logarithm of the inverse of their estimated variances yields an estimate of 0.12. Notably, these coefficients show substantial variation—their raw standard deviation equals 0.21, which is suggestive of substantial (model) uncertainty. Nonetheless, following Pesaran and Smith (1995), one can use the standard deviation of a set of comparable regression coefficients to derive a nonparametric estimate of the standard error of their mean.⁶ Applying this method delivers an estimate of the standard error of the unweighted mean effect of 0.06. Correspondingly, the mean effect is statistically significant, which is consistent with the formal meta-analysis of 68 earlier aid growth studies found in Mekasha and Tarp (2013).⁷

Interestingly, the range of results reported in table 1 is consistent with the empirical estimates originally found in RS08, which sparked the recent wave of literature. Although these authors' own results are not statistically significant at standard levels, Arndt et al. (2010) correct the treatment of unreported values for aid flows from missing to null and show that plausible modifications to RS08's empirical strategy yield positive and significant estimates with the same data set.⁸ Furthermore, Arndt et al. (2015) find comparable and significant results using data updated to 2007 (table 1, final row). It also bears remarking that alternative econometric methods suggest similar conclusions. For example, drawing on country-specific (cointegrated) time series analysis for 36 African countries, Juselius et al. (2014) find that foreign aid had a statistically significant positive effect on investment or GDP (or both) in 27 cases. In an additional seven cases the effect on investment or GDP is positive but statistically insignificant; and in only two cases (Ghana and Tanzania) does the impact of aid appear to be negative and significant.⁹

Before proceeding, we note that regression-based estimates of the marginal effect of aid on growth can be taken directly as approximations of the internal rate of return (IRR) associated with aid.¹⁰ This relationship is set out formally in Appendix A (online supplement). While we take up calculation of the IRR in further detail in the next section, this approximation implies that the weighted mean effect in table 1 roughly translates to an IRR of 12 percent. In sum, therefore, recent empirical studies provide consistent support for the view that aid has had a positive average effect on growth (and economic return) when viewed over an extended time frame. The view that aid is ineffective finds much weaker support; and the notion that aggregate aid is actively harmful on average (e.g., see Moyo 2009) finds no endorsement in recent academic research.

II. Framework

Model

The previous section highlighted a notable degree of consistency across the full range of empirical studies published since 2008. An open question is whether these results are plausible—that is, given what we know about aid volumes, is the distribution of these estimates in accordance with reasonable assumptions about how aid affects growth? A preliminary answer is provided by RS08. Using a straightforward Solow-Swan framework, the authors derive an expression for the expected marginal effect of aid on growth, which is equal to the product of the share of aid invested, the share of capital in income and the output-capital ratio. Based on rough estimates of the latter for developing economies, RS08 conclude that a plausible range for the marginal effect of aid (as a share of GDP) on growth is from 0.08 to 0.16 but could be higher (lower) if aid enhances (undermines) productivity growth.

Following RS08 and other studies (e.g., Dalgaard and Erickson 2009), we retain a highly aggregate neoclassical growth framework and extend it in three main directions. First, we focus on the dynamic behaviour of the model rather than static results. This is relevant since, under the assumption that foreign aid contributes to growth through the accumulation of stocks of physical and human capital, marginal returns to these stocks may vary over time, as does the absolute value of associated depreciation charges. Second, there is a potential lag between receiving aid and its impact on growth, which is likely to depend on the proximate goals of aid. Indeed, we show that different assumptions about the duration of this lag have important implications for calculations of the effect of aid on growth and returns to aid, especially over short time horizons.

Third, we do not rely on a simplifying assumption that aid only affects growth through accumulation of physical capital. Although analytically convenient, research suggests that the process of transition of economies from low to high incomes simply cannot be understood purely in terms of accumulation of physical capital. Upgrading of economy-wide human capital is fundamental, particularly to shift production from lower to higher value added goods and services. Moreover, inclusion of this channel accords with the explicit interest of many foreign aid donors in social policy outcomes, such as education and health.

Our complete model is summarized as follows:

$$Y_t={\theta }_t{K}_t^{\alpha }(h_t{L}_t{)}^{1-\alpha }$$

(1)

$$K_t=K_{t-1}(1-\delta )+\delta K_0+(1-\gamma -\phi )A_{t-\mu }$$

(2)

$$A_t=\left\{\begin{array}{cc}\lambda Y_t,& \text{if}1\le t\le t^{\ast }\\ 0,& \text{otherwise}\end{array}\right.$$

(3)

$$c_t=(1+\phi A_{t-1}/Y_{t-1})c_{t-1}$$

(4)

$$w_t=(c_{t-1}+14w_{t-1})/15$$

(5)

$$h_t=w_T^{\eta }$$

(6)

$$L_t=(c_t/c_{t-1}{)}^{\kappa }{L}_{t-1}$$

(7)

$${\theta }_t^{\ast }=1+\tau A_{t-1}/Y_{t-1}$$

(8)

$${\theta }_t=\left\{\begin{array}{cc}\theta {{\hspace{0.17em}}_t^{\ast }}^{t/\omega },& \text{if}t<\omega ,\tau \ne 0\\ {\theta }_t^{\ast },& \text{otherwise}\end{array}\right.$$

(9)

$$K_0=(r_0/\alpha {)}^{\frac{1}{\alpha -1}};L_0=1;w_0=1;c_0=1;0\le \gamma +\phi \le 1$$

where t denotes time (in years; $t=0$ is the initial or base year); $Y_t$ denotes real national income; ${\theta }_t$ is total factor productivity (TFP); $K_t$ is an aggregate measure of (physical) capital stock; $L_t$ is a metric of labor quantity inputs (e.g., hours worked); $h_t$ is an index of human capital quality; $A_t$ is the volume of foreign aid; $c_t$ is an index of the quality of human capital embodied in children; $w_t$ is an index of workers' human capital; $r_0$ is the initial rental rate of physical capital. All human capital variables are normalized taking a base value of one.

With respect to time invariant parameters, $\alpha >0$ is capital's share of income; $0<\delta <1$ is the rate of decay of physical capital; $\mu \ge 1$ is an integer that captures aid timing effects; $0\le \gamma \le 1$ indicates the share of aid that does not contribute to the accumulation of domestic factors of production; $0\le \phi \le 1$ gives the share of aid spent on human capital quality upgrading; $\lambda >0$ is the share of aid in income (GDP); $\eta$ reflects returns to worker human capital; $\kappa \ge 0$ indicates the impact of human capital expenditures on labor supply; $\tau$ is the terminal impact of aid on aggregate productivity; and $\omega$ denotes the period over which such productivity increases phase in (starting $t=1$⁠).

