II-6.
Externalities
Outline
Dusko Pavlovic
Introduction
(Part II: Security of Economics)
Lecture 6: Network effects and
externalities
Introduction
Positive effects
Negative effects
II-6.
Externalities
Dusko Pavlovic
Positive effects
Introduction
Negative effects
Positive network effects and self-fulfilling expectations
Dusko Pavlovic
Negative network effects and minority game
Spring 2013
Outline
II-6.
Externalities
Three witches
Dusko Pavlovic
Introduction
II-6.
Externalities
Dusko Pavlovic
Introduction
Introduction
Positive effects
Positive effects
Negative effects
Negative effects
Positive network effects and self-fulfilling expectations
Negative network effects and minority game
The Tragedy of Macbeth
II-6.
Externalities
The Tragedy of Macbeth
Dusko Pavlovic
Introduction
Introduction
Positive effects
Three witches’ prophecy
First Witch: All hail, Macbeth! Hail to thee, Thane of
Glamis!
Second Witch: All hail, Macbeth, hail to thee, Thane of
Cawdor!
Third Witch: All hail, Macbeth, thou shalt be King
hereafter!
Negative effects
II-6.
Externalities
Dusko Pavlovic
Positive effects
Self-fulfilling prophecy
1. Macbeth is just a little spooked that the witches knew
that he was Thane of Glamis.
Negative effects
The Tragedy of Macbeth
II-6.
Externalities
The Tragedy of Macbeth
Dusko Pavlovic
Introduction
Introduction
Positive effects
Self-fulfilling prophecy
Negative effects
II-6.
Externalities
Dusko Pavlovic
Positive effects
Self-fulfilling prophecy
1. Macbeth is just a little spooked that the witches knew
that he was Thane of Glamis.
1. Macbeth is just a little spooked that the witches knew
that he was Thane of Glamis.
2. Macbeth gets promoted into Thane of Cawdor by the
King — and recognizes the prophecy.
2. Macbeth gets promoted into Thane of Cawdor by the
King — and recognizes the prophecy.
Negative effects
3. Macbeth kills the King — and realizes the prophecy.
How does future forecasting work?
II-6.
Externalities
Is lying sometimes a rational strategy?
Dusko Pavlovic
Introduction
Introduction
Positive effects
Positive effects
Negative effects
Negative effects
Why do we believe in stars at 30000 light years away?
Why do we advertise?
II-6.
Externalities
Dusko Pavlovic
Is lying effective? If not, why do we lie?
II-6.
Externalities
Outline
Dusko Pavlovic
Introduction
II-6.
Externalities
Dusko Pavlovic
Introduction
Introduction
Positive effects
Positive effects
Demand
Negative effects
Externalities
Positive network effects and self-fulfilling expectations
Adoption equilibrium
Self-fulfilling
Economy of demand and intrinsic values
Economy with externalities
Adoption equilibrium
Self-fulfilling
If the market is efficient, and computes the right prices,
why is it rational to invest in advertising?
Negative network effects and minority game
Negative effects
II-6.
Externalities
Demand and valuation
Demand and valuation are inverses
Dusko Pavlovic
Market computes the demand for a product
Introduction
Market computes the demand for a product
Positive effects
demand: q(y ) = x — the quantity required at the price y
Demand
Externalities
Self-fulfilling
II-6.
Externalities
Dusko Pavlovic
Introduction
demand: consumers’ names are x ∈ [0, 1]
Demand
!
!
ordered by their valuations for the good Γ
if x purchases Γ, then
!
!
all x # ∈ [0, x ] purchase Γ,
because r (x # ) ≥ r (x ), and
Self-fulfilling
II-6.
Externalities
Intuitions
Introduction
Positive effects
Externalities
Negative effects
Dusko Pavlovic
