Soft Budget Constraints, Interregional Redistribution and Spillovers
by
a
Emilson C. D. Silvaa
School of Business, University of Alberta, Edmonton, Alberta, Canada
E-mail: [email protected]
Xie Zhub,1
b
Department of Economics, Oakland University, Rochester, MI 48309, USA
E-mail: [email protected]
Abstract: In a federal economy, we show that the occurrence of bailouts depends on whether
there are spillover effects, on whether the spillover effects are positive or negative, on their
impacts on the privately informed region and on the ability of the center of providing
interregional transfers ex-ante and ex-post. One of our main results is at odds with the “too big to
fail” argument for the occurrence of bailouts. For excessively large regional projects, the
privately informed provider prefers to report its unit cost truthfully, leading to efficient
provision. The existence of ex-ante and ex-post interregional transfers may prevent bailouts.
Keywords: Soft budget constraint, interregional spillovers, interregional income transfers,
federation, too big to fail.
JEL Classification: H71, H72, H77, D63.
1
Corresponding author: Xie Zhu, 411 Elliott Hall, Oakland University, Rochester, MI 48309,
USA; Tel: 001-248-370-2505; Fax: 001-248-370-4275; Email: [email protected].
1
1. Introduction
Soft budget constraints for subnational governments are common in many countries around the
world. For example, Rodden et al. (2003) provides case studies of the soft budget constraint
problem in eleven countries (Argentina, Brazil, Canada, China, Germany, Hungary, India,
Norway, South Africa, Ukraine, and the United States). Following the soft budget constraint
definition for firms in Maskin (1996), a soft budget constraint exists for a subnational
government if it can extract a larger grant ex-post from the central government than would have
been considered efficient ex ante. Facing soft budget constraints, subnational governments may
inefficiently expand their expenditures in the hope that the central government will help the
financing of excessive expenditures through additional transfers that bail them out ex-post.
Kornai (1979, 1980 and 1986) first introduced the concept of soft budget constraints to
describe the behavior of state owned enterprises in centrally planned economies. Dewatripont
and Maskin (1995) and Maskin (1996) are early classical works analyzing the soft budget
syndrome for firms and their creditors.2 The soft budget constraint issue in the context of
intergovernmental relationship has been examined by many authors both theoretically and
empirically.3 Here we briefly present some major theoretical works. Qian and Roland (1998),
Breuillé et al. (2006) and Breuille and Vigneault (2009) examine how fiscal competition under
factor mobility may harden or soften the budget constraints of lower-level governments. Qian
and Roland (1998) also investigate how monetary centralization may strengthen fiscal discipline
of subnational governments. Goodspeed (2002) analyzes the political economy motivation of the
2
See Kornai et al. (2003) for a survey of the theoretical literature of the soft budget constraint syndrome in firms
and governments.
3
See Vigneault (2007) for a recent survey.
2
central government to provide bailouts to gain political support. As pointed out by Akai and Sato
(2008), if subnational governments determine the local expenditure ex-ante, soft budget
constraints may lead to over spending due to the cost sharing effects (e.g., Goodspeed, 2002;
Inman 2003), whereas if subnational governments determine local taxation ex-ante, soft budget
constraint may lead the subnational governments to exert too little effort in tax collection due to
the revenue sharing effects (e.g., Köthenbürger, 2004, 2007; Wildasin 1997). Akai and Silva
(2009) demonstrate that the central government’s ability to make interregional income transfers
ex-ante and ex-post can cure the soft budget syndrome when the regional public goods do not
produce interregional spillovers. Wildasin (1997) addresses the relationship between positive
interregional externalities and bailouts.4 The author shows that a bailout can be attractive to both
the central government and the regional government if the provision of a regional public good
generates sufficiently important interregional spillover benefits (measured by high marginal
external benefits). The central government is more willing to bail out larger regional
governments than smaller ones, an example of the notorious “too big to fail” argument.
Depending on the functional specification, stronger marginal external benefits increase the size
of the bailout provided by the central government and hence strengthen the incentive of a
regional government to induce a bailout.
Like Wildasin (1997), this paper investigates central and subnational fiscal decision making
when the central government may not be able to credibly commit to a no-bailout policy and the
regional public goods produce interregional spillover effects. Motivated by the provision of
various types of regional public goods, we examine how the incentives of central and regional
governments of providing or inducing bailouts vary in the presence of positive spillovers,
negative spillovers or in the absence of spillovers. Examples of interregional spillovers
4
See Crivelli and Staal (2013) for an investigation of soft budget constraints and district size.
3
associated with the provision of regional public goods abound. Suppose, for instance, that the
public facilities are public hospitals. Everyone benefits from the total sum of hospital services
(i.e., the amount of the federal public characteristic associated with the regional public goods) in
the federation because health services may prevent the spread of contagious diseases such as the
H1N1 virus. There are also regional-specific benefits associated with provision of public
hospitals, since some of their services may benefit the regional populations only. In some other
instances, the spillover effect may be negative. Suppose, for example, that the public facilities
are public parks, that usage of public parks damages some of its flora and that such damages
generate emissions of a uniformly mixed pollutant in the atmosphere (say, carbon dioxide). This
type of pollution, therefore, represents the federal public characteristic in this case. Since public
parks also generate many amenities that are enjoyed by those who visit them only, they also
produce regional-specific benefits.
We develop a model of a two-tier federation composed of a central government and two
regional governments. Region 1’s government has private information on the unit production
cost of its regional public facility prior to the commencement of production activities (i.e., exante). Such information becomes common knowledge only after region 1 starts production of its
regional public facility (i.e., ex-post). Policy making proceeds in a sequential manner. Ex-ante,
based on the reported unit cost by region 1’s government, the central government determines the
sizes of the regional public projects, collects taxes and provides grants to the regional
governments to produce regional public facilities. If region 1’s government underreports its unit
cost, the size of the regional public facility in region 1 will be larger than the size under truthful
reporting and region 1 will need an additional grant to complete the production of its public
facility. Ex-post, the center faces the question of whether or not to provide additional funding
4
(i.e., the bailout) if underreporting occurs. We find that decision making by the central and
regional governments depends on the form of interregional spillovers associated with the
regional public facilities. If the regional public facilities do not generate interregional spillovers,
the central government always finds it socially desirable ex-post to soften the budget constraint
for region 1’s government if it underreports its unit cost. If the regional public facilities generate
spillovers, bailing out a cheating region leads to a higher level of the federal public characteristic
than the no bail-out policy. In the presence of a negative spillover, the higher amount of the
harmful external effects due to the higher level of the federal public characteristic in the event of
a bailout represents an extra disincentive to provide a bailout by the central government relative
to the situation in which no spillover is present. Depending on the functional specification, it is
shown that the central government may credibly commit to hard budget constraints in the
presence of a negative spillover because the social costs may be greater than the social benefits
of completing region 1’s public facility which causes external harms.
In the presence of a positive spillover, surprisingly, providing a bailout does not always lead
to more beneficial spillovers than shutting down region 1’s public facility does. If the
interregional spillover is positive and if the regional marginal utilities from the federal public
characteristic are positive, the larger amount of the beneficial external effects resulting from the
bailout compared with the no-bailout policy presents an extra incentive to the center to provide
the bailout relative to the situation without the spillover. If the regional marginal utilities from
the federal public characteristic are negative, the bailout which results in a higher amount of the
federal public characteristic will lower the level of aggregate beneficial external effects in the
federation compared with the no-bailout policy. The smaller beneficial external effects serve as
an extra cost to the central government to provide the bailout relative to the situation without the
5
spillover. Depending on the functional specification, the central government may still be willing
to bail out a cheating region in the presence of a positive spillover despite the possible extra
disincentive created by the positive spillover.
From region 1’s perspective, the bailout outcome yields a higher size of the regional public
facility and a higher level of the federal public characteristic than the truthful reporting outcome.
In the presence of a negative spillover, the higher amount of the harmful external effects
represents an extra disincentive to underreporting relative to the situation without spillovers.
Depending on the functional specification, region 1 may always choose to underreport the unit
cost if facing a soft budget constraint in the presence of a negative spillover, or in the absence of
spillovers. Similar to the center’s bailout decisions, the larger amount of the federal public
characteristic caused by underreporting in the presence of a positive spillover may create an extra
incentive or disincentive to cheat depending on whether the regional marginal utilities from the
federal public characteristic are positive or negative. Though strong positive marginal external
effects may promote cheating relative to the situation without spillover effects, on the other hand,
in sharp contrast to Wildasin (1997), we show that region 1’s government prefers to report
truthfully rather than underreport its unit cost if the marginal external benefits are sufficiently
strong. The rationale for this result is that strong positive marginal external effects may lead the
central government to choose too large a size of the regional public facility so that the regional
marginal utilities from using the facility become negative in region 1. In such a case, region 1
prefers to make an honest report to avoid producing an “oversized” regional public facility.
Depending on the functional specification, it can be shown that the stronger are the marginal
external benefits, the higher is the incentive of the privately informed region to report truthfully;
truthful reporting not only occurs when the regional marginal utilities from using the regional
6
public facility are negative, but also occurs when the regional marginal utilities from using the
regional public facility are positive. This interesting result indicates that the “too big to fail”
argument may not be applicable to excessively large regional public projects, since the regional
governments would find it desirable to prevent such excess by behaving honestly. The “too big
to fail” argument seems appropriate for regional projects ranging from moderate to large sizes.