At the outset, five features of this framework can be highlighted. First, we assume there is no preexisting aid in the system (⁠$A_0=0$⁠), and the domestic saving rate is permanently fixed at $\delta$⁠. Furthermore, in the absence of aid, productivity and both the quality and quantity of human capital (⁠$h_t{L}_t$⁠) are constant. This means that the economy begins in a steady state and that foreign aid is the only lever able to shift the economy from its initial equilibrium. These assumptions are adopted not for realism; rather, they are a consequence of defining a simple model that focuses on the incremental contribution of aid. It follows that a comparison of a given macroeconomic aggregate at $t>0$ relative to its value at $t=0$ will uniquely capture the total effect due to aid in a counterfactual sense.

Second, the model is framed as a closed economy that receives exogenous injections of funds in the form of foreign aid. We recognize this assumption is questionable. However, Barro et al. (1995) show that a credit-constrained neoclassical small open economy displays very similar dynamic properties to that of a closed economy. Thus, we believe critical insights are not lost by retaining a closed economy assumption. Also, we explicitly allow for human capital and productivity effects (through $\phi$ and $\tau$⁠, respectively), which are critical aggregate channels through which aid can influence growth in open economies, including via the real exchange rate. Third, capital's share of income remains constant over time, which is a direct corollary of Bowley's Law (e.g., Krämer 2011). Fourth, prices are presumed constant, normalized to one.

Fifth, a trade-off with adopting a simple model is that certain features are excluded. By definition, income (or other) shocks are ruled out, thereby ignoring the potentially important role of foreign aid in support of consumption smoothing. Also, we do not include disaggregated actors (e.g., firms, households, government) or their interactions. Certainly, such dynamics may be material in specific cases; however, they go beyond the present exercise which aims to establish broad orders of magnitude for aid's effects.¹¹ Moreover, despite the absence of utility maximizing agents, it remains appropriate to consider Solow-Swan growth models as dynamic general equilibrium frameworks (see Acemoglu 2008).

Looking in more detail, equations (1) and (2) are the basic Solow-Swan equations, expressed in aggregate (not per capita) terms and augmented by foreign aid. Equation (3) is an exogenous aid allocation rule. We treat aid as exogenous precisely because our aim is to investigate the counterfactual impact of aid on the macro-economy. Put differently, since we wish to verify the plausibility of recent econometric evidence regarding the causal impact of aid, it makes sense to undertake simulations “as if” aid were exogenous. Thus, the equation says that from $t=1$ to $t^{\ast }$⁠, the simulated economy receives aid inflows equal to a prespecified proportion of national income (⁠$\lambda$⁠). Terminating aid inflows at a specified period means that per capita growth will return to zero after some further time. In turn, this implies there is a finite horizon over which it is meaningful to study the aid-induced behaviour of the economy. As discussed below with reference to the calculation of returns to aid, this is analytically helpful since it delimits the relevant life-cycle of aid in the economy.

Equations (4) through (7) constitute the human capital block, in which the impact of aid operates through two main microchannels. At the intensive margin, we allow aid to to enhance the quality of human capital embodied in children, such as through education and nutrition interventions. As stated in equation (5), we assume one-fifteenth of the workforce is replaced in each period by the representative child, implying a long lag from aid delivery to its having a substantive effect on the quality of worker human capital that enters the production function.¹² At the extensive margin, we allow aid to increase hours worked, which can be thought of as incorporating some combination of an improvement in the disease environment and reductions in mortality. This effect is assumed to operate directly on existing workers and is cumulative. Moreover, we assume all human capital effects are permanent and thus permit the economy to reach a new steady state level of income after termination of aid flows. An implication of these two effects is that domestic savings (⁠$\delta K_0$⁠) will decline as a share of income—that is, we exclude an endogenous savings response. While this is conservative, our assumption of a zero rate of depreciation on human capital is more bullish. However, it accords with assumptions deployed by Ashraf et al. (2009) and avoids the complex question of how this should be parameterized (see McFadden 2008).

Finally, equations (8) and (9) represent a block of direct (aggregate) productivity effects. Specifically, equation (8) is a simple linear relationship that defines an aid-adjusted permanent level of TFP, which is achieved gradually as described by equation (9). These capture improvements in policies, institutions, or productive technologies, which are frequently the stated goals of donor funding. A related motivation for considering this channel is that, despite donor intentions, critics have often opined that aid may have negative consequences for aggregate productivity (e.g., Moss et al. 2006; Rajan and Subramanian 2007; Djankov et al. 2008). As such, this block can be considered an umbrella capturing a range of potentially important specific channels through which aid may affect productivity or competitiveness, which are not discussed explicitly here.¹³

Calibration

The above framework provides a basis to explore the most important generic mechanisms through which aid is often thought to have macroeconomic effects. To provide numerical insights, calibration of the model is necessary. In doing so, we stress that our objective is not to provide a single set of results that approximate a given country or circumstance. Our aim is to provide plausible bounds on the order of magnitude of the macroeconomic effects due to aid, considering individual mechanisms both separately and in conjunction.

A simulation of the model is defined by the values taken by the parameters; these determine the outcomes of interest. More formally, we define a parameter vector for simulation i as a draw from a joint distribution: $s_i\in \mathrm{\Omega }=f\{t^{\ast },\lambda ,\alpha ,\mu ,\delta ,\gamma ,\phi ,\kappa ,\eta ,\tau ,\omega ,r_0\}$⁠. Evidently, $\mathrm{\Omega }$ spans multiple dimensions. To reduce the dimensionality of the analytical problem, we keep parameters of secondary importance fixed throughout (⁠$t^{\ast }=\omega =30;\eta =0.4$⁠; see below). For the remaining parameters, insufficient prior information exists to completely specify their joint distribution. In particular, covariances and moments greater than two generally are unknown. Thus, we use existing literature to identify a plausible range of values. In most cases, we also presume an agnostic independent uniform distribution. This corresponds to Laplace's principle of insufficient reason and is intended to generate a broad prior over the parameter space.

Appendix table B1 summarizes the assumed ranges and distributions for parameters that vary between simulations. Capital's share of income ranges from 30 to 70 percent; the delay from aid delivery to it adding to physical capital is an integer uniformly distributed between one and seven periods (years). The share of income received in foreign aid is drawn from a truncated Beta distribution, taking a mean of approximately 5 percent, yielding a distribution with a large range that broadly reflects the varied pattern of historical aid flows. The duration of aid inflows (⁠$t^{\ast }$⁠) is held fixed at 30 periods, which is roughly an upper limit on the assessment window found in recent literature (see section I).

Choices regarding the rate of depreciation and marginal returns to capital are important and more controversial. Few rigorous estimates of depreciation rates are found in the literature, especially for developing economies. Bu (2006) uses data from enterprise surveys for a range of developing countries and finds aggregate capital stock depreciation rates of between 9 and 23 percent. These values compare to rates of 2 to 10 percent employed by corporate auditors in the same countries, typically also used in cross-country capital stock estimations (also Bu 2006). Faced with these ranges, we take an agnostic stance and assume depreciation rates are uniformly distributed between 2.5 and 25 percent.

