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Local Public Goods, Risk Sharing, and Private Information in Federal Systems
Journal of Urban Economics 47, 39᎐60 Ž2000. Article ID juec.1999.2132, available online at http:rrwww.idealibrary.com on
Local Public Goods, Risk Sharing, and Private Information in Federal Systems1 Richard C. Cornes Department of Economics, Keele Uni¨ ersity, Keele, Staffordshire ST5 5BG, UK
and Emilson C. D. Silva Department of Economics, Tulane Uni¨ ersity, New Orleans, Louisiana 70118-5698 Received January 29, 1997; revised January 25, 1999 We examine the implications of information asymmetries for the optimal design of interregional insurance schemes and the allocation of local public goods. We have good and bad news. The good news is that the presence of informational asymmetries is not in itself sufficient to deny federations the attainment of either efficiency or egalitarian goals. The bad news is that not only the first best may not be incentive compatible but also the incentive compatible optimum may violate participation constraints. Federations may have to content themselves with ‘‘fourth best’’ allocations as a result of strategic manipulation of information by privately informed jurisdictions. 䊚 2000 Academic Press Key Words: information asymmetry; interregional insurance; federal systems; participation constraints.
1. INTRODUCTION The role of explicit real resource transfers between the member states of federations has been widely discussedᎏsee, for example, Wildasin w16x. In some established federations, they represent a significant fraction of tax revenue, and the existing institutions and principles that determine and affect them continue to be keeny debated. Within the European Union ŽEU. also, as it moves toward a more closely knit federal structure, the design of schemes for transferring real income between regions has become a key issue in at least two respects. First, the move toward monetary 1 We thank Jan Brueckner, two anonymous referees, and the participants of the 52nd Congress of the IIPF in Tel-Aviv for numerous helpful comments.
39 0094-1190r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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CORNES AND SILVA
union implies that a major implicit insurance instrumentᎏexchange rate adjustmentᎏwill cease to be available. Second, the EU’s structural policies cannot be implemented without interregional transfers. Major financial instruments used to advance structural policies within the EUᎏe.g., the Regional Development Fund, the Social Fund, the Cohesion Fund, and the loans from the European Investment Bankᎏobey the principle of ‘‘additionality,’’ whereby the EU offers additional resources to member nations to help them finance their own structural programs. These programs are mostly intended to develop regions that are lagging behind. The resources are used to build infrastructure, to improve human resources, and to stimulate productive investment. ‘‘The resources for EU structural policies have increased substantially from 3.7 billion ECU in 1985 to 18.3 billion ECU in 1992 and to 33 billion ECU in 1999. For the period of 1994᎐99, around 170 billion ECU is available from the Community’s budget for structural policies. This represents about a third of total Community spending and 0.45% of Community GDP. Over the decade 1988᎐99, spending amounts cumulatively to 6.5% of annual Community GDP. A comparison makes its importance clear: Marshall aid to postwar Europe was equivalent to 1% of US GDP per year and amounted cumulatively Ž1948᎐51. to 4% of US GDP’’ ŽFirst Report from the Commission on Economic and Social Cohesion w8, p. 9x.. The term ‘‘fiscal equalization’’ suggests a concern with equity as a motivating force behind such transfers. In the EU, for example, economic and social cohesion is the main objective of structural policies. Article 130a of the Treaty on European Union states this objective in terms of ‘‘harmonious development’’ as follows: ‘‘reducing disparities between the levels of development of the various regions and the backwardness of the least favored regions, including rural areas.’’ In the United States, the current public education financing equalization trend can be explained by a concern of the states with equal educational opportunities. Several state aid formulas now exist to implement transfers among school districts with the objective of reducing education spending disparities Žsee, e.g., Hoxby w9x and Monk w12x.. But fiscal equalization cannot be the whole story, since it is hard to imagine a region or country agreeing to the prospect of becoming a permanent source of transfers to others. This suggests an efficiency role, and one natural role that may be filled by transfers is that of providing insurance, namely, allowing region that encounter a ‘‘bad shock’’ to augment their income through transfers from those enjoying better fortune. Suppose that, within each jurisdiction, the value of some exogenous parameter that influences the attainable per capital utility level in that jurisdiction is known only as a probability distribution. Each jurisdiction may face a different probability distribution. On the basis of these distribu-
PRIVATE INFORMATION IN FEDERAL SYSTEMS
41
tions, the jurisdictions wish to create an income redistribution program so that, in the event that the realized values of this parameter differ across jurisdictions, transfers from jurisdictions experiencing ‘‘good shocks’’ to those experiencing ‘‘bad shocks’’ reduce interregional differences in per capita utilities. Such a redistribution scheme is desirable if and only if it is individually rational for each jurisdiction to participate in an ex ante sense. If this condition is satisfied, each jurisdiction will agree to delegate to a central agency the power to effect transfers and to commit itself to remaining in the scheme after realizing its own parameter value. In the EU, the 1988 reform of the Structural Fundsᎏi.e., the creation of Objective regions, ranking from 1 to 6 according to their development priorityᎏmay have been motivated by a concern of the center with voluntary participation of all member nations in the financing of the structural policies. Objective 1 regions, for example, are regions where development is lagging behind. Just about every member nation has a fraction of its territory considered an Objective 1 region. These regions receive 70% of all resources spent by the structural policies. Hence, member nations that are now net exporters of resources may in the future be net importers. We may indeed experience such a reversal within a few years. Whereas Ireland, now a net importer of resources, is growing at a much faster rate than the Community’s average, Finland, a net exporter of resources, is growing at lower than average rate. It is quite possible that within the near future, Ireland will be a net exporter and Finland will be a net importer of resources. The current system for allocating the Structural Funds may provide member nations with insurance against hard times, and such an insurance motive may be strong enough to induce current net exporters of resources to participate in the financing of the structural policies. w11x and Costello w6x have observed, in addition to But, as Maler ¨ uncertainty about the state of nature, there may also be information asymmetriesᎏspecifically, information about local conditions is often decentralized.2 In this paper, we explore the implications of information decentralization for the allocation of local public goods and risk sharing within a federation. To keep things simple, our federation is assumed to be formed by a central government and two jurisdictional governments. Information decentralization is captured with the assumption that the central agency cannot observe the realized value of each jurisdiction’s 2
In his analysis of the European acid rain problem, Maler ¨ noted: ‘‘There is another kind of uncertainty which we must look into, however. This has to do with the information one country may have about costs and damage in other countries. In general, the control costs and environmental damage in one particular country is known to that country only’’ ŽMaler ¨ w11, p. 266x.. The first kind of uncertainty Maler ¨ referred to is uncertainty regarding both damage and control costs.
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CORNES AND SILVA
exogenous parameter, even though this realization is revealed to that jurisdiction. The optimal redistribution scheme with such private information must be individually rational ex anteᎏso that each jurisdiction finds it desirable to participate voluntarilyᎏand incentive compatible ex postᎏso that the privately informed jurisdiction finds it desirable to reveal its ‘‘type’’ truthfully. We find that the constrained efficient allocation, designed by the central agency, may involve Ž1. no binding constraints Žfirst best.; Ž2. one binding incentive compatibility constraint Žsecond best.; Ž3. one binding incentive compatibility and one binding participation constraint Žthird best.; and Ž4. one binding incentive compatibility and two binding participation constraints Žfourth best.. Several issues involving informational asymmetries between local and central authorities as well as risk sharing within federations have recently received much attention from students of fiscal federalism Žsee, e.g., Boadway, Horiba, and Jha w1x, Bordignon, Manasse, and Tabellini w2x, Bucovetsky w3x, Cornes and Silva w4᎐5x, Cremer and Pestieau w7x, Lockwood w10x, Persson and Tabellini w14x, and Raff and Wilson w15x.. Unlike this literature, we study the limits imposed by both Žex ante. participation constraints and Žex post. incentive compatibility constraints on the design of interregional income redistribution schemes and the provision of local public goods. The creation of Objective regions in the EU as well as the recent experiences with federation formation in Europe and the threat of secession in Canada seem to indicate that participation constraints deserve serious consideration. Indeed, participation constraints may lower the levels of interregional transfers and hence be harmful to the provision of interregional insurance. 2. THE MODEL Consider a federation consisting of two jurisdictions, indexed by i, i s 1, 2. Jurisdiction i possesses a population of n i immobile residents. To emphasize the crucial differences between jurisdictions, we shall assume throughout that n1 s n 2 s n. The representative consumer of jurisdiction i consumes a composite basket of goods, the level of which is denoted c i . In most of what follows, the consumption level for the representative consumer in each jurisdiction is stochastic because it depends on the realizations of two i.i.d. random variables, i , which represent the exogenous components of the unit production costs of the local public goods provided by the two jurisdictions. A vector of quantities Ž c iL, L , c iL, H , c iH , L , c iH , H . for the representative consumer in jurisdiction i represents a contingent consumption plan in which he consumes a quantity c iL, L when Ž 1 , 2 . s Ž 1L , 2L ., a quantity c iL, H when Ž 1 , 2 . s Ž 1L , 2H ., and so on. Let g Ž0, 1. denote the probability that i s i L .
43
PRIVATE INFORMATION IN FEDERAL SYSTEMS
The representative consumer in jurisdiction i is assumed to derive the expected utility over consumption of Ž c iL, L , c iL, H , c iH , L , c iH , H .,
u Ž c iL , L . q Ž 1 y . u Ž c iL , H . q Ž 1 y . u Ž c iH , L . q Ž 1 y . u Ž c iH , H . ,
i s 1, 2,
Ž 1.
where u⬘ ) 0, u⬙ - 0, and c ij, k ' x ij, k q f Ž qij .
when
Ž 1 , 2 . s Ž 1j , 2k . , i s 1, 2,
j, k s H , L.
Ž 2.
For simplicity, the consumption function c ij, k are assumed to be quasilinear. The terms x ij, k denote consumption levels of a private good Žnumeraire., and the terms f Ž qij . represent consumption levels of a local public good. The function f, which is assumed to be increasing and strictly concave, transforms qij units of the local public good into f Ž qij . units of consumption. The quasi-linear form of the consumption functions implies that income effects are captured solely by consumption of the private good. Hence, the level of the local public good in jurisdiction i depends exclusively upon the realization of i . To provide qij units of a local public good, the government of jurisdiction iᎏhenceforth called ‘‘government i’’ᎏincurs the sum of production costs, pij qij, an administrative costs, ⌿ Ž e ij ., pij qij q Ž e ij . ,
i s 1, 2,
j s H , L,
Ž 3.
pij ' i j y e ij ,
i s 1, 2,
j s H , L.
Ž 4.
where
pij
In jurisdiction i and state of nature j, denotes the price of the local public good and e ij denotes the level of cost-reducing effort exerted by government i. In most of what follows, we will assume that the random exogenous component of the local public good’s price in jurisdiction 1 is private information to that jurisdiction. Although the center knows that government 1’s effort level will be either low or high, it cannot tell a priori if a high price for the local public good in jurisdiction 1 is consequence of a bad shock or of shirking by government 1.3 In other words, government 3 We shall assume that prices and quantities of the local public goods are observable by the center. Our analysis, therefore, focuses on tangible local public goods, namely, local public goods whose quantities and prices are readily observable Že.g., local infrastructure .. Our model is not suitable to explain situations where quantities of local public goods are unobservable Že.g., local public education.. See Cornes and Silva w5x for an analysis of how privately informed jurisdictions mix unobservable quantities of local public goods in light of finance equalization schemes.
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1’s administrative effort is non-observable by the center. This assumption is consistent with Niskanen’s w13x theory of bureaucratic behavior. According to this theory, Congress Žor the central government. lacks the time or incentive to monitor the activities undertaken by the bureaucracy. As a result, there is a moral hazard problem: the center does not know the actual costs associated with programs overseen by the bureaucracy. Our jurisdictional governments are analogous to Niskanen’s bureaus. If government i reduces its local public good’s price by e ij units, it incurs an administrative cost of ⌿ Ž e ij .. We assume that ⌿ Ž0. s 0, ⌿ is increasing and strictly convex, and ⌿ approaches infinity as e ij approaches i j. With this last assumption, we guarantee that the prices of the local public goods will always be strictly positive. We assume that the center knows ⌿. This implies that the center knows the two possible levels of the administrative cost in each jurisdiction, but does not know which of these two levels has been realized. We shall refer to these costs as accounting costs in what follows. Each consumer in the federation is initially endowed with I units of the private good. In the two states of nature, the budget constraints for the representative consumer in jurisdiction i are x ij, k q t ij, k s I,
i s 1, 2,
j, k s H , L,
Ž 5.
where
t ij, k '
pij qij q ⌿ Ž e ij . q i j, k n
,
i s 1, 2,
j, k s H , L,
Ž 6.
represent his jurisdictional taxes.4 The terms x ij, k give his expenditures on private-good consumption. The terms i j, k denote the income transfers received or paid by government i. If, for example, government 1 makes a transfer to government 2 when Ž 1 , 2 . s Ž 1j, 2k ., we have lj, k ) 0 and 2j, k s y 1j, k - 0. The interregional transfer scheme, if implemented, is controlled by the central government. We will have more to say about this scheme below.
