Ecient Mechanisms for Public Goods with Use Exclusions

y

Peter Norman

May 15, 2002

Abstract

This pap er studies \voluntary bargaining agreements" in an environment where preferences

over an excludable public go o d are private information. Unlike the case with a non-excludable

public go o d, there are non-trivial conditions when there is signi cant provision in a large econ-

omy. The provision level converges in probability to a constant, which makes it p ossible to

approximate the optimal solution by a simple xed fee mechanism, whichinvolves second degree

price discrimination if identities are informative ab out the distribution of preferences. Truth-

telling is a dominant strategy in the xed fee mechanism.

Being able to limit a public go o ds' consumption do es not make it a turn-blue private

good. For what, after all, are the true marginal costs of having one extra family tune

in on the program. They are literally zero. Why then limit any family which would

receive p ositive pleasure from tuning in on the program from doing so? [Samuelson [24 ],

pp 335]

1 Intro duction

Virtually all theory on collective goods considers a \pure" public good, which is non-excludable

and non-rival in consumption. Obviously, these prop erties need not go hand in hand. In fact, it



I'm indebted to Mark Armstrong and two referees who challenged me to write a more interesting pap er than

the initial draft. I also thank Ray Deneckere, John Kennan, Bart Lipman, George Mailath, Ro dy Manuelli, Andrew

Postlewaite, Larry Samuelson, William Sandholm and participants at several conferences and academic institutions

for comments and helpful discussions. The usual disclaimer applies.

y

Department of Economics, University of Wisconsin-Madison, 1180 Observatory Drive Madison, WI 53706,

[email protected] 1

is easier to nd realistic examples of excludable public goods, by which I mean non-rival go o ds for

which it is feasible to exclude consumers from usage. To the extent that copying can b e prevented,

a recording of a song or anything else that can be stored in digital format is an almost p erfect

example. Other examples include cable TV, public facilities with controlled access and excess

capacity parks, gyms, zo os, swimming p o ols, trains, innovations, services by the p olice and the

re department, and access to databases see Brito and Oakland [3 ] for further examples.

Until recently, use exclusions were usually considered irrelevant. Excluding consumers is Pareto

inecient and rst b est can b e implemented with, say, the Lindahl equilibrium mechanism. The few

pap ers studying use exclusions in complete information mo dels therefore had to restrict the ability

to price discriminate. This is most explicit in Dr eze [10 ], who argues that individualized Lindahl

prices are never observed, and that there are go o d reasons for this, such as the lack of incentives for

truthful revelation of preferences. Imp osing a uniform price as a constraint, Dr eze shows that some

consumers are excluded from usage if the planner faces a binding budget constraint. The reason is

that exclusions serve as an imp erfect substitute for price discrimination.

Asymmetric information in itself is not sucient to rationalize uniform prices. Versions of \pivot

mechanisms" Clarke [4 ], Groves [11 ], and d'Asp eremont and Gerard-Varet [7 ] can implement rst

b est for a pure public go o d. Excluding consumers is then again a pure waste of resources.

This pap er considers an environment where rst b est cannot b e implemented. Mechanisms that

are considered implementable must, b esides incentive compatibility, also b e consistent with volun-

tary participation and b e self- nancing, constraints that make it natural to think of the setup as a

\voluntary bargaining problem". Alone, neither voluntary participation or budget balance suces

to create a role for exclusions: rst b est can be implemented with pivot mechanisms satisfying

either constraint. However, if b oth constraints are required, rst b est is unattainable for reasons

familiar from Myerson and Satterthwaite [18 ]. This creates a role for use exclusions. The basic

reason is that excluding lowtyp es ameliorates the free riding problem by making it less app ealing

for high typ es to mimic lower typ es Moulin [17 ] and Dearden [8].

The main fo cus of this pap er is the provision problem for a large economy. It is then well-

known that it is \asymptotically imp ossible" to provide a non-excludable public go o d in a voluntary

bargaining agreement Mailath and Postlewaite [16 ], Rob [22 ], and Guth  and Hellwig [12 ], whereas

many natural trading institutions for private go o ds are asymptotically rst b est ecient. 2

For an excludable public go o d the provision level is strictly p ositive with probability near one

in a large economy under non-trivial parametric conditions. Hence, exclusions not only improve

eciency, but changes qualitative results signi cantly. Excluding customers is the only way to

extract more than the lowest p ossible valuation from higher typ es in a large economy. This suggests

that private markets for public go o ds should b e able to op erate only if consumers can b e excluded,

which seems roughly consistent with the array of collective go o ds actually provided privately.

I consider b oth a binary and a continuous public go o d. Interestingly, there is almost no qual-

itative di erence between the two variants of the mo del. The reason is that the provision level

is asymptotically constant in the mo del with a quantity choice. Adjusting the quantity based on

rep orts makes it p ossible to provide more when the surplus from provision is high. Providing at

higher levels when many agents have high valuations also discourages high typ es from misrep orting.

But, b oth these considerations vanish in a large economy. The average typ e conditional on b eing

included converges in probability and the gain from using the provision rule to extract more rev-

enues from higher typ es b ecomes negligible, since agents correctly p erceivetohave little in uence

on the quantity provided see Al-Na jjar and Smoro dinsky [1] and Ledyard and Palfrey [14 ] for

discussions of in uence. Moreover, agents prefer the exp ectation for sure to a lottery, an e ect

which dominates in a large economy. The provision level therefore converges in probability.

A simple xed fee mechanism is asymptotically optimal. This mechanism sets a user fee for each

agent dep ending on identity if this provides information ab out the distribution of the valuation,

provides the good if and only if the revenues cover the costs, and allows a consumer to enjoy the

go o d if and only if she is willing to pay the fee. This may b e thought of as the obvious arrangement,

but solutions to design problems are usually not this simple. Moreover, this is not optimal if the

mechanism designer can force participation or if resources from outside the mo del are available.

The intuition is similar to the intuition for convergence in probability of the level of the public

good. The essence of the incentive problem is to discourage high typ es to mimic lowtyp es. Ecient

inclusion rules therefore only include agents with valuations ab ove a certain threshold. Typ es

who values the good more than the threshold typ e are willing to sp end more only if it increases

the exp ected consumption. But the average agent has little in uence on quantity provided, and

transfers from agents that are included are thus almost indep endent of typ e in large economies.

The eciency loss of xed fees is therefore negligible with many consumers. 3

The xed fee mechanism is not only simple and almost optimal. Truth-telling is a dominant

strategy, budget balance holds ex p ost, and participation is ex p ost individually rational, so almost

all conceivable desiderata are satis ed. The analysis thus provides some justi cation for the ap-

proach in Dr eze [10 ], Brito and Oakland [3 ] and others, and generates a limiting mo del that can

b e used for more applied questions.

The asymptotic optimality of xed fees is somewhat akin to the asymptotic eciency of simple

voting schemes in Ledyard and Palfrey [14 ],[15 ]. The di erence is that they assume equal cost

shares. Voluntary participation is then not an issue, so there is no diculty to raise sucient

revenues. Their mechanism therefore outp erforms the constrained optimal mechanism of this pap er.

An imp ortant example of an excludable public go o d is a xed cost in pro duction of a private

good. Cornelli [5 ] studies such xed costs for a pro t maximizing monop olist. Her fo cus is the

opp osite, the main insight b eing that, with a small number of p otential customers, it is optimal

to sell the good b efore pro ducing it in order to use the threat of non-pro duction to discriminate

between high and low valuation customers. However, she studies a few large economy examples

where similar di erences b etween the excludable and the non-excludable case o ccur as in this pap er.

It should be emphasized that results are sensitive to how costs of provision are treated as

the economy grows. I assume that costs increase with the number of participants. Literally,

the limiting results are therefore for joint changes in the number of participants and the cost of

provision. However, as discussed in Section 2.1, this can b e viewed as a normalization to guarantee

that signi cant p er capita contributions are needed to supply the public go o d.

2 The Mo del

2.1 Preferences and Costs

Consider an economy with a set of agents I = f 1; :::; ng bargaining over the provision of an exclud-

able public go o d. Agents di er in their valuations for the public go o d and preferences are private

information to the agents. I mo del this by assuming that the utility of an agent i 2 I is given by

v y  t ; 1

i i

where y is the quantity of the public good, v 
 is a continuous, strictly increasing and concave

function satisfying v 0 = 0; t is a transfer of \money" paid by agent i; and is the typ e of the

i i 4

agent, representing unobservable taste di erences. If the agent is excluded from usage her utility

is t : Preferences over lotteries are of exp ected utility form.

i

The typ e is distributed over  =[ ; ] in accordance with distribution F ; which is equipp ed

i i i i

i

with the continuous and strictly p ositive density f : Typ es are sto chastically indep endent, so F is

i i

1

the prior ab out for all other agents as well as for the mechanism designer. For brevity  is used

i

> 0 for all i 2 I: to denote   and to avoid trivialities I assume that

i i i

I assume that the level of the public go o d is b ounded by y and that the per unit cost of providing

y ]thus costs yC  ntoprovide. Note here that n is the the public go o d is C  n : Quantity y 2 [0;

number of agents and not the number of users. The go o d is thus fully non-rival.

The rationale for indexing costs of provision by n is that I consider sequences of economies



where the numb er of participants approach in nity. I then assume that C  n =n has limit c > 0.



The simplest example of such a sequence of cost functions is if C n = c n for each n; implying



that the cost of providing quantity y is yc n: Obviously, this means that the cost of providing even

the tiniest amount tends to in nityas n !1: However, all results in the pap er are of the form that

for a given >0 there is a nite N such that the characterization of a nite economy of size n  N

is within an  distance usually in terms of p er capita surplus from the limiting economy. For each

nite economy the costs of provision go es to zero as the provision level is approaching zero. As long

as limiting results are interpreted as an approximation of a nite economy the assumption that



lim C  n =n = c > 0isthus merely a normalization of p er capita costs to keep the provision

n!1

2

problem \signi cant" for a large economy also see Rob erts [23 ].

Hellwig [13 ] studies more or less the same mo del, except that costs are constantasn !1. This

will in some cases make exclusions irrelevant: if the level of the public go o d is b ounded by y and

costs are constant, rst b est can asymptotically be implemented under the standard additional

 0 for all i: The reason is that an arbitrarily small p er capita tax is sucient assumption that

i

3

for maximal provision.

1

Indep endence is imp ortant. With correlated values there are circumstances where the ex p ost ecient rule can

b e implemented in the pure public go o ds case Pesendorfer [21 ], which eliminates any role for exclusions.

