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

a Department of Economics, Rice University, 6100 Main, Houston, TX 77005, USAb Department of Economics, PO Box 117140, University of Florida, Gainesville, FL 32611, USA

c Department of Economics, UCLA, Los Angeles, CA 90095-1477, USAd Department of Economics, The Hebrew University of Jerusalem, Jerusalem, Israel

e Centro de Investigacion y Docencia Economicas (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

0047-2727/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.jpubeco.2005.08.004

* Corresponding author.

E-mail address: [email protected] (J.H. Hamilton).

D.L. Brito et al. / Journal of Public Economics 90 (2006) 871–895872

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.

D.L. Brito et al. / Journal of Public Economics 90 (2006) 871–895 873

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 i

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 tha

instead of being independent producers, the workers are hired by the owner of the infrastructure who acts as a profi

maximizing producer. The outcome will be constrained efficient, but the ability of the government to redistribute across

different types of workers will be limited.

f

.

t

t


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 f0. The output per worker in this

activity is w0. The utility from entering this activity is a worker’s consumption less the disutility

of effort, c0� f0.

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:

y ið Þ ¼ G z ið Þ;E; ið Þ 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:

BG z;E; 1ð ÞBz

zBG z;E; 2ð Þ

Bz: ð2Þ

Combining condition (2) with the condition that G(0,E, i)=0 implies that G(z,E, 1)zG(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). Let a(i), 0Va(i)V1, denote the fraction of type i individuals who enter the SCA. Let

M ¼P2

i¼1 a ið Þh ið Þ be the level of participation in the SCA, and let Z ¼P2

i¼1 a ið Þh ið Þz ið Þ bethe total effort in the SCA. The externality E is a nondecreasing function of these aggregates:

E ¼ e M ; Zð 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 ee aa; zzÞ:ð 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.


Using Eq. (3), the derivatives of e with respect to a(i) and z(i) are:

Bee

Bz ið Þ ¼ a ið Þh ið Þ BeBZ

ð5Þ

Bee

Ba ið Þ ¼ h ið Þ Be

BMþ Be

BZz ið Þ

� �: ð6Þ

The utility function for an individual of type i in the SCA is:

U c ið Þ; z ið Þ; ið Þ ¼ c ið Þ � f z ið Þ; ið Þ 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 z; 1ð ÞBz

bBf z; 2ð Þ

Bz: ð8Þ

Condition (8) and f(0,1)= f(0,2) imply that f(z, 1)b f(z, 2) for any z N0.

To simplify the analysis, we make two additional technical assumptions on net productivity:

BG 0;E; ið ÞBz

� Bf 0; ið ÞBz

¼l; i ¼ 1; 2 ð9Þ

9 zz such that G zz; 0; 1ð Þ � f zz; 1ð Þ N w0 � f0: 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, cp.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.


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 0ba(i)b1, 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:

Maxc ið Þ;z ið Þ;a ið Þ;c0;cp

h 1ð Þ a 1ð Þ c 1ð Þ � f z 1ð Þ; 1ð Þð Þ þ 1� a 1ð Þð Þ c0 � f0ð Þ½ � ð11Þ

s:t: a 2ð Þ c 2ð Þ � f z 2ð Þ; 2ð Þð Þ þ 1� a 2ð Þð Þ c0 � f0ð ÞzU 2 ð12Þ

cpzUp ð13Þ

X2i¼1

h ið Þ a ið Þ c ið Þ � G z ið Þ;E; ið Þð Þ þ 1� a ið Þð Þ c0 � w0ð Þ½ � þ cpV0 ð14Þ

c ið Þz0; z ið Þz0; cpz0; c0z0; 0Va ið ÞV1

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 c0 could be increased. The Pareto constraints will also hold with equality if the utility levels

are high enough that U pz 0 and U2 þ a 2ð Þf z 2ð Þ; 2ð Þ þ 1� a 2ð Þð Þf0z0. Under these condi-

tions, we can use constraints (12), (13), and (14) to eliminate c(i), c0, and cp. This reduces the

optimization problem to:

