Optimizing Finance for Development

The World Bank Group recently adopted the "cascade framework" to "maximize finance for development." The cascade recommends that reforms be tried first, followed by subsidies, and then public investments. To understand the economics of the cascade, this paper presents a model where reforms, subsidies, and public investments can be used to fill the investment gap, and computes the welfare associated with their different sequencing. The cascade is optimal when reforms increase efficiency at no cost. When they are costly, if policies can be project specific, their sequencing does not matter; if not, the cascade can be optimal if agents are myopic, but not if they are forward-looking. Tensions may thus arise between maximizing private financing and optimizing financing for development.


Introduction
In September 2015, the UN general assembly adopted a resolution laying out an ambitious agenda for sustainable development (Agenda 2030). 1 Multilateral development banks readily embraced it, but they soon realized that existing o¢ cial resources were not enough for the undertaking. 2 The challenge, to use World Bank President Jim Yong Kim's words, became that of …nding ways to "leverage the billions of dollars in o¢ cial development assistance to trillions in investment of all kinds, whether public or private, national or global." 3 Against this backdrop, in March 2017, the World Bank Group (WBG) adopted 4 the "cascade approach as a concept to guide the WBG's e¤ort to leverage the private sector for growth and sustainable development,"(World Bank, 2017b, p.1). If such language may sound a bit elusive, the guidelines on how to implement the cascade are very clear: "When a project is presented ask -'Is there a sustainable private sector solution that limits public debt and contingent liabilities?'If the answer is 'Yes' -promote such private solutions. If the answer is 'No' -ask whether it is because of: (i) Policy or regulatory gaps or weakness? If so, provide WBG support for policy and regulatory reforms. (ii) Risks? If so, assess the risks and see whether WBG instruments can address them. If you conclude that the project requires public funding, pursue that option." (World Bank, 2017b, p.2) The aim of this paper is to provide a simple framework that helps us understand better (i) how the adoption of the cascade may a¤ect the allocation of …nance across projects, and (ii) the conditions under which maximizing and optimizing …nance for development are likely to coincide, and those under which they are not.
With these goals in mind, we present a very simple model where investment projects, which create positive externalities, can be …nanced by the private or by the public sector. We then identify the set of worthy projects that the private sector should …nance (because of its e¢ ciency advantage vis-à-vis the public sector), but that it does not (because they are not commercially viable). To …ll such an investment gap, following the cascade, we consider three di¤erent government interventions: the …rst is an upstream policy reform that allows private investors to extract (part of) the externalities; the second is a public subsidy (for instance in the form of public sector co…nancing of projects or of subsidized guarantees/insurance instruments) that induces the private sector to invest; the third intervention is the direct funding of the entire project by the government.
If upstream reforms are able to crowd private investments in at no cost for the society then, by themselves, they can address the investment gap. In such a case, the cascade is clearly optimal. When instead, there are costs associated with reforms (e.g., higher private returns are the outcome Engel et al. (2014) analyze the economics and …nance of public-private partnerships. Finally, ADB (2015) discusses how public and private sources should coexist and reinforce each other in the new sustainable development agenda. While this paper builds on the …nancial incentives literature, the model we present is designed with the cascade framework in mind. This explains some of its non-standard features.
The paper is organized as follows: Section 2 describes the basic model and explains why a …nancing gap arises; Section 3 discusses the di¤erent instruments that can be used to …ll such a gap, and how they should be allocated when they can perfectly target the di¤erent projects. Section 4 deals with the more realistic case in which subsidies and reforms cannot be project speci…c, so that the sequencing of policy actions matters; it then solves for the allocations and the welfare levels associated with di¤erent policy sequencing. It also distinguishes between the case in which agents/agencies are myopic and forward looking. Finally, Section 5 discusses the critical assumptions of the model and concludes.

