Economy and network effect (externality): modeling and analysis

Revision: Network Externalities
The Economy without Network Effects
The Economy with Network Effects

Part 3: The Economy and Network Effects

Sumant Kulkarni

International Institute of Information Technology, Bangalore

Sumant Kulkarni Externalities and Network Effects 1/48


Agenda

Revision: Network Externalities.
Economy without Network Eﬀect.
Economy with Network Eﬀects.



Externality

Two parties do business willingly only if there is a profit for
both of them1 .
Business may be the exchange of financial value (like renting
house) or social welfare (like marriage).
Externality either puts some cost or provide some benefit to
the people not involved in the business.
For example:
Renting out a house to a night club in the residential complex
(assume that it is legal).
Renting out a house to a very influential politician with
mindset of helping people.
1
Kelvin Hartnall, Externalities and Network Effects


Network Externalities

We are part of many networks. For example,
Network of same company car users.
Network of same telephone service.
Network of people using same social network.
Network Externality is a phenomenon in which entry of new
user into the network, has either beneﬁt or cost to the
other user of the network.
If the entry costs something to other users, then it is
Negative Externality.
If the other users are beneﬁted, it is Positive Externality.



Network Externalities

In the networks showing network externality, the users have
two separate sub part in the value he receives by being a part
of the network2 .
1 Autarky value: The value from the product/service he is
using (consumer has paid for it). User gets this even if there is
no other person using the same product/service.
2 synchronization value: The value from the network as the
result of joining it (complementary but not optional).
The latter part of the value decides whether it is Positive
Externality or Negative Externality.

2
Network Externalities (Eﬀects) by S. J. Liebowitz and Stephen E. Margolis


Negative Externalities

If the entry of a new user into the network costs something to
other users of the network, then it is Negative Externality.
A classic example is the traffic congestion.
The negative externalities can often be seen in the later stage
of networks, where resources are finite.
What other reasons can be there for a network to have
negative externality?
Can their be a network which is having negative externality
from the first user of the network?
Is negative externality a manifestation of the indication of
resource crunch?


Positive Externalities

If the other users are benefited by the entry of a new user into
the network, then it is Positive Externality.
In positive externality, “The value of the service or product
will increase as its installed base expands3 ”.
Positive Externality is known as Network Effect.
Though many networks have Network Effect initially, once
they start facing resource crunch and once they scale above a
level, they might start showing negative externality.

3
Network Effects and the Impact of Free Goods: An Analysis of the Web
Server Market

Setup of Market
Formally Modeling the Lack of Individual Eﬀect on Market
How Market looks with no Network Eﬀects
The Equilibrium Quantity of the Good



Setup of Market

The Setup of the Economy without Network Eﬀects

We make many assumptions to simplify the complexity of the
market.
The market is for one good and has huge number of
consumers.
Basic Condition = No Network Eﬀect in the Market
Condition ⇒ Consumers do not care how many other
users of the good are there.


Setup of Market

Let us analyse the way market function with the assumption.
Assumption: Large number of potential purchasers with very
little individual share in purchase.
⇒Each user can make independent individual decisions
without affecting other users.
Example: User buying a car does it without thinking about
whether his decision affects the price of the car in market.
Real markets with finite real users do behave like this.
The effect of each individual can be very negligible on the
aggregate. Hence, we completely ignore the effect of each
individual on the market while modelling the economy.

Setup of Market

Modeling Consumers

Consumers are represented as the set of all real numbers in
the interval strictly between 0 and 1.
Q1: set of all real numbers in the interval strictly between 0
and 1 is infinite. How do we map them to finite number of
n+1 N−n+1
users (say N)? Ex. (Map(n) = N ), (Map(n) = N )
Each consumer is named after a different unique real number.


Setup of Market

Modeling Consumers

Due to uniform distribution, set of consumers between 0
and x < 1 represent x fraction of population.
We can think of this model of consumers as the continuous
approximation of market with large but ﬁnite number of
consumers.
Q2: How does the continuous model avoid having to deal
with the explicit eﬀect on any one individual on the overall
population? (May be due to the uniform distribution of the
consumers between 0 and 1. How how exactly does it help?)