Maxz ið Þ;a ið Þ

X2i¼1

h ið Þ a ið Þ G z ið Þ;E; ið Þ�f z ið Þ; ið Þð Þþ 1�a ið Þð Þ w0�f0ð Þ½ ��h 2ð ÞU 2�Up ð15Þ

s:t: 0Va ið ÞV1 and z ið Þz0

in the region defined by

Upz0 ð16ÞU2 þ a 2ð Þf z 2ð Þ; 2ð Þ þ 1� a 2ð Þð Þf0z0 ð17ÞUp þ h 2ð ÞU 2Vhþ h 1ð Þ a 1ð Þf z 1ð Þ; 1ð Þ þ 1� a 1ð Þð Þf0½ � ð18Þ

where huP2

i¼1 h ið Þ a ið Þ G z ið Þ;E; ið Þ � f z ið Þ; ið Þð Þ þ 1� a ið Þð Þ w0 � f0ð Þ½ �. Conditions (16)–(18)ensure that cp, c(i), and c0 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 Up and U2 are such

that constraints (12) and (13) hold with equality, the optimal z*(i) and a*(i) are independent ofU2 and Up, so that only c*(i),c0*, and cp* vary along the frontier. Furthermore, z*(1)N z*(2) and

either a*(2)=0 and a*(1)N0 or a*(2)N0 and a*(1)=1.


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 SCAwhile 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:

a ið Þ c ið Þ � f z ið Þ; ið Þð Þ þ 1� a ið Þð Þ c0 � f0ð Þza jð Þ c jð Þ þ G z jð Þ;E; ið Þ � G z jð Þ;E; jð Þð� f z jð Þ; ið ÞÞ þ 1� a jð Þð Þ c0 � f0ð Þ; i ¼ 1; 2; jp 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

a ið Þ c ið Þ � f z ið Þ; ið Þð Þ þ 1� a ið Þð Þ c0 � f0ð Þzc0 � f0; i ¼ 1; 2 : ð20Þ

9 When a*(2)=0,c*(2) is irrelevant and c0*=U2+ f0. When a*(2)N0, c*(2) and c0* are not separately determined with

c0* able to take any value between 0 and (U2+a*(2)f(z*(2),2)+ (1�a*(2))f0) / (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)VG(z(1),E, 1), then even if c(1)N0, 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 0ba(i)b1, 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.


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 Upz0, the ex ante Implementable Pareto Frontier can be found as the solution to

the following optimization:

Maxc0;c ið Þ;a ið Þ;z ið Þ

h 1ð Þ a 1ð Þ c 1ð Þ � f z 1ð Þ; 1ð Þð Þ þ 1� a 1ð Þð Þ c0 � f0ð Þ½ �

s:t: a 2ð Þ c 2ð Þ � f z 2ð Þ; 2ð Þ � c0 þ f0ð Þ ¼ max½0;U 2 � c0 þ f0; a 1ð Þðc 1ð Þ þ G z 1ð Þ;E; 2ð Þ� G z 1ð Þ;E; 1ð Þ � f z 1ð Þ; 2ð Þ � c0 þ f0Þ� ð21Þ

a 1ð Þ c 1ð Þ � f z 1ð Þ; 1ð Þ � c0 þ f0ð Þza 2ð Þðc 2ð Þ þ G z 2ð Þ;E; 1ð Þ � G z 2ð Þ;E; 2ð Þ � f z 2ð Þ; 1ð Þ� c0 þ f0Þ ð22Þ

X2i¼1

h ið Þ a ið Þ c ið Þ � G z ið Þ;E; ið Þð Þ þ 1� a ið Þð Þ c0 � w0ð Þ½ � þ Up ¼ 0 ð23Þ

c ið Þz0; c0z0; z ið Þz0; 0Va ið ÞV1 :

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 c0N0, 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 Up and U2, 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 f0 b

min [f(z*(1),1), f(z*(2),2)] and that aggregate income is sufficiently large.13 Then there exist

values of U2 and Up with Upz0 and which satisfy the following:

U2 þ f0z0 ð24ÞhTþ h 1ð ÞaT 1ð Þ G zT 1ð Þ;ET; 2ð Þ � G zT 1ð Þ;ET; 1ð Þ þ f zT 1ð Þ; 1ð Þ � f zT 1ð Þ; 2ð Þð ÞVU p

þ h 1ð Þ þ h 2ð Þð ÞU 2VhTþ h 1ð ÞaT 2ð ÞðG zT 2ð Þ;ET; 2ð Þ � G zT 2ð Þ;ET; 1ð Þ þ f zT 2ð Þ; 1ð Þ� f zT 2ð Þ; 2� ��

ð25Þ

13 The required bound on aggregate income is given in (A9).

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.

