Pblicgoods Economics presentations basics.pdf

Public goods

1. Public goods
Microeconomics 2
Bernard Caillaud
Master APE - Paris School of Economics
February 2, 2017 (Lecture 4)
Bernard Caillaud Public goods

2. I. The nature of goods { I.1. Examples and denitions
Suppose I grow rare owers....
... I can sell them to you: rivalry and exclusion
... I can open a ower exhibition and charge you an entry
fee for the delightful view: non-rivalry but exclusion
... I can keep them but it improves the chances that these
rare seeds continue to exist, i.e. I contribute to biodiversity:
non-rivalry and non-exclusion
In the rst case, there is a market that "works" probably
well enough
In the second case, there is some sort of a market that works
dierently
In the third case, there is (yet) no market: if I stop incurring
the cost, we (on earth) will all become "poorer" !
Bernard Caillaud Public goods

3. I.1. Examples and denitions
Denitions
- A good is
cannot be consumed by more than one person at the same time.
- A good is
ally feasible to prevent some people to consume the good.
Rival and excludable goods: private consumption goods...
we know that !
Rival and non-excludable goods: common resources,
e.g. red tuna in the sea
Non-rival and excludable goods: pay-TV, computer soft-
ware, patented knowledge - ideas
Non-rival and non-excludable goods:,
e.g. national defense, scientic knowledge - ideas, public TV
Bernard Caillaud Public goods

4. I.1. Examples and denitions
In fact, a matter of degree of rivalry and exclusion
A public good makes collective consumption possible
But the satisfaction from consuming it may depend on others
consuming it (e.g. network eects, congestion,...)
Subtle dierence: reduction of value vs destruction by con-
sumption!
Strong link between public goods and externalities
"Public" goods are not necessarily supplied by the govern-
ment: e.g. TF1, research in private universities
"Private" goods may be supplied by public rms / organi-
zations: health services, mail delivery
Bernard Caillaud Public goods

5. I.1. Examples and denitions
The intriguing example of roads
A non-toll road with uid trac is a public good: I can
drive without bothering others and I cannot be prevented
from driving on this road
Toll highways are not pure public goods, they are excludable.
Roads may also be forbidden for heavy trucks.
Paris' circular highway (Boulevard Peripherique) is packed
almost always: one additional driver prevents the others
from using this facility: the good become (almost) rival due
to extreme congestion externalities.
Bernard Caillaud Public goods

6. I.2. Road map for today
Objectives: analysis of economy with apure public good
Markets tend not to provide public goods eciently
Foundations for public / market intervention
Introduction to public economics and environmental eco-
nomics
Precise roadmap:
The basic market failure in a simple example: BLS condi-
tions for eciency, inecient private provision
Remedies: quotas, taxes, Lindhal equilibria, voting on public
good provision
Link between externalities and public goods
Why asymmetric information is a major concern
Bernard Caillaud Public goods

7. II. The basic market failure { II.1. A simple economy
with public good
We will investigate the case of apure public good, that is of a
non-rival, non-excludable good, in a simple environment
An economy withIconsumers, andH+ 1 goods
Goodsh= 1; :::; Hare standard private goods; goodh= 1
is normalized as the numeraire
Goodh= 0 is a public good: whenx0is available in the
economy, all consumers benet fromx0
Consumers' preferences are represented by:u
i
(x0; x
i
) in which
x
i
is the bundle of private good consumptionx
i
= (x
i
1
; x
i
2
; :::; x
i
H
)
u
i
(:) is assumed dierentiable, increasing and concave
Bernard Caillaud Public goods

8. II.1. A simple economy with public good
Production of the public good through a rm (or equiva-
lently a sector of identical rms) with technology:
y0f(y)
withy= (y1; y2; :::; yH) the vector of input (counted posi-
tively)
f(:) is assumed dierentiable increasing and concave
Available total initial endowments in private goods!=
(!1; !2; :::; !H)
Bernard Caillaud Public goods

9. II.2. Optimal provision of public good
We look for Pareto optima in this economy
maxu
1
(x0; x
1
) [ 1= 1]
8i6= 1; u
i
(x0; x
i
)vi [i]
y0f(y) [ ]
y0=x0and8h >0;
P
i
x
i
h
+yh=!h[h]
Usual FOC across private goods:8(h; k) non-null and8i
h
k
=MRS
i
h;k
=
@hu
i
@ku
i
=
@hf
@kf
=MRTh;k
Bowen-Lindhal-Samuelson conditions 8h; i

h
=
I
X
i=1
MRS
i
0;h
=
I
X
i=1
@0u
i
@hu
i
=
1
@hf
=MRT0;h
Bernard Caillaud Public goods