demand: consumers’ names are x ∈ [0, 1]
Demand
Adoption equilibrium
valuation: r (q(y )) = y ∈ [0, ∞] — value derived from use
Negative effects
Intuitions
Introduction
Positive effects
demand: q(r (x )) = x ∈ [0, 1] — fraction of consumers
Adoption equilibrium
valuation: r (x ) = y — the reserve price for x consumers
II-6.
Externalities
Dusko Pavlovic
Positive effects
Demand
Externalities
!
Adoption equilibrium
!
Self-fulfilling
ordered by their valuations for the good Γ
if x purchases Γ, then
Negative effects
!
!
all x # ∈ [0, x ] purchase Γ,
because r (x # ) ≥ r (x ), and
Externalities
Adoption equilibrium
Self-fulfilling
Negative effects
valuation: prices are y ∈ [0, ∞]
!
!
ordered by the demand for Γ
if y > y # then
!
!
!
Equilibrium of demand and supply
II-6.
Externalities
q(y ) < q(y # ), and
all x ∈ [0, q(y # )] will buy Γ
for r (x ) ∈ [y # , 1]
Equilibrium of demand and supply
Dusko Pavlovic
Introduction
!
Let p = y ∗ be the fixed (average) production cost.
Positive effects
Introduction
!
Let p = y ∗ be the fixed (average) production cost.
!
The products will be priced at y > y ∗ .
Demand
The products will be priced at y > y ∗ .
Adoption equilibrium
Externalities
Self-fulfilling
Negative effects
!
The buyers x <
!
!
x∗
q(y ∗ )
will purchase Γ at
Adoption equilibrium
Self-fulfilling
Negative effects
x∗
q(y ∗ )
!
The buyers x <
the prices y > y ∗ = r (x ∗ ).
!
the prices y > y ∗ = r (x ∗ ).
The market will demand x ∗ = q(y ∗ ) of Γ.
!
The market will demand x ∗ = q(y ∗ ) of Γ.
!
&x ∗ , y ∗ ' is the demand-supply equilibrium
=
Positive effects
Demand
Externalities
!
II-6.
Externalities
Dusko Pavlovic
!
=
where y ∗ = r (x ∗ )
will purchase Γ at
Social benefit at the equilibrium
II-6.
Externalities
Intrinsic values and externalities
Dusko Pavlovic
SB(x ∗ ) =
!
x∗
r (x)dx − x ∗ r (x ∗ )
Introduction
Introduction
Positive effects
Positive effects
Demand
Demand
Externalities
0
Adoption equilibrium
" x∗
Self-fulfilling
is the difference of the total utility 0 r (x)dx and the
production cost x ∗ p = x ∗ r (x ∗ ), i.e. the upper triangle in
II-6.
Externalities
Dusko Pavlovic
Negative effects
Externalities
Intrinsic values of goods are expressed through their
market prices and their production costs.
Adoption equilibrium
Self-fulfilling
Negative effects
Externalities are the values of goods taken by those who
are neither producers nor consumers of
these goods.
II-6.
Externalities
Examples of externalities
Valuations with externalities
Dusko Pavlovic
Introduction
Market adoption influences the valuation
Positive effects
Positive:
!
!
!
public health, security, education
freeware, creative commons
social adoption of shared applications
II-6.
Externalities
Dusko Pavlovic
Introduction
Positive effects
Demand
Demand
Externalities
Externalities
Adoption equilibrium
Adoption equilibrium
v (x , z) = r (x ) · f (z)
Self-fulfilling
Negative effects
Self-fulfilling
Negative effects
where
Negative:
!
!
!
!
!
pollution, environmental change
exploitation of resources (e.g. fishing)
systemic risk (e.g. in banking)
congestion
price increase due to demand
!
r (x ) is the intrinsic valuation
!
!
f (z) is the network effect
!
Valuations with positive externalities
II-6.
Externalities
x ’s reserve price if market fully adopts Γ
price change if z-part of the market adopts Γ
Network adoption equilibrium
Dusko Pavlovic
Introduction
Positive effects
!
r : [0, 1] → [0, 1] is monotone decreasing function
!
!
!
!