Another contribution we make to the literature on soft budget constraints in
intergovernmental relations is that we extend the results of Akai and Silva (2009) to the
circumstances in which regional public goods generate interregional spillovers. In our two – tier
federation, region 1’s government will choose to report its true unit cost, if the central
government which cares about interregional equality as well as efficiency is capable of
implementing income transfers across the regions both ex-ante and ex-post. The equalizing
transfers provide region 1’s government with incentives to report the true cost to maximize social
welfare – same objective of the central government. Truthful reporting also occurs in a context
of decentralized leadership, whereby the regional governments determine the sizes of the public
facilities, levy and collect taxes to finance the public expenditures, and the regional governments
make fiscal policy decisions before the central government transfers incomes across the regions.5
The paper is organized as follows. Section 2 builds the basic model. In section 3, the central
government cares only about efficiency and does not promote equalizing transfers across the
regions. We characterize the sequential policy game played by the central government and the
regional governments in subsection 3.1. By imposing additional assumptions on utility functions,
we analyze policy making when regional public projects do not generate interregional spillovers,
5
See, e.g., Köthenbürger (2004) and Akai and Sato (2008), for discussions on fiscal policy making with
decentralized leadership.
7
generate negative interregional spillovers and generate positive interregional spillovers in
subsections 3.2.1 – 3.2.3, respectively. We provide a numerical example of positive interregional
spillovers in subsection 3.3. In section 4, the central government cares about both efficiency and
equity and implements interregional income transfers. The sizes of the regional public projects
are centrally determined in subsection 4.1 and are determined in a decentralized fashion in
subsection 4.2. Section 5 concludes the paper.
2. The Model
Consider a federation with two regions, indexed by j , j  1, 2 . There are two regional
governments and one central government, which shall be referred to as “regions” and “center,”
respectively. Each region may produce a public facility of variable size. We assume that the unit
cost of producing a public facility in region 1 is high and is denoted cH . This information is
known by region 1 only prior to the commencement of production activities (i.e., ex-ante). Exante, we assume in sections 2 and 3 that the center and region 2 know that region 1’s per unit
cost can take two values - either a low one, cL , or a high one, cH . Let cL  1 and cH  c  1   ,
where   0 is an exogenous component of the high unit cost. In section 4, we assume that the
center and region 2 perceives that region 1’s per unit cost can take any value in the interval 1, c.
To keep things simple, we assume that region 2’s unit cost is low and that this piece of
information is common knowledge. This assumption enables us to focus on the game played by
region 1 and the center.
Following the soft budget literature, we assume that the center is unable to commit to a
revelation mechanism. The center and region 2 become fully informed about region 1’s cost type
sometime after this region initiates production of its public facility (i.e., ex-post). There are two
alternative events that reveal region 1’s cost type ex post: (i) completion of the public facility’s
8
production process; or (ii) region 1’s request for an additional grant to complete its production
process. The first event occurs if region 1 reveals its cost type truthfully to the center. The
second event occurs if region 1 lies to the center about its cost type (i.e., if it cheats).
In the sequential game played by the center and region 1, we assume that ex ante the center
acts on the belief that the public facilities will be completed ex post. We demonstrate that this
belief is time-consistent for the public facility produced in region 2 because the center knows its
cost type ex ante. However, the center is aware that its ex-ante belief may turn out to be timeinconsistent with respect to completion of the public facility in region 1. The center knows that it
may find it desirable, ex post, to leave the production process in region 1 unfinished.
The timing of the sequential game is as follows. In the first stage, region 1 sends a report of
its unit cost ĉ , cˆ 1, c , to the center.6 The center observes the report and in the second stage
determines the sizes of the regional public facilities, gˆ j  cˆ  , j  1, 2 , collects taxes to finance the
provision of regional grants and distributes the grants to the regions. We assume that both
regions are equally populated and normalize the population of a region to be equal to one. The
center’s ex-ante budget-balance condition is
2
2tˆa   Cˆ j ,
(1)
j 1
ˆ ˆ1  cˆ  and Cˆ 2  gˆ 2  cˆ  denote the
where tˆ a is the income tax paid by each resident, and Cˆ1  cg
grants supplied by the center to regions 1 and 2, respectively. For expositional simplicity, we
assume that the cost of changing the size of a public facility after the beginning of its production
process is prohibitively high. The grant given to a region represents a sunk cost to the center.
6
The reported unit cost ĉ will be a continuous variable with support 1, c in section 4.
9
If region 1 reports its cost type truthfully (i.e., ĉ  c ), the actual cost of production in region
1, C1  cgˆ1  cˆ   cgˆ1  c  , is equal to the amount of the ex-ante grant provided by the center, Cˆ1.
Therefore, the outcome of the sequential game is equivalent to the outcome the center obtains if
it has full information about the unit cost of production in region 1. This equivalence motivates
us to refer to the situation in which region 1 reports its unit cost truthfully as ‘the first best,’
henceforth. In the first best, the policy game ends immediately after the second stage. Both
regions complete production of the public facilities and subsequently consumption takes place.
If region 1 misreports its cost type (i.e., cˆ  1 ), the actual cost of production,
C1  cgˆ1 1  1    gˆ1 1 , is higher than the amount of the ex-ante grant, Cˆ1  gˆ1 1 , provided
that gˆ1 1  0 . Thus, region 1 needs additional funds equal to
C1  C1  Cˆ1   gˆ1 1  0
(2)
in order to complete its public facility. According to expression (2), region 1 contacts the center
to request the additional grant,  gˆ1 1 . Upon receipt of such a request, the center becomes
aware that region 1 cheated. The policy game then continues to the third stage, at which time the
center decides whether or not to provide additional funding (i.e., the bailout). If the center
decides in favor of the bailout, it collects an additional income tax, t p , from every citizen to
finance the additional expenditure. Hence, the center’s ex-post budget-balance condition is
2t p  C1 ,
(3)
where   0,1 is an indicator function:   0 if the bailout does not occur; and   1 if the
bailout occurs. If the center decides against the bailout, the public project in region 1 shuts down.
The policy game ends immediately after the third stage.
10
As we mentioned above, consumption takes place after production activities. Ex post, the
resident of region j derives utility from consumption of x j units of a private good (numeraire)
and from the potential usage of a regional public facility of size g j . In most of what follows, the
consumer who resides in region j also derives utility (due to a positive spillover) or disutility
(due to a negative spillover) from the public facility provided by region k , j , k  1, 2, j  k .
The
resident
of
region
j
derives
the
following
utility
ex
post:
U  x j , j g j , G   x j  v  j g j   su  G  , where s 0,1 , G  1 g1  2 g 2 and  j , j  1, 2, is an
indicator function. The value that  j takes is conditional on a piece of information that is
available ex post only:  j  0 if region j does not complete production of its public project and
 j  1 if region j completes production of its public project. Let q j   j g j denote the ex-post
size for the public facility in region j, j  1, 2. With this modification, the ex-post utility of
consumer j is U  x j , q j , Q   x j  v  q j   su  Q  , where Q  q1  q2 . We assume that v  0  0
and, for all q j  0 , v  q j   0 . Thus, consumer j derives no utility from usage of the public
facility in region j if it is incomplete.
The parameter s is a spillover index. If s  0, the resident of region j is the only one who
enjoys benefits from provision of region j ’s public facility. If s  1, consumer j is affected by a
(positive or negative) spillover if the public facility in region k is completed, j , k  1, 2, j  k .
We assume that u  0   0 and, for all Q  0, u  Q   0 .
We examine both positive and negative spillover effects because they may yield different
outcomes in what respects the occurrence of cheating and bailouts. We adopt the following
11
convention in what follows: The spillover is positive (respectively, negative) if and only if
u  Q   0 (respectively, u  Q   0 ) for all Q  0 in the relevant range.
It is important to note that in the presence of spillovers the provision of regional public
facilities can be interpreted as contributions to an impure federal public good or bad. Following
this interpretation, we shall refer to the quantity Q as the federal public good (bad) when the
spillover effect is positive (negative). In addition, we refer to u  Q  as the federal utility
(disutility) associated with enjoyment of the federal public good (bad).
With or without spillover effects, we assume that U  x j , q j , Q  is increasing in q j  0 in the
relevant range. With negative or no spillover effects, this assumption requires that v  q j   0 for
all q j  0 . In the presence of positive spillover effects, however, we simply require that
v  q j   0 for some sufficiently small q j  0 : the assumption is that v  q j   u   Q   0 for all
q j  0 . This is consistent with: (i) v  q j   0 and u  Q   0 , but also with either (ii) v  q j   0
and u  Q   0 provided that the federal marginal utility is greater in absolute value than the
regional marginal utility, or (iii) v  q j   0 and u  Q   0 provided that the regional marginal
utility is greater in absolute value than the federal marginal utility. The second case captures a
situation in which the public facility in region j is large from the perspective of the regional
utility it yields. The third case captures a situation in which the public facility in region j is not
as large from the perspective of the regional utility it bestows, but the federal public good is large
from the utility it generates. This last case may occur when the federal marginal utility falls
sufficiently fast. We shall refer to the second case as the situation in which the public facility in
12
region j is ‘too large for the regional taste’ and to the third case as the situation in which the
federal public good is ‘too large for the federal taste.’
3. Social Efficiency as the Sole Objective
In this section, we shall assume that the center’s actions are solely motivated by its concern for
efficiency. We will later compare the outcomes of the analysis of this section with the outcomes
of the analysis of the next section, in which the center also cares about equity. This comparison
will enable us to clearly demonstrate that interregional redistribution may be a powerful
preventive medicine – it may prevent cheating and bailouts in federations.
Suppose that the center is utilitarian. Its objective function is W U1,U 2   U1  U 2 . Suppose
also that consumer j is initially endowed with I  0 units of the numeraire good. Thus, the two
sources of asymmetry across regions are the unit costs of producing a regional public facility and
the degree at which the information about the unit cost of production is private. The ex-post
budget constraint of consumer j is
x j  tˆa  t p  I , j  1,2.
3.1.
(4)
The Sequential Policy Game
Consider the sequential policy game. We start by examining the second stage. The center
observes ĉ and then chooses  gˆ j 
and tˆ a to maximize social welfare, conditional on the unit
j 1,2
  xˆ
2
cost reported by region 1,
j 1
j