2

At the cost of some additional complexity it is p ossible to handle convex cost functions C y; n. The limiting

 

economywould then b e equipp ed with a convex p er capita cost function c y  such that lim C y; n=n = c y :

n!1

Since the provision level is almost constant in a large economyeven in the linear case this generalization is not very

interesting: it only adds an additional force in favor of a constant provision level.

P

3

>C this is obvious and rst b est can b e implemented exactly. If =0 for all agents, the

If lim

n!1

i i

i 5

The mo del b ecomes more sp eci c, but the public go o d can b e reinterpreted as a xed cost for

the pro duction of a private good. In this case y is the quality level and yC n the xed cost of

setting up a plant that pro duces quality y . For this setup to map exactly into the sp eci cation

ab ove the marginal cost of pro duction must b e zero and consumption b e binary. Positive marginal

costs can b e \netted out", but the binary demand is a tight restriction since non-linear pricing can

b e used otherwise, a margin that is imp ossible to utilize with a public go o d

2.2 The Design Problem

No matter what mechanism or bargaining institution is set up in the economy, the outcome of this

pro cess should determine:

1. the level of the public go o d,

2. which agents should b e allowed to use the public go o d,

3. how the costs of the public go o d should b e shared.

By app eal to the revelation principle I restrict attention to direct mechanisms for which truth-

telling is a Bayesian Nash equilibrium. Allowing randomizations in the inclusion/exclusion decision

a direct mechanism can be represented as a triple y; ;  , where y :  ! [0; y ] is the provision

n

n

is the cost sharing rule. I adopt the rule, :  ! [0; 1] is the inclusion rule, and  :  ! R

+

convention that  is the probability that agent i is allowed to consume the public go o d given

i

the announcement : Payo s are indep endentof   if y  =0; so I will for convenience allow

i

agents to b e included to consume nothing even if the go o d is not pro duced. In principle it is also

allowed to provide a p ositive quantity and exclude everyb o dy, but this is obviously sub optimal.

The exp osition b elow also assumes that agent i contributes    when is announced no matter

i

4

whether she consumes the public go o d or not, which is purely for notational convenience.

b b b b

The exp ected utility for agent i of typ e is   v y     ; where denotes the vector of

i i i i

rep orted typ es. For truth-telling to b e an equilibrium in the revelation game it must b e incentive

result is more subtle. Then the idea is that the necessary p er capita contribution approaches zero faster than the

probability of b eing pivotal see Hellwig [13 ].

4

The alternative istomake transfers conditional on inclusion. Due to risk-neutrality in \money" transfers this

leads to the same characterization of incentive feasible provision and inclusion rules. 6

compatible to announce the true typ e for every agent i and any p ossible typ e realization. I let E

i

denote the exp ectation op erator with resp ect to = ; :::; ; ; ::;  and express this as

i 1 i1 i+1 n

h i

b b b b

E [   v y     ]  E  ;  v y  ;    ;  8i 2 I; ; 2  : 2

i i i i i i i i i i i i i i i i i

Allo cations must also be feasible in the sense that the contributions collected cover the costs of

provision. It is not a priori obvious whether to imp ose this ex post or ex ante, but the seemingly

weaker form is the ex ante feasibility budget balance constraint,

 !

n

X

E    y   C n  0; 3

i

i=1

while the ex post version requires the argument of 3 to be p ositive for all : I use 3 in my

analysis, but by adapting an argument from Cramton, Gibb ons and Klemp erer [6 ] one shows that

the ex ante and ex post constraints are equivalent in the sense that if 3 holds, then there is a

mechanism satisfying the ex post constraints with the same provision and exclusion rules.

Finally, I assume that voluntary participation or individual rationality must be resp ected. I

assume that agents know their typ e when they decide whether to participate in the mechanism.

Hence, individual rationality is imp osed at the interim stage as,

E [   v y     ]  0 8i 2 I; 2  : 4

i i i i i i

Mechanisms that satisfy 2,3 and 4 are called incentive feasible. Note here that the con-

straints 3 and 4 makes it natural to think of the problem as a \voluntary bargaining problem".

The option to walk away from an agreement is in 4, thus capturing the notion of voluntary partic-

ipation. The constraint 3 means that private consumption must b e sacri ced to enjoy the public

good, thus making it a bargaining setup rather than a \pure" problem of preference revelation.

2.3 Preliminaries: Characterization of Constrained Optimal Mechanisms

In this section I characterize the incentive feasible mechanisms by combining 2,3 and 4 into a

single integral constraint, which eliminates transfers from the problem. This is a routine adaptation

of techniques from Baron and Myerson [2], Myerson [19 ], Myerson and Satterthwaite [18 ] and others.

5

The purp ose of the exp osition is therefore to intro duce notation and formal pro ofs are omitted.

5

Details are available in Norman [20 ]. 7

Fix an arbitrary mechanism y; ;   and let U   b e the indirect exp ected utility for an agent

i i

of typ e : De ne t   E    ; which is the exp ected transfer for agent i of typ e given

i i i i i i

truthful revelation. Similarly, de ne v   = E   v y  , which may be thought of as the

i i i i

\exp ected consumption b ene t" reduces to a scaling of the exp ected consumption when v is linear.

In a truth-telling equilibrium it must b e the case that

b b b

  ;  5  ; v y  ;  E U   = max E

i i i i i i i i i i i i

b

2

i i

b b

v   v  = v   t   8 i 2 I; 2  ; = max

i i i i i i i i i i i i

b

2

i i

where the last equality is a consequence of the revelation principle. Using routine arguments one

rst shows that.

Lemma 1 y; ;   is incentive compatible if and only if v   is increasing in for al l i and

i i i

Z

i

b b

v   d 8i 2 I; ; 2  6 U  =U  +

i i i i i i i i

b

i

This is a standard result which has nothing to do with the collective nature of the go o d. Equally

routine pro cedures using the characterization of incentive compatibility in Lemma 1 shows that:

Lemma 2 Suppose y; ;   is incentive compatible. Then,

Z Z

1 F  

i i

::: E  =  8i 7   v y    f   d U 

i i i i

k k k k

i

f  

i i

n

1

Moreover, if v   is increasing in for al l i 7 holds, then y; ;   is incentive compatible.

i i i

Lemma 2 fully characterizes the set of incentive compatible provision and exclusion rules. The

nal step is to combine incentive compatibility with the participation constraints 4 and the

feasibility requirement 3. Expressed in terms of the indirect utility function, the participation

constraints 4 are that U    0 for all i and : Since v  =E   v y    0we observe

i i i i i i i

from 6 that U   is increasing in ; so all participation constraints are ful lled if and only if

i i i

U    0: Feasibility requires that

i

i

Z Z

X

y   C n f   d ; 8 ::: E     Ey   C n=

i

k k k k

n

1

i

and combining 8 with 7 and the fact that all participation constraints hold if and only if

  0 for all i one concludes that the set of implementable provision and inclusion rules can U 

i

i 8

be characterized without direct reference to the transfers as all y;  that satis es the condition

! 

Z Z

X

1 F  

i i

  v y   y   C n  f   d  0: 9 :::

i i

k k k k

f  

i i

n

1

i

The whole discussion in this section can thus b e summed up as:

Prop osition 1 There exists a contribution scheme  such that y; ;   satis es 2,3 and 4 if

and only if v is increasing for each i and 9 holds.

i

A constrained ecient mechanism is a mechanism designed to maximize so cial surplus as p os-

sible sub ject to b eing incentive feasible. Applying Prop osition 1 that means that a constrained

ecienty;  solves

Z Z

X

  v y   y   C n  f   d 10 ::: max

i i

k k k k

n

i

y 
;f 
g

n g f

1

i=1

R R

P

1F  

i i

s.t :::   v y   y   C n  f   d  0

i i

k k k k

i

f  

n

1

i i

v  =E v y     increasing in 8i 2 I monotonicity

i i i i i

y and 0     1 8i 2 I b oundary 0  y   

i

For intuition it is useful to note that the only di erence b etween the function in the constraint

and the ob jective function of 10 is that the term 1 F   =f   replaces in the

i i i i i i

constraint. This captures that higher typ es must have no incentives to mimic typ es with lower

valuations for the public go o d. That is, there are informational rents for higher typ es that limit the

mechanism designer to extract \the virtual surplus" from each agent and typ e rather the actual

surplus.

The solution to problem 10 can be characterized by standard Lagrangian techniques which

gives some useful information for the pro ofs in the sections to follow. De ne

Z Z

X

S y;  :::   v y   y   C  n  f   d 11

i i

k k k k

i

n

1

Z Z

X

Gy;  :::   v y  x   y   C  n  f   d ; 12

i i i

k k k k

i

n

1

where

1 F  

i i

x   : 13

i i i

f  

i i

Ignoring the monotonicity constraint, the Lagrangian for 10 is Ly; ;  S y; +Gy; : If

n n n

y ;  2 arg max Ly; ;   14 9

n n n n n

and  Gy ; =0, then y ;  solves 10, given that the solution to this relaxed problem is

6

monotonic. This makes the problem very tractable since 14 is solved by p ointwise optimization.

The non-obvious part of Lemma 3 b elow is that it establishes existence of a value of the multiplier

such that it together with asso ciated maximizer of 14 is a saddle p oint of the Lagrangian, thereby

proving existence of solutions to 10 as well as giving a useful characterization.

n n n

Lemma 3 Suppose that x   is weakly increasing. Then, there exists   0 such that y ; 

i i

n

is an optimal solution to 10 if and only if for almost al l 2 

n

X

n n n

y   = arg max v y  max [0; +  x  ] 1 +   yC n ; 15

i i i

y 2[0; y ]

i=1

7

and where for al l i 2 I and al l such that y   > 0

8

>

n

<

1 if +  x    0

i i i

n

 = : 16

i

>

:

0 otherwise

The pro of follows an argument in Hellwig [13 ] closely and is omitted details on how to mo dify

the pro of with use exclusions available are in Norman [20 ]. As usual, x   is assumed to be

i i

increasing to guarantee that the unrestricted solutions are monotonic.

Certain deviations on sets of measure zero from 15 and 16 may result in an optimal solution

monotonicity requirements rule out some but not all deviations from 16. Such deviations are

irrelevant in the sense that neither the exp ected provision level or the probabilities of inclusion

change. I will therefore ignore to add quali ers ab out negligible sets in the remainder of the pap er.