10. II.2. Optimal provision of public good
dx0requires
dx0
@hf
of inputhand increases by@0u
i
dx0any
i's utility; maintainingi's utility (i6= 1) constant by reducingi's
consumption ofhbydx
j
h
=MRS
i
0;h
dx0; overall, for agent 1:
@0u
1
dx0+@hu
1
[
dx0
@hf
+
X
i6=1
MRS
i
0;h
dx0] = 0,BLS
Remark:Given FOC wrt private goods, BLS forh= 1 is su-
cient
Particular case:for separable utilities without revenue eect,
I
X
i=1
@0u
i
(x
Opt
) =
1
@1f(x
Opt
)
=mc(x
Opt
0
)
sum of marginal benets = marginal cost of public good
Bernard Caillaud Public goods

11. II.2. Optimal provision of public good
Linear quadratic example
Only two goods, the public good and the numeraire
u
i
(x0; xi) =xi
i
2
(1x0)
2
, with12:::I
f(y) = (2y)
1
2, so thatc(y0) =
1
2
y
2
0
andmc(y0) =y0
BLS conditions:
I
X
i=1
@0u
i
(x) =
I
X
i=1
mb
i
(x0) = (
I
X
i=1
i)(1x0) =x0=mc(x0)
the
sum of private marginal benets (one unit benets all!), it
must be equal to the (social) marginal cost
Pareto optimum:x
Opt
0
=
P
I
i=1
i
1+
P
I
i=1
i
Symmetric case:x
Opt
0
=
I
1+I
which goes to 1 whenI! 1
Bernard Caillaud Public goods

12. II.3. Private provision for the public good
Do market allocation mechanisms attain Pareto optimality ?
Suppose total quantity of public good = sum of all quantities
purchased individually by consumers
Each consumerichooses how much of the public goodx
i
0
to
buy, taking as given the price system AND the amount of
public good purchased by other consumers
Subscription equilibrium, i.e. private provision of public good
(x
i
0
; x
i
)
I
i=1
;(y

0
; y

);(p

0
; p

) such that:
(x
i
0
; x
i
) = arg max
x
i
0
;x
iu
i
(x
i
0
+
P
j6=i
x
j
0
; x
i
) under budget
constraintp

0
x
i
0
+p

x
i
B(p

0
; p

) andx
i
0
0; non-
negativity not trivial if others make purchases!
(y

0
; y

) = arg maxy0;y(p

0
y0p

y) withy0f(y)
Markets clear:
P
i
x
i
0
=y

0
and8h >0;
P
i
x
i
h
+y

h
=
P
i
!
i
h
Bernard Caillaud Public goods

13. II.3. Private provision for the public good
FOC:8(h; k) non-null and8i
MRS
i
h;k
=
@hu
i
@ku
i
=
p

h
p

k
=
@hf
@kf
=MRTh;k
MRS
i
0;h
=
@0u
i
+
i
@hu
i
=
p

0
p

h
=
1
@hf
=MRT0;h
with
i
0 multiplier associated to non-negativity constraint
If at the equilibrium,y

0
>0, then9^iwith a positive demand
for public good, hence
^i
= 0 andMRS
^i
0;h
=
p

0
p

h
=MRT0;h
If the public good is always a "good", positiveMRS
i
0;h
for
alliand therefore:
P
I
i=1
MRS
i
0;h
> MRT0;h
Bernard Caillaud Public goods

14. II.3. Private provision for the public good
The sum of MRS across agents not equal to MRT !.
The free-rider problem: agent ionly takes into account his own
private marginal benet of purchasing the public good, and not
positive eects on others. With concavity: demand for the public
good too low, compared to optimum
Particular separable linear case:
@0u
i
(x

0
; x
i
) =p

0
, foriwith positive demand, so:
I
X
j=1
@0u
j
(x

0; x
j
)> p

0
Under-provision of public "good", over-provision if public
"bad"
No obvious conclusion in general equilibrium (prices move)
Bernard Caillaud Public goods

