Regional state capacity and the optimaldegree of fiscal decentralization∗

Andres Bellofatto† Martin Besfamille‡

November 3, 2014

Abstract

This paper presents a model featuring a central government and local authorities,the latter being characterized by the same level of administrative and fiscal capacity.We analyze two fiscal regimes are analyzed. Under partial decentralization, regionalgovernments invest in a local public project. Depending upon their administrativecapacity, projects may remain incomplete. Under this institutional regime, regionalgovernments can rely on central bailouts to finalize them, and thus face soft budgetconstraints. Anticipating this, they may inefficiently over invest. Under full decentral-ization, regional governments are never rescued, and so face hard budget constraints.This generates the opposite type of inefficiency: regional governments under invest.The first goal of the paper is to assess which regime dominates. The second goal isto investigate how different levels of regional state capacity affect the normative com-parison between regimes. As expected, when the regional fiscal capacity is low, partialdecentralization dominates; otherwise full decentralization may dominate. But, con-trary to the common wisdom, we find that full decentralization is the preferred regimewhen regional administrative capacity is low.

Keywords: Fiscal federalism - Partial and full fiscal decentralization - Soft andhard budget constraints - State capacity.

JEL Classification: D82 - H77.

∗We thank L. Arozamena, P. Dal Bo, A. de Paula, J. Dubra, E. Espino, I. Fainmesser, N. Figueroa, F.Forges, M. Gonzalez Eiras, B. Guimaraes, C. Hevia, B. Knight, J. C. Hallak, B. Lockwood, A. Manelli,E. Mattos, P. Mongin, A. Neumeyer, E. Ornelas, H. Piffano, F. Piguillem, A. Porto, E. Rezk, C. Ruzzier,R. Serrano, W. Sosa Escudero, M. Tommasi, F. Weinschelbaum, attendants to the Annual Congress of theAssociation for Public Economic Theory (Seattle, 2014), Latin American Meeting of the Econometric Society(Santiago de Chile, 2011), Jornadas Internacionales de Finanzas Publicas (Cordoba, 2010) and ReunionAnual de la Asociacion Argentina de Economıa Polıtica (Buenos Aires, 2010), and seminar participantsat Brown, EESP-FGV, San Andres and Pontificia Universidad Catolica de Chile for useful comments andsuggestions. We are also grateful to S. Briole for his excellent research assistance. The usual disclaimerapplies.†Tepper School of Business, Carnegie Mellon University. E-mail: andresb@cmu.edu‡Corresponding author. Instituto de Economıa, Pontificia Universidad Catolica de Chile. E-mail:

mbesfamille@uc.cl

1

1 Introduction

In many countries, tax and expenditure assignments to subnational or regional governmentsare not balanced. In particular, regional governments are often in charge of delivering localpublic services, but cannot raise the needed revenues to finance these expenditures. Thissituation is known as “vertical fiscal imbalance”, and is rather prevalent in advanced and indeveloping economies. According to Eyraud and Lusinyan (2013), across OECD countries,the average share of subnational government expenditure not financed through own revenueshas been 40 percent between 1995 and 2005, with some countries having much higher sharesthan this (e.g., Belgium and Mexico, with 60 and 83 percent, respectively). Corbacho et al.(2013) document that vertical fiscal imbalances in Latin America are the highest among thedeveloping nations.

In practice, vertical fiscal imbalances are often solved by centrally provided transfers toregional governments (Boadway and Shah (2007)). In such a institutional setting, defined byBrueckner (2009) as “partial decentralization”, regional governments may face soft budgetconstraints.1 Wildasin (1997) and Goodspeed (2002) show that the willingness of the centralgovernment to bail out regional governments creates a negative externality if, as is usual,the cost of a bailout is met through increases in national taxes. As other regions willpartially finance one region’s bailout, this induces excessive expending or borrowing initially.Among others, Pettersson-Lidbom (2010) confirms this theoretical result, by estimating than,between 1979 and 1992, Swedish local governments increase their debt by more than 20percent when they expected to receive future bailouts.

In the economic and policy literature, it is widely accepted that, because of this common-pool fiscal externality, regional governments have to face hard budget constraints which, ac-cording to Weingast (2009), provide “local political officials with incentives for prudent fiscalmanagement of their jurisdiction.” Indeed, much of the literature, as referenced in Roddenet al. (2003) and Oates (2005), is concerned with the design of institutional mechanismsaimed to harden regional government’s budget constraints. One such mechanism that hasattracted quite a lot of attention is complete decentralization of taxing powers to subnationalgovernments. Qian and Roland (1998) argue that full fiscal decentralization creates tax com-petition, which in turn raises the perceived marginal cost of public funds at the regional level.This increase endogenously “hardens” regional governments’ budget constraints, and makesthem to be reluctant to bailout inefficient projects undertaken by public enterprises. Thisformal result is at the basis of the idea that “federalism preserves markets” (see Weingast(1995, 2009), Montinola et al. (1995) and Qian and Weingast (1997), among others), and hasalso been adopted by international organizations as one of the main arguments to activelypromote fiscal decentralization reforms (World Bank, (2000)).

But this conventional wisdom has been challenged on two different grounds. From atheoretical point of view, Besfamille and Lockwood (2008) show that Qian and Roland’s

1According to Kornai et al. (2004), “A budget-constrained organization faces a hard budget constraintas long as it does not receive support from other organizations to cover its deficit and is obliged to reduceor cease its activity if the deficit persists. The soft budget constraint phenomenon occurs if one or moresupporting organizations are ready to cover all or part of the deficit.”

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(1998) results were somewhat restrictive.2 In a model where regional governments exert effortto carry out a local project, they prove that, under more general parameter configurationsthan those adopted by Qian and Roland, hard budget constraints can be inefficient becausethey trigger an excessive level of effort, which in turn generates inefficient under provision ofprojects. This implies that a normative comparison between partial and full decentralization,along the abovementioned trade-offs, deserves to be undertaken.

Other authors, like Prud’homme (1995) and Bardhan and Mookherjee (2006), point outthat this pro-decentralization literature ignores relevant local, institutional characteristics.One of these characteristics is the level of state capacity, defined by Besley and Persson (2010)as the “state’s ability to implement a range of policies”. Many qualitative reviews of decen-tralization reforms (e.g., Bird (1995), Litvack et al. (1998)) and more recent quantitativeevaluations of such processes (Loayza et al. (2014)) show that when regional governmentslack these abilities, the benefits accruing from a decentralized form of government are farfrom granted. Moreover, Cabrero and Martınez-Vazquez (2000) even affirm that an ade-quate administrative capacity at the subnational level is a prerequisite for any successfuldecentralization reform. These assertions seem to suggest that one cannot compare partialand full decentralization without explicitly incorporating regional state capacities into themodel.

To our knowledge, there is no contribution to the local public finance literature thataddresses these issues theoretically. The goal of this paper is precisely to build a modelthat allows to compare, from a normative point of view, a partially and a fully decentralizedregime, and to determine how this comparison evolves with the level of regional state capacity.

In our model, regional governments decide whether to provide a discrete, local publicgood or project. The project’s initial cost is covered by their financial resources. If theproject is initiated, it is carried out by regional bureaucracies, which are characterized bya given level of administrative capacity to fulfill this task. This administrative capacity isformalized as the probability that the project’s execution ends in due time and generates asocial benefit greater than the initial cost. With the complementary probability, the project’sconstruction lasts more, and needs a second round of financing to be completed. In this lastcase, the project also generates a social benefit to the region, but, due to its delay, lower thanwhen the project is carried out on time. The model is symmetric: all governments possessthe same level of state capacity, and all projects’ costs and benefits are identical. Despitethis, depending upon the project’s execution length, outcomes can be different ex post.

In the partially decentralized regime, no regional government has tax revenues to refinanceits incomplete project. But the central government can bailout any region, via a uniformnational tax on local capital, which is imperfectly mobile. We show central bailouts do notprovide appropriate incentives for efficient initial investment. So, partial decentralizationmay lead to inefficient overprovision of public projects.

Next, we analyze the fully decentralized regime, where regional governments have to

2Although Rodden and Rose-Ackerman (1997) also argue against the predictions of Montinola et al.(1995), they do not criticize neither the logic nor the results of this model per se, but rather the fact thatits assumptions are unrealistic.

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refinance incomplete projects through a tax on capital invested in their jurisdiction, in acontext of tax competition. In this case, we assume that regional governments can collectonly a fraction of their potential tax base. This fraction describes the regional fiscal capacity,a second dimension of state capability. First, we obtain the refinancing and taxing equilibria.When all regions face an incomplete project, they refinance it with the same tax rate but, ifit is the case, bearing an implicit cost due to imperfect fiscal capacity. When there is at leastone region that does not need to refinance, the other regional governments are compelledto raise taxes that end-up being distortionary because they trigger costly capital outflows.Therefore, for some parameter values of the model, regional governments may decide not torefinance their incomplete project: an endogenous hard budget constraint, as in Qian andRoland (1998). Moving back to the initiation decision, we find the opposite inefficiency thanunder partial decentralization. Imperfect regional fiscal capacity, distortionary taxation orendogenous hard budget constraints may underincentivize regional governments to initiallyinvest, and thus the fully decentralized regime can generate inefficient underprovision ofprojects.

Then, for different pairs of administrative and fiscal capacity levels, we compare the ex-pected welfare of both institutional regimes. The main results are the following. When thelevel of regional fiscal capacity that prevails in the federation is low, refinancing incompleteprojects under full decentralization turns out to be too costly. Therefore, regardless of thelevel of regional administrative capacity, partial decentralization dominates. For higher levelsof the regional fiscal capacity, full decentralization dominates, provided the level of regionaladministrative capability is low. Although this entails that projects remain incomplete al-most surely, it also implies that distortionary refinancing under full decentralization is lesslikely. Thus, full decentralization’s distortions cost less, in welfare terms, than inefficientinvestments made under partial decentralization.

Finally, we evaluate the model’s robustness in two different ways. First, we undertakesome comparative statics exercises. When the regional capital stock (the project’s highestbenefit) increases, full (partial) decentralization dominates more often. When the number ofregions increases, each regime starts to be preferred in a parameter configuration of the modelwhere the other one used to be the best. Second, we extend the model to include distortionarynational taxation. This implies ipso facto that the central government does not bailout allincomplete projects, which corresponds to another endogenous hard budget constraint. Weshow that this extension modifies significantly some outcomes under partial decentralization,in particular less projects are inefficiently initiated, but inefficient underinvestment may alsoemerge. All this decreases the expected welfare under partial decentralization, and thus fulldecentralization is preferred more often.

The layout of the remainder of the paper is as follows. Section 2 presents the model,and Section 3 describes the efficient benchmark. In Section 4 we analyze outcomes underpartial decentralization, whereas Section 5 studies the equilibrium under full decentralization.Section 6 contains the normative comparison between partial and full decentralization, andsome results on its robustness. Section 7 discusses related literature. Finally, Section 8concludes. The main proofs are shown in the Appendix, and supplementary material appearsin an Online Appendix.

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2 The model

2.1 Preliminaries

The model has three periods t = {1, 2, 3} and L ≥ 2 regions. Each region ` ∈ {1, ..., L} hasa continuum of measure 1 of risk-neutral residents, each of whom has an endowment κ ofcapital. In the last period, each resident derives utility from consumption of a numerairegood. This good is produced in every region from the capital input by competitive firms usinga constant-returns technology, where units are chosen so that one unit of capital producesone unit of output. Following Persson and Tabellini (1992), we assume that capital is mobilebetween regions, but at a cost: specifically, a resident of one region that exports f units ofcapital to other region(s) incurs a mobility cost f 2/2. As we explain below, residents mayalso benefit from a discrete public good, or project.

There are two levels of government: central and regional. Throughout the paper, weassume that both levels of government are benevolent, i.e., they maximize the sum (oraverage) of utilities of their jurisdictions’ residents over the two last periods. For simplicity,we assume that there is no discounting of future payoffs.

2.2 Timing

The order of events, and other relevant features of the model in more detail, are as follows.

B

b

0Termination

Refinancing

t = 2 t = 3

π

π1

investment

Initial

t = 1

regime nalinstitutio

of Choice

nrealizatio

Cost

gundertakin

sProject'

Figure 1: Timing of the model

At t = 1, a political body (e.g., a Congress) chooses the institutional regime that will ruleall fiscal interactions between the central and regional governments.

At the beginning of t = 2, Nature draws projects’ cost c according to a strictly positiveprobability density function h(c) on [0, b]. Then, regional governments choose whether toinitiate a project in their region. We denote this decision by i` ∈ {I,NI}, where I (NI)stands for initiation (not initiation). Regional governments have just enough resources tofund the initial investment c.