•

•

U1

U2

Full Information Pareto Frontier _ _ _ _ _ _ _ Implementable Pareto Frontier

Fig. 1. For a fixed level of Up, 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 Up and U2 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 0ba(i)b1 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

schedulesF(z) and a(z) for those whowish to enter the SCA. Individuals select a level of effort andbuy 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

a zð Þ G z;E; ið Þ � f z; ið Þ � F zð Þð Þ þ 1� a zð Þð Þ w0 � f0 þ sð Þ ð26Þ


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, (z1,F(z1), a(z1)) and (z2, F(z2), a(z2)), where each option is intended for the respective

type. Overall, the government’s budget must balance:

X2i¼1

h ið Þa zið ÞF zið Þ ¼X2i¼1

h ið Þ 1� a zið Þð Þsþ r : ð27Þ

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)) andgiving 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)=a1 if zz z and a(z)=a2 if z b z, for some z. The optimal

choice of effort for type i can be found by solving the following two optimizations and taking

the better solution:14

Maxz

a2 G z;E; ið Þ � f z; ið Þ � T � tzð Þ þ 1� a2ð Þ w0 þ s� f0ð Þ s:t:zV zz ð28Þ

Maxz

a1 G z;E; ið Þ � f z; ið Þ � T � tzð Þ þ 1� a1ð Þ w0 þ s� f0ð Þ s:t:zz zz : ð29Þ

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, a1, a2, z, 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)N0, 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, T1 and T2, may allow more redistribution and thus

allow more of the Full Information Pareto Frontier to be achieved.

14 Since z V z is specified in (28), instead of z b z, this optimization actually finds the supremum in the region with z b z

rather than the maximum. Since z = z in (28) will be ruled out as the overall optimum, this is done at no cost to ensure that

a solution to (28) exists.


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 T1 and T2 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. T1 can differ from T2 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,

(a1, z1, F1) and (a2, z2, F

2) subject to self-selection constraints. The owner seeks to maximize

total income subject to self-selection and participation constraints.

Maxai;zi;Fi

h 1ð Þa1F1 þ h 2ð Þa2F2 þ r ð30Þ

s:t: ai G zi;E; ið Þ � f zi; ið Þ � Fi � Ti � w0 � sþ f0� �

zajðG zj;E; i� �

� f zj; i� �

� Fj

�Tj � w0 � sþ f0Þ; i ¼ 1; 2; jpi ð31Þ

ai G zi;E; ið Þ � f zi; ið Þ � Fi � Ti � w0 � sþ f0� �

z0; i ¼ 1; 2 ð32Þ

16 In effect, the government looks ahead to the owner’s choices and sets taxes T1 for those who choose the better option

for type 1’s and T2 for those who choose the better option for type 2’s.

15 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.


G zi;E; ið Þ � Fi � Tiz0; 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, f0bmin[ f(z1,1), f(z2,1)] is assumed, then constraint (33)

is implied by constraint (32) as long as c0=w0+ sz0 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 a1 and a2,

h 1ð Þa1T1 þ h 2ð Þa2T2 ¼ h 1ð Þ 1� a1ð Þ þ h 2ð Þ 1� a2ð Þð Þsþ r: ð34Þ

Deviations from the equilibrium choices by the owner in setting a1 and z1 either unbalance

the budget if T1, T2, s, and r are taken as fixed or prevent using differentiated taxes T1p T2 if

z1= z2. 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 T1, T2, 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:

Maxai;zi;Fi

h 1ð Þa1 F1 þ T 1 þ s� �

þ h 2ð Þa2 F2 þ T2 þ s� �

� h 1ð Þ þ h 2ð Þð Þs ð35Þ

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 ai and zi can be found as the solution to the following

optimization:

Maxai;zi

X2i¼1

h ið Þai G zi;E; ið Þ � f zi; ið Þ � w0 þ f0ð Þ

þ h 1ð Þa2 G z2;E; 2ð Þ � f z2; 2ð Þ � G z2;E; 1ð Þ þ f z2; 1ð Þð Þ � h 1ð Þ þ h 2ð Þð Þs ð36Þ

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 T1, T2, 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 T1, T2 or s which in turn can lead to an inefficient

outcome. When these are fixed and r adjusts, this does not occur.

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(z2,1)zG(z2,2). For 2’s who act as 1’s, G(z1,2)�F1�T1b0 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.


