Private information, Coasian bargaining, and the second welfare theorem

Journal of Public Economics 90 (2006) 871 – 895 www.elsevier.com/locate/econbase

Private information, Coasian bargaining, and the second welfare theorem Dagobert L. Brito a, Jonathan H. Hamilton b,e,*, Michael D. Intriligator c, Eytan Sheshinski d, Steven M. Slutsky b b

a Department of Economics, Rice University, 6100 Main, Houston, TX 77005, USA Department of Economics, PO Box 117140, University of Florida, Gainesville, FL 32611, USA c Department of Economics, UCLA, Los Angeles, CA 90095-1477, USA d Department of Economics, The Hebrew University of Jerusalem, Jerusalem, Israel e Centro de Investigacio´n y Docencia Econo´micas (CIDE) Mexico, D.F.

Received 16 February 2005; received in revised form 26 July 2005; accepted 18 August 2005 Available online 15 December 2005

Abstract Most of the debate about Coasian bargaining in the presence of externalities relates to the First Welfare Theorem: is the outcome under bargaining efficient? This debate has involved the definition and importance of transaction costs, the significance of private information, and the effect of entry. There has been little analysis of how Coasian bargaining relates to the Second Welfare Theorem: even if the bargaining outcome is efficient, does the process limit the set of Pareto optimal allocations which can be achieved? We consider a model in which individuals utilize a common resource and may affect each other’s output. The individuals differ in their productivities or tastes and this information is private to each of them. The government can manage the common resource and use nonlinear taxes to correct for the externality or it can turn the common resource over to a private owner who can charge individuals to utilize it with a nonlinear fee schedule. The government and the owner have the same information about tastes and productivities of the individuals. Except for the private information, there are no bargaining or administrative costs for collecting the taxes or fees. Whether there is public or private ownership, the government desires to redistribute, but it faces self-selection constraints. We show that the outcome of Coasian bargaining is constrained Pareto efficient. That is, given the information constraints, no Pareto improvement is possible. However, private ownership may limit what Pareto optimal allocations the government can achieve. The private owner in seeking to maximize profits always proposes contracts which counteract the government’s attempts to redistribute across individuals

* Corresponding author. E-mail address: [email protected] (J.H. Hamilton). 0047-2727/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jpubeco.2005.08.004

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with different characteristics. Under public management, any Pareto optimum can be sustained. In this context, private ownership, while not inefficient, does limit the government’s ability to redistribute. D 2005 Elsevier B.V. All rights reserved. Keywords: Coasian bargaining; Asymmetric information; Welfare theorems; Externalities

1. Introduction Coase (1960) demonstrated through a series of examples that bargaining among agents could lead to efficient outcomes despite the presence of externalities if private property rights were well defined and the costs of bargaining were zero. Subsequent literature has explored a variety of factors which affect whether private bargaining can eliminate inefficiencies from externalities including the nature and impact of positive transaction costs, the impact of different structures of property rights, the interaction of taxes and bargaining, the possibility that some participants possess private information, the effect of nonconvexities, and the implications of endogenous participation.1 The literature on Coasian bargaining almost entirely relates to whether a version of the First Fundamental Theorem of Welfare Economics holds with externalities — is the outcome with private ownership Pareto optimal? Whether Coasian bargaining affects the Second Fundamental Theorem of Welfare Economics has been largely unexplored.2 Does assigning property rights as a solution to an externality problem in any way limit the government’s ability to achieve different Pareto optimal allocations by engaging in redistribution? In a situation of complete information, it is straightforward to show that the Second Welfare Theorem remains valid when Coasian bargaining occurs. In this paper, we specify a model with asymmetric information and consider whether the outcome under private bargaining is unbiased, placing no restrictions on the government’s ability to redistribute. Specifically, we study a model where individuals produce output by applying their labor to a common productive asset. Individuals differ in their productivities and disutilities of effort in the commons. Only individuals know their own productivities or disutilities of effort in the commons. There is an alternative activity in which all individuals have identical productivities and disutilities. 1 Some of the many papers relating to these issues include Allen (1991), Baumol and Oates (1988), Baumol and Bradford (1972), Buchanan and Stubblebine (1962), Coase (1988), Dixit and Olson (2000), Farrell (1987), Frech (1979), Hamilton et al. (1989), Hurwicz (1999), Laffont (2000), Posner (1977), Samuelson (1985), Starrett and Zeckhauser (1974), Stiglitz (1994), Turvey (1963). 2 Baumol and Oates (1988, p. 126) at least implicitly show that a Second Welfare Theorem holds for Pigovian taxation even when there are certain types of nonconvexities. They do not consider issues of asymmetric information in this context. Ledyard (1971) considers both welfare theorems with externalities. His analysis focuses on what information firms or individuals have about the effect of their actions on aggregate outcomes given the externality. Again, asymmetric information between individuals and the government is not treated. Mas-Colell et al. (1995, Section 18.0), and Stiglitz (1994, Chapter 4) consider aspects of the Second Welfare Theorem in contexts with asymmetric information. Mas-Colell, Whinston, and Green analyze whether a planner can sustain any allocation on the full-information frontier. With discrete types, they find that only a subset of such allocations can be sustained, and with a continuum of types, only a Walrasian equilibrium can be sustained. This differs from our analysis which considers whether the government, working alone or through a private operator, can sustain any allocation on the constrained Pareto frontier. Similarly, Stiglitz argues that reaching allocations on a constrained Pareto frontier involves a mixture of distribution and efficiency concerns, thus violating the spirit of the Second Welfare Theorem which divorced allocation from distribution. He did not consider whether some methods of allocation limit the points on the constrained Pareto frontier which can be attained.

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We allow for the possibility that there are production externalities where each individual’s productivity in the commons may depend on the total effort or participation there. This externality could be beneficial or detrimental or in the borderline case in between not exist at all. The alternative activity involves no externality. Because the externality in the first activity depends only on the actions of those using it, following Brito et al. (1997), we call this a separable communal activity (SCA). One example of this structure is extracting minerals from the sea where more activity reduces everyone’s productivity. The alternative activity is mining claims on land with established property rights. Another example is the Internet where individuals have differing productivities on the network and increased usage slows everyone’s service. Uncongested alternative activities might include using regular mail or telephone service or doing library research.3 We make the standard Coasian assumption of zero transaction costs and assume that the government and a private individual would have exactly the same information when operating the SCA. Such an operator, whether a private owner or the government, can costlessly observe individual effort but cannot directly observe individual characteristics. The operator can use fully nonlinear fee or tax schedules subject to incentive compatibility constraints. Any difference between the outcomes under government and private operation, therefore, is not due to differences in instruments, information, or costs. Our main result is that under government management the Second Fundamental Welfare Theorem holds, while under profit maximizing private ownership it does not. Any Pareto optimal allocation that satisfies information constraints can be sustained by the government. Only a small (measure zero) subset of these constrained Pareto optimal allocations can be achieved with private ownership. While private ownership does not prevent redistribution between the owner and workers, it limits the ability to transfer utility between the more skilled and the less skilled workers. That only a limited subset of Pareto optimal allocations can be sustained is not because the other Pareto optimal allocations are infeasible with a private owner, but because they are not maximizing choices given the owner’s objective of maximizing profits. In order to limit any information rents which must be paid to induce revelation by workers, the owner fixes the difference in utilities achieved by the different types of workers. It is worth noting that this result is independent of whether the externality is beneficial or detrimental or is not present at all.4 The factor crucial for the failure of the Second Welfare Theorem is the presence of the asymmetric information. Having a profit maximizing private owner as an intermediary between the government and the workers in order to deal with information issues limits the ability of the government to redistribute across different types of workers. In the interpretation of the results, we tend to focus on cases with externalities since these have been widely treated in the literature and because having private bargaining seems most realistic in these cases.5 3 These examples are consistent with the simplifying assumption of the model that there is a single consumption good so that the outputs of the SCA and the alternative activity are perfect substitutes. Similar results would occur if consumption is disaggregated and the outputs of the two activities are not perfect substitutes. Any situation in which multiple workers use a common resource would then fit the model. 4 Some further results do depend upon the presence of externalities. See the discussion following Theorem 5. 5 One example without externalities is a model with workers who differ in their productivities such as Stiglitz (1982). Assume these workers utilize a common infrastructure which is never congested so no externality exists. Assume that instead of being independent producers, the workers are hired by the owner of the infrastructure who acts as a profit maximizing producer. The outcome will be constrained efficient, but the ability of the government to redistribute across different types of workers will be limited.

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Section 2 presents the basic model. Section 3 specifies the Pareto frontier when there is full information about individuals’ types. Section 4 specifies the Pareto frontier when private information exists and relates that frontier to the full information one. Section 5 analyzes government management of the SCA with linear and nonlinear taxes. Section 6 analyzes the operation of the SCA under the management of a private owner with complete property rights and considers how this affects the Second Welfare Theorem. Section 7 presents our conclusions. 2. The model Individuals have a choice of two activities to enter. In the first activity, individuals have identical productivities. They have identical utility functions over consumption and leisure, so they all choose the same level of effort with disutility equal to f 0. The output per worker in this activity is w 0. The utility from entering this activity is a worker’s consumption less the disutility of effort, c 0  f 0. In the second activity, individuals use a common resource where they differ in productivity and disutility of effort. There are two types of individuals denoted by i = 1, 2. Each type consists of a continuum of individuals with mass h(i). Let z(i) denote the effort provided by a type i individual, and let y(i) denote the output produced by that individual. The production function for output by an individual of type i is: yðiÞ ¼ Gð zðiÞ; E; iÞ

ð1Þ

where E is the externality defined below. We assume that, for each i, G is strictly concave in z and E and strictly monotonically increasing in z with G(0, E, i) = 0. Type 1’s are at least as productive as type 2’s at any z and E: BGð z; E; 1Þ BGð z; E; 2Þ z : Bz Bz

ð2Þ

Combining condition (2) with the condition that G(0, E, i) = 0 implies that G(z, E, 1) z G(z, E, 2), for all z. G may be increasing or decreasing in E, depending upon whether the externality is beneficial or detrimental. When E is zero, the marginal external effect is 0, BG(z, 0, i) / BE = 0. The externality E is atmospheric in nature; it depends upon aggregate values and not on the composition of the aggregate among individuals.6 Furthermore, the aggregates depend only on those participating in activity 2. We thus call activity 2 a separable communal activity (SCA).PLet a(i), 0 V a(i) V 1, denote the fraction of type i individuals whoP enter the SCA. Let M ¼ 2i¼1 aðiÞhðiÞ be the level of participation in the SCA, and let Z ¼ 2i¼1 aðiÞhðiÞzðiÞ be the total effort in the SCA. The externality E is a nondecreasing function of these aggregates: E ¼ eð M ; Z Þ

ð3Þ

where E(0, 0) = 0. Let a˜ and z˜ denote the vectors of participation rates and effort. Then the reduced form solution for Eq. (3) is: E ¼ e¯ ða˜ ; z˜ Þ:

ð4Þ

6 This assumption simplifies the analysis by limiting differences between individuals to productivity or preferences. As seen below, it allows the planner to achieve an undistorted outcome using relatively simple instruments. The essential nature of the results would not be altered if individuals also differed on the amount of externality they generated.