5

If initiated, a project is carried out by the regional bureaucracy, which possesses a level ofadministrative competence, or capacity, to fulfill this task. Given the level of administrativecapacity, a project yields different outcomes. With an exogenous probability π ∈ [0, 1], whichwe interpret as the regional level of administrative capacity, a project generates, at the endof the current period, a non-financial benefit B > 0 for all residents of the region. On theother hand, with probability (1 − π), the project remains incomplete and yields no benefitduring this period.3 These realized outcomes are observed by everybody.

At t = 3, central or regional governments, depending on the institutional regime in place, de-cide whether to shutdown or continue incomplete projects. In the last case, to be completed,a project requires an additional input of c of the consumption good.Under partial decentralization (PD), the central government decides on refinancing incom-plete projects, through a uniform tax τ on capital, collected by the national tax authority.Uniformity implies that, for (non-modeled) constitutional reasons, the central governmentcan neither set different tax rates contingent on which regional government has asked foradditional funds, nor make side-payments to any specific regional government.Under full decentralization (FD), each regional government decides whether to refinance itsincomplete project, using a per unit tax on capital invested in its region, at a rate τ`. Aftertaxes are set, capital owners decide to invest in the region(s) with the highest net return(s)and production takes place. Then, regional governments raise their taxes. Due to costlytax enforcement/collection or to regional governments’ inability to compel individuals topay their due taxes, regional net fiscal revenues end up being a fraction θ ∈ [0, 1] of theirpotential tax base. From now on, the parameter θ will characterize regions’ fiscal capacity,the second dimension of the state capabilities.Upon completion, the project generates a non-financial benefit b for all residents of theregion. We assume that B/2 ≤ b < B.4

2.3 Discussion

Some features of the model deserve further comments. First, we have adopted Mann’s (1984)general definition of state capacity as the infrastructural power of the state to enforce policywithin its territory, and followed Snyder (2001) and Ziblatt (2008) to apply this concept toregional governments. Specifically, we focus on the administrative and extractive dimensionsof state capacity, as defined by Hanson and Sigman (2013). First, we consider the probabilityπ to be an appropriate proxy of the region’s capacity to carry out projects in due time.5

As B > b, the higher is this probability, the higher is the regional administrative capacity.

3Patil et al. (2013) document that public projects’ delays are pervasive in Indian states, and that theirmain cause is administrative problems that arise during the land acquisition process.

4The difference between B and b reflects the negative impact of delaying the project’s construction onthe regional welfare. For example, an incomplete park may affect pedestrian movement. But the benefit’sdecrease due to the delay is lower than 50 percent of its value.

5This measure of administrative capacity is somewhat related to the “infrastructure reform” ability,quantified by Fortin (2010) in her study of state capacity in post-communist countries.

6

Regarding the second dimension of the regional state capacity, our characterization of fiscalcapability follows Arbetman-Rabinowitz et al. (2012) and Gadenne and Singhal (2013).Second, we consider a discrete, regional public project, instead of a continuous public good, asis common in most of the literature on tax competition.6 This has the following consequences.The project’s indivisibility fixes the type of competition between regions. As in Wildasin(1988), here regions compete first in refinancing decisions, and then taxes are set accordingly,in a context of imperfect capital mobility.7 Moreover, this assumption, combined with thecharacterization of administrative competence as a probability of completing projects ontime, is a simple way to analyze, via refinancing decisions, the interaction between levels ofregional state capacity and different intergovernmental fiscal arrangements.Third, we assume that, across regions, governments and projects are ex ante and interimidentical. All regional governments share the same exogenous levels of state capacity π andθ. Concerning projects, ex ante (i.e., in period 1) they are all characterized by the sameconfiguration of exogenous social benefits b and B, and by the same probability densityfunction h(c). Interim (i.e., at the beginning of period 2), the cost c is realized and appliesfor all projects in all regions. But projects can be different ex post. Indeed, at the end ofperiod 2, when π ∈ (0, 1), some regions that have initially invested end up with completeprojects, while the other face incomplete ones, that might or might not be finalized at theend of period 3. In the Conclusions, we comment more on this important feature of thismodel.Finally, we assume that c ≤ b. When c > b, both institutional regimes generate the same out-come. Therefore, we prefer to consider a parameter configuration of the model under whichthe institutional comparison between partial and full decentralization is indeed relevant.

3 First best outcomes

In order to have a benchmark, consider a social planner who makes all decisions, but cannotanticipate whether a project will be completed at the end of t = 2 (i.e., he has to carry outprojects through the regional bureaucracies). We solve his decision problem backwards.

First, as individual utilities are linear in income and the planner maximizes the sumof utilities, the refinancing decision in any region is independent of his choice in any otherregion. Thus, in any region, continuing an incomplete project is always optimal becausec ≤ b.

Moving back to the initial investment decision, the planner faces again a separable prob-lem between regions. Knowing the cost c, he initiates projects provided their expected benefit

6To the best of our knowledge, the unique contribution to the local public finance literature that dealswith discrete projects is Cremer et al. (1997). In a two-region model, the authors compare the optimal andthe (Nash) equilibrium provision of projects that can be used by residents in both regions. Although theydo not assume distortionary local taxation to finance the project (as we do), they find equilibrium projects’under provision, as expected in a model with spillovers.

7Akai and Sato (2008) and Kothenburger (2011) also analyze models with this timing, but instead considerincome or wage taxation.

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is higher than their expected cost (which includes a possible second round of financing). Let’sdenote by

c∗(π) ≡ πB + (1− π)b

2− πthe cost that makes the expected, net regional welfare from initiating a project equal tozero. Hence, if c ≤ c∗(π), initial investment is efficient in any region `. When c > c∗(π), notinvesting is the optimal decision.For a given configuration of parameters (b, B), efficient investment decisions are depicted inthe following figure. There, any point in the (π, c) plane represents a project, that costs c,in a region with administrative capacity π.

b

c

π

*I

2/b

10

)(πc

Figure 2: First best outcomes

*NI

𝜋∗

When 0 ≤ π ≤ π∗ ≡ b/B, 0 < c∗(π) ≤ b. Therefore, there exists a non-empty area NI∗,delimited by the thick curve c∗(π), where projects are optimally not initiated. Bellow thiscurve, in the area I∗, it is efficient to undertake all projects therein.Ceteris paribus, as π increases, the projects’ expected, net benefit increases. Thus c∗(π)increases as well. Therefore, when π∗ < π ≤ 1, c∗(π) > b. As regional administrativecapacities are relatively high, all projects are efficiently initiated.

4 Partial decentralization

Under this institutional framework, again incomplete projects are always refinanced by thecentral government because c ≤ b. Therefore, regional welfares may be a priori interdepen-dent, via the central government’s budget constraint. Given this, there is a simultaneousgame between regions at the beginning of the second period, when they decide on initialinvestment.

At this time, region `’s expected welfare is

EW PD` (i`, im) = κ(1− τ e) + 1I{i`=I}[πB + (1− π)b− c], (1)

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where im is the profile of investment decisions chosen by regions m 6= `, 1I{i`=I} is an indicatorfunction that takes the value of 1 if region ` has initiated the project and 0 otherwise, andτ e is the expected tax.

What is the value of the expected tax τ e? At the end of the second period, all projects’outcomes are realized. Let ω be a profile of outcomes, and let’s denote by #B(ω) the numberof completed projects in this particular realization of outcomes. For any profile ω, the centralgovernment mechanically sets a tax τω to cover, at the beginning of the third period, the costof refinancing L−#B(ω) incomplete projects. As this tax is uniform, and exporting capitalis costly, every household will invest in its own region. This implies that the tax base isLκ, and taxation is non-distortionary. Hence, under the profile ω, the central government’sbudget constraint is

τω.Lκ = [L−#B(ω)]c.

Therefore, when deciding on initial investment before the realization of projects’ outcomes,each region faces the expected tax τ e, which satisfies

τ e.Lκ =

(L∑`=1

1I{i`=I}(1− π)

).c, (2)

where the term in brackets gives the expected number of bailouts. Substituting (2) into (1)and rearranging, we obtain

EW PD` (i`, im) = κ+ 1I{i`=I}

[πB + (1− π)(b− c

L)− c

]−

(∑m6=`

1I{im=I}(1− π)

)· cL

(3)

By inspection of (3), the effect of i` on EW PD` - measured by the term in square brackets -

is independent im. So, we can analyze the choice of i` just for a representative region `.Notice that each region only pays 1/L of the cost of refinancing its incomplete project,

as this cost is shared through national taxation. Therefore, the central government’s budgetconstraint generates a common-pool fiscal externality : any resident of ` is negatively affectedby the possibility of an incomplete project in a region m 6= `.

Let

cPD(π) ≡ L[πB + (1− π)b]

L+ 1− πbe the cost that makes the expected, net regional welfare from initiating the project underpartial decentralization equal to zero. The next proposition completely characterizes regionalproject initiation decisions under this institutional regime.

Proposition 1

If c ≤ cPD(π), initial investment occurs. Otherwise, there is no initial investment in anyregion.

In the Appendix we show that cPD(π) > c∗(π). Hence, we can establish the form of theinefficiencies that emerge under this institutional regime, as follows.

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Corollary 1

Under partial decentralization, equilibrium outcomes can be inefficient. When this is so,initial investments are made when it is inefficient to do so.

Inefficiencies involve over investment. This kind of inefficiency, driven by the common-poolfiscal externality, is well known [See Wildasin (1997) and Goodspeed (2002)]. The followingfigure depicts equilibrium outcomes that emerge under partial decentralization.

b

c

π

PDI

2/b

10

)(πc

Figure 3: Partial decentralization outcomes

PDNI)(πcPD

1/ LLb

𝜋2𝑃𝐷𝜋1

𝑃𝐷

When 0 ≤ π < πPD1 ≡ bL(B−b)+b , c

PD(π) < b. Therefore, there exists a non-empty area

NIPD, delimited by the thick curve representing cPD(π), where projects are efficiently notinitiated in any region. Bellow this curve, in the area denoted by IPD, all projects areinitiated. There, in the white area, regional decisions are optimal and so, in equilibrium,each region expects to contribute an amount equal to the cost of refinancing by itself itsincomplete project. Thus, interim regional expected welfares coincide with the first bestlevel. But this is not always the case. In the shaded area, when c ∈ [c∗(π), cPD(π)], theinefficient investment decision is adopted in equilibrium by all regions.

As π increases, cPD(π) increases as well. Thus, when πPD1 ≤ π ≤ πPD2 ≡ b/B, c∗(π) ≤b ≤ cPD(π), which implies that all projects are initiated in equilibrium. Still, projects in theshaded area, whose cost satisfies c∗(π) ≤ c ≤ b, are inefficiently initiated.

Finally, when πPD2 < π ≤ 1, b < c∗(π) < cPD(π). This implies that inefficient investmentscannot emerge because the model is biased towards project initiation. Under these parameterconditions, partial decentralization replicates the first best outcome.

5 Full decentralization

In this section, we analyze the case of full decentralization, when regional governments decidenot only about initial investment, but also if they refinance their incomplete project, taxing

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capital employed in their region, in a context of imperfect capital mobility. Therefore, thereis now a three-stage game between regions, where they play simultaneously at each stage.As before, the game is solved backwards.

5.1 Capital flows

Given a profile of tax rates τ = {τ1, ..., τL} set by all regions, a household resident in region` decides where to invest its capital endowment. Let’s denote by f`m the amount of capitalthat this household invests in a region m 6= `, and by M the set of regions m 6= ` that havechosen the minimum tax rate τ` = min{τm}m6=`. The following proposition characterizes thehousehold’s investment decision.

Proposition 2

If τ` ≥ τ`,∑m∈M

f`m = τ` − τ` ≥ 0. Otherwise, f`m = 0.

As a household in region ` seeks to maximize net returns from its investments, its portfoliodecision just depends upon the comparison between τ` and τ`, and not between τ` and thewhole profile of tax rates chosen by regions m 6= `. As expected, a household in region `invests “abroad” in a region m ∈ M provided τ` > τ`. When only one region has set theminimum tax rate τ`, all capital that leaves region ` goes there. But if two or more regionshave chosen the same tax rate τ`, the value of the capital outflow f`m that goes to each ofthem is not determined: only the total amount of capital that leaves region ` is characterizedby the expression shown in the proposition, expression whose intuition is straightforward.Capital leaves region ` until the marginal saving in taxes equals the marginal mobility cost.As expected, this flow increases with the difference τ` − τ`.When the regional tax rate τ` is lower or equal than τ`, there is no capital outflow from region`: its residents invest all their endowment “at home”. Moreover, in this case, region ` receivescapital inflows from other regions. But this does not benefit directly its residents becausereturns from these investments are consumed abroad, by residents in regions m 6= `, m.Despite this fact, we will see in the next section that these capital inflows will have animportant role to play in the determination of the equilibrium tax rates.