3 A Binary Public Good

In this section I consider the case when the public good comes as a single indivisible unit. This

is a direct extension of the mo del in Mailath and Postlewaite [16 ], the only change b eing that use

exclusions are allowed. While a single indivisible unit may seem sp ecial, one of the main lessons

from the analysis of the general mo del is that, as the size of the economy increases, the design

6 0 0 0 0 n n 0 0 0 0 n 0 0

To see this, supp ose y ;  is such that S y ;  >Sy ;  and Gy ;   0: Then Ly ; ; =S y ; 

n 0 0 n n n n n n n

_

+ Gy ;  >S y ; = Ly ; ; ; contradicting that y ;  maximizes the Lagrangian:

7

The inclusion rule is irrelevant when y  =0; but it is without loss to assume that inclusions are always in

accordance to 16. 10

problem reduces to a binary problem. The material of this section is therefore of interest also for

the understanding of the general mo del.

Prop ositions 2 and 3 establish that the probability of provision in the surplus maximizing

P

 

mechanism converges to either zero or one dep ending on whether lim 1 F  =n is

n!1 i

i

i i

smaller or greater than lim C n =n: I then show in Prop osition 4 that a \ xed fee mechanism"

n!1

is asymptotically optimal.

The cost C  nisnow simply the cost of the pro ject, so the feasibility constraint 3 simpli es

P

n

to E [       C  n]  0: I write  :  ! [0; 1] for a generic random provision rule,

i

i=1

where    is the probability of providing the public go o d given announcements : The utilityofan

agentoftyp e is now t if the public go o d is consumed and t otherwise, and the exp ected

i i i i

b b b b b

utility of agent i of typ e given announcements is E     E   ; where   is the

i i i i i i i

probability of inclusion conditional on provision. The only change in preferences is thus that   

replaces v y  ; so the mo del is equivalent to a mo del with a quantity decision where v y  = y

and y = 1; a case obviously covered by the characterization in Section 2.3. We conclude that a

surplus maximizing provision-inclusion rule must solve

! 

Z Z

X

max :::    f   d 17   C  n

i i

k k k k

n


;f 
g

g f n

1

i=1

i

R R

P

  x   C n    f   d  0 s.t ::: 

i i i

k k k k

i

n

1

  =E     increasing in 8 i 2 I monotonicity

i i i i i

0      1 and 0     1 8 i 2 I b oundary

i

n

and by adapting Lemma 3 we have that asso ciated with the problem 17 there is some   0

n n n

such that  ;  is an optimal solution if and only if is given by 16 for every i and ; and

i

8

P

>

n n

<

1 8 such that max [0; +  x  ] 1 +   C  n  0

i i i

i

n

  = 18

>

:

0 otherwise

3.1 The Surplus Maximizing Mechanism in a Large Economy

Inow consider sequences of economies, where agents are added one at a time. The nth economyof

n

a sequence consists the cost of provision C n and agents f1; :::; ng . I let F =F ; :::; F  denote

1 n

the vector of indep endent distributions over typ es. An economy of size n is thus given by the

n

pair C n ;F  : For ease of exp osition I will in the remainder of the pap er assume that x  is

i i 11

strictly increasing for each agent i weak monotonicity is sucient. I de ne



= arg max 1 F   ; 19

i i i

i

2

i i

which may be interpreted as the \monop oly price", the price that a pro t maximizing monop o-

list would charge if restricted to take-it-or-leave-it o ers and costs of provision are sunk. Strict



monotonicityofx implies that is uniquely de ned, which is the reason for the assumption.

i

i

The rst result provides a condition for when it is \asymptotically imp ossible" to provide the

public go o d

1

n

Prop osition 2 Let fC n ;F g be a sequence of economies, where, for each i; distribution F

i

n=1

has a density f such that x   is strictly increasing, where f   >k for some k>0; and where

i i i i i



  for some uniform bounds 1 < < <1: Moreover, suppose there exists c such

<

i

i

P

    

: Final ly, suppose that lim [1 F  ] =n

n!1 i n!1

i

i i i

1

n n n

in 19: Then, lim E  =0 for any sequence f  ; g of feasible solutions to 17.

n!1

n=1

A pro of is in the app endix, but an informal sketch is instructive. An upp er b ound on the

provision probability is found by maximizing the ex ante probability of provision sub ject to the

constraints in 17. Since no welfare considerations enter this problem the inclusion rule maximizes

the exp ected transfer from each agent, whichisachieved by the rule

8

>



<

1 if 

i

i

n

: 20  =

i

>



:

0 if <

i

i

n n

A crucial implication of 20 is that   x   is sto chastically indep endentof   x   for

i i j j

i j

all i; j; and an application of Cheb eshevs inequality shows that

" 

n n

n  

X X

  x  

] [1 F 

i i i

i i i

lim Pr =0 21  

n!1

n n

i=1 i=1

for every  > 0: The interpretation of 21 is that the maximal revenue conditional on the

pro ject b eing implemented for sure converges in probability. The rule that maximizes the prob-

ability of provision involves a threshold k such that the public good is provided if and only if

n

P

n

  x   =n  k : The main argument in the pro of is to show that k must be set so that

i i n n

i

i

the provision probability go es to zero to satisfy the integral constraint in 17. This involves some

work, but the rough idea is if the provision probability stays b ounded away from zero, then the 12

P

n

exp ectation of   x   =n conditional on provision approaches the unconditional exp ecta-

i i

i

i

P

n

tion of   x   =n: Exp ected per capita revenues conditional on provision are thus near

i i

i

i

P

 

[1 F  ] =n; which is less than C n =n in a large economy.

i

i

i i

P

n

  

Next, I turn to the case when lim [1 F  ] =n>c , which makes it is necessary

n!1 i

i=1

i i

n n

to consider problem 17 in more detail. The rst observation is that Lemma 3 implies if  ; 

n

solves 17, then there exists some for every i 2 I such that

i

8

>

n

<

1 if 

i

i

n

: 22

 =

i

>

n

:

0 if <

i

i

Hence, the optimal inclusion rule for agent i is again indep endent of announcements by other agents,

which is crucial for the analysis b ecause it makes it p ossible to conclude that,

" 

n n

n n n

X X

  x   [1 F  ]

i i i

i i i

lim Pr   =0: 23

n!1

n n

i=1 i=1

P

n n

The interpretation of 23 is that the total revenue converges in probabilityto [1 F  ] =n

i

i

i i

P

[1 if the pro ject is undertaken for sure. Since it is feasible to set the threshold such that

i

n n n 

F  ] =n > C n=n in a large economy for example by = for all i and n the limiting

i

i i i i

result switches from asymptotic imp ossibility to provision with probability 1 in this case.

P

  

=n>c , but that al l other conditions in Propo- ] [1 F  Prop osition 3 Assume lim

i n!1

i

i i

1

n n n

sition 2 hold. Then, lim E   = 1 for any sequence f ; g of optimal solutions to

n!1

n=1

17.



It cannot be optimal to provide with probability zero since it is p ossible to use threshold

i

 

for each i and charge from each typ e  : The transfers collected from this mechanism

i

i i

exceeds the provision cost with probability converging to one, so this inclusion rule together with

the rule \always provide" is incentive feasible in a large economy and generates a strictly p ositive

surplus. Hence, if the probability of provision do es not converge to 1, there must at least be

some strictly p ositive probability of provision in the limit of any subsequence. This requires that

P

 n n

=n = c . I then construct an inclusion rule near the hyp othetical solution ] lim [1 F 

n!1 i

i

i i



that generates an exp ected per capita revenue strictly larger than c , implying that it is feasible

to provide for sure with a very small change in the inclusion rule. Increased provision is go o d for 13

almost all realizations of and the increase in the p er capita transfer is negligible, so the deviation

generates a strictly larger surplus.

The main dicultyinthe pro of is that a strict budget surplus in the limit for inclusion rules

close to the original is needed to guarantee feasibility for a large nite economy. This requires a con-

vexi cation of the achievable exp ected revenues, whichisachieved by randomizing over thresholds

n 

and :

i i

3.2 A Fixed Fee Mechanism is Almost Optimal

Taken together, Prop osition 2 and 3 give a sharp characterization of the ecient provision rule

in a large economy. However, one mayworry that unreasonably complicated transfer schemes are

required. To address this, I now consider a very simple mechanisms which is almost optimal.

De nition 1 The mechanism ; ;   is a xedfeemechanism if the inclusion rule satis es  =

i

e e e

1 if and only if  for some 2  and each i and   =   for each i and each 2 

i i i i i i i

Thresholds will in general di er across agents, so price discrimination based on observables is al-

lowed and \ xed" is relativetyp e. Not surprisingly however, there are no gains from discriminating

n n

agents with the same distributions. That is, 16 shows that = whenever F = F :

i j

i j

Inow need an additional regularity assumption. I assume that for each i; F b elongs to a nite

i

set F . This covers among other things the replica-case, in which case there is some k such that

8

F = F for all i  1. The result is:

i

i+k

P

  

[1 F  ] =n 6= c and that F 2F for every i; where Prop osition 4 Suppose that lim

i i n!1

i

i i

F is nite. Then for each  > 0 there exists some nite N such that for every n  N there is

an incentive feasible xed fee mechanism satisfying ex post budget balance such that the di erence

8

The role of the assumption is to assure that the limit of the average exp ected revenue is strictly in b etween the

limit average exp ected revenue in the optimal and revenue maximizing mechanisms if thresholds are chosen in b etween

n 

and . This was not an issue for Prop osition 3 since randomizations were used, whereas randomizations now are

i i

ruled out by de nition of a xed fee mechanism. An alternative sucient condition is to assume that 1 F   is

i i i

weakly concave. I conjecture that the necessary convexi cation can b e achieved generally by an alternative strategy

of pro of where agents pay a price that is either equal to the surplus maximizing inclusion threshold or the revenue

maximizing threshold. However, this approachintro duces some other diculties since agents in the two \groups"

can not b e picked arbitrarily. 14

in per capita surplus between this mechanism and a surplus maximizing mechanism is less than .

Moreover, truth-tel ling is a dominant strategy in the xed fee mechanism.

Hence, a xed fee mechanism can approximate the p er capita surplus of an ecient mechanism

arbitrarily well. The basic intuition is that in the large economy, the di erence between the per

capita transfer that the planner can extract using any conceivable mechanism and by using a xed

user fee b ecomes negligible since the p erceived in uence for the average agent b ecomes negligible.

Besides b eing dominant strategy implementable and simple the xed fee mechanism also satis es

ex p ost voluntary participation.

Transfer schemes are indeterminate for the surplus maximizing mechanisms, but for large n,

the set of where the provision rules di er and the set of where the inclusion rule for i di er is

i

negligible. It therefore makes some sense also to say that an ecient mechanism is close to a xed

fee mechanism.

4 Results for the More General Mo del

I now return to the more general mo del where the public go o d may be provided in any quantity

between 0 and y: In Prop osition 5 I establish that the quantity provided converges in probability

to its exp ectation and Prop osition 6 shows convergence in probability to a constant under the

additional assumption that the sequence of economies is generated by replicating a nite economy.