Setup of Market

Modeling Consumers Willingness to Buy

Assumption: Each consumer wants to buy at most one unit
of the good.
Value of that one unit of good for the consumer is determined
by the intrinsic interest of the consumer to buy it.
No other factor than the intrinsic interest motivates him to
buy the good.


Setup of Market

Reservation Prices
Let us consider how market looks with the above assumptions.
Each consumer’s interest in the good is speciﬁed by a single
price called reservation price.
Reservation price ( r(x) ) is the maximum amount the
consumer x is willing to pay for one unit of the good.
Assumption: Consumers are arranged in the decreasing order
of their reservation price between 0 and 1.
If r (x) > r (y ) then x < y .
To be more clear r (0) > r (1) and 0 < 1
Assumption: r (·) is a continuous function and no two
consumers have exactly the same reservation prize.
r (·) is strictly decreasing over the increasing interval 0 to 1.

Setup of Market

Market Price for a Unit of Good

Market Price (p) for a unit of good is the minimum price at
which the good can be bought and there will not be any unit
of the good sold below price p.
Assuming that there will not be any unit of the good sold
below price p.
Any x having r (x) >= p can buy the good and any x having
r (x) < p can not buy the good. (Why?)
At p > r (0), nobody can buy the good. (Why?)
At p <= r (1), everyone can buy the good. (Why?)


Setup of Market

Figure: When there are no network eﬀorts, the demand for a product at a
ﬁxed market price p can be found by locating the point where the curve
y = r (x) intersects the horizontal line y = p.


Setup of Market

Market Price for a Unit of Good

r (·) is strictly decreasing continuous function.
For r (·), in the region between 0 and 1 (the region of
interest), their lies a unique sweet point for which r (x) = p.
This means:
The consumers between 0 and x (including x), can buy the
good.
The consumers named greater than x can not buy the good.
Hence, x fraction of the consumers buy the product (due to
uniform distribution assumption).

Setup of Market

Market Demand for the Good

For every prize p, there will be an x, which speciﬁes the
fraction of population that will purchase at price p.
The x (determined from price), is an indicator of Market
Demand for the good.
Increase in x shows the increase in demand.

Setup of Market

Market Demand for the Good

Increase in x shows the increase in demand.
r (·) describes the inverse demand function. (Why?)
p = r (x) and hence x = r −1 (p).
This relation between number of units consumed (x) and the
price p is very interesting. To sell more units we need to
reduce the price.

Setup of Market


Assuming that constant production cost/unit of good is p ∗
Assuming that there are many producers so that no one can
inﬂuence the market.
The producers will be willing to supply any amount of goods
at the prize p ∗ per unit.
The producers will not be willing to supply any amount of
goods at the prize lesser than p ∗ per unit.
It is highly unlikely that the price of the per unit good will
remain above p ∗ (competition).
Proﬁt is ZERO.
This is due to long-run competitive supply for any good
produced by a constant-cost industry.

Setup of Market


If p ∗ (constant production cost/unit) is above r (0), no one can
buy the good.
If p ∗ is below r (1), everyone buys the good.
Thus, the interesting point is r (0) > p ∗ > r (1)
When r (0) > p ∗ > r (1):
When p ∗ = r (x ∗ ), we call x ∗ the equilibrium quantity of the
good for given reservation prices and cost p ∗ .
Usually 0 < x ∗ < 1 for a stable market for a good.


Setup of Market

The Equilibrium Quantity of the Good - Reservation Price
and Cost

Figure: When copies of a good can be produced at a constant cost p ∗
per unit, the equilibrium quantity consumed will be the number x ∗ for
which r (x ∗ ) = p ∗

Setup of Market

Equilibrium Quantity

x ∗ represents an equilibrium in the population’s consumption
of the good.
If consumption < x ∗ , then there will be “upward pressure” on
the consumption of good. (Want to Buy as r(x) = p > p ∗ )
If consumption > x ∗ , then there will be “downward pressure”
on the consumption of good. (Regret as r(x) = p < p ∗ )


Setup of Market

Social Welfare (SW)

Social Welfare is the the diﬀerence between total reservation
prices of consumers who bought a copy of the good and total
production cost of those many units of good.
For x units of goods, the Social Welfare will be possible only
when they are allocated to all consumers between 0 and x.
When x = x ∗ (equilibrium), then the Social Welfare is
maximum.