 v ˆj gˆ j  su ˆ1 gˆ1  ˆ2 gˆ 2 , subject to (1), gˆ j  0, j  1, 2, and
xˆ j  tˆa  I , j  1, 2,
(5)
where ˆ j , j  1, 2, are the beliefs held by the center concerning completion of the public
facilities. As we mentioned above, ˆj  1, j  1, 2. It is also important to note that ex post the
13
center can adjust its ex-ante choices concerning the regional amounts of the numeraire good to
be consumed because it is able to adjust its expenditure ex post. Thus, ex post, x j  xˆ j  t p .
Equation (3) indicates that: (i) if there is no bailout, t p  0 ; and (ii) if there is bailout, t p  0 .
Consequently, equations (5) are identical to equations (4). Equation (1) informs us that
2
tˆa   Cˆ j 2 . Together with (5), this result allows us to write the center’s objective function as
j 1
ˆˆ
  I  v  gˆ   su Gˆ   cg
2
j 1
j
1

 gˆ 2  2 , where Gˆ  gˆ1  gˆ 2 . The first-order conditions for an
interior solution in the second stage are7
 
(6a)
 
(6b)
v  gˆ1   2su Gˆ  cˆ,
v  gˆ 2   2su Gˆ  1.
Let gˆ j  cˆ  , j  1, 2, denote the implicit functions defined by equations (6a) and (6b). These
functions inform us that the sizes of the public facilities vary according to region 1’s reported
cost. Equations (6) are the report-constrained Samuelson conditions for the optimal provision of
the public facilities. For each region, the chosen size of the public facility is the one that equates
the marginal social benefit to the marginal social cost, given region 1’s report.
As we shall see below, the comparative statics for the system of equations (6a) and (6b) with
respect to the reported unit cost, ĉ , provide us with important pieces of information with respect
to the incentives facing region 1 in its choice of whether or not to report truthfully in the first
stage of the game. Let gˆ sj , j  1, 2, denote the center’s choices. If s  0 , we have gˆ 0j and if
s  1 , we have gˆ 1j , j  1, 2 . Suppose, initially, that there is no spillover effect. Plugging
7
We assume that each consumer’s income level, I, is sufficiently large to ensure interior solutions.
14
gˆ 0j  cˆ  , j  1, 2, into equations (6a) and (6b) and then differentiating the implied equations with
respect to ĉ yields dgˆ10 dcˆ  1 v  gˆ10   0 and dgˆ 20 dcˆ  0 . Thus, we obtain:
gˆ10 1  gˆ10  c  , gˆ 20 1  gˆ 20  c  and Gˆ 0 1  Gˆ 0 c ,
(7a)
where Gˆ 0  gˆ10  gˆ 20 . Results (7a) inform us that the center’s choice with respect to the size for
the public facility in region 1 is distorted if region 1 cheats. Relative to the first best, the center
chooses a higher size for the public facility in region 1. Region 1’s choice, however, does not
affect the size for the public facility in region 2. The aggregate level of the regional public goods
in the federation is hence higher than the first best level.
Suppose now that there are spillover effects. Let Gˆ 1  gˆ11  gˆ 12 . Plugging gˆ 1j  cˆ  , j  1, 2, into
equations
(6)
and
 
then
 
dgˆ11 dcˆ  v gˆ 12  2u Gˆ 1 D  0,
differentiating
the
 
dgˆ 12 dcˆ   2u Gˆ 1
implied
equations
yields
D  0 and dGˆ 1 dcˆ  vgˆ 12  D  0,
 
where D  v  gˆ11  v  gˆ 12   2u Gˆ 1 v  gˆ11   v  gˆ 12   0 . Hence, we have
gˆ11 1  gˆ11  c  , gˆ 12 1  gˆ 12  c  and Gˆ 1 1  Gˆ 1  c  .
(7b)
According to results (7b), if region 1 cheats, the center’s choices yield a higher size for the
public facility in region 1, a lower size for the public facility in region 2 and a higher amount of
the federal public characteristic relative to the first-best choices. From results (7a) and (7b) we
see that cheating always leads to overprovision of the total sum of regional grants.
The solution to the center’s problem yields the following utility for the resident of region j :



 