Due to convergence in probability, there is no signi cant loss to provide the good at a constant

level if providing at all. Hence, all results from the binary mo del can b e extended also to this case.

Prop osition 7 gives conditions similar to Prop osition 2 and Prop osition 3 for when the provision

level is asymptotically zero versus strictly p ositive and Prop osition 8 generalizes the asymptotic

optimality of a xed fee mechanism. Finally, Prop osition 9 generalizes the asymptotic imp ossibility

result for a non-excludable public go o d from Mailath and Postlewaite [16 ].

n

An economy is now de ned by the primitives v; [0; y ];C n ;F : The willingness to pay for

the public go o d is now v y   rather than ; but, like in the binary case, the p erceived in uence

i i

on the provision decision is on average negligible in a large economy. The ecient inclusion rule

in 16 is of the same form as in the binary case, a threshold rule that is indep endent of the realized

P

n

  x   =n values of for each agent i: Hence, 23 holds true also in this case, that is

i i i

i

i 15

P

n n n

converges in probability to 1 F   =n; suggesting that typ es  should be taxed

i i

i

i i i

n

roughly Evy   for the right to consume the public go o d.

i

The advantages of varying the level of the public go o d dep ending on the realization of are that

higher typ e realizations generate a higher so cial surplus for a given level of the public go o d and that

making the level of the public go o d increasing in the announcements is a way to improve incentives.

However, the average typ e conditional on b eing ab ove the threshold converges in probability and

incentives approach those of a xed fee mechanism, so b oth these rationales for variabilityin y are

negligible in large economies. Moreover, agents dislikevariation in y if v is strictly concave, whichI

assume. This suggests that the level of provision in an optimal mechanism converges in probability

to the exp ected provision level. Indeed:

1

n

Prop osition 5 Suppose f v; [0; is a sequenceofeconomies, where every distribu- y ];C n ;F g

n=1

tion F satis es the regularity conditions of Proposition 2, v is strictly concave and that there exists

i

  n n

c such that C n =n = c . Then, lim Pr  jy   E  y  j=0 for al l >0 and any

n!1

n n

sequenceof constrained optimal mechanisms fy ; g :

The basic structure of the pro of is straightforward. Assuming that the provision level do es not

converge in probability I consider a sequence of mechanisms which provide the exp ected level of the

public go o d from the hyp othetical optimal mechanism for sure and uses the same inclusion rules.

Using convergence in probability of the average actual and virtual surpluses and strict concavityof

v; the alternative sequence is shown to b e feasible and more desirable if n is large:

Hence, while the level of the public good is endogenous, it is asymptotically a constant. We

maythus think of the design problem as a binary one. For any given y the same arguments as in

Section 3 tells us that y can b e provided for sure and is desirable to provide for sure if

P

 

[1 F  ]

i

i

i i 

v y  lim >yc 24

n!1

n

holds. Clearly, 24 is harder to satisfy the higher is y; so the dividing line b etween the case where

n

y   converges to zero in probability and where it remains strictly p ositive can be obtained by

taking the limit of 24 as y approaches zero:

There is no sense in which the constrained optimal solutions approach rst b est eciency.

P

Except for in trivial cases where the rst b est ecient level of provision approach zero, 1

i 16

n n

F   =n must converge to zero in order for the surplus to approach rst b est eciency: But,

i

i i

P

n n n

then the per capita revenue, which is roughly v Ey   1 F   =n in a large economy,

i

i

i i

also approaches zero, whereas the p er capita costs are b ounded away from zero, violating feasibility.

4.1 The Replica Case

It is natural to ask whether the provision level converges in probability to a constant, that is,

n

whether Ey   has a well-de ned limit. This analysis intro duces some new technical issues. In

particular, I must now establish that problems with large n are \near each other" in the sense that

if average surplus S is feasible in a particular large economy, then something near S is feasible

for all suciently large economies. To deal with this I will now restrict attention to sequences of

economies that are generated by replicating a given nite economy.

1

n

De nition 2 fv; [0; is said to be a sequence of replicas of an economy with r y ];C  n ;F g

n=1

9

agents if there exists r such that F = F for al l i:

i+r i

The analytical advantage of making the restriction to replications of a nite economy is that

there is a well-de ned \limiting economy". For sequences of replicas one shows:

1

n

y ];C n ;F g is a sequence of replicas. Then there exists Prop osition 6 Suppose that f v; [0;

n=1

n  

y ] such that lim Pr [jy   y j] = 0 for any  > 0 and any sequence of some y 2 [0;

n!1

optimal solutions to 10.

To get a sense of how the pro of works, de ne

8

>

<

1 if 1  + x    0

i i i

e

 ;  : 25

i i

>

:

0 otherwise

n n n n

for every i and observe that if =  =1+ and  is the multiplier asso ciated with the optimal

n

e

mechanism, then 
;  coincides with the inclusion rule for agent i in the optimal mechanism.

i

Let r b e the size of the economy b eing replicated and de ne,

P

Z Z

r

e

 ; 

i i i

i=1

  :::  f   d : 26

k k k k

r

r

1

9

Toavoid intro ducing additional notation I continue to add agents one at a time in rather than adding r agents

at a time. 17

P

Z Z

k

e

 ; x  

i i i i

i=1

 f   d 27   :::

k k k k

r

r

1



v y [1  +  ] yc : 28 Q  arg max

y ] y 2[0;



The basic idea is that v Q   Q c is a go o d approximation of the p er capita surplus in

a large economy and that the constraint to the programming problem is well approximated with

1

 n n n n

v Q    Q c ; where is the limiting value of =  =1 +   and f  g is the

n=1

sequence of Lagrange multipliers asso ciated with a sequence of optimal solutions to the problem.

Using Prop osition 6 and doing a Taylor approximation of the constraintwe get a characterization

of when provision is zero in the limit and when provision is strictly p ositive.



Prop osition 7 For each i; let be de ned by 19. Moreover, suppose that:

i

P

n

0    n

1. lim v 0 1 F  =n < c : Then lim Pr [y    ]=0 for any >0 and

n!1 i n!1

i=1

i i

any sequenceof feasible solutions to 10

0

P

v 

r

n 0   

2. lim y ];C n ;F g is a sequence = 1 or if v 0 1 F  =n > c ; andf v; [0;

!0 i

i=1

i i



n 

of replicas. Then y   converges in probability to some y > 0 for any sequence of optimal

solutions to 10.

The result is proved by translating the problem to a binary problem by making a linear ap-

proximation of the constraint. This relationship b etween the binary mo del and the setup with a

quantity decision also allows us to extend the approximate eciency of xed fee mechanisms to

the setup with a quantity dimension. The arguments are very similar to previous pro ofs, so I have

omitted the formal pro of. All there is to verify is that there exists a provision rule that takes on



two values, 0 and something close to y ; that is b oth feasible and generates a surplus that can b e

made arbitrary close to that of the optimal mechanism for n large. This is straightforward since a



small decrease of y from y generates a strict budget surplus by strict concavityofv: One can then

app eal directly to the result for the binary mo del and conclude:

n

Prop osition 8 Suppose fv; [0; y ];C n ;F g is a sequence of replicas. Then, for each >0 there

exists a sequence of xedfeemechanisms and some N such that the di erenceinper capita surplus

between the xed fee mechanism and a constrained optimal mechanism is less than  for every

n  N . Moreover, truth-tel ling is a dominant strategy in the xed feemechanism. 18

4.2 Comparison with the Pro t Maximizing Mechanism

Consider rst the case with a binary public go o d. A pro t maximizing monop olist then faces the

same constraints as in 17. The ob jective function is exp ected revenues net of costs of provision,

which is the left hand side of the integral constraint of the problem. Hence the pro t maximizing

provider would maximize the value of this integral of the sum of virtual valuations net of the costs,

sub ject only to the b oundary constraints. It is intuitive that the inclusion threshold will be set



to de ned in 19 for each i. Average exp ected virtual valuations conditional on provision

i

P

n

 

1 F  =n and arguments along the same lines as for the still converge to lim

i n!1

i=1

i i

surplus maximizing case establish that the ex ante probability of provision converges to zero or one

P

n

  

dep ending on whether lim 1 F  =n is greater than or smaller than c .

n!1 i

i=1

i i

Hence, for a large economy the provision rule is almost identical to that of a b enevolent planner.

 n

for in a constrained ecient mechanism is strictly less than However the inclusion threshold

i i



each i: This can b e seen from observing that 1 F    is single-p eaked with maximum at and

i i i

i

that the optimal mechanism satis es the feasibility constraint with equality. Budget balance will b e



achieved bylowering the threshold relative since the so cial surplus is decreasing in the inclusion

i

thresholds. We conclude that the pro t maximizing mechanism is inecient due to charging to o

high a price, which excludes to o many p otential customers from usage, but that the provision

decision is not distorted in the binary case.

In the case with a quantitychoice, over-exclusions o ccur for the same reasons as in the binary

n

case: it is always b ene cial for a pro t maximizer to raise the inclusion threshold from to

i

 n

for any xed provision rule y   : In terms of the provision decision this is like switching

i

from a weighted average of per capita surplus and virtual surplus to virtual surplus alone in the

determination of the level of provision in 28. The exp ected virtual valuation is always b elow the

exp ected valuation, so in general this leads to under-provision as well as over-exclusions compared

with the constrained ecient mechanism. The prop erty that the only ineciency is that a to o high

price discourages to o many agents to participate is thus an artefact of the binary mo del and the

usual monop oly result Guth  and Hellwig [12 ] applies in the more general mo del. 19

4.3 Comparison with a Non-Excludable Public Good

It is of course an easy matter to remove the use exclusions from the mo del. In the binary case

this reduces to exactly the mo del studied in Mailath and Postlewaite [16 ], but for the case with

a quantity dimension to the problem this actually lls a gap in the literature. Without exclusion

p ossibilities the set of feasible provision rules consists of all y :! R for which E v y   is

+ i

weakly increasing and

! 

Z Z

X

1 F  

i i

y   C n  f   d  0: 29 ::: v y  

i

k k k k

f  

i i

n

1

i

The following generalization of the asymptotic imp ossibility result in Mailath and Postlewaite result

is then very easy to prove:

P

n

0 n

v 0 Prop osition 9 Suppose lim =n lim C  n =n < 0 and that f y g is a sequence

n!1 n!1

i=1

n

of incentive feasible provision rules. Then Ey   ! 0 as n !1:

Making the typical assumption that = 0 is thus sucient for the exp ected provision level to

i

b e near zero in a large economy. This demonstrates that the asymptotic imp ossibility for voluntary

agreements in a large group to realize large p otential gains is not an artefact of the collective

decision b eing a binary choice.