f : [0, 1] → [0, 1] is monotone increasing function
!
e.g. f (z) = z
!
!
∗
r (0) = 1: Γ is not valued at ∞ by anyone
r (1) = 0: Γ has no value for some consumers
f (0) = 0: Γ has no value if no adoption
f (1) = 1: Γ has full value with full adoption
[0, 1] represents the price interval [0, ∞].
Let p∗ be the fixed (average) production cost.
Introduction
Positive effects
Demand
Externalities
Externalities
Self-fulfilling
∗
!
Demand
Adoption equilibrium
e.g. r (x ) = 1 − x
II-6.
Externalities
Dusko Pavlovic
Negative effects
Adoption equilibrium
Self-fulfilling
Negative effects
Network adoption equilibrium
II-6.
Externalities
Network adoption equilibrium
Dusko Pavlovic
!
Let p∗ be the fixed (average) production cost.
Introduction
!
Let p∗ be the fixed (average) production cost.
!
Suppose that x knows that
Positive effects
!
Suppose that x knows that
Demand
!
z -part of the market has adopted Γ
!
Self-fulfilling
!
II-6.
Externalities
∗
z -part of the market has adopted Γ
*
for all x # holds x # ∈ [0, z ∗ ] ⇐⇒ x # has bought Γ
Network adoption equilibrium
Dusko Pavlovic
!
Let p∗ be the fixed (average) production cost.
Introduction
Suppose that x knows that
Demand
!
Let p∗ be the fixed (average) production cost.
!
Suppose that x knows that
!
!
z ∗ -part of the market has adopted Γ
*
for all x # holds x # ∈ [0, z ∗ ] ⇐⇒ x # has bought Γ
*
for all x # holds x # ∈ [0, z ∗ ] ⇐⇒ r (x # )f (z ∗ ) ≥ p∗
!
Self-fulfilling
Negative effects
!
!
II-6.
Externalities
z ∗ -part of the market has adopted Γ
*
for all x # holds x # ∈ [0, z ∗ ] ⇐⇒ x # has bought Γ
*
for all x # holds x # ∈ [0, z ∗ ] ⇐⇒ r (x # )f (z ∗ ) ≥ p∗
⇓
r (x )f (z ∗ ) ≥ p∗ ⇐⇒ x ∈ [0, z ∗ ]
Network adoption equilibrium
Dusko Pavlovic
!
Let p∗ be the fixed (average) production cost.
Introduction
Suppose that x knows that
Demand
!
Let p∗ be the fixed (average) production cost.
!
Suppose that x knows that
!
!
!
!
z ∗ -part of the market has adopted Γ
*
for all x # holds x # ∈ [0, z ∗ ] ⇐⇒ x # has bought Γ
*
for all x # holds x # ∈ [0, z ∗ ] ⇐⇒ r (x # )f (z ∗ ) ≥ p∗
⇓
r (x )f (z ∗ ) ≥ p∗ ⇐⇒ x ∈ [0, z ∗ ]
*
x will buy Γ ⇐⇒ x ≤ z ∗
Demand
Adoption equilibrium
Self-fulfilling
Negative effects
II-6.
Externalities
Introduction
Positive effects
Externalities
!
Introduction
Dusko Pavlovic
Positive effects
!
II-6.
Externalities
Externalities
Adoption equilibrium
!
Network adoption equilibrium
Self-fulfilling
Negative effects
Positive effects
Externalities
!
Adoption equilibrium
Dusko Pavlovic
Positive effects
!
Demand
Externalities
Adoption equilibrium
Negative effects
Network adoption equilibrium
Introduction
Positive effects
Externalities
∗
II-6.
Externalities
Dusko Pavlovic
Demand
Externalities
Adoption equilibrium
!
Self-fulfilling
Negative effects
!
!
!
!
!
z ∗ -part of the market has adopted Γ
*
for all x # holds x # ∈ [0, z ∗ ] ⇐⇒ x # has bought Γ
*
for all x # holds x # ∈ [0, z ∗ ] ⇐⇒ r (x # )f (z ∗ ) ≥ p∗
⇓
r (x )f (z ∗ ) ≥ p∗ ⇐⇒ x ∈ [0, z ∗ ]
*
x will buy Γ ⇐⇒ x ≤ z ∗
&z ∗ , p∗ ' is the network adoption equilibrium
!
where p∗ = r (z ∗ )f (z ∗ )
Adoption equilibrium
Self-fulfilling
Negative effects
II-6.
Externalities
Calculating equilibria
Dynamics of market adoption
Dusko Pavlovic
Given
Introduction
!
fixed production price p∗
!
reserved price function r (z) = 1 − z
!
!
z ∈ [0, z # ): v (z) < p∗ causes z / 0
!
z = z # : v (z) = p∗ makes z stable
!
z ∈ (z # , z ## ): v (z) > p∗ causes z 0 z ##
Positive effects
Demand
network effect f (z) = z