For rates of return to physical capital, we rely on the calculations Caselli and Feyer (2007). Their baseline estimates point to an average marginal product of capital, which is equivalent to the rental rate under the assumption of constant returns to scale and competitive capital markets, of 0.27 with a standard deviation of 0.09 among low-income countries. Notably, this estimate does not take into account “correction factors” such as differences in prices of capital goods and the absence of complementary factors. To some extent, these items can be considered as contributing to an effective depreciation rate. Here, the initial net rate of return $r_0-\delta$ takes a mean of 13.2 percent and standard deviation of 11.0 percent, which indeed is much closer to the corrected rates of return estimated by the same authors.

The magnitude of health and education impacts on aggregate income, as well as the effects of foreign aid on these proximate drivers, remain contested. With respect to the latter, Arndt et al. (2015) estimate that sustained aid inflows of around 5 percent are associated with an increase in completed years of schooling among adults of between 2 and 3 years on average. Global estimates of schooling provided by Barro and Lee (2013) indicate that improvements of this magnitude over a generation (30 years) are reasonable, especially for developing countries starting from a low educational base. For instance, average years of schooling (for the population 15+) in Botswana increased from 2.14 to 8.69 years from 1970 to 2000. Correspondingly, equation (4) indicates that if one half of a sustained aid inflow equal to 5 percent of GDP is spent on human capital upgrading (⁠$\phi =0.5$⁠,$\lambda =0.05$⁠), child human capital quality would roughly double over a generation.

With respect to the economic impact of such investment, our model defines marginal returns to worker human capital quality as: $\mathrm{\partial }{Y}_t/\mathrm{\partial }{w}_t=\eta (1-\alpha )Y_t/w_t$⁠. Thus, as a proportion of gross income, an upper bound on these returns equals $\eta (1-\alpha )$⁠. For simplicity, to investigate the behaviour of the model we fix $\eta =0.4$ throughout, implying a maximum per-period marginal return of 28 percent to increments in worker quality (⁠$0.3\le \alpha \le 0.7$⁠). While this is larger than typical macro-Mincerian returns to education (see Lange and Topel 2006), it represents a strict upper bound and mechanically declines as w grows. Moreover, to the extent that upgrading of human capital generates economy-wide positive externalities, this calibration may prove conservative.

Aggregate income effects due to changes in morbidity or mortality are perhaps even more controversial. They are important, however, because demographic changes associated with large public health interventions can generate complex general equilibrium effects (see the discussion in Bleakley 2010). Ashraf et al. (2009) use burden of disease estimates (years lost to disability) to capture the effective loss of working time associated with poor health, ignoring premature mortality. Per capita figures compiled by the World Health Organization suggest this equals around 11 percent in low-income countries.¹⁴ Years lost to premature mortality are substantially larger, implying large health interventions could generate even more substantial labor supply effects over time. With this in mind, we set $\kappa \in [0,0.15]$⁠, implying a maximum labor force quantity increment of around 22 percent over a generation at the mean of simulated aid inflows.¹⁵

Lastly, the productivity parameter $\tau$ (equation [9]) is restricted to vary between zero and two. At the mean simulated value of aid inflows this implies a maximum terminal productivity increment equal to 10 percent of initial income, holding labor supply fixed. Since we assume this increment is phased-in over 30 periods, it equates to productivity growth of around 0.3 percentage points per year. Effects of this magnitude are not implausible. Long-run estimates of annual labor productivity growth in the US cluster around one percentage point on average (Nordhaus 2002). Moreover, in low-income countries, estimates of efficiency losses from factor misallocation are typically an order of magnitude larger (Vollrath 2009). That is, these estimates suggest large one-off gains might be achieved by suitable policy interventions.

For analytical purposes, we exclude negative values for $\tau$⁠, thereby imposing a positive correlation between productivity and aid inflows. We recognize this assumption is questionable; we adopt it because the rate of return calculations become highly unstable in the absence of positive cash flows. Nonetheless, because the first order effects of $\tau$ are symmetric around the origin (on average), our Monte Carlo simulation results can be used to consider the (average) magnitude of $\tau$ necessary for aid-induced productivity losses to fully dominate capital accumulation effects. In this light, it should also be pointed out that theoretical and empirical estimates of the impact of aid on competitiveness (e.g., via Dutch Disease) typically only point to weaker relative growth of manufacturing industries (Rajan and Subramanian 2011) and, at worst, stagnant aggregate income effects (Adam and Bevan 2006). As we show in sections III and IV, impacts of this kind are entirely consistent with negative values of $\tau$ within the mirror of the stipulated parameter range.

Outcomes

To analyze numerical outcomes from the model, two main indicators are in focus. The first is the marginal effect of aid on growth calculated over different periods. For run i and period length J, this is defined as:

$$g_{iJ}=\frac{1}{\lambda J}\sum _{j=1}^J{\stackrel{.}{y}}_{i,t+j}$$

(10)

where ${\stackrel{.}{y}}_t=(Y_t/Y_{t-1})(L_{t-1}/L_t)-1$⁠, which can be thought of as a lower bound on the per capita growth income rate—that is, it is binding if effects due to equation (7) operate exclusively through improvements in mortality. This indicator corresponds to the principal focus of the empirical literature discussed in section I. In line with these studies we report results for 5, 10, and 30 periods. Thus, results for different window lengths provide insight as to how alternative econometric approaches to the problem may behave.¹⁶

The second indicator is the internal rate of return (IRR), which is calculated from the profile of domestic cash flows (net additional income) generated by aid. Under the maintained hypothesis that the economy is in a “no-aid” steady state at $t=0$⁠, the real value of the per-period cash flow attributable to aid is given by:

$$C{F}_t=Y_t-(Y_0+A_t)=Y_t(1-\lambda )-Y_0$$

(11)

This says that the aggregate economic cash flow attributable to aid in a given year is the difference between aggregate income (with aid) at time t minus the sum of the counterfactual income (without aid) and the effective cost of aid in that period. This equation makes explicit our model assumption that all changes in the economy, including productivity and human capital effects, can only be induced by aid inflows.

From a project finance perspective, returns to investment depend on both the magnitude and time profile of such cash flows. For instance, although cash flows are expected to be negative in early years, viable investments should yield a positive net present value (NPV) of flows over the life cycle of the project. In other words, the key concern is whether the discounted sum of cash flows is positive.¹⁷ To calculate the NPV, the discount factor must be specified ex ante. If unknown, the IRR can be used to identify the discount factor which yields an NPV of zero. Intuitively, the IRR indicates the maximum cost of capital required for a project to break even. IRR calculations from simulations of our model thus indicate the maximum effective financing costs of foreign aid inflows that would generate no net losses. This interpretation suits our model as we assume all income gains associated with aid are available for consumption or payment of financing costs. For instance, if the IRR is found to be 10 percent, then the effective costs of financing aid over the lifetime of the project must be below 10 percent for such aid to have a positive overall net income effect.