The Basic Model
We assume that there is a continuum of investment projects, k = fx i ; j g 2 K. All projects are of the same size, which we normalize to 1, they are indivisible, and they can be undertaken either by the private, P , or by the public sector, G. Investment projects di¤er with respect to the returns they generate. The returns associated with project k, are R l k = r l ka + r l ke , where r l ka denotes appropriable returns (e.g., pro…ts), and r l ke non-appropriable returns (e.g., externalities). Superscript l, l 2 [P; G], denotes whether a project is undertaken by the private or by the public sector.
The existence of a di¤erence in the returns associated with privately and publicly …nanced investment projects is what motivates the cascade approach. The private sector's appropriable returns are denoted by r P ia = 1 + x i , and the public sector's by r G ia = 1. To focus on the policy relevant trade-o¤s, most of the analysis features the case in which the private sector has an e¢ ciency advantage vis-à-vis the public sector, that is, when x i 0.
We further assume that the private sector's e¢ ciency advantage is project speci…c, and that it does not extend to externalities. Non-appropriable returns are thus the same, independently of whether the project is undertaken by the public or the private sector. We thus set r G je = r P je = j , with j 0. Hence, a project k is characterized by an x i and a j that we assume to be independently and uniformly distributed on the interval [0; 1], so that K = [0; 1] 2 . As per the costs associated with the di¤erent projects, we assume that one unit of capital is needed to …nance a project. The cost of such unit of capital, c 1, the cost of funding hereinafter, is the same for the public and the private sector; 6 therefore, the total cost of the project, c, is also the same. Without great loss of generality, we assume that c 2 (1; 2).
In the simple set-up we just described, it is easy to identify the set of welfare improving projects that the private sector should, but does not …nd it pro…table to …nance. Indeed, the private sector, which maximizes private returns, R P i c, …nds it pro…table to invest in a project i¤ 1 + x i c, or However, since the private sector does not internalize the externality, j , its participation constraint (1) is stricter than the condition that insures that the project is welfare improving. Such a condition can be written as Our basic set-up is summarized in Figure 1a, where, setting c = 3 2 , we order the projects'space according to the associated private sector's e¢ ciency advantage and externalities, that is, in the (x i ; j ) space. Region D, is the locus of the projects that are quintessentially private. They are characterized by high private sector e¢ ciency advantage (x i > x P ), and they always satisfy the private sector participation constraint (1). We will disregard such projects in the remaining of the paper, as we will disregard the projects in region C, with x i < 0, where the public sector has an e¢ ciency advantage and, thus, it should always undertake them. Hence, the area of interest for policy makers, and the focus of this paper, is Region B, where x i 2 [0; x P ], and j P . Projects that belong to such a region should, on welfare grounds, be undertaken by the private sector. Having disregarded regions C and D, the parameter space that is pertinent to the analysis is the one depicted in Figure 1b. In the …gure, we split region B in two subregions, B 0 and B 00 . In Region B 0 , we have that > c 1 G ; and hence public investments are welfare improving. However, on e¢ ciency grounds, it would be preferable if projects were undertaken by the private sector. Instead, in region B 00 , condition (3) does not hold so that the only projects that are welfare improving are those undertaken by the private sector. This being said, since the private sector does not internalize the positive externality, and projects do not generate positive net returns-condition (1) is not met-, the private sector is not willing to invest in region B. Finally, projects in region A are not worth …nancing by either the private or the public sector. However, we cannot ignore them altogether because, as we will see later, some of them could end up being …nanced if the government is not able to target subsidies to speci…c projects.
3 How to …ll the investment gap?
Before discussing issues related with the sequencing of interventions, it is useful to introduce the three di¤erent policies that, in our framework, can …ll the investment gap: reforms, subsidies, and public investments, separately. We then compute the conditions under which each of them is optimal, provided that the government can perfectly tailor policies according to the characteristics of every speci…c project.

Reforms
We start by assuming that the government can undertake policy reforms that allow …rms to appropriate part of the (previously) non-appropriable returns. More precisely, we assume that the government can increase private appropriable returns by j , with 2 [0; ], at a cost (1 ) j in terms of non-appropriable returns. This may sound quite abstract. So, what is the kind of reforms we have in mind? A good example would be a change in the regulatory framework that allows or facilitates the outsourcing of infrastructure investments to the private sector, which then can charge a usage fee. Through the fee, investors could extract some of the externalities/users' surplus generated by the infrastructure; however, when < 1, this entails a net e¢ ciency loss, think, for instance, of the classical Harberger's triangle. Other examples of "costly" reforms could be a strengthening of property rights, a change in the regulatory framework of utilities, a weakening of antitrust enforcement, etc.
Notice that, while the analysis focuses on the more interesting case of costly reforms, there is no reason to rule out e¢ ciency generating reforms, where 1. As examples of such reforms, consider measures aiming at reducing corruption, at cutting red tape, or at simplifying bureaucracy. More generally all the measures that increase the "size of the pie"where the pie, here, includes all externalities belong to this category. Notice, however, that, when they exist, e¢ ciency generating reforms would, by themselves alone, …ll the investment gap, and the cascade algorithm, which in this case collapses to "just do reforms," is clearly optimal.
While, in our framework, we could allow the government to choose any reform in the [0; ] interval, in all possible scenarios, the welfare associated with reforms increases monotonically with . Hence, without loss of generality, we only consider the case = . That is, denoting the reform scenario by tilde, we assume that if reforms crowd in private investments the associated returns are given byr r P ke = 0: In such a case, the private sector …nds it pro…table to invest if 1 + x i + c, or Notice that, under assumption (4), private and total welfare 7 associated with reform induced private sector's investments are exactly the same. This means that the private sector participation constraint, and the condition that insures that the private sector's projects are welfare improving, coincide. This condition can be written as Notice that, when = 1, by the sole use of reforms the government is able to induce the private sector to …nance all welfare improving projects at no cost for the society as a whole. Moreover, if > 1, reforms increase the set of welfare improving projects B. For smaller values of , a set of valuable projects (those below R in Figure 2a) that are not …nanced by the private sector remains; in the special case of = 0, no additional project is …nanced. Figure 2a illustrates how the introduction of reforms changes the private sector's incentives to invest for di¤erent values of .

