WPS8320
Policy Research Working Paper 8320
Optimizing Finance for Development
Tito Cordella
Development Economics Vice Presidency
Strategy and Operations Team
January 2018
Policy Research Working Paper 8320
Abstract
The World Bank Group recently adopted the “cascade associated with their different sequencing. The cascade
framework” to “maximize finance for development.” The is optimal when reforms increase efficiency at no cost.
cascade recommends that reforms be tried first, followed When they are costly, if policies can be project specific,
by subsidies, and then public investments. To understand their sequencing does not matter; if not, the cascade can
the economics of the cascade, this paper presents a model be optimal if agents are myopic, but not if they are for-
where reforms, subsidies, and public investments can be ward-looking. Tensions may thus arise between maximizing
used to fill the investment gap, and computes the welfare private financing and optimizing financing for development.
This paper is a product of the Strategy and Operations Team, Development Economics Vice Presidency. It is part of a larger
effort by the World Bank to provide open access to its research and make a contribution to development policy discussions
around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author
may be contacted at tcordella@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Optimizing Finance for Development
Tito Cordella
The World Bank
I started this project with Oliver Masetti. Unfortunately, he got too busy implementing the cascade and had to
quit. I thank him for his initial contribution to the paper, and for the many long discussions we have had. I would
also like to thank Shanta Devarajan, Marianne Fay, Ana Paula Fialho Lopes, Joaquim Levy, David Rosenblatt,
Anushka Thewarapperuma, and Mike Toman for constructive comments and suggestions. Woori Lee provided out-
standing research assistance. The usual disclaimers apply.
1 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) adopted4 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
1
Available at http://www.un.org/ga/search/view_doc.asp?symbol=A/RES/70/1&Lang=E
2
See https://sustainabledevelopment.un.org/
3
Address at the Third International Conference on Financing for Development Addis Ababa, Ethiopia, June
13, 2015. Available at http://documents.worldbank.org/curated/en/144851468190446079/pdf/101985-WP-
Box393267B-PUBLIC-2015-07-13-JK-Billions-To-Trillions-Ideas-to-Actions.pdf
4
World Bank (2017a).
1
of e¢ ciency gains but also of higher fees that restrict access) trade-o¤s between the three di¤erent
policy instruments arise. To get a better understanding of such trade-o¤s, we consider …rst the
case in which government interventions can be perfectly targeted to speci…c projects. When this
is the case, the government has the ability to o¤er subsidies that are project speci…c and to decide
the set of projects to which a particular reform applies or not. We then consider the case in which
the subsidy should be the same for all projects and, if reforms are implemented, they a¤ect all
(un…nanced) projects. In both cases, we compute the allocation and the welfare levels associated
with any of the possible sequences of policies: reforms …rst, subsidy second, public project third;
subsidy …rst, reforms second, public project third, and so forth.
Our main …ndings are the following: (i) when reforms and subsidies can be perfectly targeted
to projects, the sequencing of policies does not matter; (ii) when they cannot, the sequencing does
matter. If the agents/agencies in charge of a speci…c policy are “myopic,” that is, they do not
anticipate the policies implemented by other agencies, there are situations in which the cascade
sequencing can be optimal; (iii) when agencies are forward-looking and anticipate the e¤ects of their
own policies on downstream policy interventions (and reforms are part of the optimal solution) then
the cascade is never optimal and the optimal sequencing requires subsidies/guarantees to be o¤ered
…rst.
While the paper focuses on a very speci…c problem: the economic underpinnings of the cascade
framework recently adopted by the WBG, the issues it touches relate to a wide body of literature.
With respect to the Agenda 2030, Schmidt-Traub and Sachs (2015) provide an assessment of public
and private investment needs to achieve the SDGs, while Samans (2016) analyzes the investment
barriers that the private sector faces in emerging markets and looks at the role that blended …nance
can play in overcoming them.