5.2 Equilibrium in tax rates

Anticipating households’ portfolio decisions and taking other regional tax rates as given, eachregional government sets its tax rate to raise enough resources to refinance its incompleteproject, if it has decided to do so.

To obtain the equilibrium tax rates, in the Appendix we derive region `’s reaction functionτ` (τm), where τm denotes the profile of tax rates chosen by regions m 6= `. We have justseen that, depending upon the whole profile of tax rates, capital may leave region ` or maycome there, from other regions. Despite these different possibilities, region `’s after-tax,private-good consumption monotonically decreases with τ`, and so does the regional welfare.Hence, for any profile τm, the tax rate chosen in region ` should be the lowest tax rate that,

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given its fiscal capacity and the capital invested there, enables the regional government toraise c.

The reaction function τ` (τm) is build around the value c/θκ, which is the tax rate that aregional government would choose in autarchy. Moreover, τ` (τm) “decreases” in the followingsense.When the profile τm is such that all tax rates τm < c/θκ, region `’s optimal response isto tax strictly above the minimum tax τ`. Despite the fact that this decision will trigger acapital outflow, this is the only way to ensure the project’s refinancing. When all tax ratesτm are set equal to c/θκ, region `’s optimal response is to replicate this level. Due to theway we model imperfect capital mobility, the tax collection’s elasticity with respect to τ` islower than one. This, combined with the fact that region ` needs to collect enough revenuesto refinance its incomplete project, makes tax undercutting not a profitable deviation in thisparticular case. Finally, when the profile of tax rates is such that τ` > c/θκ or τ` = c/θκ butat least one region n 6= ` has chosen a tax rate τn > c/θκ, region `’s optimal response is totax strictly below τ`. This decision generates an inflow of capital that enables the governmentto raise sufficient revenues to pursue its incomplete project, moderating the tax burden onits residents.Another important feature of region `’s reaction function is that it is non-continuous. Whenτ` converges to c/θκ from below, τ` (τm) converges to this limit from above. But whenthe distribution of taxes is such that τ` = c/θκ but at least one region n 6= ` has chosenτn > c/θκ, tax undercutting is a profitable deviation in this case. Despite this fact, we cancharacterize the Nash equilibrium of this subgame, as follows.

Proposition 3

When all regions have decided to refinance their incomplete project, the unique symmetricNash equilibrium in pure strategies is characterized by τ` = c/θκ.When there is at least one region that does not refinance, region ` has to set the tax rate

τ` (0) =1

2

[κ−

√κ2 − (4c/θ)

]to refinance its incomplete project.

Consider the tax competition subgame that emerges when all regions have decided to refi-nance their incomplete project. The equilibrium tax rate τ` increases with the project cost c.Moreover, as in equilibrium nobody invests abroad, regions tax their own capital endowmentwithout bearing any deadweight loss due to its mobility. Thus, the higher this endowment,the lower the equilibrium tax rate. Similarly, the higher the fiscal capacity θ, the lower theequilibrium tax τ`. In this case, in equilibrium, region `’s welfare (net of the initial cost c)is W FD

` = κ + b− cθ. When θ < 1, imperfect fiscal capacity implies that regions do not pay

only the technical cost of completing the project c, but a higher, effective refinancing costc/θ. The difference between these values corresponds to the tax collection’s cost.The proposition also shows that, when at least one region does not refinance (in which caseit does not need to tax its population), region ` has to set the tax rate τ` (0) to pursueits ongoing project. Thus, asymmetric taxation emerges as a possible equilibrium, as in

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Bucovetsky (1991) and Wilson (1991). The main difference with their result is the following:here, tax differences are not the consequence of an ex ante regional endowment asymmetry,but rather occur because some regions en up with incomplete projects ex post. The tax rateτ` (0) also decreases with the capital endowment κ and the fiscal capacity θ. The intuition issimilar in both cases. Ceteris paribus, an increase in κ or θ enlarges the net fiscal revenues,which lowers the tax rate needed to raise the amount c. On the other hand, the tax rateincreases with c because the fiscal need increases. With this tax rate, the resulting capitaloutflow is

∑m∈M

f`m = τ` (0) and the equilibrium regional welfare (net of the initial cost c)

ends up being

W FD` = (κ−

∑m∈M

f`m) (1− τ`) +∑m∈M

f`m − 12(∑m∈M

f`m)2 + b

= κ+ b− T (c, θ),

where

T (c, θ) =c

θ+

[τ` (0)]2

2

measures the total refinancing cost. It comprises the effective refinancing cost c/θ, plus thedeadweight loss [τ` (0)]2/2 of financing project’s continuation through a distortionary tax.This distortion is due to mobility costs incurred by owners of capital seeking to avoid taxationin region `. Therefore, distortionary taxation only emerges in some final nodes of the taxcompetition subgame, when at least one region does not refinance. More importantly, thelikelihood of these final nodes depends upon the level of regional administrative capacity π.This observation will have an important consequence, as we shall see below.

5.3 Refinancing

At the beginning of the third stage, the strategy for region ` is r` ∈ {R,NR}, where R (NR)denotes “refinancing” (“not refinancing”). Conditional on i = (i1, ..., iL), and given rm, theprofile of refinancing decisions chosen by regions m 6= `, region `’s welfare is{

W FD` (r` = R, rm, i) = κ+ 1I`{I,inc}{[b− 1Im{I,inc,R}

cθ− (1− 1Im{I,inc,R})T (c, θ)]− c}

W FD` (r` = NR, rm, i) = κ− 1I`{I,inc}c

where 1I`{I,inc} is an indicator function that takes the value of 1 if region ` initiated a projectthat has remained incomplete at t = 2, and 0 otherwise. Also 1Im{I,inc,R} is an indicatorfunction that takes the value of 1 if all regions m 6= ` initiated their project, have notcompleted it in due time but decided to refinance them in t = 3, and 0 otherwise.

Let’s denote by c1 the value of c that makes the total refinancing cost T (c, θ) equal tothe benefit b, and by c2 > c1 the value of c that makes the effective refinancing cost c/θequal to the benefit b. The following proposition characterizes the Nash equilibria of thissubgame.

13

Proposition 4

When all regions face an incomplete project, they refinance them in equilibrium providedc ≤ c2. Otherwise, no region refinances.When there is at least one region that does not need refinancing, region ` refinances itsincomplete project provided c ≤ c1.

In the Appendix we show that there always exists two strictly positive thresholds c1 and c2

such that c ≤ c1 (c ≤ c2) implies T (c, θ) ≤ b (c/θ ≤ b). When all regions failed to completetheir project in due time, the Nash equilibria of this refinancing subgame depend upon thecost c. When c < c1, refinancing is a dominant strategy. But when c1 ≤ c ≤ c2, the gamebecomes a standard coordination one, with two Nash equilibria. The first one is trivial:despite the fact that c > c1, when all regions decide to refinance, these strategies form aNash equilibrium because, as there will be no capital flows (and thus no deadweight loss dueto distortionary taxation), only the fact that c/θ ≤ b matters. But if at least one regionhas decided to not refinance, the other regions also shutdown their project because c > c1.In this case, these regions face an endogenous hard budget constraint due to other regions’decisions, as in Qian and Roland (1998). As the equilibrium where all regions refinance isstrong (Aumann (1959)) and coalition-proof (Berheim et al. (1987)), we select it as theNash equilibrium. Finally, when c2 ≤ c, not refinancing is a dominant strategy. Regionsface again an endogenous hard budget constraint, but this time as a consequence of theirimperfect regional fiscal capacity. As c ≤ b, this outcome is inefficient.

In all other subgames, when there is at least one region that does not need refinancingbecause it executed its project in due time, region ` refinances its incomplete project providedc ≤ c1. If c > c1, the total cost from refinancing is higher than the benefit b, pushing region `to shutdown its incomplete project. But again, as c ≤ b, this is clearly an inefficient outcome.

5.4 Project initiation

Anticipating refinancing equilibria, regional governments simultaneously adopt the projectinitiation decision. Let cFDR (π) (cFDNAR(π)) [cFDNR(π)] be the cost that makes the expected,net regional welfare from initiating the project under full decentralization equal to zero,when incomplete projects are always refinanced (when all regions refinance their incompleteproject) [when incomplete projects are never refinanced]. In the next proposition, we presentthe investment equilibria under this regime.

Proposition 5

When 0 ≤ π ≤ πFD1 , the symmetric Nash equilibria under full decentralization is as follows. Ifc ≤ cFDR (π), initial investment occurs in all regions. Otherwise, there is no initial investmentin equilibrium in any region.When πFD1 < π ≤ πFD2 , the symmetric Nash equilibria under full decentralization is asfollows. If c ≤ cFDNAR(π), initial investment occurs in all regions. Otherwise, there is noinitial investment in equilibrium in any region.

14

When πFD2 < π ≤ πFD3 , the symmetric Nash equilibria under full decentralization is asfollows. If c ≤ cFDNR(π), initial investment occurs in all regions. Otherwise, there is no initialinvestment in equilibrium in any region.When πFD3 < π ≤ 1, the symmetric Nash equilibria under full decentralization is thatinvestment occurs in all regions.

In the Appendix we characterize the probability thresholds πFD1 , πFD2 and πFD3 , and weshow that cost thresholds cFDR (π), cFDNAR(π) and cFDNR(π) are lower than c∗(π). Hence, we canestablish the form of the inefficiencies that emerge under this institutional regime, as follows.

Corollary 2

Under full decentralization, equilibrium outcomes can be inefficient. If this is so, either (i)initial investments are not made in equilibrium when it is efficient to do so; (ii) initial invest-ments are made in equilibrium and are efficient, but incomplete projects are not refinancedwhen it is efficient to do so; or (iii) initial investments are made in equilibrium and areefficient, but incomplete projects are refinanced in a distortionary way.

The following figure depicts equilibrium outcomes that emerge under full decentralization.

b

c

π

2/b

10

)(πc

Figure 4: Full decentralization outcomes

FDNI

2c

1c

)(πcFDR

)(πcFDNAR

)(πcFDNR

FDI

𝜋3𝐹𝐷𝜋2

𝐹𝐷𝜋1𝐹𝐷

θb/θ +1

In the non-empty area NIFD, delimited from below by the thick curves representing thethree abovementioned cost thresholds, projects are never initiated. In the complementaryarea, denoted by IFD, all projects are initiated.When 0 ≤ π ≤ πFD1 , two different types of inefficiency emerge. First, in the blue area whenc ∈ [0, cFDR (π)], initiation and continuation decisions are optimal, but refinancing is done

15

bearing deadweight losses generated by imperfect regional fiscal capacity or distortionarycapital taxation. Second, as cFDR (π) < c∗(π), the condition for project initiation is stricterwith full decentralization than for the social planner. Therefore, in the shaded area when c ∈[cFDR (π), c∗(π)], investments are not initiated in equilibrium, despite the fact that it is efficientto do so. Underinvestment is due to i) distortionary refinancing, when c ∈ [cFDR (π), c1], andii) endogenous hard budget constraints, when c ∈ [c1, c

∗(π)]. The former finding is analog ofthe Zodrow and Mieszkowski’s (1986) result, while the latter has been analyzed by Besfamilleand Lockwood (2008), in a setting where an exogenous hard budget constraint is imposedto regional governments.When πFD1 < π ≤ πFD2 , a new type of inefficiency emerges. In the yellow area, whenc ∈ [c1, c2], projects are initiated but only refinanced when all regions do so. Again, there isinefficient underinvestment when c ∈ [cFDNAR(π), c∗(π)], but only caused by endogenous hardbudget constraints.When πFD2 < π ≤ πFD3 , the last type of inefficiency emerges. In the green area whenc ∈ [c2, c

FDNR], projects are initiated but never refinanced. There is still a shaded area where,

because of endogenous hard budget constraints, projects are inefficiently not initiated.8

Finally, when π ≥ πFD3 , the model is biased towards project initiation. Incomplete projectsare either refinanced in a distortionary way, shutdown in some terminal nodes of the taxcompetition subgame or never finalized.The figure illustrates that efficient decisions are only adopted in the area where projectsare not initiated. In the remainder areas, either the projects’ initiation and continuationdecisions are downwardly distorted or refinancing is done bearing deadweight losses. Thus,in these areas, expected welfare is below the first best level.