Public sector intervention: Subsidies and public investments
Let us now consider the situation in which the government is willing to use subsidies (guarantees) or public projects to …ll the …nancing gap. We start with the subsidies, s i , that the government can o¤er to the private sector at a cost cs i . Since the government looks at total welfare, and s i is a pure transfer from the government to the private sector, the welfare cost of a subsidy s i , is s i (c 1),which is equal to the government's cost of funding. Under such a policy, the private sector …nds it pro…table to invest if 1 + x i + s i c, or Expressing the minimum subsidy that makes project x i pro…table as and substituting it into (8), the latter becomes Notice that, in this framework, the government o¤ers a di¤erent subsidy to each individual …rm. The alternative policy that the government can put in place is that of directly …nancing the project. As we already discussed, such a policy is welfare improving as long as: Figure 2b-c describes the projects that are worth …nancing with well targeted subsidies and with public investments.

Optimal sequencing of interventions
From the previous analysis, it is clear that a government can rely on di¤erent instruments to …ll the investment gap. The question that follows is which instrument has to be preferred and under what circumstances. This is what we analyze next, under the assumption that the government can perfectly tailor policies according to the characteristics fx i ; j g of any speci…c project k. In other words, the government can implement reforms that only apply to projects with given values of x i and j , and the same is true for subsidies and public investments. 8 In what follows, we characterize the allocation of instruments that maximizes total welfare. We denote such allocation as the optimal policy allocation, OP A. Lemma 1 below fully characterizes such an allocation as a function of the the degree of private sector advantage (x i ) and the externalities ( j ) associated with each speci…c project.
Lemma 1 (1):If reforms are not e¢ cient enough ( < 1 c ): (i) if j is su¢ ciently low, no investment inducing policy is welfare improving; (ii) for su¢ ciently high values of j , and low values of x i , it is optimal for the government to implement public projects; (iii) for su¢ ciently high values of j and x i , subsidies are instead optimal.
(2): If reforms are e¢ cient enough ( 1 c ):(i) if j is su¢ ciently low, no investment inducing policy is welfare improving; (ii) for high j and low x i , it is optimal for the government to implement public projects; (iii) for projects with intermediate j and su¢ ciently high x i , reforms are optimal; while (iv) for su¢ ciently large values of j and x i , subsidies are optimal.
Proof: See Appendix.  9 The …rst, easy, takeaway from the …gure is that, no matter how e¢ cient reforms are, when dealing with high externalities/low private sector advantage-type of projects, the best option is public …nancing. When private sector advantage is high, instead, the best option is a subsidy. The reason for the optimality of a subsidy is that, when x i is large, a very small subsidy is su¢ cient to crowd in private investors. Hence, the costs of the subsidy are of second order (b s i tends to zero when x i tends to c 1), while those associated with public investments and reforms are always of …rst order (except in the limit cases in which x i = 0, or = 1). Thus, while there are always situations in which public investments or subsidies are optimal, for reforms to have a place in an OP A, they should be e¢ cient enough, that is, we need that > 1 c . E¢ ciency here is measured against the costs c associated with the subsidy. In addition, the appeal of reforms weakens in the presence of large externalities because of the deadweight loss associated.
A pretty trivial, but important point, often forgotten in the cascade debate, is that Remark 1 If the government is able to perfectly target the policy instruments, the sequencing of interventions does not matter.
Indeed, if the government can associate an instrument to any fx i , j g, it will perfectly de…ne the scope of each policy so that the sequencing of the interventions does not matter. In other words, the fact that, in order to …ll the investment gap, the government starts with reforms, subsidies, or public investments, is irrelevant; what is done …rst does not pose limits to what can be done next. In such a world, there is no need for a cascade, or for any other sequencing algorithm.

