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.

# 2 The Basic Model

We assume that there is a continuum of investment projects, k = fxi ; �jg 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 Rk<sup>l=r</sup> ka<sup>l+ r</sup> ke<sup>l,wherer</sup> ka<sup>l</sup> denotes appropriable returns (e.g., pro...ts), and rke<sup>lnon-appropriablereturns(e.g.,externalities).</sup> 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 ria<sup>P=1 + xi,andthepublicsector’sbyr</sup> ia<sup>G=1.Tofocusonthepolicy</sup> 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 xi � 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 rje<sup>G=r</sup> je<sup>P=</sup> �j, with �j � 0. Hence, a project k is characterized by an xi and a �j that we assume to be independently and uniformly distributed on the interval [0; 1], so that K = [0; 1]<sup>2</sup> . 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;<sup>6</sup> therefore, the total cost of the project, c, is also the same. Without great loss of generality, we assume that c 2 (1; 2).

> 6 Of course, the private sector’s cost of borrowing can di¤er from the public sector’s. However, in our simple framework, this would be equivalent to a horizontal shift in the support of xi, towards the left when the public sector has a cost advantage, towards the right when the private sector has it.

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