Of course, the discussion on the role that public and private investment play in the process
of economic development predates the Agenda 2030. Khan and Reinhart (1990), and Khan and
Kumar (1997) study the contribution of investments, public and private, on long term growth
in developing countries and conclude that the contribution of private investments is larger. The
problem is that, in such countries, private investments often fail to take o¤. The reasons can be
many, ranging from property rights and institutional quality (Banerjee et al., 2006) to political
uncertainty (Rodrik, 1991).
Whatever the structural reasons that deter private investments in developing countries are,
public policies can be part of the solution. OECD (2004) provides an overview of the existing
sources of …nance for developing countries and discusses how to mobilize new ones, including for
the provision of global public goods. OECD (2015) looks at the di¤erent …nancing instruments that
can support infrastructure investments in the sectors or regions that need them the most. Klemm,
(2010) provides a comprehensive overview of existing tax incentives for business investments,5 while
5
The fact that tax incentives are very common instruments used to attract investments does not mean that
they are uncontroversial. According to Pennings (2000) and Yu et al. (2007) investment subsidies are a better in-
strument than tax cuts to foster private investments. However, Danielova and Sarkar (2011) show that when debt
…nancing is possible, a combination of tax reduction and investment subsidy is optimal.
2
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 ; 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 Rk l = r l + r l , where r l
ka ke ka
l
denotes appropriable returns (e.g., pro…ts), and rke 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
P
returns are denoted by ria = 1 + xi , and the public sector’ G
s by ria = 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 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 G = rP =
je
j , with j 0 . Hence, a project k is characterized by an xi and a j that we assume to be
2
independently and uniformly distributed on the interval [0; 1], so that K = [0; 1] . 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).
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.
3
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, Ri P c, …nds it pro…table to invest in a project i¤ 1 + xi c, or
xi c 1 xP : (1)
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 1 + xi + j c, or
j c 1 xi P
: (2)
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
(xi ; j ) space. Region D, is the locus of the projects that are quintessentially private. They are
characterized by high private sector e¢ ciency advantage (xi > xP ), 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 xi < 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 xi 2 [0; xP ], 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, B0 and B00 . In
Region B0 , we have that
>c 1 G
; (3)
and hence public investments are welfare improving. However, on e¢ ciency grounds, it would be
00
preferable if projects were undertaken by the private sector. Instead, in region B , 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
4
Figure 1: Projects and …nancing gap
optimal, provided that the government can perfectly tailor policies according to the characteristics
of every speci…c project.
3.1 Reforms
We start by assuming that the government can undertake policy reforms that allow …rms to ap-
propriate 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; ]
5
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 by
P
~ka
r = 1 + xi + j; (4)
P
~ke
r = 0:
In such a case, the private sector …nds it pro…table to invest if 1 + xi + c, or
xi c 1 j xR : (5)
Notice that, under assumption (4), private and total welfare7 associated with reform induced pri-
vate sector’s investments are exactly the same. This means that the private sector participation
s projects are welfare improving,
constraint, and the condition that insures that the private sector’
coincide. This condition can be written as
c 1 xi
j R
: (6)
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
s incentives to invest for di¤erent values of
the introduction of reforms changes the private sector’
.
3.2 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, si , that the government
can o¤er to the private sector at a cost csi . Since the government looks at total welfare, and si
is a pure transfer from the government to the private sector, the welfare cost of a subsidy si , is
si (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 + xi + si c, or
xi c 1 si xs : (7)
7
Throughout the paper, we assume that the public sector maximizes total returns, that is, the pro…ts and exter-
nalities associated with the projects.
6
Figure 2: Filling the gap
Since subsidies are costly, the set of welfare improving private sector projects shrinks when compared
to the benchmark case; hence condition (2) becomes
j c 1 xi + si (c 1): (8)
Expressing the minimum subsidy that makes project xi pro…table as
bi = c
s 1 xi ; (9)
and substituting it into (8), the latter becomes
j (c 1 xi )c S
: (10)
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:
j c 1 G
.
Figure 2b-c describes the projects that are worth …nancing with well targeted subsidies and with
public investments.
7
3.3 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 fxi ; 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 xi 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 (xi ) 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 invest-
ment inducing policy is welfare improving; (ii) for su¢ ciently high values of j , and low values of
xi , it is optimal for the government to implement public projects; (iii) for su¢ ciently high values
of j and xi , subsidies are instead optimal.