4 In our analysis of private information, the actual administrative costs in jurisdiction 1 will be unobservable. The center will not observe the actual jurisdictional taxes in such jurisdiction. The center, however, does observe this jurisdiction’s accounting administrative costs, which may or may not be the same as the actual costs. As a result, the center observes this jurisdiction’s taxes based on accounting rather than actual costs.
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PRIVATE INFORMATION IN FEDERAL SYSTEMS
Combining Eqs. Ž2. through Ž6., we can rewrite the consumption functions as follows: c ij, k ' c Ž qij , pij , i j, k ; i j . ' I q f Ž qij . y
pij qij q ⌿ Ž i j y pij . q i j, k n i s 1, 2,
j, k s H , L.
,
Ž 7.
To keep things simple, we shall assume henceforth that the induced consumption functions Ž7. are strictly concave in the first two arguments. This assumption implies that 5 nf ⬙ Ž qij . ⌿⬙ Ž i j y pij . q 1 - 0,
i s 1, 2,
j s H , L,
Ž 8.
A. The Decentralized Equilibrium: No Risk Sharing To motivate our analysis of risk sharing under private information, we shall first consider the allocation where the jurisdictions do not share risk. The central government does not intervene, and the jurisdictions do not transfer income between themselves on their own. The jurisdictional governmentsᎏassumed to be benevolent and utilitarian ᎏplay a Nash noncooperative game. Since i j, k s 0, government 1 chooses q1j , p1j 4jsH , L to maximize
u IqfŽ
ž
q1L
p1L q1L q ⌿ Ž 1L y p1L .
.y
/
n
qŽ 1 y . u I q f Ž
ž
q1H
.y
p1H q1H q ⌿ Ž 1H y p1H . n
/
,
Ž 9a .
taking the decisions of government 2 as given. Similarly, government 2 chooses q2j , p 2j 4jsH , L to maximize p 2L q2L q ⌿ Ž 2L y p 2L .
u I q f Ž q2L . y
ž
n
q Ž 1 y . u I q f Ž q2H . y
ž
/
p 2H q2H q ⌿ Ž 2H 2 y p 2H . n
/
,
Ž 9b .
taking the decisions of government 1 as given. Direct inspection of objective functions Ž9. reveals that the Nash equilibrium will be degenerate, since the welfare of each jurisdiction is unaffected by the other jurisdiction’s choices. 5
Conditions Ž8. guarantee the uniqueness of the decentralized equilibrium.
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CORNES AND SILVA
The decentralized equilibrium is characterized by the following equations: nf ⬘ Ž qij . s pij ,
i s 1, 2,
⬘ Ž x ij y pij . s qij ,
i s 1, 2,
j s H , L, j s H , L.
Ž 10 . Ž 11 .
Equations Ž10. are the Samuelson conditions that determine the efficient provision levels for the local public goods. They demonstrate that the local public goods should be provided at levels which equate each jurisdiction’s marginal social benefits to marginal production costs. At each state of nature, jurisdiction i’s marginal production cost depends on the optimal level of effort. Equations Ž11. show that the optimal prices for the local public good are derived from equalization of the marginal administrative costs to the marginal cost savings which result from exerting effort. Our assumption that the consumption functions Ž7. are strictly concave implies that the Nash equilibrium is unique. Let qij0 , p j0 . i 4jsH , L denote government i’s optimal solution. Straightforward comparative statics yield dpij0 d ii dqij0 d i
j
s s
nf ⬙ ⌿⬙ nf ⬙ ⌿⬙ q 1 ⌿⬙ nf ⬙ ⌿⬙ q 1
) 0,
i s 1, 2,
j s H , L,
Ž 12 .
- 0,
i s 1, 2,
j s H , L.
Ž 13 .
Given Ž8., the common denominator on the right sides of Eqs. Ž12. and Ž13. is strictly negative. To save space, we shall simplify our model even further by assuming from this point on that 2H s 2L and 1H ) 2H s 2L ) 1L ) 0. In words, the exogenous parameter in jurisdiction 2 takes the same value in both good and bad states of nature. This constant value is assumed to lie between the values taken by the exogenous parameter in jurisdiction 1. These assumptions lead to no significant loss of generality. We can still examine the efficiency gains associated with risk sharing subject to private information regarding 1. Since 2H s 2L and 1H ) 2H s 2L ) 1L ) 0, conditions Ž12. imply that p1H 0 ) p 2H 0 s p 2L0 ) p1L0 and conditions Ž13. imply that q1L0 ) q2L0 s q2H 0 ) q1H 0 . Government 2 provides the same bundle in both states of nature because its exogenous cost parameter takes the same value in both states. Equations Ž10. and Ž11., therefore, essentially yield three systems, each of two equations and two variables.
47
PRIVATE INFORMATION IN FEDERAL SYSTEMS
Let government i’s ex post, indirect utility be denoted as follows: u ij0 ' u Ž c ij0 . ' u Ž c Ž qiL0 Ž i j . , piL0 Ž i j . , 0 . . ,
i s 1, 2,
j s H , L.
Since du ij0 d i j
sy
u⬘ Ž c ij0 . ⌿⬘ n
- 0,
i s 1, 2,
j s H , L,
it follows that u1L0 ) u 2L0 s u 2H 0 ) u1H 0 . Henceforth, the consumption functions, transfers, and all other stochastic variables will possess only one superscript, which denotes the state of nature in jurisdiction 1. In the decentralized equilibrium, the level of welfare in jurisdiction 1 is higher than the level of welfare in jurisdiction 2 in the good state, but the ordering is reversed in the bad state. This implies that the federation faces macroeconomic risk and that consumers will not be fully insured against risk. B. Risk Sharing with Public Information: The First Best Let us now examine the ideal setting, where the jurisdictions share risk and information about the realization of 1 is public Ži.e., common knowledge.. We assume that a benevolent and utilitarian central government wishes to design an interregional insurance scheme in which jurisdiction 1 receives from jurisdiction 2 an income transfer in the bad state of natureᎏi.e., 1H - 0ᎏwhile jurisdiction 2 receives from jurisdiction 1 an income transfer in the good state of natureᎏi.e., 1L ) 0. Remember that the system is closed in the sense that 2j s y 1j, j s H, L. Government 2 now cares about the realization of the random variable 1. The ex post welfare level in jurisdiction 2 will be higher Žlower. in the good Žbad. state of nature than in the decentralized equilibrium. The income transfers enable the jurisdictions to smooth their residents’ consumption levels over the two possible states. Such a consumption smoothing is highly desirable because consumers are risk averse. Ex ante social preferences are assumed to be represented by the expected social welfare function,
u Ž c Ž q1L , p1L , 1L ; 1L . . q u Ž c Ž q2 , p 2 y 1L ; 2 . . q Ž 1 y . u Ž c Ž q1H , p1H , 1H ; 1H . . q u Ž c Ž q2 , p 2 , y 1H ; 2 . . , Ž 14 . where, to simplify notation hereafter, 2 ' 2H s 2L , p 2 ' p 2H s p 2L , and q2 ' q2L s q2H.
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CORNES AND SILVA
Our benchmark allocationᎏdenoted ‘‘first best’’ᎏis the allocation that maximizes the expected social welfare function Ž14. subject to the requirement that each jurisdiction receives at least as much expected utility in the first best as in the decentralized equilibrium, namely,
u Ž c Ž q1L , p1L , 1L ; 1L . . q Ž 1 y . u Ž c Ž q1H , p1H , 1H ; 1H . . G u1L0 q Ž 1 y . u1H 0 ,
Ž 15a .
u Ž c Ž q2 , p 2 , y 1L ; 2 . . q Ž 1 y . u Ž c Ž q2 , p 2 , y 1H ; 2 . . G u 20 , Ž 15b . where u 02 ' u 2L0 s u 2H 0 denotes the expected utility level for the representative consumer of jurisdiction 2 in the decentralized equilibrium. We impose the participation constraints Ž15. because we wish to provide a rationale for the ¨ oluntary formation of a federal insurance scheme in which jurisdictions share risks associated with jurisdictional shocks. To highlight the limits imposed by informational asymmetries, when information about 1 is private, we assume that both participation constraints are satisfied slack in the first best. As we shall see in our analysis of private information, the optimal mechanism may be characterized by the typical distortion of effort for the ‘‘high-cost type,’’ in addition to distortionary income transfers. With such distortions, the expected welfare level in one or both jurisdictions will necessarily be lower than in the first best. As a result, one or both participation constraints may bind in the constrained efficient allocation. Given our assumptions, the central government’s problem simplifies to choosing q1L , q1H , q2 , p1L , p1H , p 2 , 1L , 1H 4 to maximize Ž14.. The Žinterior. solution to this problem is given by conditions Ž10., Ž11., Ž15. and the equations below 6 : u⬘ Ž c 2j . y u⬘ Ž c1j . s 0 « c 2j s c1j « u1j s u 2j ,
j s H , L.
Ž 16 .
Equations Ž16. determine the first best transfers. These transfers are chosen to equalize marginal social utilities of income between the two jurisdictions in each state of nature. Therefore, the first best, being the efficient allocation where all individuals in the federation receive the same amount of utility, leads to no conflict between efficiency and equity. For implementation purposes, it is worth noting that risk sharing is the sole difference between the decentralized and first best allocations. There are at least two alternative ways of implementing the first best. Let 6
See Appendix A.
PRIVATE INFORMATION IN FEDERAL SYSTEMS
49
q1L1 , q1H 1, q21 , p1L1 , p1H 1, p12 , 1L1, 1H 1 4 denote the first best quantities. First, suppose that the central government controls all relevant variables and offers take-it-or-leave-it contracts Ž q1L1, p1L1, 1L1 ., Ž q1H 1, p1H 1 , 1H 1 .4 , and q21 , p12 , 1L1, 2H 14 to governments 1 and 2, respectively. Since these contracts are individually rational, both jurisdictional governments will accept them. Second, and more interestingly, suppose that the central government controls the interregional transfer scheme only and offers a take-it-orleave-it contract 1L1 , 1H 14 to both jurisdictional governments. Now, by correctly anticipating that the decentralized equilibrium will implement the other first best quantities, both jurisdictional governments will accept this insurance contract. The level of expected welfare in each jurisdiction will then correspond to the first best one, which is strictly greater than each jurisdiction’s level of expected welfare in the decentralized equilibrium. C. Risk Sharing with Pri¨ ate Information: Constrained Efficiency Let us now consider the setting with private information. Henceforth, jurisdiction 1 is the sole observer of the realization of the random variable 1. Being privately informed about its typeᎏi.e., ‘‘lost cost’’ if 1 s 1L or ‘‘high cost’’ if 1 s 1H ᎏjurisdiction 1 will be induced to reveal its type truthfully. The central government needs this piece of information to make the appropriate interregional transfer. The game played between central and jurisdictional governments can be described as follows. There are two stages. Stage 1 takes place before the realization of the random variable I . In this stage, the central government offers two contracts: contract Ž q1L , p1L , 1L ., Ž q1H , p1H , 1H .. to government 1 and contract q2 , p 2 , 1L , 1H 4 to government 2. The jurisdictional governments must decide, prior to the realization of the random variable, whether or not to accept such contracts. If both accept, a federal insurance scheme will then be created. If at least one of the jurisdictional governments refuses its contract, the federal insurance scheme will not be createdᎏi.e., no risk sharing will take place. Stage 2 occurs after the realization of the random variable 1. The subgame played in this stage depends on whether or not the federal insurance scheme is created. If not, the jurisdictional governments play a Nash game, and we obtain the decentralized equilibrium described above. If the federal insurance scheme is created, we have a ‘‘revelation game’’ whereby government 1 reveals its type by choosing either the contract tailored to the low-cost type or the contract tailored to the high-cost type. Let Ž q1L , p1L , 1L . and Ž q1H , p1H , 1H . be the contracts tailored to the low-cost and high-cost types, respectively. Incentive compatibility amounts
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CORNES AND SILVA
to u Ž c Ž q1L , p1L , 1L ; 1L . . G u Ž c Ž q1H , p1H , 1H ; 1L . . ª c Ž q1L , p1L , 1L ; 1L . G c Ž q1H , p1H , 1H ; 1L . , Ž 17a . u Ž c Ž q1H , p1H , 1H ; 1H . . G u Ž c Ž q1L , p1L , 1L ; 1H . . ª c Ž q1H , p1H , 1H ; 1H . G c Ž q1L , p1L , 1L ; 1H . .