Imperfect targeting
Let us now remove the assumption that the government can o¤er subsidies or reforms that are project speci…c. Consequently, we assume that x i , and j are observable but not contractible. This implies that the government can decide which speci…c projects to …nance directly, but it can only o¤er one subsidy, the same one to all investors. With respect to reforms, they apply to all projects that have not yet been …nanced, and only to those ones. This means that reforms do not a¤ect the returns of the projects that they are unable to crowd in and that are later …nanced through a subsidy. 10 We also assume that, once a project is …nanced, the source of …nancing cannot be modi…ed. This means that if an investor decides to …nance a project applying for particular subsidy, it cannot later on renounce the subsidy and take advantage of a reform; similarly, if an investor decides to …nance a project taking advantage of a particular reform, then it cannot "renounce the reform" and apply for a subsidy.
It is clear that, now, the order of interventions does matter. If the government starts with reforms, all projects that have been …nanced through reforms cannot later be …nanced by subsidies or public investment. The same argument applies to public investment and subsidies. Of course, the optimal allocation of policies is not the same under perfect or imperfect targeting. The constrained optimal policy allocation (COP A)-that is, the policy allocation that is optimal when the government is not able to target policies according to the characteristics of the speci…c projects-should equalize the marginal returns associated with each policy (at least when all policy instruments are utilized in equilibrium). Asking the reader to be patient, and to wait until Section 4.4 for a characterization of the COP A, we illustrate such an allocation in Figure 3b. Unsurprisingly, given that subsidies cannot be properly targeted, and that they end up …nancing a number of projects that are not worth …nancing (those under the 45 degree P line), the government will …nd it optimal to o¤er smaller subsidies under the COP A than under the OP A.

Myopic beliefs
Having set the optimal benchmark as a yardstick to compare di¤erent policies, we now assume that, when implementing a policy, whatever it be (reform, subsidies, public investments), the agent, or better the governmental agency in charge, does not anticipate that other agencies will adopt other policies at a later stage. When this is the case, each agency maximizes social returns assuming no other policy will ever be put in place.
The reason we focus on such myopic beliefs is that, in our reading, these are the ones implicitly assumed by the cascade approach. Indeed, the algorithm we discussed in the introduction requires that upstream policy reforms be tried …rst, public coinvestment or risk-sharing next, and …nally, only if both reforms and subsidies are insu¢ cient to close the …nancing gap, public investments should be pursued. The assumption of myopic beliefs will be removed in Section 4.5, where we allow the di¤erent agencies to be forward looking and implement each single policy anticipating how it will a¤ect the policies that will be put in place at a later stage by other agencies.
Starting with the case of myopic beliefs, we analyze the policy allocations and the welfare associated with the di¤erent sequencing of the policies. Having three policies, R; S; G, where R stands for reforms, S for subsidies, and G for public (government funded) investments, we have 3!=6 di¤erent sequences. Following the order of the interventions, we denote the cascade approach by RSG, the "anticascade" scenario (that with the opposite sequencing as the cascade) by GSR, and so forth.
In the main text, we provide a sketch on how to compute policy assignments and welfare under the cascade approach, and we compare the cascade with the other scenarios. We refer the reader to the Appendix for the formal derivation of all the di¤erent cases.

The Cascade Approach (RSG)
If the government follows the cascade approach, it starts by implementing reforms, it then moves to subsidies and, if worthy projects remain un…nanced, it …lls such a gap with public investments. The welfare gains W R C associated with reforms (R) under the cascade approach (C) are given by: wherex fx : R = 1g = c 1 . Let us now introduce subsidies and compute the additional associated welfare W S C . We should distinguish between two cases. In the …rst, c 1, and thus R < 1 for all x i 2 [0; 1]. When this condition is veri…ed, the problem of the government is that of …nding the optimal subsidy s (it can be zero) that maximizes When, instead, < c 1, the problem of the government is that of …nding the optimal subsidy s that maximizes Finally, the government directly …nances through public investments all projects in the domain have not been crowded in by the reforms and/or the subsidies put in place. We refer the reader to the Appendix for the full derivation of the results. The …ndings of our analysis are summarized in Figure 3c. When we compare the allocations obtained using the cascade algorithm with the constrained optimal ones, unsurprisingly, we …nd that the cascade framework pushes reforms far beyond, and reduces public investments far below, what is optimal from a welfare perspective. Interestingly, since the relative cost of subsidies increases with the degree of e¢ ciency of the reforms, 11 it is when reforms are less e¢ cient that subsidies become increasingly generous and, thus, they are more likely to crowd out more e¢ cient public sector projects.

