Social Interactions and Endogenous Association - Department of ...

The parameter Φ is analogous, but reflects the effect of associates’ characteristics on the
utility of associating. For instance, people likely obtain utility from associating with
people who are attractive or athletic and people of the same race or ethnicity.
Once people are viewed as choosing to associate with a portion of their groups, it
becomes important to consider how changes in the number of associates affect the
strength of interactions. One view, which comes closest to the literature, is that new
associations completely crowd out existing ones, leaving the total effect of associates
fixed, that α = 0 . One might think that people who have many associates are affected by
their associates (as a whole) more than people with few associates, that α > 0 .
II.B. Solving the Model
Differentiation with respect to y gives the person’s optimal action,
A A
( θy
+ γx
) α + ε
y = β ′ x + T
. (*)
This is the standard linear-in-means behavioral equation, where the total social effect will
vary across people if α ≠ 0 . We consider a model with a binary, scalar characteristic
(Weinberg [2005] discusses a model with continuous characteristics), and begin by
eliminating the action to focus on associations on characteristics. We assume that the
characteristic takes on values x ∈{ −1,1 }
low and high types. Let
i
, and refer to people with these characteristics as
π denote the share of the group with x = i ∈{ −1,1 }
Φ = 1 and that the other social utility parameters (θ, ψ, ω, and γ) are all zero.
A
Let π [ 0,
π ]
person with characteristic
i| j
∈
i
denote the measure of people with characteristic
i
. We assume
x with whom a
x
j
associates, which must be non-negative and is constrained
by group compostion. With Φ > 0 and the values of the characteristic taking on opposite
signs, people only associate with people of the same type, so that a type-i person chooses
A
i i
[ 0 π ]
π
|
∈ , . The first order condition for active associations is,
i
7 The cost could be negative if the opportunity cost of time is low and people like interacting.
7