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Using Eq. (3), the derivatives of e¯ with respect to a(i) and z(i) are: B¯e Be ¼ aðiÞhðiÞ BzðiÞ BZ   B¯e Be Be ¼ hð i Þ þ zðiÞ : BaðiÞ BM BZ

ð5Þ ð6Þ

The utility function for an individual of type i in the SCA is: U ðcðiÞ; zðiÞ; iÞ ¼ cðiÞ  f ð zðiÞ; iÞ

ð7Þ

where c(i) is consumption and f(z(i), i) is the disutility of effort function which is strictly convex and increasing in z(i). The disutility of zero effort is zero, f(0, i) = 0. Since U is quasilinear, the marginal utility of consumption is constant across types. The disutility of effort, however, varies with type: Bf ð z; 1Þ Bf ð z; 2Þ b : Bz Bz

ð8Þ

Condition (8) and f(0, 1) = f(0, 2) imply that f(z, 1) b f(z, 2) for any z N 0. To simplify the analysis, we make two additional technical assumptions on net productivity: BGð0; E; iÞ Bf ð0; iÞ  ¼ l; i ¼ 1; 2 Bz Bz 9 ˆz such that Gðzˆ ; 0; 1Þ  f ðzˆ ; 1Þ N w0  f0 :

ð9Þ ð10Þ

Condition (9) is an Inada condition which insures that it is socially desirable to have positive effort from any individual assigned to the SCA. Condition (10) says the SCA is sufficiently productive relative to the alternative activity that it is socially desirable to have at least some individuals assigned to the SCA. We assume that neither the type nor the output of an individual in the SCA is observable, but entry into the SCA and effort, z(i), are observable.7 We model the government and the private rights holder as having identical information. They observe the same variables and have full knowledge of the structure of the economy. Neither faces administrative or monitoring costs in collecting taxes or fees. Finally, there is a single non-worker in the economy who is the potential resource owner under private management. This agent’s utility equals her consumption, c p.8 3. The ex ante full information Pareto frontier As a benchmark, consider the Pareto problem for a social planner who has full information about every individual’s type and who can directly assign each individual’s activity, effort, and 7 It is important that we assume that the commodity causing the externality, input levels z(i), is observable and hence directly controllable. Without significantly changing the results, we could assume that the externality is caused by observable output y(i). Assuming y(i) were observable while unobservable z(i) causes the externality would create additional second best issues. See Plott (1966). Note, we cannot assume both y(i) and z(i) are observable since this could allow an individual’s type to be directly inferred from his choices. 8 Were the owner to work in the commons, the incentive compatibility constraints would differ between public and private management.

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consumption. In choosing who enters the SCA, the planner may optimally assign only a fraction a(i) of the individuals of a particular type there in order to limit the externality they cause. We assume this assignment is done randomly, so that if 0 b a(i) b 1, individuals of type i face a lottery. The ex ante Pareto frontier depends on individuals’ expected utilities. To find the frontier, one can maximize the expected utility of type 1’s subject to a Pareto constraint on type 2’s expected utility, a Pareto constraint on the utility of the potential resource owner, and the aggregate resource constraint. Solving the following maximization for all allowable utility levels determines the full information Pareto frontier: Max

cðiÞ;zðiÞ;aðiÞ;c0 ;cp

hð1Þ½að1Þðcð1Þ  f ð zð1Þ; 1ÞÞ þ ð1  að1ÞÞðc0  f0 Þ

ð11Þ

s:t: að2Þðcð2Þ  f ð zð2Þ; 2ÞÞ þ ð1  að2ÞÞðc0  f0 ÞzU 2

ð12Þ

cp zU p

ð13Þ

2 X

hðiÞ½aðiÞðcðiÞ  Gð zðiÞ; E; iÞÞ þ ð1  aðiÞÞðc0  w0 Þ þ cp V0

ð14Þ

i¼1

cðiÞz0; zðiÞz0; cp z0; c0 z0; 0VaðiÞV1 where E is defined by Eq. (4). In this problem, at the optimum, constraint (14) must hold with equality since if not, either c(i) or c 0 could be increased. The Pareto constraints will also hold with equality if the utility levels are high enough that U p z 0 and U 2 þ að2Þf ð zð2Þ; 2Þ þ ð1  að2ÞÞf0 z0. Under these conditions, we can use constraints (12), (13), and (14) to eliminate c(i), c 0, and c p. This reduces the optimization problem to: Max

zðiÞ;aðiÞ

s:t:

2 X

hðiÞ½aðiÞðGð zðiÞ; E; iÞf ð zðiÞ; iÞÞþð1aðiÞÞðw0 f0 Þhð2ÞU 2 U p

ð15Þ

i¼1

0VaðiÞV1 and zðiÞz0

in the region defined by U p z0 U 2 þ að2Þf ð zð2Þ; 2Þ þ ð1  að2ÞÞf0 z0 U p þ hð2ÞU 2 Vh þ hð1Þ½ að1Þf ð zð1Þ; 1Þ þ ð1  að1ÞÞf0 

ð16Þ ð17Þ ð18Þ

P where hu 2i¼1 hðiÞ½aðiÞðGð zðiÞ; E; iÞ  f ð zðiÞ; iÞÞ þ ð1  aðiÞÞðw0  f0 Þ. Conditions (16)–(18) ensure that c p, c(i), and c 0 are nonnegative after the substitutions. Note that we can eliminate four unknowns using three equations (defined from constraints (12)–(14)) holding with equality) because utility is quasilinear so only expected consumption enters expected utility. The solution to problem (15) is characterized in the following result. Theorem 1. At any point on the Full Information Pareto Frontier at which U p and U 2 are such that constraints (12) and (13) hold with equality, the optimal z*(i) and a*(i) are independent of U 2 and U p , so that only c*(i),c 0*, and c p* vary along the frontier. Furthermore, z*(1) N z*(2) and either a*(2) = 0 and a*(1) N 0 or a*(2) N 0 and a*(1) = 1 .

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Proof. See the Appendix for all proofs. The Full Information Pareto Frontier is found by having the planner choose the a(i) and z(i) to maximize aggregate welfare and redistribute among individuals just by varying their consumptions. At the optimum, some or all type 1’s are assigned to the SCA while type 2’s may or may not be assigned there. When some individuals of type 2 are assigned to the SCA, all type 1’s participate and each type 1 puts in more effort than a type 2.9 4. The ex ante Implementable Pareto Frontier The Pareto optimal allocations found in Section 3 may not be implementable for two reasons. First, the planner does not know the type of any individual, so it must offer bundles or lotteries that induce individuals to reveal their types truthfully. To ensure this, the bundle designed for each type must yield that type a higher expected utility than the bundle intended for the other type: aðiÞðcðiÞ  f ð zðiÞ; iÞÞ þ ð1  aðiÞÞðc0  f0 Þzað jÞðcð jÞ þ Gð zð jÞ; E; iÞ  Gð zð jÞ; E; jÞ  f ð zð jÞ; iÞÞ þ ð1  að jÞÞðc0  f0 Þ; i ¼ 1; 2; j p i :

ð19Þ

The term G(z( j), E, i)  G(z( j), E, j) on the right hand side of constraint (19) is present because the planner cannot observe output. Thus, a type 1 who acts as a type 2 produces more output than a true type 2 and can consume the incremental output without the planner’s knowledge. Conversely, a type 2 who acts as a type 1 does not produce as much as the type 1 does, and so must cut consumption below c(1) to provide the required output to the government.10 Second, the planner’s assignments may not be decentralizable if individuals who are supposed to work in the SCA prefer working in the alternative activity instead. The following participation constraints rule this out:11 aðiÞðcðiÞ  f ð zðiÞ; iÞÞ þ ð1  aðiÞÞðc0  f0 Þz c0  f0 ; i ¼ 1; 2 :

9

ð20Þ

When a*(2) = 0,c*(2) is irrelevant and c*0 = U 2 + f 0. When a*(2) N 0, c*(2) and c*0 are not separately determined with c 0* able to take any value between 0 and (U 2 + a*(2)f(z*(2), 2) + (1  a*(2))f 0) / (1  a*(2)). 10 There is an issue of how to deal with the possible negativity of c(1) + G(z(1), E, 2)  G(z(1), E, 1) in (19) for i = 2. Since G(z(1), E, 2) V G(z(1), E, 1), then even if c(1) N 0, this term might be negative. If so, then a type 2 who misrevealed as a type 1 would be assigned negative consumption which is impossible. It would be infeasible for a type 2 to act as a type 1 and consume the adjusted type 1 bundle. Simply truncating the consumption as the maximum of 0 and c(1)  G(z(1), E, 1) + G(z(1), E, 2) would be problematic. A truthfully revealing type 1 would then have an incentive to say he was a type 2 who produced less and was not able to turn over as much as had been committed to the planner. In effect, the truthful type 1 as well as the misrevealing type 2 would renege on the contract with the planner. An attempt to renege must in some way be punished. The type 2 who misrevealed cannot be fined monetarily since consumption is already zero. We assume there is a fine in the form of a hassle cost imposed on someone who reneges. They are compelled to expend effort to explain their reneging which lowers their utility. To simplify matters, we assume the hassle cost exactly equals the consumption shortfall. This is sufficient to make (19) for i = 2 hold with strict inequality so not be a binding constraint. A larger hassle cost would add nothing further. Formally, we impose nonnegativity only for c(1) and c(2) and not for the consumptions of those who misreveal. 11 Note, if (20) holds with a strict inequality when 0 b a(i) b 1, the ability to decentralize the allocation will be incomplete. All type 2’s will strictly prefer the SCA but only some are assigned there. A weaker version of this problem arises if (20) holds with equality since then individuals are indifferent to where they are assigned. Some control mechanism must exist to ensure that only the desired number of type 2’s enter the SCA. This could be accomplished in a variety of ways such as having sign up on the internet where the first ones to sign up are chosen. Individuals do not observe where they are in the queue, so they face a lottery. Individuals are allowed to sign up for one of the three choices (the SCA as type 1, the SCA as type 2, or the alternative activity) and cannot switch if they lose a lottery.

