Marie-Laure Breuillé
email: breuille@wzb.eu

Thierry Madiès
email: thierry.madies@unifr.ch

Emmanuelle Taugourdeau
email: emmanuelle.taugourdeau@ens-cachan.fr

Hubert Kempf

Fiscal Federalism and Soft Budget Constraint: Does the nature of public spending matter?

Keywords: JEL Classi…cation: E62; H7 Soft Budget Constraint, Fiscal Federalism, Tax Competition, Public Input

This paper analyses the impact of both the nature of regional public spending and the federal government's …scal tools on the softness of the regional budget constraint and the regional provision of public good. We show that i) whatever the nature of regional public spending, the regional budget constraint is harder when the federal government can no longer manipulate its lump sum tax and ii) under the assumption that the federal government can no longer manipulate its lump sum tax, the federal bailout is lower when the region provides a public input rather than a public good but the regional budget constraint can be either softer or harder.

Introduction

How does the nature of regional public spending …nanced by borrowing a¤ect intergovernmental relationships in a federation? Does the implementation of a golden rule harden the budget constraint of subnational jurisdictions? We contribute to this debate by analyzing the e¤ect on the soft budget constraint (SBC) phenomenon of a golden rule, which limits subnational governments'borrowing for …nancing investment.

In the absence of market discipline borrowing autonomy of subnational governments has been identi…ed empirically as one of the main determinants of the emergence of a soft budget constraint1 in intergovernmental relationships [START_REF] Rodden | Fiscal Decentralization and the Challenge of Hard Budget Constraints[END_REF]). Swedish municipalities that faced no borrowing restrictions or balanced budget rules were repeatedly bailed out over the period 1974-1992[START_REF] Dahlberg | An empirical approach for evaluating Soft Budget Constraints[END_REF] show that over this period, if the decision maker expected a bailout with a probability of 1 rather than a probability of 0, the municipal debt increased by 13%. Similarly, in Germany in 1992, the federal government's lack of control over the borrowing of the Länder, combined with their low level of autonomy in setting their tax rates, led to the German Constitutional Court's decision to support the Saarland and Bremen's demand for …nancial assistance from the federal government to cope with their high debt. The recent attempt of the heavily indebted city of Berlin to extract a massive bailout from the federal government indicates the persistence of this evil which threatens Germany's ability to stay within the general government de…cit limit imposed by Stability and Growth Pact.

This empirical evidence raises the question of what policies should be introduced to curb the soft budget constraint problem associated with the borrowing autonomy of subnational governments. Apart from reinforcing market discipline and implementing administrative controls, a set of budgetary rules, such as balanced budget rules, can be conceived to reduce the borrowing autonomy of subnational governments. The balanced budget rules currently implemented vary in several dimensions, i.e. the budget period (annual or multi-annual) and the target (current budget balance, current budget balance and capital account). As shown by the OECD Economic Studies (2005), the golden rule of public …nance, which targets the current budget alone and allows subnational governments to borrow only to …nance public investment, has become more widespread among countries. To give just one example, it has been implemented in Sweden since 2000. The debate about the performance of the golden rule, which has been particularly intense in Europe with regard to the reform of the Stability and Growth Pact, has been mainly about procyclicity, intergenerational equity, or biases in public spending, but it does not address the issue of the soft budget constraint.

On the theoretical side, few recent papers deal with both public …nance and the soft budget constraint. Exceptions are papers by [START_REF] Wildasin | Externalities and bailouts : hard and soft budget constraints in intergovernmental …scal relations[END_REF], [START_REF] Qian | Federalism and the Soft Budget Constraint[END_REF], [START_REF] Goodspeed | Bailouts in a Federation[END_REF] and Köthenburger (2004). A key point of almost all these papers is that the soft budget constraint is formulated in the context of a sequential game where the …rst move is made by regional governments which generally borrow, the federal government has the second move and, at that point, the costs to the federal government of not providing additional funds may exceed these of providing them. However, the authors give di¤erent reasons for the emergence of the soft budget constraint issue. [START_REF] Wildasin | Externalities and bailouts : hard and soft budget constraints in intergovernmental …scal relations[END_REF] suggests that large sub-national jurisdictions are more likely to be rescued by the federal government than smaller ones, because, if they were to fail, there would be negative externalities for other jurisdictions (here the well-known "too big to fail" argument of the banking literature). [START_REF] Qian | Federalism and the Soft Budget Constraint[END_REF] show that …scal decentralization, together with taxbase mobility may serve as a commitment device for hardening budget constraints of state-owned enterprises by increasing the opportunity cost of bailouts. However, in their paper the federal government does not act as a player, and the hardness or softness of the budget constraints is derived exogenously. [START_REF] Goodspeed | Bailouts in a Federation[END_REF] demonstrates that transfers from higher levels of government to lower ones generally involve a "common pool" e¤ect, since part of the bailout must be paid for through increased taxes and then shared by all the regions. He derives the soft budget constraint bailout behaviour endogenously, but ignores tax interactions among governments, since transfers are …nanced through an immobile and exogenous tax base. Conversely, Köthenbürger (2004), in a static model, explicitly introduces capital mobility among regional governments. He shows that an ex post federal transfer policy results in the internalisation of the impact of interregional tax competition, but does so by bringing in a new source of ine¢ ciency that causes a deterioration in welfare.

Our paper is mainly in line with the paper by [START_REF] Goodspeed | Bailouts in a Federation[END_REF]. However, unlike in his paper, we assume that (i) there is horizontal tax competition and (ii) the public spending can be either a public good or a public input. We set up a simple model of federal government transfer decisions with inter-temporal regional spending choices when there are horizontal tax externalities. The externalities result from horizontal tax competition among regions aimed at attracting mobile capital. Furthermore, we assume that the public spending in the …rst period can be either a public good or a public input. The provision of a public good is the most standard choice for modelling. There are few papers that deal with both kinds of public spending and the resulting ine¢ ciency in relation to the optimal composition of public expenditure (the papers by [START_REF] Zodrow | Pigou, Tiebout, Property Taxation and the Underprovision of Local Public Goods[END_REF], Keen and[START_REF] Keen | Fiscal competition and the pattern of public spending[END_REF]Matsumoto (2000)) are exceptions). Our choice of modelling is, however, relevant for at least two reasons. First, our model has connections with recent literature (see Justman, Thisse andvan Ypersele (2002, 2004)) which shows that regional governments are likely to compete in terms of the quality of their infrastructures, and that this may have an impact on the intensity of tax competition and then on equilibrium tax rates. Second, in our model we assume that the public input (e.g an investment in infrastructure or human capital) is …nanced by borrowing in the …rst period, whereas the public good is …nanced through taxation in the second period. This assumption is clearly in line with the golden rule of public …nance and the reform proposals of the European Stability and Growth Pact. In addition it is noteworthy that tax competition models do not use this assumption although it …ts nicely with the public accountability rules in most countries, since they assume that both kinds of public spending are …nanced through taxation.

In our two-period framework, we show that the pattern of public spending crucially in ‡uences the softness of the regional budget constraint: the public input provision can limit or even cancel the softness of the regional budget constraint. Furthermore, in line with the standard literature on optimal taxation, we analyze our results with regard to the distortive/non distortive e¤ect of the federal …nancing of the bailout.

Our paper is organized as follows. Section 2 presents the model and section 3 solves it when a centralization scheme is assumed. In section 4, the main features of the model are presented, in a context of decentralized choices, when a public good is provided in both periods. Section 5 looks at how the introduction of a public input in the …rst period modi…es the previous results. Section 6 concludes.

The model

The economy consists of a federation run by a federal government with n identical regions governed by regional governments. Suppose, …rst, that a regional public good is provided for households in each period.

Consumers

The representative consumer of each region derives a utility U (c i1 ; G i1 ; c i2 ; G i2 ) from the consumption of the public good (G i1 ; G i2 ) and the private good (c i1 ; c i2 ) over two periods2 :

U (c i1 ; G i1 ; c i2 ; G i2 ) = u (c i1 ) + v (G i1 ) + c i2 + v (G i2 ) (1)
where the utility functions u and v are increasing in every argument, twice di¤erentiable and concave.

In period 1, each representative consumer is endowed with w units of the goods that are allocated between private consumption and savings:

c i1 = w S i (2)
where S i = s i i + P j6 =i s j i with s i i 0 and P j6 =i s j i 0. s i i stands for investments in the home region while P j6 =i s j i represents investments made in foreign regions. Savings invested in a region j are remunerated at the before-tax interest rate r j on which a regional tax j is levied according to the source principle3 . In period 2, the representative agent consumes the proceeds of her savings plus a rent i earned in her jurisdiction once the lump sum tax has been paid to the federal government:

c i2 = n X j=1 (1 + r j j ) s j i + i (3)
In this way, we implicitly assume that the consumer owns the …rm located in her region.

Regional governments

Each regional government aims at maximizing the utility of the representative consumer located in its region. The regional public good (G i1 ) provided in the …rst period is …nanced by an exogenous federal transfer (T i1 ) and regional borrowing4 (B i1 ):

G i1 = T i1 + B i1 (4)
The region has to maintain a balanced budget over the two periods. As a result, the repayment of the debt reduces the budgetary room for manoeuvre in the second period, ceteris paribus. The regional public good5 (G i2 ) provided in the second period, and the repayment of the debt, are …nanced by both a federal transfer (T i2 ) and the revenue from regional taxation on capital ( i K i ):

G i2 = T i2 + i K i (1 + r i ) B i1 ; (5)
where K i stands for the amount of capital available in region i.

Federal government

The federal government, which is assumed to be benevolent, maximizes the aggregated utility of citizens

W i = n P i=1 U i :
The transfers granted to the regions in the second period are …nanced by a lump sum tax , identical in each region:

n X i=1 T i2 = n (6)
2.4 Capital market F (K i ), the output in region i, is a function of the capital K i located in this region. The production function is assumed to be monotonously increasing in capital ( @F @K i > 0 ) with decreasing marginal product (

@ 2 F @K 2 i < 0 ). All rents arising in region i i = F (K i ) r i K i
are assumed to accrue to the representative consumer. Firms'pro…t maximizing behaviour implies the following familiar condition for marginal factor productivity: F 0 (K i ) = r i 8i, which determines the demand for capital K i (r i ) in each region i. The demand for capital K i (r i ), as well as the rents i (r i ), are decreasing functions of the interest rate r i , i.e.