If costs are kept constant as n go es out of b ounds, Hellwig [13 ] shows that the asymptotic

prop erties dep end critically on whether the level of the public good is b ounded or not. Ex p ost

eciency can be achieved in the limit if the quantity is b ounded. This is b ecause the necessary

p er capita contribution is of order 1=n whereas the pivot probability in a mechanism that provides

the ecient level if and only if at least m agents announce that they have a valuation ab ove 

p p

n: Hence it is p ossible to induce payments of order 1= n from each agent with is of order 1=

p

valuation ab ove some , so the aggregate transfers are of order n; which is sucienttocover cost

for a large economy.

For the same reasons, the constrained ecient level of the public good approaches in nity

if the ecient level is unb ounded. However, the ratio of the so cial surplus for the constrained

ecient outcome and that of the ex p ost ecient rule converges to zero, thus providing an analogue

Prop osition 9. In a sense, the assumption that the ecient level of the public go o d is unb ounded is



similar to the assumption that lim C n =n = c > 0. Both assumptions ensure that transfers

n!1

p

n are insucient to generate anything close to eciency. of order 20

5 Discussion

5.1 Related Literature

The mechanism design literature on public goods provision is enormous, but the literature on

excludable public go o ds is rather limited. The most obvious exceptions are Cornelli [5], Dearden

[8 ], Hellwig [13 ], and Moulin [17 ]. I have already discussed the work of Cornelli and Hellwig

elsewhere in the pap er, so I will here fo cus on the other two pap ers.

Dearden [8 ] considers a slightly more general mo del than the binary version of the mo del

considered here. The mo del allows for crowding, but even without crowding he nds that use

exclusions will help overcome the free riding problem. The pap er also contains some results for a

large economy, but costs are held constant and exclusions are irrelevant for large economies.

Moulin [17 ] studies exclusions in the context of strategy-pro of implementation where a class of

\serial cost sharing" mechanisms fully characterizes the set of mechanisms that satisfy voluntary

participation, anonymity and strategy pro ofness see also Dearden [9 ]. The mechanism works as

follows. Agents are ordered according to their announced demands, the cost to provide the lowest

announced demand is divided equally among all agents, the incremental cost between the lowest

and the second lowest announced demand are split among the remaining agents i.e., all agents

except the lowest demand, the incremental cost between the second and third lowest are split

among the remaining agents and so on. Within this class, the p ossibility of exclusion from the

public go o d helps in alleviating the free-rider problem for similar reasons as in this pap er. It is also

notable the mechanism has a similar avor with the mechanisms considered in this pap er in that

the lowest typ e among those included determines the level of the public go o d.

5.2 Copyright Protection

Taking for granted that the collective good must be handled by a private market arrangement,

the sharp contrast b etween an excludable and a non-excludable public go o d provides an eciency

rationale for establishing \rights to exclude". Clearly, this logic can be applied to copyright pro-

tection, patents, discussions ab out whether software companies should be forced to publish their

co de in an intelligible way and many other issues. In particular this seems relevant for discussions

ab out intellectual prop erty rights relating to copyright protection, an issue which seems critical for 21

the music industry due to recent innovations in computer technology.

5.3 Time Consistency

There is a time inconsistency problem with the surplus maximizing mechanism, similar to that of

a durable good monop olist. Once contributions are collected a b enevolent planner has incentives

to make the public go o d available to everyone, and use exclusions would therefore b e non-credible.

The conclusion of this would be that, unless the planner can commit, one is back in the dismal

outcome of the pure public go o ds case.

This issue deserves more attention and I will only p oint out that commitment need not be a

crazy assumption. In many cases transfers actually o ccur ex post think ab out tollways. Requiring

the mechanism to balance the budget presumably means that something bad happ ens if costs are

not covered, such as a budget de cit or even a default on loans to pay for the construction.

6 Concluding Remarks

The basic conclusions from this pap er are that it is p ossible for a \private market" to provide

nontrivial amounts of public go o ds, as long as it is p ossible to exclude consumers, and that there

are essentially no gains to exploit b eyond second degree price discrimination.

However, to the extent one thinks that governments are able to comp el participation one could

argue that the normative recommendation still would be to let governments handle goods of col-

lective nature and avoid exclusions completely by using pivot mechanisms that makes some agents

worse o , but are b ene cial for the collective. The cheap way to dismiss this argument would of

course b e to argue that governments seem to have little to do with welfare maximization. A more

interesting approachwould b e to build a mo del that generates a disadvantage for the government

for some more fundamental reason. I b elieve that one p otentially fruitful way to think ab out this

would b e to assume that there is an additional informational problem, where some \innovator" has

a b etter idea ab out the p otential value of an excludable public go o d. 22

A App endix: Pro ofs

R R R

in the integral expressions in the pro ofs To conserve space I often write rather than :::

2

n

1

n

that follow, except when this may create confusion. I also use dF   as shorthand notation for

n

 f   d :

k k

k =1

A.1 Prop osition 2

P

  

[1F  ] c lim

n!1

i

i i

i

Pro of. Let  = > 0: An upp er b ound on the ex ante probability that

2

R

n n

   dF   sub ject to the constraints in 17. the go o d is provided is found by maximizing

2

n

It is straightforward to adapt the argument in Lemma 3 to conclude that there exists  such that

n n

 ;  solves the problem if and only if

8 8

P

> >

n n 

< <

1 if 1 +     x   C  n  0 1 if 

i i i

i

i i

n n

  = A1  =

i

> >



: :

0 < 0 otherwise

i

i

n n

Dividing by  n and de ning k C n =n 1=n we can express the ex ante probability of

n

P

n n

provision in economy n as E  =Pr[   x   =n  k ] :

i i n

i

i

P

  n  

CASE 1: Supp ose k 

[1 F  ]=n + : We note that E   x  = [1 F  ] and

n i i i i

i

i i i i i

n

n n

that f is a sequence of n indep endent random variables, where   x   g   x   2 [0; ]

i i i i

i i

i=1

2 n

for every i and n: Hence there exists some  < 1 such that the variance of   x   is less

i i

i

2

than  for all i and Chebyshevs inequality implies that

" 

P

 

n 2

X X

  x   

i i

i

i n  

Pr  k A2  Pr   x   [1 F  ]  n 

n i i i

i i i

2

n  n

i i

P P

  n

CASE 2: Next, supp ose k < [1 F  ]=n +  and de ne H  n=f j   x   =n

n i i i

i i

i i i

P P P

  n  

> [1 F  ]=n +  g and L n=f 2 j k    x   =n  [1 F  ]=n +  g.

i n i i i

i i i

i i i i i

n

Since   = 1 if and only if 2 H  n [ L  n the integral constraint evaluated at the optimal

rule is

P P

Z Z

n n

  x   C n C  n   x  

i i

i i

i n n i

dF  + dF   0 

n n n n

2Ln 2H n

 !

X

 

 C  n =n Pr  H n + [1 F  ]=n +  C n =n Pr L n ; A3

i

i i

i

P P

 n 

]=n +  for all 2 L n and that after observing that   x   =n  [1 F 

i i i

i i

i i i

n

for all i and : By the same application of Chebyshevs inequality as in A2,

  x   

i i i

i 23

P

2



 

Pr H n  [1 F  ]=n C n =n = 2 and lim C n =n = . Moreover, lim

i n!1 n!1 2

i

i i

 n

P



   

and C n =n  c  for c , so there exists N such that [1 F  ]=n +  C  n =n 

i

i

i i

2



2

2 c +

 



for n  N: n  N: Combining with A3 and rearranging shows that Pr  L n 

2



 n

Hence,

2



 n

c +  = : A4 E   = Pr H  n+PrL n  1+2

2

 n

CASE 1 and CASE 2 are exhaustive, so A2 and A4 implies that there is N < 1 such that

" 

2

2

 

n 

E    max ; 1+2 c +  = 8n  N A5

2 2

 n  n

n

Since the right hand side of A5 go es to zero as n !1 it follows that lim E  =0:

n!1

A.2 Prop osition 3

n n n

Lemma A1 Let  ;  bea sequenceoffeasible mechanisms, where for al l n and i there exists

i

n n n n n

such that  =1if  and  =0for < : Moreover, let  be the best provision rule

i i i i

i i i i

P

n n n n 

associated with for every n. Then, 1 E   ! 1 as n !1if lim [1 F  ]=n > c ;

n!1 i

i

i i

P

 n n n

]=n < c : [1 F 

2 E   ! 0 as n !1 if lim

i n!1

i

i i

P

n n

Pro of. CASE 1For any xed it is ex p ost optimal to provide if and only if   

i i

i

i

P

n

  x    C n and C n : Since  x   it follows that   =1 for all such that

i i i i i i i

i

P

n

  x  

i i i i

i

P P

Z Z

n n

  x   C n   x   C  n

i i i i

i i

n n n

i i

   dF    dF  

n n

2 2

n n n n

X X

[1 F  ] C n [1 F  ]

i i

i i i i 

! lim c > 0 A6

=

n!1

n n n

i i

n n

The integral constraint of 17 thus holds strictly for n large enough, so  ;  is feasible. Since

P P

n n n

 x   it follows that 1 E  =Pr[    C  n]  Pr[   x    C  n].

i i i i i i i i

i i

i i

n n

By hyp othesis, for each >0 there exists N such that C  n =n  [1 F  ]=n  for every

i

i i

n  N: An application of Chebyshevs inequality completes the pro of.

n

CASE 2 The probability of provision for the b est rule conditional on is b ounded ab ove by

n

the maximal probabilityof provision conditional on : Although the inclusion rules are di erent

n 0

from the ones in Prop osition 2,  still has the same form as in A1. Replacing  with  =

P

n n 

[1 F  ]=n one can pro ceed as in the pro of of Prop osition 2. 1=2c lim

i n!1

i

i i 24

n n

Pro of of Prop osition 3. Supp ose for contradiction that f ; g is a sequence of solutions to

P

1

n

n n n

be- 17 such that E   do es not converge to 1. Every element of f [1 F  ]=ng

i

i=1

i i

n=1

n n

longs to a compact set, since [1 F  ] < ; and optimality of the inclusion rules requires

i

i i

n n

that [1 F  ]  0 see 16. The provision rule in the solution to 17 must optimize the

i

i i

ob jective function conditional on the inclusion rule in the optimal solution. Taking a subsequence if