Considering that estimates of the marginal effect of aid on growth represent approximations to the IRR (see section I; also Appendix A), one might query the added value of undertaking exact estimates of the IRR according to equation (11). In addition to the gain in precision, our motivation is that the approximation is only likely to be reasonable under highly restrictive conditions. To see this, note that the approximation assumes that the marginal aid-growth effect is constant and contributes both fully and immediately to growth in the first period aid flows arrive, which broadly corresponds to the basic model in RS08. It follows that once we allow aid to affect the economy through more complex (time-varying) channels, the quality of the approximation is likely to decline. Ultimately, however, this is an empirical question.

To complement the IRR, we also calculate two additional metrics. The first is the average ratio of net cash flow to initial income, denoted CFY. This is equivalent to the NPV calculated with a discount factor of zero, divided by the product of the initial income level and the number of periods in the simulation. Values greater than zero imply that the (undiscounted) cumulative cash flow from the project is positive—that is, the real value of income generated by aid is greater than the value of aid received, regardless of the time profile of positive and negative cash flows (to which donors may be indifferent). The second indicator reports the first time period when cumulative cash flows are positive, denoted PCCF. It gives the time horizon necessary for aid to generate cumulative income gains equal in value to the amount of aid given, again ignoring time discounting or financing costs.¹⁸ This captures the degree of patience required by donors to observe positive overall returns to aid outflows.

III. Impact Channels

We proceed with the analysis in two main steps. In this section, we examine the general behaviour of the model for a number of selected parameter vectors – i.e., individual simulations. These are deliberately chosen to isolate aggregate mechanisms through which aid may affect the macro-economy. The distribution of outcomes is considered in the next section.

Physical capital accumulation

To facilitate comparison with previous studies (e.g., RS08), our first simulation only permits aid to affect the economy via investment in physical capital. Productivity, labor supply and human capital quality are all held fixed at their initial levels (⁠$\mathrm{\forall }t:{\theta }_t=1,L_t=L_0,W_t=W_0;\phi =0;\tau =0$⁠); also, for now, we assume all aid is invested without waste (⁠$\gamma =0$⁠). A principal implication of these assumptions is that there can be no persistent or permanent effects of aid on income since all aid-induced increments to physical capital depreciate over the long-run. We consider this our baseline simulation, against which other trajectories can be compared.

The time path of selected economic aggregates from this simulation are illustrated in figure 1, which is based on an aid inflow equal to 5 percent of GDP (received over 30 periods) and setting all other nonzero parameters to their medians (for chosen parameter values refer to table B2). Panel 1(a) shows the path of aid flows; panel 1(b) indicates the volume of additional capital stock attributable to these aid inflows (net of depreciation), reported as a percent of its base (steady state) level. Income growth effects per labor input unit attributable to aid are illustrated in panel 1(c), reported in percentage points; and panel 1(d) illustrates the cash flow associated with aid.

Figure 1.

Paths of Selected Variables Under Baseline (physical capital) Simulation Threshold

Source: Author's calculations.

Three points can be highlighted from the figure. First, due to the lag between receiving aid and additional capital entering productive use, positive impacts on growth and physical capital are first registered at $t=1+\mu =5$⁠. Similarly, although there are no aid inflows after $t=30$⁠, new aid-financed productive capital continues to be added to the economy until $t=34$⁠. The sharp discontinuities at $t=5$ and $t=35$ in both the capital stock and income growth directly reflect discontinuities in the supply of aid. Second, following equation (2), depreciation charges are made to the capital stock in each period. Once aid inflows terminate, these ensure the economy gradually returns to its steady state income position. Indeed, the capital stock peaks at $t=34$ and suffers a large write-down due to depreciation charges in the next period, generating negative income growth. Finally, cash flows only enter positive territory at $t=10$⁠. This captures the point that it takes time for returns to aid to exceed the cost of new aid inflows (see equation [11]).

Row 1 of table 2 summarizes the primary outcomes of interest from the same simulation. The first three columns report the marginal effect of aid on growth per unit of labor over alternative windows (see equation [10]), which for presentational purposes are all calculated assuming the “marginal” unit of aid equals 10 percent of GDP. As we find throughout, the marginal aid-growth results are sensitive to the length of the assessment window and, due to aid delivery lags, tend to be lowest for the shortest window. Nonetheless, it is notable that even for this simple simulation all these estimates are broadly similar to those reported in table 1. The associated IRR results, calculated over the full project lifetime (here 78 periods), equals 7.2 percent. Thus, if the aid inflow were to be debt financed it would be unsustainable at a cost of financing above this threshold.

The remaining outcome metrics of table 2 confirm positive and modest effects of aid on the economy. For instance, it takes 22 periods for the cumulative value of cash flows (PCCF) to turn positive. We also see that while income per unit of labor is around 10 percent higher than in the base line after 30 periods, as expected this has returned to zero after the full life-time of the project. In turn, this explains that cash flows as a share of base income (CFY) are only marginally above zero, on average.

Table 2.

Summary of Estimates from Selected Model Runs

Row	Channel	Growth outcomes			Returns outcomes			$\mathrm{\Delta }y$%
		5	10	30	IRR	CFY	PCCF	30	End
1	Physical capital	0.53	1.10	0.63	7.2	1.8	22.0	9.9	0.0
2	Consumption	0.16	0.34	0.19	−4.7	−0.9	–	2.9	0.0
3	Human capital	0.23	0.58	1.06	9.1	18.1	26.0	17.2	26.5
4	Human capital+	−0.24	0.05	0.50	13.7	27.6	17.0	7.8	15.8
5	Productivity	1.17	1.75	1.31	16.4	10.3	13.0	21.6	10.0

Notes: Cells report outcomes from individual simulations of the model, chosen to illustrate the behaviour of specific impact channels; row names indicate the channel simulated; column headers indicate the outcomes; $\mathrm{\Delta }y$% is the difference in income per unit of labour as a percentage of initial income evaluated after 30 periods and at the end of the project life cycle (“End”); growth outcomes are the marginal effect of aid on growth, calculated for units of aid equal to 10% of GDP and normalized by the quantity of labour inputs; IRR estimates are calculated on an aggregate cash flow basis; see Appendix table B2 for parameter values used in each simulation.

Sources: Authors' calculations.

Consumption

In the second simulation, entitled “consumption” we relax the assumption that all aid represents a productive transfer to the economy. For purposes of comparison we retain the previous (baseline) parameters, set $\gamma =0.7$⁠, and rerun the model. Summary results are reported in row 2 of table 2. By construction, higher values of $\gamma$ diminish the volume of aid that enters the system with no other side effects. Thus, all marginal growth outcomes are proportionally lower, by a factor of around 0.7. The IRR outcome falls below zero, and even applying a discount factor of zero, the average net present value of cash flows attributable to aid is negative. The rather trivial implication is that, in order to induce positive macroeconomic effects, aid must embody a productive transfer to the recipient economy.

Human Capital

The next two simulations consider the role of human capital upgrading. In the first instance, we ignore labor supply effects (⁠$\kappa =0$⁠) and restrict attention to ‘pure quality’ effects that operate through investments in child human capital (see equation [4]). For comparative purposes, we assume 80 percent of aid goes toward human capital upgrading and the remaining share goes to physical capital. Thus for this simulation, entitled “human capital,” we take the baseline parameter values and simply set $\phi =0.8$⁠.