Uˆ sj cˆ   I  v gˆ sj cˆ   su Gˆ s cˆ   cˆgˆ1s cˆ   gˆ 2s cˆ  2 , j  1, 2, s  0,1.
(8)
Anticipating the possibility of receiving an additional grant from the center ex post, region 1
makes its decision in the first stage by comparing its ex-post utility level from being honest with
15
its ex-post utility level from cheating in each of the two possible ex-post outcomes produced by
cheating. For s 0,1 , let U sE
denote region j ’s ex-post utility when region 1 reports its unit
j
cost truthfully, U sB
denote region j ’s ex-post utility when region 1 cheats and the center
j
subsequently provides a bailout and U sN
denote region j ’s ex-post utility when region 1 cheats
j
and the center decides against the bailout. If region 1 reports truthfully, the ex-post tax t p is zero
and  j  ˆj  1, j  1, 2. Then, the ex-post utility levels for both regions are the same as the exante utility levels described by expressions (8):
sE
sE
sE
ˆs
U sE
2, j  1, 2, s 0,1 ,
j  U j  c   I  v  g j   su  G   C
(9a)
sE
ˆs
 g1sE  g 2sE , and C sE  cg1sE  g 2sE .
where g sE
j  g j  c  , j  1, 2, G
If region 1 cheats and subsequently the center favors the bailout, the ex-post tax t p is equal to
C1 2 . The ex-post utility level for the resident in region j is
p
sB
sB
sB
ˆs
U sB
2, j  1, 2, s 0,1 ,
j  U j 1  t  I  v  g j   su  G   C
where
ˆs
g sB
j  g j 1 ,
j  1, 2,
G sB  g1sB  g 2sB
and
C sB  cg1sB  g 2sB .
(9b)
Let
F  cˆ; c   cgˆ1s  cˆ   gˆ 2s  cˆ  . Note that C sE  F  c; c  and C sB  F 1; c  . Differentiating F  cˆ; c 
with
respect
to
ĉ
yields
dF dcˆ  c dgˆ1s dcˆ  dgˆ 2s dcˆ   dgˆ1s dcˆ  dGˆ s dcˆ  0.
Hence,
C sB  C sE . This result, together with (7a) and (7b), leads us to conclude that:
Proposition 1: Relative to the first best, the bailout outcome is characterized by a higher size of
the public facility in region 1, a lower (in the presence of spillovers) or equal (in the absence of
spillovers) size of the public facility in region 2, a higher amount of the federal characteristic
(good or bad) and a higher total public expenditure.
16
Let R sB  U1sB  U1sE be the rent that region 1 obtains when it cheats and the center provides
the bailout ex post. Anticipating the bailout, region 1 cheats if and only if R sB  0 . Using (9a)
and (9b), we obtain
R sB  v  g1sB   v  g1sE   s u  G sB   u  G sE   C sE  C sB  2 .
(9c)
Since C sB  C sE , the third bracketed term of expression (9c) is negative. If s  0, the first
bracketed term is positive because g10 B  g10 E and v  0. The sign of R 0B depends on the
comparison between the gain from cheating, as it is described in the first bracketed term, and the
cost of cheating, as it is captured in the third bracketed term. If s  1 , cheating leads to a greater
level of the federal characteristic than truthful reporting because G1B  G1E . If the spillover is
negative, we must have v  0 . This guarantees that the first bracketed item is positive. The
second bracketed term is negative, however. Thus, the cost of cheating in the presence of a
negative spillover is the sum of the last two bracketed terms. If s  1 and the spillover is positive,
we must have u  0. The first and the second bracketed terms may be positive or negative, but
both cannot be negative.

We consider three possibilities.8 First, v  0 and u  0 for



gˆ11 cˆ   gˆ11 c , gˆ11 1 and Gˆ 1 cˆ   Gˆ 1 c , Gˆ 1 1 . The gain from cheating is the sum of the first two
positive bracketed terms. Second,

v  0 and u  0
for


gˆ11 cˆ   gˆ11 c , gˆ11 1
and

Gˆ 1 cˆ   Gˆ 1 c , Gˆ 1 1 . The federal public good is too large for the federal taste. The gain from
cheating is given by the first bracketed term and the cost of cheating is the sum of the last two
8
In addition to the three possibilities we discuss here, there are situations in which the signs of v and u change




over gˆ11cˆ  gˆ11c, gˆ111 and Gˆ 1 cˆ   Gˆ 1 c , Gˆ 1 1 and the signs of the first and the second bracketed terms are not
directly clear.
17
bracketed terms, as when the spillover effect is negative.



9
Third, v   0 and u  0 for

gˆ11 cˆ   gˆ11 c , gˆ11 1 and Gˆ 1 cˆ   Gˆ 1 c , Gˆ 1 1 . The first bracketed term is negative and the second
bracketed term is positive. The public facility in region 1 is too large for the regional taste. The
gain from cheating is given by the second bracketed term and the cost of cheating is the sum of
the first and third bracketed terms. As we show in Appendix A, the cost of cheating outweighs
the gain from cheating in this case. Hence, U 1E  U 1B and R1B  0. Region 1 prefers to report
truthfully. This last result is novel and thus merits to be formally recorded:
Proposition 2: Region 1 prefers to report its cost truthfully in the presence of positive spillover
effects whenever the public facility in this region is too large for the regional taste (i.e., the
regional marginal utility is negative).
Proof. See Appendix A.
If region 1 cheats and subsequently the center decides against the bailout, region 1 is unable
to complete production of its public facility. Since, in this case, 1  0 and  2  1 , we have
1  ˆ1 and 2  ˆ2 . The ex-post utility levels are
U1sN  I  su  G sN   C sN 2,
(10a)
U 2sN  I  v  g 2sN   su  G sN   C sN 2,
(10b)
where g 2sN  gˆ 2s 1, G sN  g 2sN and C sN  gˆ1s 1  g2sN . For future reference, note that C sB  C sN .
Let R sN  U1sN  U1sE denote the rent that region 1 obtains when it cheats and the center does not
9
In the first and second cases, if ĉ were a continuous variable with support 1, c , region 1 would choose to cheat if
the center favored the bailout, as we discuss in Appendix A; in the two-type situation, subsection 3.2.2 informs us
that region 1 may find it advantageous to report truthfully.
18
provide the bailout ex post. Anticipating that the center will not provide the bailout, region 1
cheats if and only if R sN  0. Utilizing (10a) and (10b), we have
       

R sN  v g1sE  s u G sN  u G sE  C sE  C sN 2.
  
(10c)

If s  0, R0 N  v g10 E  cg10 E  gˆ10 1 2 because g 20 E  g 20 N . Adding and subtracting cg10 E 2
on the right side of (10c) yields
 
 

R0 N   v g10 E  cg10 E  cg10 E  gˆ10 1 2  0,
(10d)
since the first bracketed term in the right side of (10d) is non-negative due to the fact that g10 E


maximizes v  g10 E   cg10 E if s  0 and the second bracketed term is positive because g10 E  0
and gˆ10 1  0. Thus, we conclude that
Proposition 3: In the absence of spillovers, region 1 does not cheat if it anticipates that the
center will not favor the bailout ex post.
If s  1, R1N may be positive or negative. However, some conclusions are forthcoming if c
is close to one. Remember that c  1   . Thus, from equations (6) we can conclude that, for
sufficiently small  , gˆ j 1     gˆ j 1 ,
j  1, 2. This implies that G1E  G1N  g11B  0 and
C1E  C1N . It follows that R1N is negative if the spillover is positive and the federal public good
is not too large for the federal taste, for sufficiently small  . It also follows that R1N is positive
if the spillover is negative, or if the spillover is positive and the federal public good is too large
for the federal taste, for sufficiently small  , whenever  u  G1E   u  G1N   v  g11E  . The left
side of this inequality is positive because G1E  G1N for sufficiently small  and u  0. This
positive quantity represents the opportunity gain from cheating in these two cases. When region
1 cheats instead of telling the truth, the public facility in region 1 is shut down rather than being
19
completed. Relative to the first best, the amount of the federal public characteristic is lower – it
equals G1N if region 1 cheats and G1E if region 1 reports truthfully. The right side of the
inequality above is the opportunity cost of cheating faced by region 1 in these two cases. This is
the utility consumer 1 must sacrifice when region 1 cheats instead of reporting truthfully. If the
federal utility or disutility associated with the federal public characteristic is sufficiently strong,
the opportunity gain from cheating outweighs the opportunity cost of doing so. Cheating is
beneficial to region 1 because it is an effective strategy to reduce the level of the federal public
bad, or to increase the level of the federal public good. In sum,
Proposition 4: For sufficiently small  and in the presence of positive spillover effects where
the federal public good is not too large for the federal taste, region 1 does not cheat if it knows
that the center will decide against the bailout. For sufficiently small  and in the presence of
negative spillover effects, or in the presence of positive spillover effects where the federal public
good is too large for the federal taste, however, region 1 cheats if the federal utility or disutility
is sufficiently strong and if it knows that the center will decide against the bailout.
2

sN
Let  s  W sB  W sN   U sB
j U j
j 1

denote the (ex-post) net social benefit of the bailout.
This benefit equals the ex-post level of social welfare in the presence of the bailout minus the expost level of social welfare in the absence of the bailout. The bailout occurs if and only if
 s  0. 10 Note that
 s  v  g1sB   2s u  G sB   u  G sN    g1sB .
(11a)
In writing equation (11a), we utilized the facts that g1sB  gˆ1s 1 and g2sB  gˆ 2s 1  g2sN . If s  0,
expression (11a) can be rewritten as
10
We assume that the center will provide a bailout ex-post if the net social benefit of the bailout is zero.
20
 0  v  g10 B   cg10 B   g10 B  0 .
(11b)
We have  0  0 because g10 B  0 and the bracketed term in the right side of (11b) is non-


negative due to the fact that g10 B maximizes v  g10 B   cg10 B when s  0 . Hence, we obtain the
following result:
Proposition 5: In the absence of spillovers, the center always provides the bailout ex post if
region 1 cheats ex ante.
Suppose now that s  1 . The amount of the federal public characteristic (good or bad) is
larger when the center favors the bailout than when it decides against it, since
G1B  G1N  g11B  0. When the federal public good is too large for the federal taste (or when the


spillover effects are negative), u  0 for Gˆ 1 cˆ   Gˆ 1 c , Gˆ 1 1 . This implies that the bracketed
term in the right side of expression (11a) is negative, representing thus an extra disincentive to
provide the bailout relative to the situation without spillover effects. In such circumstances, the
center’s opportunity gain from favoring the bailout is simply v  g11B  . The opportunity cost of