Ž 17b . Constraint Ž17a. states that the low-cost type receives at least as much utility from accepting the low-cost contract as it does by accepting the high-cost contract. Similarly, constraint Ž17b. states that the high-cost type can do no better than accepting the contract designed for it. From our previous analysis, it should be clear that the high-cost type has no incentive to masquerade as low cost. Such a lie will bring no benefit, but only the cost of having to behave consistently with the lieᎏbesides having to provide the bundle Ž q1L , p1L ., which will be different from its most desirable bundle, the high-cost type has to give up an amount of income transfer equal to 1L y 2H , since it must pay 1L and not receive y 1H. Hence, we can safely conclude that constraint Ž17b. will always be satisfied by the menu of optimal contracts. Note that the incentive compatibility constraint Ž17a. can be rewritten as follows:
c1L y c1H G
⌿ Ž 1H y p1H . y ⌿ Ž 1L y p1H . n
.
Ž 18 .
To obtain Ž18., add ⌿ Ž 1H y p1H .rn to both sides of Ž17a. and rearrange the terms. The left side of inequality Ž18. reveals the Žper capita. opportunity cost faced by the low-cost type of pretending to be high cost. The right side of Ž18. gives us the Žper capita. opportunity benefit of such pretending, namely, the savings in effortᎏin monetary unitsᎏinvolved with providing the high-cost bundle. The right side of Ž18., therefore, defines the Žper capita. informational rent received by the low-cost type when it pretends to be high cost:
R Ž p1H ; 1L , 1H , n . '
⌿ Ž 1H y p1H . y ⌿ Ž 1L y p1H . n
) 0.
Ž 19 .
PRIVATE INFORMATION IN FEDERAL SYSTEMS
51
The inequality in Ž19. follows from ⌿⬘ ) 0 and 1H ) 1L . Given Ž19., we can rewrite Ž18. as follows: c1L y c1H G R Ž p1H ; 1L , 1H , n . .
Ž 20 .
Inequality Ž20. tells us that incentive compatibility can be maintained if and only if the difference between low- and high-cost Žper capita. consumption levels is no less than the Žper capita. informational rent. In addition, since this informational rent is always positive, the low-cost Žper capita. consumption level must always exceed the high-cost Žper capita. consumption levelᎏi.e., c1L ) c1H. Since ⌿⬙ - 0, we also know that R⬘ Ž
p1H ;
1L ,
1H ,
n. ' y
⌿⬘ Ž 1H y p1H . y ⌿⬘ Ž 1L y p1H . n
- 0. Ž 21 .
Let us now consider the central government’s problem. The central government chooses q1L , q1H , q2 , p1L , p1H , p 2 , 1L , 1H 4 to maximize Ž14. subject to Ž15. and Ž20.. The incentive compatibility constraint Ž20. is the sole difference between this problem and the problem that the center faces when it is perfectly informed about government 1’s type. In other words, if we ignored constraint Ž20. in the maximization problem above, we would obtain the maximization problem that yields the first best. Hence, if constraint Ž20. does not bind in the solution to the problem above, this solution will be identical to the first best. In such a case, the first best is incentive compatible. It is only when the first best is not incentive compatible that the center should worry about constraint Ž20.. The first best is not incentive compatible if c1L1 y c1H 1 - R Ž p1H 1 ; 1L , 1H , n . .
Ž 22 .
Let 1 , 2 , and denote the nonnegative Lagrangian multipliers associated with constraints Ž15a., Ž15b., and Ž20., respectively. Assume that inequality Ž22. holds. This implies that ) 0. When the incentive compatibility constraint Ž20. binds, the constrained efficient allocation is characterized by conditions Ž10. and Ž15. and7 c1L y c1H s R Ž p1H ; 1L , 1H , n . , ⌿⬘ Ž 7
See Appendix B.
1L
y
p1L
.s
q1L ,
Ž 23 . Ž 24a .
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CORNES AND SILVA
⌿⬘ Ž 1H y p1H . s q1H q
Ž 1 q 1 . u⬘ Ž c1H . y Ž 1 q 2 . u⬘ Ž c2H . nR⬘ Ž p1H ; 1L , 1H , n . , Ž 1 q 2 . u⬘ Ž c2H . Ž 24b . ⌿⬘ Ž 2 y p 2 . s q2 ,
Ž 24c .
Ž 1 q 2 . u⬘ Ž c2L . y Ž 1 q 1 . u⬘ Ž c1L . s Ž 1 q 1 . u⬘ Ž c1H . y Ž 1 q 2 . u⬘ Ž c2H . s
,
1y
Ž 25a . ,
Ž 25b .
1 u Ž c Ž q1L , p1L , 1L ; 1L . . q Ž 1 y . u Ž c Ž q1H , p1H , 1H ; 1H . . y u1L0 y Ž 1 y . u1H 0 s 0, Ž 26a .
2 u Ž c Ž q2 , p 2 , y 1L ; 2 . . q Ž 1y . u Ž c Ž q2 , p 2 , y 1H ; 2 . . yu 20 s 0.
Ž 26b . Equation Ž23. represents the binding incentive compatibility constraint. Equations Ž24a. and Ž24c. are the first best first-order conditions with respect to effort levels for the low-cost government 1 and government 2, respectively. Equation Ž24b. is the first-order condition with respect to the price of the local public good faced by the high-cost government 1. It states that effort for the high-cost type must be distorted to reduce the informational rent enjoyed by the low-cost type. Equations Ž25a. and Ž25b. are the first-order condition with respect to the transfers. The optimal transfers now depend upon incentive compatibility and participation incentives. Unlike in the first best, the optimal transfers will not equalize jurisdictional Žper capita. consumption levels in each state of nature because ) 0: incenti¨ e compatibility limits the degree of risk sharing in each state of nature. Finally, Eqs. Ž26a. and Ž26b. are complementary slackness conditions. The following thought experiment provides a straightforward intuition for the constrained efficient allocation when ) 0. Consider the Žinfeasible. first best. Departing from this allocation, an easy way of satisfying Ž23. is to adjust the transfers, while maintaining the other quantities at their first best levels. The adjustment should be made to enlarge the left side of Ž22. until both sides of this inequality are equated. This can be done by either reducing the size of the transfer made by government 1 in the good state of nature or by reducing the size of the transfer received by government 1 in the bad state of nature. But, since the utility function is
PRIVATE INFORMATION IN FEDERAL SYSTEMS
53
strictly concave, a mixture of both of these measures strictly dominates each single measure. The actual amount of reduction in each transfer depends on the likelihood of each state of nature, as conditions Ž25. reveal. An alternative way of satisfying Ž23. is to distort the price of the local public good in jurisdiction 1 in the bad state of nature, while keeping all the other quantities at their first best levels. Since R⬘ - 0, an increase in this price from its first best level reduces the Žper capita. informational rent received by the low-cost type. But, this measure also enlarges the left side of Ž23. because it reduces the level of Žper capita. consumption in jurisdiction 1 in the bad state of nature. Close inspection of conditions Ž24b. and Ž25. reveals that, rather than distorting one policy instrument only, it is socially efficient to combine the two types of distortions, namely, to distort both the transfers and the price of the local public good in jurisdiction 1 in the bad state of nature. The combination of both distortions prevents a dramatic reduction in the level of risk sharing as well as an excessive ex post ‘‘punishment’’ to jurisdiction 1 when it is confronted with the bad state of nature. It is important to note that the constrained efficient allocation may be characterized by slack participation constraints even when ) 0. Although it is certainly true that the level of expected welfare in the federation is lower in the presence of the distortions than without them, which implies that the level of expected welfare in at least one of the jurisdictions must be strictly lower than in the first best, it does not necessarily follow that the level of expected welfare in either jurisdiction will be lower than in decentralized equilibrium. Given this remark, we shall define the ‘‘second best’’ as the constrained efficient allocation in which ) 0 and 1 s 2 s 0. The second best, however, may be infeasible because it leads to a violation of one, but only one, of the participation constraints Ž15.. In such a case, we have an allocation that we call ‘‘third best.’’ That is, the constrained efficient allocation in which ) 0 and either 1 ) 0 or 2 ) 0, but not both, is denoted third best. Finally, the third best may also be infeasible because both participation constraints bind. Hence, the constrained efficient allocation in which ) 0 and 1 ) 0 and 2 ) 0 is denoted ‘‘fourth best.’’ Since the fourth best yields the same levels of expected welfare as the decentralized equilibrium, there is no reason to design a mechanism in which the jurisdictions share risk. It is easy to verify that some of the constrained efficient quantities correspond to their first best counterparts. For jurisdiction 2, both the price and quantity levels for the local public good in the constrained efficient allocation are identical to their first best counterparts. For jurisdiction 1, the same correspondence is true, but only in the good state of nature. Also, given our discussion above, it should be clear that relative
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to the first best, the second best leads to lower transfers and a higher price for the local public good provided by government 1 in the bad state of nature. A higher price for the local public good in jurisdiction 1, in turn, implies that government 1 will provide a lower quantity of the local public good in the bad state of nature, relative to the first best. Proposition 1 summarizes these results. PROPOSITION 1. Let q1L 2 , q1H 2 , q 22 , p 1L 2 , p 1H 2 , p 22 , 1L 2 , 1H 2 4 and q1L3 , q1H 3, q23 , p1L3 , p1H 3, p 23 , 1L3 , 1H 34 denote the second and third best quantities, respecti¨ ely. We obtain q1L3 s q1L2 s q1L1 ;
q23 s q22 s q21 ;
p1L3 s p1L2 s p1L1 ;
p 23 s p 22 s p12 ;
q1H 1 ) q1H 2 ;
1L1 ) 1L2 ) 0;
Ž 27 .
p1H 2 ) p1H 1 ;
Ž 28 .
0 ) 1H 2 ) 1H 1 .
Ž 29 .
Proof. To prove Ž27., one simply needs to notice that conditions Ž10., Ž24a., and Ž24c. are satisfied in the first, second, and third best. From our discussion above, we know that p1H 2 ) p1H 1. Since f ⬙ - 0 and nf ⬘Ž q1H . s p1H , we have q1H 1 ) q1H 2 . The results in Ž29. follow from our previous discussion. How do the third best quantities compare with their second best counterparts? In light of the many potential ambiguities that may plague such a comparison, we sill state our reasoning concerning this exercise as a conjecture rather than as a formal result. Let us start our discussion by assuming that the constrained efficient allocation involves ) 0, 1 ) 0, and 2 s 0. We obtain this third best allocation because the following inequalities hold in the second best:
u1L q Ž 1 y . u1H - u1L0 q Ž 1 y . u1H 0 ,
Ž 30a .
u 2L q Ž 1 y . u 2H ) u 02 .
Ž 30b .
As we shall attempt to demonstrate, Ž30b. holds in the second best because it also holds in the third best. The level of expected welfare in jurisdiction 2 should be lower in this third best allocation than in the second best. Departing from the Žinfeasible . second best, the participation constraint Ž15a. can be satisfied if the central government adjusts the second best transfers to increase c1L or c1H or both. Since utility is strictly concave, it is optimal to adjust both transfers rather than simply one of them, as Eqs. Ž25. demonstrate. Relative to the second best levels for the transfers, the adjustment should reduce the transfer made by government 1 in the good
PRIVATE INFORMATION IN FEDERAL SYSTEMS
55
state of nature and increase the transfer received by this government in the bad state of nature. These measures clearly increase the levels of both c1L and c1H and decrease the levels of both c 2L and c 2H relative to their second best ones. This implies that the level of expected utility in jurisdiction 2 will be lower in the third best than in the second best. Hence, if Ž30b. holds in the third best, it must also hold in the second best. As Eq. Ž24b. reveals, the central government should also adjust the price level for the local public good provided by government 1 in the bad state of nature. This price should be reduced from its second best level. This measure should have two main effects: Ž1. it allows the low-cost type to enjoy a higher informational rent, and Ž2. it allows the high-cost type to provide a higher level of Žper capita. consumption to its representative residentᎏi.e., c1H increases. By Ž23., an increase in R must be accompanied by an increase in c1L y c1H. It follows that the level of risk sharing should be lower in this third best allocation than in the second best. Now, since c1H increases, c1L should also increaseᎏin fact, the latter should increase more than the former! Both effects, therefore, should lead to an increase in the expected utility enjoyed by government 1 relative to its second best level. Suppose now that ) 0, 1 s 0, and 2 ) 0. We obtain this third best allocation because the following inequalities hold in the second best:
u1L q Ž 1 y . u1H ) u1L0 q Ž 1 y . u1H 0 ,
u12
q Ž1 y .
u 2H
-
u 02 .