6 Regional state capacity and the optimal institutional

regime

6.1 The main result

In the initial period, there is an institutional choice between partial and full decentraliza-tion, made under uncertainty. At this stage, the Congress observes projects’ benefits b, B,the regional capital endowment κ and state capacities (π, θ), and knows that the cost c isdistributed according to a strictly positive probability density function h(c) on [0, b]. TheCongress computes, under each institutional regime IR ∈ {PD,FD}, the expected welfareof a region

EW IR =

bˆ

0

EW IR` h(c)dc

and chooses the regime that maximizes it.

8Figure 4 depicts full decentralization outcomes when θ < 1. If θ = 1, c2 = b and πFD2 = πFD

3 : the areawhere projects are never refinanced vanishes.

16

As one of the goals of this paper is to evaluate how this choice is affected by the level ofregional state capacity, we first characterize the relation between EW IR and the pair (π, θ).Taking into account equilibrium decisions and outcomes, in the Online Appendix we showthat, when L ≤ L (a threshold defined in the Appendix), EW FD is a continuous, every-where differentiable, increasing, convex function of the administrative capacity π. Moreover,we also show that it increases with the fiscal capacity θ, except when π = 1. Therefore,under full decentralization, an increase in either π or θ increases EW FD, as suggested bythe decentralization literature mentioned in the Introduction. But this assertion does notnecessarily imply that full decentralization dominates for high levels of state capacity. Whynot? Because we also prove that EW PD is a continuous, increasing, convex function of theadministrative capacity π, with a kink at π = πPD1 .9 Hence, the comparison between bothregimes is not a priori evident. The following proposition presents the result that comes outthis normative comparison, for every pair (π, θ) ∈ [0, 1]2.

Proposition 6Assume an intermediate number of regions L and a uniform distribution for the cost c.10

When the regional fiscal capacity θ ≤ θ0 ≡ 2L/(1 + L2), partial decentralization domi-nates for all values of the regional administrative capacity π ∈ [0, 1). When the regionalfiscal capacity θ ≥ θ0, there exists a unique threshold π(θ) such that, when the regionaladministrative capacity π ≤ π(θ), full decentralization dominates. Otherwise, partial decen-tralization dominates. When the regional administrative capacity π is equal to one, bothregimes are efficient.

The following figure illustrates this result. There, each point in the (θ, π) plane representsthe regional state capacity that prevails in the federation. We point out the area where eachinstitutional regime dominates.

9When π ≥ πPD1 , EWPD increases linearly with the administrative capacity π.

10The technical conditions on L are given in the Appendix.

17

π

θ0θ

)(ˆ θπ

Full Decentralization

PartialDecentralization

10

1

PartialDecentralization

Figure 5: Partial vs. full decentralization

𝑏/𝐵

Partial Decentralization = Full Decentralization

PartialDecentralization

PartialDecentralization

When the regional fiscal capacity θ ≤ θ0, refinancing incomplete projects under full decen-tralization is too costly. Therefore, regardless the level of regional administrative capacityπ ∈ [0, 1), partial decentralization dominates. This result supports Gadenne and Singhal’s(2014) empirical finding that there is relatively less fiscal decentralization and larger fiscalimbalances in developing countries, most of them characterized by low levels of regional fiscalcapacity. But this partial decentralization’s dominance does not hold for all other values ofθ. As expected, full decentralization may dominate when the regional fiscal capacity is rela-tively high. What is more surprising is that this indeed occurs11, but for relatively low levelsof the regional administrative capacity, i.e., when π ≤ π(θ). The intuition for this result isthe following. When π is relatively low, it is very likely that projects will remain incomplete,and thus refinancing them is the issue to deal with. When is refinancing a problem under fulldecentralization? When there is at least one region that has carried out its project in duetime, and thus the refinancing region needs to tax capital in a distortionary way. But whenπ is low, the likelihood of this event is also low. Thus, full decentralization’s distortions costless than inefficient investments made under partial decentralization. In the Appendix, weshow that π(θ) is unique and increases with θ.12 Indeed, the higher the fiscal capacity θ, thehigher the administrative capacity π for which this abovementioned argument holds.

11As L ≥ L, θ0 ≤ 0.6. So full decentralization dominates in a non negligible area of the figure. This resultis in some way unexpected because, in this model, the fully decentralized regime generates qualitatively moredistortions than partial decentralization.

12In order to prove π(θ)’s unicity, we have imposed some conditions on the number of regions. When thesesufficient conditions do not hold, the model becomes analytically untractable and thus π(θ)’s unicity cannotbe ensured. In order to verify whether our results would hold under parameter configurations of the modelthat do not satisfy those conditions, we have simulated the model and replicated Figure 5. All simulationsconfirmed the results presented in Proposition 6, and are available upon request from the authors.

18

When π is above b/B, the model is biased towards project initiation. Under partialdecentralization, outcomes are efficient. On the other hand, under full decentralization, thelikelihood of facing capital mobility costs or the project’s shutdown is very high. Thus,partial decentralization dominates. But if π increases further and converges to one, thelikelihood and welfare cost of full decentralization’s distortions decrease, attenuating partialdecentralization’s dominance. In the limit, when π is equal to one, both regimes yield optimaloutcomes.

These results clarify how the level of regional state capacity prevalent in a federationaffects the trade-off between partial and full decentralization. On the one hand, we confirmthat a high level of fiscal capacity is a necessary condition for full decentralization to be theoptimal institutional regime. But, on the other hand, we also caution against the position,held by some contributors to the literature on decentralization, that affirm the necessity ofhigh levels of regional state capacity for successful decentralization reforms. First, our modelshows that high levels of regional administrative or fiscal capacity do not always imply thatfull decentralization should dominate. Second, low levels of administrative capacity do notprevent full decentralization’s dominance. Under this last circumstance, the outside option(or the status quo regime) may generate more important distortions.

6.2 Robustness

In this section we undertake some sensitivity analysis of the model, and then we extend it,to evaluate its robustness.

6.2.1 Comparative statics

Next paragraphs explain in detail, with reference to their corresponding figures, how changesin some parameters of the model modify the way by which regional state capacity affects thechoice between partial and full decentralization.

π

θ0θ

)(ˆ θπ

Full Decentralization

PartialDecentralization

10

1

Figure 6a: Increase in capital endowment κ

π

θ0θ

)(ˆ θπ

Full Decentralization

PartialDecentralization

10

1

Figure 6b: Increase in project’s benefit B

π

θ0θ

)(ˆ θπ

Full Decentralization

PartialDecentralization

10

1

Figure 6c: Increase in the number of regions L

19

Change in the capital endowment κ Figure 6a shows that an increase in κ favors fulldecentralization, implying that this regime dominates in larger area of the plane (θ, π). Theintuition of this result hinges on two facts. First, the cost of distortions generated underpartial decentralization do not depend upon the value of κ. Second, an increase in theregional stock of capital increases expected welfare under full decentralization, because, asthe tax τ` (0) decreases with κ, so does capital mobility costs.

Change in the highest project’s benefit B Figure 6b depicts that an increase in Balways favors partial decentralization, implying that this regime dominates in a larger areaof the plane (θ, π). The reason is as follows: when B increases, the welfare loss due tooverinvestment (underinvestment) under partial (full) decentralization decreases (increases).

Change in the number of regions L Contrary to the previous comparative static exer-cises, Figure 6c shows that an increase in L has two effects. On the one hand, for relativelyhigh levels of the regional fiscal capacity, partial decentralization dominates in an area wherefull decentralization used to be the optimal regime. On the other hand, for relatively low val-ues of the regional administrative capacity, full decentralization dominates in an area wherepartial decentralization used to be the optimal regime. The intuition for this result hingeson the relative costs of the consequences of L’s increase. On the one hand, the common-pool fiscal externality under partial decentralization gets larger. On the other hand, underfull decentralization, the likelihood that a given region has to bear deadweight losses dueto capital mobility or to shutdown its project increases as well. When the administrativecapacityπ is relatively low, the cost of the latter is lower; the opposite holds for relativelyhigh values of π.

6.2.2 An extension: distortionary national taxation.

So far, the tax set by the central government to bailout regions under partial decentralizationwas non-distortionary. This is of course a strong assumption: usually national taxes alsogenerate deadweight losses. If so, assuming (as we did) non-distortionary national taxationunderestimates the cost of the common-pool fiscal externality. To address this issue, letus model distortions in national taxation in the simplest possible way, by assuming thatthe cost of bailing out an incomplete project under partial decentralization is c + λ, whereλ > 0.13 Although this change does not affect the fully decentralized regime, outcomes inthe other regime are substantially modified, as shown by the following figure, which depictsequilibrium outcomes that emerge under partial decentralization with distortionary nationaltaxation.14

13This assumption is a shortcut that captures, for example, cost differences in complying with centraland regional tax systems. As stressed by Slemrod and Venkatesch (2002), these differences are substantial:nearly 70% of their compliance spending was devoted by firms to federal government’s compliance, whereasalmost 25% was spent on regional and local compliance. If, for the sake of simplicity, we normalize regionalcompliance costs to zero, λ measures the cost difference in complying with the central government.

14Proofs that correspond to this section are in the Online Appendix.

20

b

c

π10

Figure 7: Outcomes under partial decentralizationwith distortionary national taxation

λPDNI ,

𝜋2𝑃𝐷,𝜆𝜋1

𝑃𝐷,𝜆 𝜋3𝑃𝐷,𝜆

𝑏 − 𝜆

No bailout

Bailout

𝑐∗(𝜋)

𝑐𝑃𝐷,𝜆 (𝜋)

𝑐𝑃𝐷(𝜋)

𝛼

λPDI ,

Now, the central government refinances less projects than before. Indeed, only when c ≤b − λ < b, incomplete projects are bailout; otherwise, the central government faces anendogenous hard budget constraint. When 0 ≤ π ≤ πPD,λ1 ≡ b−λL

L(B−b)+b , despite the fact thatthe central government bailouts all projects with costs lower than b−λ, only those for which

c ≤ cPD,λ(π) ≡ cPD(π)− (1− π)λ

L+ 1− p

are initiated and refinanced. Otherwise, projects are not undertaken, because refinancingthem is too costly. As cPD,λ(π) < cPD(π), the condition for project initiation is stricter thanunder the same regime with non-distortionary national taxation. Hence, in the shaded area,less projects are inefficiently initiated than in Section 4. As π increases, the cost intervalwhere these inefficient projects are initiated when the central government refinances vanishes.And when the central government does not bailout regions, no project is undertaken inequilibrium, as shown when b−λL

L(B−b)+b < π ≤ πPD,λ2 ≡ b−λB

. Observe that, for π ∈ [α, πPD,λ2 ],

cPD,λ(π) < c∗(π). This implies the opposite project’s distortion: in the shaded area, regionsinefficiently underinvest. Then, as b−λ

B≤ π ≤ πPD,λ3 ≡ b/B, all projects that would be

refinanced are initiated. But this does not hold for projects where c > b− λ. These projectsare undertaken provided c ≤ πB; otherwise they are not initiated. Again, there is inefficientunderinvestment in the shaded area. Finally, when πPD,λ3 ≤ π ≤ 1, all projects are initiated,despite the fact that some with high cost would not be refinanced if they remain incompletein the second period.

How this new regime compares with full decentralization? The following propositionpresents the result.

Proposition 7

When the value of the deadweight loss λ increases, full decentralization dominates in a largerarea of the plane (θ, π). relative to a situation with a non-distorted national tax system.

21

Clearly, departing from λ = 0, increasing λ favors full decentralization. Although ex-pected, the intuition of this result should not be based on the mere assertion that, byincreasing the bailout cost, expected welfare under partial decentralization should automat-ically decrease. In fact, when λ increases, this institutional regime does generate bailoutsmore costly than before, inefficient project’s shutdown and underinvestment. But when theadministrative capacity is relatively low, a higher λ also reduces the size of the cost intervalwhere projects are inefficiently initiated. Despite these countervailing effects, we can showthat the last one is dominated by the former three, and thus expected welfare under thisregime decreases with the deadweight loss λ.