Negative effects
z = z ## : v (z) = p∗ makes z stable
the equilibria &ẑ, p∗ ' satisfy ẑ − ẑ 2 = p∗ .
!
z ∈ (z ## , 1]: v (z) < p∗ causes z / z ##
II-6.
Externalities
Tipping point
Dusko Pavlovic
The Secret of Network Startups
Demand
The unstable equilibrium
!
!
is a tipping point:
The Silicon Valley Imperative (Brian Arthur)
Externalities
Self-fulfilling
Push down z # :
!
!
Negative effects
lower the price p∗ (free trials . . . )
widen the parabola v (z) by speeding up f (z)
If the adoption is pushed past z # , the demand will
grow to z”.
II-6.
Externalities
Self-fulfilling expectation equilibrium
Dusko Pavlovic
!
The Silicon Valley Imperative (Brian Arthur)
Positive effects
!
!
⇓
The adoption attractor z” will go up.
Positive effects
Demand
Externalities
Negative effects
lower the price p∗ (free trials . . . )
widen the parabola v (z) by speeding up f (z)
Introduction
Externalities
Self-fulfilling
Push down
Let p∗ be the fixed (average) production cost.
Demand
Adoption equilibrium
z#:
II-6.
Externalities
Dusko Pavlovic
Introduction
!
Demand
Adoption equilibrium
!
If the adoption is not pushed to z # , the demand will
drop to 0.
Tipping point
!
Positive effects
Externalities
Self-fulfilling
Negative effects
II-6.
Externalities
Introduction
Adoption equilibrium
z#
Self-fulfilling
Dusko Pavlovic
Introduction
Positive effects
Adoption equilibrium
Negative effects
!
Tipping point
Demand
Externalities
Adoption equilibrium
valuation v (z) = z(1 − z) = z − z 2
!
Introduction
Positive effects
Externalities
Self-fulfilling
II-6.
Externalities
Dusko Pavlovic
Adoption equilibrium
Self-fulfilling
Negative effects
Self-fulfilling expectation equilibrium
II-6.
Externalities
II-6.
Externalities
Self-fulfilling expectation equilibrium
Dusko Pavlovic
!
Let p∗ be the fixed (average) production cost.
Dusko Pavlovic
Introduction
!
Positive effects
Introduction
Let p∗ be the fixed (average) production cost.
Positive effects
Demand
!
Suppose that x believes that z-part of the market has
adopted Γ (which may not be true).
Demand
Externalities
Adoption equilibrium
!
Self-fulfilling
Suppose that x believes that z-part of the market has
adopted Γ (which may not be true).
Negative effects
!
Self-fulfilling expectation equilibrium
II-6.
Externalities
Let p∗ be the fixed (average) production cost.
x purchases Γ ⇐⇒ r (x )f (z) ≥ p∗
Suppose that x believes that z-part of the market has
adopted Γ (which may not be true).
Dusko Pavlovic
Introduction
!
Positive effects
Introduction
Let p∗ be the fixed (average) production cost.
Positive effects
!
!
x purchases Γ ⇐⇒ r (x )f (z) ≥ p∗
*
# p∗ $
x purchases Γ ⇐⇒ x ≤ r −1 f (z)
Demand
Externalities
Adoption equilibrium
Self-fulfilling
II-6.
Externalities
Self-fulfilling expectation equilibrium
Demand
!
Adoption equilibrium
Negative effects
Dusko Pavlovic
!
Externalities
!
Self-fulfilling
Suppose that x believes that z-part of the market has
adopted Γ (which may not be true).
Negative effects
!
!
!
Externalities
Adoption equilibrium
Self-fulfilling
Negative effects
x purchases Γ ⇐⇒ r (x )f (z) ≥ p∗
*
# p∗ $
x purchases Γ ⇐⇒ x ≤ r −1 f (z)
The true market adoption (depending on the belief z) is
% ∗ &
p
g(z) = q
f (z)
because r −1 = q.
II-6.
Externalities
Example of adoption function
Given
!
Introduction
!
fixed production price p∗
!
reserved price r (z) = 1 − z, demand q(z) = r −1 (z) = 1 − z
Positive effects
!
Demand
Externalities
Adoption equilibrium
!
network effect f (z) = z