Alternative sequencing
In the same way as we did for the cascade, we can compute the allocations that are associated with the …ve other possible sequences of policies (see Appendix). The results are summarized in Figure 3d-h. If we compare the cascade, RSG, with the RGS sequence, reforms necessarily lead to the …nancing of the same set of projects. This, in turn, implies that results are identical when reforms are so e¢ cient that subsidies are never implemented (high values of ). However, they di¤er substantially for low values of . When this is the case, public investments replace subsidies.
Let us now consider the SRG sequence. Of course, in this case, subsidies play a substantial role. However, di¤erently from the cascade, they are implemented when they are comparatively more e¢ cient, namely when the private sector e¢ ciency advantage is substantial (high values x i ). In addition, while for low values of subsidies completely crowd out reforms, for intermediate values of both policies coexist at equilibrium. When we move to the SGR sequencing, the scope for subsidies remains the same. However, now, public investments replace reforms completely, for intermediate values, and partially for high values of .
When we move to the "anticascade"sequence GSR, public investments are implemented when-ever they yield positive returns (including the externality), while subsidies crowd in the remaining projects where the private sector has a substantial e¢ ciency advantage; reforms, when su¢ ciently e¢ cient, play the residual role. Lastly, in the GRS scenario, reforms completely crowd out subsidies for intermediate and high values of , but not for lower ones.

Optimal myopic sequencing
In the previous section, we illustrated the policy allocations associated with the di¤erent possible policy sequences. The obvious question that remains to be answered is which sequence is superior on welfare grounds, and under which circumstances. By a simple comparison of the di¤erent policy allocations with the COP A, it is evident that the latter coincides with SRG and SGR for low values of and with SGR alone for intermediate values. Instead, for high values of , the cascade, or the identical RGS scenario, are the ones that resemble more to the COP A. To …nd a de…nitive answer to the question we compute, for all values of and c, the welfare functions derived in the Appendix (integrating over x i and j ), we compare them all and, for each combination of the parameters, we compute the policy allocation(s) that leads to higher welfare levels. We …nd that: Proposition 1 (i) When reform e¢ ciency is low (low values of ) the optimal myopic sequencing is SRG or SGR (which are identical). (ii) When reform e¢ ciency is high, the optimal myopic sequencing is RSG (the cascade) or RGS (which are identical).

Proof: In Appendix
The …ndings of our analysis are illustrated in Figure 4, which fully characterizes the optimal myopic policy allocations for any value of the parameters, that is in the ( , c) space. From the discussion in the previous section, and from an even cursory inspection of the …gure, it is evident that, for low values of , the optimal (myopic) policy sequencing are SRG and SGR-which in this case coincide, since no reform is undertaken-that mimic the COP A. When reforms are very e¢ cient, the cascade (RSG) or RGS-which coincide since reforms completely crowd out subsidiesare instead the optimal allocations. For intermediate values of , the optimal allocation depends on the value of c. If c is small, so that subsidy costs are limited, reforms will never be implemented and, again, SRG and SGR will be identical, and constitute the preferred option. For intermediate values of c, reforms are implemented and the larger is c, the more likely it is that reforms are a better instrument to …ll the …nancing gap than public investments.

Optimal forward-looking sequencing
To understand the results derived in the previous section, it is useful to think of a government with three agencies dealing with the investment gap, one in charge of reforms, one of subsidies, and one of public investments. Each of the agencies is given the mandate of maximizing total The problem of the government is that of deciding which agency moves …rst, which second and which third. The setback (and additional source of ine¢ ciency) here is that no agency anticipates that its own decisions a¤ect the decisions of the other agencies. This is the reason why we called such behavior myopic.
We now relax such an assumption and let each agency maximize total welfare (that is, the returns from investments triggered by the policies of all three agencies) fully anticipating how its own policies a¤ect the behavior of the other agencies. Again, the order of the "moves"does matter because reforms and subsidies cannot be perfectly targeted. When this is the case, the optimal forward-looking sequencing is the constrained optimal policy allocation (COPA), we depicted in Figure 3c. More precisely, we have that Proposition 2 The COPA requires that subsidies be o¤ ered …rst and reforms and public investment later (the order does not matter). When agents are forward looking, the cascade sequencing coincides with the COPA, only when reforms do not belong to it.

