The Economics of Social Networks
Paolo Pin
Universit`
a di Siena
Siena – May 5th, 2015
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Examples of networks
Examples of networks
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Examples of networks
Basic definitions
Nodes and links
Nodes: a set of elements that may pairwise have or not a specific
relations
Links: there is a link between nodees i and j if there is this relation
between them
Links could be directed or undirected, may have different weights,
may or may not be reflexive, let us stick with the simple intuition
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Examples of networks
Exogenously designed netwoks
Saccharomyces cerevisiae
Nodes are the proteins of the monocellular Saccharomyces cerevisiae, there
is a link between two proteins if they interact in at least one chemical
reaction in the cell
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Examples of networks
Exogenously designed netwoks
Hardware networks
Nodes are computers, printers, servers. . . there is a link if they are directly
connected.
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Examples of networks
Endogenous networks
The World Wide Web
Nodes are webpages in the internet, links are (directed) web-links
Webpages .edu in the US
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Examples of networks
Endogenous networks
Coautorships
Nodes are researchers, if they wrote a paper together there is a link
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Examples of networks
Endogenous networks
Airports
Nodes are airports, if there is a regular flight there is a link
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Examples of networks
Endogenous networks
Co–board memberships
Nodes are Italian firms in 1972, there is a link if there is at least one
common member in the boards (right is FIAT alone)
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Examples of networks
Endogenous networks
International trade
Nodes are countries, there is a link if there are reciprocal exports above a
certain threshold
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Examples of networks
Endogenous networks
Friendships
Nodes are students from a US High School, there is a link if in a survey
one cites the other as his/her friend
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Examples of networks
Addhealth Data
AddHealth data
The National Longitudinal Study of Adolescent Health (Add Health)
1994 survey in 84 American high schools and middle schools, by the
UNC Carolina Population Center
They could nominate their friends from a list of all the other students
in their school
We consider a link whenever at least one of the two students
nominate the other one
Students were self–reporting a lot of other information in a huge
survey
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Examples of networks
Hybrid cases
A genealogic tree
Nodes are historical characters, a link is a parental relation or a marriage.
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Examples of networks
Hybrid cases
Citations
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Examples of networks
Common characteristics
What do these things have in common?
The examples we have seen span very different contexts, and there are
many more (Facebook, Twitter, Youtube. . . )
Nevertheless, the share many common characteristics (expecially the
endogenously generated ones):
There are a few nodes with many friends, and many with a few friends
Two friends of mine are likely to be friends together
There is always a short path between any two nodes
If nodes are heterogeneous for exogenous reasons, it is more likely
that “similar” nodes are friends
Hubs are likely to be connected together
So, understanding the structure becomes important to study and
understand all of them: Complex Social Networks
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Importance of networks
Importance of networks
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Importance of networks
Networks as a constraint
Standard economic assumptions
Completeness of markets: anyone can trade with anyone else
but most markets are not centralized
there are communication costs and asymmetric information risks when
changing partners
this is not only the case in undeveloped markets
but it is also not the case in international economics
Efficiency of markets: prices summarize all the information
the 2007/2009 crises is a good example of how limited communication
flows moved prices away from the fundamentals
Homo economicus
in taking decisions, economic agents are not influenced by peers
Externalities: public goods and bads are global concepts
most of the externalities are instead local: because of geographic, but
also cultural, distances
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Importance of networks
Networks as a constraint
Only frictions?
We observe networks in trading, information passing. . .
But is a just by chance?
Is the cost of changing partners just a negligible friction?
Can we exclude any path dependence from the links that have been in
place in the past?
If the answer to those questions is always ‘yes’, then networks are not
important for economic theory
If some answer are ‘no’, then it is important to study how this constraints
work, and what are their implications
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Importance of networks
Networks as a constraint
A classical historical example: centrality
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Importance of networks
Networks as a constraint
Peer effects: apps
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Importance of networks
Networks as a constraint
A recent macro example: path dependence
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Homophily
Homophily
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Homophily
Empirical facts
Illustrations of homophily
National Sample: only 8% of people have any people of another race
that they “discuss important matters” with (Marsden 87)
Interracial marriages U.S.: 1% of white marriages, 5% of black
marriages, 14% of Asian marriages (Fryer 07)
In middle school, less than 10% of “expected” cross- race friendships
exist (Shrum et al 88)
Closest friend: 10% of men name a woman, 32% of women name a
man (Verbrugge 77)
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Homophily
Empirical facts
Back to the school example: AddHealth and a Dutch
school
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Homophily
Evidence
Chicago
New York
Los Angeles
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Homophily
Evidence
Reasons for homophily
Idea of reduced form
communication costs
frictions
opportunities
choices
It is important to distinguish between choices and opportunities
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Games on networks
Games on networks
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Games on networks
Introduction
Why networks games?