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The Implementable Pareto Frontier is found by adding Eqs. (19) and (20) to the optimization defined by problem (11) with constraints (12)–(14). As shown in Lemma 1, this problem can be significantly simplified. Lemma 1. For U p z 0, the ex ante Implementable Pareto Frontier can be found as the solution to the following optimization: Max

c0 ;cðiÞ;aðiÞ;zðiÞ

hð1Þ½að1Þðcð1Þ  f ð zð1Þ; 1ÞÞ þ ð1  að1ÞÞðc0  f0 Þ

s:t: að2Þðcð2Þ  f ð zð2Þ; 2Þ  c0 þ f0 Þ ¼ max½0; U 2  c0 þ f0 ; að1Þðcð1Þ þ Gð zð1Þ; E; 2Þ  Gð zð1Þ; E; 1Þ  f ð zð1Þ; 2Þ  c0 þ f0 Þ ð21Þ að1Þðcð1Þ  f ð zð1Þ; 1Þ  c0 þ f0 Þzað2Þðcð2Þ þ Gð zð2Þ; E; 1Þ  Gð zð2Þ; E; 2Þ  f ð zð2Þ; 1Þ  c0 þ f 0 Þ ð22Þ 2 X

hðiÞ½aðiÞðcðiÞ  Gð zðiÞ; E; iÞÞ þ ð1  aðiÞÞðc0  w0 Þ þ U p ¼ 0

ð23Þ

i¼1

cðiÞz0; c0 z0; zðiÞz0; 0VaðiÞV1 : Condition (21) combines the Pareto, self-selection, and participation constraints for the type 2 individuals, at least one of which must hold with equality. If c 0 N 0, then it can be shown further that one of the Pareto and self-selection constraints must hold with equality. Condition (22) is the self-selection constraint for type 1’s. Condition (23) requires that production exactly equal consumption at the implementable optimum. As shown in the next result, the Implementable Pareto Frontier coincides with the Full Information Frontier over a range of values. We show this under additional assumptions that individuals feel less disutility working in the alternative activity than working in the SCA at the efficient level and that total output produced exceeds some bound. These additional assumptions simplify the derivation and affect the specific bounds on U p and U 2, but altering them would not change the qualitative nature of the results.12 Theorem 2. Fix z(i) and a(i) at the values z*(i) and a*(i) which solve Eq. (15). Assume that f 0 b min [f(z*(1),1), f(z*(2),2)] and that aggregate income is sufficiently large. 13 Then there exist values of U 2 and U p with U p z 0 and which satisfy the following: ð24Þ U 2 þ f0 z0 p hT þ hð1ÞaTð1ÞðGð zTð1Þ; ET; 2Þ  Gð zTð1Þ; ET; 1Þ þ f ð zTð1Þ; 1Þ  f ð zTð1Þ; 2ÞÞVU þ ðhð1Þ þ hð2ÞÞU 2 VhT þ hð1ÞaTð2ÞðGð zTð2Þ; ET; 2Þ  Gð zTð2Þ; ET; 1Þ þ f ð zTð2Þ; 1Þ   ð25Þ  f zT ð2Þ; 2

12

These bounds arise in large part from the linearity of the utility function. Our assumptions ensure both that nonnegativity of consumption holds for all types and determine the set of relevant constraints on the upper and lower bounds on utilities that can be assigned to different types. 13 The required bound on aggregate income is given in (A9).

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879

U1 Full Information Pareto Frontier

_ _ _ _ _ _ _ Implementable Pareto Frontier

•

•

U2

Fig. 1. For a fixed level of U p, this figure shows the relation of the Full Information and Implementable Pareto Frontiers.

The optimized value of the objective function in the optimization in Lemma 1 is the same as that found in the full information optimization (15) if and only if U p and U 2 satisfy the restrictions in conditions (24) and (25). The relationship between the two Pareto frontiers is shown in Fig. 1. In the presence of an externality, if the SCA is not managed by the government or a private owner with individuals independently choosing their activities there, the outcome will generally not be on the ex ante Implementable Pareto Frontier. Individuals will maximize their utilities G(z(i), E, i)  F(z(i), i)  T i , where T i is a tax (depending only on type), taking E as fixed since each single individual is small and has no effect on E. Free riding takes place and inefficiency results. In the next two sections, we consider government management with taxation and private ownership as ways of correcting for this inefficiency. 5. Private choice with nonlinear taxation The optimization specified above to find the Implementable Pareto Frontier relied on a command economy approach with the planner selecting all consumption levels. Such allocations can be sustained by a more decentralized approach. Consider a private choice economy with government taxation and government control of the SCA. Once taxes are announced, the individuals independently choose where to supply effort. However, if 0 b a(i) b 1 for some i, then the government must still limit entry to the SCA. The government’s instruments are a subsidy s for workers in the alternative activity, a subsidy r for the nonworker, and nonlinear fee and lottery schedules F(z) and a(z) for those who wish to enter the SCA. Individuals select a level of effort and buy a ticket in advance to expend that effort. For each z requested, the government will sell a ticket with probability a(z). An individual of type i will choose a z(i) which maximizes að zÞðGð z; E; iÞ  f ð z; iÞ  F ð zÞÞ þ ð1  að zÞÞðw0  f0 þ sÞ

ð26Þ

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where each individual treats E as given in this maximization since there is a mass of individuals of each type and each individual decides independently. Since no monotonicity or continuity restrictions have been placed on F(z) or a(z), the government can select these schedules so that only two different values are potentially acceptable. In effect, individuals will choose between two options, (z 1,F(z 1), a(z 1)) and (z 2, F(z 2), a(z 2)), where each option is intended for the respective type. Overall, the government’s budget must balance: 2 X i¼1

hðiÞaðzi ÞF ðzi Þ ¼

2 X

hðiÞð1  aðzi ÞÞs þ r :

ð27Þ

i¼1

Given these instruments, the government can sustain any point on the Implementable Pareto Frontier. Theorem 3. Any point on the Implementable Pareto Frontier can be achieved in a private choice economy with the government managing the SCA using nonlinear schedules (F(z), a(z)) and giving subsidies s and r to workers in the alternative activity and to the nonworker, respectively. The government can actually accomplish much with a simpler set of fee and participation schedules for the SCA. Consider a linear fee schedule where F(z) = T + tz and a step function participation schedule where a(z) = a 1 if z z zˆ and a(z) = a 2 if z b zˆ , for some zˆ . The optimal choice of effort for type i can be found by solving the following two optimizations and taking the better solution:14 Max a2 ðGð z; E; iÞ  f ð z; iÞ  T  tzÞ þ ð1  a2 Þðw0 þ s  f0 Þ

s:t:z V zˆ

ð28Þ

Max a1 ðGð z; E; iÞ  f ð z; iÞ  T  tzÞ þ ð1  a1 Þðw0 þ s  f0 Þ

s:t:z z zˆ :

ð29Þ

z

z

With such schedules and with subsidies to workers in the alternative activity and to the nonworker, the government can sustain some implementable points on the Full Information Pareto Frontier. Theorem 4. Under a linear fee schedule, there exist values of T, t, a 1 , a 2 , zˆ, s, and r which lead individuals to choose allocations that yield utilities on the Full Information Pareto Frontier. If a*(2) = 0, then any point on the Full Information Pareto Frontier which is also on the Implementable Pareto Frontier can be sustained by some choice of these instruments while, if a*(2) N 0, only a subset of such points can be sustained. Several factors allow the government to achieve points on the full information frontier using a linear schedule. First, the externality is atmospheric so marginal harm is constant across types. The same marginal tax rate will induce each type to choose the appropriate effect level. Second, utility varies monotonically across types. Only one instrument, T, is needed to induce individuals to make the appropriate participation decision. However, when both types participate in the SCA, using differentiated lump sum taxes, T 1 and T 2, may allow more redistribution and thus allow more of the Full Information Pareto Frontier to be achieved. 14 Since z V zˆ is specified in (28), instead of z b zˆ , this optimization actually finds the supremum in the region with z b zˆ rather than the maximum. Since z = zˆ in (28) will be ruled out as the overall optimum, this is done at no cost to ensure that a solution to (28) exists.