1
(2): If reforms are e¢ cient enough ( c ):(i) if j is su¢ ciently low, no investment inducing pol-
icy is welfare improving; (ii) for high j and low xi , it is optimal for the government to implement
public projects; (iii) for projects with intermediate j and su¢ ciently high xi , reforms are optimal;
while (iv) for su¢ ciently large values of j and xi , subsidies are optimal.
Proof: See Appendix.
Figure 3a summarizes our results for di¤erent values of .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 xi is large, a very small subsidy is su¢ cient to crowd in private investors. Hence,
the costs of the subsidy are of second order (sbi tends to zero when xi tends to c 1), while those
associated with public investments and reforms are always of …rst order (except in the limit cases
in which xi = 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
8
For public investments this is always the case since the government decides which projects to …nance.
9
Also in this …gure, we set c = 3=2. Hence, in the …rst two panels of Figure 3a, we have that < 1 c
, and 1
c
in the last one.
8
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 fxi , 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.
4 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 xi , 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 re-
forms, 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 govern-
ment 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 charac-
terization 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.
10
This is the case when, for instance, a reform allows the private sector to invest in an infrastructure project
with a cost plus fee contract. If there is no interest for the private sector to invest under such conditions, but there
is with a subsidy, then the fact that the reform occurred does not a¤ect the returns when the subsidy is o¤ered. In
the concluding section, we discuss how the relaxation of such assumption may a¤ect our results.
9
Figure 3: Alternative policy sequences
10
11
4.1 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.
4.2 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 WC R associated with reforms (R) under the cascade approach (C ) are given by:
xP
Z Z 1
R
WC = (1 + xi + j c)d j dxi ; (11)
R
xg
M axf0;~
where x~ fx : R = 1g = c 1 .
Let us now introduce subsidies and compute the additional associated welfare WC S . We should
distinguish between two cases. In the …rst, c 1, and thus R < 1 for all xi 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
12
Z
xP
Z
R
Sa
WC = (1 + xi + j s(c 1) c)d j dxi : (12)
0
M axf0;c 1 s)g
When, instead, S > RS , so that reforms
are never optimal. Subsidies are, instead, welfare improving if > S , and they are preferred to
public investment if
1 + xi + + s b(c 1) c > 1 + c; (14)
which can be re-written as
(1 c)2
xi > xSG : (15)
c
(ii) Public investments are thus optimal if > G
, and x < xSG .
(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
1+x+ c>1+x+ c b(c
s 1); (16)
or
(c 1)(c 1 x)
< RS
: (17)
1
We further have that > 1 c implies that for all x < xP , 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¤
1+x+ c>1+ c; (18)
or
x
< RG
. (19)
1
Notice that
RG
> R
() xi > (c 1)(1 ) xRG : (20)
Hence, for reforms to be optimal we need that < M inf RS ; RG g, and xi > xRG .
(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 xi > xSG .
(ii) For public investments to be welfare improving, we need that > G , for public investment
to be preferred to reforms that > RG , and x < xSG for public investments to be preferred to
subsidies.
18
6.2 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 Wz . We start with the cascade, RSG .
6.2.1 RSG (z = 1).
R associated with reforms (R) under the cascade approach (z = 1) are given
The welfare gains W1
by
ZxP Z 1
R
W1 = (1 + xi + j c)d j dxi ; (21)
R
xg
M axf0;~
~
where x fx : R
= 1g = c 1 . Solving for (21) we have that
( 2
R 6 , if 3c 1 . When this is the case, the problem of the government is that of
19
maximizing
xP Z
Z
Sa2 R
W1 = (1 + xi + j s(c 1) c)d j dxi : (27)
0
c 1 s
Di¤erentiating (27) with respect to s, we have that
Sa2
@W1 (1 + 3 c)s2 1
= < 0 () > : (28)
@s 2 2 3c 1
Hence, > 3c1 1 =) s = 0.