Ž 31a . Ž 31b .
Inequality Ž31a. holds in the second best because the level of expected utility enjoyed by government 1 in the second best is larger than in the third best. Departing from the Žinfeasible . second best, the central government obtains this third best by adjusting the transfers and the distorted public good’s price. The adjustment for the transfers should now go in the direction opposite that in the previous case. Relative to the second best, the transfer paid by government 1 in the good state of nature should be increased, and the transfer received by this government in the bad state of nature should be lowered. Relative to the second best, in each state of nature, we should observe an increase in the Žper capita. consumption level in jurisdiction 2 and a decrease in the Žper capita. consumption level in jurisdiction 1. This implies that Ž31a. holds in the second best, as we claimed above. The adjustment for the distorted price should also go in the direction opposite that of the adjustment before. All else held constant, an increase in the distorted price from its second best level should lead to Ž1. a reduction in R and Ž2. a reduction in c1H . By Ž23., a reduction in R must
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be accompanied by a reduction in c1L y c1H. As a result, the level of risk sharing should be higher in this third best allocation than in the second best. Now, since c1H decreases, c1l should also decreaseᎏthe latter should decrease more than the former. Both effects, therefore, should reinforce the transfer policies in that the expected utility enjoyed by government 1 should be lowered relative to its second best level. In sum, we conjecture that the comparison between second and third best yields
1L2 ) 1L3 ) 0; 0 ) 1H 2 ) 1H 3 ;
p1H 2 ) p1H 3 ; if 1 ) 0,
1L3 ) 1L2 ) 0; 0 ) 1H 3 ) 1H 2 ;
p1H 3 ) p1H 2 ; if 1 s 0,
q1H 3 ) q1H 2
2 s 0,
Ž 32 .
q1H 2 ) q1H 3
2 ) 0.
Ž 33 .
As before, the results regarding the comparison between publish good quantities follow immediately from the facts that f ⬙ - 0 and that the constrained efficient allocation satisfies the Samuelson conditions Ž10.. From our analysis, one can safely conclude that the level of risk sharing will be higher in the first best than in any constrained efficient allocation characterized by a binding incentive compatibility constraint. It is also safe to say that risk sharing will be lowest in the fourth best, since the fourth best is equivalent to the decentralized equilibrium with no risk sharing. Furthermore, if Ž32. and Ž33. hold, the level of risk sharing will be higher in the second best than in the third best allocation characterized by a binding participation constraint for government 1. But the level of risk sharing should be lower in the second best than in the third best allocation characterized by a binding participation constraint for government 2. The risk sharing results implied by our comparison of second and third best are intuitive. As Eq. Ž23. demonstrates, risk sharing is negatively related to the Žper capita. informational rent that accrues to the low-cost government 1. The Žper capita. informational rent, in turn, decreases with the price of the local public good in the bad state of nature. If it is the participation constraint for government 1 that is violated in the second best, the center should reduce the power of the incentive contract from its second best level, by lowering the price of the local public good price in the bad state of natureᎏthis price in the third best should be between its levels in the first and second bestᎏand, hence, allow the low-cost type to enjoy a higher Žper capita. informational rent. Risk sharing decreases, but the measure leads to an increase in welfare in jurisdiction 1 in each state
PRIVATE INFORMATION IN FEDERAL SYSTEMS
57
of nature. If, on the other hand, it is the participation constraint for government 2 that is violated in the second best, the center should increase the power of the incentive contract from its second best level, by increasing the price of the local public good in the bad state of nature. Relative to the second best, the Žper capita. informational rent should be lower, risk sharing should be higher, and the welfare in jurisdiction 2 should be higher in each state of nature. In sum, when a participation constraint is violated in the second best, the center should react by redistributing the social surplus from the jurisdiction that enjoys Žex ante. good fortune toward the jurisdiction that Žex ante. finds it undesirable to participate in the interregional insurance scheme. We can present our results regarding the ranking of risk sharing across the four types of allocations as follows. PROPOSITION 2. Pro¨ ided Ž32. and Ž33. hold, the le¨ el of risk sharing will be Ž1. higher in the first best than in the second best; Ž2. higher in the second best than in the third best if Ž15a. binds in the third best; Ž3. lower in the second best than in the third best if Ž15b . binds in the third best; Ž4. lowest in the fourth best. 3. CONCLUSION We have examined the implications of informational asymmetries, where the central agency cannot observe the realization of jurisdiction 1’s exogenous component of the unit cost of producing a local public good, for the allocation of interregional insurance and local public goods. We have good news and bad news. The good news is that the presence of informational asymmetriesᎏof the kind we assumed in this paperᎏis not in itself sufficient to deny federations the attainment of either efficiency or egalitarian goals. We demonstrated that the unconstrained centralized allocation can satisfy incentive compatibility for the jurisdiction endowed with private information. The bad news is that the unconstrained centralized allocation may violate not only the incentive compatibility for the jurisdiction endowed with private information but also participation constraints for both jurisdictions. Our federation, for example, may have to content itself with a fourth best allocation as a result of strategic manipulation of information by the privately informed jurisdiction. It is noteworthy that interregional insurance will be underprovided whenever the first best is not incentive compatible and that the comparison between levels of risk sharing in the second and third best appears to depend on which of the two participation constraints binds in the third best. It is also important to note that, although the Samuelson conditions for optimal provision of the local
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public goods are not affected by informational asymmetries,8 both second and third best allocations will involve, in the bad state of nature, levels of the local public good which differ from the first best one in the jurisdiction endowed with private information. APPENDIX A First note that, since Ž14. is strictly concave, the first-order conditions are necessary and sufficient for a unique and global maximum. Assuming the solution is interior, the first-order conditions yield the following equations:
u⬘ Ž c1L . nf ⬘ Ž q1L . y p1L s 0 « nf ⬘ Ž q1L . s p1L ,
Ž A1.
Ž 1 y . u⬘ Ž c1H . nf ⬘ Ž q1H . y p1H s 0 « nf ⬘ Ž q1H . s p1H ,
Ž A2.
u⬘ Ž c 2L . q Ž 1 y . u⬘ Ž c 2H .
nf ⬘ Ž q2 . y p 2 s 0 « nf ⬘ Ž q2 . s p 2 ,
Ž A3. u⬘ Ž c1L . ⌿⬘ Ž 1L y p1L . y q1L s 0 « ⌿⬘ Ž 1L y p1L . s q1L , Ž A4.
Ž 1 y . u⬘ Ž c1H . ⌿⬘ Ž 1H y p1H . y q1H s 0 « ⌿⬘ Ž 1H y p1H . s q1H , Ž A5. u⬘ Ž c 2L . q Ž 1 y . u⬘ Ž c 2H .
⌿⬘ Ž 2 y p 2 . y q2 s 0
« ⌿⬘ Ž 2 y p 2 . s q2 , u⬘ Ž c1j .
⭸ c1j
ž / ⭸ 1j
q u⬘ Ž c 2j .
⭸ c 2j
ž / ⭸ 1j
« u⬘ Ž c 2j . s u⬘ Ž c1j . ,
Ž A6. s
u⬘ Ž c2j . y u⬘ Ž c1j .
j s H , L.
n
s0
Ž A7.
Equations ŽA1. ᎐ ŽA3. correspond to Eqs. Ž10.. Equations ŽA4. ᎐ ŽA6. correspond to Eqs. Ž11.. Since u⬙ / 0, the last two equations above imply that c 2j s c1j . Hence, u 2j ' uŽ c2j . s uŽ c1j . ' u1j . This proves Ž16..
8
Our result that, in the bad state of nature, the Samuelson condition for provision of the local public good in jurisdiction 1 remains unchanged under each possible allocation is sensitive to our assumptions that preferences are quasi-linear and separable. If these assumptions are relaxed, our result will not hold. See Lockwood Ž1999..
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APPENDIX B Assuming an interior solution, the first-order conditions yield Eqs. ŽA3. and ŽA6. and
Ž 1 q 1 . u⬘ Ž c1L . q
nf ⬘ Ž q1L . y p1L s 0 « nf ⬘ Ž q1L . s p1L , Ž B1 .
Ž 1 y . Ž 1 q 1 . u⬘ Ž c1H . y nf ⬘ Ž q1H . y p1H s 0,
Ž B2 .
Ž 1 q 1 . u⬘ Ž c1L . q ⌿⬘ Ž 1L y p1L . y q1L s 0 « ⌿⬘ Ž 1L y p1L . s q1L ,
Ž B3 .
Ž 1 y . Ž 1 q 1 . u⬘ Ž c1H . y ⌿⬘ Ž 1H y p1H . y q1H s n R⬘ Ž p1H ; 1L , 1H , n . ,
Ž B4 .
Ž 1 q 2 . u⬘ Ž c 2L . y Ž 1 q 1 . u⬘ Ž c1L . s ,
Ž B5 .
Ž 1 y . Ž 1 q 1 . u⬘ Ž c1H . y Ž 1 q 2 . u⬘ Ž c2H . s ,
Ž B6 .
c1L y c1H y R Ž p1H ; 1L , 1H , n . s 0,
Ž B7 .
1 u Ž c Ž q1L , p1L , 1L ; 1L . . q Ž 1 y . u Ž c Ž q1H , p1H , 1H ; 1H . . y u1L0 y Ž 1 y . u1H 0 s 0, Ž B8 .
2 u Ž c Ž q2 , p 2 , y 1L ; 2 . . q Ž 1y . u Ž c Ž q2 , p 2 , y 1H ; 2 . . yu 20 s0.
Ž B9 . Equations ŽB5. and ŽB6. yield Eqs. Ž25a. and Ž25b., respectively. Equations ŽB8. and ŽB9. correspond to Eqs. Ž26a. and Ž26b., respectively. Since ) 0, Eq. ŽB7. implies Eq. Ž23.. Using condition ŽB6., we can rewrite Eqs. ŽB2. and ŽB4., respectively, as follows:
Ž 1 y . Ž 1 q 2 . u⬘ Ž c2H . nf ⬘ Ž q1H . y p1H s 0 « nf ⬘ Ž q1H . s p1H , Ž B10 . Ž 1 q 2 . u⬘ Ž c2H . ⌿⬘ Ž 1H y p1H . y q1H s Ž 1 q 1 . u⬘ Ž c1H . y Ž 1 q 2 . u⬘ Ž c 2H . nR⬘ Ž p1H ; 1L , 1H , n . .
Ž B11 . Equation ŽB11. implies Ž24b. immediately. Now, note that Ž1. Eqs. ŽA3., ŽB1., and ŽB10. correspond to Eqs. Ž10.; and Ž2. Eqs. ŽB3. and ŽA6. correspond to Eqs. Ž24a. and Ž24c., respectively.