@K i @r i = K ri = 1 F K i K i < 0 and @ i (:) @r i = ri = K i < 0.
For each region i, the amount of capital K i equalizes the total amount of savings n P j=1 s i j invested in the region. Capital is mobile across regions without cost so that the capital relocates until it earns the same post-tax return in each region:

= r i i = r j j 8 i; j. (7)
The amount of savings S i ( ), which satis…es the …rst-order condition @u @c i1 = (1 + ) from the consumer's program, increases with this net return , i.e.

S 0 i = 1 @ 2 u @c 2 i1 > 0.
The capital market clearing condition of the federation

n X i=1 K i ( + i ) = n X i=1 S i ( )
implicitly de…nes the net return which is a decreasing function of the regional tax rate:

d d j = K r j n P i=1 S 0 i n P i=1 K r i 2 [ 1; 0].
The interest rate consequently moves as follow:

dr j d j = 1 + d d j 2 [0; 1] and dr j d i = d d i 2 [ 1; 0]
Finally, in line with empirical …ndings, we postulate that the elasticity of the regional tax base with respect to the regional tax rate, denoted by

" i @K i @ i i K i = K r i dr i d i i K i belongs to the interval [ 1; 0].
3 Centralization as a benchmark case: the hard budget constraint policy

As mentioned by [START_REF] Kornai | Understanding the Budget Constraint[END_REF], "although the intuitive meaning of SBC was reasonably clear from the outset, there is still no consensus on a precise de…nition". Based on their long discussion about the SBC syndrome, we adopt the following de…nition: the SBC phenomenon occurs when a "BC-organization" (organization which faces a budget constraint) can manipulate a "S-organization" (solving organization) in order to cover all, or part, of its de…cit. Conversely a "BC organization faces a Hard BC as long as it does not receive outside support to cover its de…cit and is obliged to reduce or cease its activity if the de…cit persists" [START_REF] Kornai | Understanding the Budget Constraint[END_REF]). According to this de…nition, the regional budget constraint is hard, by de…nition, as soon as the federal government does not react to an increase in the regional borrowing, which is typically the case when there is centralization.

So we assume in a …rst step that decisions are taken by a benevolent social planner. The SBC phenomenon -linked to the inability of the federal government to refuse to bailout ex post -is totally ine¤ective in this context. Simultaneous decisions are su¢ cient to o¤set the perverse incentives of the regions. By the way, the regional budget constraints are hard.

The social planner maximizes the aggregated utility of citizens located in the federation with respect to the vector of transfers T 2 , the lump sum tax , the vector of borrowing B 1 and the vector of regional tax rates , subject to the budget constraints:

M ax T 2 ; ;B 1 ; n X i=1 [u (c i1 ) + v (G i1 ) + c i2 + v (G i2 )] s:t: c i1 = w S i c i2 = (1 + )S i + i n X i=1 T i2 = n G i1 = T i1 + B i1 G i2 = T i2 + i K i (1 + r i ) B i1
At the optimum, the sharing of the transfers satis…es the following condition:

@v @G i2 = @v @G k2 8i; k => G i2 = G k2 8i; k; (8)
and @v @G i2 = 1 8i:

The social planner allocates the optimal transfers so as to equalize the marginal utilities of the regional public good among the regions. The marginal rate of substitution between the private and the public goods in the second period is equal to unity, which is a common result in the literature on taxation. Under a hard budget constraint policy, the optimal level of borrowing satis…es the following condition:

@v @G i1 @v @G i2 = (1 + r i ):
The region's opportunity cost of borrowing is equal to the cost of the debt repayment (1 + r i ). Due to the distortive e¤ect of the regional tax competition, the optimal regional tax rate is equal to zero. The public good is optimally provided in the second period and is …nanced by the non-distortive federal taxation. This outcome will be contrasted with the outcome under a soft budget constraint policy.

4 Budgetary decisions under a SBC policy

Sequence of budgetary decisions

Each level of government interacts in a two-period model. In the …rst period, regional governments play as Nash competitors when choosing their level of the public good (G i1 ) and their level of debt (B i1 ). They also play a Nash game when choosing their tax rate level ( i ) and their level of regional public good (G i2 ) in period 2. Conversely, they play as Stackelberg leaders6 towards the federal government when determining how much they borrow, whereas they play as Nash competitors with the federal government when choosing their tax rate ( i ). In other words, the level of the regional debt depends on the credibility of the federal government's commitment not to bail out. Finally, consumers, who are supposed to be immobile, choose their level of savings and consumption. We solve the model by backward induction.

The incentives to bail out

The federal government maximizes the aggregated utility of citizens located in the federation with respect to the vector of transfers T 2 and the lump sum tax , subject to the budget constraints of consumers, regions and its own budget constraint:

M ax T 2 ; n X i=1 [u(c i1 ) + v (G i1 ) + c i2 + v (G i2 )]
s:t: (2), (3), (4), ( 5) and ( 6)

The …rst-order conditions are equivalent to conditions ( 8) and ( 9) of the centralization case. The federal government aims at implementing inter-individual equalization across the whole territory and chooses the lump sum tax such that the marginal rate of substitution between private and public goods equals the unity in the second period.

A key issue of the paper consists in determining the reaction function of the federal government, i.e. how it will react in terms of transfers granted to regions in the second period, following an increase in region i's borrowing in the …rst period. Di¤erentiating the …rst-order conditions ( 8) and ( 9) with respect to T i2 , T j2 8j 6 = i, B i1 and gives

dT i2 dB i1 = (1 + r i ) , dT j2 dB i1 = 0 and d dB i1 = (1 + r i ) n : (10)
The federal government is always inclined to increase the region i's second-period transfer, when that region increases its borrowing, in order to help the regional government to repay its debt in the second period ((1 + r i ) dB i1 ) while maintaining an optimal level of regional public good in every region. As a consequence, an increase in the regional debt does not tighten the regional budget constraint because the additional cost of the debt repayment is entirely compensated by an extra transfer from the federal government. The inability of the federal government to commit dynamically to not rescuing the region (because of its aim to equalize marginal utilities of the public good across the territory as a whole) always goes towards softening the regional budget constraint. Once the regional governments have played, the cost of not bailing out dT i2 dB i1 = 0 is from the federal government's point of view ex post higher than the cost of bailing out. Indeed, a rise in the …rst-period borrowing lowers the regional public good provision ceteris paribus, which no longer satis…es the equalization condition in the second period. The bailout is entirely …nanced by an increase in the federal tax rate levied on consumers. Note that the bailout provided for the region i does not a¤ect the amount of transfers provided for the other regions.

The opportunistic behaviour of the region

The opportunistic regional policy maker makes its budgetary decisions expecting the federal government to be unable to commit dynamically to not bailing out. The regional Stackelberg leader thus maximizes the intertemporal utility of the representative household taking into account the reaction function of the federal government:

M ax B i1 ; i u(c i1 ) + v (G i1 ) + c i2 + v (G i2 ) s:t: c i1 = w S i c i2 = (1 + )S i + i G i1 = T i1 + B i1 G i2 = T i2 + i K i (1 + r i ) B i1
and expecting dT i2 dB i1 ; d dB i1 :

The …rst-order conditions with respect to i and B i1 are respectively:

d d i S i K i dr i d i @v @G i2 dr i d i B i1 K i i K r i dr i d i = 0 (11) and @v @G i1 + @v @G i2 (1 + r i ) + dT i2 dB i1 d dB i1 = 0 (12)
which boils down to

@v @G i1 = 1 n (1 + r i )
after integrating the values of dT i2 dB i1 and d dB i1 .

A main point of our paper is to analyze the impact of the soft budget constraint phenomenon on the regional budgetary decisions. We evaluate how the expected reaction of the federal government modi…es the regional opportunity cost of borrowing, i.e. the regional incentives to borrow. As shown before, the federal government entirely compensates the additional cost of the debt repayment (1 + r i ) through extra transfers. Thus, the regional government only bears the cost linked to the increase in the federal tax rate. Compared to the centralized case, the opportunity cost of borrowing is much more lower since for decentralized choices, the regional government is able to manipulate B i1 and then to share out the cost of the bailout with the other regions.

Intergovernmental budgetary interactions with a pure redistribution scheme

Suppose now that the federal government is no longer able to manipulate its lump sum tax and that it can only redistribute7 : transfers granted to some regions are …nanced by contributions made by the other regions. As a consequence, the condition (9) disappears.

Di¤erentiating the …rst-order conditions ( 8) with respect to T i2 , T j2 8j 6 = i and B i1 , summing them and using the federal budget constraint, leads to the following reaction function of the federal government:

dT i2 dB i1 = (1 + r i ) | {z } (A i ) 1 n (1 + r i ) | {z } (B i ) and dT j2 dB i1 = 1 n (1 + r i ) 8j 6 = i (13)
The federal government is still always inclined to increase the region i's second-period transfer when that region increases its borrowing, but to a lesser extent than previously. The region i is entirely compensated for the additional cost of the debt repayment (A i ) but at the same time bears a part of the …nancing (B i ) of the bailout as all the regions do. The federal government makes all the regions including i bear the burden resulting from the extra transfer granted to region i. As a result, the larger the number of regions, the higher the bailout granted to the region i and this softens the budget constraint. This result is simply due to the fact that a larger number of regions allows the federal government to …nance the bailout from a larger …scal base and this limits the reduction of transfers to each region. The reallocation mechanism clearly generates negative externalities for the other regions because they contribute to …nancing the bailout. Note that the regional public good is still equally provided over the whole territory but to a lesser extent than when the federal government was able to manipulate its lump sum tax . The regional public good provided in a region j is now altered by an increase in the region i's borrowing.

The regional marginal cost of borrowing and the marginal cost of public funds respectively become:

@v @G i1 @v @G i2 = 1 n (1 + r i ) (14)
and

@v @G i2 = 1 (1 + " i ) dr i d i B i1 K i > 1: (15)
The opportunity cost of borrowing boils down to the part of the bailout …nanced by the region itself. The larger the number of the regions in the federation, the lower the burden borne by each of them, and this encourages the region i to borrow more than the e¢ cient level. As the federal government will bail out the region and thus compensate most of the cost of the debt repayment, it generates opportunistic behaviour at the regional level.