7 Related Literature

Related literature is as follows. First, other contributions have analyzed the trade-off be-tween partial and full decentralization. Brueckner (2009) presents a Tiebout-type model,where local governments exert effort to reduce the cost of a local public good, private de-velopers build houses and heterogeneous consumers decide on their location. Under partialdecentralization, the federal government taxes the population and transfers the tax collec-tion to jurisdictions, on an equally per capita basis. Then, local governments choose thelevel of effort and local public good provision, and finally Tiebout sorting occurs. Under fulldecentralization, the unique difference is that each local government sets a non-distortionaryproperty tax to cover the public good’s cost. Brueckner (2009) finds that full decentralizationalways dominates, because the transfer’s uniformity under partial decentralization generatesless variety of local public goods, and thus a worse preference matching. Partial decentral-ization can be optimal, provided local governments are of a Leviathan type. Here, in ourmodel, we do not find a complete regime dominance. Peralta (2011) compares both regimes,focusing on accountability issues at the local level. She presents a political economy model,with benevolent and self-interested, rent-seeking local politicians, who tax their jurisdictionand provide local public goods, in a context of double asymmetric information because vot-ers observe neither the politician’s type nor the local public good’s cost. Two equilibriaemerge: a pooling equilibrium, when the rent-seeker politician mimics a benevolent one, anda separating equilibrium, when he extracts rents, and then he is replaced in the followingelection. Peralta (2011) shows that partial decentralization improves politician’s selection(i.e., voting out rent-seekers) whereas full decentralization fosters discipline (i.e., giving in-centives to rent seekers to behave as benevolents). This last regime dominates when theproportion of rent seekers is low. Other authors have adopted different definitions of partialdecentralization. Janeba and Wilson (2011) and Hatfield and Padro i Miquel (2012) definepartial decentralization when a subset of public goods are exclusively funded and providedby local governments. These authors find that devolving some public goods to local authori-ties is always optimal. Janeba and Wilson (2011) trade-off inefficient provision decided by aminimum winning coalition at the central level against distortionary taxation and low publicgood provision in a context of capital tax competition under full decentralization. In a votingmodel, Hatfield and Padro i Miquel (2012) obtain that devolution serves as a commitment

22

device against excessive capital taxation chosen when individuals vote on the central provi-sion of public goods. Joanis (2014) defines partial decentralization as “shared responsibility”,an institutional regime where both the central and the regional government participate inthe funding of a given public good. In his model, central and local, rent-seeking, politicianssimultaneously decide how much fiscal resources invest in the provision of this public good.Despite their preferences for rents, they also invest in such provision because their aim isto manipulate their reelection probability, in a context of weak accountability, where theelectorate is unable to assess the contribution of each level of government to the public goodprovision. Partial decentralization obtains, balancing loses in productive complementaritiesbetween both levels of government and the lack of accountability at the local level. Themain differences between our paper and these contributions hinges on the fact that we studya different trade-off between partial and full decentralization, namely inefficient bailouts andprojects overprovision vs. capital tax competition and projects underprovision. Moreover,these articles do not incorporate regional state capacity into the analysis, and thus they donot consider it as a relevant factor affecting the trade-off.

The paper is also related to an important set of contributions that analyze the prosand cons of different types of regional budget constraints in federations. On the one hand,the optimality of hard budget constraints has been studied by Qian and Roland (1998)and Inman (2003); whereas the possibility that they may be inefficient has been raised byBesfamille and Lockwood (2008). The main differences between Besfamille and Lockwood(2008) and this paper are the following. First, they do not describe the institutional regimethat hardens regional government’s budget constraints; they simply assume that the federalgovernment is able to impose, exogenously and at no cost, a hard budget constraint to localgovernments. Here, we analyze full decentralization as the institutional regime that hardensregional governments’ budget constraint, but endogenously and costly. Second, the authorscompare, from a normative point of view, soft and hard budget constraints at the interimstage (in other words, project by project) whereas we take an ex ante perspective. Finally,they do not consider how different levels of regional administrative capacity affect the trade-off between partial and full decentralization, which is one of our main concerns. On the otherhand, Wildasin (1997), Goodspeed (2002), Akai and Sato (2008) and Crivelli and Staal (2013)describe how bailouts in federations distort, via a common-pool fiscal externality, decisions atthe regional level. Finally, Silva and Caplan (1997), Caplan et al. (2000) and Kothenburger(2004) claim that, under some conditions, a regime with decentralized leadership, wherethe central government sets intergovernmental transfers after regional governments haveadopted their own policy, may give a more efficient outcome than a regime with hard budgedconstraints. This result relies on a second best argument, and thus needs some pre-existingdistortion in the form of local public goods or tax spillovers to hold.

Finally, the paper is related to a recent literature that studies empirically how, in contextsof decentralized regimes, local state capacity impacts on public outcomes. Steiner (2010)measures local governments’ capacity using an index of resources available to local govern-ments, and another one that captures the level of technical and administrative capacity ofdistrict governments. She finds evidence that both household consumption and school en-rollment are positively related with the level of capacity of district governments. Loayza et

23

al. (2014) evaluate how budget size and allocation process, local capacity, local needs, andpolitical economy considerations (four factors that are usually considered important featuresthat affect the effectiveness of decentralization reforms) affect municipal budget executionrate in Peru. The authors find that budget size and local capacity are the statistically mostimportant constraints that explain municipal budget execution rates. Bandyopadhyay andGreen (2012) show that higher percentage of residents from centralized ethnic groups implyhigher development indicators at the local level in Uganda. Finally, Acemoglu et al. (2014)study the impact of municipal state capacity on public goods provision in Colombia. An im-portant feature of their paper is the consideration of spillovers: when a municipality investsin its state capacity, it also generates positive effects on neighboring municipalities. Theyempirically confirm that state capacity decisions are indeed strategic between municipali-ties, and with large effects on local prosperity. All these papers take the intergovernmentalinstitutional setting as given, and thus do not analyze different regimes, as we do.

8 Conclusions

This paper presents a model featuring a central government and regional authorities. Thelatter are characterized by the same level of administrative and fiscal capacity. We analyzetwo fiscal regimes. Under partial decentralization, regional governments invest in a localpublic project but, if it remains incomplete, it is the central government that bailouts theregions. Thus, regions face soft budget constraints and inefficiently overinvest in local publicprojects. On the other hand, under full decentralization, regional governments also decideon initial investment but cannot rely on central bailouts. Thus, regional governments facehard budget constraints because capital tax competition increases the marginal cost of publicfunds. This implies the opposite type of inefficiency: they inefficiently inderinvest.

The first goal of the paper is to evaluate the normative comparison between these distor-tions, and to assess which regime dominates. The second goal is to investigate how differentlevels of regional state capacity affect this comparison. As expected, when the regional fis-cal capacity is low, partial decentralization dominates; otherwise full decentralization maydominate. But, contrary to the common wisdom, we find that full decentralization shouldbe the preferred regime when the regional administrative capacity is low.

The model can be generalized and extended in several directions. In particular, we wouldlike to consider ex ante asymmetries between regions, either in capital endowments or in statecapacities. This is an essential “input” if one would like to analyze how regional governmentsinvest in state capacity under different fiscal regimes. Also, performing this extension wouldbe crucial for the study of the long term trade-off between partial and full decentralization.

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9 Appendix

9.1 Proof of Proposition 1

The government of region ` anticipates that its expected, net welfare from investing in theproject is

κ+ 1I{i`=I}

[πB + (1− π)(b− c

L)− c

]−∑m6=`

1I{im=I}(1− π) · cL, (4)

28

whereas its expected, net welfare from not investing is

κ−∑m 6=`

1I{im=I}(1− π) · cL.

So, for any region `, initiating the project is a dominant strategy if

c ≤ cPD(π) ≡ L[πB + (1− π)b]

L+ 1− π.

It is straightforward to show that cPD(π) > c∗(π), ∂∂πcPD(π) = LL(B−b)+B

(L+1−p)2 > 0, and that

cPD = b when π = bL(B−b)+b

9.2 Proof of Proposition 2

Given a profile of tax rates τ = {τ1, ..., τL}, a household resident in region ` decides whereto invest its capital endowment, by solving the following problem:

maxh`,{f`m}m 6=`

h` (1− τ`) +∑m6=`

f`m (1− τm)− 1

2(∑m6=`

f`m)2

subject to its portfolio constraint

h` +∑m 6=`

f`m = κ

and (L− 1) non-negativity constraints

f`m ≥ 0 ∀m 6= `,

where h` is capital invested in region `, and f`m is capital invested in another region m 6= `.Let’s denote by λ`m the multipliers associated with the non-negativity constraints. Usingthe portfolio constraint to replace h` in the maximand of the household’s problem, we obtainthe first-order conditions for f`m and the complementary slackness conditions

τ` − τm + λ`m =∑m 6=`

f`m

λ`mf`m = 0 λ`m ≥ 0 ∀m 6= `.

The proof of the proposition uses the following two lemmas.

Lemma 1 Assume that there are two regions m,n 6= `, with τm > τn. Then f`m = 0.

Proof. Subtracting m’s first-order condition from n’s first-order condition, we obtain

τm − τn + λ`n = λ`m.

29

As τm > τn and λ`n ≥ 0, λ`m > 0. Hence, by the corresponding complementary slacknesscondition, f`m = 0

Let’s denote by M the set of regions m 6= ` that have chosen the minimum tax rateτ` = min{τm}m6=`. Then, as an immediate consequence of Lemma 1, for all regions m 6= `, m,f`m = 0. Hence, tax rates τm become redundant; from now on, pertinent comparisons shouldbe done only between τ` and τ`.

Lemma 2 Assume that τ` ≥ τ`. If m ∈ M then λ`m = 0.

Proof.i) First, assume that Card{M} = 1 and consider the first-order condition

τ` − τ` + λ`m = f`m. (5)

If τ` > τ`, as λ`m ≥ 0, then f`m > 0. Thus, by the complementary slackness condition,λ`m = 0. If τ` = τ`, (5) becomes

λ`m = f`m.

If λ`m > 0, then f`m > 0, which implies, by the complementary slackness condition, thatλ`m = 0, which is a contradiction. Hence λ`m = 0.ii) Now assume that Card{M} ≥ 2. First-order conditions that characterize flows f`m are

τ` − τ` + λ`m =∑m∈M

f`m.

In order to satisfy these first-order conditions, all multipliers λ`m should have the samevalue. If they were all strictly positive, then, by their corresponding complementary slacknesscondition, all outflows f`m should be equal to 0, implying that

∑m∈M

f`m = 0. But, as τ` ≥ τ`,

all first-order conditions would yield a contradiction. Hence, all multipliers λ`m = 0

Therefore, from any first-order condition that characterizes a flow to a region m ∈ M,we obtain ∑

m∈M

f`m = τ` − τ` (6)

9.3 Proof of Proposition 3

Due to specific features of this model, we cannot apply Wildasin’s (1988) methodology. So,to obtain the equilibrium tax rates, first we derive region `’s reaction function.

9.3.1 Scenario 1: All regions have decided to refinance their incomplete project

Let’s denote by τm the profile of tax rates chosen by regions m 6= `. For any profile τm, weneed to consider three cases.

30

1. If the regional government of ` plans to set its tax rate strictly above τ`, there will becapital outflows to regions m ∈ M . Hence, the regional welfare would be

W FD` = (κ−

∑m∈M

f`m) (1− τ`) +∑m∈M

f`m (1− τ`)−1

2(∑m∈M

f`m)2 + b.

By the Envelope Theorem, ∂W FD` /∂τ` = −(κ−

∑m∈M

f`m) < 0. So the regional govern-

ment of ` should set the lowest tax rate that satisfies its budget constraint

θτ`(κ−∑m∈M

f`m) = c. (7)

Using (6), the smallest root of (7) is given by

τ ` =1

2

[κ+ τ` −

√(κ+ τ`)

2 − 4c

θ

]. (8)

Throughout the paper, we assume that κ is sufficiently large so that this square rootalways exists (see footnote 15).

2. If the regional government of ` plans to set its tax rate strictly below τ`, there will beno capital outflows to regions m 6= `. Thus, the regional welfare would be

W FD` = κ (1− τ`) + b.

Again, by the Envelope Theorem, ∂W FD` /∂τ` = −κ < 0. So, the regional government

of ` should choose the lowest tax rate that satisfies its budget constraint

θτ`(κ+∑m 6=`

fm`) = c, (9)

where, by a symmetric application of (6),

fm` = τm − τ` (10)

represents a capital inflow from a region m 6= ` to region `. Rearranging terms, thesmallest root of (9) is given by

τ ` =1

2

1

L− 1(κ+

∑m6=`

τm)−√

1

(L− 1)2(κ+

∑m 6=`

τm)2 − 4c

θ(L− 1)

. (11)

3. If the regional government of ` plans to replicate τ`, there will be capital outflows fromregions m /∈ M to regions m ∈ M. As now ` ∈ M , the regional welfare of ` would be

W FD` = κ (1− τ`) + 1I{ETC≥c}b,

31

where ETC denotes the effective tax collection

θτ`(κ+ α`∑m/∈M

∑m∈M

fmm),

and α`, 0 < α` < 1, represents the fraction of capital outflows that leave regions m /∈ Mand move to region `.