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6. Private ownership The Coasian approach to an externality problem is to establish well-defined property rights and then to allow private agents to reach whatever agreements they choose. Here, it is sufficient for defining property rights to make a single individual the owner of the SCA. This owner has complete authority over the SCA subject to the same information restrictions as the government — the owner can only observe effort and not an individual’s type directly. The owner can set entry conditions and charge individuals fees based on their effort levels. Consistent with zero transaction costs, no administrative costs or other factors bar using a nonlinear fee schedule. Given the nonlinearity of the fees, in effect, the owner not only determines who enters the SCA but also what effort they put forth. The entry fee alone may not be adequate to assure entry of the desired number by a particular type if the owner chooses only a fraction of that type to participate. Thus, the owner must have direct control of entry as well. Although the government has no direct control over the operation of the SCA, it is still responsible for redistribution. Consistent with the way in which redistribution is modeled in an Arrow–Debreu model, the government initially sets taxes to adjust initial endowments and then private individuals make their decisions. In this context, even though not operating the SCA, the government is able to observe who participates in the SCA and what their level of effort is. Thus, the government can initially announce taxes or subsidies for participants in the SCA that depend on what type they ultimately reveal themselves to be. Since the government is presumed not to be attempting to operate the SCA even indirectly, these redistributive taxes may vary with the type revealed through individuals’ choices of effort but do not vary marginally with effort levels.15 Formally, the government initially announces taxes T 1 and T 2 for the participants of each type in the SCA and subsidies s and r for workers in the alternative activity and for the owner of the SCA. T 1 can differ from T 2 provided the owner sets different effort levels for the two types when using the SCA.16 After the government announces its policies, the owner (as the government did when managing the SCA) announces nonlinear participation and fee schedules a(z) and F(z) and individuals then choose whether to participate in the SCA and what level of effort to engage in if they participate. Given the nonlinearity, this is equivalent to the owner offering two options, (a 1, z 1, F 1) and (a 2, z 2, F 2) subject to self-selection constraints. The owner seeks to maximize total income subject to self-selection and participation constraints. Max hð1Þa1 F 1 þ hð2Þa2 F 2 þ r

ai ;zi ;Fi

s:t:

      ai Gðzi ; E; iÞ  f ð zi ; iÞ  F i  T i  w0  s þ f0 zaj ðG zj ; E; i  f zj ; i  F j

T j  w0  s þ f0 Þ; i ¼ 1; 2; jpi   ai Gðzi ; E; iÞ  f ðzi ; iÞ  F i  T i  w0  s þ f0 z0; i ¼ 1; 2 15

ð30Þ

ð31Þ ð32Þ

If the government did impose taxes which varied with effort levels, it would in effect be controlling the use of the SCA which would then only nominally be privately operated. 16 In effect, the government looks ahead to the owner’s choices and sets taxes T 1 for those who choose the better option for type 1’s and T 2 for those who choose the better option for type 2’s.

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Gðzi ; E; iÞ  F i  T i z0; i ¼ 1; 2

ð33Þ

where E is given by Eq. (4) based on the owner’s choices. Conditions (33) specify that c(i) cannot be negative. If, as in Theorem 2, f 0 b min[ f(z 1, 1), f(z 2, 1)] is assumed, then constraint (33) is implied by constraint (32) as long as c 0 = w 0 + s z 0 holds. Hence, constraint (33) can be dropped from consideration.17 The actions of the owner may depend upon government policies. The government must choose its policies so that, looking ahead, its budget is balanced. That is, for the owner’s choice of a 1 and a 2, hð1Þa1 T 1 þ hð2Þa2 T 2 ¼ ðhð1Þð1  a1 Þ þ hð2Þð1  a2 ÞÞs þ r:

ð34Þ

Deviations from the equilibrium choices by the owner in setting a 1 and z 1 either unbalance the budget if T 1, T 2, s, and r are taken as fixed or prevent using differentiated taxes T 1 p T 2 if z 1 = z 2. One approach is to only require equilibrium budget balance. The government would announce policies, the owner would respond, and only policies which induce a balanced budget would be allowed. This equilibrium would not in general be on the Implementable Pareto Frontier. If such a point were the equilibrium, the government budget must be balanced there. Then, the owner, acting myopically with respect to government policies, could gain by deviating in a way which induces a government deficit and infeasibly raises the owner’s income. The alternative, which we assume, is to require that the budget be balanced on and off the equilibrium path. We assume that the government holds constant T 1, T 2, and s but adjusts r to maintain budget balance in all circumstances.18 The owner recognizes that a deviation from the anticipated action automatically causes a change in r to maintain budget balance. Thus, Eq. (34) is a definition of r which can be substituted into problem (30) to yield the following objective function for the owner:     Maxi hð1Þa1 F 1 þ T 1 þ s þ hð2Þa2 F 2 þ T 2 þ s  ðhð1Þ þ hð2ÞÞs ð35Þ ai ;zi ;F

which is maximized subject to Eqs. (31) and (32). This can be simplified as given by the next Lemma. Lemma 2. The private owner’s choice of a i and z i can be found as the solution to the following optimization: Max ai ;zi

2 X

hðiÞai ðGðzi ; E; iÞ  f ðzi ; iÞ  w0 þ f0 Þ

i¼1

þ hð1Þa2 ðGðz2 ; E; 2Þ  f ðz2 ; 2Þ  Gðz2 ; E; 1Þ þ f ðz2 ; 1ÞÞ  ðhð1Þ þ hð2ÞÞs

ð36Þ

17 Nonnegativity of consumption for individuals who misreveal is not imposed. It follows for type 1’s who would act as 2’s since G(z 2, 1) z G(z 2, 2). For 2’s who act as 1’s, G(z 1, 2)  F 1  T 1 b 0 is allowed based on the idea that the owner, as the government above, can impose hassle costs on those who seek to violate their contracts by paying less than the agreed upon fee. 18 See Hamilton and Slutsky (2004) for a discussion of off-equilibrium path balanced budget requirements in an optimal tax model with a finite number of individuals. The other instruments T 1, T 2, or s could be altered to maintain budget balance but these are paid by agents who are small in the economy and who individually will not assume that changes in their actions will cause the government to alter its policies. The owner of the SCA is not small and will recognize that this will occur. If r is fixed, the owner can gain by inducing changes in T 1, T 2 or s which in turn can lead to an inefficient outcome. When these are fixed and r adjusts, this does not occur.

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s:t: a1 ðGðz1 ; E; 1Þ  f ðz1 ; 1Þ  Gðz1 ; E; 2Þ þ f ðz1 ; 2ÞÞza2 ðGðz2 ; E; 1Þ  f ðz2 ; 1Þ  Gðz2 ; E; 2Þ þ f ðz2 ; 2ÞÞ

ð37Þ

and the optimal fees are given by:   a1 F 1 ¼ a1 Gðz1 ; E; 1Þ  f ðz1 ; 1Þ  T 1  s  w0 þ f0  a2 ðGðz2 ; E; 1Þ  f ðz2 ; 1Þ  Gðz2 ; E; 2Þ þ f ðz2 ; 2ÞÞ  a2 F 2 ¼ a2 Gðz2 ; E; 2Þ  f ðz2 ; 2Þ  T 2  s  w0 þ f0

ð38Þ ð39Þ

Let a¯1 and ¯z1 denote the solution to the problem in Lemma 2. This solution, which is independent of s, is valid for all s such that c 0 = w 0 + s and c p (which equals the value of the objective function) are nonnegative. These hold if  ðhð1Þ þ hð2ÞÞw0 Vðhð1Þ þ hð2ÞÞsV

2 X

hðiÞa¯i ðGðz¯i ; iÞ  w0 þ f0 Þ

i¼1

      þ hðiÞa¯ 2 G z¯ 2 ; E¯ ; 2  f ðz¯ 2 ; 2Þ  G z¯ 2 ; E¯ ; 1 þ f ðz¯ 2 ; 1Þ :

ð40Þ

We can now relate the solution to the private owner’s optimization to implementable Pareto optimal allocations. Let U¯ 1(T 1, T 2, s), U¯ 2(T 1, T 2, s), and U¯ p(T 1, T 2, s) be the utilities associated with the solution to the private owner’s optimization for any T 1, T 2, and s. ¯ 1 (T 1 , T 2 , s), U ¯ 2 (T 1 , T 2 , s), U ¯ p (T 1 , T 2 , s)) is on the Theorem 5. For any T 1 , T 2 and s,(U Implementable Pareto Frontier. If the economy is such that a*(2) = 0 in the solution to the ¯ 1 (T 1 , T 2 , s), U ¯ 2 (T 1 , T 2 , s), U ¯ p (T 1 , T 2 , s)) is also on Full Information Pareto optimization, then (U 1 1 2 2 1 ¯ ¯ ¯ p (T 1 , T 2 , s)) the Full Information Pareto Frontier. If a*(2) N 0, then (U (T , T , s), U (T , T 2 , s), U is not on the Full Information Pareto Frontier. Two important implications follow from this theorem. First, in this context, the First Fundamental Welfare Theorem holds when Coasian bargaining is used to resolve the problems of externalities. The equilibrium is constrained Pareto optimal. Given the information limits, no reallocation is possible that will improve everyone’s welfare. When both types participate in the SCA, then Pareto superior allocations do exist but these can be achieved only if individuals’ types are common knowledge and not private information. The presence of externalities affects the nonlinear fee schedules. A standard result in private information models without externalities is that one type receives an undistorted bundle (Sadka (1976)). That is, the FOC for that type’s bundle is the same as under full information. With the externality, this would include a bPigovian taxQ correction that would be the same for all types with an atmospheric externality. When the impact of the externality on output depends on type, there is an additional term in the FOC which depends on the differential effect of the externality on output by the two types at the less able type’s effort level.19 Second and most significantly, the Second Fundamental Welfare Theorem fails here. Consider the case with a*(2) N 0. No set of redistributive taxes lead the equilibrium to be on the

19

This follows from using Eqs. (A28) and (A3) to find the differences between the information-constrained and fullinformation FOC for the more able workers’ effort in the SCA. This additional effect is analogous to the adjustment terms in Boadway and Keen (1993) and Cremer et al. (2001) where one consideration in choosing the total level of the public good (or the externality-producing good) is relaxing self-selection constraints between types.