In addition, the distortive e¤ects of the regional taxation, together with the ine¢ ciently high level of the debt induce an underprovision of the regional public good in the second period.

The ability of the federal government to manipulate its lump sum tax plays a crucial role in the soft budget constraint problem, as stated by the following proposition:

Proposition 1 When the federal government can no longer manipulate its lump sum tax, it tends to harden the regional budget constraint. The additional cost of the bailout is now …nanced by a cut in federal transfers to all regions.

Proof. Directly from equations ( 10), ( 12) to ( 15)

Provision of a public input

In this section, we consider that a public input (I i ), which increases the marginal productivity of the capital, is produced in the …rst period instead of the public good, in order to determine the impact of the nature of the public spending on the soft budget constraint problem. This public input can be seen as public infrastructure or human capital.

Consumers

The utility of the representative consumer in the region i becomes:

U (c i1 ; c i2 ; G i2 ) = u(c i1 ) + c i2 + v (G i2 ) (16)
The expressions of the …rst-period and second-period private consumption are unchanged.

Regional and federal governments

In the …rst period, the regional government now …nances a public input (I i ) with the exogenous federal transfer (T i1 ) and regional borrowing (B i1 ):

I i = T i1 + B i1 (17)
Note that, according to this budget constraint, the decision to borrow and to produce the public input are strictly linked. In this way, our paper di¤ers from those of Keen and Marchand (1998) or [START_REF] Zodrow | Pigou, Tiebout, Property Taxation and the Underprovision of Local Public Goods[END_REF] since the provision of the public input alters the provision of the regional public good in the second period. The regional and federal budget constraints remain unchanged in the second period.

Capital market

The public input is assumed to increase both the production8 and the marginal productivity of capital, i.e. @F @I i = F I i > 0 and @ 2 F @K i @I i = F K i I i > 0. Rents arising in region i are given by i = F (K i ; I i ) r i K i :

The demand for capital resulting from the pro…t maximization K i (r i ; I i ) and the rents i (r i ; I i ) positively depend on the public input, i.e. @K i @I i = K I i =

F K i I i F K i K i > 0 and @ i @I i = F I i > 0. The capital market clearing condition n X i=1 K i (r i ; I i ) = n X i=1 S i ( )
allows us to determine

d dI j = K I j n P i=1 S 0 i n P i=1 K r i > 0 .
The decision sequence of the game remains unchanged. I i is determined in the …rst period, together with B i1 .

6 Impact of the public input provision on the SBC

The incentives to bail out

The federal government still maximizes the aggregated utility of citizens located in the federation with respect to both the vector of transfers T 2 and the federal lump sum tax : M ax

T 2 ; n X i=1 [u(c i1 ) + c i2 + v (G i2 )] s:t: c i1 = w S i c i2 = (1 + )S i + i n X i=1 T i2 = n I i = T i1 + B i1 G i2 = T i2 + i K i (1 + r i ) B i1
The …rst-order conditions are unchanged, that is, the federal government chooses the allocation of transfers so as to equalize the marginal utilities of the regional public good provision, i.e. @v @G i2 = @v @G j2 => G i2 = G j2 8i; j, and the marginal rate of substitution is equal to the unity.

Di¤erentiating the …rst-order conditions with respect to T i2 , T j2 8j 6 = i, B i1 and , leads to the following reaction function of the federal government following an increase in the region i's borrowing:

dT i2 dB i1 = (1 + r i ) i K I i + d dI i [B i1 i K r i ] (18)
and

dT j2 dB i1 = d dI i B j1 j K r j (19) and d dB i1 = 1 n [(1 + r i ) i K I i ] + 1 n n X j=1 d dI i B j1 j K r j Proposition 2
The funding of a public input rather than a public good through borrowing i) increases the amount of the bailout for i K

I i < d dI i [B i1 i K r i ] ii) reduces the amount of the bailout for i K I i > d dI i [B i1 i K r i ]
The federal government also increases the transfers granted to the other regions in order to compensate the negative externalities arising from the public input provision in the region i.

Proof. Directly from the comparison of (10), ( 18) and ( 19).

An increase in the public input provision in the region i, unlike the public good provision, generates negative externalities towards the other regions. By increasing the net return of capital , it makes debt repayment higher and reduces the regional tax base in every region. In order to maintain an optimal provision of the regional public good in each region in the second period, the federal government is inclined to provide extra transfers for an amount d dI i [B k1 k K r k ] to each region k. These extra transfers are entirely …nanced through a rise in the federal lump sum tax. Compared with the provision of a public good in the …rst period, an increase in the public input provision can either increase or reduce the amount of the bailout, depending on the comparison between the negative externalities d dI i [B i1 i K r i ] and the additional amount of …scal revenues i K I i .

The opportunistic behaviour of the region

The regional policy maker takes the budgetary decisions which maximize the intertemporal utility of the representative household, expecting the bailout from the federal government and the associated increase in the federal tax:

M ax B i1 ; i u(c i1 ) + c i2 + v (G i2 ) s:t: c i1 = w S i c i2 = (1 + )S i + i I i = T i1 + B i1 G i2 = T i2 + i K i (1 + r i ) B i1
and expecting dT i2 dB i1 ; d dB i1 :

The mechanisms that determine the equilibrium level of the regional tax rate are not altered by the provision of the public input so that the condition (11) still holds. In contrast, the condition which determines the equilibrium level of the borrowing, and thus the public input provision, changes as follows:

d dI i S i + @ i @r i d dI i + @ i @I i d dB i1 + @v @G i2 i K I i + i K r i d dI i d dI i B i1 (1 + r i ) + dT i2 dB i1 = 0
(20) which boils down to:

@ i @I i = d dB i1 () F I i = 1 n [(1 + r i ) i K I i ] 1 n n X j=1 d dI i j K r j B j1 (21)
The amount of the regional debt thus depends on the relative extent of the marginal gain in rents ( @ i @I i ) and the marginal tax cost d dB i1

following a change in the regional borrowing.

Budgetary decisions with a pure redistribution scheme

When the federal lump sum is …xed, the federal government's reaction function to regional borrowing becomes (see appendix for more detailed calculus):

dT i2 dB i1 = 2 4 (n 1) n (1 + r i ) i K I i + d dI i B i1 i K r i d dI i + d dI i 1 n X j6 =i j K r j B j1 3 5 which gives dT i2 dB i1 = (n 1) n ((1 + r) K I ) and dT j2 dB i1 = 1 n ((1 + r) K I ) (22)
in a perfectly symmetric case where the tax rates and the levels of debt are identical ( i = j = and B i1 = B j1 = B 1 ).

Proposition 3 Under the assumption that the federal government can no longer manipulate its lump sum tax, the funding of a public input rather than a public good through borrowing reduces the amount of the bailout. The transfers granted to the other regions are cut in order to …nance the bailout.

Proof. From equations ( 13) and ( 22). The federal government is not always willing to bailout the region i when that region increases its borrowing. Its behaviour depends on the relative weight of the increasing cost of the debt repayment and the expected additional tax revenues. As the federal government can no longer use the lump sum tax, the only way to …nance the bailout is the redistribution mechanism between regions, which forces the federal government to ask other regions for …nancing the bailout. If the additional cost of the debt repayment is higher than the additional receipts, the federal government helps the region i to provide a level of regional public good which satis…es its aim of equalization. Otherwise, it reallocates the additional revenues of the region among the federation. Note that, for the particular case of (1 + r) = K I , the allocation of transfers is not altered ex post by an increase in the region i's borrowing, as it is under a hard budget constraint policy. At the regional level, the condition ( 15) is still at work and characterizes an underprovision of the public good. When solving the regional government program with respect to B i1; the marginal cost of borrowing at the symmetric equilibrium becomes:

F I i @v @G 2 = 1 n [(1 + r) K I ] d dI i ( K r B 1 ) : (23)
Proposition 4 Under the assumption that the federal government can no longer manipulate its lump sum tax, the funding of a public input rather than a public good through borrowing tends to harden (soften) the regional budget constraint for

K I < (>) d dI [B 1 K r ]
Proof. By comparing the terms of the right-hand sides of equations ( 14) and ( 23).

If the negative impact of the rise in the interest rate ( d dI [B 1 K r ]) dominates the increase in the regional tax revenues ( K I ), following a rise in the public input provision, the regional budget constraint tends to be harder. This tends to support the idea that a golden rule would increase the soundness of the public …nance.

Conclusion

The aim of this paper is to study how the nature of public spending a¤ects the softness of the regional budget constraint. More precisely, we aim to evaluate the impact of the golden rule, which targets the current budget alone by allowing regions to borrow only to …nance public investments, by using the yardstick of the soft budget constraint problem. We are able to state that, when a public good is provided in each period, the regional budget constraint is always soft. In contrast, the provision of a public input can crucially modify this result, depending on the comparison between the negative impact of the rise in the interest rate and the increase in the regional tax revenues following a rise in the public input provision. As a result, …scal discipline is altered by the pattern of the public spending. We also show that, when the federal government can no longer manipulate its lump sum tax for …nancing the regional transfers, the regional budget constraints tend to be harder.

The concept of soft budget constraint was originally introduced by[START_REF] Kornai | Resource-Constrained versus Demand-Constrained Systems[END_REF][START_REF] Kornai | The Soft Budget Constraint[END_REF] to characterize the opportunistic behavior of state-owned enterprises in socialist economies. These state-owned enterprises distorted their choices ex ante, expecting to be rescued by the State ex post in the case of …nancial di¢ culty. The inability of the State to commit dynamically to not bailing out induced …rms to misbehave. This concept has recently been extended to a number of di¤erent institutional settings including …scal federalism. See the survey on the soft budget constraint by[START_REF] Kornai | Understanding the Budget Constraint[END_REF].

[START_REF] Keen | Does Federalism Lead to Excessively High Taxes?[END_REF], preferences with respect to ci2 are linear, this ensures that savings only depend on the net return of capital.

There are two polar principles of inter-jurisdictional taxation: the residence (of the taxpayer) principle and the source (of income) principle. The source principle implies that all incomes originating in a region are taxed in this region regardless of the region of residence of the taxpayers.