Row 3 of table 2 summarizes the outcomes from this simulation. Two key differences relative to the baseline simulation (“physical capital”) stand out. First, growth outcomes are larger and take longer to materialize. In the baseline (row 1), the marginal growth effect after 30 periods is lower than after 10 periods. This is due to the fact that marginal returns to additional capital stock investment quickly decline and depreciation costs mount. Shifting most investment from physical to human capital generates a very different pattern of outcomes. The lag from aid delivery to changes in the quality of workers takes at least 15 periods. Moreover, even when child human capital quality is constant (on termination of aid flows), worker quality approaches the same index value only asymptotically (see equation [5]). Consequently, growth impacts are much more spread out over time and continue after 30 periods have elapsed.

This difference can be seen graphically. Figure B1 plots the paths of income per unit of labor for the physical capital (symbol $80$⁠) and basic human capital (symbol $\mathrm{\Delta }$⁠) simulations over 60 periods. It shows that income rises more quickly when channelled to physical capital accumulation. For evaluation windows of less than 20 periods the physical capital channel dominates in all respects. As the window of evaluation is expanded, we see that the level of income is significantly higher and growth is sustained over a longer time frame under the human capital simulation.

The second difference is that rates of return are larger versus the baseline. This reflects the permanent nature of human capital upgrading and, thus, the higher level of steady state income that such investments entail. The present simulation, which is located at the most favourable end of the spectrum of such effects, suggests that sustained aid inflows of 5 percent of GDP per annum over 30 periods yield long-run income gains of around 25 percent. Although large, this is not inconsistent with the orders of magnitude derived from large-scale micro-studies regarding returns to preschool interventions such as worm eradication or nutrition enhancement (Bleakley 2010). The corresponding IRR is 9.1 percent, which is only 1.9 points larger than under the baseline simulation. This arises because income growth in the early periods is lower than under the baseline (see figure B1). Even for small positive discount rates, these early periods receive a much higher weight and the later relative gains from human capital investment are significantly discounted. This underlines the sensitivity of the rate of return calculations to the timing of cash flows.

The above findings are considerably nuanced when we allow human capital upgrading to generate positive labor supply effects. To consider this, we set $\kappa =0.15$⁠, its maximum. Outcomes from this simulation, denoted “Human capital+,” are reported in row 4 of table 2 and the time path of income per unit of labor is illustrated in figure B1 (symbol $\times$⁠). The marginal effect of aid on growth is now significantly lower. This reflects that after 30 periods, labor supply has increased around 19 percent versus the baseline. Moreover, as the labor supply effects are realised sooner than the effect of quality enhancements, income per unit of labor in fact declines over the short run.¹⁹ However, these initial effects wash-out and net income gains to human capital upgrading are positive, equalling around 16 percent of initial income in the long-run.

A further critical distinction is that the rate of return estimates are larger with labor supply effects than in their absence because returns metrics only consider aggregate returns not gains per unit of labor. Indeed, under this simulation, gross income is almost 30 percent higher after 30 periods versus its initial value. Moreover, since labor supply effects also take effect relatively soon in time (see equation [7]), they are particularly substantive from the perspective of the IRR metric. In sum, for this simulation, the IRR is almost double the baseline estimate at 13.7 percent.

Productivity

Direct productivity effects due to aid are captured in a simple fashion under the present model.²⁰ The illustrative simulation for this channel is derived from the baseline vector of parameters and setting $\tau =2$ (the maximum permitted). As noted in section II, for aid inflows of 5 percent of GDP over 30 periods this yields a long-run increase in steady state income of 10 percent, ceteris paribus. Outcomes from this simulation are reported in table 2 row 5, and the corresponding income path is shown graphically in figure B2. Broadly, all outcomes are approximately double (twice as good as) those of the baseline scenario, the difference being uniquely attributable to the productivity channel.

Figure B2 also shows the income path for an equivalent simulation with $\tau =-2$⁠. This has symmetric long-run effects, generating a 10 percent fall in steady state income. More importantly, average income growth from $t=1$ to $t=30$ is approximately zero under this simulation (−0.05 percentage points). This suggests that aid-induced productivity losses of roughly this order of magnitude would be necessary to fully wipe out the income gains associated with capital accumulation. Indeed, recall that this simulation assumes all aid is invested in physical capital.

IV. Outcome Distributions

The final simulation of the previous section illustrated that the effect of aid on growth due to one mechanism (e.g., investment in physical capital) can be modified by other mechanisms (e.g., productivity effects). It follows that richer insights may be gained from an investigation of the same outcomes over a much wider range of parameter combinations. There are three more specific motivations for this kind of (Monte Carlo) distributional analysis. First, any presumption that structural economic parameters are broadly stable, either between or within countries over time, is heroic. Thus, an appreciation of outcomes across the space of parameters enables us to consider the potential heterogeneity of aid outcomes. Second, without prior knowledge of which combinations of channels produce which effects, this analysis can place bounds on the magnitude of macroeconomic effects that may be plausibly attributed to aid. Third, variation in outcomes can be used to identify particular parameter combinations that are consistent with zero or even negative aggregate long-run macroeconomic effects. Put more simply, this approach avoids reliance on one or a few potentially ad hoc calibrations of the model. It is also consistent with our objective of quantifying the overall impact of aid through multiple channels working at the same time.

We examine the distributions of outcomes via Monte Carlo simulation. Specifically, we take 5,000 independent random draws of the relevant vectors from the parameter space (⁠$\mathrm{\Omega }$⁠; see table B1). For each draw we simulate the model and estimate key outcomes, as before. For each outcome, the complete set of estimates directly represents its simulated empirical distribution. The primary outcome distributions are those derived from Monte Carlo simulations in which all parameters are permitted to vary simultaneously.²¹ For comparison, we compute the distributions of outcomes for the individual channels investigated in section III. These are calculated from the same parameter matrix used to generate our full Monte Carlo simulation, the only exception being that specific parameters are now set to zero, thereby “turning off” specific channels. For example, to simulate outcome distributions for the physical capital channel we fix $\phi =\gamma =\tau =0$ and only allow parameters that directly moderate the impact of physical capital investments to vary.

Table 3 reports the empirical mean and standard deviation of outcomes from these Monte Carlo simulations. The first row, denoted “All,” refers to the full simulation and therefore gives no primacy to any specific impact channel. Put differently, these results are agnostic as to which particular channel dominates.²² Five points merit attention. First, contrary to the ambiguous pattern of results from the individual simulations (table 2), the magnitude of the marginal effect of aid on growth rises as the length of the assessment window increases. Following previous insights, this reflects the specific dynamics associated with investments in human capital. This point is confirmed by the empirical distributions for the same outcomes for the channels reported in the remainder of table 3. Here we see that the marginal effect of aid on growth only unambiguously increases with the assessment window for the two human capital channels.

Table 3.