P

n

n n 

necessary, Lemma A1 implies that lim [1 F  ]=n = c and that there is some >0

n!1 i

i=1

i i

n n n

and N < 1 such that E   < 1  if E   fails to converge to unity: lim E  =1 if

n!1

P P

n n

n n  n n n 

lim [1 F  ]=n > c and lim E   = 0 if lim [1 F  ]=n < c :

n!1 i n!1 n!1 i

i=1 i=1

i i i i

The latter cannot be optimal since the surplus converges to zero and the b est mechanism with

n 

= for every i and n generates a strictly p ositive surplus. Pick some  > 0 and partition the set

1

i i

P P

n n

of for which provisions o ccur into H n=f j  =1 and   =n  E   =n +  g

i i 1

i i

i i

P P

n n n n

and Ln=f j   = 1 and   =n < E   =n +  g : Let S  ;  denote the as-

i i 1

i i

i i

so ciated p er capita surplus in the nth economy, which satis es

P P

Z Z

n n

[   C n] [   C n]

i i

i i

n i n n n i

dF  + dF   S  ;  =

n n

2L n 2H n

P

n

E   C  n C  n

i

i

i

 Pr  H n +PrL n +  A7

1

n n n

An application of Chebyshevs inequality shows that lim PrH n = 0 for any  > 0 and

n!1 1

Pr L n  Pr  L n + Pr  H n  1 : Hence, for any  ; > 0 there is a nite N such that

1 2

P

n

C n E  

i

i

n n i

+  +  : A8 S  ;   1  

1 2

n n

n n

b b

Consider an alternative sub- sequence of mechanisms f ; g where

8

>



>

>

1 if 

i

>

i

>

<

n n



n 

b b

8i 2 I; n; A9   = 1 8 2   =

if   1 i

i

i

i i

2

>

>

>

>

>

:

0 otherwise



and is the threshold that extracts the maximal transfer from the agent de ned in 19. The

i

exp ected p er capita surplus from this mechanism is

R

P P

1 



i n

f   d C n E   +



i i i i i

i i

i

2 2

n n

i

b b

S  ; = : A10

n

0 n n n n

b b

Together, A8 and A10 imply that there exists N such that S  ;  < S  ;  for every 25

P P

n n    0

[1 F  ]=n = c ; [1 F  ]=n > lim n  N : Moreover, since lim

i i n!1 n!1

i i

i i i i

 !

Z

n n

X X

b b

  x   C n C n E   x  

i i i i

n n

i i

b

A11    dF  

=

n n n

2

i i

i h

P

1 

P



n n  

 

[1 F  ] + [1 F  ]

i i

i

i i i i C n  [1 F  ]

2 2

i

i

i i 

c lim

! =

n!1

n n 2 n

00 n n 00

b b

as n !1: Hence there is N such that  ;  is feasible for n>N : We conclude that mechanism

0 00

A9 is feasible and b etter than the hyp othetical optimal mechanism for n  max fN; N ;N g.

A.3 Prop osition 4

P

  

Pro of. If lim [1 F  ]=n < c , Prop osition 4 is trivial since the per capita sur-

n!1 i

i

i i

P



plus converges to zero in the ecient mechanism. Hence, I now assume that lim [1

n!1

i

i

  n n  n

e

F  ]=n > c : Let 2 0; 1 and de ne +1  ; where is the threshold from the

i

i i i i i



surplus maximizing mechanism for every i and n and is de ned in 19 for every i: Consider a

i

n o

n n n

b

b b

sequence  ; ;   of xed fee mechanisms where for each n

8 8

P

> >

n n

e e

< <

b

   C n 1 if  1 if

i i

i

i i

n n

b b

: A12   =   =

i

> >

: :

0 otherwise 0 otherwise

8

P

>

n n n

e e e

<

b

if  and    C n

i j

j

i i j

n

b

   =

i

>

:

0 otherwise

The feasibility constraint 3 holds also ex p ost by construction and ispayo from announcement

b b

 ; is

i i

8

>

n n

b b b b e

<

b

  ;  if 

i i i i i

i

n n n

b b b b b b b

b b

: A13   ;   ;    ;  =

i i i i i i i

i i

>

:

0 otherwise

The participation constraint 4 is thus satis ed and truth-telling is a dominant strategy and

n n n

b

b b

therefore also incentive compatible in the Bayesian sense. Mechanism  ; ;  isthus incentive

h i

P

n n

e

b

feasible for every n; with probability of provision given byPr   =n  C n =n . For each

i

i i

n

n let  b e the Lagrange multiplier asso ciated with the surplus maximizing mechanism.

n n

CASE 1: Supp ose that taking a subsequence if necessary lim  =1 +   = 1: Then we

n!1

 n  n

e

for any 2 [0; 1] : = for every i, implying that lim = see from 16 that lim

n!1 n!1

i i i i

P

n n   

e e

[1 F  ]=n = lim [1 F  ]=n > c and there exists  > 0 and Hence, lim

i n!1 i n!1

i

i i i i 26

P

n n

e e

nite N such that [1 F  ]=n C  n =n   for all n  N: The probabilityof provision

i

i

i i

i h

P

n n n n n n

e e e e

b b

  =n  C  n =n and E   = [1 F  ]; so another for mechanism A12 is Pr

i

i

i i i i i i

application of Chebyshevs inequality implies that

"  " 

P P

n n n

e e e

X

b

  1 F  

i i

i i

n n

i i i

e

b b

1 E    = Pr    C n  Pr 

i

i

n n

i

 "

2

X X



n n n

e e e

b

: A14  n    1 F    Pr

i i

i i i

2

 n

i i

n n n 

e

b

Thus, lim E    = 1. Since lim = lim = it is easy to check that the per

n!1 n!1 n!1

i i i

capita surplus in the xed fee mechanism approaches that of the optimal mechanism for any :

n n

CASE 2: Supp ose instead taking a subsequence if necessary that lim  =1 +  = <1:

n!1

With some abuse of notation, let  b e the inclusion threshold from 16 asso ciated with mul-

i

R

P

n  i

tiplier : We notice that lim =  < : Therefore, lim f   d =n = lim

n

n!1 i i i i i

i

i i

n!1 n!1

i

R R R

P P P

i i i

f   d =n and lim f   d =n = lim f   d =n

n n  n 

i i i i i i i i i i i i

i i i

 +1  +1 

n!1 n!1

i i i i i

for any . Moreover, the second limit converges to the rst as ! 1: Together, this implies that

for any >0 there exists <1 and N < 1 such that

Z Z

X X

i i

f   d =n f   d =n<=2: A15

i i i i i i i i

n n 

+1 

i i i

i i

n n  

for all n  N: Next, = and = for any i; j  such that F = F and F 2F for any i;

i j i

i j i j

0 n n  n

e

where F is nite. Hence there exists >0 and N < 1 such that  =1   >

i i i i

0 

for n  N . Since 1 F   is strictly increasing on [ ; ] this in turn implies that there

i i i

i

i

n n n n

e e

]+2 : Because f  ; ::::;  g takes on at most [1 F  ]  [1 F  is  > 0 such that

i 1 n i i i

i i i i

e

as many values as the cardinality of F this in turn establishes existence of some > 0 such that

P

n n 

e

   for all i: Lemma A1 implies that lim [1 F  ]=n  c since otherwise there

i n!1 i

i

i i

P

00 n n

e e

e

is no provision in the limit. Hence there exists N such that [1 F  ]=n  C n =n + 

i

i

i i

00 n

b

for every n  N : Applying A14 again we conclude that lim E    = 1 also along such

n!1

n n n n

b b

subsequence. Let S  ;  and S  ;  b e the p er capita surplus generated by the optimal and

xed fee mechanisms resp ectively. Since the probability of provision converges to one in b oth

000

mechanisms it is easy to verify that there exists N such that

Z Z

X X

i i

n n n n

b b

S  ;  S  ;   f   d =n f   d =n + =2; A16

i i i i i i i i

n n 

+1 

i i i

i i

n n n n

b b

and using A15 this implies that S  ;  S  ;  <:Since  was arbitrary the result follows. 27

A.4 Prop osition 5

For the pro ofs of Prop osition 5,6 and 7 it is convenient to de ne the functions

P

Z Z

n n

y   C  n   x  

i i

i

n n n n

i

 f   d A17 v y   ::: G y ; 

k k k k

n n

n

1

P

Z Z

n n

  v y   y   C n

i i

i

n n n n

v y   ::: S y ;   f   d : A18

k k k k

n n

n

1

n n n

Multiplying everything with 1=n do esn't change the programming problem, so S y ; may be

n n n

taken as the ob jective function to 17 and the constraint is satis ed if and only if G y ;   0:

1

n n

bea sequence of incentive feasible mechanisms, where for each n and Lemma A2 Let fy ; g

i

n=1

n n

i  n, is a threshold rule with inclusion threshold independent of : Then, for each >0

i

i i

there exists some nite N such that

P

n n

1 F   C  n

i

i

n n

i i

v Ey    Ey    8n  N: A19

n n

P P

n n n

[1 F  ]=n >  g ; where   x   =n Pro of. Fix  > 0 and let H n = f j

i i i

i i

i i i

n n n

 = =2v  y  +1 > 0: Decomp ose G y ;  in A17 as

P

Z

  x  

i i i

i

n n n n n

dF   A20 G y ;  = v y  

n

2H n

P

Z

  x   C n

i i i

i

n n n

+ v y   dF   Ey  

n n

2nH n

P P

n n

Observing that 2 nH n implies that   x   =n  [1 F  ]=n +  wehave that

i i i i

i i

i i

the second term in the right hand side of A20 satis es

P

Z

  x  

i i i

i

n n

dF   A21 v y  

n

2nH n

P

Z

n n

[1 F  ]

i

i

n n i i

+  v y  dF   

n

2nH n

P P

Z

n n n n

[1 F  ]

[1 F  ]

i i

i i

i i n n n i i

 +  +  v y  dF    v Ey  

n n

2

where the last inequality comes from concavity of v: Since  > =2v y  and v is increasing, so



n

vEy    vy   : Combined with A21 this implies that

2

P

Z

n n

[1 F  ]    x  

i i i i

i

n n i i n

dF    v Ey   + A22 v y  

n n 2

2nH n 28

n

Finally,we note that v y    v y  for every 2  and   x    for every i; so

i i i

P

Z Z

  x  

i i i

i

n n n

v y   y  dF  =v y  Pr H  n ; A23 dF    v 

n

2H n 2H n

P P

n n n

[1 F  ]=n >  ] : By Chebyshevs inequality   x   =n where Pr H  n = Pr [

i i i

i i

i i i

there exists N such that Pr  H n < for n  N and since <=2v  y  wehave that

P

Z

  x   

i i i

i

n n

v y   dF   < A24

n 2

2H n

for every n  N: The conclusion follows by substituting A22 and A24 back into A20 and

n n n

noting that G y ;   0 for feasibility.