We assume here that the regional debt is held by foreign investors, however this does not a¤ect the generality of our results.

Contrary to Wildasin (1997) we assume that there are no externalities of production or consumption.

The fact that the regional governments act as Stackelberg leaders vis-à-vis the federal government creates a favourable environment for the emergence of a soft budget constraint problem.

Germany and Canada, for example, have such a redistribution.

This assumption is based on a review of the empirical evidence on aggregate production relationships that suggests that public infrastructure has, indeed, almost always been found to be complementary to private capital (seeSturn and al (1996)).

APPENDIX

1) Let us consider the (n 1) …rst-order conditions of the federal government's program for the region i:

Di¤erentiating these …rst-order conditions (FOCs) with respect to T i2 , T j2 8j 6 = i and B i1 leads to

Summing these expressions gives:

By di¤erentiating the federal government's budget constraint (6), given a …xed , we obtain:


Environmental Innovation and the Cost of

Keywords: innovation, pollution abatement cost, production function, environmental regulation JEL codes: H23, L51

We show that the common assumption that innovation reduces the marginal abatement cost is wrong. We draw some implications about the incentives to innovate under environmental regulation. In particular, we find that adopting an environmental friendly technology may lead to more pollution and less profit at the firm level.

Introduction

It is widely acknowledged in the literature that environmental innovation reduces marginal pollution abatement costs. For example, Palmer et al. [START_REF] Palmer | Tightening the environmental standards: the benefit-cost or the no-cost paradigm?[END_REF] claim that new pollution abatement technology reduces the marginal abatement cost at all pollution levels. More recently, Jaffe et al. [START_REF] Jaffe | A tale of two market failures: technology and environmental policy[END_REF] wrote that "technology innovations (...) typically reduce the marginal cost of achieving a given unit of pollution reduction". The same argument can also be found in Requate and Unold [START_REF] Requate | Environmental policy incentives to adopt advanced abatement technology: will the true ranking please stand up?[END_REF], Fischer et al. [START_REF] Fischer | Instrument choice for environmental protection when technological innovation is endogenous[END_REF], Montero [START_REF] Montero | Permits, standards and technology innovation[END_REF] and Xepapadeas [START_REF] Xepapadeas | Advanced Principles in Environmental Policy[END_REF], among other papers. Graphically, this is reflected by a decrease of the slope of the marginal abatement cost function (see Palmer et al., [START_REF] Palmer | Tightening the environmental standards: the benefit-cost or the no-cost paradigm?[END_REF]). Hence, when an emission fee is imposed, it is intuitively expected that the innovator will pay a lower tax amount and bear a lower total abatement cost. These two arguments thus provide an unambiguous incentive for polluters to adopt environmentally friendly technologies, and there exists an extensive literature comparing policy instruments with regard to their relative incentive to innovate, taking for granted the assumption that innovation reduces marginal abatement costs.

The objective of our paper is to question this assumption. In particular, we will show that the effects of innovation on profits may be ambiguous. Environmental innovation does not necessarily reduce the marginal cost of pollution abatement at all pollution levels. It may happen that the innovating firm pollutes more than the non-innovating one, depending on the policy instrument imposed by the regulator. The effect on profits can also be unexpected. As a consequence, some well-established implications concerning the incentive for innovation under environmental regulation should be reconsidered. This article extends the McKitrick's analysis (McKitrick [START_REF] Mckitrick | A derivation of the marginal abatement cost curve[END_REF]) of the analytical properties of marginal abatement cost functions in the case of environmental innovation.

The paper is organized as follows. In the next Section we present the firm and its technology, and we define environmental innovation. In Section 3 we set up the marginal abatement cost function and analyze how environmental innovation shapes this function. Section 4 discusses the implications under two environmental regulation regimes (command-and-control and tax). Our conclusions are presented in Section 5.

Marginal pollution abatement cost and innovation

Let us consider a firm producing an amount of output Y by using energy,E, and a vector of non-energy inputs, denoted X. We assume a well-behaved production function (increasing, concave and verifying the Inada conditions) Y = F (X, E). The output is the numeraire. Pollution P results from energy combustion through a function h(.) such that P = h(E), where h(.) is a twice continuously differentiable function with h(0) = 0 and h (.) > 0. We now consider environmental innovation.

Definition 1 Environmental innovation is defined as a new pollution function, h(.), such that h (E) < h (E), ∀E ≥ 0. This function has the following properties:

h(0) = 0 and h (E) > 0. It follows that h(E) < h(E), ∀E.
This definition of environmental innovation is a very general one. It encompasses all particular specifications, such as the commonly used linear emission function. By adopting the innovation h(.) the firm reduces the pollution level of its energy consumption at the margin. This can be interpreted as adopting a new capital vintage with a lower pollution intensity or as slightly decreasing the pollution intensity of the whole energy consumption on all machines. From this definition the property that h(E) < h(E) stems naturally, which means that, on average, the pollution rate per unit of energy is also reduced, although to a lesser extent than at the margin.

When pollution is taxed (see below), this specification is consistent with the standard assumption made in the industrial organization literature that innovation (broadly defined) reduces the marginal cost of production. In order to focus on the incentive to innovate, we will assume that innovation has no fixed costs.

Under perfect competition on all markets the program of the firm consists of maximizing its profit π(X, E) = F (X, E) -p X X -p E E where p X and p E stand for input prices. The first-order conditions of this program give the optimal levels of inputs, X * and E * , and the optimal level of pollution, P * .

We define the marginal abatement cost (MAC, hereafter) function as the loss of profit when pollution is reduced by one unit, all other things being equal (see McKitrick [START_REF] Mckitrick | A derivation of the marginal abatement cost curve[END_REF]). By substituting E by the inverse function h -1 (P ) in the firm's program we obtain the profit level as a function of P , ∀P < P * . The MAC function is given by the derivative of this function and it writes,

M AC(P ) ≡ ∂π(X * (E), E) ∂P = 1 h (E) (F E -p E ) (1)
where X * (E) stands for the optimal vector of inputs resulting from the maximization problem for any level of energy consumption. At the firm's optimum, (1) equals zero since F E = p E , and P = P * . If a constraint P were to be imposed on pollution, 0 < P < P * , then we would have E < E * and F E -p E > 0 and the firm would bear a profit loss, the marginal loss being given by [START_REF] Th | Environmental performance and equilibrium[END_REF]. The slope of this function is

M AC(P ) ∂P ≡ ∂ 2 π ∂P 2 = 1 [h (E)] 2 F EE -(F E -p E ) h (E) h (E) (2)
If h(.) is convex, then ∂ 2 π/∂P 2 < 0 and the marginal abatement cost function is strictly decreasing. If h(.) is concave, the marginal abatement cost function decreases from a positive value (since lim E→0 ∂π/∂P > 0) to zero where its slope is given by lim

E→E * ∂ 2 π/∂P 2 = F EE /[h (E)] 2 < 0.
These two equations allow us to see how environmental innovation shapes the MAC function. Firstly, innovation reduces the optimal pollution level in the laissez-faire: since h(E) < h(E), ∀E > 0, then P = h(E * ) < P * = h(E * ), where P is the pollution level solution of the firm's maximization problem after innovation. This gives the motivation of the firm to innovate, lowering its pollution level, but notice that this is not necessarily the case for all energy consumption level. Secondly, the slope of the MAC function is also modified. At the optimal pollution level this slope is unambiguously greater (in absolute terms) after innovation than before since

lim E→E * ∂ 2 π/∂P 2 = F EE /[h (E)] 2 > lim E→E * ∂ 2 π/∂P 2 = F EE /[ h (E)] 2 .
However, because of the cross-elasticities among production factors, this does not necessarily hold for every pollution or energy level lower than E * . It may be that, for some pollution levels, the slope of the MAC function after innovation is greater than before innovation, or the reverse. In other words, the innovating firm may face higher marginal abatement costs if it has to reduce pollution below its optimal level in the absence of environmental regulation, although this may change for further abatements.

The following proposition summarizes the impact of environmental innovation on the MAC function in the general case.

Proposition 2 Adopting environmental innovation leads to a lower optimal pollution level in the absence of environmental regulation and a steeper marginal abatement cost curve at this pollution level.

Thus, the common assumption that environmental innovation reduces the marginal abatement cost is far from always being true. Moreover, the usual graphical representation of the impact of innovation on a MAC function, as proposed by many articles in the literature, is misleading. We illustrate this with two examples. the marginal abatement cost is higher after innovation. In the non-linear case, the functions cross twice. Innovation increases (in absolute terms) the slope of the MAC function when pollution is close to its optimal level without regulation, but it reduces it (in absolute terms) for lower pollution levels.

The marginal abatement cost is higher after innovation for medium pollution levels.

Policy implications: incentives to innovate under pollution regulation

In this section we compare the incentives for a firm to adopt a more environmentally friendly technology when pollution is regulated. To this end we consider two firms with the same production function F (X, E) but two different pollution functions, h(.) and h(.). We shall naturally call the former firm the dirty firm and the latter one the clean firm. We still assume that the fixed cost of innovation is zero. Calling the situation without environmental regulation laissez faire we successively analyze the effects of command-andcontrol and tax regulation. 1

Command-and-control regulation

Let us first consider the case of a quota on pollution. If this quota is expressed with respect to the firm's unregulated pollution level, then the abatement cost will unambiguously always be higher for the clean firm than for the dirty one. This holds regardless of whether the quota is expressed in relative terms (e.g. a 10 pc reduction w.r.t. laissez faire) or in absolute terms (e.g. x tons reduction). This results from the fact that, starting from a lower pollution 1 One straightforward implication of our results concerns the choice of policy instrument under uncertainty (prices versus quantities), as shown by Weitzman [START_REF] Xepapadeas | Advanced Principles in Environmental Policy[END_REF], since innovation modifies the slope of the marginal abatement cost curve. This will not be discussed here.

level due to a more efficient technology, the clean firm faces higher marginal abatement costs than the dirty one when further abatement is imposed. In this case, there is a clear disincentive for the firm to innovate. If the pollution quota is set at the same level for the two firms, independently of the pollution level under laissez-faire, then the clean firm will lose less profit than the dirty one. The main conclusion is that command-and-control measures based on current practices (such as benchmarking procedures or grand-fathering)

clearly provide no incentive for environmental innovation.2

Tax regulation

Let us now consider a uniform tax τ on pollution. The firm's profit (e.g.

for the dirty firm) writes

π τ = F (X, E) -p X X -p E E -τ h(E).
The firstorder condition on E for the two firms becomes F E -τ h (E τ ) = p E and

F E -τ h ( E τ ) = p E
, where E τ and E τ denote the optimal energy demand under the tax regulation. Since the two firms face the same energy price and since h (.) < h (.), ∀E, E τ > E τ if F EE -τ h (E) < 0. This holds under two conditions: firstly, if the pollution function h(.) is not too convex 3 , and, secondly, if the tax rate τ is not too high. This condition requires the profit function under the tax regime to be concave with respect to pollution. What this condition reveals is that, in the general (non-linear) case, the shape of the profit function, and therefore the shape of the MAC functions, depends on the tax level. Let us use our examples to highlight some implications.