Decisions to be made: not just diffusion – not just updating
Games with mutliplayer agents are defined naturally
So what is the role of networks?
“Strategic” Interplay: – Inter-dependencies are shaped by the network
others’ behaviors affect my utility/welfare
others’ behaviors affect my decisions, actions, consumptions,
opinions...
others’ actions affect the relative payoffs to my behaviors
Or, in more complicated games: information passes through the network
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Games on networks
{0, 1} actions
Preliminary definitions for 2–actions games
Each player is a node of the network
Each player i chooses action xi in {0, 1}
Payoff will depend on
how many neighbors a player has di
how many neighbors choose action 1: mi
So ui = (xi , di , mi )
Nash equilibrium: Every player’s action is optimal for that player given the
actions of others
Often look for pure strategy equilibria – May require some mixing
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Games on networks
{0, 1} actions
Some examples
Coordination games
u(0, di , mi ) = di − mi
and u(1, di , mi ) = mi
The efficient equilibria are those where all do the same
Network Prisoners’ Dilemma
u(0, di , mi ) = 2 ∗ mi
and u(1, di , mi ) = 2 ∗ mi − di
Only one Nash equilibrium: all 0’s
Anything can be extended from a classical 2 × 2 game: stag–hunt, BofS. . .
What is the 2 × 2 payoff matrix of the two cases above?
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Games on networks
{0, 1} actions
Simple complements
Example: agent i chooses action 1 iff at least t of her neighbors do the
same
u(0, di , mi ) = 0 and u(1, di , mi ) = mi − t
Two equilibria:
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Games on networks
One shot game
Local public goods
An exogenous network, in a pure–strategy Nash equilibrium each node i
(Ni is the neighborhood) plays xi ∈ {0, 1} according to best–response
1 if xj = 0 for all j ∈ Ni ,
bi (~x ) =
0 otherwise.
Example: Five drivers and a network of relations – car sharing
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Games on networks
One shot game
Multiple equilibria
The number of contributors may vary a lot
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Games on networks
One shot game
Multiple equilibria
The number of contributors may vary a lot... even on a regular network
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Games on networks
One shot game
Maximal Independent sets
Independent Set: a set S of nodes such that no two nodes in S are
linked,
Maximal: every node in N is either in S or linked to a node in S
The equilibria of the best shot game are maximal independent sets
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Complements and substitutes
Complements and substitutes
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Complements and substitutes
Two important classes
Definition
Strategic complements – for all d, m ≥ m0
Increasing differences:
u(1, d, m) − u(0, d, m) ≥ u(1, d, m0 ) − u(0, d, m0 )
Strategic substitutes – for all d, m ≥ m0
Decreasing differences:
u(1, d, m) − u(0, d, m) ≤ u(1, d, m0 ) − u(0, d, m0 )
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Complements and substitutes
Two important classes
Examples
Strategic complements
Choice to take an action by my friends increases my relative payoff to
taking that action (e.g., friend learns to play a video game)
education decisions – care about number of neighbors, access to jobs,
etc. – invest if at least k neighbors do
smoking & other behavior among teens, peers...
technology adoption – how many others are compatible...
learn a language...
cheating, doping
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Complements and substitutes
Two important classes
Examples
Strategic substitutes
Choice to take an action by my friends decreases my relative payoff to
taking that action (e.g., roommate buys a stereo/fridge)
information gathering – e.g., payoff of 1 if anyone in neighborhood is
informed, cost to being informed (c < 1)
local public goods (shareable products.. )
local municipalities: library, swimming pools...
competing firms (oligopoly with local markets)
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Complements and substitutes
Informal job market
Drop out decision – A case of strategic complements
Labor Participation Decisions (Calvo–Armengol & Jackson 04,07,09)
Value to being in the labor market depends on number of friends in
labor force
Drop out if some number of friends drop out
Participate if at least some fraction of friends do
Some heterogeneity in threshold (different costs, natural abilities...)
Homophily – segregation in network
Different starting conditions: history...