Consider now the case > a, then the government would set s1 = c 1 i¤
xP Z
Z
Sa3 R
W1 js=c 1 = (1 + xi + j (c 1)2 c)d j dxi + (29)
0
c 1
c Z1 aZ
R
+ (1 + xi + j (c 1)2 c)d j dxi =
0
0
2 3c(c 1)(2c 3) + (1 3(2 c)c
> 0 () < b,
6
p
3(2 c)c 1 1+3c(3c 4)(c2 2)
with b 2 , and s1 = 0, otherwise.
We now consider the case > c 1. In this case, the the welfare associated with subsidies is
given by:
ZxP Z
Sb R (1 (1 3c))s2
W1 = (1 + xi + j s(c 1) c)d j dxi = : (30)
0 2 2
0
1
I¤ 3c 1 , (30) is increasing in s, and it is positive at s = c 1. Hence, we have s = c 1,
1
if 3c 1 , and s = 0 otherwise. Summarizing our …ndings about the optimal subsidy, we have
that 8 9
>
> s1 if < a , >
>
>
> >
>
>
> if < 1 =
>
> c 1, if a < b, 3c 1 ;
>
< if
>
s1 = >
> (31)
> if 1 ;
>
> 0, 3c 1 ; )
>
>
>
> c 1, if < 1
,
>
> 3c 1 if c 1:
: 0, if 1
,
3c 1
20
Putting all the pieces together, using (26), (29), and (30), we have that:
8 3 (c+1)(1 (1 3c)
9
>
> , if < a , ) >
>
>
> 24(2c 1) >
>
>
> +3c(c 1)(2c 3)+ (1 3(2 c)c
if < 1 =
>
> 6 , if a < b, 3c 1 ;
>
< if
>
S
W1 = >
>
> g if 1 ;
>
> 0,
> 3c 1 )
>
> (c 1)3 (1+ (1 3c)) 1
>
> 6 2 , if < 3c 1 ,
>
: 1
if c 1:
0, if 3c 1 ,
(32)
Let us now look at the welfare associated with public investments. Using (31), it is easy to verify
that
8 9
>
> c R1 s1 R >
>
>
> 1
(1 + c )d dx if < 1
&& < , >
>
>
> c 1 j j i 3c 1 a >
>
>
> 0 >
>
>
> >
>
>
> 0, if a < b, , >
=
>
>
>
> (1 R)( c 1) R < c 1;
>
> R >
< c 1 (1 + j c)d j dxi + >
>
G c 1 a >
>
W1 = otherwise, >
>
>
>
c R1 R 1 >
>
>
> (1 + c) d dx i , >
>
>
> c 1 j j ;
>
> 0
>
> )
>
> 0, if < 3c1 1 ,
>
>
>
> (1 R )(c 1) R c 1;
>
> 1
: R
c 1 (1 + j c)d j dxi if 3c 1 ,
0
(33)
where (1 )(c 1) = :f R
=c 1g . Now, working through the algebra we have that
8 9
> (2 c)2 (4c2 +1+ (6+ )c) 1
>
> 4(2c 1) , if < 3c 1 && < a , >
=
>
>
>
>
G (c 1)3 (1 )3 ;
W1 = , otherwise, (34)
>
> 6 2 )
>
> 0, if < 1
,
>
> 3c 1
c 1:
>
: (c 1)3 (1 )3 1
6 2 , if 3c 1 ,
R + WS + WG .
Finally, we have that, W1 = W1 1 1
6.2.2 RGS (z = 2)
As in the cascade, we have that the welfare associated with reforms is given by:
( 2
R 6 , if
>
>
< c
R
1 (1 + j c)d j dxi + c 1 (1 + j c)d j dxi if
>
(c 1)(1
R )R
>
: R
c 1 (1 + j c)d j dxi , if c 1;
0
or, solving, (
(2 c)2 (3+ (3 )c)
G 6 if 0. In addition, we have that
1
c 1 s2 < (1 )(c 1) () < ; (40)
3c 1
1
so that s2 = s2 , if < 3c 1 . Substituting s2 into (38), we have that
Sa (c 1)3 (3 (1 + c))(1 (1 3c))
W2 = : (41)
24(2c 1)
Assume now that the optimal subsidy s2 is such that c 1 s2 > (1 )(c 1). Then, it should
be the one that maximizes
xP Z
Z
R
Sb
W2 = (1 + xi + j s(c 1) c)d j dxi . (42)
0
c 1 s
Di¤erentiating (42) with respect to s, we have that
@W2Sb (1 + 3 c)s2 1
= 0 () ; (43)
@s 2 2 3c 1
22
1
which implies that s2 = 0 if 3c 1 . Hence, we have that
(
(c 1)3 (3 (1+c))(1 (1 3c)) 1
S 24(2c 1) , if < 3c 1 ,
W2 (44)
0, otherwise.