15. II.3. Private provision for the public good
Linear quadratic example
For anyi,

i
+i(1x

0) =
cons:
p

0=
prod:
x

0
Only one agent can have a positive demand at equilibrium,
agent 1 who has the largest private marginal benet: hence,
x
1
0
=
1
1+1
=x

0
while fori >1,x
i
0
= 0
Under-provision:x

0
=
1
1+1
<
P
I
i=1
i
1+
P
I
i=1
i
=x
Opt
0
In the symmetric case, equilibrium isx

0
=

1+
, shared in
any away among agents: insensitive toI, while Pareto opti-
mumx
Opt
0
!1 whenIincreases: large ineciency in large
economies, the free-rider problems become more serious
Bernard Caillaud Public goods

16. II.3. Private provision for the public good Public good x
0
mb
1
mb
2
mb
3
prices
p
0
1
Demand curve
marg. cost
x
0
Opt
x
0
*
Social
willingness
to pay for
public good
Bernard Caillaud Public goods

17. II.3. Private provision for the public good
Graphical analysis in the separable linear case withL= 1:
To determine the aggregate demand for a private good, we sum
marginal benet curves
To determine the social marginal benet for the public good,
which then dictates Pareto optimality characterization, we have
to sum marginal benet curves
Yet, in the equilibrium of private provision of the public good,
only the private marginal benet curve for the highest marginal
benet agent matters.
Bernard Caillaud Public goods

18. III. Remedies { III.1. Government regulation and taxes
As for externalities, the problem of public good provision opens
the door for government intervention !
First solution: regulation(quotas)
Government (central planner) directly manages the provi-
sion of public good, imposing a levelx
0
0
Through government-owned public rm / service (Defense,
Meteo)
Through government regulation of private rm / service
(Water treatment, Bus in local community)
Concessions, Delegation of Public Service: mandatory ser-
vice explicit in contract between public authority and rm
Informationally demanding and benevolent government
Bernard Caillaud Public goods

19. III.1. Government regulation and taxes
Second solution: a tax (or subsidy)on private purchases of
public good
Consumerisubject to (personalized) tax on public good
purchasesi=
P
j6=i
@0u
j
(x
Opt
)
@1u
j
(x
Opt
)
(subsidy to encourage de-
mand)
New budget constraint: (p

0
+si)x
i
0
+p

x
i
B(p

0
; p

)
FOC involving the public good and the numeraire:8i
MRS
i
0;1=p

0+si,
@0u
i
(x
i
0
+
P
j6=i
x
j
0
; x
i
)
@1u
i
(x
i
0
+
P
j6=i
x
j
0
; x
i
)
+
X
j6=i
@0u
j
(x
Opt
)
@1u
j
(x
Opt
)
=p

0
There exists an ecient equilibrium
Informationally demanding and high cost of implementation
Bernard Caillaud Public goods

20. III.1. Government regulation and taxes
Linear quadratic example
For anyi, the public good purchase is taxed (in fact subsi-
dized, as is intuitive given the under-provision without in-
tervention) at ratesi=
P
j6=i
j
1+
P
j
j
<0
FOC for equilibrium with taxes are: for alli,
i(1x
i
0
X
j6=i
x
j
0
) +
P
j6=i
j
1 +
P
j
j
=p

0=x0
x0=x
0
0
=
P
j
j
1+
P
j
j
solves this system of equations: i.e. the
Pareto optimal allocation
Individual purchases are undetermined: there are multiple
equilibria that yield same global public good level !
Bernard Caillaud Public goods

21. III.2. Lindhal equilibria
Pure market solution: one market per consumerifor the
good benets experienced by consumeri!
i's consumption of public good is a distinct commodity with
own market and pricep
i
0
ichooses his
subscription eqlb!) and private consumptions, given prices
Lindhal equilibrium
(x
i
0
; x
i
);(y
i
0
; y