s:t: a1 G z1;E; 1ð Þ � f z1; 1ð Þ � G z1;E; 2ð Þ þ f z1; 2ð Þð Þza2ðG z2;E; 1ð Þ � f z2; 1ð Þ� G z2;E; 2ð Þ þ f z2; 2ð ÞÞ ð37Þ

and the optimal fees are given by:

a1F1 ¼ a1 G z1;E; 1ð Þ � f z1; 1ð Þ � T1 � s� w0 þ f0

� �� a2 Gðz2;E; 1ð Þ � f z2; 1ð Þ

� G z2;E; 2ð Þ þ f z2; 2ð ÞÞ ð38Þa2F

2 ¼ a2 G z2;E; 2ð Þ � f z2; 2ð Þ � T2 � s� w0 þ f0� �

ð39Þ

Let a1 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 c0=w0+ s and cp (which equals the value of the objective

function) are nonnegative. These hold if

� h 1ð Þ þ h 2ð Þð Þw0V h 1ð Þ þ h 2ð Þð ÞsVX2i¼1

h ið Þai G zi; ið Þ � w0 þ f0ð Þ

þ h ið Þa2 G z2; E; 2� �

� f z2; 2ð Þ � G z2; E; 1� �

þ f z2; 1ð Þ� �

: 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(T1,T2, s), and Up(T1,T2, s) be the utilities associated

with the solution to the private owner’s optimization for any T1, T2, and s.

Theorem 5. For any T1, T2 and s,(U1(T1,T2, s), U2(T1,T2, s), Up(T1,T2, s)) is on the

Implementable Pareto Frontier. If the economy is such that a*(2)=0 in the solution to the

Full Information Pareto optimization, then (U1(T1,T2, s), U2(T1,T2, s), Up(T1,T2, s)) is also on

the Full Information Pareto Frontier. If a*(2)N0, then (U1(T1,T2, s), U2(T1,T2, s), Up(T1,T2, s))

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)N0. 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 full

information 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.

-


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 T1 and T2 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 sð Þ ¼ a2 G z2; E; 1� �

� f z2; 1ð Þ � G z2; E; 2� �

þ f z2; 2ð Þ� �

þ w0 � f0 þ s

U 2 sð Þ ¼ w0 � f0 þ s

Up ¼

X2i¼1

h ið Þai G zi; E; i� �

� f zi; ið Þ � w0 þ f0� �

þ h ið Þa2 G z2; E; 2� �

� f z2; 2ð Þ � G z2; E; 1� �

þ f z2; 1ð Þ� �

� h 1ð Þ þ h 2ð Þð Þs :

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 a2(G(z2, E, 1)� f(z2,1)�G(z2, E, 2)+ f(z2,2)). Any attempt to

change their relative utilities by changing the taxes T1 and T2 is completely negated by exactly

offsetting changes in the owner’s fees F1 and F2.22 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 redistributeto 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 U1–U2 is

constant is due to the quasi-linearity of utility. With a more general specification, U1–U2 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.

UP = 0

UP > 0

U2_ f0

U1

Fig. 2. When a*(2)N0 the dashed line is the set of achievable utilities under private ownership. U2 has a lower bound o

� f0.


f

when a2N0. 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

SCAwould 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


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.

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.

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.


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 U2 and

Up which enter the objective function only as additive constants. Consider any values of U2 and

Up at which the inequalities which guarantee nonnegative consumptions are satisfied at the

solution values z*(i) and a*(i). For such U2 and Up, 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)U1, with respect to z(i) and a(i).

h 1ð Þ BU1

Bz ið Þ ¼ h ið Þa ið Þ BG z ið Þ;E; ið ÞBz ið Þ � Bf z ið Þ; ið Þ

Bz ið Þ

�

þX2j¼1

h jð Þa jð Þ BG z jð Þ;E; jð ÞBE

Be

Bz ið Þ ; i ¼ 1; 2 ðA1Þ

h 1ð Þ BU1

Ba ið Þ ¼ h ið Þ G z ið Þ;E; ið Þ � f z ið Þ;E; ið Þ � w0 � f0ð Þ½ �

þX2j¼1

h jð Þa jð Þ BG z ið Þ;E; jð ÞBE

Be

Ba ið Þ ; i ¼ 1; 2 : ðA2Þ

From the Inada conditions in (9), at the optimum z*(i)N0 and BU1 /Bz*(i) in Eq. (A1) must

equal 0. Substituting Eq. (5), this reduces to:

a ið Þ BG z ið Þ;E; ið ÞBz ið Þ � Bf z ið Þ; ið Þ

Bz ið Þ þ Be

BZ

X2j¼1

h jð Þa jð Þ BG z ið Þ;E; ið ÞBE

� !¼ 0; i ¼ 1; 2

ðA3ÞConsider 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) wereto be increased above 0. This assumption can be justified on the same grounds as used to justify subgame perfect equilibrium
.