    

doing so is 2 u G1N  u G1B  g11B . If, on the other hand, u  0 for Gˆ 1 cˆ   Gˆ 1 c , Gˆ 1 1 , the
bracketed term in equation (11a) is positive. The center has an extra incentive to provide the
bailout relative to the situation in which no spillover effects are present. Its opportunity gain
from favoring the bailout is v  g11B   2 u  G1B   u  G1N  and its opportunity cost of doing so is
 g11B . In sum, in the presence of spillover effects the center may be less inclined to favor the
bailout than in the absence of spillover effects if the spillover effects are negative or the federal
public good is too large for the federal taste. The center may be more inclined to favor the
21
bailout in the presence of positive spillover effects than without spillover effects if the federal
public good is not too large for the federal taste.
3.2. Quadratic Utilities
To gain more insight into the incentives underlying the decision making of the center and region
1, we shall assume for the remainder of this section that the sub-utility functions u and v are
quadratic as follows: u  G   dG  G2 2 and vg j   ag j  g 2j 2 , j  1,2, where a  0 and d is
either zero to characterize negative spillovers or a sufficiently large positive quantity (to be
precisely defined below) to characterize positive spillovers. Since v  g j   a  g j represents
consumer j ’s marginal willingness to pay for the regional characteristic of the public facility
produced in region j , the parameter a measures the highest marginal willingness to pay for
such characteristic. Similarly, one can interpret d as each consumer’s highest marginal
willingness to pay for the federal public characteristic. To guarantee that the center finds it
desirable to provide positive grants in the second stage of the game, we assume that a    3c ,
where   0 ; namely, the highest marginal willingness to pay associated with the regional
characteristic is larger than three times the highest unit cost.
Given the quadratic specifications, the first-order conditions (6a) and (6b) become
a  gˆ1  2s d  Gˆ  cˆ,


(12a)


(12b)
a  gˆ 2  2s d  Gˆ  1.
The interior solution to the system of equations (12a) and (12b) satisfies
gˆ1s cˆ   a  2s1  d   (1  2s)cˆ 1  4s,
(13a)
gˆ 2s cˆ   a  2scˆ  d   (1  2s) 1  4s.
(13b)
Hence, the amount of the federal public characteristic is
22
Gˆ s cˆ   2a  4sd  cˆ  1 1  4s .
(13c)
Differentiating equations (13a) and (13b) with respect to  and  results in
gˆ sj cˆ   0, and gˆ sj cˆ   0, cˆ 1, c , s 0,1, j  1, 2.
(13d)
Results (13d) inform us that the sizes of the regional public facilities are increasing in both 
and . Utilizing equations (13a) – (13c), we can write the net social benefit of the bailout and
region 1’s cheating rents as follows:
s4d sd 2s  1  2sa  3  a  1  c4s  1  8c  2aa  6  4c   14
2
21  4s 
aa  2  2c   2c  3

,
21  4s 2
s 
R sB 

(14a)

s 2d  4s  11  c   4s c  2  c   4a  c  1  1  c  6  c   4a  c  1  5  c  1
2 1  4s 
2
s2d 2s 3a  1  sd   a  1  4s4a  c4a  2  c   1
2
21  4s 
sc8a  2  c   a5a  6  2 aa  c   a  1


.
21  4s 2
21  4s 2
, (14b)
R sN  
(14c)
3.2.1. No Interregional Spillovers
Suppose that s  0. Remembering that c  1   and a    3 1    , equations (14a), (14b)
and (14c) become
 0  4  14  4  3     6      2  0,
(15a)
R 0 B   2  0,
(15b)
R 0 N   15  5  6      6      8 2  0.
(15c)
The center decides in favor of the bailout – see (15a). Region 1 anticipates this outcome and
cheats – see (15b). If the center was able to commit to not providing the bailout, region 1 would
23
not cheat – see (15c). This is the standard soft budget syndrome. We summarize the results in
this subsection in the following proposition:
Proposition 6: For quadratic utilities, if there is no spillover effect, region 1 cheats because the
center decides in favor of the bailout ex post. Were the center able to commit to a policy of never
providing bailouts, region 1 would not cheat and subsequently the allocation would be first best.
3.2.2. Negative Interregional Spillovers
Suppose now that s  1 and d  0 . This characterizes a situation in which the spillover effects
are negative. Equations (14a) – (14c) give us
1  32   46  8  3     3  22    50,
(16a)
R1B   65  20  55   50  0,
(16b)
R1N  16   27  11     6  34    44  50.
(16c)
According to (16a), the center decides in favor of the bailout if and only if  is no larger than



  46  8  46  8 2  1232  3 2  22 
12
 6  16  3.
(16d)
If 0     , region 1 cheats – see (16b). The center does not favor the bailout if and only if
   . In this case, despite the center’s commitment to a hard budget constraint, region 1 still
finds it desirable to cheat – indeed, the right side of (16c) is positive when    . We gather the
main results in this subsection in the following proposition:
Proposition 7: For quadratic utilities, if the regional public facilities generate negative
interregional spillovers, region 1 always cheats. The center favors the bailout if and only if
0     , where   16  3 , in which case the level for the federal public bad is higher than
in the first best. The center decides against the bailout if and only if    , in which case the
level for the federal public bad is lower than in the first best.
24
3.2.3. Positive Interregional Spillovers
Suppose that s  1 and d     2    3  3  0, where   0 . This value for d guarantees
that uG   dG  G 2 2  0 in the relevant range. A higher value of  indicates a higher value of
d and hence higher marginal willingness to pay for the federal public good. Equations (14a) –
(14c) yield
 1  36  120  60  180  75  100  350  25  150  200 150  0, (17a)
R1B  135  50  175  30   150,
(17b)
R1N   36  450  150  480  270  375  975  100  550  700 450  0.
(17c)
If the center was able to commit to a policy of not providing bailouts, region 1 would not cheat –
see (17c). However, equation (17a) shows that the center decides in favor of the bailout.
Knowing this, region 1 decides whether or not to cheat according to (17b). The fact that this rent
may be negative is the most significant difference between this case and the previous two cases
where the cheating rent associated with the bailout is always positive. Equation (17b) shows that
the cheating rent associated with the bailout is decreasing in  , and will become negative if  is
sufficiently large – that is, if  is larger than   27  10  35 6. To understand the
relationship between R1B and  , first note that equations (13a) and (13b) inform us that a small
change in  has the same impact on gˆ 1j c  and gˆ 1j 1, i.e., gˆ 1j c   gˆ 1j 1   2 5 , j  1,2.
Hence, Gˆ 1 c    Gˆ 1 1   4 5 . Combining these results and equations (9a), the impact on
U11E of a small change in  is
 
 

U11E   v g11E 2 5  u G1E 4 5  G1E  c  1 5.
Combining these results and equations (9b), the impact on U11B of a small change in  is
25
(18a)
 

 
U11B   v g11B 2 5  u G1B 4 5  G1B  c  1 5.
(18b)
The first bracketed terms on the right hand sides of equations (18a) and (18b) represent how a
small change in  affects region 1’s benefits from enjoying the regional and federal public
characteristics when it reports the true unit cost and when it cheats, respectively. Since
   