Effect of Network Effects on Purchase of a Good
Equilibria with Network Effects




The price of an unit of good is determined by
Intrinsic interest - (own reservation price) - r (x)
Number of consumers already using the good - z
There will be two independent factors inﬂuencing the
reservation price (RP) of a consumer - r (·) and f (·).
r (x) - Intrinsic interest for the consumer x.
f (z) - Beneﬁt to each consumer for having z fraction of
population using the good.
RP(x) = r (x) · f (z)
Multiplication means, increase in the fraction of people using
the good (z) increases the reservation price even when there
is no change in the intrinsic interest.
Increase in any or both increases the reservation price.


Pricing the Good

Assumption: f(0) = 0 ⇒ If no units of good purchased, then
no one will buy it.
The consumer willingness to pay depends on the fraction of
population using the good.
The prediction of fraction of population (z) using the good is
very important.
Assumption: p ∗ is the price of one unit the good and z
fraction of population will use the good, then
A consumer x will buy the good only if r (x)f (z) p∗



Self-fulfilling Expectations Equilibrium (SFEE)

Assumption: The prediction about number of users of the
good (z) is always correct.
What do we mean by always correct (exact) prediction of z?
“the consumers as a whole form a shared expectation that the
fraction of population using the good is z and buy it. Due to
this, the exact fraction of population reaches z fraction”.
This is self-fulfilling expectations equilibrium for the
quantity of purchasers z.
In self-fulfilling expectations equilibrium every purchaser of z
fraction predicts that z fraction of population is using it.
The expectation in turn is fulfilled by people’s own behavior.
Coming slides assume that the prediction about number of
users of the good is always correct. and Network Effects
Sumant Kulkarni Externalities 29/48


When Do We See Self-fulﬁlling Expectations Equilibrium

We observe 2 self-fulﬁlling expectations equilibria.
When z = 0 (i.e. When no fraction of populations has bought
the good.)
When 0 < z < 1 (i.e. When some fraction of populations has
bought the good.)
Assumption: Price/Unit of good = p ∗ > 0.



Self-fulﬁlling Expectations Equilibrium When z = 0 (i.e.
None of the populations has bought the good.)
When z = 0
No one will buy the good.
Reservation Price of any consumer x can be given as
RP(x) = r (x) · f (0) = 0
RP(x) < p ∗ (price of the good)
No one will buy the good as the reservation price is 0
(RP(x) < p ∗ ).
The state persists (no increase or decrease in the fraction of
consumers).
Hence, this is an equilibrium.


Self-fulﬁlling Expectations Equilibrium When 0 < z < 1
(i.e. Some of the populations have bought the good.)

If exactly z fraction of population can buy the good, then
If any x, where x <= z can buy the good. Hence, only the
consumers between 0 and z can buy the good.
The price p ∗ at which these consumers want to purchase the
good can be determined as below.
z has lowest reservation price of all the people who can buy.
This is because only consumers who can buy are 0 to z.
Hence RP(z) = p ∗
The Reservation price of z = RP(z) = r (z) · f (z)
p ∗ = r (z)f (z)
As only z fraction can purchase at this price, this is a SFEE.


Characteristics of SFEE with an Example

To conﬁrm whether Self-fulﬁlling Expectations Equilibria exists, we
need to know both r (·) and f (·).
Consider an example where,
r (x) = 1 − x
f (z) = x.
p = r (z)f (z) = (1 − z)z = z(z − 1) = z 2 − z
This is a quadratic equation representing a parabola.



SFEE - Example

Figure: The plot of the consumer Fraction z versus the Price (p). The
quadratic equation p = z(z − 1) = z 2 − z is represented in the plot.



SFEE - Example

The curve reaches its maximum (i.e. p ∗ = 1 ) when z =
4
1
2

There is no equilibrium for p ∗ > 4 .
1

This means, if p ∗ > 1 , then the good is considered too
4
expensive and no one will buy it.