Means and Standard Deviations of Empirical Distributions

	Growth Outcomes			Returns Outcomes
Channel	5	10	30	IRR	NPV	PCCF
All	0.51	0.76	0.89	11.24	97.12	21.22
	(0.39)	(0.35)	(0.33)	(3.56)	(16.73)	(6.24)
Physical capital	0.65	1.08	0.72	7.46	63.80	33.76
	(0.72)	(0.48)	(0.41)	(7.23)	(48.06)	(21.81)
Consumption	0.48	0.79	0.53	3.07	39.32	27.45
	(0.55)	(0.39)	(0.32)	(7.86)	(48.85)	(9.85)
Human capital	0.41	0.77	0.95	8.99	95.06	25.27
	(0.42)	(0.32)	(0.30)	(3.26)	(21.67)	(5.94)
Human capital+	0.26	0.60	0.77	10.65	97.30	22.03
	(0.46)	(0.38)	(0.33)	(3.25)	(16.21)	(5.67)
Productivity	0.98	1.42	1.06	13.12	91.82	18.17
	(0.75)	(0.52)	(0.46)	(6.82)	(27.41)	(8.84)

Notes: Cells report empirical means and standard deviations of the Monte Carlo distribution of marginal aid-growth outcomes and returns estimates; rows indicate the specific channel simulated, where “all” denotes that all parameters are allowed to vary (else, all other channels are switched off); growth outcomes are the marginal effect of aid on growth, calculated for units of aid equal to 10% of GDP and normalized by the quantity of labour inputs; IRR estimates are calculated on an aggregate cash flow basis; NPV denotes the share of simulations for which the net present value of cash flows is positive.

Source: Authors' calculations.

Second, the standard deviation of the aid-growth outcomes also declines as the window length increases. This demonstrates that a more coherent or precise pattern emerges as more time elapses. Moreover, this finding is consistent across all simulations reported in table 3 and therefore can be considered robust to the specific channels through which aid effects the economy.

Third, a corollary of the previous two findings is that whilst the 99 percent confidence interval for aid-growth estimates enter a negative domain when viewed over either a 5- or 10-year horizon, the same interval is strictly positive for a 30 year window (spanning 0.16 to 1.85). This is illustrated graphically in figure 2, panel (a), which shows boxplots of the simulated distribution of aid-growth effects for the three assessment windows, calculated from the same simulations. As the length of the window increases, the central tendency of the distribution also increases while its variance declines.

Figure 2.

Summary of Aid-Growth and IRR Distributions, Full Model

Notes: Horizontal axis in panel (a) is the marginal effect of aid on growth, calculated for units of aid equal to 10% of GDP; horizontal axis in panel (b) is the internal rate of return associated with aid. Growth outcomes are normalized by the quantity of labor inputs; IRR estimates are calculated on an aggregate cash flow basis.

Source: Authors' calculations.

Fourth, both the average and range of simulated estimates of the marginal aid-growth effect, considered over 30 periods, are extremely close to the findings from recent empirical studies (see table 1). We therefore conclude that the order of magnitude implied by these is entirely plausible. Moreover, whilst there is some variation in the means and distributions of the aid-growth outcomes when we isolate individual channels, there is substantial overlap in all cases. This can be seen from figure B3, which plots the distribution of the 30 period marginal aid-growth effects for individual channels. Thus, the average effect derived from recent empirical studies would appear robust to the inclusion or exclusion of specific mechanisms.

Fifth, the IRR results from the full simulation take a mean of 11.24 percent and a 99 percent confidence interval that is strictly in the positive domain (spanning 2.62 to 22.34 percent; see figure 2 panel b). Notably, the variance (range) of this outcome is considerably tighter here relative to the variance of the IRR for individual channels. For instance, the same confidence interval calculated for the physical capital channel ranges from −11.54 to 30.67 percent. As table 3 and figure B4 show, smaller variance estimates for the returns to aid are specifically associated with effects operating via human capital investments. In turn, this channel has a critical influence on the assessment of the long-run returns to aid.

We highlight that the mean overall IRR corresponds very closely to the approximate IRR calculated from the studies in table 1. Indeed, as should be expected, we find a strong positive correlation between our calculated IRRs and the corresponding simulated marginal effect estimates (which are approximate IRRs). However, a comparison of the IRRs and 30 year growth outcomes in table 3 indicates there is no 1:1 mapping between these estimates. More importantly, it indicates the two deviate systematically when full human capital and productivity effects due to aid are permitted. For instance, considering the final row of the table, the approximate IRR taken from the mean marginal effect is 10.6 percent while the mean calculated (exact) IRR equals 13.1 percent. In keeping with the discussion of section II, the approximation becomes less precise when complex dynamics due to aid are allowed.

As a final analytical exercise, we consider the sensitivity of outcomes to changes in individual parameters. As an approximation, this can be inferred from a linear regression employing parameter values from the full Monte Carlo simulation as the explanatory variables, respecified as deviations from their means. The various outcomes are employed as dependent variables whose variance is to be explained. Results from this exercise are reported in table B3. By construction, estimates of the intercepts replicate the means reported in table 3. Coefficient estimates for individual parameters (in the rows) can be interpreted as the expected change in the mean outcome for a 1 unit (100 percentage point) change in the parameter, holding all else fixed. The bottom panel of the table calculates so-called critical values, which are the parameter values required to shift the mean outcome to zero, evaluated holding all other parameters at their means.²³ In other words, approximately all aid would need to be consumed (⁠$\gamma >1$⁠) for all the IRR and 30 year aid-growth outcomes to be zero in expectation.

What do we learn from this exercise? First, note that coefficient estimates for $\alpha$ are negative. This means that economies with a higher initial share of capital in income are generally associated with lower aid effectiveness (evaluated either in terms of marginal growth impacts or by aggregate returns). We also see that for the 5- and 10-period windows, the coefficient estimates on $\phi$ are negative, implying that increasing the share of aid to human capital investments reduces their near-term impact on growth. Neither of these results should be taken to imply that physical capital investments are always expected to generate higher returns. Rather, this result simply underlines the different dynamic profiles of aggregate effects due to aid from physical as opposed to human capital investments. According to our model, the latter emerge much more slowly, taking more than a generation to fully translate into economic returns. The marginal growth outcome calculations reported here, which encompass 30 periods at most, do not capture the complete economic contribution of aid-financed investments in human capital. The IRR results, by definition, also tend to place a higher weight on income effects realized in the shorter run. Consequently, “quick returns” to physical capital investments receive a larger weight. At the same time, it should not be forgotten that human capital investments contribute to a higher permanent counterfactual level of income, while aid-induced physical capital investments eventually depreciate away (see table 2).