 
 
g11B  g11E , G1B  G1E , v   0 and u  0, we have v g11E  v g11B and u G1E  u G1B .
Utilizing these conditions and substituting the expressions of G1E and G 1B in equation (13c) into
equations (18a) and (18b) respectively, we find that the first bracketed term on the right hand
side of equation (18a) is positive and greater than its counterpart which is also positive in
equation (18b). Therefore, a higher  increases region 1’s federal and regional benefits to a
greater extent if it reports truthfully than if it cheats. On the tax payment side, an increase in 
leads to the same increase in the tax borne by region 1’s resident if it reports truthfully or if it
cheats, since the second bracketed term in equation (18a) is the same as the second bracketed
term in equation (18b). Thus, we have R1B   U11B   U11E   0, from equations (18a)
and (18b).
In sum, with relatively small marginal willingness to pay for the federal public good, i.e.,
0     , equation (17b) states that R1B  0 and region 1 will cheat anticipating the ex-post
bailout provided by the center. As  increases, U11E and U11B move closer and R1B  0 when
   . If the marginal willingness to pay for the federal public good is large (i.e.,    ), region
1 prefers to be honest, since R1B  0 according to equation (17b).
The incentive of being honest gets stronger with higher marginal willingness to pay for the
federal public good because R1B   0. If   35  10  39  6 , we have v  0 , u  0 ,




and v  u  0, for gˆ11 cˆ   gˆ11 c , gˆ11 1 and Gˆ 1 cˆ   Gˆ 1 c , Gˆ 1 1 . This is the truthful
26
reporting situation described by Proposition 2. If the high marginal willingness to pay for the
federal public good makes the center choose a public facility for region 1 whose size is too large
for the regional taste, region 1 prefers to be honest. In addition, equation (17b) informs us that a
negative regional marginal utility is sufficient but unnecessary condition for truthful reporting.
For example, for     35  10  39  6 , region 1 reports the true unit cost even
 
though v g11E  0. 11, 12 We summarize the main results of this subsection as follows:
Proposition 8: For quadratic utilities, if the regional public facilities generate positive
interregional spillovers, the center always favors the bailout. Region 1 cheats if and only if
0     , where    27   10  35 6. Region 1 reports its true unit cost if and only if    .
With honest reporting, the allocation is first best. Were the center able to commit to a policy of
never providing bailouts, region 1 would not cheat and subsequently the allocation would be
first best.
Figure 1 illustrates Proposition 8 with   6  4.5 at   0.1. When   0.1, the area
above and including the line   6  4.5 in Figure 1 (excluding the vertical axis since   0 )
shows the combinations of  and  that induce region 1 to report truthfully. The area below the
line   6  4.5 (excluding the vertical and horizontal axes since   0 and   0 ) shows the
combinations of  and  that lead region 1 to cheat. As  increases, the curve   6  4.5
shifts upward with higher constant terms, implying that higher marginal willingness to pay for
11
We assume that region 1 chooses to report truthfully when it is indifferent between being honest and being
dishonest.
12
As we pointed out in footnote 8 and will demonstrate in Appendix A, truthful reporting does not occur if
 
v g11E  0 and the reported unit cost is a continuous variable.
27
the federal public good is needed for region 1 to be honest. One particular example of  we
show in the figure is   10.5 for   0.1 and   1.
3.3. Numerical Examples
We provide two numerical examples in this subsection to illustrate the analysis in the presence of
positive interregional spillovers. In both examples,   0.1. We let   0.1 in example 1 and
  1 in example 2. Hence, c  1.1 and a  3.4 in example 1; and c  2 and a  6.1 in example
2. In each example, we consider three values of  :    ,    and    . In example 1,
  6.45 and we let  to be 4.2, 6.45 and 9.2, respectively. The values of the parameter
d    2    3  3 are 5, 7.25 and 10, respectively. In example 2, with a higher value of  ,
  10.5 and we let  to be 8.3, 10.5 and 13.3, respectively. The values of the parameter d are
hence 10, 12.2 and 15, respectively. We summarize the results on the equilibrium values of the
variables in these two examples in Table 1. Consistent with Proposition 8,  1 is always positive
and R1N is always negative in Table 1. Furthermore, R1B  0 if    , R1B  0 if    , and
R1B  0 if    .
The equilibrium values in Table 1 also allow us to better understand why R1B is decreasing
in  . As  rises, it has the same impact on gˆ 1j c  and gˆ 1j 1 , j  1, 2. For example, if 
increases from 4.2 to 6.45 in example 1, both gˆ11 c  and gˆ11 1 increase by 0.9, and both gˆ 12 c 
and gˆ 12 1 increase by 0.9. We can also see that as  grows, C 1E and C 1B increase by the same
   
   
amount, the increase in v g11E  u G1E is greater than the increase in v g11B  u G1B , and hence
U11E grows faster than U11B . For example, when  increases from 4.2 to 6.45 in example 1, both
   
   
C 1E and C 1B increase by 1.89, while v g11E  u G1E increases by 14.13 and v g11B  u G1B
increases
by
a
smaller
amount
14.08.
28
Hence,
in
example
1,
though
U11E  I  15.21  U11B  I  15.25 at   4.2, we have U11E  U11B  I  28.39 at   6.45
because U11E increases by 13.18 and U11B increases by 13.14 when  increases from 4.2 to 6.45.
When  grows from 6.45 to 9.2, U11E increases by 21.62 and U11B increases by 21.56 so that we
have U11E  I  50.01  U11B  I  49.95 at   9.2.
4. The Center Cares About Both Efficiency and Equity
Suppose that the center cares about both efficiency and equity. Its objective function is
W U1,U2   U1   U2 , where   0 and   0. By assuming that the center has the
ability to implement interregional income transfers both ex ante and ex post, we demonstrate that
such income transfers can cure the soft budget constraint syndrome that may occur when the
regional public facilities generate interregional spillovers. Instead of assuming that the reported
unit cost ĉ can take two values as in sections 2 and 3, we assume that ĉ is a continuous variable
with support 1, c in this section. Our results thus extend the positive results of Akai and Silva
(2009), which shows that such income transfers solve the soft budget problem in the absence of
interregional spillovers.
The budget constraint of the resident in region j with income transfers is:
x j  tˆa  t p  I  ˆ aj   pj , j  1,2,
(19)
where ̂aj is the ex-ante income transfer and  pj is the ex-post income transfer received (if
positive) or paid (if negative) by the resident in region j. Since the transfers are redistributive,
2
we have
ˆaj  0 and
j 1
2

j 1
p
j
 0.
29
4.1. Centrally Determined Regional Public Facilities
In this subsection, we maintain the assumption that the center determines the sizes of the regional
public facilities. In addition, the center collects taxes to finance the regional public facilities and
is in charge of ex-ante and ex-post interregional income redistribution. Region 1 decides whether
or not to report its unit cost truthfully. We examine a sequential game with the similar timing of
policy making as in section 3.
If the game reaches the third stage, the center knows that region 1 cheated and decides
whether or not to favor the bailout and chooses the amounts of income that need to be
redistributed across regions. We assume that the center decides to provide an additional grant to
region 1 whenever cheating occurs. Hence,   1 and  j  1, j  1,2. Later, we will demonstrate
that this decision rule is indeed socially optimal. We thus write the ex-post budge balance
condition as follows:
2t p  C1.
(20)
The center chooses  pj j 1, 2 to maximize
2


p
p
  Uˆ   j  t ,
j 1
(21a)
2
subject to the ex-post budget balance condition (20) and   pj  0. As we shall demonstrate
j 1
below, the center’s ex-ante income transfers equalize the two regions’ ex-ante utilities, which are
denoted Û for both regions. This implies that there is no need to redistribute income across
regions ex post:
 pj  0, j  1,2.
(21b)
30
Results (21b) follow from the fact that the center wishes to achieve horizontal equity ex-post,
i.e., equalization of the two regions’ ex-post utilities. Since the center already achieves horizontal
equity ex ante, and since the ex-post grant to region 1 is equally borne by the residents of the two
regions, there is no need to implement transfers to obtain horizontal equity ex post. With
equations (21b), the ex-post utility of each resident can be denoted as U  Uˆ  C1 2 . The
center’s objective function (21a) hence can be written as 2 U , which tells us that maximization
of social welfare is equivalent to the maximization of the ex-post utility of each resident.
In the second stage of the game, the center acts on the beliefs ˆ j  1, j  1,2. Given region 1’s
reported cost ĉ, the center chooses ˆ aj , gˆ j j 1, 2 and tˆ a to maximize

 
  xˆ j  vgˆ j   su Gˆ ,
2
j 1
(22a)
2
subject to the ex-ante budget balance condition (1),
ˆ
j 1
a
j
 0 and
xˆ j  tˆa  I  ˆ aj , j  1,2.
(22b)
2
Equation (1) informs us that tˆa   Cˆ j 2, which, together with (22b), allows us to write the
j 1
center’s objective function as



a
  I  vgˆ j   su Gˆ  ˆ j  cˆgˆ1  gˆ 2  2 .
2
j 1
The first-order conditions for ˆ aj j 1, 2 satisfy
(22c)
2
ˆ
j 1
a
j
   