);(p
i
0
; p

) fori= 1;2; :::Isuch that:
(x
i
0
; x
i
) maximizesu
i
(x
i
0
; x
i
) under budget constraint
p
i
0
x
i
0
+p

x
i
B(p

0
; p

)
(y
i
0
; y

) maximizes the rm's prot
P
I
i=1
p
i
0
y
i
0
p

yun-
der thejoint productiontechnological constrainty
1
0
=
y
2
0
=y
I
0
=f(y)
Markets clear:8i; x
i
0
=y
i
0
and8h;
P
i
x
i
h
+y

h
=
P
i
!
i
h
Bernard Caillaud Public goods

22. III.2. Lindhal equilibria
The rm in fact maximizes (
P
I
i=1
p
i
0
)f(y)p

y, hence:
@hf
@kf
=
p

h
p

k
and
1
@hf
=
P
I
i=1
p
i
0
p

h
Consumers' immediate FOC:
@hu
i
@ku
i
=
p

h
p

k
and
@0u
i
@hu
i
=
p
i
0
p

h
Hence the BLS conditions:
I
X
i=1
@0u
i
@hu
i
=
P
I
i=1
p
i
0
p

h
=
1
@hf
In a Lindhal-version of the economy, the competitive equilibrium
yields the Pareto optimal allocation
Bernard Caillaud Public goods

23. III.2. Lindhal equilibria
No surprise ! Lindhal economy has only private goods and com-
plete markets under perfect competition: so, equilibria are ef-
cient. Moreover Pareto optima in the Lindhal economy cor-
respond to Pareto optima in the original economy. The joint
production aspect is inconsequential
Critical assumption 1: public good must be,
otherwise consumer would not buy the public good for his
own experience, he would free-ride on others' purchases
Critical assumption 2: only
individualized markets for public good!
tition assumption is not tenable
Lindhal equilibria: ne theoretical but unrealistic solution
Market solutions not convincing for public goods (contrast
with localized externalities)
Bernard Caillaud Public goods

24. III.3. Political economy equilibria
From the Lindhal equilibrium theory, individualized lump sum
taxesti=p
i
0
x
0
0
, with consumers choosing their private con-
sumption on private markets, enable the government to produce
and nance the optimal quantity of public good (but informa-
tion...!)
We now go further in the direction of formalizing public nance
by looking at government budget constraint for nancing the
public good
Same economy, but for simplicity, public good produced
from the numeraire only
A
by lump sum transfers from consumers: (x0;ft
i
g
I
i=1
) such
that
P
I
i=1
t
i
=f
1
(x0), wheret
i
is paid (input supplied) by
consumeri
Bernard Caillaud Public goods

25. III.3. Political economy equilibria
A political economy equilibrium
(p

;(x

0
;ft
i
g
I
i=1
);fx
i
g
I
i=1
) withx

0
=f(
P
I
i=1
t
i
) such that:
lettingX
i
(p; t
i
; x0) denotei's demand functions under con-
straintpx
i
+t
i
p!
i
Consumers maximize utility and pay taxes: x
i
=
X
i
(p

; t
i
; x

0
)
Markets clear:8h >1;
P
I
i=1
x
i
h
=
P
I
i=1
!
i
h
and
P
I
i=1
x
i
1
=
P
I
i=1
!
i
1

P
I
i=1
t
i
(taxes in the numeraire)
andthere exists no budget(x0;ft
i
g
I
i=1
) withx0=
f(
P
i
t
i
), that improves all agents' welfare, i.e. such
thatu
i
(x0; X
i
(p

; t
i
; x0))u
i
(x

0
; x
i
) (one at least strict)
Idea is to study mode of nancing (government budgets) that
cannot be unanimously defeated by agents
Bernard Caillaud Public goods

26. III.3. Political economy equilibria
Optimality of political economy equilibrium
A political economy equilibrium is a Pareto optimum
If not,9(x0;x
i
),x0=f(
P
i
(!
i
1
x
i
1
)), that dominates i.e.:
u
i
(x0;x
i
)u
i
(x

0
;x
i
)
Consider budget (x0;t
i
) witht
i
=p

(!
i
x
i
)
It nances the public good, i.e.:
P
i
t
i
=f
1
(x0) since
P
i
!
i
h
x
i
h
= 0 and
P
i
!
i
1
x
i
1
=f
1
(x0) (by feasibility)
It is preferred by all, since maxu
i
(x0; x
i
) under constraint
p

x
i
p

!
i
t
i
=p

x
i
necessarily yields (weakly) larger
maximum thanu
i
(x0;x
i
), hence thanu
i
(x