Now, in order to completely characterize the reaction function τ` (τm), we need to identifyprofiles of tax rates τm for which τ `, τ ` and τ` are region `’s best responses.

Consider first that, facing a profile of tax rates τm, region ` wishes to set τ `. It isstraightforward to show that, for any change in one particular tax rate τm, m 6= ` that doesnot modify τ`,

∂τ `∂τm

< 0.

Therefore, given τ`, the highest tax rate τ ` < τ` is set when all regions m 6= ` have chosenτm = τ`. Let’s denote by τ `(τ`) this tax rate, which is given by

τ `(τ`) =1

2

κ

L− 1+ τ` −

√(κ

L− 1+ τ`

)2

− 4c

θ(L− 1)

.We can show that

∂τ `(τ`)

∂τ`< 0

andlim

τ`↓c/θκτ `(τ`) = c/θκ.

If τ` ≤ c/θκ, τ `(τ`) ≥ τ`, contradicting the definition of τ `(τ`) as the highest tax rate thatregion ` can set strictly below τ`. Therefore, if τ` ≤ c/θκ, the government in region ` cannotset a tax rate lower than τ` while, at the same time, satisfying its budget constraint.

Now consider that, facing a profile of tax rates τm, region ` wishes to set τ ` > τ`. It isstraightforward to show that

limτ`→0

τ ` =1

2

[κ−

√κ2 − 4c

θ

]≡ τ`(0) > 0,

∂τ `∂τ`

< 0,∂2τ `∂τ`2

> 0

andlim

τ`↑c/θκτ ` = c/θκ.

32

If τ` ≥ c/θκ, τ `(τ`) ≤ τ`, which again contradicts the definition of τ ` as the lowest tax ratethat region ` can set strictly above τ`. Hence, if τ` ≥ c/θκ, the government in region ` cannotset a tax higher than τ` while, at the same time, satisfying its budget constraint.

Two more cases remain to be analyzed. First, consider a profile of tax rates τm suchthat τ` = c/θκ but when at least one region n 6= ` has set τn > c/θκ. We have just provedthat region ` cannot tax strictly above c/θκ. If region ` chooses τ` = c/θκ, there are nooutflows f`m. Thus, its welfare amounts to W FD

` = κ − c. But region ` receives a capitalinflow α`

∑m/∈M

∑m∈M

fmm, and so its tax collection is

θc

θκ(κ+ α`

∑m/∈M

∑m∈M

fmm) > c.

Therefore, there is room for a decrease in τ`. Indeed, region ` can set its tax rate τ ` < c/θκ,satisfying its budget constraint and increasing its welfare (with respect to the choice ofτ` = c/θκ).

Finally, when all regions m 6= ` have set τm = c/θκ, we have already proved that region` cannot tax strictly above or below c/θκ. Hence, the optimal reaction is to set τ` = c/θκ.

Region `’s reaction function is thus characterized as follows:

τ` (τm) =

12

[κ+ τ` −

√(κ+ τ`)

2 − 4cθ

]if τm < c/θκ ∀m 6= `

c/θκ if τm = c/θκ ∀m 6= `

12

κ+∑m 6=`

τm

L−1−

√√√√(κ+∑m 6=`

τm

L−1

)2

− 4cθ(L−1)

if τm ≥ c/θκ ∀m 6= `

and ∃n 6= ` : τn > c/θκ.

(12)

The reaction function is build around the value c/θκ. When the profile of tax rates τm issuch that all tax rates τm < c/θκ, region `’s optimal response is to tax strictly above theminimum tax τ`. In this case, τ` (τm) depends only upon τ`, the unique pertinent tax ratethat defines the size of the capital outflow. When the profile of tax rates changes, so thatthe minimum tax rate τ` increases and converges to c/θκ from below, τ` (τm) decreases andconverges to c/θκ from above.When all tax rates τ` are equal to c/θκ, region `’s optimal response is to replicate this level.Finally, when the profile of tax rates is such that τm ≥ c/θκ but at least one region n 6= `has chosen a tax rate τn > c/θκ, region `’s optimal response is to tax strictly below τ`. Inthis case, τ` (τm) depends upon the whole profile τm, because each tax rate τm determinesthe size of the capital inflow that moves from a region m 6= ` to region `.

Region `’s reaction function is non-continuous. When τ` converges to c/θκ from below,τ` (τm) converges to this limit from above. But when the distribution of taxes is such thatτ` = c/θκ but at least one region n 6= ` has set τn > c/θκ, the optimal response is to setτ` (τm) < c/θκ.

33

Clearly, τ1 = .... = τL = c/θκ is a Nash equilibrium of this subgame because it is afixed point of the best reply correspondence. To prove uniqueness, we proceed in two steps.First, by a simple inspection of (12), it is immediate to notice that no asymmetric choice oftaxes can be a Nash equilibrium. Second, we show that there cannot be another symmetricequilibrium. Assume the contrary: let τ ′` = τ ′m and, without any loss of generality,

τ ′` = τ ′m =c

θκ+ ε, (13)

with ε 6= 0 be another symmetric equilibrium. By substituting (13) into (12), we obtain acontradiction

9.3.2 Scenario 2: At least one region has decided not to refinance

In this case, to refinance its incomplete project, the regional government of ` has to set thetax rate τ`(0). This tax rate is obtained replacing τ` by 0 in the definition of τ `.

15

9.4 Proof of Proposition 4

First, we prove the existence of c1 and c2. When there is at least one region that does notneed refinancing, the total cost from completing a project in any region T (c, θ) is a strictlyincreasing and convex function of c, that satisfies

limc→0

T (c, θ) = 0 and limc→b

T (c, θ) =b

θ+

[limc→b

τ`(0)]2

2> b.

Hence, by Bolzano’s Theorem, there exists a threshold 0 < c1 < b such that, when c ≤ c1,b− T (c, θ) ≥ 0. Replacing τ`(0) in T (c, θ), and then equaling to b, we obtain

c1 = θ(2b−H),

where H = κ2 −√κ2(κ2 − 2b). Also, as c/θ > c, there trivially exists a threshold c2 such

that c2/θ = b. Moreover, as c/θ < T (c, θ), c1 < c2.

Then, consider the first case, when all regions face an incomplete project. When 0 ≤ c ≤c1,

W FD` (r` = R, rm, i) ≥ W FD

` (r` = NR, rm, i).

Refinancing is a dominant strategy, and so all regions refinance in equilibrium. When c1 <c ≤ c2,

W FD` (r` = R, rm = R, i) = b− c/θ ≥ W FD

` (r` = NR, rm = R, i) = 0 ∀m 6= `

15A sufficient condition for the existence of the square root in (8) and to ensure that τ` < 1 is κ ≥Max{2

√b/θ, b/θ}.

34

but

W FD` (r` = R, rm = NR, i) = b− T (c, θ) ≤ W FD

` (r` = NR, rm = NR, i) = 0 ∀m 6= `.

These payoffs define a coordination (sub)game between regions, with two Nash equilibria.In the first equilibrium all regions refinance; while in the second one no region refinances.

In fact, we prove that the first equilibrium is both the strong (Aumann, 1959) and thecoalition-proof Nash equilibrium (Berheim et al., 1987), as follows.

1. As both equilibria are Nash equilibria, no region can do better by unilaterally changingits equilibrium strategy. Then consider L−regions coalitions, with 1 < L < L. If theother regions refinance (not refinance), the L regions do not want to deviate becauseb − c/θ ≥ 0 (b − T (c, θ) < 0). Finally, consider the L-regions coalition. If all regionsrefinance, they do not want to deviate since b−c/θ ≥ 0. But, when no region refinances,they all wish to deviate because the first Nash equilibrium is Pareto optimal. Hence,the unique equilibrium that is strong is the first one.

2. Again, as both equilibria are Nash equilibria, no region can do better by unilaterallychanging its equilibrium strategy. Then consider L−regions coalitions, with 1 < L < L.If the other regions refinance, the coalition of L regions can jointly decide to notrefinance. Although this deviation is self-enforcing, it is not worthy because b−c/θ ≥ 0.But if the other regions do not refinance, the unique available deviation to the coalitionis to refinance. But this deviation is not self-enforcing because b−T (c, θ) < 0. Finallyconsider the L-regions coalition. If all regions refinance, they can jointly decide to notrefinance. Although this deviation is self-enforcing, it is not worthy because b− c/θ ≥0. But when no region refinances, they all wish to deviate because the first Nashequilibrium is Pareto optimal. Hence, the unique coalition proof Nash equilibrium isthe first one.

Therefore, when c1 < c ≤ c2, we select the equilibrium where all regions refinance as theNash equilibrium of this subgame. Finally, when c2 < c ≤ b,

W FD` (r` = R, rm, i) ≤ W FD

` (r` = NR, rm, i).

Not refinancing is a dominant strategy, and thus no region refinances in equilibrium.

The proof of the second part of the proposition is immediate, and thus omitted

9.5 Proof of Proposition 5

First, we proceed to evaluate expected, net regional welfares under different parameter con-ditions. If c ≤ c1, region `’s expected, net welfare is:

EW FD` (i` = I, im = I) = κ− c+ πB + (1− π)L[b− c

θ]

−(1− π)(1− (1− π)L−1)[b− T (c, θ)] if im = I ∀m 6= `

EW FD` (i` = I, im = NI) = κ− c+ πB + (1− π)[b− T (c, θ)] if ∃m 6= ` : im = NI`

EW FD` (i` = NI, im) = κ ∀im ∈ {I,NI}

35

If c1 < c ≤ c2, region `’s expected, net welfare is:

EW FD` (i` = I, im = NI) = κ− c+ πB + (1− π)L[b− c

θ] if im = I ∀m 6= `

EW FD` (i` = I, im = NI) = κ− c+ πB if ∃m 6= ` : im = NI`

EW FD` (i` = NI, im) = κ ∀im ∈ {I,NI}

When c2 < c ≤ b, region `’s expected, net welfare is

EW FD` (i` = I, im) = κ− c+ πB ∀im ∈ {I,NI}

EW FD` (i` = NI, im) = κ ∀im ∈ {I,NI}

In order to characterize the Nash equilibria, we need to evaluate these different levels ofexpected, net regional welfare. As their comparison is cumbersome, we proceed in an indirectway, as follows. First, let’s define

∆∗(c) ≡ (1− π)[B − (b− c)] :

it is an increasing, linear function of c that satisfies

limc→0

∆∗(c) = ∆∗(0) ≡ (1− π)[B − b] and limc→b

∆∗(c) = ∆∗(b) ≡ (1− π)B.

Then, when c ≤ c1, we can show that:

1. EW FD` (i` = I, im = I) ≥ EW FD

` (i` = NI, im)⇔ ∆FDR (c) ≤ B − c, where

∆FDR (c) ≡ (1− π)

[B − (1− π)L−1(b− c

θ)− (1− (1− π)L−1)(b− T (c, θ))

]is a continuous, increasing, convex function of c that satisfies

limc→0

∆FDR (c) ≡ ∆FD

R (0) = (1− π)[B − b] = ∆∗(0), ∆FDR (c) > ∆∗(c)

andlimc→c1

∆FDR (c) = ∆FD

R (c1) ≡ (1− π)[B − (1− π)L−1(b− c1

θ)]< ∆∗(b).

2. EW FD` (i` = I, im = NI) ≥ EW FD

` (i` = NI, im)⇔ ∆′(c) ≤ B − c, where

∆′(c) ≡ (1− π)[B − (b− T (c, θ))]

is another continuous, increasing, convex function of c that satisfies

limc→0

∆′(c) = (1− π)[B − b] = ∆∗(0), ∆′(c) > ∆FDR (c) > ∆∗(c)

andlimc→c1

∆′(c) = ∆′(c1) ≡ (1− π)B = ∆∗(b).

36

When c1 < c ≤ c2, we can also show that:

1. EW FD` (i` = I, im = I) ≥ EW FD

` (i` = NI, im)⇔ ∆FDNAR(c) ≤ B − c, where

∆FDNAR(c) ≡ (1− π)

[B − (1− π)L−1(b− c

θ)]

is a continuous, increasing, linear function of c that satisfies

limc→c1

∆FDNAR(c) = ∆FD

R (c1), ∆FDNAR(c) > ∆∗(c) and lim

c→c2∆FDNAR(c) = ∆FD

NAR(c2) ≡ (1−π)B = ∆∗(b).

2. EW FD` (i` = I, im = NI) ≥ EW FD

` (i` = NI, im)⇔ ∆′′(c) ≤ B − c, where

∆′′(c) ≡ (1− π)B

satisfies∆′′(c) = ∆FD

NAR(c2) = ∆′(c1) = ∆∗(b).