SFEE - Example

When z = 0,
There price of the good for any consumer x is
r (x)f (z) = r (x)f (0) = 0
No one will buy the good as p ∗ > 0 i.e p ∗ > reservation price
There will not be any increase in z due to this.
Hence, z = 0 is an equilibrium.


SFEE - Example

When 0 < z < 1, there are two equilibria
Earlier Equilibrium z
Later Equilibrium z



Pressures in the SFEE

Any fraction other than z = 0, z = z and z = z do not form
SFEE.
Hence, if the fraction z is any point other than z = 0, z = z
and z = z , then there will be diﬀerent kinds of pressures
acting in the system.



If z is between 0 and z
The actual fraction of purchasers (z) is lesser than the SFEE
fraction z .
(p = r (z)f (z)) < (p ∗ = r (z )f (z ))
Value of the purchased good due to z fraction consumers is
less than the price of the good p ∗ .
People have got good which has lower value than their RP.
People feel that they should not have bough that good.
Number of people buying reduces, pushing the demand down.
This is Downward Pressure on consumption.



If z is between z and z
The actual fraction of purchasers (z) is greater than the SFEE
fraction z .
(p = r (z)f (z)) > (p ∗ = r (z )f (z ))
more than the price of the good p ∗ .
People have got good which has more value than their RP.
People feel good about it and more people start buying.
Number of people buying increases, pushing the demand up.
This is Upward Pressure on consumption.



If z is between z and 1
The actual fraction of purchasers (z) is greater than the SFEE
fraction z .
(p = r (z)f (z)) < (p ∗ = r (z )f (z ))
less than the price of the good p ∗ .
People have got good which has lower value than their RP.
People feel that they should not have bough that good.
Number of people buying reduces, pushing the demand down.
This is Downward Pressure on consumption.


Stability of z

More stable Equilibrium.
If z crosses z , downward pressure will pull it towards z .
If z is between z and z upward pressure will push it to z .



Stability of z

If z is between 0 and z - approach 0.
If z is between z and z - approach z .
Highly unstable.
It is known as critical point or tipping point.
If z crosses z , it will increase till z .
If z is lesser than z , then the fraction of people consuming
the good will reduce and becomes 0.
Very important to cross the z , to success in the business.



SFEE - Some Observations

Two important observation (for the same good, at the same price):

Effect of consumer confidence.
Effect of relationship between the price and the equilibrium
quantity.



Effect of Consumer Confidence.

Self-fulfilling expectations equilibrium corresponds to
consumer confidence.
If no consumer confidence in the success of the good, no
success will be seen.
If population is confident in the success of the good, the
success will be seen.
Multiple Equilibria are due to the different characteristics of
the markets in which NE works. The number of equilibria will
depend on the type of Market.



Eﬀect of the Relationship between the Price and the
Equilibrium Quantity

In the market with NE, the relationship between price and
equilibrium quantity is more complicated than that with out
NE.
For example, as p ∗ goes on dropping below 1 , z moves to the
4
left and z moves to the right. This means that each
equilibria move away from each other. This means that, z
moves towards smallerKulkarni
Sumant fractionExternalities and moves towards the
while z Network Eﬀects 46/48


References

Network, Crowds and Markets - book
Network Effects, http://www.moreno.marzolla.name/
teaching/CS2011/NetworkEffects.pdf
The Economy with Network Effects,
http://www.systems.ethz.ch/education/fs11/
struct-social-inf-networks/lectures/Lecture%209.
pdf
Reverse Network Effect,
http://www.readwriteweb.com/archives/is_there_a_
reverse_network_effect_with_scale.php



Network effects: related pages
Barriers to entry - Anything that makes it difficult for a new entrant to
break into a market.
First mover advantage - The competitive advantage that the first
company to launch a new type of product should have over those that
start later.
Natural monopoly - A monopoly that arises from the nature of the
industry, rather than being imposed by law or resulting from
anti-competitive practices.
Product differentiation - Making a product or service look different in the
eyes of consumers.
Submarine patent - A patent that is deliberately kept quite, in the hope
of extracting money later from those who use an idea believing it not to
be patented.
Cross licensing - Exchange rights to patent portfolios, which reduces

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