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full information part of the Implementable Pareto Frontier. No matter how the taxes are set, the owner always adjusts fees to limit the information rents that must be paid to the type 1’s to induce them to reveal truthfully their type. Conversely, when a*(2) = 0 and only type 1’s are in the SCA, the information distorted portions of the Implementable Pareto Frontier cannot be reached.20 In fact, in either case, only a small (measure zero) set of the points on the Implementable Pareto Frontier can be achieved. The next result specifies the achievable utility levels for the three types. Theorem 6. The attainable utility levels when there is private ownership are independent of the taxes T 1 and T 2 on the workers in the SCA and depend only on the subsidy (or tax) s given to workers in the alternative activity. For s satisfying the bounds in condition (40), the attainable utilities are:       U¯ 1 ðsÞ ¼ a2 G z¯ 2 ; E¯ ; 1  f ðz¯ 2 ; 1Þ  G z¯ 2 ; E¯ ; 2 þ f ðz¯ 2 ; 2Þ þ w0  f0 þ s U¯ 2 ðsÞ ¼ w0  f0 þ s p U¯ ¼

2 X

    hðiÞa¯ i G z¯ i ; E¯ ; i  f ðz¯ i ; iÞ  w0 þ f0

i¼1

      þ hðiÞa¯ 2 G z¯ 2 ; E¯ ; 2  f ðz¯ 2 ; 2Þ  G z¯ 2 ; E¯ ; 1 þ f ðz¯ 2 ; 1Þ  ðhð1Þ þ hð2ÞÞs : Having a private owner of the SCA drastically limits the ability of the government to redistribute. Note that redistributing between the owner and workers is not constrained. By raising the subsidy to workers in the alternative activity, the government can reduce the utility of the owner and raise the utility of all workers. The consumption and utility of the owner can be reduced to zero.21 The government is limited in its ability to redistribute between the different types of workers. No matter what policies the government chooses, the difference between the utilities of types 1 and 2 equals a¯ 2( G(z¯ 2, E¯ , 1)  f(z¯ 2, 1)  G(z¯ 2, E¯ , 2) + f(z¯ 2, 2)). Any attempt to change their relative utilities by changing the taxes T 1 and T 2 is completely negated by exactly offsetting changes in the owner’s fees F 1 and F 2.22 Fig. 2 shows the set of achievable utilities 20 This limit on the ability of the government to redistribute is not due to the assumption that the government moves first. Were the private owner to set entry fees and effort levels before the government sets taxes for the two types of participants and the nonparticipants, the government would have to choose taxes that satisfy the incentive compatibility and participation constraints or pooling of types would occur. A complete solution to this sequential move game would require specifying a welfare function so that the owner could predict the government’s choice of taxes. Suppose the owner chose the allocation such that type 2’s participation constraint and type 1’s incentive compatibility constraint bind with a zero subsidy to nonparticipants. Then if the government attempts to tax type 2 workers to subsidize type 1 workers, all type 2s will choose the bundle intended for type 1. This would frustrate the governmentTs plan to redistribute to type 1 workers. Hence, portions of the constrained Pareto frontier are unattainable, regardless of the order of moves. 21 Thus, auctioning off the right to own or operate the SCA as proposed by Demsetz (1968) for monopolies would be irrelevant in this context. 22 While the result that private ownership reduces the range of Pareto efficient allocations is general, that U 1–U 2 is constant is due to the quasi-linearity of utility. With a more general specification, U 1–U 2 in the private owner solution would depend on the government’s choice to the subsidy to nonparticipants. The owner would continue to make type 2 workers’ participation constraint and type 1 workers’ incentive compatibility constraint bind. For any level of the subsidy, this would be a single allocation, so the governmentTs ability to redistribute under private ownership would remain quite limited.

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U1

UP = 0

UP > 0 _f

0

U2

Fig. 2. When a*(2) N 0 the dashed line is the set of achievable utilities under private ownership. U 2 has a lower bound of f 0.

when a¯ 2 N 0. In general, for almost all social welfare functions, the social optimum will not be in this achievable set. The government is only limited in redistributing across individual types. It is not limited in redistributing across individuals by name independent of type. A tax on an individual independent of which activity the individual entered or which type the individual acts as in the SCA would appear on both sides of the participation and self-selection constraints and cannot be offset by changes in the fees set by the owner. For such redistribution to be desirable, there must be some basis for differentiating between individuals other than their types. 7. Conclusions We specified a model with private information and congestion and considered both governmental and private ownership mechanisms to resolve inefficiencies that might arise due to the externality. Both approaches yielded outcomes which were Pareto optimal given the limits imposed by the information constraints. The governmental solution in some circumstances could even utilize a relatively simple linear Pigovian tax structure. Thus, a version of the First Fundamental Theorem of Welfare Economics holds for both the tax mechanism and Coasian bargaining. Significantly, the Second Fundamental Theorem of Welfare Economics holds for the governmental mechanism using nonlinear taxes but does not hold for Coasian bargaining. Using nonlinear taxes, the government can achieve any Pareto optimal allocation consistent with information constraints. When property rights are assigned to a private individual, only a limited subset of the Pareto Frontier can be achieved. The Coasian approach, while yielding an efficient outcome, limits the government’s ability to redistribute. Two factors are important for this result. One is the presence of private information. If full information about types is available to both the government and the private owner, then

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the Second Welfare Theorem would hold under Coasian bargaining. The owner would select the efforts and participation rates to operate the SCA efficiently and the government would redistribute in any way it wanted by using lump sum taxes on each individual by name. Private information alone is not the complete explanation for the failure of the Second Welfare Theorem with Coasian bargaining. The government has the same information restrictions yet can achieve any point on the implementable frontier with nonlinear taxes. It would be feasible for the private owner to implement any outcome that the government can implement by using a nonlinear fee schedule. The problem is that it is not in that owner’s interest to do this. Thus, the second factor is that the private owner seeks to maximize profits in operating the SCA. In selecting policies to maximize profits, the owner both counteracts attempts by the government to redistribute between worker types and selects efforts and participation which are consistent with only some points on the Implementable Pareto Frontier. If both types participate, the owner will always select a distorted outcome off the full information frontier in order to minimize the information rents it must pay the more productive individuals to reveal their type. Maximizing social welfare and maximizing profits yield different outcomes because the information rents that must be paid to induce truthful revelation enter into social welfare but not into profits.23 The failure of the Second Welfare Theorem under private ownership does not imply that allocation and distribution decisions cannot be separated from each other. When there is government operation of the SCA, the government could decentralize into allocation and distribution branches where the allocation branch operates the SCA just to achieve efficiency without knowing the government’s distributional preferences. However, the allocation branch would maximize not profits but some more complicated objective function.24 With a private owner, the government could try to mimic this function by imposing complicated nonlinear taxes on the owner but this would really be equivalent to having the government manage the SCA. That redistribution is limited under private ownership may be a significant factor in explaining support for and opposition to privatization in some circumstances beyond the more standard question of whether private operation is more or less efficient than government operation. Opposition to privatization may come from those who view the ability to redistribute as an important role for government. They might support government operation of some activity even if they believed a private owner would manage the activity more efficiently so that redistribution could occur. On the other hand, opponents of redistribution, either on philosophical grounds or because of other costs of redistribution which we have not modeled, might support privatization as a way of committing the government not to engage in redistribution. They might take this position even though government operation yielded larger net output either because it did not induce distortions to minimize information rents or because the government’s powers of compulsion might allow it to acquire more information about types then would be possible for a private owner. That private ownership is not an unbiased mechanism by itself may not lend support for or opposition to the use of private ownership. 23 In Hamilton et al. (1989), in an externality model with a variable number of firms, profit maximization by a private owner can lead to a failure of the First Welfare Theorem due to a nonconvexity in the profit function. This contrasts with the welfare maximization problem where this nonconvexity may not arise. 24 See Hamilton and Slutsky (2000) for a general analysis of when the government can separate into allocation and distribution branches which operate independently but which do limited transfers of information between themselves.

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Acknowledgements We thank two anonymous referees, Ted Bergstrom, John Ledyard, Paul Rubin, David Sappington, and audiences at Florida, IDEI (Toulouse), IAE (Barcelona), the Association for Public Economic Theory Meeting, Midwest Economic Theory Meeting, Public Choice Society Meeting, the Southeastern Economic Theory Meeting, the Southern Economic Association Meeting, and the Winter Econometric Society Meeting for comments. Hamilton and Slutsky thank the Public Utility Research Center and the College of Business Administration of the University of Florida for research support. Appendix A Proof of Theorem 1. The values of z(i) and a(i) which solve (15) are independent of U 2 and U p which enter the objective function only as additive constants. Consider any values of U 2 and U p at which the inequalities which guarantee nonnegative consumptions are satisfied at the solution values z*(i) and a*(i). For such U 2 and U p, z*(i) and a*(i) are the Pareto optimal values. To characterize the optimal z*(i) and a*(i), consider the derivatives of the objective function, denoted h(1)U 1, with respect to z(i) and a(i).  BU 1 BGð zðiÞ; E; iÞ Bf ð zðiÞ; iÞ  ¼ hðiÞaðiÞ hð 1Þ BzðiÞ BzðiÞ BzðiÞ þ

2 X j¼1

hð 1Þ

hð jÞað jÞ

BGð zð jÞ; E; jÞ Be¯ ; BE BzðiÞ

i ¼ 1; 2

BU 1 ¼ hðiÞ½Gð zðiÞ; E; iÞ  f ð zðiÞ; E; iÞ  ðw0  f0 Þ BaðiÞ 2 X BGð zðiÞ; E; jÞ Be¯ ; i ¼ 1; 2 : þ hð jÞað jÞ BE BaðiÞ j¼1

ðA1Þ

ðA2Þ

From the Inada conditions in (9), at the optimum z*(i) N 0 and BU 1 / Bz*(i) in Eq. (A1) must equal 0. Substituting Eq. (5), this reduces to: ! 2  BGð zðiÞ; E; iÞ Bf ð zðiÞ; iÞ Be X BGð zðiÞ; E; iÞ  þ að i Þ hð jÞað jÞ ¼ 0; BzðiÞ BzðiÞ BZ j¼1 BE

i ¼ 1; 2

ðA3Þ Consider the values of z*(1) and z*(2) which make the term multiplying a (i) in Eq. (A3) equal zero for i = 1 and 2.25 From (2) and (8), z*(1) = z*(2) cannot hold. Strict concavity of G(z, E, i)  f(z, i) then implies z*(1) N z*(2). 25

We assume this value of z*(i) even when a(i) = 0 since this would be the effort assigned to a type i individual if a(i) were to be increased above 0. This assumption can be justified on the same grounds as used to justify subgame perfect equilibrium.