Other estimates from table B3 corroborate previous insights. The critical values associated with the relative magnitude of productivity effects due to aid (⁠$\tau$⁠) range from −1.61 to −2.70 for the growth outcomes. To fully undermine the (average) positive human and physical capital accumulation effects associated with aid, any simultaneous productivity effects must be substantial and negative. For example, for an inflow of aid equal to 5 percent of GDP, productivity must fall by around 13.5 percent over the long-run to yield a net growth outcome of zero. By way of comparison, Rajan and Subramanian (2011) estimate that exporting industries grow 0.5 percentage points slower in a counterfactual sense for each 1 percentage point increase in Aid/GDP. Thus, assuming an Aid/GDP inflow equal to 5 percent and that all other sectors grow at 1.8 percentage points per annum with or without aid (the mean for their sample), their results require that exports contribute at least 72 percent of value added to generate an aggregate growth rate of zero.²⁴

These less sanguine results regarding aid corroborate the more general point that large negative productivity effects are needed to wipe out, as opposed to just dampen, growth. To be clear, this does not rule out that negative productivity effects may arise in specific cases (for a whole gamut of potential reasons), but rather they would have to be large to outweigh the positive impacts through human and physical capital accumulation. Moreover, if such large negative effects had frequently resulted from past aid flows, they would be inconsistent with the regression estimates summarized in table 1.

Lastly, table B3 indicates that outcomes are particularly sensitive to small changes in $\kappa$⁠, suggesting that small sustained aid-induced changes to labor supply (e.g., via mortality) can have considerable long-run effects. Equally, while assumptions regarding the physical capital investment lag $\mu$ become increasingly small as a longer evaluation horizon is chosen, assumptions regarding the rate of depreciation on physical capital are more crucial. Nonetheless, it would take a depreciation rate of approximately 0.62 to nullify the 30 year aid-growth results, holding all other parameters at their simulation means.

V. Conclusion

This article took as its point of departure the substantial degree of consistency across the full range of recently published empirical studies that investigate the long-run marginal effect of aid on growth. Employing different methods and data sets, the large majority of these studies place this effect in the positive domain. The weighted average result from these studies indicates that a sustained inflow of foreign aid equivalent to 10 percent of GDP is expected to raise growth rates per capita by about one percentage point on average.

The first objective of this study was to investigate the coherence of these empirical results using numerical simulations. To do so, we proposed a dynamic general equilibrium model that captures a range of aggregate channels likely to influence the size and direction of the macroeconomic effects of aid. These included physical capital investment, consumption of aid, human capital upgrading—including quantity and quality mechanisms—and direct productivity effects. We then simulated the model to capture the effects of these channels, both individually and in combination. Selected simulation results were used to understand the behaviour of the model. These were augmented by Monte Carlo simulations, based on 5,000 random draws from the distributions of the core model parameters (constrained over reasonable ranges). These simulations were used to approximate the empirical distributions of key outcomes, principally the marginal impact of aid on growth and its internal rate of return (IRR).

When assessed over a short time horizon (5 years), the simulations revealed that the marginal effect of aid on growth is negative over a nonnegligible share of observations and has large variance. This reflects sensitivity to the lag structure through which aid effectively contributes to the economy. When assessed over the long-run (30 years), the macroeconomic effects of aid are consistently positive and distributed more tightly, especially from the point of view of the IRR. More specifically, both the mean and broad range of the simulated marginal effects of aid are highly comparable to findings from recent empirical studies. We also found that when aid is allowed to affect human capital quality, a time frame of a generation may not be sufficient to fully capture the economic contributions of aid.

The basic upshot of our analysis is that the overall picture given by the aid-growth empirics of table 1 is reasonable. It is compatible with a range of plausible parameterizations of a simple yet general model of the growth process. The central tendency of our simulations approaches the mean of recent empirical aid-growth studies, particularly when moderate positive effects from aid on human capital and/or aggregate productivity are permitted. In sum, both the latest empirical evidence and numerical simulations point to positive average growth returns to foreign aid when viewed over long time frames. The notion that aid has broadly harmed development performance receives essentially no empirical support.

The IRR calculations associated with the same simulations require careful interpretation. On the one hand, these results might be considered fairly moderate. For instance, if only physical capital accumulation effects are permitted, as per our baseline model, the average IRR over 5,000 model runs is equal to 7.5 percent (table 3, row 2), while the corresponding 99 percent confidence interval ranges from −11.5 to 30.7 percent. Also, across all simulations the cumulative value of cash flows attributable to aid only turns positive after around 20 periods (at least). This reflects the lag structure of the model and that aid contributes only slowly to income via additions to stocks of capital. Simply put, long lags between the injection of funds and realization of economic benefits naturally militate against high IRRs.

From the perspective of both marginal growth effects and economic returns, these simulations underline the need for realistic expectations of foreign aid. Certainly, aid cannot be expected to deliver the kinds of high returns sometimes implied by two gap model exercises, which underpinned projections of rapid economic development in the early days of foreign assistance—a point vividly made by Easterly (1999; also Dalgaard and Erickson 2009). Similarly, some aid proponents have suggested that aid may yield very large economic returns when used to escape poverty traps. As Kraay and Raddatz (2007) note, however, these hopes appear not to have materialized. Observed reality, which is consistent with our simulation results, is that the journey from low- to middle-income status typically remains long and arduous, even when supported by effective aid inflows.

At the same time, predictions from poverty trap or basic two gap models are not required to provide an economic rationale for foreign aid. Our results imply that a sufficient criterion is its macroeconomic rate of return. The distribution of returns summarized in table 3 (also fig. 2), and which should be viewed as ex post, cluster around commonly used rates of return in ex ante project analysis such as the 10 percent cut-off applied by the World Bank (Pohl and Mihaljek 1992), or the seven percent cut-off employed by the United States Office of Management and Budget (Powers 2003). Recent growth empirics combined with a simple model of growth point to rates of return to aid that lie in a range commonly associated with successful long-run public sector investments.

Additionally, productivity and human capital accumulation effects appear to be important, if not fundamental, mechanisms through which aid can affect the macroeconomy. This is especially true when viewed over the long-run. For example, for moderate choices of the partial correlation between aid and the final level of TFP, the simulated distribution of the long-run marginal aid-growth effect more closely approximates the weighted average from recent empirical studies and the corresponding average IRR is significantly higher, at 13.1 percent. Allowing all parameters to vary simultaneously yields a mean IRR of 11.2 percent (table 3, row 1). Thus, whatever the combination of channels or their relative weight, it remains appropriate to view aid as a long-term investment whose benefits cumulate slowly over long periods. Additionally, rates of return of the order of magnitude found here continue to confirm a valid role for aid as a concessionary source of financing.

So, why did it take so long to figure all this out? Both aid volumes and their associated impacts are not so large as to be easily identifiable in macroeconomic data. The simulation modeling presented here underscores that long time frames are required to detect a growth impact. This reflects lags in the realization of benefits and the relatively moderate contribution of aid to the overall growth rate. In reality, detecting the contribution of aid is further complicated by large fluctuations in growth that have been an inherent part of the experience of nearly all developing countries. On top of this, observations of both the flow of aid funds to developing countries and their growth rates are known to be imperfect. For these reasons, it is not surprising that the economics profession has only recently converged on the more consistent range of estimates presented in table 1.