Proof: In Appendix
In the Appendix, together with the proof, we provide the full derivation of the COPA. Here, we will spare the reader the technical details and just focus on the intuition behind Proposition 2. As we already mentioned, when the e¢ ciency advantage of the private sector is substantial, the costs associated with the implementations of subsidies are of second order. Since the marginal gains associated with reforms or public investment do not depend on x i , it is always better to implement subsidies …rst, to avoid that the most e¢ cient subsidies be crowded out by reforms or public investments. Since the government can perfectly target public investments, the order in which it implements public investments and reforms really does not matter (as the choice will only be based on their relative e¤ectiveness). Using the cascade sequencing, if reforms are part of the policy menu, they will necessarily crowd out more e¢ cient subsidies. Hence, it is only when no reforms are part of the COPA that the cascade is as good as any other sequencing, since, in this case, it does not matter whether to implement reforms or public investments …rst, since reforms will not be implemented in either case.

Concluding Remarks
According to a recent McKinsey report (McKinsey Global Institute, 2016), to keep current growth trajectories, the world needs to invest about $3.3 trillion, 3.8 percent of GDP a year, in economic infrastructure, about 60 percent of which in developing and emerging economies. The estimated …nancing gap for the world as a whole is about $350 billion a year, a …gure that should be multiplied by three, if one budgets in new global commitments. The available public and o¢ cial resources alone clearly do not su¢ ce, so that the success of the Agenda 2030 critically depends on private sector participation; this is not only because of the …nancial resources but also because of the expertise the private sector can bring to the table.
The cascade framework, adopted by the WBG, is an attempt to crowd in private resources by using public funds as a last resort. In adopting the framework, the WBG wants to signal that the timing of "using public …nance till there is money and only then consider private sector options"had to be replaced by that of "trying everything and only if nothing works use public funds." 12 While the motivations and the political economy behind the cascade framework are easy to understand, the economic underpinnings of the algorithm are less clear. This is what drove us to "write a model." The model we presented is the simplest one-at least the simplest one we could come up withthat can describe the main trade-o¤s associated with the cascade framework. We like simple models; however, it is important to discuss how reasonable our simplifying assumptions, and more importantly how robust our results, are. This is what we try to do next.
First, discussing the sequencing of policies, we assumed that when investors are o¤ered a subsidy they do not wait for a more pro…table reform (at a later stage) to invest, or if o¤ered a subsidy they do not wait for a more pro…table reform. While such behavior could be seen as a consequence of myopic behavior (similar to the one of the government agencies), it can also be seen as the outcome of a competitive environment: if one investor passes on a pro…table opportunity, another will take advantage of it.
Second, the main rationale behind the cascade framework is that public resources are scarce, and this is why they should be used only as a last resort. In our model, we do not have a public resource constraint, and we further assume that the marginal cost of public funds is constant. However, such assumptions are not critical. Consider the case in which the government has a hard budget constraint or faces an increasing marginal cost of funding. This does a¤ect the amount of subsidies it can o¤er or the number of projects it can …nance, but not the optimal sequence of policies, both when agencies are myopic and forward looking.
Third, we assumed that the government cannot perfectly target subsidies and reforms. Such an assumption, as we already mentioned, is critical for the sequencing of policies to matter, and thus should be part of any analysis of the cascade. Our modeling of subsidies is pretty standard and, were the government able to o¤er project speci…c subsidies, this would make the case for "subsidies …rst" even stronger. As per the reforms, we assumed that if a reform does not attract a particular investment project, it does not a¤ect the returns of the same project if the project is …nanced through subsidies. This is a reasonable assumption if one thinks of regulatory reforms such as those that allow the private sector to bid for concessions. However, it is a less reasonable one if reforms have a cost in terms of externalities. Think, for instance, of a reform that reduces some environmental standards, or one that strengthens patent rights. Once such reforms are undertaken, they apply to all projects, already, and yet to be …nanced. When this is the case, and agents are myopic, the appeal of subsidies vis-à-vis reforms decreases and the cascade sequencing is optimal for a larger set of parameters. When, instead, agents are forward-looking, then the sequencing becomes irrelevant; the incentives of both subsidies and reforms coexist in any single project; however, the overall appeal of reforms decreases as the associated costs in term of lost externalities now a¤ect all projects that end up being …nanced by the private sector.
While the previous discussion suggests that our main conclusions should hold true in a more general framework, we do not claim that the subsidy …rst policy should replace the cascade algorithm under all possible circumstances. For instance, one could add government failures to the model, and this can tilt the decision tree in di¤erent directions.
There is, however, one general lesson that can be learned from our analysis. The lesson is that the objective of maximizing private …nance for development may con ‡ict with the objective of optimizing …nance for development; this means that policy-makers should carefully weigh the di¤erent trade-o¤s if they want to use the scarce existing resources, which are vital to achieve the Sustainable Development Goals, in the most e¤ective way. 6 Appendix 6.1 Proof of Lemma 1 (1):(i) < 1 c implies that S < R . Hence, < M inf G ; S g is a necessary and su¢ cient condition for no investment inducing policy to be welfare improving.
(iii) We further have that < 1 c implies that, for all x < x P , R > S > RS , so that reforms are never optimal. Subsidies are, instead, welfare improving if > S , and they are preferred to public investment if which can be re-written as (ii) Public investments are thus optimal if > G , and x < x SG .
(2):(i) > 1 c implies that R < S . Hence, < M inf G ; R g is a necessary and su¢ cient condition for no investment inducing policy to be welfare improving.
(iii) Notice that for reforms to be preferred to subsidies we need that We further have that > 1 c implies that for all x < x P , RS > R , so that there is a non empty set of values in which reforms are preferred to subsidies. Finally, reforms are preferred to public investments i¤ Notice that Hence, for reforms to be optimal we need that < M inf RS ; RG g, and x i > x RG .
(iv) For subsidies to be preferred to reforms we need that > RS . Notice that, since RS > S , such a condition also insures that subsidies are welfare improving. Finally, subsidies are preferred to public investments if x i > x SG .
(ii) For public investments to be welfare improving, we need that > G , for public investment to be preferred to reforms that > RG , and x < x SG for public investments to be preferred to subsidies.