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Complements and substitutes
Informal job market
Real data
Drop-Out Rates & Labor Force Participation Rate
(Chandra (2000), DiCecio et al (2008) – males 25 to 55)
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Complements and substitutes
Informal job market
Example with homophily
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Continuous actions
Continuous actions
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Continuous actions
Bramoull´
e and Kranton (2007)
Again public good contribution: strategic substitutes
xi ∈ [0, 1] (information acquisition)
Payoff:


Πi = f xi +
X
xj  − cxi
with f concave
j∈Ni
Equilibrium: minimum xi such that
X
xi +
xj ≥ x ∗ where
f 0 (x ∗ ) = c
j∈Ni
Two types of equilibria
distributed: where for some i we have 0 < xi < x ∗
specialized: where for each i either xi = 0 or xi = x ∗
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Continuous actions
Bramoull´
e and Kranton (2007)
Two types of equilibria
distributed: where for some i we have 0 < xi < x ∗
specialized: where for each i either xi = 0 or xi = x ∗
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Linear quadratic models
Linear quadratic models
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Linear quadratic models
Ballester, Calvo–Armengol and Zenou (2006)
A case of strategic complements
There it is applied to contagious behaviors (smoking, studying. . . )
X
b
ui (xi , ~x¬i ) = axi − xi2 +
wij xi xj
2
Best response to ~x¬i is
xi∗
=
a+
P
wij xj
b
In matricial form (if Iˆ − w
ˆ /b is invertible)
~x =
−1
1 ~
a1 + w~
ˆ x = Iˆ − w
ˆ /b
b
a~
1
b
Note: you can obtain the same best responses from many other utility
functions (question: what is it that characterizes a game?)
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Linear quadratic models
Bramoull´
e, Kranton & D’Amours (2014)
Does a solution exist?
−1 a
~1
~x = Iˆ − w
ˆ /b
b
Remember that xi ∈ [0, 1], or at least xi > 0, so the solution to this may
not be feasible.
What happens if not?
solution may explode
multiple equilibria, where some agents contribute 0
Bramoull´e et alii (2014): there is an interior equilibrium only if
−1
Iˆ − w
ˆ /b
is positive definite ⇔ minimal eigenvalue |λmin (w
ˆ )| < b
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Linear quadratic models
Bramoull´
e, Kranton & D’Amours (2014)
What is minimal eigenvalue about?
Left–network has λmin = 3, right–network has λmin ' 2.24
More local links (i.e. higher clustering, longest paths. . . ) make it easier to
have unique interior equilibria
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The case of incomplete information
The case of incomplete information
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The case of incomplete information
GGJVY (2010)
How can we analyze things?
If all agents know the complete network, things can be computationally
hard
each agent has to compute spectral analysis of the network
it is easily the case that equilibria are multiple
The solution is obtained reducing knowledge
the agents know only some statistics about the network
typically degree distribution – but also degree correlation
and then they play a Bayesian game
another interpretation is that first they choose the action, and then a
random network is realized
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The case of incomplete information
GGJVY (2010)
Results
In the case of no degree correlation – under some specific assumptions
In the case of strategic complements
action increases with degree
In the case of strategic substitutes
action decreases with degree
When there is degree correlation (assortativity) the trend depends on the
sign of the externality: positive or negative
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Repeated games on networks
Repeated games on networks
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Repeated games on networks
Prisoners’ dilemma – Dall’Asta, Marsili & Pin (2012)
2 players case
Example:
prisoners’ dilemma game
c
d
c
d
1,
1 0, 1 + x
1 + x, 0 x,
x
,
discount rate δ
2 players tit–for–tat: iterated cooperation possible if
∞
X
1
1
1+
δt x = 1 +
x≤
⇒ x ≤δ
1−δ
1−δ
t=0
What happens with three nodes in a complete network?
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Repeated games on networks
Prisoners’ dilemma – Dall’Asta, Marsili & Pin (2012)
Networks make a difference
What happens with three nodes in a complete network?
Start from full cooperation
suppose that k deviates, because x ≤ δ
is the punishment by one from i and j always credible?
what if:
1
1
1≥1+
2x ⇒ x ≤ 2δ ?
1−δ
1−δ
If δ ≤ x ≤ 2δ we have that a coalition of only two cooperators is
stable under tit–for–tat
This can be extended to a general network: cluster of cooperators satisfy
geometric conditions
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Market applications
Market applications
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Market applications
What is the value of a network
How big are online social networks?
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Market applications
What is the value of a network
What can a network do?
Indignados
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Market applications
What is the value of a network
Can we detect it?
Other media detect too late
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Market applications
What is the value of a network
A network helps coordination
Arab spring
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Market applications
What is the value of a network
A network can be used
Electoral campaigns
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Market applications
What is the value of a network
What is the value of a network?
It is priced by the markets
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Market applications
What is the value of a network
Decentralized markets
Many markets are not centralized:
not only small trading – over–the–counter markets are huge
General economic theory assumes completeness of markets:
everyone can trade with everyone else
but also Walrasian assumptions are against decentralized trading
because price is decided ex–ante by the Walrasian auctioneer
in equilibrium, all prices must be the same if the network is connected
(why?)
but the path to equilibrium can pass from different prices
This is a fundamental aspect that is still poorly studied
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Market applications
Bertrand competition – Nermuth et alii (2013)
Competition on a network
The model
one good,
n homogeneous firms,
constant marginal cost (c = 0)
mass 1 of buyers, one unit of good per capita
q1 of them see a single price at random
q2 of them sample two prices, . . .
q
n are totally informed
P
n
~ is the search distribution
q
i=1 qi = 1,
consumers buy at lowest price, if not above r
firms compete on prices
~ (marketing surveys)
they know only distribution q
any p ≥ 0 is in the strategy profile
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Market applications
Bertrand competition – Nermuth et alii (2013)
Extreme cases
Lemma 1 (Bertrand competition)
If q1 = 0, each firm will set p = 0.