R + WG + WS .
Finally, we have that, W2 = W2 2 2
6.2.3 SRG (z = 3)
Starting with subsidies, the government has to maximize
xP Z
Z 1
Sa
W3 = (1 + xi + j s(c 1) c)d j dxi : (45)
0
c 1 s
Di¤erentiating (45) with respect to s, we have that a necessary and su¢ cient condition for an
internal maximum is
@W3 S 1
= 0 () s = s3 . (46a)
@s 2(2c 1)
p
Notice, further, that c 1 s3 > 0 () c > 3+4 5 ca 1:3. If c < ca , the government will put
subsidies in place if it derives positive utility with s = 1 c, that is, if
xP Z
Z 1
Sb 2c2 + 5c 3 3
W3 = (1 + xi + j (c 1)2 c)d j dxi = >0,c< ; (47)
0 2 2
0
which is always the case. Hence, we have that
(
1 c, if c < ca ;
s3 = (48)
s3 , if c ca ;
and ( (c 1)(3 2c)c
S 2 , if c < ca ,
W3 = 1 (49)
8(2c 1) , if c ca .
Let us now move to reforms. When c < ca , there is no space for reforms as subsidies completely
crowd out any other policy. When, instead, c ca , as always, we have to distinguish between the
case in which is smaller or larger than c 1. When < c 1, reforms occur if s3 < , that is,
1
> 2(2c 1) . Consequently, the utility associated with reforms is given by
23
8 o
>
> 0, if c < ca ;
>
>
> 9
>
>
> 0, if < 1
2(2c 1) , ) >
< >
>
R 1 s3 R
c R if
=
W3 = 1
(1 + xi + c)d j dxi , if 1
,
>
> j 2(2c 1) if c ca;
>
> c 1 R
>
>
> 1 s3 R
c R o >
>
>
> 1 >
;
: (1 + xi + j c)d j dxi , if c 1;
0 R
(50)
or, doing the algebra,
8 o
>
> 0, if c < ca ;
>
> 9
>
> )
>
< 0 1 >
>
if < , >
>
R
W3 = 2(2c 1)
if ((4c 2) 1)3 1
>
> 1)3
, if , if c ca ;
>
>
48 (2c 2(2c 1) o >
>
>
> , if c 1; >
>
: 48 (2c 1)3 ;
with (1 6c + 4c2 )(7 + 12 2 (1 2c)2 30c + 8c2 (7 2(3 c)c) + 6 (3 4c(3 2(2 c)c))).