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REFERENCES 1. R. Boadway, I. Horiba, and R. Jha, The design of conditional grants as a principal-agent problem, unpublished manuscript Ž1995.. 2. M. Bordignon, P. Manasse, and G. Tabellini, Optimal regional redistribution under asymmetric information, unpublished manuscript Ž1995.. 3. S. Bucovetsky, Federalism, equalization and risk aversion, Journal of Public Economics, 67, 301᎐328 Ž1998.. 4. R. C. Cornes and E. C. D. Silva, Local public goods, inter-regional transfers and private information, unpublished manuscript Ž1996.. 5. R. C. Cornes and E. C. D. Silva, Public good mix in a federation with incomplete information, unpublished manuscript Ž1998.. 6. D. Costello, Fiscal federalism in theory and practice, in ‘‘The Economics of Community Public Finance, European Economy: Reports and Studies,’’ No. 5. Directorate-General for Economic and Financial Affairs, Brussels Ž1993.. 7. H. Cremer and P. Pestieau, Income redistribution in an economic union: The trade-off between inter- and intra-national redistribution, International Tax and Public Finance, 4, 325᎐335 Ž1997.. 8. Commission on Economic and Social Cohesion, First Report from the Commission on Economic and Social Cohesion, COM Ž96. 542r2 ŽEUROP., Brussels Ž1997.. 9. C. M. Hoxby, Are efficiency and equity in school finance substitutes or complements? Journal of Economic Perspecti¨ es, 10, No. 4, 51᎐72 Ž1996.. 10. B. Lockwood, Inter-regional insurance, Journal of Public Economics, 72, 1᎐37 Ž1999.. 11. K.-G. Maler, Environmental issues in the new Europe’’ in ‘‘Economics for the New ¨ Europe’’ ŽA. B. Atkinson and R. Brunetta, Eds.., New York Univ. Press, New York Ž1991.. 12. D. H. Monk, ‘‘Educational Finance: An Economic Approach,’’ McGraw-Hill, New York Ž1990.. 13. W. A. Niskanen, Jr., ‘‘Bureaucracy and Representative Government,’’ Aldine, Chicago Ž1971.. 14. T. Persson and G. Tabellini, Fiscal federal constitutions. Part I: Risk sharing and moral hazard, Econometrica, 64, 623᎐646 Ž1996.. 15. H. Raff and J. D. Wilson, Income redistribution with well-informed local governments, International Tax and Public Finance 4, 407᎐427 Ž1997.. 16. D. E. Wildasin, ‘‘Urban Public Finance,’’ Harwood, Chur, Switzerland Ž1986..
Local Public Goods, Risk Sharing, and Private Information in Federal Systems1 Richard C. Cornes Department of Economics, Keele Uni¨ ersity, Keele, Staffordshire ST5 5BG, UK
and Emilson C. D. Silva Department of Economics, Tulane Uni¨ ersity, New Orleans, Louisiana 70118-5698 Received January 29, 1997; revised January 25, 1999 We examine the implications of information asymmetries for the optimal design of interregional insurance schemes and the allocation of local public goods. We have good and bad news. The good news is that the presence of informational asymmetries is not in itself sufficient to deny federations the attainment of either efficiency or egalitarian goals. The bad news is that not only the first best may not be incentive compatible but also the incentive compatible optimum may violate participation constraints. Federations may have to content themselves with ‘‘fourth best’’ allocations as a result of strategic manipulation of information by privately informed jurisdictions. 䊚 2000 Academic Press Key Words: information asymmetry; interregional insurance; federal systems; participation constraints.
1. INTRODUCTION The role of explicit real resource transfers between the member states of federations has been widely discussedᎏsee, for example, Wildasin w16x. In some established federations, they represent a significant fraction of tax revenue, and the existing institutions and principles that determine and affect them continue to be keeny debated. Within the European Union ŽEU. also, as it moves toward a more closely knit federal structure, the design of schemes for transferring real income between regions has become a key issue in at least two respects. First, the move toward monetary 1 We thank Jan Brueckner, two anonymous referees, and the participants of the 52nd Congress of the IIPF in Tel-Aviv for numerous helpful comments.
39 0094-1190r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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union implies that a major implicit insurance instrumentᎏexchange rate adjustmentᎏwill cease to be available. Second, the EU’s structural policies cannot be implemented without interregional transfers. Major financial instruments used to advance structural policies within the EUᎏe.g., the Regional Development Fund, the Social Fund, the Cohesion Fund, and the loans from the European Investment Bankᎏobey the principle of ‘‘additionality,’’ whereby the EU offers additional resources to member nations to help them finance their own structural programs. These programs are mostly intended to develop regions that are lagging behind. The resources are used to build infrastructure, to improve human resources, and to stimulate productive investment. ‘‘The resources for EU structural policies have increased substantially from 3.7 billion ECU in 1985 to 18.3 billion ECU in 1992 and to 33 billion ECU in 1999. For the period of 1994᎐99, around 170 billion ECU is available from the Community’s budget for structural policies. This represents about a third of total Community spending and 0.45% of Community GDP. Over the decade 1988᎐99, spending amounts cumulatively to 6.5% of annual Community GDP. A comparison makes its importance clear: Marshall aid to postwar Europe was equivalent to 1% of US GDP per year and amounted cumulatively Ž1948᎐51. to 4% of US GDP’’ ŽFirst Report from the Commission on Economic and Social Cohesion w8, p. 9x.. The term ‘‘fiscal equalization’’ suggests a concern with equity as a motivating force behind such transfers. In the EU, for example, economic and social cohesion is the main objective of structural policies. Article 130a of the Treaty on European Union states this objective in terms of ‘‘harmonious development’’ as follows: ‘‘reducing disparities between the levels of development of the various regions and the backwardness of the least favored regions, including rural areas.’’ In the United States, the current public education financing equalization trend can be explained by a concern of the states with equal educational opportunities. Several state aid formulas now exist to implement transfers among school districts with the objective of reducing education spending disparities Žsee, e.g., Hoxby w9x and Monk w12x.. But fiscal equalization cannot be the whole story, since it is hard to imagine a region or country agreeing to the prospect of becoming a permanent source of transfers to others. This suggests an efficiency role, and one natural role that may be filled by transfers is that of providing insurance, namely, allowing region that encounter a ‘‘bad shock’’ to augment their income through transfers from those enjoying better fortune. Suppose that, within each jurisdiction, the value of some exogenous parameter that influences the attainable per capital utility level in that jurisdiction is known only as a probability distribution. Each jurisdiction may face a different probability distribution. On the basis of these distribu-
PRIVATE INFORMATION IN FEDERAL SYSTEMS
41
tions, the jurisdictions wish to create an income redistribution program so that, in the event that the realized values of this parameter differ across jurisdictions, transfers from jurisdictions experiencing ‘‘good shocks’’ to those experiencing ‘‘bad shocks’’ reduce interregional differences in per capita utilities. Such a redistribution scheme is desirable if and only if it is individually rational for each jurisdiction to participate in an ex ante sense. If this condition is satisfied, each jurisdiction will agree to delegate to a central agency the power to effect transfers and to commit itself to remaining in the scheme after realizing its own parameter value. In the EU, the 1988 reform of the Structural Fundsᎏi.e., the creation of Objective regions, ranking from 1 to 6 according to their development priorityᎏmay have been motivated by a concern of the center with voluntary participation of all member nations in the financing of the structural policies. Objective 1 regions, for example, are regions where development is lagging behind. Just about every member nation has a fraction of its territory considered an Objective 1 region. These regions receive 70% of all resources spent by the structural policies. Hence, member nations that are now net exporters of resources may in the future be net importers. We may indeed experience such a reversal within a few years. Whereas Ireland, now a net importer of resources, is growing at a much faster rate than the Community’s average, Finland, a net exporter of resources, is growing at lower than average rate. It is quite possible that within the near future, Ireland will be a net exporter and Finland will be a net importer of resources. The current system for allocating the Structural Funds may provide member nations with insurance against hard times, and such an insurance motive may be strong enough to induce current net exporters of resources to participate in the financing of the structural policies. w11x and Costello w6x have observed, in addition to But, as Maler ¨ uncertainty about the state of nature, there may also be information asymmetriesᎏspecifically, information about local conditions is often decentralized.2 In this paper, we explore the implications of information decentralization for the allocation of local public goods and risk sharing within a federation. To keep things simple, our federation is assumed to be formed by a central government and two jurisdictional governments. Information decentralization is captured with the assumption that the central agency cannot observe the realized value of each jurisdiction’s 2
In his analysis of the European acid rain problem, Maler ¨ noted: ‘‘There is another kind of uncertainty which we must look into, however. This has to do with the information one country may have about costs and damage in other countries. In general, the control costs and environmental damage in one particular country is known to that country only’’ ŽMaler ¨ w11, p. 266x.. The first kind of uncertainty Maler ¨ referred to is uncertainty regarding both damage and control costs.
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exogenous parameter, even though this realization is revealed to that jurisdiction. The optimal redistribution scheme with such private information must be individually rational ex anteᎏso that each jurisdiction finds it desirable to participate voluntarilyᎏand incentive compatible ex postᎏso that the privately informed jurisdiction finds it desirable to reveal its ‘‘type’’ truthfully. We find that the constrained efficient allocation, designed by the central agency, may involve Ž1. no binding constraints Žfirst best.; Ž2. one binding incentive compatibility constraint Žsecond best.; Ž3. one binding incentive compatibility and one binding participation constraint Žthird best.; and Ž4. one binding incentive compatibility and two binding participation constraints Žfourth best.. Several issues involving informational asymmetries between local and central authorities as well as risk sharing within federations have recently received much attention from students of fiscal federalism Žsee, e.g., Boadway, Horiba, and Jha w1x, Bordignon, Manasse, and Tabellini w2x, Bucovetsky w3x, Cornes and Silva w4᎐5x, Cremer and Pestieau w7x, Lockwood w10x, Persson and Tabellini w14x, and Raff and Wilson w15x.. Unlike this literature, we study the limits imposed by both Žex ante. participation constraints and Žex post. incentive compatibility constraints on the design of interregional income redistribution schemes and the provision of local public goods. The creation of Objective regions in the EU as well as the recent experiences with federation formation in Europe and the threat of secession in Canada seem to indicate that participation constraints deserve serious consideration. Indeed, participation constraints may lower the levels of interregional transfers and hence be harmful to the provision of interregional insurance. 2. THE MODEL Consider a federation consisting of two jurisdictions, indexed by i, i s 1, 2. Jurisdiction i possesses a population of n i immobile residents. To emphasize the crucial differences between jurisdictions, we shall assume throughout that n1 s n 2 s n. The representative consumer of jurisdiction i consumes a composite basket of goods, the level of which is denoted c i . In most of what follows, the consumption level for the representative consumer in each jurisdiction is stochastic because it depends on the realizations of two i.i.d. random variables, i , which represent the exogenous components of the unit production costs of the local public goods provided by the two jurisdictions. A vector of quantities Ž c iL, L , c iL, H , c iH , L , c iH , H . for the representative consumer in jurisdiction i represents a contingent consumption plan in which he consumes a quantity c iL, L when Ž 1 , 2 . s Ž 1L , 2L ., a quantity c iL, H when Ž 1 , 2 . s Ž 1L , 2H ., and so on. Let g Ž0, 1. denote the probability that i s i L .
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PRIVATE INFORMATION IN FEDERAL SYSTEMS
The representative consumer in jurisdiction i is assumed to derive the expected utility over consumption of Ž c iL, L , c iL, H , c iH , L , c iH , H .,
u Ž c iL , L . q Ž 1 y . u Ž c iL , H . q Ž 1 y . u Ž c iH , L . q Ž 1 y . u Ž c iH , H . ,
i s 1, 2,
Ž 1.
where u⬘ ) 0, u⬙ - 0, and c ij, k ' x ij, k q f Ž qij .
when
Ž 1 , 2 . s Ž 1j , 2k . , i s 1, 2,
j, k s H , L.
Ž 2.
For simplicity, the consumption function c ij, k are assumed to be quasilinear. The terms x ij, k denote consumption levels of a private good Žnumeraire., and the terms f Ž qij . represent consumption levels of a local public good. The function f, which is assumed to be increasing and strictly concave, transforms qij units of the local public good into f Ž qij . units of consumption. The quasi-linear form of the consumption functions implies that income effects are captured solely by consumption of the private good. Hence, the level of the local public good in jurisdiction i depends exclusively upon the realization of i . To provide qij units of a local public good, the government of jurisdiction iᎏhenceforth called ‘‘government i’’ᎏincurs the sum of production costs, pij qij, an administrative costs, ⌿ Ž e ij ., pij qij q Ž e ij . ,
i s 1, 2,
j s H , L,
Ž 3.
pij ' i j y e ij ,
i s 1, 2,
j s H , L.
Ž 4.
where
pij
In jurisdiction i and state of nature j, denotes the price of the local public good and e ij denotes the level of cost-reducing effort exerted by government i. In most of what follows, we will assume that the random exogenous component of the local public good’s price in jurisdiction 1 is private information to that jurisdiction. Although the center knows that government 1’s effort level will be either low or high, it cannot tell a priori if a high price for the local public good in jurisdiction 1 is consequence of a bad shock or of shirking by government 1.3 In other words, government 3 We shall assume that prices and quantities of the local public goods are observable by the center. Our analysis, therefore, focuses on tangible local public goods, namely, local public goods whose quantities and prices are readily observable Že.g., local infrastructure .. Our model is not suitable to explain situations where quantities of local public goods are unobservable Že.g., local public education.. See Cornes and Silva w5x for an analysis of how privately informed jurisdictions mix unobservable quantities of local public goods in light of finance equalization schemes.