Even though the clean firm's pollution is lower than that of the dirty firm under laissez-faire it may become higher if the tax on pollution is high enough. In the linear case (Example 1) the threshold tax rate is given by the intersection of the two MACs ( τ in Fig. 1). Despite this potential adverse effect (it means that the clean firm pays more tax than the dirty one), in the linear case the clean firm always gains from innovating because its total profit under the tax regime is always higher than that of the dirty firm. This shows that the firms' ranking in terms of environmental performance is influenced by the policy regime (see Bréchet and Michel [START_REF] Th | Environmental performance and equilibrium[END_REF] for a general discussion of environmental performance in an equilibrium setting). Things become much trickier in the non-linear case (Example 2). If the function h(.) is highly convex then the fact that the clean firm gets a lower profit than the dirty one under the tax regime cannot be ruled out. Actually, when computing total profits with a tax on pollution in our example we find that the clean firm has a higher profit than the dirty one only for medium tax levels, i.e.

τ ∈ (0.5, 1). Furthermore, the non-linear example reveals that introducing a tax on pollution not only moves the curve downwards (as in the linear case) but also alters its whole shape. As a consequence, the graphical analysis in which the optimal pollution level is given by the intersection between the tax rate and the MAC curve without tax (as in the linear case) is no longer valid.

Conclusion

In this paper we have shown that environmental innovation does not necessarily decrease the marginal cost of pollution abatement, as widely assumed in the literature. Actually, the marginal abatement cost function in the general case before and after innovation cannot be compared because the whole shape of this function is altered by policy instruments. Thus, the simple graphical analysis based on marginal abatement costs functions may lead to false conclusions 4 . Importantly, by comparing command-and-control regulation and emission fees we have shown that, in some cases, the incentive for environmental innovation is ambiguous in the sense that profit levels can 4 Some applied studies also make use of MAC functions in the analysis of the carbon market, since the seminal paper by Ellerman and Decaux [START_REF] Ellerman | Analysis of post-Kyoto CO 2 emissions trading using marginal abatement curves[END_REF]. The MAC functions considered at the macro-level represent the marginal GDP loss incurred for any abatement constraint imposed on the whole economy and evaluated in a general equilibrium setting.

Each point of the macro-MAC curve corresponds to a particular counterfactual general equilibrium of the economy expressed with respect to the initial equilibrium regarded as the business-as-usual scenario. Such functions should not be interpretated as micro-level abatement cost functions. All the studies involving applied macro-MAC functions assume that technological progress reduces the marginal abatement cost.

be lower after innovation. When a tax is imposed on pollution, it may be that the innovating firm (the clean one) gets a lower profit than the noninnovating firm (the dirty one). In the linear case, this cannot happen, but the clean firm may pollute more than the dirty one for strong abatement levels. Clearly, having reconsidered the effect of environmental innovation on pollution abatement costs yields unexpected results and opens the door for further researches.

Example 3 :

 3 the linear case. Let the production function be y = √ x and pollution given by p = αy, with α = 1 and α = 0.5. All prices are equal to unity. Figure 1 displays the two MAC functions. Example 4 : the non-linear case. Let the production function be y = √ x and pollution given by p = y α , with α = 1.5 and α = 1.1. All prices are equal to unity. Figure 2 displays the two MAC functions. ** Insert Figure 1 and Figure 2 ** In the linear case (Fig. 1), the two MAC functions have the following properties. They cross each other once and the areas below these two functions are equal (they represent the firm's total profit). Innovation increases (in absolute terms) the slope of the MAC function. For low pollution levels,

Figure 2 : 2 . 1 .

 221 Figure 2: MAC functions in the non-linear case

6 -

 6 Figure 1: MAC functions in the linear case 0 ................................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	∂π/∂P
	2 1 τ	∂ π/∂P H H H H A H A H A H A H H A A A A A A A A A A A A A A H H A A A	H	H	H	∂π/∂P H H H H H H H
		P	1/4		1/2	P
		6
							-

Another well-known results is that, with heterogeneous firms, command-and-control regulation is not cost-efficient.

For most pollutants and technologies this condition on h(.) will hold. For carbon dioxide, for example, this function is linear, if single-fuel based, or concave, if multi-fuel based.


Marie-Laure Breuillé
email: breuille@wzb.eu

Thierry Madiès
email: thierry.madies@unifr.ch

Emmanuelle Taugourdeau
email: emmanuelle.taugourdeau@ens-cachan.fr

Funding source and soft budget constraint

Keywords: JEL Classi…cation: E62; H7 Tax Competition, Soft Budget Constraint, Fiscal Federalism

The paper analyses the impact on the softness of the regional budget constraint of three di¤erent funding sources of the bailout: a horizontal equalization scheme, a federal non distortive tax on rents and a federal tax on the capital which generates vertical tax externalities. We show that the funding source of the bailout has a considerable impact both on the credibility of the federal government's commitment and on the regional opportunistic behaviour. By choosing the adequate tax tool, the federal government is able to harden the regional budget constraint.

Introduction

The soft budget constraint problem 1 in intergovernmental relationships is, to some extent, present in all countries but "its severity and the proposed mechanisms to handle it depend on each country's institutions" as emphasized by [START_REF] Rodden | Fiscal Decentralization and the Challenge of Hard Budget Constraints[END_REF] in their case studies. Transition economies or developing countries such as Brazil, India and Argentina (Jones, Sanguinetti, and Tommasi (1999), [START_REF] Dillinger | Can …scal stability coexist with decentralization? The case of LAC[END_REF]), as well as established federations such as the United States, Canada, Sweden (Pettersson-Lidbom and Dahlberg (2003)) and Germany [START_REF] Seitz | Fiscal Policy, De…cits, and Politics of Subnational Governments: The Case of the German Länder[END_REF]) have recently experienced …scal crisis linked to the soft budget constraint phenomenon which put substantial stress on macro-…scal stability. Fundamentally, the combination of the opportunistic behaviour of subnational governments, whose aim is to extract additional transfers from the federal government, and the federal government's inability to deny transfers ex post is the heart of the soft budget constraint problem. By extracting bailouts from the federal government, the subnational government shifts part of its spending to the national taxpayers and, generally, this turns out to be politically advantageous. While the federal government …nds it optimal ex ante to deny additional transfers in order to maintain …scal discipline in the federation, for a variety of reasons it often …nds it optimal ex post to bail out the subnational jurisdiction. The most common reasons are insurance against …scal shocks, political reasons, a large vertical …scal imbalance or equalization.

The recent theoretical contributions mainly focus on the determinants of the soft budget constraint problem in intergovernmental relationships. [START_REF] Wildasin | Externalities and bailouts : hard and soft budget constraints in intergovernmental …scal relations[END_REF] establishes a positive correlation between the size of the subnational jurisdiction and the amount of the bailout, i.e. the well-known "too big to fail" argument, in the presence of inter-regional spillovers. But the willingness of the federal government to provide a bailout turns out to depend negatively on the size of the subnational jurisdiction, in the presence of economies of scale in the public good provision, as shown by [START_REF] Crivelli | Size and Soft Budget Constraints[END_REF]. Another reason of the inability of the federal government to commit dynamically not to bail out, which has been implicitly put forward by [START_REF] Goodspeed | Bailouts in a Federation[END_REF], is the existence of constitutional obligations. For instance, the constitution may impose on the federal government to ensure equal opportunities for citizens across the country, as it is the case in Germany. The recent attempt of the heavily indebted city of Berlin to extract a massive bailout from the federal government shows that this implicit equalization objective strongly contributes to generate bailout expectations. [START_REF] Goodspeed | Bailouts in a Federation[END_REF] stresses that transfers from the federal level to the regional level generally involve a "common property problem", since the increasing federal tax rate levied to …nance the bailout reduces the opportunistic behaviour of the regional government by increasing its opportunity cost of the borrowing. In practice, in unitary states or federations, the same tax base is often taxed by several levels of government, which generates both horizontal tax externalities among jurisdictions of the same hierarchical level -which compete to attract the mobile base -and vertical tax externalities between di¤erent hierarchical levels of government due to the co-occupation of the same tax base [START_REF] Wilson | Theories of Tax Competition[END_REF], [START_REF] Wildasin | Fiscal Competition[END_REF] for a survey). Since horizontal and vertical tax externalities are a characteristic feature of most countries, understanding the impact of the tax structures and, more particularly, of the funding source of the bailout on the soft budget constraint problem is of considerable importance to the theory of …scal federalism. Do some tax architectures reinforce the softness of the budget constraint of subnational jurisdictions? Does tax competition serve as a commitment2 or disciplinary device to harden the budget constraint of subnational jurisdictions?

The purpose of the paper is to determine how federal transfers should be …nanced in order to eliminate or at least reduce the soft budget constraint phenomenon in intergovernmental relationships, by giving the subnational governments the incentives to behave …scally responsibly and by strengthening the federal government's commitment not to bail out. We consider three di¤erent funding sources for the bailout: a horizontal redistribution scheme 3(i.e. the bailout is …nanced by contributions made by the other regions), a federal non distortive tax on rents arising in each region, and a federal distortive tax on the capital which generates vertical tax externalities due to the co-occupation of this endogenous tax base with regions, as modelled by [START_REF] Keen | Does Federalism Lead to Excessively High Taxes?[END_REF]. The basic setting is a two-period model similar to [START_REF] Goodspeed | Bailouts in a Federation[END_REF]. Regions are allowed to borrow in the …rst period in order to …nance a public good, but the regional budget constraint must be balanced at the end of the second period. Regional governments act as Nash competitors with each other, but play as Stackelberg leaders vis-à-vis the federal government. Regional leadership is particularly appropriate for characterizing the intergovernmental relationships in a bottom-up system, and even more in a system of multiple mandates. The fact that the regional decision-maker is also a Member of Parliament enables him to expect perfectly the federal reaction function.