 Uˆ1   Uˆ 2 ,
 0 and
(22d)
which leads to
Uˆ1  Uˆ 2 and ˆ1a  ˆ a2  vgˆ 2   vgˆ1  2.
31
(22e)
The first-order conditions for gˆ j j 1, 2 yield the Samuelson conditions (6a) and (6b), which
implicitly define gˆ j cˆ , j  1,2. Equation (22d) informs us that the center implements ex-ante
income transfers so that the marginal social utilities of income are equalized across regions.
Since  is strictly concave, equation (22d) implies that the two regions’ ex-ante utilities are
equalized, as we pointed out in the third stage of the game. Utilizing equations (6) and (22e), the
level of the utility for the resident in region j implied by the solution to the center’s problem in
the second stage can be written as follows:
 
Uˆ j cˆ   Uˆ cˆ   I  su Gˆ cˆ   vgˆ1 cˆ   vgˆ 2 cˆ  2  cˆgˆ1 cˆ   gˆ 2 cˆ  2 , j  1,2.
(23)
In the first stage of the game, region 1 decides whether to report truthfully. It knows that its
ex-ante utility level will be given by equation (23). It also knows that the center has the power to
implement income transfers and make cost adjustment ex post. Hence, its ex-post utility level to
be determined by the center is:
 
U cˆ; c   Uˆ cˆ   C1 2  I  su Gˆ cˆ   vgˆ1 cˆ   vgˆ 2 cˆ  2  cgˆ1 cˆ   gˆ 2 cˆ  2 .
(24)
Region 1 chooses cˆ  1, c to maximize U cˆ; c . Since U cˆ; c  is the objective function for both
the center in the third stage and region 1 in the first stage of the game, region 1 chooses to report
cˆ  c in the first stage of the game just like the center would have done under a similar
circumstance (please refer to Appendix B for the technical proof of cˆ  c ).
With truthful reporting by region 1 in the first stage, the center’s beliefs ˆ j  1, j  1,2, in the
second stage are confirmed in equilibrium. The sizes of the regional public facilities determined
by equations (6) are at their first best levels. The subgame perfect equilibrium allocation
corresponds to the social optimum with perfect information. The center’s decision to provide
32
additional fiscal assistance ex post is hence socially optimal, though no cost adjustment actually
occurs ex post. Thus, we have the following proposition:
Proposition 9: With centrally determined regional public facilities, the subgame perfect
equilibrium of the sequential policy game in which the center implements interregional income
transfers ex ante, and is capable of adjusting cost and redistributing income ex post, corresponds
to the social optimum with perfect information.
4.2. Decentralized Leadership
In this subsection, the center’s only role is to transfer incomes across regions. The federation is
characterized by decentralized leadership. The regions are in charge of determining the sizes of
their public facilities. The sequential game played ex-ante has now three stages. In the first stage,
region 1 sends a report about its unit cost. In the second stage, the regions determine the sizes of
their public facilities and collect lump sum taxes to finance their expenditures. In the third stage,
the center redistributes income across regions. If region 1 cheats, the game reaches the fourth
stage. In this stage, the center redistributes income across regions and region 1 collects additional
taxes from its resident to finance the additional cost of provision. Hence, t p  C1 for consumer
1 and t p  0 for consumer 2.
Since the ex-ante income transfers equalize ex-ante utilities, we can write the ex-post utility
of consumer 1 as Uˆ   1p  C1 and the ex-post utility of consumer 2 as Uˆ   2p . If the game

 

reaches the fourth stage, the center chooses  pj j 1, 2 to maximize  Uˆ   1p  C1   Uˆ   2p ,
2

subject to
j 1

p
j
 0. The first-order conditions are the constraint and
 

  Uˆ   1p  C1    Uˆ   2p ,
(25a)
33
where equation (25a) indicates that the transfers should equate the two regions’ ex-post utilities.
Hence,
Uˆ   1p  C1  Uˆ   2p and 1p  2p  C1 2 .
(25b)
To achieve horizontal equity ex post, the transfers make the two regions to equally share the
costs of refinancing region 1’s public facility. With equations (25b), the ex-post utility of each
resident can again be denoted U  Uˆ  C1 2 , as in subsection 4.1. Thus, the center again
wishes to maximize the ex-post utility of each resident.
In the third stage of the game, having observed region 1’s reported unit cost, ĉ, and the
choices of ĝ1 and ĝ 2 made by the two regional governments, the center chooses ˆ aj j 1, 2 to


 
 I  v  gˆ j   su Gˆ  ˆaj  Cˆ j , subject to
2
maximize
j 1
2
ˆ
j 1
a
j
 0. The first-order conditions
2
satisfy ˆ aj  0 and (22d), which leads to
j 1


 
Uˆ1  Uˆ 2 and ˆ1a  ˆ a2  vgˆ 2   Cˆ 2  vgˆ1   Cˆ1 2 .
(26)
Given equations (26), the level of the utility for consumer j , j  1,2, implied by the solution
to
the
center’s
problem
in
the
third
stage
can
be
written
as