0
; x
i
)
Contradiction
Bernard Caillaud Public goods

27. III.3. Political economy equilibria
Very heavy procedure to attain a non-unanimously-dominated
budget; in particular, assume that agents reveal their preferences
to block budget proposal, although they may reduce their taxes
by misrepresenting their preferences
Describe a more realistic mode of political decision process: vot-
ing procedures to determine a budget
Linear quadratic example
With initial endowment!
i
= 1 in the numeraire andIodd
Constitution: egalitarian nancing of the public good, at
leveltper capita, simple majority vote ont
Using budget constraintx
i
= 1tandx0=f(
P
i
t
i
) =
f(It) = (2It)
1=2
, agents' indirect utility is:
v
i
(t; i) = 1t
i
2
(1(2It)
1=2
)
2
Bernard Caillaud Public goods

28. III.3. Political economy equilibria
Linear quadratic example, cont'd
Note:@tv
i
=1 +i(
q
I
2t
I)
v
i
(:) concave,@tv
i
positive fort!0, negative fort!1,
maximum atti=
I
2
i
2(1+Ii)
2: unimodal
Median voter theorem: with unimodal preferences, the me-
dian voter preferred policy is a Condorcet winner, i.e. wins
in a simple majority vote
Let, the median coecient, the levelt

=
I
2
2(1+I)
2is
adopted, which yieldsx0=
I
1+I
6=x
Opt
0
=
P
I
i=1
i
1+
P
I
i=1
i
In general inecient, except if the median equals the mean
coecient (e.g. under symmetry): ineciency in very non-
equalitarian economies
Bernard Caillaud Public goods

29. III.3. Political economy equilibria
Political economy equilibria with very sophisticated public
decision procedures lead to eciency, but are also quite un-
realistic
More realistic procedures miss eciency
Moreover, there are no well-behaved model of voting for
more complicated situations of public good decision
Going further in this direction requires full courses in public -
nance and in the theory of collective choices
Bernard Caillaud Public goods

30. IV. Public goods and multilateral externalities
Consider air pollution / foul air:
Non-source specic externality from factories on people
Foul air as a public "bad", non-rival and non-excludable !
Multilateral externalities
ternalities. They have a lot in common with public goods
Market-based solutions are less convincing and quotas / taxes
more appropriate for public goods than for bilateral externalities:
so what in the case of multilateral externalities ?
Bernard Caillaud Public goods

31. IV. Public goods and multilateral externalities
Multilateral
ence of the externality by one agent reduces the amount felt by
another agent
E.g. dumping of garbage on people's property
Characteristics of a private good (garbage oni's land does
not aecti
0
)
Market solutions appropriate: property rights + trade
Multilateral
Air pollution, smog through automobile use, congestion
Close to public goods
Quotas / taxes more appropriate
Bernard Caillaud Public goods

32. IV. Public goods and multilateral externalities
Very simple partial equilibrium model in a much reduced form:
Through producing,Jrms generate a negative externality
onIconsumers
Emitting an externalityzjcorresponds to a prot for rm
jequal toj(zj),j(:) concave: prot maximization yields

0
j
(z

j
) = 0
Externality is homogenous: total externality isz=
P
j
zj
When experiencing externalityyiand consuming the amount
of numerairexi, consumerigets quasi-linear utilityxi+
ui(yi),ui(:) decreasing (negative externality) concave
Bernard Caillaud Public goods

33. IV. Public goods and multilateral externalities
Pareto optimum withdepletable multilateral externality:
max
(y;z)
X
j
j(zj) +
X
i
ui(yi)
s:t:
X
j
zj=
X
i
yi
FOC: for alli; j,u
0
i
(y
Opt
i
) =
0
j
(z
Opt
j
)
Conditions similar to optimality in a one-good economy with

0
j
(:) as rmj's marginal cost of producing the externality
With well-dened property rights and large number of par-
ticipants, market solutions likely to be eective
Bernard Caillaud Public goods

34. IV. Public goods and multilateral externalities
Pareto optimum withnon-depletable multilateral external-
ity:
max
z
X
j
j(zj) +
X
i
ui(
X
j
zj)
FOC:
P
i
u
0
i
(
P
j
z
Opt
j
) =
0
j
(z
Opt
j
);8j
Condition is analogous to the BLS conditions with
0
j
(:) as
rmj's marginal cost of production
By analogy with public goods, a market for the externality
will not restore eciency:
Bernard Caillaud Public goods

