Lecture 15,public goods and
externality
Content
? Public goods
– Subscription
– Efficient provision of public goods
– Public choice
? Externality
– What’s externality?
– How to solute it?
– What’s the efficient condition with externality,
Public goods
? Subscription
Excludable Non-excludable
Rival Private goods Regulated goods(?)
Non-rival Club goods Public goods
Public goods
? Subscription
– Excludability is the property problem
– Rivality is the physical problem,
– Y.Barzel(1989),Public domain
– Price determined by the property,
Public goods
? Efficient provision of public goods
? Discrete case,
– xi, private goods
– G,public goods
– gi,i’s contribution to public goods
– ui(G,xi),utility function
12
12
1 i f
0 i f
g g c
G
g g c
???
? ?
???
Public goods
? Discrete case,benchmark
? If providing the public goods will Pareto
dominate non,So
? ri,the reservation price of i,
1 1 1 1 1
2 2 2 2 2
( 1,) (0,)
( 1,) (0,)
u w g u w
u w g u w
??
??
( 1,) ( 0,)i i i i iu w r u w??
Public goods
? Then will get,
? The sum of reservation price exceeds the
cost of public goods,it’s efficient to
provide the public goods,
1 2 1 2r r g g c? ? ? ?
Public goods
? But,can private provide it?
? Reflection function,
? If,
? Non cooperation equilibrium (0,0)
ijg c g??
(,) (,)i i i i i ju G w g u G w c g? ? ? ?
12
21
r c g
r c g
??
??
,0i j jg c r g? ? ?
Public goods
? Voting,
Public goods
? Continuous case,
? Social welfare function,
12()G f g g??
12( ( ),)i i iu f g g w g??
12( 1 )W a u a u? ? ?
Public goods
? Continuous case,
? Benchmark,
12
12,m a x ( 1 )gg a u a u??
1 2 1
1
1 2 2
2
( 1 )
( 1 ) ( 1 )
u u u
a a a
G G x
u u u
a a a
G G x
? ? ?
? ? ?
? ? ?
? ? ?
? ? ? ?
? ? ?
1 1 2 2
12
1u u u uG x G x? ? ? ?? ? ?? ? ? ?
Public goods
? Samuelson condition,
1
( ) ( )
I
i
i
u G c G??
?
????
Public goods
? Private provision
? By Kuhn-Tucker first order condition,
See the fig,
1
1 1 2 1 1
1
m a x (,)
,, 0
g
u g g w g
s t g
??
?
11
1
1
1,f o r 0uu gGx?? ????
11
1
1
1,f o r 0uu gGx?? ????
Public goods
? A Nash equilibrium,
? The equation holds for one i at least,
? If the utility is quasi-linear,then we have,
? If,then 2 to be the free-rider,
1
(,) 1,i i iu G x u
Gx
????
???
1 1 2( ) 1,u g g??? ?? 2 1 2( ) 1,u g g??? ??
12( ) ( ) u G u G???
Public goods
? When there is no free-rider?
? For 1,
? Or
11
1 1 2 1
,
1
1 1 1
m a x (,)
,, 0
gx
u g g x
s t g
w g x
?
?
??
1
11
,
2
1 2 1
m a x (,)
,, G
Gx
u G x
s t g
w g G x
?
? ? ?
Public goods
? Let be the demand for public goods
of his wealth,We have,
? And
? in the same way,we got,
1()Dw
1 2 2m a x { ( ),}G D w g g??
1 2 1 2 2m a x { ( ),0 }g G g D w g g? ? ? ?? ? ? ? ?
2 1 1 1m a x { ( ),0 }g D w g g? ? ?? ? ?
Public goods
? If utility is quasi-linear,
? If,then and
? If,then
()iiD w g?
1 1 2m a x {,0 }g g g????
2 2 1m a x {,0 }g g g????
12gg? 11gg? ? 2 0g ? ?
12gg? 12 0gg????
Public goods
? Insufficient provision of public goods
– While,
– Let
– Then,
– For
– So
( ),t h e e q u a t i o n h o l d s f o r 0iiu G p g? ? ?? ??
( ),t h e e q u a t i o n h o l d s f o r 0c G p G? ? ?? ??
**1,w h e n 0,a n d 0,w h e n 0i i i igg??? ? ? ?
1
[ ( ) ( ) ] 0
I
ii
i
u G c G? ??
?
?????
1 1 1
( ) ( ),f o r 1
I I I
i i i i
i i i
u G u G I?? ??
? ? ?
????? ? ?
1
( ) ( )
I
i
i
u G c G??
?
????
Public goods
? Voting,
– Paradox of voting (Condorcet paradox)
– Single-peaked preference (i pay fraction si for
G),has a unique maximum,
– Majority rule and median voter,
– Bowen equilibrium
()iiu G s G?
()mmu G s? ?
Public goods
? Lindahl allocation,
– Everybody have a competitive market to buy
the public goods,
– First order condition,
,
m a x (,)
,,
i
iiGx
i i i
u G x
s t w p G x??
ii
i
i
uu p
Gx
?? ?
??
Public goods
? Revealing mechanism of demand for
public goods,
– How to reveal the pi?
? The Groves-Clarke Mechanism
– Each agent report a bid for the public good,bi,
– The public goods is provided if
– Each agent i receives a side-payment equal
to the other bids,if the public goods
is provided,
0i
i
b ??
0j
ji
b
?
??
Public goods
? The Groves-Clarke Mechanism
if 0
p a y o f f to
0 if 0
i j i j
j i j i
ij
ji
v b b b
i
bb
??
?
? ? ? ?
?
? ?
???
?
??
?
Externality
? What’s externality?
? Private cost and social cost
See the fig,
1 m a x ( )x px c x? ??
2 ()ex? ??
Externality
? Integration,internalize the externality,
? Pigovian taxes
m a x ( ) ( )
x
p x c x e x? ? ? ?
m a x ( )
x
px c x t x? ? ? ?
()t e x??
Externality
? The first Coase theorem,
? The first order condition,
? It’s indifferent for or !
1 m a x ( ) ( )x p x c x r x? ? ? ?
2 m a x ( ) ( )x r x e x? ??
( ) ( )p c x e x????
( ) 0rx? ? ( ) 0rx? ?
Externality
? The second Coase theorem,
? If there is a transaction cost,
( ) 0Tx? ?
1 m a x ( ) ( ) ( )x p x c x r x T x? ? ? ? ?
2 m a x ( ) ( )x r x e x? ??
Externality
? A compensation mechanism,
? The firms announce a Pigovian tax ti
? Firm 1 produces x and pays t2x tax,firm 2
receives t1x compensation,And both pay
penalty on the difference between their tax
rate,say 212()tt?
2
1 2 1 2m a x ( ) ( ) ( )x p x c x t x t t? ? ? ? ? ?
2
2 1 1 2m a x ( ) ( ) ( )x t x e x t t? ? ? ? ?
Externality
? For this game have a Subgame perfect
equilibrium if and only if,
? Second stage,firm 1,
? First stage,firm 1
? Firm 2,
? So we get,
12tt?
22( ) ( )p c x t x x t?? ? ? ?
1 1 1 2( ) 2( ) 0t t t? ? ? ? ? ?
2 2 1 2 1 2( ) ( ( ) ) ( ) 2( ) 0t t e x x t t t? ? ??? ? ? ? ?
( ) ( )p c x e x????
Public goods
G
p
1MU
2MU
Public goods
Public goods
Private goods
Public goods
Private goods
g1
g2
x1 w1
g2
w1
Private cost and social cost
()cx?
( ) ( )c x e x???
x?x x