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1’s administrative effort is non-observable by the center. This assumption is consistent with Niskanen’s w13x theory of bureaucratic behavior. According to this theory, Congress Žor the central government. lacks the time or incentive to monitor the activities undertaken by the bureaucracy. As a result, there is a moral hazard problem: the center does not know the actual costs associated with programs overseen by the bureaucracy. Our jurisdictional governments are analogous to Niskanen’s bureaus. If government i reduces its local public good’s price by e ij units, it incurs an administrative cost of ⌿ Ž e ij .. We assume that ⌿ Ž0. s 0, ⌿ is increasing and strictly convex, and ⌿ approaches infinity as e ij approaches i j. With this last assumption, we guarantee that the prices of the local public goods will always be strictly positive. We assume that the center knows ⌿. This implies that the center knows the two possible levels of the administrative cost in each jurisdiction, but does not know which of these two levels has been realized. We shall refer to these costs as accounting costs in what follows. Each consumer in the federation is initially endowed with I units of the private good. In the two states of nature, the budget constraints for the representative consumer in jurisdiction i are x ij, k q t ij, k s I,
i s 1, 2,
j, k s H , L,
Ž 5.
where
t ij, k '
pij qij q ⌿ Ž e ij . q i j, k n
,
i s 1, 2,
j, k s H , L,
Ž 6.
represent his jurisdictional taxes.4 The terms x ij, k give his expenditures on private-good consumption. The terms i j, k denote the income transfers received or paid by government i. If, for example, government 1 makes a transfer to government 2 when Ž 1 , 2 . s Ž 1j, 2k ., we have lj, k ) 0 and 2j, k s y 1j, k - 0. The interregional transfer scheme, if implemented, is controlled by the central government. We will have more to say about this scheme below.
4 In our analysis of private information, the actual administrative costs in jurisdiction 1 will be unobservable. The center will not observe the actual jurisdictional taxes in such jurisdiction. The center, however, does observe this jurisdiction’s accounting administrative costs, which may or may not be the same as the actual costs. As a result, the center observes this jurisdiction’s taxes based on accounting rather than actual costs.
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PRIVATE INFORMATION IN FEDERAL SYSTEMS
Combining Eqs. Ž2. through Ž6., we can rewrite the consumption functions as follows: c ij, k ' c Ž qij , pij , i j, k ; i j . ' I q f Ž qij . y
pij qij q ⌿ Ž i j y pij . q i j, k n i s 1, 2,
j, k s H , L.
,
Ž 7.
To keep things simple, we shall assume henceforth that the induced consumption functions Ž7. are strictly concave in the first two arguments. This assumption implies that 5 nf ⬙ Ž qij . ⌿⬙ Ž i j y pij . q 1 - 0,
i s 1, 2,
j s H , L,
Ž 8.
A. The Decentralized Equilibrium: No Risk Sharing To motivate our analysis of risk sharing under private information, we shall first consider the allocation where the jurisdictions do not share risk. The central government does not intervene, and the jurisdictions do not transfer income between themselves on their own. The jurisdictional governmentsᎏassumed to be benevolent and utilitarian ᎏplay a Nash noncooperative game. Since i j, k s 0, government 1 chooses q1j , p1j 4jsH , L to maximize
u IqfŽ
ž
q1L
p1L q1L q ⌿ Ž 1L y p1L .
.y
/
n
qŽ 1 y . u I q f Ž
ž
q1H
.y
p1H q1H q ⌿ Ž 1H y p1H . n
/
,
Ž 9a .
taking the decisions of government 2 as given. Similarly, government 2 chooses q2j , p 2j 4jsH , L to maximize p 2L q2L q ⌿ Ž 2L y p 2L .
u I q f Ž q2L . y
ž
n
q Ž 1 y . u I q f Ž q2H . y
ž
/
p 2H q2H q ⌿ Ž 2H 2 y p 2H . n
/
,
Ž 9b .
taking the decisions of government 1 as given. Direct inspection of objective functions Ž9. reveals that the Nash equilibrium will be degenerate, since the welfare of each jurisdiction is unaffected by the other jurisdiction’s choices. 5
Conditions Ž8. guarantee the uniqueness of the decentralized equilibrium.
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CORNES AND SILVA
The decentralized equilibrium is characterized by the following equations: nf ⬘ Ž qij . s pij ,
i s 1, 2,
⬘ Ž x ij y pij . s qij ,
i s 1, 2,
j s H , L, j s H , L.
Ž 10 . Ž 11 .
Equations Ž10. are the Samuelson conditions that determine the efficient provision levels for the local public goods. They demonstrate that the local public goods should be provided at levels which equate each jurisdiction’s marginal social benefits to marginal production costs. At each state of nature, jurisdiction i’s marginal production cost depends on the optimal level of effort. Equations Ž11. show that the optimal prices for the local public good are derived from equalization of the marginal administrative costs to the marginal cost savings which result from exerting effort. Our assumption that the consumption functions Ž7. are strictly concave implies that the Nash equilibrium is unique. Let qij0 , p j0 . i 4jsH , L denote government i’s optimal solution. Straightforward comparative statics yield dpij0 d ii dqij0 d i
j
s s
nf ⬙ ⌿⬙ nf ⬙ ⌿⬙ q 1 ⌿⬙ nf ⬙ ⌿⬙ q 1
) 0,
i s 1, 2,
j s H , L,
Ž 12 .
- 0,
i s 1, 2,
j s H , L.
Ž 13 .
Given Ž8., the common denominator on the right sides of Eqs. Ž12. and Ž13. is strictly negative. To save space, we shall simplify our model even further by assuming from this point on that 2H s 2L and 1H ) 2H s 2L ) 1L ) 0. In words, the exogenous parameter in jurisdiction 2 takes the same value in both good and bad states of nature. This constant value is assumed to lie between the values taken by the exogenous parameter in jurisdiction 1. These assumptions lead to no significant loss of generality. We can still examine the efficiency gains associated with risk sharing subject to private information regarding 1. Since 2H s 2L and 1H ) 2H s 2L ) 1L ) 0, conditions Ž12. imply that p1H 0 ) p 2H 0 s p 2L0 ) p1L0 and conditions Ž13. imply that q1L0 ) q2L0 s q2H 0 ) q1H 0 . Government 2 provides the same bundle in both states of nature because its exogenous cost parameter takes the same value in both states. Equations Ž10. and Ž11., therefore, essentially yield three systems, each of two equations and two variables.
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PRIVATE INFORMATION IN FEDERAL SYSTEMS
Let government i’s ex post, indirect utility be denoted as follows: u ij0 ' u Ž c ij0 . ' u Ž c Ž qiL0 Ž i j . , piL0 Ž i j . , 0 . . ,
i s 1, 2,
j s H , L.
Since du ij0 d i j
sy
u⬘ Ž c ij0 . ⌿⬘ n
- 0,
i s 1, 2,
j s H , L,
it follows that u1L0 ) u 2L0 s u 2H 0 ) u1H 0 . Henceforth, the consumption functions, transfers, and all other stochastic variables will possess only one superscript, which denotes the state of nature in jurisdiction 1. In the decentralized equilibrium, the level of welfare in jurisdiction 1 is higher than the level of welfare in jurisdiction 2 in the good state, but the ordering is reversed in the bad state. This implies that the federation faces macroeconomic risk and that consumers will not be fully insured against risk. B. Risk Sharing with Public Information: The First Best Let us now examine the ideal setting, where the jurisdictions share risk and information about the realization of 1 is public Ži.e., common knowledge.. We assume that a benevolent and utilitarian central government wishes to design an interregional insurance scheme in which jurisdiction 1 receives from jurisdiction 2 an income transfer in the bad state of natureᎏi.e., 1H - 0ᎏwhile jurisdiction 2 receives from jurisdiction 1 an income transfer in the good state of natureᎏi.e., 1L ) 0. Remember that the system is closed in the sense that 2j s y 1j, j s H, L. Government 2 now cares about the realization of the random variable 1. The ex post welfare level in jurisdiction 2 will be higher Žlower. in the good Žbad. state of nature than in the decentralized equilibrium. The income transfers enable the jurisdictions to smooth their residents’ consumption levels over the two possible states. Such a consumption smoothing is highly desirable because consumers are risk averse. Ex ante social preferences are assumed to be represented by the expected social welfare function,
u Ž c Ž q1L , p1L , 1L ; 1L . . q u Ž c Ž q2 , p 2 y 1L ; 2 . . q Ž 1 y . u Ž c Ž q1H , p1H , 1H ; 1H . . q u Ž c Ž q2 , p 2 , y 1H ; 2 . . , Ž 14 . where, to simplify notation hereafter, 2 ' 2H s 2L , p 2 ' p 2H s p 2L , and q2 ' q2L s q2H.
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Our benchmark allocationᎏdenoted ‘‘first best’’ᎏis the allocation that maximizes the expected social welfare function Ž14. subject to the requirement that each jurisdiction receives at least as much expected utility in the first best as in the decentralized equilibrium, namely,
u Ž c Ž q1L , p1L , 1L ; 1L . . q Ž 1 y . u Ž c Ž q1H , p1H , 1H ; 1H . . G u1L0 q Ž 1 y . u1H 0 ,
Ž 15a .
u Ž c Ž q2 , p 2 , y 1L ; 2 . . q Ž 1 y . u Ž c Ž q2 , p 2 , y 1H ; 2 . . G u 20 , Ž 15b . where u 02 ' u 2L0 s u 2H 0 denotes the expected utility level for the representative consumer of jurisdiction 2 in the decentralized equilibrium. We impose the participation constraints Ž15. because we wish to provide a rationale for the ¨ oluntary formation of a federal insurance scheme in which jurisdictions share risks associated with jurisdictional shocks. To highlight the limits imposed by informational asymmetries, when information about 1 is private, we assume that both participation constraints are satisfied slack in the first best. As we shall see in our analysis of private information, the optimal mechanism may be characterized by the typical distortion of effort for the ‘‘high-cost type,’’ in addition to distortionary income transfers. With such distortions, the expected welfare level in one or both jurisdictions will necessarily be lower than in the first best. As a result, one or both participation constraints may bind in the constrained efficient allocation. Given our assumptions, the central government’s problem simplifies to choosing q1L , q1H , q2 , p1L , p1H , p 2 , 1L , 1H 4 to maximize Ž14.. The Žinterior. solution to this problem is given by conditions Ž10., Ž11., Ž15. and the equations below 6 : u⬘ Ž c 2j . y u⬘ Ž c1j . s 0 « c 2j s c1j « u1j s u 2j ,
j s H , L.
Ž 16 .
Equations Ž16. determine the first best transfers. These transfers are chosen to equalize marginal social utilities of income between the two jurisdictions in each state of nature. Therefore, the first best, being the efficient allocation where all individuals in the federation receive the same amount of utility, leads to no conflict between efficiency and equity. For implementation purposes, it is worth noting that risk sharing is the sole difference between the decentralized and first best allocations. There are at least two alternative ways of implementing the first best. Let 6
See Appendix A.