When c2 < c ≤ b, we show that EW FD` (i` = I, im) ≥ EW FD

` (i` = NI, im)⇔ ∆FDNR(c) ≤ B−c,

where∆FDNR(c) ≡ (1− π)B = ∆′′(c).

These functions ∆∗(c), ∆FDR (c), ∆′(c), ∆FD

NAR(c) and ∆FDNR(c) are (weakly) increasing in c,

while B−c decreases with c. Therefore, at some point, these five functions intersect B−c. Wecharacterize these intersections as cost thresholds. Then, we divide the cost range [0, b] in sub-intervals, according to these thresholds, and we find the Nash equilibria in each correspondingsub-interval. Moreover, as these five functions depend upon π, the cost thresholds alsodepend upon this parameter.

1. 0 ≤ π ≤ πFD1

Let πFD1 be implicitly defined by ∆FDR (c1) = B − c1. As ∆′(c) ≥ ∆FD

R (c) ≥ ∆∗(c), theintersection between ∆′(c) and B − c defines a threshold c′(π) ≤ c∗(π), whereas theintersection between ∆FD

R (c) and B− c defines another threshold cFDR (π) that satisfiesc′(π) ≤ cFDR (π) ≤ c∗(π). The Nash equilibria are the following:16

(a) When 0 ≤ c ≤ c′(π), ∆FDR (c) ≤ ∆′(c) ≤ B − c, which implies that

EW FD` (i` = I, im = I) ≥ EW FD

` (i` = NI, im)

andEW FD

` (i` = I, im = NI) ≥ EW FD` (i` = NI, im).

Hence all regions initiate their project and, if it remains incomplete at the end oft = 2, they refinance it in t = 3.

16We present here the complete proof for this case. We omit those for the remaining cases because theyare identical.

37

(b) When c′(π) < c ≤ cFDR (π), ∆FDR (c) ≤ B − c < ∆′(c), which implies that

EW FD` (i` = I, im = I) ≥ EW FD

` (i` = NI, im)

butEW FD

` (i` = I, im = NI) ≤ EW FD` (i` = NI, im).

So two Nash equilibria emerge: i) all regions initiate their project and, if it remainsincomplete at the end of t = 2, they refinance it in t = 3, or ii) no region invests.As the first equilibrium is strong and coalition-proof, we select it as the Nashequilibrium.

(c) When cFDR < c, B − c < ∆FDR (c) < ∆′(c), which implies that

EW FD` (i` = I, im = I) ≤ EW FD

` (i` = NI, im)

andEW FD

` (i` = I, im = NI) ≤ EW FD` (i` = NI, im).

Hence, no region initiates a project.

2. πFD1 < π ≤ πFD2

Let πFD2 ≡ c1/B and cFDNAR(π) ≤ c∗(π) be defined by the intersection between ∆FDNAR(c)

and B − c. The Nash equilibria are the following:

(a) When 0 ≤ c ≤ c1, all regions initiate their project and, if it remains incompleteat the end of t = 2, they refinance it in t = 3.

(b) When c1 < c ≤ cFDNAR(π), all regions initiate their project and, if it remainsincomplete at the end of t = 2, they refinance it in t = 3, provided all otherregions do the same.

(c) When cFDNAR(π) < c ≤ b, no region initiates a project.

3. πFD2 < π ≤ πFD3 ≡ b/B

Let cFDNR(π) ≡ πB ≤ c∗(π) be defined by the intersection between ∆FDNR(c) and B − c.

The Nash equilibria are the following:

(a) When 0 ≤ c ≤ c1, all regions initiate their project and, if it remains incompleteat the end of t = 2, they refinance it in t = 3.

(b) When c1 < c ≤ c2, all regions initiate their project and, if it remains incompleteat the end of t = 2, they refinance it in t = 3, provided all other regions do thesame.

(c) When c2 < c ≤ cFDNR(π), all regions initiate their project but, if it remains incom-plete at the end of t = 2, they do not refinance it.

(d) When cFDNR(π) < c ≤ b, no region initiates a project.

38

4. πFD3 < π ≤ 1

The Nash equilibria are the following:

(a) When 0 ≤ c ≤ c1, all regions initiate their project and, if it remains incompleteat the end of t = 2, they refinance it in t = 3.

(b) When c1 < c ≤ c2, all regions initiate their project and, if it remains incompleteat the end of t = 2, they refinance it in t = 3, provided all other regions do thesame.

(c) When c2 < c ≤ b, all regions initiate their project but, if it remains incompleteat the end of t = 2, they do not refinance it

9.6 Proof of Proposition 6

The proof of this proposition uses the expected welfare expressions derived in the OnlineAppendix, and many intermediate results.

9.6.1 Analysis of expected welfares at the extreme values of π

π=0

When π = 0, c∗(0) = b/2, cPD(0) = LbL+1

and cFDR (0) = θb1+θ

. Hence, expected welfares are

EW PD(π, θ) = κ+bL

(1 + L)2and EW FD(π, θ) = κ+

θb

2(1 + θ).

Let θ0 ≡ 2L1+L2 be the regional fiscal capacity that equalizes these expected welfares. Hence,

when θ ≥ θ0, EW FD ≥ EW PD. Otherwise, partial decentralization dominates.

Last, we analyze the value of ∂EWFD(π,θ)∂π

∣∣∣π=0

as a function of the fiscal capacity θ. For

any pair (b, θ), we can always find a real number β and write κ = β√

4b1+θ

. Computing the

abovementioned derivative, we obtain

−θL[−12bB(1 + θ) + 8(L− 1)βb2(β2 − 1)32 + b2(9− 12β2 + 8β4 + 12θ + L(−8β4 + 2β2 − 3))]

12(1 + θ)2.

The value of L that makes this expression equal to zero is

L(θ) = 1 +3[4B(1 + θ)− 2b(1 + 2θ)]

b[−8β4 + 12β2 + 8β(β2 − 1)32 − 3]

≥ 1.

The fraction’s denominator converges fast to zero, i.e., for values of β below 1. But recallthat in footnote 15 we have imposed κ to be sufficiently high; so β should be in fact a real

39

number higher than 1. Thus, L(θ) turns out to be a big number. Moreover, L(θ) increaseswith θ. Hence, if L ≤ L ≡ L(0)

∂EW FD(π, θ)

∂π

∣∣∣∣π=0

≥ 0,∀θ ∈ [0, 1]

and it increases with θ. As we prove in the Online Appendix that EW FD(π, θ) is convex, wecan conclude that EW FD(π, θ) is an increasing function of the administrative capacity π.

πPD2 = πFD3 = b/B ≤ π ≤ 1

When b/B ≤ π < 1, partial decentralization replicates the first best outcomes, whereas fulldecentralization’s outcomes are highly distorted. Hence, partial decentralization dominates.But if we compute

∂

∂π

[EW FD(π, θ)− EW PD(π)

]and we take the limit, we obtain

limπ→1

∂

∂π

[EW FD(π, θ)− EW PD(π)

]=

1

b

bˆ

0

[b− c]dc−c1ˆ

0

[b− T (c, θ)]dc

.

As c1 < b and T (c, θ) > c, this limit is strictly positive. So, when π converges to 1, the fulldecentralization expected welfare converges, from below, to the first best level.

9.6.2 Comparison of expected welfares when θ ∈ [θ∗, 1]

First, assume that θ = 1. When π = 0,

EW FD(0, 1) =bL

4,

and when π = πPD1 ,

EW PD(πPD1 ) =bL(2B − b)

2[BL− b(L− 1)].

If L ≥ L ≡ 3 + BB−b , EW

FD(0, 1) ≥ EW PD(πPD1 ). Hence, EW FD(π, θ) lies everywhere above

EW PD(π) when π ∈ [0, πPD1 ] because the former is an increasing function of π. Therefore, asEW FD(π, θ) must converge to EW PD(π) from below when π converges to one, EW FD(π, θ)has to cross EW PD(π) in its linear part, from above. Let’s denote by π(1) the administrativecapacity level that corresponds to this intersection.17 In fact, π(1) is unique. If this werenot the case, EW FD(π, θ) would cross again EW PD(π), from below. But, if such secondintersection occurs, by convexity, EW FD(π, θ) could not converge to EW PD(π) at π = 1.

When θ decreases, EW FD(π, θ) decreases as well, while EW PD(π) remains constant. Bycontinuity, we know that there exists θ∗, a value of the regional fiscal capacity such that

17As partial decentralization dominates when π ≥ πPD2 , πPD

1 ≤ π(1) < πPD2 .

40

EW FD(π, θ∗) lies above EW PD(π) everywhere when π ∈ [0, πPD1 ]. Hence, when θ ∈ [θ∗, 1],EW FD(π, θ) intersects EW PD(π) in its linear part. Using the same geometrical argument aswe did above, we know that both expected welfares cross only once, at π(θ). Applying theImplicit Function Theorem, we show that

∂π(θ)

∂θ= − ∂EW FD(π(θ), θ)/∂θ

∂EW FD(π(θ), θ)/∂π − ∂EW PD(π(θ), θ)/∂θ> 0

because, at π(θ), EW FD(π, θ) crosses EW PD(π) from above. Thus, π(θ) increases with θ.When θ < θ∗, we cannot a priori ensure that EW FD(π, θ) crosses EW PD(π) only once.

The goal of the following paragraphs is to prove that this is indeed the case.

9.6.3 Comparison of expected welfares when θ ≤ θ0

When θ = θ0,∂EW PD(π)

∂π

∣∣∣∣π=0

=L[4(L+ 1)B + (L2 − 5L− 2)b]

2(L+ 1)3> 0

because L2 − 5L− 2 > −8.25 and L ≥ L ≥ 3. Also

∂EWFD(π,θ)∂π

∣∣∣π=0

= 1b

´ θ0b1+θ0

0 [B − b+ L cθ0

+ (1− L)T (c, θ0)]dc

< 1b

{´ θ0b1+θ0

0 [B − b]dc+´ θ0b

1+θ00 T (c, θ0)dc

}

< 1b

´ θ0b1+θ0

0 [B − b]dc+ θ0b2(1+θ0)

= (2B−b)L(1+L)2

,

because T (c, θ0) ≤ b. Hence,(∂EW PD(π)

∂π− ∂EW FD(π, θ0)

∂π

)∣∣∣∣π=0

>bL2(L− 3)

2(L+ 1)3> 0.

Next, we prove that, when π ∈ [0, πPD1 ], EW PD(π) > EW FD(π, θ0). To do so, we first findan upper-bound for the expected welfare under full decentralization. If πPD1 ≥ πFD2 ,

EW FD(πPD1 , θ0) = κ+ 1b

{´ cFDNR(π)

0[πPD1 B − c]dc+

´ c20

(1− πPD1 )L(b− cθ0

)]dc

+´ c1

0(1− πPD1 )(1− (1− πPD1 )L−1)(b− T (c, θ0))dc

}

41

where cFDNR = πPD1 B and c2 = θ0b. So

EW FD(πPD1 , θ0) = κ+ bB2

2[b+L(B−b)]2 +bL(1− b

b+L(B−b) )L

1+L2 + 1b

´ c10

(1− πPD1 )(1− (1− πPD1 )L−1)(b− T (c, θ0))dc

< κ+ bB2

2[b+L(B−b)]2 +bL(1− b

b+L(B−b) )L

1+L2 + 1b

´ c10

(1− πPD1 )(1− (1− πPD1 )L−1)(b− cθ0

)dc

= κ+ bB2

2[b+L(B−b)]2 +bL(1− b

b+L(B−b) )

1+L2 ≡ A.

If πPD1 < πFD2 ,

EW FD(πPD1 , θ0) = κ+ 1b

{´ cFDNAR(π)

0[πPD1 B − c+ (1− πPD1 )L(b− c

θ0)]dc

+´ c1

0(1− πPD1 )(1− (1− πPD1 )L−1)(b− T (c, θ0))dc

}< κ+ 1

b

{´ c20

[πPD1 B − c+ (1− πPD1 )L(b− cθ0

)]dc

´ c10

(1− πPD1 )(1− (1− πPD1 )L−1)(b− T (c, θ0))dc}< A.

Hence A is one upper bound for EW FD(πPD1 , θ0). Now, let’s define

Γθ0(πPD1 ) ≡ EW PD(0) + ∂EWPD(π)

∂π

∣∣∣π=0

.πPD1 = κ− bL[bL(L+ 5)− 2B(L+ 1)(L+ 2)]

2(1 + L)3[BL+ b(L− 1)].