In contrast to [START_REF] Goodspeed | Bailouts in a Federation[END_REF], we consider an endogenous (rather than an exogenous) tax base co-occupied by the two layers of government, which generates horizontal and vertical externalities, and we explicitly derive the expressions of the bailout and the opportunity cost of borrowing for the di¤erent funding sources.

We show that the funding source of the intergovernmental transfers has a considerable impact both on the federal government's propensity to bail out and on the regional governments' strategic behaviour. By choosing the adequate tax instrument, the federal government turns out to be able to harden the regional budget constraint. The federal government's incentives for bailing out stem from its aim of equalizing the marginal utilities derived by citizens from the regional public good provision in the second period. Due to their decentralized leadership, the regional governments expect that the federal government will adjust both the amount and allocation of transfers in the second period following an increase in their borrowing in the …rst period. An expected bailout always encourages the regional government to borrow more than the e¢ cient level, certainly, but this opportunistic behaviour is deterred by expected negative tax externalities generated by the increase in the federal tax rate.

A horizontal redistribution scheme organized by the federal government, with no federal taxation, is always pro…table to the region which increases its borrowing, in spite of the fact that the region itself bears a part of the cost of the bailout which is proportional to its relative valuation of the public good. The amount of the bailout is even higher when …nancing by a non-distortive tax on rents, since it entirely compensates the additional cost of the debt repayment, but the reduction in the net rents, following an increase in the federal tax rate, deters the region from borrowing excessively. We explicitly derive the conditions under which the absence of funding of the bailout through taxation serves as a disciplinary device to harden the regional budget constraint with respect to the funding through a nondistortive tax on rents. It can be shown that a horizontal redistribution scheme organized by the federal government, with no federal taxation, reduces the regional …scal irresponsibility in a perfectly symmetric setting. Finally, the impact of the vertical tax competition on the softness of the regional budget constraint crucially depends on the distortive e¤ect of the taxation on capital. The case where the federal government punishes the borrowing region by reducing its transfers in order to compensate the vertical tax externalities borne by the other regions occurs under certain conditions. When hardening the regional budget constraint, the vertical tax competition is virtuous.

The paper is organized as follows. Section 2 presents the two-period model with regional borrowing. Section 3 analyses the federal government's incentives to bail out regions with respect to the funding source. Section 4 studies the impact of the tax structure on the regional incentives to borrow.

The model

Consider a federation with n regions, each of which consists of a representative citizen and a representative …rm, both immobile. The framework is a simple two-period intertemporal model. Regions have the possibility of borrowing in the …rst period, but the regional budget must be balanced at the end of the second period.

The representative citizen

The representative citizen located in region i derives utility from the consumption of private goods (c i1 ; c i2 ) and regional public goods (g i1 ; g i2 ) over the two periods 4 :

U i (c i1 ; g i1 ; c i2 ; g i2 ) = u i (c i1 ) + v i (g i1 ) + c i2 + z i (g i2 ) ; (1)
where u i (:), v i (:) and z i (:) are strictly increasing, twice di¤erentiable, and concave. Each citizen has an initial amount of income w i which is split between private consumption c i1 and savings

S i = s i i + P j6 =i s j i : c i1 = w i S i : (2)
Savings s j i invested5 in region j by the representative citizen of region i are remunerated at the before-tax interest rate r j minus the sum of a regional tax rate j and a federal tax rate c . Rents i arising in region i from the industrial activity are fully transferred to the representative citizen, for example, in the form of dividend, and can be taxed at the rate c by the federal government. The level of private consumption in the second period thus corresponds to the principal and interest on savings plus net rents:

c i2 = n X j=1 (1 + r j j c ) s j i + (1 c ) i : (3)

The capital market

The representative …rm located in region i produces an output F (K i ) from the capital

K i = n P j=1
s i j invested in the region, where F (:) is strictly increasing, twice di¤erentiable and concave. The maximization of the pro…t i = F (K i ) r i K i de…nes the demand for capital K i (r i ) in region i, where K 0 i = 1 F 00 < 0. Rents i (r i ) arising in region i are thus strictly decreasing with respect to the interest rate r i , i.e. 0 i = K i < 0. Capital is perfectly mobile across regions so that it relocates until it earns the same post-tax return in each region:

= r i i c = r j j c 8 i; j . ( 4
)
The amount of savings S i ( ), which satis…es the …rst-order condition @u i @c i1 = (1 + ) from the citizen's program, depends positively on this net return, i.e.

S 0 i = @S i @ = 1 @ 2 u i @c 2 i1 > 0. The market-clearing condition n X i=1 S i ( ) = n X i=1 K i (r i )
implicitly de…nes the net return ( 1 ; :::; n ; c ), so that

@ @ i = K 0 i n P j=1 S 0 j n P j=1 K 0 j 2 [ 1; 0],
and

@ @ c = n P i=1 K 0 i n P j=1 S 0 j n P j=1 K 0 j 2 [ 1; 0]; implying @r i @ i = 1 + @ @ i > 0 and @r i @ c = 1 + @ @ c > 0.
Horizontal and vertical tax externalities coming from the sharing of an endogenous tax base go through the net return of capital .

From the horizontal perspective, a tax rate reduction in region i, which increases the net return in that region, leads to a capital out ‡ow in other regions, ceteris paribus. From the vertical perspective, a tax rate reduction in region i increases the tax base shared with the federal level, ceteris paribus.

The regional government

The regional government aims at maximizing the utility U i (:) of the representative citizen located in its region i. A balanced budget is only required at the end of the second period which allows some intertemporal ‡exibility. The regional government is able to borrow an amount B i1 in the …rst period in order to …nance more of the public good g i1 , but the principal and interest of the debt (1 + r i ) B i1 must be repaid at the end of the second period 6 .

This debt variable turns out to be the instrument of the regional opportunistic behaviour.

Each regional government is encouraged to behave strategically in order to receive additional transfers from the federal government. Let T i1 denote the initial level of the federal transfer, which is exogenous to the game to be played, and T i2 the level of the federal transfer received by region i (or paid if negative) in the second period. The public good provision g i2 and the debt repayment are …nanced by the transfer T i2 and the taxation at the rate i of the capital

K i = n P j=1 s i
j invested in the region. Regional budget constraints of a region i are thus given by:

g i1 = T i1 + B i1 ; (5)
and

g i2 = T i2 + i K i (r i ) (1 + r i ) B i1 : (6)

The federal government

The federal policy 7 in terms of transfers is motivated by the maximization of an utilitarian utility function n P i=1 U i . In order to evaluate the impact of the source of the funding of the federal transfer 8 on the soft budget constraint issue, let us consider three cases: a) the federal government has no tax power:

n X i=1 T i2 = 0: (7a)
Transfers granted to a region k (T k2 > 0) are balanced by contributions made by the other regions (T k2 < 0). The federal government only proceeds to a revenue reallocation among regions in the second period.

b) the federal government levies a non-distortive tax c on rents arising in each region:

n X i=1 T i2 = c n X i=1 i (r i ) : (7b)
Note that bottom-up tax externalities arise from the regional taxation because this modi…es the interest rate and thus the level of the rents taxed by the federal government. In contrast, there are no top-down tax externalities due to the fact that the variation of c does not alter the regional tax base. c) the federal government levies a distortive tax c on capital:

n X i=1 T i2 = c n X i=1 K i (r i ) : (7c)
This source of funding generates both top-down and bottom-up tax externalities 9 . Due to the co-occupation of the endogenous tax base, i.e. the capital, a change in the tax policy decided either by the regional or federal governments generates vertical tax externalities by altering the tax revenue of both layers.

Our work aims at evaluating the e¤ects of the source of funding of the transfers on the federal government's incentives to bail out and also on the regional government's opportunistic behaviour.

The sequence of budgetary interactions

Regional governments act together as Nash competitors. Each region takes the other regions' borrowing and tax rate as given. In contrast, regional governments are assumed to act as Stackelberg leaders vis-à-vis the federal government. Each regional government knows how the federal government will react in second period, i.e. how it will modify the allocation of transfers, and takes into account the reaction function of the federal government in its choice of borrowing in the …rst period. The level of the regional borrowing will thus depend on the credibility of the federal government's commitment not to bail out in the second period. In addition, regional and federal governments also interact in the second period, but simultaneously, when choosing their tax rates. Finally, the citizens choose their level of savings and consumption. We solve the model by backward induction.

The federal government' s incentives to bail out

The federal government's best reply to a change in region k's borrowing is a key determinant of the softness or hardness of the budget constraint at the regional level. The inability of the federal government to commit dynamically not to bail out may indeed induce regional governments to misbehave, distorting their …scal choices ex ante, if they expect additional transfers from the federal government ex post. The federal commitment to deny a bailout is credible for @T k2 @B k1 = 0 whereas @T k2 @B k1 > 0 generates opportunistic expectations at the regional level. To derive the federal government's best reply to a change in region k's borrowing, let us …rst derive the sharing rule for regional transfers.

The federal policy is motivated by the maximization of the aggregated utility in the federation. Given regional budgetary choices, the federal government solves max

T 2 ; c; c n X i=1 [u i (c i1 ) + v i (g i1 ) + c i2 + z i (g i2 )]
s:t:

c i1 = w i S i ( ) ; c i2 = (1 + )S i ( ) + (1 c ) i (r i ); g i1 = T i1 + B i1 ; g i2 = T i2 + i K i (r i ) (1 + r i ) B i1 ; n X i=1 T i2 = c n X i=1 i (r i ) + c n X i=1 K i (r i ) :
In order to evaluate the impact of the source of funding of the transfers on the federal government's incentives to bail out, let us successively consider the case without tax power, the case with federal taxation on rents, and …nally the case with vertical tax competition.