Finally, let us now consider public investments. The associated utility is given by
8 o
>
> 0, if c < ca ;
>
> 9
>
>
>
> >
>
>
> 1 s3 R
c R 9 >
>
>
> 1 > >
>
> 1
> >
>
>
> c 1 (1 + j c)d j dxi if < 2(2c 1) , >
> >
>
>
>
> 0 >
> >
>
>
< c R
1 s3 R = >
>
R >
>
G
W3 = c 1 (1 + j c)d j dxi + if c 1 1 >
>
>
> if , >
> if c ca ;
>
>
c R1 R1 2(2c 1) >
> >
>
> ; >
>
> c 1 (1 + j c)d j dxi , >
>
>
> 0 >
>
>
> 1 s5 R
c R o >
>
>
> >
>
>
> c
R
1 (1 + j c)d j dxi , if c 1; >
>
>
> >
>
>
:
0 >
;
(52)
or, doing the algebra,
8 o
>
> 0, if c < ca ,
>
> 9
>
> )
>
< (2 c)2 (c2 6c+1) , 1 >
>
if < , >
>
G
W3 = 4(2c 1) 2(2c 1)
if
> , if 1
, if c ca ;
>
> (48 2 (2c 1)3 ))( 1+6 ( 1+c)( 1+2c) 2(2c 1) o >
>
> >
>
>
: 48 2 (2c 1)3 ( 1+6 ( 1+c)( 1+2c)
, if c 1; >
;
(53)
where,
1 6 (c 1)(2c 1) + 8 3 (2c 1)3 (3c 5) + 12(c 1)(2 c )2 (c(23 2c(9 2c) 7) ,
24
(1 6c + 4c2 )(7 30c + 8c2 (7 2(3 c)c) + 12 2 (1 3c + 2c2 )2 6 (1 c)(2c 1)(3 6c + 4c2 )).
Finally, we have that, W3 = W3 S + W R + W G.
3 3
6.2.4 SGR (z = 4)
As in the previous case, the welfare associated with subsidies is given by
( (c 1)(3 2c)c
S 2 , if c < ca ,
W4 = 1 (54)
8(2c 1) , if c ca .
Let us now consider public investments, welfare is given by
8
>
< 0, if c < ca ; ,
G 1 s3 R
c R
W4 1 2 (4c2 6c+1) (55)
>
: c 1 (1 + j c)d j dxi = (c 2)4(2 c 1) if c ca .
0
After subsidies and (eventually) public investments have been implemented, there is a residual
space for reforms i¤,
1
x: = (c 1)(1 ) = d , (56)
R 2 6c + 4c2
where (c 1)(1 ) = xi : 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 > s3 , hence
8 o
>
> 0 if c < ca ;
>
>
< )
R
W4 0 < d , (57a)
>
> c R
1 s5 Rc , if c ca ;
>
>
1
(1 + x + c)d j dxi ; , ,
: R
j d
(c 1)(1 )
or, doing the algebra,
8 o
>
> 0 if c < ca ;
< )
R
W4 0, < d , (58)
>
> , if c ca .
: (a(2 6c+4c2 ) 1)3
, ,
48 (2c 1)3 d
S + W G + W R.
Finally, we have that, W4 = W4 4 4
25
6.2.5 GRS (z = 5)
Starting with public investments, the associated welfare gains are given by
xP Z
Z 1
G (2 c)2 (c 1)
W5 = (1 + j c)d j dxi = : (59)
c 1 2
0
If the government implements reforms after public investments, the associated utility is given by
xP
Z Z c 1
R (c 1)3 2
W5 = (1 + xi + j c)d j dxi = : (60)
6
R
(1 )(c 1)
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
xP
Z Z
R
Sa
W5 = (1 + xi + j s(c 1) c)d j dxi + (61)
0
(1 )(c 1)
(1 Z)(c 1)Z
c 1
(1 + xi + j s(c 1) c)d j dxi .
0
c 1 s
Di¤erentiating (61) with respect to s, we have that
@W5Sa ((c 1) + 1)(c 1)
= 0 () s = s5 : (62)
@s 2(2c 1)
It is easy to verify that c 1 s5 > 0; it remains to verify that c 1 s5 < (1 )(c 1). It is
easy to show that
1
c 1 s5 < (1 )(c 1) () < .
3c 1
When, instead, c 1 s > (1 )(c 1), the problem is the same as in (42) and, again, we obtain
1
that s5 = 0 if 3c 1 . Hence, we have that
(
Sa = (c 1)3 (3 (1+c))((1 (1 3c)) 1
S W5 24(2c 1) , if < 3c 1 ,
W5 1
(63)
0, if 3c 1 .
G + WR + WS .