Alternative sequencing (myopic beliefs)
In this section, we compute the allocations corresponding to each of the di¤erent policy sequences z, z = f1; : : : ; 6g, and the associated welfare W z . We start with the cascade, RSG . (z = 1).

RSG
The welfare gains W R 1 associated with reforms (R) under the cascade approach (z = 1) are given by wherex fx : R = 1g = c 1 . Solving for (21) we have that , if c 1: Let us now introduce subsidies, s; we …rst consider the case c 1, so that R < 1 for all x i 2 [0; 1]. If the optimal subsidy s 1 (we denote the optimal subsidy by " " hereinafter) is greater than , it is the one that maximizes Di¤erentiating (23) with respect to s, we have that @W Sa 1 1 @s = 1 + (c 1) (4c 2)s 2 ; (24) so that a necessary condition for an internal maximum is Substituting this value into (23) we have that Notice that, for s 1 to be the optimal subsidy, we need that s 1 , that is, 1 3c 1 , and c 1 s 0, that is, 2(2c 1) 1 c 1 a . Assume now that > 1 3c 1 . When this is the case, the problem of the government is that of Di¤erentiating (27) with respect to s, we have that Hence, > 1 3c 1 =) s = 0. Consider now the case > a , then the government would set s 1 = c 1 i¤ , and s 1 = 0, otherwise.
We now consider the case > c 1. In this case, the the welfare associated with subsidies is given by: I¤ 1 3c 1 , (30) is increasing in s, and it is positive at s = c 1. Hence, we have s = c 1, if 1 3c 1 , and s = 0 otherwise. Summarizing our …ndings about the optimal subsidy, we have that Putting all the pieces together, using (26), (29), and (30), we have that: (32) Let us now look at the welfare associated with public investments. Using (31), it is easy to verify that (33) where (1 )(c 1) = : f R = c 1g . Now, working through the algebra we have that otherwise, Finally, we have that,