No federal tax power ( c = 0, c = 0)

When the federal government has no tax power, the …rst-order conditions are:

@z i @g i2 = @z j @g j2 8i; j; (8) n X i=1 T i2 = 0: (9)
According to the condition (8), the federal government allocates transfers so as to equalize the marginal utilities of public consumption across citizens in the second period.

The marginal utility derived by the public good consumption must be the same for each citizen, independently of the region in which he or she resides. Note that, due to the concavity of z i (:) 8i, regions with high-valuation citizens receive more transfers from the federal government than regions with low-valuation citizens.

The federal government's best reply to a change in region k's borrowing is derived by di¤erentiating the …rst-order conditions ( 8) and ( 9) with respect to T k2 , T j2 8j 6 = k, and B k1 , which leads to the following Result:

Result 1: Without tax power, the federal government always bails out the region k, through lump-sum taxes levied on the other regions, following a rise in region k's borrowing:

@T nt k2 @B k1 = (1 + r k ) + 1 (n 1) X j6 =k @ 2 z j @g 2 j2 @ 2 z k @g 2 k2 @T nt j2 @B k1 > 0; @T nt j2 @B k1 < 0 8j 6 = k:
Proof: From the di¤erentiation of the …rst-order conditions (8) with respect to T i2 8i 6 = k,

T j2 8j 6 = k: @ 2 z i @g 2 i2 dT nt i2 = @ 2 z j @g 2 j2 dT nt j2 8i 6 = k; j 6 = k,
it follows that dT nt i2 8i 6 = k and dT nt j2 8j 6 = k have the same sign. In addition, the di¤erentiation of the federal government's budget constraint:

n X i=1 dT nt i2 = 0;
implies that dT nt k2 and P j6 =k dT nt j2 have the opposite sign. Using the expression of @T nt k2 @B k1 and the di¤erentiated federal budget constraint yields

X j6 =k 0 @ 1 + 1 (n 1) @ 2 z j @g 2 j2 @ 2 z k @g 2 k2 1 A @T nt j2 @B k1 = (1 + r k ) < 0 which implies that @T nt k2 @B k1 > 0 and @T nt j2 @B k1 < 0 8j 6 = k. Q:E:D:
A rise in region k's borrowing lowers the regional public good provision g k2 by

(1 + r k ) dB k1 ceteris paribus, which no longer satis…es the federal government's aim of equalization given by the condition [START_REF] Keen | Does Federalism Lead to Excessively High Taxes?[END_REF]. Once the regional government has played, the cost of not bailing out ex post becomes higher from the federal government's perspective than the cost of bailing out, which explains why the federal government always intervenes. The best reply of the federal government with respect to B k1 , denoted T nt k2 = ' nt k (B k1 ; B k1 ) in the absence of federal tax power, consists of two e¤ects. Firstly, the federal government compensates the additional cost of the debt repayment (1 + r k ). Secondly, the equalization of marginal utilities implies that the cost of the bailout is split among all the regions, including region k, accounting for the asymmetry between them. Note that @T nt k2 @B k1 = (n 1) n (1 + r k ) and

@T nt j2 @B k1 = 1 n (1 + r k ) 8j 6 = k in a symmetric setting.
Federal taxation on rents ( c = 0)

When the federal government only taxes rents arising in each region, the …rst-order conditions are:

@z i @g i2 = @z j @g j2 8i; j; ( 10
) n X i=1 i + @z i @g i2 n X i=1 i = 0; (11) n X i=1 T i2 = c n X i=1 i : (12)
The federal government's budgetary choices are still guided by the aim of equalization across the federation in the second period. Its best replies in terms of transfers and taxation to a change in region k's borrowing, respectively denoted T rt k2 = ' rt k (B k1 ; B k1 ) and rt c = rt k (B k1 ; B k1 ), are derived by di¤erentiating the …rst-order conditions ( 10), ( 11) and ( 12) with respect to T k2 , T j2 8j 6 = k, c and B k1 , which leads to the following Result:

Result 2: When the federal government is able to tax the rents, it always bails out the region k through additional federal tax revenues, without altering the transfers granted to other regions, following a rise in region k's borrowing:

@T rt k2 @B k1 = (1 + r k ) ; @T rt j2 @B k1 = 0 8j 6 = k and @ rt c @B k1 = (1 + r k ) n P i=1 i :
Contrary to the previous case where the federal government had no tax power, region k's bailout now amounts to the entire additional cost of the debt repayment (1 + r k ). The bailout is completely …nanced by an increase in the tax rate c levied on rents, which implies that the other regions'transfers are not a¤ected by the region k's level of debt. Whatever the level of the regional debt in the …rst period, the federal government ensures an optimal level of the public good such that @z i @g i2 = 1 8i. Let us now determine whether this result still applies with vertical tax competition.

Vertical tax competition ( c = 0)

When the federal government taxes the capital invested in each region, assuming an interior solution, the …rst-order conditions are 10 :

@z i @g i2 = @z j @g j2 8i; j; ( 13
) n X i=1 K i + @z i @g i2 n X i=1 ( i + c ) K 0 i @r i @ c @r i @ c B i1 + K i = 0; (14) n X i=1 T i2 = c n X i=1 K i : (15)
The federal government still allocates transfers among regions so as to equalize the marginal utilities of public consumption across citizens in the second period. From the condition ( 14), the optimal level of the federal tax c crucially depends on the extent of negative vertical tax externalities which go through the variation of the net return of capital . Following a rise in c , the net return decreases, which both reduces the rents ( 0 i @r i @ c < 0), and the common tax-base K 0 i @r i @ c < 0 , and weighs down the debt repayment @r i @ c B i1 > 0 . The optimal tax rate is such that the marginal bene…t of public consumption o¤sets these negative e¤ects. Note that we cannot rule out the particular case c = 0 which arises for considerable negative tax externalities 11 . In such a case, transfers granted to some regions are balanced by contributions made by other regions as studied above. 10 Without any rearrangement, the …rst-order condition with respect to c is

n X i=1 @u i @c i1 S 0 i @ @ c + (1 + ) S 0 i @ @ c + @ @ c S i + 0 i @r i @ c + @z i @g i2 i K 0 i @r i @ c @r i @ c B i1 + @z i @g i2 " n X i=1 K i + c n X i=1 K 0 i @r i @ c # = 0:
The expression ( 14) is obtained by using 0 i = K i , the …rst-order condition @ui @ci1 = (1 + ) from the consumer's program and the market-clearing condition. 11 An interior solution implies

n P i=1 h ( i + c ) K 0 i @ri @ c @ri @ c B i1 + K i i > 0:
Since @z i @g i2 is higher than one for all regions from the conditions ( 13) and ( 14), we derive the …nding that top-down tax externalities induce an underprovision of the regional public good in each region in the second period:

@z i @g i2 = n P i=1 K i n P i=1 h ( i + c ) K 0 i @r i @ c @r i @ c B i1 + K i i > 1;
which is a standard implication of tax competition.

The federal government's best replies in terms of transfers and taxation to a change in region k's borrowing, respectively denoted T vt k2 = ' vt k (B k1 ; B k1 ) and vt c = vt k (B k1 ; B k1 ), are also a¤ected by the vertical tax competition. By di¤erentiating the …rst-order condition [START_REF] Köthenburger | Tax competition in a …scal union with decentralized leadership[END_REF] with respect to T k2 , T j2 8j 6 = k, c and B k1 and summing across j 8j 6 = k, we obtain the following transfer scheme: Result 3: With vertical tax competition, the federal transfers'adjustment following a change in region k's borrowing satis…es:

@T vt k2 @B k1 = (1 + r k ) + 1 (n 1) X j6 =k @ 2 z j @g 2 j2 @ 2 z k @g 2 k2 @T vt j2 @B k1 1 (n 1) @ 2 z k @g 2 k2 X j6 =k @ @ c @z k @g k2 @z j @g j2 @ vt c @B k1 ; @ 2 z i @g 2 i2 @T vt i2 @B k1 + @ @ c @z i @g i2 @ vt c @B k1 = @ 2 z j @g 2 j2 @T vt j2 @B k1 + @ @ c @z j @g j2 @ vt c @B k1 8i 6 = k; 8j 6 = k:
Both the amount of the bailout and the sharing of federal transfers are modi…ed by the vertical tax externalities arising from the tax base sharing. The aim of equalization across the federation requires an adjustment of the allocation of the transfers which depends on the relative impact of the vertical tax competition on the marginal utilities of the public good consumption in the second period. The amount of the bailout @T vt k2 @B k1 thus crucially depends on the asymmetry among regions and on the federal government's best reply in terms of taxation to a change in region k's borrowing @ vt c @B k1 . The case where @T vt k2 @B k1 < 0 and @ vt c @B k1 > 0 cannot be excluded. By di¤erentiating the federal government's budget constraint with respect to T i2 8i, c and B k1 , we establish that:

@ vt c @B k1 = n P i=1 @T vt i2 @B k1 (1 + " c ) n P i=1 K i () @ vt c @B k1 (1 + " c ) n X i=1 K i = n X i=1 @T vt i2 @B k1 (16)
where

" c = @ n P i=1 K i @ c c n P i=1 K i
is the elasticity of the federal tax base with respect to the federal tax rate. Assuming " c 2 [ 1; 0] in line with empirical …ndings 12 , aggregated transfers move in the same way as the federal tax rate following a change in region k's borrowing. The value of @ vt c @B k1 (and thus the amount n P i=1 @T vt i2 @B k1 ) obtained by di¤erentiating the …rst-order conditions ( 13), ( 14) and ( 15) depends on the speci…cation of the model and, more particularly, on the distortive impact of vertical tax competition 13 . As regards sharing of regional transfers, @T vt k2 @B k1 (resp. @T vt j2 @B k1 8j 6 = k) can either be positive or negative, depending on the respective valuations for the public good provided in the second period and the value of @ vt c @B k1 . However, by considering the perfectly symmetric case, we are able to state the following results:

Result 4: With vertical tax competition, in a perfectly symmetric setting, the federal transfer scheme simpli…es as follow:

@T vt k2 @B k1 = (1 + r k ) + @T vt j2 @B k1 ; @T vt i2 @B k1 = @T vt j2 @B k1 8i 6 = k; 8j 6 = k;
which implies that:

-for 0 < @ vt c @B k1 (1 + " c ) n P i=1 K i < (1 + r k )
, the federal government always bails out the region k, through both additional federal tax revenues and lump-sum taxes levied on the other regions, following a rise in region k's borrowing 14 .