PRIVATE INFORMATION IN FEDERAL SYSTEMS
49
q1L1 , q1H 1, q21 , p1L1 , p1H 1, p12 , 1L1, 1H 1 4 denote the first best quantities. First, suppose that the central government controls all relevant variables and offers take-it-or-leave-it contracts Ž q1L1, p1L1, 1L1 ., Ž q1H 1, p1H 1 , 1H 1 .4 , and q21 , p12 , 1L1, 2H 14 to governments 1 and 2, respectively. Since these contracts are individually rational, both jurisdictional governments will accept them. Second, and more interestingly, suppose that the central government controls the interregional transfer scheme only and offers a take-it-orleave-it contract 1L1 , 1H 14 to both jurisdictional governments. Now, by correctly anticipating that the decentralized equilibrium will implement the other first best quantities, both jurisdictional governments will accept this insurance contract. The level of expected welfare in each jurisdiction will then correspond to the first best one, which is strictly greater than each jurisdiction’s level of expected welfare in the decentralized equilibrium. C. Risk Sharing with Pri¨ ate Information: Constrained Efficiency Let us now consider the setting with private information. Henceforth, jurisdiction 1 is the sole observer of the realization of the random variable 1. Being privately informed about its typeᎏi.e., ‘‘lost cost’’ if 1 s 1L or ‘‘high cost’’ if 1 s 1H ᎏjurisdiction 1 will be induced to reveal its type truthfully. The central government needs this piece of information to make the appropriate interregional transfer. The game played between central and jurisdictional governments can be described as follows. There are two stages. Stage 1 takes place before the realization of the random variable I . In this stage, the central government offers two contracts: contract Ž q1L , p1L , 1L ., Ž q1H , p1H , 1H .. to government 1 and contract q2 , p 2 , 1L , 1H 4 to government 2. The jurisdictional governments must decide, prior to the realization of the random variable, whether or not to accept such contracts. If both accept, a federal insurance scheme will then be created. If at least one of the jurisdictional governments refuses its contract, the federal insurance scheme will not be createdᎏi.e., no risk sharing will take place. Stage 2 occurs after the realization of the random variable 1. The subgame played in this stage depends on whether or not the federal insurance scheme is created. If not, the jurisdictional governments play a Nash game, and we obtain the decentralized equilibrium described above. If the federal insurance scheme is created, we have a ‘‘revelation game’’ whereby government 1 reveals its type by choosing either the contract tailored to the low-cost type or the contract tailored to the high-cost type. Let Ž q1L , p1L , 1L . and Ž q1H , p1H , 1H . be the contracts tailored to the low-cost and high-cost types, respectively. Incentive compatibility amounts
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CORNES AND SILVA
to u Ž c Ž q1L , p1L , 1L ; 1L . . G u Ž c Ž q1H , p1H , 1H ; 1L . . ª c Ž q1L , p1L , 1L ; 1L . G c Ž q1H , p1H , 1H ; 1L . , Ž 17a . u Ž c Ž q1H , p1H , 1H ; 1H . . G u Ž c Ž q1L , p1L , 1L ; 1H . . ª c Ž q1H , p1H , 1H ; 1H . G c Ž q1L , p1L , 1L ; 1H . .
Ž 17b . Constraint Ž17a. states that the low-cost type receives at least as much utility from accepting the low-cost contract as it does by accepting the high-cost contract. Similarly, constraint Ž17b. states that the high-cost type can do no better than accepting the contract designed for it. From our previous analysis, it should be clear that the high-cost type has no incentive to masquerade as low cost. Such a lie will bring no benefit, but only the cost of having to behave consistently with the lieᎏbesides having to provide the bundle Ž q1L , p1L ., which will be different from its most desirable bundle, the high-cost type has to give up an amount of income transfer equal to 1L y 2H , since it must pay 1L and not receive y 1H. Hence, we can safely conclude that constraint Ž17b. will always be satisfied by the menu of optimal contracts. Note that the incentive compatibility constraint Ž17a. can be rewritten as follows:
c1L y c1H G
⌿ Ž 1H y p1H . y ⌿ Ž 1L y p1H . n
.
Ž 18 .
To obtain Ž18., add ⌿ Ž 1H y p1H .rn to both sides of Ž17a. and rearrange the terms. The left side of inequality Ž18. reveals the Žper capita. opportunity cost faced by the low-cost type of pretending to be high cost. The right side of Ž18. gives us the Žper capita. opportunity benefit of such pretending, namely, the savings in effortᎏin monetary unitsᎏinvolved with providing the high-cost bundle. The right side of Ž18., therefore, defines the Žper capita. informational rent received by the low-cost type when it pretends to be high cost:
R Ž p1H ; 1L , 1H , n . '
⌿ Ž 1H y p1H . y ⌿ Ž 1L y p1H . n
) 0.
Ž 19 .
PRIVATE INFORMATION IN FEDERAL SYSTEMS
51
The inequality in Ž19. follows from ⌿⬘ ) 0 and 1H ) 1L . Given Ž19., we can rewrite Ž18. as follows: c1L y c1H G R Ž p1H ; 1L , 1H , n . .
Ž 20 .
Inequality Ž20. tells us that incentive compatibility can be maintained if and only if the difference between low- and high-cost Žper capita. consumption levels is no less than the Žper capita. informational rent. In addition, since this informational rent is always positive, the low-cost Žper capita. consumption level must always exceed the high-cost Žper capita. consumption levelᎏi.e., c1L ) c1H. Since ⌿⬙ - 0, we also know that R⬘ Ž
p1H ;
1L ,
1H ,
n. ' y
⌿⬘ Ž 1H y p1H . y ⌿⬘ Ž 1L y p1H . n
- 0. Ž 21 .
Let us now consider the central government’s problem. The central government chooses q1L , q1H , q2 , p1L , p1H , p 2 , 1L , 1H 4 to maximize Ž14. subject to Ž15. and Ž20.. The incentive compatibility constraint Ž20. is the sole difference between this problem and the problem that the center faces when it is perfectly informed about government 1’s type. In other words, if we ignored constraint Ž20. in the maximization problem above, we would obtain the maximization problem that yields the first best. Hence, if constraint Ž20. does not bind in the solution to the problem above, this solution will be identical to the first best. In such a case, the first best is incentive compatible. It is only when the first best is not incentive compatible that the center should worry about constraint Ž20.. The first best is not incentive compatible if c1L1 y c1H 1 - R Ž p1H 1 ; 1L , 1H , n . .
Ž 22 .
Let 1 , 2 , and denote the nonnegative Lagrangian multipliers associated with constraints Ž15a., Ž15b., and Ž20., respectively. Assume that inequality Ž22. holds. This implies that ) 0. When the incentive compatibility constraint Ž20. binds, the constrained efficient allocation is characterized by conditions Ž10. and Ž15. and7 c1L y c1H s R Ž p1H ; 1L , 1H , n . , ⌿⬘ Ž 7
See Appendix B.
1L
y
p1L
.s
q1L ,
Ž 23 . Ž 24a .
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CORNES AND SILVA
⌿⬘ Ž 1H y p1H . s q1H q
Ž 1 q 1 . u⬘ Ž c1H . y Ž 1 q 2 . u⬘ Ž c2H . nR⬘ Ž p1H ; 1L , 1H , n . , Ž 1 q 2 . u⬘ Ž c2H . Ž 24b . ⌿⬘ Ž 2 y p 2 . s q2 ,
Ž 24c .
Ž 1 q 2 . u⬘ Ž c2L . y Ž 1 q 1 . u⬘ Ž c1L . s Ž 1 q 1 . u⬘ Ž c1H . y Ž 1 q 2 . u⬘ Ž c2H . s
,
1y
Ž 25a . ,
Ž 25b .
1 u Ž c Ž q1L , p1L , 1L ; 1L . . q Ž 1 y . u Ž c Ž q1H , p1H , 1H ; 1H . . y u1L0 y Ž 1 y . u1H 0 s 0, Ž 26a .
2 u Ž c Ž q2 , p 2 , y 1L ; 2 . . q Ž 1y . u Ž c Ž q2 , p 2 , y 1H ; 2 . . yu 20 s 0.
Ž 26b . Equation Ž23. represents the binding incentive compatibility constraint. Equations Ž24a. and Ž24c. are the first best first-order conditions with respect to effort levels for the low-cost government 1 and government 2, respectively. Equation Ž24b. is the first-order condition with respect to the price of the local public good faced by the high-cost government 1. It states that effort for the high-cost type must be distorted to reduce the informational rent enjoyed by the low-cost type. Equations Ž25a. and Ž25b. are the first-order condition with respect to the transfers. The optimal transfers now depend upon incentive compatibility and participation incentives. Unlike in the first best, the optimal transfers will not equalize jurisdictional Žper capita. consumption levels in each state of nature because ) 0: incenti¨ e compatibility limits the degree of risk sharing in each state of nature. Finally, Eqs. Ž26a. and Ž26b. are complementary slackness conditions. The following thought experiment provides a straightforward intuition for the constrained efficient allocation when ) 0. Consider the Žinfeasible. first best. Departing from this allocation, an easy way of satisfying Ž23. is to adjust the transfers, while maintaining the other quantities at their first best levels. The adjustment should be made to enlarge the left side of Ž22. until both sides of this inequality are equated. This can be done by either reducing the size of the transfer made by government 1 in the good state of nature or by reducing the size of the transfer received by government 1 in the bad state of nature. But, since the utility function is
PRIVATE INFORMATION IN FEDERAL SYSTEMS
53
strictly concave, a mixture of both of these measures strictly dominates each single measure. The actual amount of reduction in each transfer depends on the likelihood of each state of nature, as conditions Ž25. reveal. An alternative way of satisfying Ž23. is to distort the price of the local public good in jurisdiction 1 in the bad state of nature, while keeping all the other quantities at their first best levels. Since R⬘ - 0, an increase in this price from its first best level reduces the Žper capita. informational rent received by the low-cost type. But, this measure also enlarges the left side of Ž23. because it reduces the level of Žper capita. consumption in jurisdiction 1 in the bad state of nature. Close inspection of conditions Ž24b. and Ž25. reveals that, rather than distorting one policy instrument only, it is socially efficient to combine the two types of distortions, namely, to distort both the transfers and the price of the local public good in jurisdiction 1 in the bad state of nature. The combination of both distortions prevents a dramatic reduction in the level of risk sharing as well as an excessive ex post ‘‘punishment’’ to jurisdiction 1 when it is confronted with the bad state of nature. It is important to note that the constrained efficient allocation may be characterized by slack participation constraints even when ) 0. Although it is certainly true that the level of expected welfare in the federation is lower in the presence of the distortions than without them, which implies that the level of expected welfare in at least one of the jurisdictions must be strictly lower than in the first best, it does not necessarily follow that the level of expected welfare in either jurisdiction will be lower than in decentralized equilibrium. Given this remark, we shall define the ‘‘second best’’ as the constrained efficient allocation in which ) 0 and 1 s 2 s 0. The second best, however, may be infeasible because it leads to a violation of one, but only one, of the participation constraints Ž15.. In such a case, we have an allocation that we call ‘‘third best.’’ That is, the constrained efficient allocation in which ) 0 and either 1 ) 0 or 2 ) 0, but not both, is denoted third best. Finally, the third best may also be infeasible because both participation constraints bind. Hence, the constrained efficient allocation in which ) 0 and 1 ) 0 and 2 ) 0 is denoted ‘‘fourth best.’’ Since the fourth best yields the same levels of expected welfare as the decentralized equilibrium, there is no reason to design a mechanism in which the jurisdictions share risk. It is easy to verify that some of the constrained efficient quantities correspond to their first best counterparts. For jurisdiction 2, both the price and quantity levels for the local public good in the constrained efficient allocation are identical to their first best counterparts. For jurisdiction 1, the same correspondence is true, but only in the good state of nature. Also, given our discussion above, it should be clear that relative
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to the first best, the second best leads to lower transfers and a higher price for the local public good provided by government 1 in the bad state of nature. A higher price for the local public good in jurisdiction 1, in turn, implies that government 1 will provide a lower quantity of the local public good in the bad state of nature, relative to the first best. Proposition 1 summarizes these results. PROPOSITION 1. Let q1L 2 , q1H 2 , q 22 , p 1L 2 , p 1H 2 , p 22 , 1L 2 , 1H 2 4 and q1L3 , q1H 3, q23 , p1L3 , p1H 3, p 23 , 1L3 , 1H 34 denote the second and third best quantities, respecti¨ ely. We obtain q1L3 s q1L2 s q1L1 ;
q23 s q22 s q21 ;
p1L3 s p1L2 s p1L1 ;
p 23 s p 22 s p12 ;
q1H 1 ) q1H 2 ;
1L1 ) 1L2 ) 0;
Ž 27 .
p1H 2 ) p1H 1 ;
Ž 28 .
0 ) 1H 2 ) 1H 1 .
Ž 29 .
Proof. To prove Ž27., one simply needs to notice that conditions Ž10., Ž24a., and Ž24c. are satisfied in the first, second, and third best. From our discussion above, we know that p1H 2 ) p1H 1. Since f ⬙ - 0 and nf ⬘Ž q1H . s p1H , we have q1H 1 ) q1H 2 . The results in Ž29. follow from our previous discussion. How do the third best quantities compare with their second best counterparts? In light of the many potential ambiguities that may plague such a comparison, we sill state our reasoning concerning this exercise as a conjecture rather than as a formal result. Let us start our discussion by assuming that the constrained efficient allocation involves ) 0, 1 ) 0, and 2 s 0. We obtain this third best allocation because the following inequalities hold in the second best:
u1L q Ž 1 y . u1H - u1L0 q Ž 1 y . u1H 0 ,
Ž 30a .
u 2L q Ž 1 y . u 2H ) u 02 .
Ž 30b .