35. IV. Public goods and multilateral externalities
With non-depletable externality, market-based solutions are du-
bious. Rather rely on quotas (imposez
Opt
j
, possibly as a ceiling
quota) and taxes (tax externality att=
P
I
i=1
u
0
i
(
P
j
z
Opt
j
))
The market can still be used withglobal quota and permits:
distribute permitszjwith
P
j
zj=
P
j
z
Opt
j
that are trad-
able on a permit market at equilibrium pricepz
The equilibrium is such that:
0
j
(zj) =p

z,
P
j
zj=
P
j
zj=
P
j
z
Opt
j
and necessarilyp

z=
P
I
i=1
u
0
i
(
P
j
z
Opt
j
)
There exists an equilibrium that implements the optimum
Relax the informational burden on the government
Bernard Caillaud Public goods

36. V. Public goods and asymmetric information
In many of the remedies, the government has to know much about
the economy and the agents: unrealistic
Build or not build under asymmetric information
Iconsumers, the numeraire, binary public goodz2 f0;1g
Building the public good costsc
Consumeri's utility:xi+iz, endowed with!iin the
numeraire, such that
P
I
i=1
!i> c
cis publicly known,i2Risi's private information
Ex post eciency: build whenever
P
I
i=1
ic
Take the asymmetry of information seriously: using quotas and
taxes, can the governement achieve eciency for all realizations
of (1; :::; I), i.e.?
Bernard Caillaud Public goods

37. V. Public goods and asymmetric information
The egalitarian, but naive procedure...
Agents are asked to report simultaneously their valuations
i: let
a
i
denote the announcements
Then,z= 0 if
P
I
i=1

a
i
< cand no transfers,z= 1 otherwise
with equal nancing by agents
The dierence between the two outcomes is that ify= 1,i
gets!i+i
c
I
while ify= 0, he simply gets!i
So, ifi> c=I,ishould maximally over-report, while other-
wise he should maximally under-report
The outcome is (a.a.) inecient and the government does
not extract the relevant information
Bernard Caillaud Public goods

38. V. Public goods and asymmetric information
A procedure with the same decision rule and where agents pay
what they claim the public good is worth for them, wheny= 1
Each agentigetsi
a
i
on top of!i, in all circumstances
in whichy= 1
So, by announcing
a
i
=i, they never get more than!i
Each agent will shade his value and announce
a
i
< i, so as
to get a positive rent, even though it may happen less often
The outcome therefore involves systematic under-evaluation
of the public good benets, hence ineciency
The government may possibly infer the relevant information
about theis (by inverting the Bayesian equilibrium strate-
gies,
a
i
=m

i
(i)), but the procedure does not allow it to
use this information
Bernard Caillaud Public goods

39. V. Public goods and asymmetric information
The Groves mechanism
The procedure characterized by the same decision rule and, when
y= 1,i's payment equal toc
P
j6=i

a
j
, induce all agents to
report their parameteritruthfullyas a dominant strategy,
and it leads to the ecient decision.
Supposei+
P
j6=i

a
j
c; then ifireports so that
P
I
j=1

a
j
<
c, he gets!i, if he reports so that
P
I
j=1

a
j
c, he gets
!i+i+
P
j6=i

a
j
c!i; among latter optimal choices,
truthful revelation !
Supposei+
P
j6=i

a
j
< c; then ifireports so that
P
I
j=1

a
j
<
c, he gets!i, if he reports so that
P
I
j=1

a
j
c, he gets
!i+i+
P
j6=i

a
j
c < !i; among former optimal choices,
truthful revelation
Bernard Caillaud Public goods

40. V. Public goods and asymmetric information
Suppose that
P
I
i=1
j> c, theny= 1 (eciently) and the
public budget is:
I
X
i=1
[c
X
j6=i
j]c=(I1)[
I
X
j=1
jc]<0
The government does not collect enough to nance the public
good
Adding oni's transfer a term that only depends on the
others' announcements preserves truth-telling and eciency,
and ensures a non-negative budget
But in general impossible to ensure
Going further requires advanced methodology in dealing with
asymmetric information (later in this course !)
Bernard Caillaud Public goods

41. Required reading
* Bergstrom, T., L. Blume and H. Varian (1986),Journal of
Public Economics, 29, 25-49.
Hardin, G. (1968),Science, 162, 1243?1248.
Laont, J.J. (1988),Fundamentals in Public Economics, MIT
Press.
* MC-W-G, Ch 11 C-D
Milleron, J.C. (1972),Journal of conomic Theory, 5, 419-
477.
Ostrom, E. (1990),Governing the commons: The evolution
of institutions for collective action, Cambridge Univ. Press.
Bernard Caillaud Public goods