-for @ vt c @B k1 (1 + " c )

n P i=1 K i > (1 + r k )
, the federal government always bails out the region k, as well as the other regions, following a rise in region k's borrowing.

12 See Chirinko, Fazzari, Steven and Meyer (1999) for instance. 13 Computations on request. 14 Note that Goodspeed (2002) only considers two possibilities, i.e. P j6 =k @Tj2 @B k1 > 0 or P j6 =k @Tj2 @B k1 = 0, while our model also leads to envisaging P j6 =k @Tj2 @B k1 < 0.

-for @ vt c @B k1 (1 + " c ) n P i=1 K i < (n 1) (1 + r k ), the federal government reduces all the transfers, following a rise in region k's borrowing. Vertical tax competition serves as a commitment device for hardening the regional budget constraint.

Proof: Inserting the expressions of @T vt k2 @B k1 and @T vt j2 @B k1 in the condition [START_REF] Rodden | Fiscal Decentralization and the Challenge of Hard Budget Constraints[END_REF] gives:

@T vt k2 @B k1 = (n 1) n (1 + r k ) + 1 n (1 + " c ) n X i=1 K i @ vt c @B k1 ; @T vt j2 @B k1 = 1 n " (1 + " c ) n X i=1 K i @ vt c @B k1 (1 + r k )
# which allows us the derive the previous results. Q:E:D:

The relative extent of the additional cost of the debt repayment (1 + r k ) and the additional federal tax revenues following a change in region k's borrowing @ vt c @B k1 (1 + " c )

n P i=1 K i ,
crucially determines whether a threat from the federal government not to bail out is credible or not. Note that for (1 + " c )

n P i=1 K i @ vt c @B k1 = (1 + r k ) (resp. @ vt c
@B k1 = 0), the results are the same as those with a taxation on rents (resp. no taxation). Let us now evaluate the impact of the funding source of the bailout on the regional opportunistic behaviour. 4 The regional government' s opportunistic behaviour

The regional policy maker of the region k, which is motivated by the maximization of the utility of the representative citizen located in its region, solves max

B k1 ; k u k (c k1 ) + v k (g k1 ) + c k2 + z k (g k2 ) s:t: c k1 = w k S k ( ) c k2 = (1 + )S k ( ) + (1 c ) k (r k ) g k1 = T k1 + B k1 g k2 = T k2 + k K k (r k ) (1 + r k ) B k1
expecting the federal government's reaction function, respectively

T nt k2 = ' nt k (B k1 ; B k1 ) (17a)
T rt k2 = ' rt k (B k1 ; B k1 ) and rt c = rt k (B k1 ; B k1 ) (17b)
T vt k2 = ' vt k (B k1 ; B k1 ) and vt c = vt k (B k1 ; B k1 ) (17c)
when the federal government has no tax power (17a), when it levies a tax c on rents (17b)

and when it levies a tax c on capital (17c).

In order to evaluate the impact of the funding source of the bailout on the regional government's opportunistic behaviour, let us successively consider the …rst-order conditions without federal tax power, with federal taxation on rents and …nally with vertical tax competition.

No federal tax power ( c = 0, c = 0)

@v k @g k1 + @z k @g k2 @T nt k2 @B k1 (1 + r k ) = @v k @g k1 + @z k @g k2 (n 1) X j6 =k @ 2 z j @g 2 j2 @ 2 z k @g 2 k2 @T nt j2 @B k1 = 0 (18a) @ @ k S k + 0 k @r k @ k + @z k @g k2 K k + k K 0 k @r k @ k @r k @ k B k1 = 0 Federal taxation on rents ( c = 0) @v k @g k1 + @z k @g k2 @T rt k2 @B k1 (1 + r k ) @ rt c @B k1 k = @v k @g k1 (1 + r k ) k n P i=1 i = 0 (18b) @ @ k S k + (1 c ) 0 k @r k @ k + @z k @g k2 K k + k K 0 k @r k @ k @r k @ k B k1 = 0 Vertical tax competition ( c = 0) @v k @g k1 + @z k @g k2 @T vt k2 @B k1 (1 + r k ) + @ @ c S k + 0 k @r k @ c + @z k @g k2 k K 0 k @r k @ c @r k @ c B k1 @ vt c @B k1 = 0 (18c) @ @ k S k + 0 k @r k @ k + @z k @g k2 K k + k K 0 k @r k @ k @r k @ k B k1 = 0
Regardless of the funding source of the bailout, the equilibrium tax rate results from a trade-o¤ between the reduction in the utility derived from the private good consumption and the increase in the utility derived from the public good consumption in the second period.

On the one hand, the rise in the regional tax rate reduces the net return of savings, i.e. @ @ k S k < 0, and the rent accruing to the citizen through an increase in the cost of capital, i.e. 0 k @r k @ k < 0, which reduces the private consumption c i2 . On the other hand, the rise in the regional tax rate generates additional tax revenues, which are higher than the additional cost of the debt repayment for an interior solution, and this allows the regional government to produce more of the public good.

In the absence of decentralized leadership, the region would choose a level of debt which satis…es the following condition:

@v k @g k1 (1 + r k ) @z k @g k2 = 0 , @v k @g k1 @z k @g k2 = (1 + r k ) :
The price faced by the regional government when it borrows would amount exactly to the marginal cost of the debt repayment (1 + r k ).

Given that 0 < @T nt k2 @B k1 < (1 + r k ), in the case of the absence of funding through taxation, the inability of the federal government to commit not to bail out always reduces the regional opportunity cost of borrowing. From the moment the federal government levies taxes in order to …nance the additional transfers, the increase in the federal tax pressure reduces, ceteris paribus, the regional incentives to borrow. As a result, the decentralized leadership does not always give a strategic advantage to the regional government over the federal government.

It turns out, from the comparison between (18a) and (18b), that:

Result 5: The funding of the bailout via the horizontal redistribution scheme rather than via the tax on rents serves as a disciplinary device to harden the regional budget constraint when: @z k @g k2 (n 1) X j6 =k @ 2 z j @g 2 j2 @ 2 z k @g 2 k2 @T nt j2 @B k1 < (1 + r k )

k n P i=1 i ;
which is always the case in a perfectly symmetric setting, since @z nt k @g k2 > 1.

As for the impact of the funding through taxation on capital, which generates vertical tax externalities, on the regional incentives to borrow, it depends on the respective extent of @z k @g k2 @T vt k2 @B k1

(1 + r k ) and h @ @ c S k + 0 k @r k @ c + @z k @g k2 k K 0 k @r k @ c @r k @ c B k1 i @ vt c @B k1 . The term in square brackets being negative, a rise in c following an increase in the region k's borrowing always dissuades the region from borrowing because it reduces the proceeds of citizens' savings @ @ c S k , the rents 0 k @r k @ c , the regional tax revenues k K 0 k @r k @ c and weights the debt repayment @r k @ c B k1 . So, when the federal tax rate reacts positively (resp. negatively) to an increase in borrowing, the region is less (resp. more) incited to borrow. In addition, the less the region values an additional unit of public good with respect to the others regions, the lower its additional transfer, which reduces its incentives to borrow.

Result 6: The vertical tax competition may serve as a disciplinary device to harden the regional budget constraint as well as it may encourage a regional opportunistic behaviour.

Proof: The vertical tax competition clearly serves as a disciplinary device (resp. encourages a regional opportunistic behaviour) when @ vt c @B k1 > 0 and @T vt k2 @B k1 < 0 (resp. @ vt c @B k1 < 0 and @T vt k2 @B k1 > 0). Q:E:D:

The vertical tax competition may thus be virtuous, by hardening the regional budget constraint. In accordance with the Result 3, it clearly occurs when @T vt k2 @B k1 < 0 following a rise in the federal tax rate on capital, i.e. when the vertical tax externalities borne by the other regions j 6 = k are much more higher than the cost borne by the region k. To some extent, the federal government punishes the borrowing region by reducing its transfers in order to compensate the vertical tax externalities borne by the other regions.

The concept of soft budget constraint was originally introduced by[START_REF] Kornai | The Soft Budget Constraint[END_REF] to describe the behaviour of state-owned enterprises in socialist economies that got into huge debt, expecting the rescue by the State ex post. This concept has recently been extended to a number of di¤erent institutional settings including …scal federalism.

[START_REF] Qian | Federalism and the Soft Budget Constraint[END_REF] show that …scal decentralization, together with horizontal tax-base mobility among the rescuers, may serve as a commitment device for hardening the budget constraints of state-owned enterprises, as this increases the opportunity cost of the bailout.

If the German Constitutional Court would have decided in 2006 to support the demand by Berlin for …nancial assistance to cope with its high debt, wealthy Länder would have contributed, according to the "Länder…nanzausgleich".

As[START_REF] Keen | Does Federalism Lead to Excessively High Taxes?[END_REF], preferences with respect to c i2 are linear, this ensures that savings only depend on the net return of capital.

Capital is freely mobile so that citizens are free to invest their savings anywhere.

By simpli…cation, the regional debt is held by foreign investors.

By simpli…cation, no public good is provided at the federal level as this would not modify the strategic interactions between the federal and regional governments given our additive utility function.

T i2 can also be negative which amounts to taxing region i.

Top-down (resp. bottom-up) tax externalities are caused by the higher (lower) level of government.

Concluding remarks

The funding source of the intergovernmental transfers turns out to have a considerable impact both on the federal government's propensity to bail out and on the regional governments' …scal responsibility. From a simple decentralized leadership model over two periods, we compare the hardening impact of three funding sources of the bailout: a horizontal equalization scheme with no federal taxation, a federal non distortive tax on rents, and a federal tax on the capital which generates vertical tax externalities. Especially in a symmetric setting, the funding of the bailout via an horizontal equalization scheme organized by the federal government, with no federal taxation, improves the …scal discipline with respect to the funding via a non-distortive tax. Interestingly, the vertical tax competition may be virtuous, by hardening the regional budget constraint.