Given the bounds on a(i), the first order condition with respect to a(i) is

BU1 /Ba(i)V0 if a(i)=0, BU1 /Ba(i)=0 if 0ba(i)b1, and BU1 /Ba(i)z0 if a(i)=1, where

BU1 /Ba(i) is defined from Eq. (A2) after substituting Eq. (6). To analyze this, consider

BU1 /Ba(1)�(BU1 /Ba (2))(h(1) /h(2)):

BU1

Ba 1ð Þ �h 1ð Þh 2ð Þ

BU1

Ba 2ð Þ ¼ G z 2ð Þ;E; 1ð Þ � f z 2ð Þ; 1ð Þ � G z 2ð Þ;E; 2ð Þ � f z 2ð Þ; 2ð Þð Þ½ �

þ G z 1ð Þ;E; 1ð Þ � f z 1ð Þ; 1ð Þ þ z 1ð Þ BeBZ

X2i¼1


" #

��G z 2ð Þ;E; 1ð Þ� f ðz 2ð Þ; 1Þþ z 2ð Þ Be

BZ

X2j¼1


�: ðA4Þ

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)NG(z(2),E, 2)� f(z(2),2). Consider the function G(z,E*,1)� f(z, 1)+ zb where E* is

fixed at e(z*, a*) and b is a constant set equal to BeTBZ

P2j¼1 h jð ÞaT jð Þ BG zT jð Þ;ET; jð Þ

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, theoptimal z*(1) also maximizes it. Therefore,

G zT 1ð Þ;ET; 1ð Þ � f zT 1ð Þ; 1ð Þ þ zT 1ð Þ BeTBZ

X2j¼1

h jð ÞaT jð Þ BG zT jð Þ;ET; jð ÞBE

NG zT 2ð Þ;ET; 1ð Þ � f zT 2ð Þ; 1ð Þ þ zT 2ð Þ BeTBZ

X2j¼1

h jð ÞaT jð Þ BG zT jð Þ;ET; jð ÞBE

:

Combining these inequalities with Eq. (A4) yields BU1 /Ba(1)N (h(1) /h(2))(BU1 /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)N0 then

implies a*(1)=1 as required. 5

Proof of Lemma 1. First, (19) and (20) can be simplified by subtracting c0� f0 from both sides

of each inequality:

a ið Þ c ið Þ � f z ið Þ; ið Þ � c0 þ f0ð Þza jð Þðc jð Þ þ G z jð Þ;E; ið Þ � G z jð Þ;E; jð Þ � f z jð Þ; ið Þ� c0 þ f0Þ; i ¼ 1; 2; jpi ðA5Þ

aðiÞðcði Þ � f ðzðiÞ; iÞ � c0 þ f0Þz0; 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)zG(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 c0 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 Upz0, (13) must hold with equality. If not, cp could be

reduced leading (14) to hold with strict inequality which leads to a contradiction. Then cp=Up

can be substituted into (14) yielding (23).


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)N0, 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 cp=Up 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):

G zT 1ð Þ;E; 1ð Þ � f zT 1ð Þ; 1ð Þ � G zT 2ð Þ;E; 1ð Þ þ f zT 2ð Þ; 1ð Þ

¼Z zT 1ð Þ

zT 2ð Þ

BG z;E; 1ð ÞBz

� Bf z; 1ð ÞBz

� dzN

Z zT 1ð Þ

zT 2ð Þ

BG z;E; 2ð ÞBz

� Bf z; 2ð ÞBz

� dz

¼ G zT 1ð Þ;E; 2ð Þ � f zT 1ð Þ; 2ð Þ � G zT 2ð Þ;E; 2ð Þ þ f zT 2ð Þ; 2ð Þ: ðA7Þ

Hence, G (z*(1), E , 1)�G (z*(1), E , 2) + f (z*(1), 2)� f (z*(1), 1) NG (z*(2), E , 1)�G(z*(2),E, 2)+ f(z*(2), 2)� f(z*(2),1). Since a*(1)za*(2) and a*(1)N0 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 ofU2 and Up exist which satisfy both inequalities. To satisfy all the restrictions, the

lower bounds on Up and U2 must be consistent with the upper bound in (25). That is:

� h 1ð Þ þ h 2ð Þð Þf0bhTþ h 1ð ÞaT 2ð ÞðGT zT 2ð Þ;ET; 2ð Þ � G zT 2ð Þ;ET; 1ð Þ þ f zT 2ð Þ; 1ð Þ� f zT 2ð Þ; 2ð ÞÞ: ðA8Þ

Substituting for h* and combining terms, (A8) is satisfied if the following equality holds:

X2i¼1

h ið Þ aT ið ÞG zT ið Þ;ET; ið Þ þ 1� aT ið Þð Þw0� �

Nh 1ð ÞaT 1ð Þ f zT 1ð Þ; 1ð Þ � f0ð Þ

þ h 2ð ÞaT 2ð Þ f zT 2ð Þ; 2ð Þ � f0ð Þ þ h 1ð ÞaT 2ð ÞðG zT 2ð Þ;ET; 1ð Þ � G zT 2ð Þ;ET; 2ð Þþ f zT 2ð Þ; 2ð Þ � f zT 2ð Þ; 1ð ÞÞ: ðA9Þ