Idea of the proof
to charge alone maximal price p¯ gives null profit
at least two charge the same price p
suppose p > 0 ⇒ p − is better
Lemma 2 (Monopoly) If q1 = 1, each firm will set p = r .
The interesting case is 0 < q1 < 1
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Market applications
Bertrand competition – Nermuth et alii (2013)
Price distribution
~ with q1 ∈ (0, 1):
Proposition 3 Take q
no equilibrium in pure strategies
unique symmetric equilibrium in mixed strategies
price distribution function f~q (p) R
cumulative distribution F~q (p) = f~q (p) dp
support on [pmin , r ]
q1 r
pmin = Pn
j=1 jqj
Fq~ (p) = 1 −
Paolo Pin
Φ−1
~
q
q1 r
p
, with
Φq~ (x) ≡
Microeconomics II – Lecture 18
n
X
iqi x i−1 .
i=1
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Market applications
Bertrand competition – Nermuth et alii (2013)
Comparative statics (1/2)
Prop. 5: if we let some uninformed buyers search more
more search is beneficial for all other buyers
more competition → smaller expected prices
q0 r
expected profit of firms decreases ( n1 < qn1 r )
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Market applications
Bertrand competition – Nermuth et alii (2013)
Comparative statics (2/2)
Proposition 6
~
take a search distribution q
0 = q − ∆ and q 0
~ 0 where qm
consider q
m
m+1 = qm+1 + ∆ (m ≥ 2)
there is a i ∗ ≥ 2
such that pi0 > pi for i < i ∗ , and pi0 ≤ pi for i ≥ i ∗ .
Idea of the proof there is no stochastic dominance
we apply Riemann definition of integrals
p10 > p1 , pi0 /pi decreases in i
and limi→∞ pi0 /pi < 1
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Market applications
Bertrand competition – Nermuth et alii (2013)
The digital divide
From Prop. 6, if we let some already informed buyers search more
firms will simultaneously move probability weight on extrema
monopoly price p = r and competitive price p = pmin
even if the average of charged prices increase, more sampling is
beneficial from a certain point on
the externality of more search is bad for low–informed and good for
highly–informed
the aggregate expected profits of firms will not change
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Market applications
Goods with peer effects
Consumers’ choices
Consider an exogenous directed (maybe weighted) network gˆ of consumers
they buy continuous quantities ~x of a single good
that has complementary peer effects (e.g. a videogame)
a monopolist can perfectly price discriminate
the utility for consumer i is


X
gij xj  − pi xi
ui (xi , ~x¬i , pi ) = ai xi − bi xi2 + xi 
j
if, for all i, 2bi >
P
j
gij (in the discrete case: degree), then
xi =
ai − pi
1 X
+
gij xj
2bi
2bi
j
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Market applications
Goods with peer effects
Monopoly on a network – Candogan et alii (2013)
Optimal pricing for the monopolist
cal Λij = 2bi if i = j, and 0 otherwise
the optimal pricing scheme for the monopolist is
t
ˆ − gˆ )(Λ
ˆ − gˆ + gˆ )−1 ~a
p~ = ~a − (Λ
2
2
It follows, that if gˆ is symmetric (i.e. gˆ = gˆ t ), then
the optimal pricing scheme is
p~ =
~a
2
there is no discrimination due to peer effects
there is no gain for the monopolist in knowing the network
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Lab experiments
Lab experiments
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Lab experiments
Charness et alii (2014)
Theory works
All results are confirmed in the lab
when the network is known
and in best shot games, the refinement of stochastic stability also
works
when only degree distribution is known
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A field experiment in Ghana
A field experiment in Ghana
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A field experiment in Ghana
Microfinance
A field experiment in Ghana
Microfinance has proven itself not to help innovative entrepreneurship:
too risky?
not enough diverse social contacts?
We run a controlled field experiment in rural Ghana,
under a more ambitious microfinance program for entrepreneurs,
run by a local NGO
We use the optimal local internet infrastructure. . .
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A field experiment in Ghana
Microfinance
The treatment
. . . to provide an enlarged virtual social network with an ad–hoc App
with the help of Facebook, Airtel and NGO Village Exchange Ghana
The pilot (financed by CEPR) is starting now
with 200 subjects (50 under treatment) in 10 villages
The ERC grant is intermediate to reach National Ghanian level
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