Uˆ j  Uˆ  I  su Gˆ  vgˆ1   vgˆ 2  2  cˆgˆ1  gˆ 2  2 , j  1,2.
In the second stage, the regions choose the sizes of their public facilities conditional on the
report made by region 1 to maximize Û . Anticipating the center’s actions, which equalize
utilities ex ante and ex post (if necessary), the regional governments make choices that
internalize interregional spillovers. Thus, the equations that determine the sizes of the public
34
facilities are again the report-constrained Samuelson conditions (6), which implicitly define
gˆ j cˆ , j  1,2, and allow us to write Û as Uˆ cˆ  described by equations (23).
It is now clear that region 1’s objective function in the first stage is the same as the ex-post
objective function of the center, i.e., U cˆ; c  described by equations (24). Hence, as in the
centralized case in subsection 4.1, region 1’s best strategy is to report the true unit cost. The
following proposition is now immediate:
Proposition 10: With decentralized leadership, the subgame perfect equilibrium of the
sequential policy game in which the center implements interregional income transfers ex-ante
and has the ability of doing so ex-post corresponds to the social optimum with perfect
information.
5. Conclusions
We investigate the soft budget syndrome in a hierarchical fiscal system in which one region is
privately informed about its cost of building a public facility. We find that the shape of policy
making in such a federation depends fundamentally on whether the regional public facilities
yield interregional spillovers and, if so, on whether the spillover effects are positive or negative.
In the presence of positive spillover effects, the center has an extra motivation to favor a bailout
if the privately informed region cheats relative to a situation in which no spillover effects are
present whenever the federal public good is not too large for the federal taste. However, if the
federal public good is too large for the federal taste, the center has an extra disincentive to favor
the bailout relative to a situation in which no spillover effects are present. Hence, we can only
partially support the rationale for the ‘too big to fail’ argument advanced in the literature.
Interestingly and counter-intuitively, the privately informed region may decide to report its true
unit cost in the presence of positive spillover effects. We show that a sufficient condition for
35
honest reporting in the presence of positive spillover effects is that the size chosen by the center
for the public facility located in the privately informed region is too large for the regional taste.
In the presence of negative spillover effects, the center has an extra incentive to refuse a
bailout if the privately informed region cheats relative to a situation in which no spillover effects
are present. The privately informed region, however, may have an extra incentive to cheat under
these circumstances, since cheating may trigger refusal of the bailout by the center and
subsequently the production of the public facility in the privately informed region shuts down.
The amount of the federal public bad resulting from such an outcome is simply the amount
contributed by the other region. This quantity is lower than the quantity the center chooses under
honest reporting.
We also find that if the central government has the ability to make interregional income
transfers ex ante and ex post, the privately informed regional government reports its true unit
cost in the presence or in the absence of spillover effects, thus avoiding the soft budget
syndrome. Truthful reporting occurs irrespective of whether the center government or the regions
determine the sizes of the regional public facilities.
References
Akai, N., Sato, M., 2008. Too big or too small? A synthetic view of the commitment problem of
interregional transfers, Journal of Urban Economics 64, 551-559.
Akai, N., Silva, E.C.D., 2009. Interregional redistribution as a cure to the soft budget syndrome
in federations, International Tax and Public Finance 16, 43-58.
Breuillé, M.-L., Vigneault, M., 2010. Overlapping soft budget constraints, Journal of Urban
Economics 67 (3), 259-269, doi:10.1016/j.jue.2009.09.011 .
36
Breuillé, M.-L., Madiès, T., Taugourdeau, E., 2006. Does tax competition soften regional budget
constraints? Economics Letters 90 (2), 230–236.
Crivelli, E., Staal, K., 2013. Size, spillovers and Soft budget constraints, International Tax
and Public Finance 20 (2), 338 - 356.
Dewatripont, M., Maskin, E., 1995. Credit and efficiency in centralized and decentralized
economies, Review of Economic Studies 62, 541-555.
Inman, R., 2003. Transfers and bailouts: enforcing local fiscal discipline with lessons from US
federalism, in: Rodden, J., Eskeland, G., Litvak, J. (Eds.), Fiscal Decentralization and the
Challenge of Hard Budget Constraints. Cambridge, MA: The MIT Press.
Kornai, J., 1979. Resource-constrained versus demand-constrained systems, Econometrica 47,
801-819.
Kornai, J., 1980. Economies of Shortage. Amsterdam: North-Holland.
Kornai, J., 1986. The soft budget constraint, Kyklos 39, 3-30.
Kornai, J., Maskin, E., Roland, G., 2003. Understanding the soft budget constraint, Journal of
Economic Literature 41, 1095-1136.
Köthenbürger, M., 2004. Tax competition in a fiscal union with decentralized leadership, Journal
of Urban Economics 55, 498-513.
Köthenbürger, M., 2007. Ex-post redistribution in a federation: Implications for corrective
policy, Journal of Public Economics 91, 481-496.
Maskin, E. S., 1996. Theories of the soft budget-constraint, Japan World Economy 8, 125-133.
Qian, Y., Roland, G., 1998. Federalism and soft budget constraint, American Economic Review
88, 1143-1162.
37
Rodden, J. A., Eskeland, G. S., Litvack, J., 2003. Fiscal Decentralization and the Challenge of
Hard Budget Constraints. Cambridge, MA: The MIT Press.
Vigneault, M., 2007. Grants and soft budget constraints, in: Boadway, R., Shah, A. (Eds.),
Intergovernmental Fiscal Transfers: Principles and Practice. World Bank, Washington, DC.
Wildasin, D. E., 1997. Externalities and bailouts: hard and soft budget constraints in
intergovernmental fiscal relations, World Bank Policy Research Working Paper Series 1843,
World Bank, Washington, DC.
38
  6  4.5
Figure 1: Cheating and no cheating areas when   0.1 in subsection 3.2.3.
39
Table 1: Equilibrium values in example 1 with     0.1, cL  1, cH  1.1, and a  3.4; and in
example 2 with   0.1,   1, cL  1, cH  2, and a  6.1.
    0 .1
  0.1,   1

4.2
6.45
9.2
8.3
10.5
13.3
d
5
7.25
10
10
12.2
15
gˆ11 1
2.48
3.38
4.48
5.02
5.9
7.02
gˆ 12 1
2.48
3.38
4.48
5.02
5.9
7.02
Gˆ 1 1
4.96
6.76
8.96
10.04
11.8
14.04
gˆ11 c 
2.42
3.32
4.42
4.42
5.3
6.42
gˆ 12 c 
2.52
3.42
4.52
5.42
6.3
7.42
Gˆ 1 c 
4.94
6.74
8.94
9.84
11.6
13.84
d  Gˆ 1 1
0.04
0.49
1.04
-0.04
0.4
0.96
d  Gˆ 1 c 
0.06
0.51
1.06
0.16
0.6
1.16
C 1B
5.21
7.10
9.41
15.06
17.70
21.06
C 1E
5.18
7.07
9.38
14.26
16.90
20.26
   
17.86
31.94
54.66
68.02
92.93
130.22
v g11E  u G1E
   
17.80
31.93
54.70
67.18
92.53
130.38
U11B
I+15.25
I+28.39
I+49.95
I+60.49
I+84.08 I+119.69
U 21B
I+15.25
I+28.39
I+49.95
I+60.49
I+84.08 I+119.69
U11E
I+15.21
I+28.39
I+50.01
I+60.05
I+84.08 I+120.25
v g11B  u G1B
40
U 21E
I+15.3
I+28.40
I+49.9
I+61.23
I+84.38 I+119.43
U11N
I+6.84
I+15.41
I+30.28
I+32.58
I+48.68
I+73.64
U 21N
I+12.20
I+21.19
I+35.48
I+50.60
I+67.26
I+91.82
1
11.46
20.18
34.14
37.80
52.22
73.92
R1B
0.05
0
-0.06
0.44
0
-0.56
R1N
-8.36
-12.98
-19.72
-27.47
-35.40
-46.61
41
Appendix A: Proof of Proposition 2


Consider v  0, and u  0 so that v  u  0, for gˆ11 cˆ   gˆ11 c , gˆ11 1


 

and

Gˆ 1 cˆ   Gˆ 1 c , Gˆ 1 1 . Let U1s cˆ; c   I  v gˆ1s cˆ   u Gˆ s cˆ   F cˆ; c  2 . Note that U1sE  U1s c; c 
and U1sB  U1s 1; c  . Differentiating U1s cˆ; c  with respect to ĉ yields

 


 



dU1s dcˆ  v gˆ1s cˆ   u Gˆ s cˆ   c 2 dgˆ1s dcˆ  u Gˆ s cˆ   1 2 dgˆ 2s dcˆ .
(A1)
In the presence of a positive spillover, the first order condition (6a) can be written as
vgˆ cˆ  uGˆ cˆ  cˆ 2 uGˆ cˆ  cˆ 2  0.
1
1
Since
1

v  0
1
(A2)

vgˆ cˆ  uGˆ cˆ  cˆ 2  0 and
for gˆ11 cˆ   gˆ11 c , gˆ11 1 , we must have
1
1
1
uGˆ cˆ  cˆ 2  0. The inequality vgˆ cˆ  uGˆ cˆ  cˆ 2  0 indicates that
vgˆ cˆ  uGˆ cˆ  c 2  0 for cˆ  1, c.
The inequality uGˆ cˆ   cˆ 2  0 indicates that
uGˆ cˆ  1 2  0 for cˆ  1, c.
1
1
1
1
1
1
1
(A3)
1
1
Combining dgˆ11 dcˆ  0, dgˆ 12 dcˆ  0

and
inequalities
(A3)
and
(A4)
(A4)
yields

dU11 dcˆ  0 for Gˆ 1 cˆ   Gˆ 1 c , Gˆ 1 1 , which indicates that U11E  U11B .
If


v gˆ11 c   0,
equation
(6a)
evaluated
vgˆ c  uGˆ c  c 2  0 and uGˆ c  c 2  0 .
1
1
1
1
at
cˆ  c
informs
us
that
In this case, dU11 dcˆ in (A1) is negative
at cˆ  c . If ĉ is a continuous variable and cˆ  1, c, region 1 will choose to underreport its unit
cost when facing a soft budget constraint.
42
Appendix B: Proof of cˆ  c in Subsection 4.1
In the first stage of the game, region 1 chooses cˆ  1, c to maximize U cˆ; c  described by
equations (24), taking into account the center’s ex ante and ex post reactions. Differentiating
U cˆ; c  with respect to ĉ yields
  

  

dU dcˆ  su Gˆ cˆ   vgˆ1 cˆ  2  c 2 dgˆ1 dcˆ   su Gˆ cˆ   vgˆ 2 cˆ  2  1 2 dgˆ 2 dcˆ . (B1)
Equation (6b) allows us to write equation (B1) as
  

dU dcˆ  su Gˆ cˆ   vgˆ1 cˆ  2  c 2 dgˆ1 dcˆ .
(B2)
Equation (6a) allows us to write equation (B2) as
dU dcˆ  cˆ  c 2dgˆ1 dcˆ.
(B3)
Since dgˆ1 dcˆ  0, equation (B3) informs us that dU dcˆ  0 for any cˆ  1, c ; and dU dcˆ  0 at
cˆ  c. Hence, the solution to region 1’s first stage problem is to report cˆ  c.
43