The right hand side of (A9) is positive since f0b (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 U2

and UP 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 f0b f (z*(2), 2)

implies (17). Condition (18) can be rewritten as:

UP þ h 1ð Þþ h 2ð Þð ÞU 2VhTþ h 1ð Þ aT 1ð Þ f zT 1ð Þ; 1ð Þ þ 1� aT 1ð Þð Þf0½ � þ h 1ð ÞU 2: ðA10ÞIf the upper bound in (A10) is at least as great as the upper bound in (25), then any values of

U2 and UP satisfying (25) will also satisfy (18). This will hold if:

U2 þ f0zaT 2ð Þ G zT 2ð Þ;ET; 2ð Þ � G zT 2ð Þ;ET; 1ð Þ þ f zT 2ð Þ; 1ð Þ � f zT 2ð Þ; 2ð Þð Þ

þ aT 1ð Þ f0 � f zT 1ð Þ; 1ð Þð Þ: ðA11Þ

Since the right hand side of (A11) is negative, (24) implies (A11) and hence (18).


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):

aT 2ð Þ cT 2ð Þ � f zT 2ð Þ; 2ð Þ þ f0 � c0Tð Þ ¼ U2 þ f0 � c0T: ðA12Þ

Eq. (A12) can be substituted into (21), (22), and (23). Then (21) will be satisfied if:

U2 � c0Tþ f0z0 ðA13Þ

U 2 � c0Tþ f0zaT 1ð Þ cT 1ð Þ þ G zT 1ð Þ;ET; 2ð Þ � G zT 1ð Þ;ET; 1ð Þ � f zT 1ð Þ; 2ð Þ � c0Tþ f0ð ÞðA14Þ

while (22) becomes:

aT 1ð Þ cT 1ð Þ � f zT 1ð Þ; 1ð Þ � c0Tþ f0ð ÞzU 2 þ f0 � c0T

þ aT 2ð Þ G zT 2ð Þ;ET; 1ð Þ � G zT 2ð Þ;ET; 2ð Þ þ f zT 2ð Þ; 2ð Þ � f zT 2ð Þ; 1ð Þð Þ: ðA15Þ

Eq. (23) can be solved for a*(1)c*(1)+ (1�a*(1))c0* which can then be substituted into

(A14) and (A15) yielding the two inequalities in (25). Condition (A13) plus nonnegativity of c0*

is equivalent to (24). Note that for a*(2)N0, c0* can be any value in a range above 0. Hence, if

(24) holds, then allowable values of c0* 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: c0, c(i), a(i), andz(i) Set zi= z(i), a(zi)= a(i), f(zi)=G(z(i),E,i)� c(i), r=UP, and s= c0�w0. 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� ��

w0 þ s� f0ð Þza z j� �

G z j;E; i� ��

� f z j; i� �

� F z j� �Þ þ 1� a z j

� �� w0 þ s� f0ð Þ; i ¼ 1; 2; jpi : ðA16Þ

Thus zi maximizes (26) for type i over the schedule offered for the SCA. Similarly, (21), (22),

and (A16) imply that:

a zi� �

G zi;E; i� �

� f zi; i� �

� F zi� ��

þ 1� a zi� ��

w0 þ s� f0ð 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

Proof of Theorem 4. Set a1 ¼ aT 1ð Þ; a2 ¼ aT 2ð Þ; zz ¼ zT 1ð Þ; and t ¼ � BeTBZ

� �P2j¼1 h jð ÞaT jð Þ

BG zT jð Þ;ET; jð ÞBE

. Consider (28) and (29) for i =2 If the bounds on z are ignored, then the solutions to

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 z. 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 z= z*(1). The overall solution will be z*(2) provided:

aT 2ð Þ G zT 2ð Þ;ET; 2ð Þ � f zT 2ð Þ; 2ð Þ � T � tzT 2ð Þ � w0 � sþ f0ð ÞzaT 1ð Þ G zT 1ð Þ;ET; 2ð Þð� f zT 1ð Þ; 2ð Þ � T � tzT 1ð Þ � w0 � sþ f0Þ: ðA18Þ