1

n

beasequenceofthreshold inclusion rules. Then there Lemma A3 Fix any >0 and let f g

n=1

n n n n 0n n

exists some nite N such that S y ;  >S y ;  for every n  N and any provision rules

n 0n

y ;y satisfying

R

P

i

f   d

n

i i i i

i

C n

n 0n n 0n

i

Evy   Evy   Ey   Ey   >: A25

n n

o n

R

P P

n

i

f   d =n  ; where   =n  Pro of. Fix  > 0 and de ne H n =

n

i i i i i

i i

i

i

n n

 = =2v  y  +1: For mechanism y ;  the p er capita surplus can b e decomp osed as

P

Z

n n

  v y  

i

i

n n n n i

dF   A26 S y ;  =

n

2H n

P

Z

n n

  v y   C n

i

i

n n

i

+ dF   Ey  

n n

2nH  n

 !

Z Z

X

i

C n f   d

i i i i

n n n

  v y  dF   Ey  

n

n n

2H n

i

i

where

Z Z

n n n n n

v y  dF   = Evy   v y  dF   A27

2H n 2nH n

n

y 1 Pr H  n :  Evy   v 

Together, A27 and A26 imply

1 0

R

i

f   d

n X

i i i i

C n

C B

n n n n n

i

 y 1 Pr H  n] Ey   A28 S y ;   [Evy   v 

@ A

n n

i 29

n o

R

P P

n

i

  =n  Let L n= f   d =n +  : A symmetric argument shows

n

i i i i i

i i

i

i

 !

Z Z

X

i

n 0n n 0n n

S y ;   f   d =n +  v y  dF   A29

i i i i

n

2Ln

i

i

P

v y  C  n

i

i

0n

+ [1 Pr L  n] Ey  

n n

! 

P

Z

X

i

v y  C  n

i

i

0n n

f   d =n +  Evy   + [1 PrLn] Ey   : 

i i i i

n

n n

i

i

R

P P

n

i

f   d =n; so, by Chebyshevs inequality, there exists Now, >0 and E f   g =

n

i i i i i

i i

i

i

N such that Pr  H n  1  for all n  N and Pr  L n  1  for all n  N: Together with

y  + 1 this implies that A28,A29,A25 and our choice of  = =2v 

n n n n 0n n

S y ;  S y ;  A30

R

P

i

f   d

n

i i i i

i

C n

n 0n n 0n

i

 Evy   Evy   Ey   Ey  

n n

{z } |

> by A25

Z

X

i

n

 [ Evy   v  v  f   d =n y 1 Pr H n ] y 1 Pr H  n

i i i i

n

| | {z } {z }

i

i

<

| {z }

y Pr H n ]< v y  < [v 

<

P

v y 

i

i

0n

Evy    vy   vy  vy   vy  0: [1 Pr  L n]

> =

| {z } {z } |

n

{z } |

<

< v y 

Since  was arbitrary, the result follows.

Lemma A4 Suppose v is strictly concave. Then, for each  ; > 0 there exists some >0 such

1 2

n n n n n

that v  Ey    Evy  + for every y 
 such that Pr jy   E y  j    :

1 2

Pro of. Omitted.

n n

Lemma A5 Consider a sequence of incentive feasible mechanisms f y ; g : Suppose there are

n n

 ; > 0 and N such that Prj y   Ey  j      for every n  N . Consider the

1 2 1 2

n n n n

alternative sequence f y ; g where y  =Ey   for al l 2  and every n and the inclusion

0 n n 0

rules are unchanged: Then, there exists N such that f y ; g is incentive feasible for every n  N .

n n

Pro of. By Lemma A4 there exists >0 such that v Ey    Evy   + : Moreover under

0 n 0

the hyp othesis of the Lemma there exists  such that, Ey   > for all n  N: Applying Lemma 30

P

n n

A2 this implies that there exists K > 0 such that [1 F  ]=n  K for all n  N since

i

i

i i

n

e

otherwise Ey   ! 0 since A19 would be violated otherwise. De ne  = K= vy  > 0 and

o n

P P

n n n n

e

e

[1 F  ]=n +  . A decomp osition of G in A17   x   =n  let H  n = j

i i i

i i

i i i

n n n

evaluated at y ;  along the same lines as the decomp osition of S in Lemma A3 yields

P

n n

C  n [1 F  ]

i

i

n n n n n i i

e

e

+  + v y  PrH n Ey   : A31 G y ;   Evy  

n n

n n

For the mechanism f y ; g , a direct calculation shows that

P

n n

C  n [1 F  ]

i

i

n n n n n i i

Ey   : A32 G y ; =v Ey  

n n

P P

n n n

E   x  = [1 F  ]; so an application of Chebyshevs inequality shows that there

i i i

i i

i i i

0 n n

e

e

exists N  N < 1 such that PrH  n < : Using this and that v Ey    Evy   +

0

 together with A32 shows that for n  N

P

n n

] [1 F 

i

i

n i i n n n n n n

e

e

G  y ;   G y ; + v y  PrH  n Evy  

| | {z } {z }

n

| {z }

e

<

>K

n n n n n n

e

> G y ; +K vy 1 + =G y ;   0; A33

n n n n

where the nal inequality follows b ecause y ;  is incentive feasible. Hence  y ;  is also

0

incentive feasible for n  N .

Pro of of Prop osition 5. If the prop osition would fail, then taking a subsequence if necessary

n n

there exists some  > 0 and  > 0 such that Pr jy   Ey  j    for all n: Consider

1 2 1 2

n n

y ; g ; where for each n the only di erence with the a sequence of alternative mechanisms f

n n

initial mechanism is that y   = Ey   : By Lemma A4 we have that there exists  such that

n n

v Ey    Evy  + holds for every n in the sequence. Moreover, for the same reasons as

R

P

i

in Lemma A5 there exists K>0 such that f   d =n  K: Hence,

n

i i i i

i

i

R

P

i

f   d

n

i i i i

i

C n

n n n n

i

y   Evy    E y   Ey   Ev >K > 0; A34

n n

n n

y ;  generates a which by application of Lemma A3 implies that there exists N such that 

n 0 n n

y; : By Lemma A5 there exists N such that y ;  is incentive feasible higher surplus than 

0 n

for n  N : Hence wehave contradicted that f y; g is a sequence of optimal mechanisms. 31

A.5 Prop osition 6

Lemma A6 
; 
 and Q
 de ned in 26, 26 and 28 satisfy the fol lowing properties:

1. 
; 
 and Q
 are continuous functions of

2.   >   for every 2 [0; 1]

3. 
; 
 and Q
 are weakly decreasing in

 0 00 0

4.  v Q  Q c is weakly decreasing in : Moreover, if < is such that Q  >

00 0 0 0  00 00 00 

Q ; then  v Q  Q c >  v Q  Q c .

Pro of. PART1 1  + x   is strictly increasing in ; implying that there is a unique

i i i i

e e

threshold   2 [ ; ] such that 1  + x    0 if and only if   : Moreover,

i i i i i i i

i

e e

1  + x   < 0 for every <   and 1  + x   > 0 for every >  ; so

i i i i i i i i i i

n n n

e

e e

 ;  =  ;  for all 6=  : if f g is a sequence with lim ! ; then lim

i i i i i i n!1 n!1

Continuityof
 and 
 follows by Leb esgues' monotone convergence theorem and the theorem

of the maximum guarantees that Q
 is an upp er-hemi-continuous corresp ondence. Strict concavity

of v implies that Q  is unique for every 2 [0; 1] ; implying that Q
 is a continuous function.

e

e

PART2 Each inclusion rule 
;  can b e characterized by an inclusion threshold   p ossibly

i i

 for each : Since x   is continuous and x  equal to = 1 F  =f  = there

i i i i i i i i i i

i

0 0 0

e e

such that x   > 0 for all  and    1  for every 2 [0; 1] : Hence, exists <

i i i i i i

i i i

P

Z Z

r

e

 ;  x  

i i i i i

i=1

n

dF   A35     = :::

r

r

1

Z Z

r r

X X

i i

1 1

=  x   f   d = 1 F   d > 0

i i i i i i i i i

r r

e e

   

i i

i=1 i=1

0 00

PART3 The pro of which is omitted see Norman [20 ] uses that Q  is b etter than Q  for

0 00

= and vice versa for = together with PART 2 of the claim.



PART4 To show that  v Q  Q c is weakly decreasing, supp ose for contradiction that

0 00 0 0 0  00 00 00 

there exists < such that  v Q  Q c <  v Q  Q c : Then,

 

0 0 0 0 0 0 

v Q  1  +   Q c A36

00 00 00  0 0 0 0

<  v Q  Q c + v Q     

| {z }

<0 32

00 00 00  00 0 0 0

  v Q  Q c + v Q     

 

00 0 0 0 0 00 

= v Q  1  +   Q c ;

0 0 0 00

contradicting the assumption that Q  solves 28 for = : Finally, if Q  > Q  then

0 0 0 0 00 0 0 0

v Q      

diction also if from a weak inequality,thus validating the nal claim.

r

Lemma A7 Consider a sequence of replications of a given nite economy fv; [0; y ];C  r  ;F g :

1

n n n

Let fy ; g bea sequence of optimal mechanisms and  be the Lagrange multiplier associated

n=1

n



n n n

with the mechanism y ;  and = for each n: Then, for any subsequence fn g such that

n

k

1+

n

k

lim = we have that

n !1

k

P

Z Z

n

k

 

i

i

i

 f   d =   A37 lim :::

k k k k

n !1

n k

k

n

1

k

P

Z Z

n

k

  x  

i i

i

i

::: lim  f   d =   A38

k k k k

n !1

k n

k

n

1

k

Pro of. Omitted see Norman [20 ].

n

k

Lemma A8 Suppose the hypotheses of Lemma A7 are ful l led. Then lim Ey  = Q 

n !1

k

n

k

for any subsequence such that lim =

n !1

k

Pro of. De ne  ; =1  + x   and let

i i i i i

P

n n

n

k k

k

   ;  yC n 

v y E 

i i

k

n

i i=1

k

Y arg max A39

n

y 2[0; y ]

k

P

n n

n

k k

k

;  yC  n     v y 

i i

k

n

i i=1

k

; y   = arg max

n

y ] y 2[0;

k

n n

k k

where Y is well-de ned for every n by strict concavityofv and y   is an alternativewayto

k

express the provision rule in 15 for economy n : Pick an arbitrary >0 and let m =2y +1 < 1:

k

n

k

  in the parameters of the problem the theorem of the maximum there exists By continuityofy

n n 

k k

>0 such that jy   Y j < =m for all such that jC n =n c j and

P P

n n n n

n n

k k k k

k k

   ;  E    ; 

i i i i

i=1 i i=1 i

  A40

n

k

n

n

k

k

Moreover, f  is a sequence of indep endent random variables with b ounded variance, ; g

i i

i=1

so an application of Chebyshevs inequality to implies that there exists N such that