While growth is clearly important, income is not the only metric that can or should be employed to evaluate aid. As noted in the Introduction, other metrics largely support the case for aid. For example, aid has been shown to contribute to accumulation of important elements of human capital, particularly improved educational attainment and improved health. These can be considered merit goods with intrinsic worth. Humanitarian assistance is given with the expectation of saving lives rather than making a growth contribution.²⁵ Finally, under the assumption of diminishing marginal utility of income, the chasm that separates the living standards of citizens of recipient countries and the living standards of citizens of donor countries should enter the calculus. In income terms, our analysis indicates that the monetary gains to recipients are somewhat larger than the monetary costs imposed on donors (depending on the assessed opportunity cost of the aid to the donor). It follows that a utility lens would greatly magnify the assessment of the returns to aid.

To close, we note that simulation is a powerful and flexible tool employed in a wide array of disciplines to shed light on empirical phenomena. In a widely cited article, Alroy (2001) employed simulation to better understand the end-Pleistocene mass extinction of large species (mega-fauna). He finds that across a wide array of plausible parameter values, human population growth and hunting almost invariably leads to a major mass extinction. Here, we employed simulation to better understand the returns to aid investments over the past 40 years. Our findings substantiate the need to evaluate aid over long periods and indicate that recent empirical results are coherent with plausible parameterizations of a simple yet general growth model. Across a wide array of plausible parameter values, the evidence points to a material contribution of aid to achieving development objectives. Calls for the extinction of aid and its associated institutions on the basis of poor or negative returns are unjustified. At the same time, our simulations illustrate the scope for substantial heterogeneity in the aggregate effects of aid across different economies and different points in time.

Footnotes

1

See Dalgaard and Hansen (2005) for estimates of the marginal productivity of investment associated with aid.

2

More extensive overviews of the aid effectiveness literature can be found elsewhere. For example, see references in table 1, also Temple (2010); Arndt et al. (2010); Roodman (2007); Dalgaard et al. (2004).

3

It follows from these criteria that the coefficients reported in table 1 exclude: “naïve” OLS estimates, (even where reported in included studies); results that focus exclusively on selected regions, disaggregated aid measures or alternative outcomes; and estimates from (recent) working papers (such as Galiani et al. 2014).

4

This value is chosen as an approximation to the average annual value of aid received by developing countries over the last thirty years.

5

The data underlying the first of these negative results (NDHKM12) has been reexamined and found wanting by Lof et al. (2015). The second negative result (HM13) derives from an estimation model that controls for aggregate investment, implying the estimated effect of aid on output is restricted to noninvestment channels.

6

This estimator assumes the coefficients (observations) are independent; thus the present results must be seen as approximate.

7

These authors find a positive and significant weighted average partial correlation between aid and growth, which is robust to adjustments addressing publication bias and moderator effects. Note that the estimates in Mekasha and Tarp (2013) are partial correlations (effect sizes) and thus are not directly comparable in magnitude to the estimates in table 1.

8

In fact, the correction from missing to null is sufficient to achieve statistical significance.

9

Juselius et al. (2013) review these two cases in further detail considering country-specific historical factors, leading to a different conclusion.

10

We thank an anonymous referee for suggesting this connection.

11

See IMF (2008) for examples of country-specific aid modeling exercises that include greater sectoral detail.

12

We recognize these demographic dynamics are crude. However, adding richer structure adds little analytical meat and therefore is excluded for parsimony.

13

Appreciation of the real exchange rate is one example. See Arndt et al. (2015) for further discussion and Berg et al. (2010) for a microfounded dynamic general equilibrium framework that includes some of these complex features.

14

See: http://www.who.int/healthinfo/global_burden_disease/estimates_regional/en/index1.html.

15

This is calculated using equation (7) as follows: $(1+\phi \lambda {)}^{30\kappa }=(1+0.9\times .05{)}^{4.5}=1.219$⁠.

16

Panel estimators consider multiple consecutive windows across various countries. Even so, typically these rely on identification via first differences—that is, they focus on changes in growth relative to changes in aid inflows. Viewed in this way, only the initial period of the simulations relative to the baseline would be informative as this provides the primary variation in aid inflows.

17

We define the project life-cycle as the period from the beginning of aid inflows (⁠$t=1$⁠) to the period at which growth returns to a zero trend. For simulations where aid shifts the steady state income level, this termination rule ignores the ongoing (infinite) stream of benefits due to the changed equilibrium level of income. Note that for all simulations we impose a minimum life cycle of 60 periods, which can be thought of as comprising 30 years of aid inflows followed by a minimum of 30 additional years for impacts to ‘work through’ the economy.

18

If the project generates no positive cash flows, then the last period is used.

19

This is consistent with the simulation model of Ashraf et al. (2009).

20

As already noted, productivity effects represent an umbrella of complex channels of aid impacts, which are not modeled explicity.

21

As before, specific parameters are held constant throughout, namely: $t^{\ast }=\omega =30;\eta =0.4$⁠.

22

In order to satisfy the restriction $0\le \phi +\gamma \le 1$⁠, values for the share of aid allocated to human capital accumulation are given by: ${\phi }^{\ast }=\phi (1-\gamma )$⁠, where the parameters on the RHS are raw draws from their distributions.

23

These are derived as the negative of the value of the ratio of the intercept to the regression coefficient of interest. Standard errors are calculated via the delta method.

24

That is, defining x as the contribution of export industries to value added and g as the economy-wide per annum growth rate, then: $g=[0.018(1-x)+(0.018-5\times 0.005)x]\le 0\text{iff}x\ge 0.72$⁠. Note that we do not include Rajan and Subramanian (2011) in the summary of studies discussed in Section I since their dependent variable of interest is the average rate of growth of value added in specific industrial sectors.

25

A contribution to growth may nevertheless occur. For example, Hoddinott et al. (2008) find that improvements in early childhood nutrition contribute to adult labor productivity and hence to long term economic growth. To the extent that humanitarian assistance prevents major declines in early childhood nutrition, it may also improve the adult labor productivity of cohorts born during or shortly before times of crisis relative to a no assistance counterfactual.

Author notes

Channing Arndt is senior research fellow at UNU-WIDER ([email protected]). Sam Jones is associate professor at the University of Copenhagen ([email protected]). Finn Tarp (corresponding author) is director of UNU-WIDER and professor at the University of Copenhagen; telephone: [phone]; fax: [phone]). The authors report no conflicts of interest. The authors are grateful to the WBER editor and two anonymous referees for guidance and very helpful comments. They have also benefited from the advice of Tony Addison, Richard Carey, Carl-Johan Dalgaard, John Hudson, Richard Manning, Jean-Philippe Platteau, Lant Pritchett, Roger Riddell, L. Alan Winters and Adrian Wood, as well as participants at the 17th IEA World Congress (Jordan, 2014) and the Nordic Conference on Development Economics (Helsinki, 2014). The usual caveats apply. Funding to pay the Open Access publication charges for this article was provided by UNU-WIDER. The supplemental appendices to this article are available at http://wber.oxfordjournals.org/.