0
the true adoption is '
z = g(z) = 


1 −
!
Self-fulfilling
Negative effects
if z ≤ p∗
p∗
z
otherwise
II-6.
Externalities
Finding self-fulfilling equilibrium
Dusko Pavlovic
!
!
Dusko Pavlovic
g(z) = '
z ≤ z ∈ [0, z # ): v ('
z ) < p∗ causes '
z/0
g(z) = '
z = z # : v ('
z ) = p∗ makes '
z stable
#
##
∗
g(z) = '
z ≥ z ∈ (z , z ): v ('
z ) > p causes '
z0z
g(z) = '
z = z ## : v ('
z ) = p∗ makes '
z stable
Introduction
Positive effects
Demand
##
g(z) = '
z ≤ z ∈ (z ## , 1]: v ('
z ) < p∗ causes '
z / z ##
for g(z) as in the example
Externalities
Adoption equilibrium
Self-fulfilling
Negative effects
Finding self-fulfilling equilibrium
!
!
!
!
!
g(z) = '
z ≤ z ∈ [0, z # ): v ('
z ) < p∗ causes '
z/0
g(z) = '
z = z # : v ('
z ) = p∗ makes '
z stable
g(z) = '
z ≥ z ∈ (z # , z ## ): v ('
z ) > p∗ causes '
z 0 z ##
g(z) = '
z = z ## : v ('
z ) = p∗ makes '
z stable
II-6.
Externalities
Self-fulfilling equilibrium when f (0) > 0
Dusko Pavlovic
II-6.
Externalities
Dusko Pavlovic
Introduction
Introduction
Positive effects
Positive effects
Demand
Demand
Externalities
Externalities
Adoption equilibrium
Adoption equilibrium
Self-fulfilling
Self-fulfilling
Negative effects
Negative effects
g(z) = '
z ≤ z ∈ (z ## , 1]: v ('
z ) < p∗ causes '
z / z ##
for general g(z)
Self-fulfilling equilibrium when f (0) > p∗
II-6.
Externalities
II-6.
Externalities
Summary
Dusko Pavlovic
Introduction
Positive effects
Dusko Pavlovic
Introduction
Why do we lie?
Positive effects
Demand
Demand
Externalities
Adoption equilibrium
z#
Externalities
!
If you convince >
people that you are King,
!
then they will help you to sujugate z” people.
Self-fulfilling
Negative effects
Outline
II-6.
Externalities
Dusko Pavlovic
Introduction
Positive network effects and self-fulfilling expectations
Negative network effects and minority game
Adoption equilibrium
Self-fulfilling
El Farol Bar, Santa Fe NM
Negative effects
II-6.
Externalities
Dusko Pavlovic
Introduction
Introduction
Positive effects
Positive effects
Negative effects
Negative effects
El Farol Problem: Minority Game
II-6.
Externalities
II-6.
Externalities
Minority Game
Dusko Pavlovic
!
capacity: 60 places
!
!
Introduction
Introduction
Positive effects
Positive effects
Negative effects
Negative effects
!
players: i = 1, 2, . . . , 100
attraction: music nights
!
moves: Ai = {Y , N}, for all i
customers: 100 music fans
!
payoffs:
!
!
!
Dusko Pavlovic
# visitors ≤ 60 =⇒ pleasant
# visitors > 60 =⇒ unpleasant




1
ui (a) = 


−1
goal of the game: visit El Farol when # visitors ≤ 60
Minority Game
II-6.
Externalities
if #{k|ak = ai } ≤ 60
if #{k|ak = ai } > 60
II-6.
Externalities
Minority Game
Dusko Pavlovic
Dusko Pavlovic
Introduction
Introduction
Positive effects
Positive effects
Negative effects
Exercise
!
Analyze Nash equilibria in this game.
The members of the majority have a joint incentive to
switch.
!
!
Recall: Network effects
II-6.
Externalities
Negative effects
Negative feedback
"No one goes to El Farol. It’s too busy."
The Nash equilibria are unstable.
II-6.
Externalities
Negative network effects
Dusko Pavlovic
!
Let p∗ be the fixed (average) production cost.
!
Suppose that x believes that z-part of the market has
adopted Γ (which may not be true).
!
The true market adoption (depending on the belief z) is
% ∗ &
p
g(z) = q
f (z)
because r −1 = q.
Dusko Pavlovic
Introduction
Introduction
Positive effects
Positive effects
Negative effects
Negative effects
Given
!
fixed production price p∗
!
reserved price r (z) = 1 − z, demand q(z) = r −1 (z) = 1 − z




if z ≤ .6
z
network effect f (z) = 


1 − z if z > .6
!



0





'
the true adoption is z = g(z) = 
1−





1 −
if z ≤ p∗
p∗
z
p∗
1−z
p∗ < z ≤ .6
.6 < z
Dynamics of El Farol Bar
II-6.
Externalities
Dusko Pavlovic
Introduction
Positive effects
Negative effects
ongoing research