Finally, we have that, W5 = W5 5 5
26
6.2.6 GSR (z = 6)
Starting with public investments, as in the previous case, the associated welfare gains are given by
Z Z 1
G (2 c)2 (c 1)
W6 = (1 + j c)d j dxi = : (64)
c 1 2
0
Let us now move to subsidies, the government has to maximize
xP Z
Z c 1
S
W6 = (1 + xi + j s(c 1) c)d j dxi : (65)
0
c 1 s
Di¤erentiating (65) with respect to s, a necessary and su¢ cient condition for an internal maximum
is
@W1 S c 1
= 0 () s = s6 ; (66)
@s 2(2c 1)
and substituting this expression in (64), we have that
S (c 1)3
W6 = : (67)
8(2c 1)
Let us now consider reforms. After public investment and subsidies, there is a residual space for
reforms if, and only if,
1
x: R
= (c 1)(1 ) if < 2(2c 1) ,
R 1 s6
c R
W6 Rc 1 (c 1)3 ((4c 2) 1)3 1
(69)
>
: (1 + xi + j c)d j dxi = 48 (2c 1) , if 2(2c 1) .
R
(1 )(c 1)
G + W S + W R.
Finally, we have that, W6 = W6 6 6
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
b is such that
allocation of reforms and public investments at the later stage. If the optimal subsidy s
(c 1 s b >1 ), where 1 = xi : RG = 1, and thus in the interval [1 ;c 1 s b] reforms
are always preferred to public investments, and the optimal subsidy is the one that maximizes
27
(c 1)(1
Z )Z
1
Wb
3
= (1i + j c)d j dxi +
c 1
0
Z
1 Z 1 Z
1 Z
RG
+ (1i + j c) c)d j dxi + (1 + xi + j c)d j dxi + (70)
RG R
(c 1)(1 ) (c 1)(1 )
c Z1 sZ xP Z
Z
1 1
+ (1 + xi + j c)d j dxi + (1 + xi + j s(c 1) c)d j dxi :
R
0
(1 ) c 1 s
Notice that a necessary condition for c 1 b >1
s to hold is that >2 c. Di¤erentiating
(70) with respect to s, we have that
a
@Wb p
3
ba =
= 0 () s 2 (c 1) + (1 + (3 + 4(c 2)c): (71)
@s
p
2+(5 2 ) + (1 ) (1+4(2 ) )
In addition, we can show that c 1 ba > 1
s () c > 4 +1 cb .
ba in (70), we obtain
Substituting s
a 1 p
Wb
3
= (((2 c)2 (2c 1) + 2 (2c 1)(7 2c(5 2c)) 2 3 =2
(3 2(2 c)c) +
6
p p p p p p p
+2 (1 + 3S 3c ) + (12 4 c(24 c(3 c + 12 )))) ,
with 1 + (3 4(2 c)c).
Assume now that c < cb , so that (c 1)(1 ) (c
s 1)(1 ) () c > 1+4 .
28
bb in (72), we obtain
Substituting s
b 4
p p
Wb
3
= 1=6( 2 1)3 + 2 3 (1 3c + 4c3 ) + (c 1)(3(c 2)2 + 6
(2c 2c )+ (74)
2
p p p p
2 (2c 1)(4 c(4 c 2 )) (2 c(9 + 3(c 4)c + 4 )));
with (1 ) (c(4 + 2 c) 3 2 (2c 1)2 ). Let now consider the case in which c < 3+2 ,
1+4
so that public investments are always preferred to reforms because of (20). Total welfare is given
by
c Z1 sZ ZxP Z
1 1
c
Wb3
= (1i + j c)d j dx i + 1 + xi + j s(c 1) c)d j dxi : (75)
c 1 0
0 c 1 s
Di¤erentiating (75) with respect to s, we have that
@W (3 c)(c 1)
= 0 () s = bc :
s (76)
@s 4c 2
It is also easy to verify that c 1 bc in (76) we obtain
bc > 0. Substituting s
s
c (c 1)(c(63 + c( 43 + 9c)) 25)
Wb
3
= : (77)
8(2c 1)
Summarizing our …ndings, we have that:
8 )
> c , 3+2
< Wb
> 3
if c < 1+4 ,
if c < cb ;
c b , 3+2
Wb = Wb if c c 1+4 ,
3 > 3 o
>
: W ,
b
3
if c cb .