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Given the bounds on a(i), the first order condition with respect to a(i) is BU 1 / Ba(i) V 0 if a(i) = 0, BU 1 / Ba(i) = 0 if 0 b a(i) b 1, and BU 1 / Ba(i) z 0 if a(i) = 1, where BU 1 / Ba(i) is defined from Eq. (A2) after substituting Eq. (6). To analyze this, consider BU 1 / Ba(1) (BU 1 / Ba (2))(h(1) / h(2)): BU 1 hð1Þ BU 1  ¼ ½Gð zð2Þ; E; 1Þ  f ðzð2Þ; 1Þ  ðGð zð2Þ; E; 2Þ  f ð zð2Þ; 2ÞÞ Bað1Þ hð2Þ Bað2Þ " # 2 Be X BGð zð jÞ; E; jÞ hð jÞað jÞ þ Gðzð1Þ; E; 1Þ  f ð zð1Þ; 1Þ þ zð1Þ BZ i¼1 BE   2 X Be BGð zð jÞ; E; jÞ : ðA4Þ  Gð zð2Þ; E; 1Þ f ðzð2Þ; 1Þþ zð2Þ hð jÞað jÞ BZ j¼1 BE

From G(0, E, 1)  f(0, 1) = G(0, E, 2)  f(0, 2) and assumptions (2) and (8), G(z(2), E, 1)  f(z(2), 1) N G(z(2), E, 2)  f(z(2), 2). Consider the function G(z, E*, 1)  f(z, 1) + zb where E* is P2 BGðzTð jÞ; ET; jÞ . fixed at e¯ (z˜ *, a˜*) and b is a constant set equal to BeT j¼1 hð jÞaTð jÞ BZ BE This function is strictly concave in z by assumption. The first order condition for maximizing it is the same as the condition that the term in Eq. (A3) multiplying a(1) equal 0. Hence, the optimal z*(1) also maximizes it. Therefore, 2 BeT X BGð zTð jÞ; ET; jÞ Gð zTð1Þ; ET; 1Þ  f ð zTð1Þ; 1Þ þ zTð1Þ hð jÞaTð jÞ BZ j¼1 BE NGð zTð2Þ; ET; 1Þ  f ð zTð2Þ; 1Þ þ zTð2Þ

2 BeT X BGð zTð jÞ; ET; jÞ : hð jÞaTð jÞ BZ j¼1 BE

Combining these inequalities with Eq. (A4) yields BU 1 / Ba(1) N (h(1) / h(2))(BU 1 / Ba(2)) From (10) and the assumption that BG(z, 0, i) / BE = 0, a*(1) = a*(2) = 0 is ruled out. At least one a*(i) must be positive. From the first order conditions with respect to a(i), a*(2) N 0 then implies a*(1) = 1 as required. 5 Proof of Lemma 1. First, (19) and (20) can be simplified by subtracting c 0  f 0 from both sides of each inequality: aðiÞðcðiÞ  f ð zðiÞ; iÞ  c0 þ f0 Þzað jÞðcð jÞ þ Gð zð jÞ; E; iÞ  Gð zð jÞ; E; jÞ  f ð zð jÞ; iÞ  c0 þ f0 Þ; i ¼ 1; 2;

ðA5Þ

jpi

aðiÞðcði Þ  f ðzðiÞ; iÞ  c 0 þ f0 Þz0;

i ¼ 1; 2

ðA6Þ

Condition (A5) for i = 1 is (22). Second, (A6) for i = 2 and (A5) for i = 1 together with f(z(2), 1) b f(z(2), 2) and G(z(2), E, 1) z G(z(2), E, 2) imply (A6) for i = 1 which can therefore be eliminated from the optimization. Third, at the optimum, (14) must hold with equality. To see this, assume (14) holds with strict inequality. Consider increasing c(1), c(2) and c 0 all by the same small amount. This change would be feasible since it would not affect (A5) or (A6), would help (12) be satisfied, and would not violate (14) for a small change. It would increase the objective function (11), a contradiction of being at an optimum. Fourth, it then follows that for U p z 0, (13) must hold with equality. If not, c p could be reduced leading (14) to hold with strict inequality which leads to a contradiction. Then c p = U p can be substituted into (14) yielding (23).

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Fifth, assume (A5) for i = 2, (12), and (A6) for i = 2 all hold with strict inequality at the optimum. If a(2) = 0, then (A6) for i = 2 holds with equality. If a(2) N 0, then a small reduction in c(2) is feasible and raises the objective function. Thus, the solution to maximizing (11) subject to (12)–(14), (19), and (20), also must satisfy (21), (22), and (23). Conversely, it is straightforward that c p = U p and (21)–(23) imply (12)–(14), (19) and (20). Hence, the optimization problem in the Lemma is equivalent to the problem of finding the Implementable Pareto Frontier. 5 Proof of Theorem 2. First, we show that the system of inequalities given can be satisfied so that the result is not vacuous. From (2), (8), and the result in Theorem 1 that z*(1) N z*(2): Gð zTð1Þ; E; 1Þ  f ð zTð1Þ; 1Þ  Gð zTð2Þ; E; 1Þ þ f ð zTð2Þ; 1Þ Z zTð1Þ  Z zTð1Þ  BGð z; E; 1Þ Bf ð z; 1Þ BGð z; E; 2Þ Bf ð z; 2Þ   ¼ dz N dz Bz Bz Bz Bz zTð2Þ zTð2Þ ¼ Gð zTð1Þ; E; 2Þ  f ð zTð1Þ; 2Þ  Gð zTð2Þ; E; 2Þ þ f ð zTð2Þ; 2Þ:

ðA7Þ

Hence, G (z *(1), E , 1)  G (z *(1), E , 2) + f (z *(1), 2)  f (z *(1), 1) N G (z *(2), E , 1)  G(z*(2),E, 2) + f(z*(2), 2)  f(z*(2), 1). Since a*(1) z a*(2) and a*(1) N 0 from Theorem 1, this implies a* (1) ( G (z * (1), E, 1)  G(z *(1), E, 2) + f(z *(1), 2)  f(z *(1), 1)) N a * (2)( G(z * (2), E, 1)  G(z*(2), E, 2) + f(z*(2), 2)  f(z*(2), 1)). Therefore, the upper bound in (25) exceeds the lower one, so values of U 2 and U p exist which satisfy both inequalities. To satisfy all the restrictions, the lower bounds on U p and U 2 must be consistent with the upper bound in (25). That is:  ðhð1Þ þ hð2ÞÞf0 bhT þ hð1ÞaTð2ÞðGTð zTð2Þ; ET; 2Þ  Gð zTð2Þ; ET; 1Þ þ f ð zTð2Þ; 1Þ  f ð zTð2Þ; 2ÞÞ: ðA8Þ Substituting for h* and combining terms, (A8) is satisfied if the following equality holds: 2 X

  hðiÞ aTðiÞGð zTðiÞ; ET; iÞ þ ð1  aTðiÞÞw0 Nhð1ÞaTð1Þð f ð zTð1Þ; 1Þ  f0 Þ

i¼1

þ hð2ÞaTð2Þð f ð zTð2Þ; 2Þ  f0 Þ þ hð1ÞaTð2ÞðGð zTð2Þ; ET; 1Þ  Gð zTð2Þ; ET; 2Þ þ f ð zTð2Þ; 2Þ  f ð zTð2Þ; 1ÞÞ:

ðA9Þ

The right hand side of (A9) is positive since f 0 b (z*(i), i) by assumption. If the left hand side (which is aggregate income) is sufficiently large, then (A9) will hold and the restrictions on U 2 and U P can all be satisfied. Second, note that the nonnegativity conditions (16)–(18) for problem (15) to be valid are all satisfied. Condition (16) holds by assumption. The combination of (24) and f 0 b f (z*(2), 2) implies (17). Condition (18) can be rewritten as: U P þ ðhð1Þþ hð2ÞÞU 2 VhT þ hð1Þ½ aTð1Þf ð zTð1Þ; 1Þ þ ð1  aTð1ÞÞf0  þ hð1ÞU 2 :

ðA10Þ

If the upper bound in (A10) is at least as great as the upper bound in (25), then any values of U 2 and U P satisfying (25) will also satisfy (18). This will hold if: U 2 þ f0 zaTð2ÞðGð zTð2Þ; ET; 2Þ  Gð zTð2Þ; ET; 1Þ þ f ð zTð2Þ; 1Þ  f ð zTð2Þ; 2ÞÞ þ aTð1Þðf0  f ð zTð1Þ; 1ÞÞ: Since the right hand side of (A11) is negative, (24) implies (A11) and hence (18).

ðA11Þ

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Finally, if the solution to the problem in (15) is feasible in the problem in Lemma 1 given the restrictions in (24) and (25), then it is also optimal there since both problems have the same objective function while the problem in (15) has fewer constraints. From (12) holding with equality at the solution to (15): aTð2ÞðcTð2Þ  f ð zTð2Þ; 2Þ þ f0  cT0 Þ ¼ U 2 þ f0  cT0 :

ðA12Þ

Eq. (A12) can be substituted into (21), (22), and (23). Then (21) will be satisfied if: U 2  cT0 þ f0 z0

ðA13Þ

U 2  cT0 þ f0 zaTð1ÞðcTð1Þ þ Gð zTð1Þ; ET; 2Þ  Gð zTð1Þ; ET; 1Þ  f ð zTð1Þ; 2Þ  cT0 þ f0 Þ ðA14Þ while (22) becomes: aTð1ÞðcTð1Þ  f ð zTð1Þ; 1Þ  cT0 þ f0 ÞzU 2 þ f0  cT0 þ aTð2ÞðGð zTð2Þ; ET; 1Þ  Gð zTð2Þ; ET; 2Þ þ f ð zTð2Þ; 2Þ  f ð zTð2Þ; 1ÞÞ:

ðA15Þ

Eq. (23) can be solved for a*(1)c*(1) + (1  a*(1))c*0 which can then be substituted into (A14) and (A15) yielding the two inequalities in (25). Condition (A13) plus nonnegativity of c*0 is equivalent to (24). Note that for a*(2) N 0, c*0 can be any value in a range above 0. Hence, if (24) holds, then allowable values of c*0 exist for which (A13) holds. Thus (24) and (25) ensure that the solution to (15) satisfies (A13)–(A15) and hence (21)–(23). 5 Proof of Theorem 3. Consider any solution to the optimization in Lemma 1: c¯ 0, c¯ (i), a¯(i), and z¯ (i) Set z i = z¯ (i), a(z i ) = a¯ (i), f(z i ) = G(z¯ (i),E¯ ,i)  c¯ (i), r = U P, and s = c¯ 0  w 0. Substituting these values into (23) yields Eq. (27). Substituting into (21) and (22) yields:                a zi G zi ; E; i  f zi ; E; i  F zi þ 1  a zi ðw0 þ s  f0 Þza z j G z j ; E; i         f z j ; i  F z j Þ þ 1  a z j ðw0 þ s  f0 Þ; i ¼ 1; 2; j pi : ðA16Þ Thus z i maximizes (26) for type i over the schedule offered for the SCA. Similarly, (21), (22), and (A16) imply that:            a z i G z i ; E; i  f zi ; i  F z i þ 1  a z i ðw0 þ s  f0 Þzw0 þ s  f0 ; i ¼ 1; 2: ðA17Þ Individuals prefer to approach the SCA than to go directly to the alternative activity. Thus, the solution to the Lemma 1 optimization is sustained by the nonlinear schedules a(z) and F(z). 5   P2 Proof of Theorem 4. Set a1 ¼ aTð1Þ; a2 ¼ aTð2Þ; zˆ ¼ zTð1Þ; and t ¼  BeT j¼1 hð jÞaTð jÞ BZ BGðzTð jÞ;ET; jÞ . Consider (28) and (29) for i = 2 If the bounds on z are ignored, then the solutions to BE both are the same and, since Eq. (A3) is satisfied, equal z*(2). Since z*(1) N z*(2), the solution to (28) is z*(2) since it satisfies the constraint z*(2) b zˆ . Since the objective function in (29) is strictly concave and the constraint is violated at z*(2), the solution is the closest feasible value of z to z*(2), which is zˆ = z*(1). The overall solution will be z*(2) provided: aTð2ÞðGð zTð2Þ; ET; 2Þ  f ð zTð2Þ; 2Þ  T  tzTð2Þ  w0  s þ f0 ÞzaTð1ÞðGð zTð1Þ; ET; 2Þ  f ð zTð1Þ; 2Þ  T  tzTð1Þ  w0  s þ f0 Þ: ðA18Þ

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For the type 2 individual to prefer approaching the SCA than going directly to the alternative, the following must hold: aTð2ÞðGð zTð2Þ; ET; 2Þ  f ð zTð2Þ; 2Þ  T  tzTð2Þ  w0  s þ f0 Þz0:

ðA19Þ

For i = 1, from Eq. (A3), the solution to both (28) and (29) ignoring the bound on z is z*(1). This is then the solution to both (28) and (29). Since a*(1) z a*(2), the solution to (29) is the overall optimum if aTð1ÞðGð zTð1Þ; ET; 1Þ  f ð zTð1Þ; 1Þ  T  tzTð1Þ  w0  s þ f0 Þz0:

ðA20Þ

Obviously, this also implies that type 1’s will prefer to participate in the SCA over the alternative activity. If a*(2) = 0 then (A19) is automatic and (A18) and (A20) are the restrictions on T and s for the linear system to sustain the full information allocations, z*(i) and a*(i). If a*(2) N 0, G (z *(1), E *, 1)  f (z *(1), 1)  T  tz *(1) N G (z*(2), E *, 1) f (z *(2), 1)  T  tz*(2) N G(z*(2), E*, 2)  f(z*(2), 2)  T  tz*(2) where the first inequality follows since z*(1) maximizes G(z, E*, 1)  f(z, 1)  T  tz and the second follows from assumptions (2) and (8). These combined with (A19) imply (A20). Hence, (A18) and (A19) are the needed restrictions to sustain the full information allocations. Consider the case when a*(2) = 0. Using the government’s budget constraint that h(a)a(1)(T + z*(1)) = r + (h(1)(1  a*(1)) + h(2))s to eliminate T, (A18) and (A20) reduce to: hð1ÞaTð1Þ½Gð zTð1Þ; ET; 2Þ  f ð zTð1Þ; 2Þ  w0 þ f0   rVðhð1Þ þ hð2ÞÞsVhð1ÞaTð1Þ½Gð zTð1Þ; ET; 1Þ  f ð zTð1Þ; 1Þ  w0 þ f0   r:

ðA21Þ

Consider any U P z 0 and U 2 + f 0 z 0. When a*(2) = 0, c*0 = U 2 + f 0 so to sustain this allocation, r = U P and s = U 2 + f 0  w 0 must hold. Substituting these into (A21) yields (25). Hence, from Theorem 2, the linear tax sustains any point where the two frontiers coincide. For a*(2) N 0, the government budget balance equation is h (1)(T + tz *(1)) + h(2)a*(2)(T + tz*(2))= r + h(2)(1  a*(2))s since a*(1) must equal 1. Using this to substitute for s in (A18) and (A19) yields: r  ðhð1Þ þ hð2ÞÞtzTð1Þ þ hð2Þ½Gð zTð1Þ; ET; 2Þ  f ð zTð1Þ; 2Þ  aTð2ÞðGð zTð2Þ; ET; 2Þ  f ð zTð2Þ; 2ÞÞ þ ð1  aTð2ÞÞðw0  f0 ÞVðhð1Þ þ hð2ÞÞT Vr  hð1ÞtzTð1Þ  hð2ÞaTð2ÞtzTð2Þ þ hð2Þð1  aTð2ÞÞðGð zTð2Þ; ET; 2Þ  f ð zTð2Þ; 2Þ  w0 þ f0 Þ ðA22Þ Since z*(1) maximizes G(z, E*, 1)  f(z, 1)  tz, G(z*(1), E*,1)  f(z*(1), 1)  tz*(1) N G(z*(2), E*, 1)  f(z*(2), E*, 1)  tz*(2) and therefore the upper bound of (A22) exceeds the lower bound. On the full information frontier, h (1)U 1 + h (2)U 1 + U P = h*, so h(1)c*(1) = h(1)[U 1 + f(z*(1), 1)] = h*  h(2)U 2  U P +h(1)f(z*(1),1), while under the linear tax, h(1)c*(1) = h(1)[( G(z*(1), E*, 1)  T  tz*(1))]. For these to be equal, hð1ÞT ¼ U P þ hð2ÞU 2  hð1ÞtzTð1Þ  hð2Þ½aTð2ÞðGð zTð2Þ; ET; 2Þ  f ð zTð2Þ; 2ÞÞ þ ð1  aTð2ÞÞðw0  f0 Þ:

ðA23Þ

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Substituting this in Eq. (A23) yields: hT þ hð1Þ½Gð zTð1Þ; ET; 1Þ  Gð zTð1Þ; ET; 1Þ  f ð zTð1Þ; 2Þ þ f ð zTð1Þ; 1ÞVU P þ ðhð1Þ þ hð2ÞÞU 2 VhT þ hð1Þ½Gð zTð2Þ; ET; 2Þ  Gð zTð2Þ; ET; 1Þ þ f ð zTð2Þ; 1Þ  f ð zTð2Þ; 2Þ þ hð1Þ½Gð zTð2Þ; ET; 1Þ f ð zTð2Þ; 1Þ tzTð2ÞGð zTð1Þ; ET; 1Þ þ f ð zTð1Þ; 1Þþ tzTð1Þ: ðA24Þ P

2

Values of U and U on the Full Information Pareto Frontier satisfying (A24) can be sustained by the linear fee schedule provided nonnegativity also holds. c*(1) = G(z*(1), E*, 1)  T  tz*(1) and c*(2) = G(z*(2), E*, 2)  T  tz*(2) will be nonnegative from (A20) and (A21) if c 0* = w 0 + s z 0 since f 0 b min [ f(z*(1), 1), f(z*(2), 2)] is assumed. To consider w 0 + s z 0 substitute for s from the government budget constraint and then substitute T from combining U 1 = G(z*(1), E*, 1)  f(z*(1), 1)  T  tz*(1) given the linear fee schedule and h(1)U 1 = h ( 1 ) [ G ( z * ( 1 ) , E * , 1 )  f ( z * ( 1 ) , 1 ) ] + h ( 2 ) [ a* ( 2 ) ( G ( z * ( 2 ) , E * , 2 )  f ( z * ( 2 ) , 2 ) ) + (1  a*(2))(w 0  f 0)]  h(2)U 2  U P. This yields the following condition for w 0 + s z 0: U P þ ðhð1Þ þ hð2ÞÞU 2 zhð1Þ½Gð zTð2Þ; ET; 2Þ  f ð zTð2Þ; 2Þ  Gð zTð1Þ; ET; 1Þ   þ f ð zTð1Þ; 1Þ þ tzTð1Þ  tzTð2Þ þ hT  ½ð1  aTð2ÞÞ=aTð2Þhð1Þ U 2 þ f0 :