As we shall attempt to demonstrate, Ž30b. holds in the second best because it also holds in the third best. The level of expected welfare in jurisdiction 2 should be lower in this third best allocation than in the second best. Departing from the Žinfeasible . second best, the participation constraint Ž15a. can be satisfied if the central government adjusts the second best transfers to increase c1L or c1H or both. Since utility is strictly concave, it is optimal to adjust both transfers rather than simply one of them, as Eqs. Ž25. demonstrate. Relative to the second best levels for the transfers, the adjustment should reduce the transfer made by government 1 in the good
PRIVATE INFORMATION IN FEDERAL SYSTEMS
55
state of nature and increase the transfer received by this government in the bad state of nature. These measures clearly increase the levels of both c1L and c1H and decrease the levels of both c 2L and c 2H relative to their second best ones. This implies that the level of expected utility in jurisdiction 2 will be lower in the third best than in the second best. Hence, if Ž30b. holds in the third best, it must also hold in the second best. As Eq. Ž24b. reveals, the central government should also adjust the price level for the local public good provided by government 1 in the bad state of nature. This price should be reduced from its second best level. This measure should have two main effects: Ž1. it allows the low-cost type to enjoy a higher informational rent, and Ž2. it allows the high-cost type to provide a higher level of Žper capita. consumption to its representative residentᎏi.e., c1H increases. By Ž23., an increase in R must be accompanied by an increase in c1L y c1H. It follows that the level of risk sharing should be lower in this third best allocation than in the second best. Now, since c1H increases, c1L should also increaseᎏin fact, the latter should increase more than the former! Both effects, therefore, should lead to an increase in the expected utility enjoyed by government 1 relative to its second best level. Suppose now that ) 0, 1 s 0, and 2 ) 0. We obtain this third best allocation because the following inequalities hold in the second best:
u1L q Ž 1 y . u1H ) u1L0 q Ž 1 y . u1H 0 ,
u12
q Ž1 y .
u 2H
-
u 02 .
Ž 31a . Ž 31b .
Inequality Ž31a. holds in the second best because the level of expected utility enjoyed by government 1 in the second best is larger than in the third best. Departing from the Žinfeasible . second best, the central government obtains this third best by adjusting the transfers and the distorted public good’s price. The adjustment for the transfers should now go in the direction opposite that in the previous case. Relative to the second best, the transfer paid by government 1 in the good state of nature should be increased, and the transfer received by this government in the bad state of nature should be lowered. Relative to the second best, in each state of nature, we should observe an increase in the Žper capita. consumption level in jurisdiction 2 and a decrease in the Žper capita. consumption level in jurisdiction 1. This implies that Ž31a. holds in the second best, as we claimed above. The adjustment for the distorted price should also go in the direction opposite that of the adjustment before. All else held constant, an increase in the distorted price from its second best level should lead to Ž1. a reduction in R and Ž2. a reduction in c1H . By Ž23., a reduction in R must
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be accompanied by a reduction in c1L y c1H. As a result, the level of risk sharing should be higher in this third best allocation than in the second best. Now, since c1H decreases, c1l should also decreaseᎏthe latter should decrease more than the former. Both effects, therefore, should reinforce the transfer policies in that the expected utility enjoyed by government 1 should be lowered relative to its second best level. In sum, we conjecture that the comparison between second and third best yields
1L2 ) 1L3 ) 0; 0 ) 1H 2 ) 1H 3 ;
p1H 2 ) p1H 3 ; if 1 ) 0,
1L3 ) 1L2 ) 0; 0 ) 1H 3 ) 1H 2 ;
p1H 3 ) p1H 2 ; if 1 s 0,
q1H 3 ) q1H 2
2 s 0,
Ž 32 .
q1H 2 ) q1H 3
2 ) 0.
Ž 33 .
As before, the results regarding the comparison between publish good quantities follow immediately from the facts that f ⬙ - 0 and that the constrained efficient allocation satisfies the Samuelson conditions Ž10.. From our analysis, one can safely conclude that the level of risk sharing will be higher in the first best than in any constrained efficient allocation characterized by a binding incentive compatibility constraint. It is also safe to say that risk sharing will be lowest in the fourth best, since the fourth best is equivalent to the decentralized equilibrium with no risk sharing. Furthermore, if Ž32. and Ž33. hold, the level of risk sharing will be higher in the second best than in the third best allocation characterized by a binding participation constraint for government 1. But the level of risk sharing should be lower in the second best than in the third best allocation characterized by a binding participation constraint for government 2. The risk sharing results implied by our comparison of second and third best are intuitive. As Eq. Ž23. demonstrates, risk sharing is negatively related to the Žper capita. informational rent that accrues to the low-cost government 1. The Žper capita. informational rent, in turn, decreases with the price of the local public good in the bad state of nature. If it is the participation constraint for government 1 that is violated in the second best, the center should reduce the power of the incentive contract from its second best level, by lowering the price of the local public good price in the bad state of natureᎏthis price in the third best should be between its levels in the first and second bestᎏand, hence, allow the low-cost type to enjoy a higher Žper capita. informational rent. Risk sharing decreases, but the measure leads to an increase in welfare in jurisdiction 1 in each state
PRIVATE INFORMATION IN FEDERAL SYSTEMS
57
of nature. If, on the other hand, it is the participation constraint for government 2 that is violated in the second best, the center should increase the power of the incentive contract from its second best level, by increasing the price of the local public good in the bad state of nature. Relative to the second best, the Žper capita. informational rent should be lower, risk sharing should be higher, and the welfare in jurisdiction 2 should be higher in each state of nature. In sum, when a participation constraint is violated in the second best, the center should react by redistributing the social surplus from the jurisdiction that enjoys Žex ante. good fortune toward the jurisdiction that Žex ante. finds it undesirable to participate in the interregional insurance scheme. We can present our results regarding the ranking of risk sharing across the four types of allocations as follows. PROPOSITION 2. Pro¨ ided Ž32. and Ž33. hold, the le¨ el of risk sharing will be Ž1. higher in the first best than in the second best; Ž2. higher in the second best than in the third best if Ž15a. binds in the third best; Ž3. lower in the second best than in the third best if Ž15b . binds in the third best; Ž4. lowest in the fourth best. 3. CONCLUSION We have examined the implications of informational asymmetries, where the central agency cannot observe the realization of jurisdiction 1’s exogenous component of the unit cost of producing a local public good, for the allocation of interregional insurance and local public goods. We have good news and bad news. The good news is that the presence of informational asymmetriesᎏof the kind we assumed in this paperᎏis not in itself sufficient to deny federations the attainment of either efficiency or egalitarian goals. We demonstrated that the unconstrained centralized allocation can satisfy incentive compatibility for the jurisdiction endowed with private information. The bad news is that the unconstrained centralized allocation may violate not only the incentive compatibility for the jurisdiction endowed with private information but also participation constraints for both jurisdictions. Our federation, for example, may have to content itself with a fourth best allocation as a result of strategic manipulation of information by the privately informed jurisdiction. It is noteworthy that interregional insurance will be underprovided whenever the first best is not incentive compatible and that the comparison between levels of risk sharing in the second and third best appears to depend on which of the two participation constraints binds in the third best. It is also important to note that, although the Samuelson conditions for optimal provision of the local
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public goods are not affected by informational asymmetries,8 both second and third best allocations will involve, in the bad state of nature, levels of the local public good which differ from the first best one in the jurisdiction endowed with private information. APPENDIX A First note that, since Ž14. is strictly concave, the first-order conditions are necessary and sufficient for a unique and global maximum. Assuming the solution is interior, the first-order conditions yield the following equations:
u⬘ Ž c1L . nf ⬘ Ž q1L . y p1L s 0 « nf ⬘ Ž q1L . s p1L ,
Ž A1.
Ž 1 y . u⬘ Ž c1H . nf ⬘ Ž q1H . y p1H s 0 « nf ⬘ Ž q1H . s p1H ,
Ž A2.
u⬘ Ž c 2L . q Ž 1 y . u⬘ Ž c 2H .
nf ⬘ Ž q2 . y p 2 s 0 « nf ⬘ Ž q2 . s p 2 ,
Ž A3. u⬘ Ž c1L . ⌿⬘ Ž 1L y p1L . y q1L s 0 « ⌿⬘ Ž 1L y p1L . s q1L , Ž A4.
Ž 1 y . u⬘ Ž c1H . ⌿⬘ Ž 1H y p1H . y q1H s 0 « ⌿⬘ Ž 1H y p1H . s q1H , Ž A5. u⬘ Ž c 2L . q Ž 1 y . u⬘ Ž c 2H .
⌿⬘ Ž 2 y p 2 . y q2 s 0
« ⌿⬘ Ž 2 y p 2 . s q2 , u⬘ Ž c1j .
⭸ c1j
ž / ⭸ 1j
q u⬘ Ž c 2j .
⭸ c 2j
ž / ⭸ 1j
« u⬘ Ž c 2j . s u⬘ Ž c1j . ,
Ž A6. s
u⬘ Ž c2j . y u⬘ Ž c1j .
j s H , L.
n
s0
Ž A7.
Equations ŽA1. ᎐ ŽA3. correspond to Eqs. Ž10.. Equations ŽA4. ᎐ ŽA6. correspond to Eqs. Ž11.. Since u⬙ / 0, the last two equations above imply that c 2j s c1j . Hence, u 2j ' uŽ c2j . s uŽ c1j . ' u1j . This proves Ž16..
8
Our result that, in the bad state of nature, the Samuelson condition for provision of the local public good in jurisdiction 1 remains unchanged under each possible allocation is sensitive to our assumptions that preferences are quasi-linear and separable. If these assumptions are relaxed, our result will not hold. See Lockwood Ž1999..
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APPENDIX B Assuming an interior solution, the first-order conditions yield Eqs. ŽA3. and ŽA6. and
Ž 1 q 1 . u⬘ Ž c1L . q
nf ⬘ Ž q1L . y p1L s 0 « nf ⬘ Ž q1L . s p1L , Ž B1 .
Ž 1 y . Ž 1 q 1 . u⬘ Ž c1H . y nf ⬘ Ž q1H . y p1H s 0,
Ž B2 .
Ž 1 q 1 . u⬘ Ž c1L . q ⌿⬘ Ž 1L y p1L . y q1L s 0 « ⌿⬘ Ž 1L y p1L . s q1L ,
Ž B3 .
Ž 1 y . Ž 1 q 1 . u⬘ Ž c1H . y ⌿⬘ Ž 1H y p1H . y q1H s n R⬘ Ž p1H ; 1L , 1H , n . ,
Ž B4 .
Ž 1 q 2 . u⬘ Ž c 2L . y Ž 1 q 1 . u⬘ Ž c1L . s ,
Ž B5 .
Ž 1 y . Ž 1 q 1 . u⬘ Ž c1H . y Ž 1 q 2 . u⬘ Ž c2H . s ,
Ž B6 .
c1L y c1H y R Ž p1H ; 1L , 1H , n . s 0,
Ž B7 .
1 u Ž c Ž q1L , p1L , 1L ; 1L . . q Ž 1 y . u Ž c Ž q1H , p1H , 1H ; 1H . . y u1L0 y Ž 1 y . u1H 0 s 0, Ž B8 .
2 u Ž c Ž q2 , p 2 , y 1L ; 2 . . q Ž 1y . u Ž c Ž q2 , p 2 , y 1H ; 2 . . yu 20 s0.
Ž B9 . Equations ŽB5. and ŽB6. yield Eqs. Ž25a. and Ž25b., respectively. Equations ŽB8. and ŽB9. correspond to Eqs. Ž26a. and Ž26b., respectively. Since ) 0, Eq. ŽB7. implies Eq. Ž23.. Using condition ŽB6., we can rewrite Eqs. ŽB2. and ŽB4., respectively, as follows:
Ž 1 y . Ž 1 q 2 . u⬘ Ž c2H . nf ⬘ Ž q1H . y p1H s 0 « nf ⬘ Ž q1H . s p1H , Ž B10 . Ž 1 q 2 . u⬘ Ž c2H . ⌿⬘ Ž 1H y p1H . y q1H s Ž 1 q 1 . u⬘ Ž c1H . y Ž 1 q 2 . u⬘ Ž c 2H . nR⬘ Ž p1H ; 1L , 1H , n . .
Ž B11 . Equation ŽB11. implies Ž24b. immediately. Now, note that Ž1. Eqs. ŽA3., ŽB1., and ŽB10. correspond to Eqs. Ž10.; and Ž2. Eqs. ŽB3. and ŽA6. correspond to Eqs. Ž24a. and Ž24c., respectively.
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