For the type 2 individual to prefer approaching the SCA than going directly to the alternative,

the following must hold:

aT 2ð Þ G zT 2ð Þ;ET; 2ð Þ � f zT 2ð Þ; 2ð Þ � T � tzT 2ð Þ � w0 � sþ f0ð 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)za*(2), the solution to (29) is the

overall optimum if

aT 1ð Þ G zT 1ð Þ;ET; 1ð Þ � f zT 1ð Þ; 1ð Þ � T � tzT 1ð Þ � w0 � sþ f0ð 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). Ifa*(2) N0, G (z*(1), E*, 1)� f (z*(1), 1)�T � tz*(1) NG (z*(2), E*, 1)�f (z*(2), 1)�T �tz*(2)NG(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:

h 1ð ÞaT 1ð Þ G zT 1ð Þ;ET; 2ð Þ � f zT 1ð Þ; 2ð Þ � w0 þ f0½ � � rV h 1ð Þðþ h 2ð ÞÞsVh 1ð ÞaT 1ð Þ G zT 1ð Þ;ET; 1ð Þ � f zT 1ð Þ; 1ð Þ � w0 þ f0½ � � r: ðA21Þ

Consider any UPz0 and U2+ f0z0. When a*(2)=0, c0*=U2+ f0 so to sustain this allocation,

r =UP and s=U2+ f0�w0 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) N0, 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 � h 1ð Þ þ h 2ð Þð ÞtzT 1ð Þ þ h 2ð Þ G zT 1ð Þ;ET; 2ð Þ � f zT 1ð Þ; 2ð Þ � aT 2ð Þ G zT 2ð Þ;ET; 2ð Þð½� f zT 2ð Þ; 2ð ÞÞ þ 1� aT 2ð Þð Þ w0 � f0ð Þ�V h 1ð Þ þ h 2ð Þð ÞTVr � h 1ð ÞtzT 1ð Þ� h 2ð ÞaT 2ð ÞtzT 2ð Þ þ h 2ð Þ 1� aT 2ð Þð Þ G zT 2ð Þ;ET; 2ð Þ � f zT 2ð Þ; 2ð Þ � w0 þ f0ð 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)[U1+ f(z*(1), 1)]=h*�h(2)U2�UP+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,

h 1ð ÞT ¼ UP þ h 2ð ÞU 2 � h 1ð ÞtzT 1ð Þ � h 2ð Þ aT 2ð Þ G zT 2ð Þ;ET; 2ð Þ � f zT 2ð Þ; 2ð Þð Þ½þ 1� aT 2ð Þð Þ w0 � f0ð Þ�: ðA23Þ


Substituting this in Eq. (A23) yields:

hTþ h 1ð Þ G zT 1ð Þ;ET; 1ð Þ � G zT 1ð Þ;ET; 1ð Þ � f zT 1ð Þ; 2ð Þ þ f zT 1ð Þ; 1ð Þ½ �VUP þ h 1ð Þðþ h 2ð ÞÞU 2VhTþ h 1ð Þ G zT 2ð Þ;ET; 2ð Þ � G zT 2ð Þ;ET; 1ð Þ þ f zT 2ð Þ; 1ð Þ � f zT 2ð Þ; 2ð Þ½ �þ h 1ð Þ G zT 2ð Þ;ET; 1ð Þ� f zT 2ð Þ; 1ð Þ� tzT 2ð Þ�G zT 1ð Þ;ET; 1ð Þþ f zT 1ð Þ; 1ð Þþ tzT 1ð Þ½ �:

ðA24Þ

Values of UP and U2 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

c0*=w0+ sz0 since f0bmin [ f(z*(1), 1), f(z*(2), 2)] is assumed. To consider w0+ sz0

substitute for s from the government budget constraint and then substitute T from combining

U1=G(z*(1),E*, 1)� f(z*(1), 1)�T� tz*(1) given the linear fee schedule and h(1)U1=

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))(w0� f0)]�h(2)U2�UP. This yields the following condition for w0+ sz0:

UP þ h 1ð Þ þ h 2ð Þð ÞU 2zh 1ð Þ G zT 2ð Þ;ET; 2ð Þ � f zT 2ð Þ; 2ð Þ � G zT 1ð Þ;ET; 1ð Þ½þ f zT 1ð Þ; 1ð Þ þ tzT 1ð Þ � tzT 2ð Þ� þ hT� 1� aT 2ð Þð Þ=aT 2ð Þ½ �h 1ð Þ U2 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 c0* 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 UP and � f0 for U2 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 a1=0, then, since G(z2,E, 1)� f(z2,1)NG(z2,E, 2)� f(z2,2), (31) can only hold

if a2=0. All of (31) and (32) then hold with equality. Consider a2N0. 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, F1 could be increased without violating any

constraint. If a2=0 then (32) for i=2 will hold with equality as required. Consider a2N0 and

assume (32) for i=2 holds with strict inequality. Consider small increases in both F1 and F2 such

that a1F1�a2F

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 U1 with Pareto constraints on U2 and UP the

Implementable Pareto Frontier can be found by maximizing UP subject to Pareto constraints on

U1 and U2 as well as the resource constraint (14), the self-selection constraints (19), the

participation constraints (20), and nonnegativity conditions. Dropping the Pareto constraints and


solving the problem in their absence yields the maximum possible value of UP on the frontier.