RGS (z = 2)
As in the cascade, we have that the welfare associated with reforms is given by: , if c 1: The welfare associated with public investments, G, which are implemented after, is instead given by or, solving, Let us now consider subsidies. If the optimal subsidy s 2 is such that c 1 s 2 < (1 )(c 1), it is the one that maximizes (38) Di¤erentiating (38) with respect to s, we have that It is easy to verify that c 1 s 2 > 0. In addition, we have that so that s 2 = s 2 , if < 1 3c 1 . Substituting s 2 into (38), we have that Assume now that the optimal subsidy s 2 is such that c 1 s 2 > (1 )(c 1). Then, it should be the one that maximizes , if < 1 3c 1 , 0 , otherwise.
Finally, we have that,

SGR (z = 4)
As in the previous case, the welfare associated with subsidies is given by Let us now consider public investments, welfare is given by After subsidies and (eventually) public investments have been implemented, there is a residual space for reforms i¤, where (c 1)(1 ) = x i : R < (c 1), and hence there is space for subsidies after public investments have been put in place. Notice that for c 2 [1; 2], d > s 3 , hence or, doing the algebra, Finally, we have that,

GRS (z = 5)
Starting with public investments, the associated welfare gains are given by If the government implements reforms after public investments, the associated utility is given by Let us now consider subsidies. If the optimal subsidy s is such that c 1 s < (1 )(c 1), it is the one that maximizes Di¤erentiating (61) with respect to s, we have that @W Sa 5 @s = 0 () s = ((c 1) + 1)(c 1) 2(2c 1) It is easy to verify that c 1 s 5 > 0; it remains to verify that c 1 s 5 < (1 )(c 1). It is easy to show that When, instead, c 1 s > (1 )(c 1), the problem is the same as in (42) and, again, we obtain that s 5 = 0 if 1 3c 1 . Hence, we have that Finally, we have that, W 5 = W G 5 + W R 5 + W S 5 .

GSR (z = 6)
Starting with public investments, as in the previous case, the associated welfare gains are given by Let us now move to subsidies, the government has to maximize Di¤erentiating (65) with respect to s, a necessary and su¢ cient condition for an internal maximum is and substituting this expression in (64), we have that Let us now consider reforms. After public investment and subsidies, there is a residual space for reforms if, and only if, x : R = (c 1)(1 ) < c 1 s 6 () 1 2(2c 1) .
Hence, we have that Finally, we have that, 6.3 Forward-looking agents: The COPA (z = b 3) We denote this case by b 3 (since the sequence is the same as for SRG, and the "b " denotes the forward-looking scenario. In this case the government chooses the optimal subsidy, given the optimal allocation of reforms and public investments at the later stage. If the optimal subsidy b s is such that (c 1 b s > 1 ), where 1 = x i : RG = 1, and thus in the interval [1 ; c 1 b s] reforms are always preferred to public investments, and the optimal subsidy is the one that maximizes Notice that a necessary condition for c 1 b s > 1 to hold is that > 2 c. Di¤erentiating (70) with respect to s, we have that In addition, we can show that c 1 b s b > (c 1)(1 ) () c > 3+2 1+4 .

Proof of Proposition 1
(i) For low enough values of , no reforms are undertaken under GSR, SRG, SGR, while they always are under RSG, RGS, and GRS. We also have that, at x = x P , the welfare changes associated with having reforms, instead of subsidies, is given by Di¤erentiating (78) with respect to x i , we have that @DW SR @x i = s(2(c 1 x i ) + s) 2 > 0: By plotting the di¤erent expressions, it is immediate to verify that the optimal myopic sequencing is SRG or SGR (which are identical), when the cost of resources c is low (c < 3+

Proof of Proposition 2
The proof that subsidies should be implemented before reforms is along the same lines as the proof of part (i) of Proposition 1. If a subsidy crows out public investments, the change in welfare is given by Di¤erentiating (84) with respect to x i , we have that @DW SG @x i = s > 0: Thus, a necessary condition for subsidy to dominate public investments is that DW SG j x i =c 1 > 0.
In addition, we have that @DW SG @s j x=c 1 = 2s 3 + c(4(1 s) c) 2 , and Lim s!0 @DW SG @s j x=c 1 = (3 c)(c 1) 2 ; so that a small subsidy strictly improves welfare, notwithstanding the fact that it crowds out public investments. This implies that subsidies should be implemented before public investments. Finally, the only situation in which the cascade and the COP A coincide is when reforms are not part of the COP A. When this is the case, we necessarily have that no reforms are undertaken in the cascade. The reason is simple. Assume that reforms were undertaken, then a positive reform will improve welfare given the optimally chosen subsidies and investments. But, if this were the case, the COP A would be strictly dominated by another allocation. A contradiction.