ðA25Þ

It is straightforward that the upper bound in (A24) exceeds the lower bounds in (A24) and (A25) and is less than the upper bound in (25). The lower bound in (A25) is below the lower bound in (A24) for a*(2) near zero but above when a*(2) is near one. When it is above, nonnegativity of c 0* restricts the set of utilities at which a linear schedule can sustain the full information optimum. In any case, since the upper bound in (A24) is below that in (25), with a linear schedule the government can sustain fewer points on the full information frontier than with a nonlinear schedule. In addition, the lower bounds of 0 for U P and  f 0 for U 2 must be consistent with the upper bound in (A24), while a tighter condition than (A9) can be satisfied if productivity is sufficiently great. 5 Proof of Lemma 2. In any circumstance, (31) for i = 1 and (32) for i = 2 will hold with equality at the optimum. If a 1 = 0, then, since G(z 2, E, 1)  f(z 2, 1) N G(z 2, E, 2)  f(z 2, 2), (31) can only hold if a 2 = 0. All of (31) and (32) then hold with equality. Consider a 2 N 0. Condition (32) for i = 1 is implied by (31) for i = 1 and (32) for i = 2 and therefore can be dropped. Then, (31) for i = 1 must hold with equality at the optimum since, if it did not, F 1 could be increased without violating any constraint. If a 2 = 0 then (32) for i = 2 will hold with equality as required. Consider a 2 N 0 and assume (32) for i = 2 holds with strict inequality. Consider small increases in both F 1 and F 2 such that a 1F 1  a 2F 2 remains constant. This change would be feasible since (31) would not be affected and (32) for i = 2 would continue to hold but the objective function would increase. Hence, (32) for i = 2 must hold with equality. Then solving (32) for i = 2 holding with equality yields Eq. (39). Substituting Eq. (39) into (31) for i = 1 and solving that condition holding with equality yields Eq. (38). Substituting Eqs. (38) and (39) into (35) and (31) for i = 2 yields (36) and (37), respectively as required. 5 Proof of Theorem 5. Instead of maximizing U 1 with Pareto constraints on U 2 and U P the Implementable Pareto Frontier can be found by maximizing U P subject to Pareto constraints on U 1 and U 2 as well as the resource constraint (14), the self-selection constraints (19), the participation constraints (20), and nonnegativity conditions. Dropping the Pareto constraints and

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solving the problem in their absence yields the maximum possible value of U P on the frontier. Clearly, (14) must hold with equality since if not c p could always be increased. Then (14) can be solved for c p and substituted in the objective function. Among the constraints, (20) for i = 1 is implied by (19) for i = 1 and (20) for i = 2, so they can be dropped. Consider (19) for i = 1 at the optimum. If a(1) N 0 and that condition holds with strict inequality, then c(1) could be reduced leading to an increase in c p, a contradiction. If a(1) = 0 then a(2) = 0 must also hold in order for (19) for i = 1 and (20) for i = 2 to both hold. When a(1) = a(2) = 0, (19) for i = 1 holds with equality. Next consider (20) for i = 2 at the optimum. If a(2) = 0, this holds with equality. If a(2) N 0, a small decrease in c(1) and c(2) holding a(2)c(2)  a(1)c(1) constant would be feasible and would allow c p to increase. Then (19) for i = 1 and (29) for i = 2 holding with equality can be solved for c(1) and c(2) and substituted into the objective function and (19) for i = 2. This yields the optimization: 2 X Max hðiÞaðiÞðGð zðiÞ; E; iÞ  f ð zðiÞ; iÞ þ f0  w0 Þ þ hð1Það2ÞðGð zð2Þ; E; 1Þ i¼1

 Gð zð2Þ; E; 2Þ þ f ð zð2Þ; 2Þ  f ð zð2Þ; 1ÞÞ þ ðhð1Þ þ hð2ÞÞðw0  c0 Þ ðA26Þ s:t: að1ÞðGð zð1Þ; E; 1Þ  Gð zð1Þ; E; 2Þ  f ð zð1Þ; 1Þ þ f ð zð1Þ; 2ÞÞzað2ÞðGð zð2Þ; E; 1Þ  Gð zð2Þ; E; 2Þ  f ð zð2Þ; 1Þ þ f ð zð2Þ; 2ÞÞ:

ðA27Þ

Since c 0 = w 0 + s, this optimization is identical to that in Lemma 2. Hence, the solution to the owner’s optimization is a point on the Implementable Pareto Frontier. Assume that a(2) = 0 in the solution to the Full Information optimization. Set a 2 = 0 in (36) and (37). Then (37) holds for all a 1, z 1, and z 2 so can be dropped. The first order conditions with respect to a 1, z 1, and z 2 for (15) given in the Proof of Theorem 1 are the same as those for (36). Hence, given a 2 = 0, the other variables will be at their full information values. At those values, the constraint in (37) holds with strict inequality at a 2 = 0. Hence, the first order condition with respect to a 2 is also the same in the full information and private owner optimizations. Finally, consider the case when a(2) N 0. Assume a i = a*(i) since if not, the solution for (36) and (37) differs from that in the full information optimization. Since a 1 N a 2 then holds, (37) holds with strict inequality so can be deleted. Take the first order conditions with respect to z 1 and z 2: "   2 X BG zj ; E; j Be BU P BGðz1 ; E; 1Þ Bf ðz1 ; 1Þ ¼ hð1Þa1  þ hð jÞaj Bz1 Bz1 BZ Bz1 BE j¼1   BGðz2 ; E; 2Þ BGðz2 ; E; 1Þ Be  þ hð1Þa2 ¼0 ðA28Þ BE BE BZ "   2 X BG zj ; E; j Be BU P BGðz2 ; E; 2Þ Bf ðz2 ; 2Þ ¼ hð2Þa2  þ hð jÞaj BE Bz2 Bz2 BZ Bz2 j¼1   BGðz2 ; E; 2Þ BGðz2 ; E; 1Þ Be  þ hð1Þa2 BE BE BZ   BGðz2 ; E; 2Þ Bf ðz2 ; 2Þ BGðz2 ; E; 1Þ Bf ðz2 ; 1Þ þ hð1Þa2   þ ¼ 0: ðA29Þ Bz2 Bz2 Bz2 Bz2 If the full information optimum is to be sustained, then Eq. (A3) must hold. If it does, then Eqs. (A28) and (A29) reduce to:  BGðz2 ; E; 2Þ BGðz2 ; E; 1Þ Be  ¼0 ðA30Þ hð1Þa2 BE BE BZ

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BGðz2 ; E; 2Þ BGðz2 ; E; 1Þ Be  hð1Þa2 BE BE BZ  BGðz2 ; E; 2Þ Bf ðz2 ; 2Þ BGðz2 ; E; 1Þ Bf ðz2 ; 1Þ þ hð1Þa2   þ ¼ 0: Bz2 Bz2 Bz2 Bz2

ðA31Þ

z2 ;E;2Þ ðz2 ;2Þ z2 ;E;1Þ ðz2 ;1Þ  Bf Bz  BGðBz þ Bf Bz b0, then both Eqs. (A30) and (A31) cannot Since BGðBz 2 2 2 2 hold. Hence, the full information optimum cannot be sustained when a*(2) N 0. 5

Proof of Theorem 6. That T 1 and T 2 do not affect utilities follows immediately from Lemma 2. These taxes on the participants in the SCA do not appear in (36) or (37), thus do not affect the solution. The values of the utilities as functions of s follow from substituting Eqs. (38) and (39) into the definitions of U i uai ðGðzi ; E; iÞ  f ðzi ; iÞ  F i  T i Þ þ ð1  ai Þðw0 þ s  f0 Þ. 5 References Allen, D., 1991. What are transaction costs? Research in Law and Economics 14, 1 – 18. Baumol, W., Bradford, D., 1972. Detrimental externalities and non-convexity of the production set. Economica 39, 160 – 176. Baumol, W., Oates, W., 1988. The Theory of Environmental Policy, 2nd ed. Cambridge University Press, New York. Boadway, R., Keen, M., 1993. Public goods, self-selection and optimal income taxation. International Economic Review 34, 463 – 478. Brito, D., Intriligator, M., Sheshinski, E., 1997. Privatization and the distribution of income in the commons. Journal of Public Economics 64, 181 – 205. Buchanan, J., Stubblebine, W., 1962. Externality. Economica 29, 371 – 384. Coase, R., 1960. The problem of social cost. Journal of Law and Economics 3, 1 – 44. Coase, R., 1988. The Firm, the Market, and the Law. University of Chicago Press, Chicago. Cremer, H., Gahvari, F., Ladoux, N., 2001. Second-best pollution taxes and the structure of preferences. Southern Economic Journal 68, 258 – 280. Demsetz, H., 1968. Why regulate utilities? Journal of Law and Economics 11, 55 – 68. Dixit, A., Olson, M., 2000. Does voluntary participation undermine the Coase Theorem? Journal of Public Economics 3, 309 – 335. Farrell, J., 1987. Information and the Coase Theorem. Journal of Economic Perspectives 1, 113 – 129. Frech III, H., 1979. The extended Coase Theorem and long run equilibrium: the nonequivalence of liability rules and property rights. Economic Inquiry 17, 254 – 268. Hamilton, J., Slutsky, S., 2000. Decentralizing allocation and distribution by separation with information transfers. Journal of Public Economic Theory 2, 289 – 318. Hamilton, J., Slutsky, S., 2004. Optimal Nonlinear Taxation with a Finite Population, working paper, Department of Economics, University of Florida. Hamilton, J., Sheshinski, E., Slutsky, S., 1989. Production externalities and long-run equilibria: bargaining and Pigovian taxation. Economic Inquiry 27, 453 – 471. Hurwicz, L., 1999. Revisiting externalities. Journal of Public Economic Theory 1, 225 – 245. Laffont, J.-J., 2000. Incentives and Political Economy. Oxford University Press, New York. Ledyard, J.O., 1971. The relation of optima and market equilibrium with externalities. Journal of Economic Theory 3, 54 – 65. Mas-Colell, A., Whinston, M., Green, J., 1995. Microeconomic Theory. Oxford University Press, New York. Plott, C., 1966. Externalities and corrective taxes. Economica N.S. 33, 84 – 87. Posner, R., 1977. Economic Analysis of Law, 2nd ed. Little Brown, Boston. Sadka, E., 1976. On income distribution, incentive effects and optimal income taxation. Review of Economic Studies 43, 261 – 268. Samuelson, W., 1985. A comment on the Coase Theorem. In: Roth, A. (Ed.), Game Theoretic Models of Bargaining. Cambridge University Press, New York.

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Starrett, P., Zeckhauser, R., 1974. Treating external diseconomics — markets or taxes? In: Pratt, J. (Ed.), Statistical and Mathematical Aspects of Pollution Problems. Marcel Dekker, New York, pp. 65 – 84. Stiglitz, J., 1982. Self-selection and Pareto efficient taxation. Journal of Public Economics 17, 213 – 240. Stiglitz, J., 1994. Whither Socialism? MIT Press, Cambridge. Turvey, R., 1963. On divergences between social cost and private cost. Economica 30, 309 – 313.