Clearly, (14) must hold with equality since if not cp could always be increased. Then (14) can be

solved for cp 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)N0 and that condition holds with strict inequality, then c(1) could be reduced

leading to an increase in cp, 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)N0, a small

decrease in c(1) and c(2) holding a(2)c(2)�a(1)c(1) constant would be feasible and would allowcp 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:

MaxX2i¼1

h ið Þa ið Þ G z ið Þ;E; ið Þ � f z ið Þ; ið Þ þ f0 � w0ð Þ þ h 1ð Þa 2ð Þ G z 2ð Þ;E; 1ð Þð

� G z 2ð Þ;E; 2ð Þ þ f z 2ð Þ; 2ð Þ � f z 2ð Þ; 1ð ÞÞ þ h 1ð Þ þ h 2ð Þð Þ w0 � c0ð Þ ðA26Þs:t: a 1ð Þ G z 1ð Þ;E; 1ð Þ � G z 1ð Þ;E; 2ð Þ � f z 1ð Þ; 1ð Þ þ f z 1ð Þ; 2ð Þð Þza 2ð Þ G z 2ð Þ;E; 1ð Þð� G z 2ð Þ;E; 2ð Þ � f z 2ð Þ; 1ð Þ þ f z 2ð Þ; 2ð ÞÞ: ðA27Þ

Since c0=w0+ 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 a2=0 in (36)

and (37). Then (37) holds for all a1, z1, and z2 so can be dropped. The first order conditions with

respect to a1, z1, and z2 for (15) given in the Proof of Theorem 1 are the same as those for (36).

Hence, given a2=0, the other variables will be at their full information values. At those values,

the constraint in (37) holds with strict inequality at a2=0. Hence, the first order condition with

respect to a2 is also the same in the full information and private owner optimizations.

Finally, consider the case when a(2)N0. Assume ai =a*(i) since if not, the solution for (36) and(37) differs from that in the full information optimization. Since a1Na2 then holds, (37) holds

with strict inequality so can be deleted. Take the first order conditions with respect to z1 and z2:

BUP

Bz1¼ h 1ð Þa1

BG z1;E; 1ð ÞBz1

� Bf z1; 1ð ÞBz1

þX2j¼1

h jð ÞajBG zj;E; j� �BE

Be

BZ

"

þ h 1ð Þa2BG z2;E; 2ð Þ

BE� BG z2;E; 1ð Þ

BE

� Be

BZ

�¼ 0 ðA28Þ

BUP

Bz2¼ h 2ð Þa2

BG z2;E; 2ð ÞBz2

� Bf z2; 2ð ÞBz2

þX2j¼1

h jð ÞajBG zj;E; j� �BE

Be

BZ

"



BE

� Be

BZ

�


Bz2� Bf z2; 2ð Þ

Bz2� BG z2;E; 1ð Þ

Bz2þ Bf z2; 1ð Þ

Bz2

� �¼ 0: ðA29Þ

If the full information optimum is to be sustained, then Eq. (A3) must hold. If it does, then

Eqs. (A28) and (A29) reduce to:

h 1ð Þa2BG z2;E; 2ð Þ


BE

� Be

BZ¼ 0 ðA30Þ


h 1ð Þa2BG z2;E; 2ð Þ


BE

� Be

BZ


Bz2� Bf z2; 2ð Þ

Bz2� BG z2;E; 1ð Þ

Bz2þ Bf z2; 1ð Þ

Bz2

� ¼ 0: ðA31Þ

SinceBG z2;E;2ð Þ

Bz2� Bf z2;2ð Þ

Bz2� BG z2;E;1ð Þ

Bz2þ Bf z2;1ð Þ

Bz2b0, then both Eqs. (A30) and (A31) cannot

hold. Hence, the full information optimum cannot be sustained when a*(2)N0. 5

Proof of Theorem 6. That T1 and T2 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 Uiuai G zi;E; ið Þ � f zi; ið Þ � Fi � Tið Þ þ 1� aið Þ w0þð s� f0Þ. 5

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Documents

Private information, Coasian bargaining, and the second welfare theorem