Finally, we evaluate

Γθ0(πPD1 )− A = H(b, B, L) ≡ 1

2b[ −B2

[b+L(B−b)]2 −L[bL(L+5)−2B(L+1)(L+2)]

(1+L)3[BL+b(L−1)]−

2L(1− bb+L(B−b) )

1+L2 ].

As L ≥ L, we can write L = 2 + n + BB−b , for n ≥ 1. The solution to the equation

H(b, B, L) = 0 in B can be expressed in the form B = α(n).b. Although it is not possibleto obtain a close form expression for α(n), the following figure depicts the curve α(n), whenn ∈ {0, ..., 100}.

42

As α(1) = 2.6817 and B/2 < b, H(b, B, L) > 0 for any n ∈ {1, ..., 100}.18 Therefore, weconclude that EW FD(πPD1 , θ0) < Γ0(πPD1 ).

These results enable us to assert that, when π ∈ [0, πPD1 ], EW FD(π, θ0) lies below Γθ0(π), astraight line that takes the values EW PD(0) and Γθ0(π

PD1 ) when π = 0 and πPD1 , respectively.

In fact, Γθ0(π) is the tangent of EW PD(π, ) at π = 0. Hence, as EW PD(π) is convex,EW FD(π, θ0) < EW PD(π) everywhere on π ∈ [0, πPD1 ]. We depict this result in the followingfigure.

0

PD

1

𝔼𝑊𝐹𝐷 𝜋, 𝜃0

𝔼𝑊𝑃𝐷(𝜋)

Γ𝜃0(𝜋)𝔼𝑊𝐹𝐷(𝜋1𝑃𝐷, 𝜃0)

Γ𝜃0(𝜋1𝑃𝐷)

Figure 9: Expected welfares when 𝜃 = 𝜃0 and 𝜋 ∈ [0, 𝜋1𝑃𝐷]

Moreover, when π ∈ (πPD1 , 1], EW FD(π, θ0) cannot cross EW PD(π). If this were the case,the former curve would intersect the latter from bellow, in its linear part. But then, byconvexity, it could not converge towards EW PD(π) from below at π = 1.

Hence, when θ = θ0, EW FD(π, θ0) < EW PD(π), except at π = 0 when they coincide.Therefore, as EW FD(π, θ) increases with θ, EW FD(π, θ) < EW PD(π) for all values of πwhenever θ ≤ θ0.

9.6.4 Comparison of expected welfares when θ ∈ [θ0, θ∗]

We will prove, using a series of geometrical arguments, that when θ ∈ [θ0, θ∗], EW FD(π, θ)

intersects EW PD(π) exactly once.Assume that we can find θ = θ0 + ε such that

EW FD(0, θ0 + ε) > EW PD(0) and EW FD(πPD1 , θ0 + ε) < Γθ0(πPD1 ) < EW PD(πPD1 ).

Clearly, EW FD(π, θ0+ε) intersects the line Γθ0(π) once, from above. Hence, as EW FD(π, θ0+ε) and EW PD(π) are both convex in π, the former has to cross the latter only once, fromabove. This argument can be replicated until θ = θ1 > θ0, which is implicitly defined by

EW FD(πPD1 , θ1) = Γθ0(πPD1 ).

18This result also holds for n > 100.

43

When θ = θ1, there can be two possibilities. When π converges to πPD1 , either 1) EW FD(π, θ1)converges to Γθ0(π

PD1 ) from bellow, in which case the previous intersection argument applies,

or 2) this convergence is from above. This means that

∂EW FD(π, θ1)

∂π

∣∣∣∣π=πPD1

< slope Γθ0(π).

Hence, EW FD(π, θ1) has to cross EW PD(π) from above, only once. So, for any θ ∈ [θ0, θ1],EW FD(π, θ) intersects EW PD(π) exactly once.

Now let’s construct Γθ1(π): it’s a line that has the same slope than Γθ0(π) and is defined byΓθ1(0) = EW FD(0, θ1). By construction, Γθ1(π

PD1 ) > EW FD(πPD1 , θ1). Now assume that we

can find θ = θ1 +ε′ such that EW FD(0, θ1 +ε′) > Γθ1(0) and EW FD(πPD1 , θ1 +ε′) < Γθ1(πPD1 ).

Clearly, EW FD(π, θ1 + ε′) crosses only once Γθ1(π), from above. Hence, as EW FD(π, θ1 + ε)and EW PD(π) are both convex in π, the former has to cross the latter only once, from above.This argument can be replicated until θ = θ2 > θ1, which is implicitly defined by

EW FD(πPD1 , θ2) = Γθ1(πPD1 ).

Now let’s construct an increasing sequence θn, ∀n ≥ 0 defined byEW FD(0, θn+1) > Γθn(0)

EW FD(πPD1 , θn+1) = Γθn(πPD1 ).

This sequence can behave in two different ways: either there exists N ∈ N such that, forall n ≥ N, θn ≥ θ∗; or θn < θ∗ for all n ∈ N. In the first case, the previous geometricarguments apply, and thus we can assert that EW FD(π, θ) crosses EW PD(π) only once, fromabove, when π ∈ [0, πPD1 ]. Let’s we prove that the second case can be ruled out. Assumethat θn < θ∗ for all n ∈ N. As the sequence θn is increasing and bounded (by θ = 1), itmust converge. Let’s denote this limit by θ. As the functions that define the sequence arecontinuous, they also converge towards EW FD(π, θ) and Γθ(π). These limit functions haveto satisfy

EW FD(0, θ) = Γθ(0)

EW FD(πPD1 , θ) = Γθ(πPD1 ).

Let’s define θ = θ + ε. Clearly, as EW FD(π, θ) increases with θ, EW FD(0, θ) > Γθ(0). Butif this were the case, we can construct Γ

θ(π) > Γθ(π), which is a contradiction. Hence, the

sequence θn does not converge to a value θ < θ∗.We conclude that, when θ ∈ [θ0, θ

∗], EW FD(π, θ) intersects EW PD(π) exactly once,from above, at π(θ). Applying again the Implicit Function Theorem, we can show that∂π(θ)/∂θ > 0.

44

9.7 Comparative statics

9.7.1 Increase in the capital endowment κ

Applying the Implicit Function Theorem, we can compute

∂π(θ)

∂κ=∂EW FD(π(θ), θ)/∂κ− ∂EW PD(π(θ))/∂κ

∂EW PD(π(θ))/∂π − ∂EW FD(π(θ), θ)/∂π.

This derivative has the sign of the numerator because the denominator is always positive atπ(θ). Therefore, as

∂EW PD(π(θ))

∂κ= 1

and

∂EW FD(π(θ), θ)

∂κ=

1− (1−π)(1−(1−π)L−1)b

´ cFDR (π)

0

∂T (c, θ)

∂κ︸ ︷︷ ︸<0

dc when 0 ≤ π ≤ πFD1

1− (1−π)(1−(1−π)L−1)b

´ c10

∂T (c, θ)

∂κ︸ ︷︷ ︸<0

dc when πFD1 ≤ π ≤ 1,

the numerator is strictly positive, except at π(θ0) = 0 when it is equal to zero. Hence,∂π(θ)/∂κ ≥ 0.

9.7.2 Increase in the project’s benefit B

Applying the Implicit Function Theorem, we can compute

∂π(θ)

∂B=∂EW FD(π(θ), θ)/∂B − ∂EW PD(π(θ))/∂B

∂EW PD(π(θ))/∂π − ∂EW FD(π(θ), θ)/∂π.

Again, this derivative has the sign of the numerator. Let’s compute

∂EW PD(π(θ))

∂B=

πbcPD(π)Φ(π) when 0 ≤ π ≤ πPD1

π when πPD1 ≤ π ≤ 1

where

Φ(π) ≡ 2 + (L− 2)π

L+ 1− π,

45

and

∂EW FD(π(θ), θ)

∂B=

πbcFDR (π) when 0 ≤ π ≤ πFD1

πbcFDNAR(π) when πFD1 ≤ π ≤ πFD2

πbcFDNR(π) when πFD2 ≤ π ≤ πFD3

π when πFD3 ≤ π ≤ 1.

When πPD1 ≤ π(θ) ≤ πPD2 ≡ b/B,

∂EW FD(π(θ), θ)

∂B<∂EW PD(π(θ)

∂B.

So ∂π(θ)∂B

< 0.

When π(θ) = 0,∂EW FD(π(θ), θ)

∂B=∂EW PD(π(θ))

∂B= 0.

So ∂π(θ)∂B

= 0.To show what happens when π(θ) is close to zero, we need to analyze how the difference

c(π)− cPDR (π)Φ(π) (14)

evolves on the locus (π(θ), θ) when θ0 is close to zero. Let’s study the sign of

∂∂θ

[cFDR (π(θ))− cPDR (π(θ))Φ(π(θ))

]∣∣θ=θ0

=[∂cFDR∂π−(∂cPD

∂πΦ(π) + cPD ∂Φ(π)

∂π

)]∂π(θ)∂θ

+∂cFDR∂θ

.

As

∂cFDR∂π

∣∣∣∣θ=θ0

=B − b+ L

cFDR (π(θ))

θ0− (L− 1)T (cFDR (π(θ)), θ0)

1 + 1θ0

<2L [(L+ 1)2(B − b) + (L2 + 1)b]

(L+ 1)4

because T (cFDR (π(θ)), θ0) > cFDR (π(θ))/θ0,[∂cFDR∂π−(∂cPD

∂πΦ(π) + cPD

∂Φ(π)

∂π

)]< −b(L+ 1)2L2

(L+ 1)4< 0.

Now, we analyze

∂π(θ)

∂θ

∣∣∣∣θ=θ0

=∂EW FD(π(θ), θ)/∂θ

∂EW PD(π(θ))/∂π − ∂EW FD(π(θ), θ)/∂π

∣∣∣∣θ=θ0

.

46

The derivatives are∂EW FD(π(θ), θ)

∂θ

∣∣∣∣θ=θ0

=b(L2 + 1)2

2(L+ 1)4,

∂EW PD(π(θ))

∂π

∣∣∣∣θ=θ0

=L[4(L+ 1)B − (5L+ 2− L2)b]

2(L+ 1)3

and∂EWFD(π(θ),θ)

∂π

∣∣∣θ=θ0

= 1b

{´ θ0b1+θ0

0

[B − b+ T (c, θ0)− L(T (c, θ0)− c

θ0)]dc

}.

As c/θ0 < T (c, θ0) ≤ b,

∂EW FD(π(θ), θ)

∂π

∣∣∣∣θ=θ0

≥ 2L(B − b)(L+ 1)2

+Lb(L2 + 1)

(L+ 1)4− 2L3b

(L+ 1)4

and thus∂π(θ)

∂θ

∣∣∣∣θ=θ0

≥ (L2 + 1)2

L2(L+ 1)2.

As

∂cFDR∂θ

∣∣∣∣θ=θ0

=b(L+ 1)2

(L+ 1)4,

we finally obtain

∂

∂θ

[cFDR (π(θ), θ)− cPDR (π(θ))Φ(π(θ)

]∣∣∣∣θ=θ0

< −b(L+ 1)2L2

(L+ 1)4.

(L2 + 1)2

L2(L+ 1)2+b(L+ 1)2

(L+ 1)4,= 0.

So, at π(θ0) = 0, the derivative of (14) is negative, implying that when θ is slightly above

θ0, ∂π(θ)∂B

< 0.

9.7.3 Increase in the number of regions L

Applying the Implicit Function Theorem, we can compute

∂π(θ)

∂L=∂EW FD(π(θ), θ)/∂L− ∂EW PD(π(θ))/∂L

∂EW PD(π(θ))/∂π − ∂EW FD(π(θ), θ)/∂π.

Again, this derivative has the sign of the numerator. Let’s compute

∂EW PD(π(θ))

∂L=

− (L−1)(1−π)2(πB+(1−π)b)2

b(L+1−π)3when 0 ≤ π ≤ πPD1

0 when πPD1 ≤ π ≤ 1

47

and

∂EW FD(π(θ), θ)

∂L=

1b(1− π)Lln(1− π)

´ cFDR (π)

0(T (c, θ0)− c

θ0)dc when 0 ≤ π ≤ πFD1

1b(1− π)Lln(1− π)

[´ cFDNAR(π)

0(b− c

θ0)dc−

´ c10

(b− T (c, θ0))dc]

when πFD1 < π ≤ πFD2

1b(1− π)Lln(1− π)

[´ c20

(b− cθ0

)dc−´ c1

0(b− T (c, θ0))dc

]when πFD2 < π ≤ 1.

So, when π(θ) = 0, ∂π(θ)∂L

> 0. On the other hand, when πPD1 ≤ π(θ) ≤ 1, ∂π(θ)∂L

< 0

48