6.4 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 = xP , the welfare changes
associated with having reforms, instead of subsidies, is given by
xP Z
Z xP Z
Z
1 1
DWSR = (1 + xi + j s(c 1) c)d j dxi (1 + xi + j c)d j dxi . (78)
0 R
xP s xP s
Di¤erentiating (78) with respect to xi , we have that
@DWSR s(2(c 1 xi ) + s)
= > 0: (79)
@xi 2
29
Thus, a necessary condition for DWSR > 0 is that DWSR jxi =c 1 > 0. In addition, we have that
@DWSR 1 s2
jx=c 1 = (1 4(c 1)s ), (80)
@s 2
and
@DWSR 1
Lim jx=c 1 = >0 (81)
s!0 @s 2
so that a small subsidy strictly improves welfare, notwithstanding the fact that it crowds out
reforms. Hence, for small values of , we necessarily have that SGR RSG and SRG RGS .
Also, since SGR = SRG, we necessarily have that SGR = SRG M axfRSG; RGS g.
The last step is to show that SGR GSR or, since reforms are never implemented for small
enough values of , that SG GS . From (54), (55), (64), (67) after some algebra, we obtain that:
( (c 1)(c(43 24c) 15
p
3+ 5
8(2c 2) , if c < 4 ;
W (SG) W (GS ) = (2 c)(3 c c2 )
p
3+ 5
(82)
8(2c 2) , if c 4 ;
expression that is positive for c 2 (1; 2) .
(ii) Let us now consider the case ! 1. Now, there are no costs associated with reforms,
and they can …nance the entire infrastructure gap. Consequently, we have that W (RGS ) =
W j !1 (RSG) is necessarily the best choice.
(iii) For intermediate values of , let us take =1
2 . In this case
((c 1)( 15+c(46+c( 35+8c))))
W !1=2 (GSR) = 8(2 c) ;
1
W !1=2 (GRS ) = 24( 1+c)(49+c( 50+13c)) ;
(
(c 1)(14+c( 16+5c))
12 , c< 3
2;
W !1=2 (RSG) = ( 55+2c(48+c( 27+5c)))
24 , c 32;
(
(c 1)(14+c( 16+5c))
12 , c< 3
2;
W !1=2 (RGS ) = ( 55+2c(48+c( 27+5c)))
24 , c 32; (83)
8 p
> (c 1)c(2c 3) 3+ 5
>
< 2 , c< 4 ; p
11+2c( 51+2c(84+c( 135+2c(57+4( 6+c)c)))) 3 3+ 5
W !1=2 (SRG) = 12(2c 1)3
, 2 >c 4 ;
>
>
: ( 55+3=(1 2c)2 1=( 1+2c)3 +2c(48+c( 27+5c)))
, 3
c 2;
24
8 p
> (c 1)c(2c 3) 3+ 5
>
< 2 , c< 4 ; p
9+2c( 28+c(41 22c+4c2 )) 3 3+ 5
W !1=2 (SGR) = 8(2 c) , 2 >c 4 ;
>
> 27+c( 276+c(1026+c( 1815+2c(795 330c+52c2 ))))
: , 3
c 2.
24(2c 1)3
30
By plotting the di¤erent expressions, it is immediate to verify that the optimal myopic
p
sequenc-
3+ 5
ing is SRG or SGR (which are identical), when the cost of resources c is low (c < 4 ), SGRfor
intermediate values of c, and SRG when c is large.
6.5 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
Zxi Z 1 Zxi Z 1
DWSG = (1 + xi + j s(c 1) c)d j dxi (1 + j c)d j dxi . (84)
0 c 1
xi s xi s
Di¤erentiating (84) with respect to xi , we have that
@DWSG
= s > 0: (85)
@xi
Thus, a necessary condition for subsidy to dominate public investments is that DWSG jxi =c 1 > 0.
In addition, we have that
@DWSG 2s 3 + c(4(1 s) c)
jx=c 1 = , (86)
@s 2
and
@DWSG (3 c)(c 1)
Lim jx=c 1 = ; (87)
s!0 @s 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.
31
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