" 

n n

k k

X X



n n

n n

k k

k k

: A41 Pr    ;  E

   ;  <  n

i i i i

k

i i

m

i=1 i=1 33

 

n n

k k

for every n  N: It follows that y    Y with probability of at least 1 ; so

k

m m

  

n n n n

k k k k

+    Y Y for n  N . Symmetrically, y with probabilityofat  1 Ey

k

m m m

   

n n n

k k k

least 1 and y    Y + + y for all ; so Ey  1 y for all n  N: Since

k

m m m m

m =2y +1wehave that

   

n n n n n

k k k k k

Y , Ey Y A42 Y 1  Ey  1

m m m m

  

n n

k k

 Y  Y 

m 2y +1 2

    

n n n n n

k k k k k

Ey  1 y , Ey Y  y Y +1 Y + +

m m m m m

  

 y +1  y +1    ;

m 2y +1 2



n n

k k

for all n  N: To complete the argumentwe observe that given a subsequence so jEy Y j

k

2

n n

k k

f g such that ! as n !1 wemay apply Lemma A7 to conclude that

k

P

n n

n

k k

k

E

   ; 

i i

i=1 i

lim =1  +   A43

n !1

k

n

k



and lim C  n  =n = c by assumption. The theorem of the maximum assures that there

n !1

k k

k

0 n 0 00 0

k

exists some nite N such that jY Q j  =2 for n  N : Picking N = max fN; N g the

k

n

k

triangle inequality implies that jEy Q j: Since >0was arbitrary the result follows.

n n n

k k k

Lemma A9 Suppose the hypotheses of Lemma A7 are ful l led. Then lim S y ;  =

n !1

k

 n

k

 v Q  Q c for any subsequence f n g such that lim =

n !1

k

k

Pro of. Pick an arbitrary >0 and let

P

Z

n

k

C n   

i

k

i

n n n n

i

e

k k k

dF   Ey   A44 S = v  Ey  

n n

2

k k

Combining A37 in Lemma A7 with Lemma A8 we conclude there exists some nite N such

n 

e

k

that 1 + v y  < 1: Conti- S  v Q  Q c < =2 for all n  N: Let m = 2

k



n

k

nuity of v implies that there exists  > 0 such that jv Ey   v  y j < for all y such that

m

n 0

k

jy Ey  j: Moreover, Prop osition 5 guarantees that for every >0 there exists N such that



0 n n n n

k k k k

for all n  N : For each n let H n =f j y    Ey    g Pr jy   Ey j  

k k k

m

P

n

k

 

i



i

i

and  and observe that Pr  H n   1 for every 2 ; so

k

m n

k

P P P

Z Z Z

n n n

k k k

     

i i i

i i i

n n n

i i i

dF   = dF   dF  

n n n

2H n  2 2nH n 

k k k

k k

P

Z

n

k

  

i

i

n

i

A45 dF   

n m

2

k 34

n n 00 0

k k

Since y    Ey    for every 2 H n  it follows that for all n  N = max f N; N g ;

k k

P

Z

n

n n

k

k k

  v y   y   C  n 

i

k

i

n n n n

i

k k k k

S y ;  = dF   A46

n

2

k

P

Z

n

k

C  n 

 

i

k

i

n n n

i

k k k

dF   Ey    v Ey    

n n

2H n 

k k

k

P

 

Z

n

k

 C n   

i

k

i

n n n

i

k k k

 v Ey   dF   Ey  

m n n

2H n 

k k

k

" 

P

 

Z

n

k

  C n 

 

i

k

i

n n n

i

k k k

 v Ey   dF   Ey  

m n m n

2

k k

 !

P

Z

n

k

   

i

i

n n n

i

e

k k k

v Ey   dF  + = S

m n m

2

k

  [1 + v y ]

n n

e e

k k

 S = S ;

m 2

n n

k k

where the last inequality comes from m =2 1 + v y . Next, let L n =f j y    Ey +  g :

k

P

n



00 n

k

k

Observing that Pr L  n   1    ; y for n  N y for all ;and that   =n 

i

k k k

i

i

m

for all wehave that

P

Z

n

n n

k

k k

  v y   y   C n 

i

k

i

n n n n

i

k k k k

S y dF ;   A47  =

n

2

k

P

Z

n

k

 

i

i

n n

i

k k

dF    v  Ey  +  

n

2Ln 

k

k

P

Z

n

k

C n   

i

k

i

n n

i

k k

+v  y  dF   Ey  

n n

2nLn 

k k

k

P

 

Z

n

k

 

 C  n  v y 

i

k

i

n n n

i

k k k

dF  +  v Ey   + Ey  

m n m n

2

k k

P

Z

n

k

 1 + v y    

i

i

n n n

i

e e

k k k

= S + v y   S + dF  +

m n m

2

k



n

e

k

 S + :

2

 

00 n  n n n n

e e

k k k k k

for all n  N and since for S [ v Q  Q c ] < Hence, S  y ;  S 

k

2 2

 n n 00 n

k k k

 [ v Q  Q c ]j < ; y all n  N it follows from the triangle inequality that jS

k

: Since  was arbitrary the result follows.

 n 

Pro of of Prop osition 6. If there exists no y such that lim Ey  =y Lemma A8 implies

n!1

1

n n

that f g cannot b e a converging sequence. Since each 2 [0; 1] ; a compact set, it must then

n=1

n n

L H

k L k H 

exist at least two accumulation p oints ; and corresp onding subsequences f g ; f g

n

J L H L H L H

k J 

= for J = L; H where Q  6= Q : Lab el ; such that < with lim

n !1

k J  35

L H

and observe that parts 3 and 4 of Lemma A6 imply that Q  >Q  and

L L L  H H H 

 v Q  Q c >  v Q  Q c : A48

 H L

Pick some  2 0; 1 and let Q = Q +1  Q . Consider a sequence of mechanisms

1

n n

fy ; g , where for each n

n=1

n  n n L

y  =Q for all 2   =  ;  for all 2  A49

i

i i

This mechanism generates a p er capita surplus

P

Z Z

n L

 ;  C n

i

i

n i n n  n 

S  ::: y ; =v Q  dF   Q ; A50

n n

n

1

n n n  L  

implying by Lemma A7 that lim S  y ; =v Q   Q c ;while

n!1

H H H  n n n

k H  k H  k H 

= v Q  Q c : A51 ; y lim S

n !1

k H 

Strict concavity implies that

h i

 L   H L L  

v Q   Q c >  v Q  + 1 v Q    Q c A52

h i h i

L H H  L L L 

=   v Q  Q c +1   v Q  Q c

i i h . . h

L L L  L H H H H 

+1   v Q  Q c         v Q  Q c

H H H 

  v Q  Q c ;

n n n

k H  k H  k H 

where the nal inequality comes from A48. Hence there exists N such that S  y ;  >

n n n

n n

k H  k H  k  H 

y ;  for large S y ;  for all n  N: Finally, we will establish feasibility of 

k H 

n n n n

k L k L k H  k  H 

enough n: Since f y ; g and fy ; g are sequences of optimal mechanisms, they

must also b e sequences of feasible mechanisms, whichby Lemma A2 requires that for every > 0

there exists a nite N such that A19 holds for j = L; H and every n  N: From A38 in

k J 

n n

k J  k J 

1F  

i

n

J J

i i

k J 

Lemma A7 we have that lim =   and lim Ey = Q 

n !1 n !1

k J  k J 

n

k J 

by virtue of Lemma A8. Combining this with A19 it follows that a necessary condition for

n n n n

k H  k H  k  L k L

g to satisfy feasibilityeverywhere in each sequence is that ; g and fy ; fy

J J J 

v Q     Q c A53

n n n

holds for J = L; H : The mechanism S  y ;  is feasible if

P

Z

n L

 ; x   C n

i i

i

i  n 

 v Q dF   Q  0; A54

n n

36

R R

P

n L

:::  ; x   =n f   d = where A38 in Lemma A7 implies that lim

i i n!1

k k k k

i

i

n

1

L L H

  and     ; so from A53 it follows that

H L H H H  L L L 

v Q     v Q     Q c and v Q     Q c A55

L H L L L 

  > v Q   +1  v Q    v Q



H  L   

 Q c +1  Q c = Q c :

0

Using A55 we see that the limit of the left hand side in A54 is strictly p ositive, so there exists N

n n n

n n n 0

k H  k H  k H 

such that S   ;  is b oth feasible and b etter y ;  is feasible for n  N : Hence S y

00 0

than the hyp othetical optimal solution for n  N = max f N; N g : The result follows.

k H 

A.6 Prop osition 7

n n n n n

Pro of. PART 1 Any feasible mechanism y ;  satis es Gy ;   0 and v y   

0 n 0 n

v 0 + v 0y  = v 0y  by concavityofv: Hence,

P

Z

n n

y   C n   x  

i i

i

n 0 n i n n

dF   v 0 y   Gy ;  

n n

2

1 1

n n  

e

De ning p   = y   = y , C n = C n ; and c = c we conclude that a necessary

0 0

v 0 v 0

R

P

n n n

e

condition for the constraint to b e satis ed is that   x   C n p   dF    0

i i

i

i

2

P

  

1 F  =n < c : Prop osition 2 can be thus be applied to conclude that and lim

i n!1

i

i i

n n n

yE p   and y is nite. Ep   ! 0as n !1. The conclusion follows since Ey   

P

r

0 0   

PART 2-SKETCH No matter whether v  !1 as  ! 0orif v 0 1 F  =n > c

i

i=1

i i

P

r

0    n

there exists  > 0 such that v   1 F  =n > c . If the hyp othesis is false y  

i

i=1

i i



converges in probability to y = 0; in which case the per capita surplus go es to zero as n ! 1:

n 

A feasible mechanism is to provide y  =  for all and include agent i if and only if  :

i

i

Plugging into the constraint and taking limits we see that the constraint is satis ed for n large

enough and since this generates a p ositive per capita surplus it is b etter than the hyp othetical

optimal mechanism.

A.7 Prop osition 9

n

Pro of. Set   = 1 for all i and all and pro ceed as in Part 1 of the pro of of Prop osition 7.

i 37

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