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Non-Market Interactions
Non-Market Interactions1
Edward Glaeser
Department of Economics
Harvard University
email: [email protected]
José A. Scheinkman
Department of Economics
Princeton University
email: [email protected]
May 3, 2002
1
Invited lecture, Econometric Society World Congress in Seattle (2000). We
thank Roland Benabou, Alberto Bisin, Avinash Dixit, Steve Durlauf, James Heckman and Eric Rasmusen for comments, Marcelo Pinheiro for research assistance,
and the National Science Foundation for research support. We greatly benefitted
from detailed comments by Lars Hansen on an earlier version.
1
Introduction
Why are there stock market crashes? Why is France overwhelmingly Christian and Thailand overwhelmingly Buddhist? Why did the great depression
occur? Why do crime rates vary so much over time and space? Why did the
adoption of hybrid corn follow an s-shaped curve? Why is there racial segregation? Why do mass cultural phenomena like the Hula Hoop and Harry
Potter occur?
This bizarre juxtaposition of questions are bound together by one common element. Over the past 30 years, economists have suggested that models
of social interactions provide the answer to every one of these questions.1
In most cases, the relevant social interactions are non-market interactions,
or interactions between individuals which are not regulated by the price
mechanism.2
Many models of non-market interactions exhibit strategic complementarities, which occur when the marginal utility to one person of undertaking
an action is increasing with the average amount of the action taken by his
peers. Consequently, a change in fundamentals has a direct effect on behavior and an indirect effect of the same sign. Each person’s actions change not
only because of the direct change in fundamentals, but also because of the
change in the behavior of their neighbors. The result of all these indirect
effects is the social multiplier. When this social multiplier is large, we expect
to see the large variation of aggregate endogenous variables relative to the
variability of fundamentals, that seem to characterize stock market crashes,
religious differences, the great depression, wildly different crime rates and
the Hula Hoop. Furthermore, for large levels of social interaction, multiple
equilibria can occur - that is one may observe different outcomes from exactly the same fundamentals. The existence of multiple equilibria also helps
us to understand high levels of variance of aggregates.
But non-market interactions models don’t just predict high levels of
variance. When individuals can choose locations, the presence of these interactions often predicts segregation across space. Cities exist because of
1
The connection between stock market crashes and social interaction models is discussed in Shiller [2000]. Becker and Murphy [to apppear] discusses social interactions and
religion. Cooper and John [1986] connect non-market interactions and business cycles.
Glaeser, Sacerdote and Scheinkman [1996] argue that social interactions can explain the
high variance of crime rates over space. Ellison and Fudenberg [1993] think about technology adoption and local interactions. Schelling [1971,1972] pioneered social interaction
models by discussing segregation. Bickchandani, Hirshleifer and Welch [1992] model fads
and fashion.
2
Non-market interactions are thus particular forms of externalities.
1
agglomeration economies which are likely to come from non-market complementarities3 . Indeed, the selection of like types into the same neighborhoods
is often what makes measuring social interactions quite difficult. In dynamic settings, social interactions can produce s-shaped curves which help
to explain the observed time series patterns of phenomena as disparate as
telephones and women in the workplace.
The potential power of these models explains the explosion of work in
this area over the past few years. In this essay, we explore the common
mathematical structure of these models. While there has been a great deal
of work in this area, there has been less effort to understand the common
elements between the disparate models. Instead of discussing the literature
broadly, we will start with a particularly general social interactions model
and examine the implications of this model. Several of the better known
models in this area can be seen as special cases of this more general model.
In this general model, the utility function includes an individual’s action
(or consumption), the actions of flexibly defined peer or reference groups,
personal characteristics (including income) and common prices. People are
arranged by social proximity and the reference groups may include only
one’s closest neighbor, or the entire city. One controversy in the literature
is whether social interactions are best thought of as being local (i.e. people
are really affected only by their closest neighbor) or global (i.e. people are
affected by the average behavior of people in a large community). Our
framework is sufficiently flexible to examine these extremes.
Although there are several examples in the literature that deal with
discrete choice, many others deal with variables such as education, which
are more naturally thought of as continuous (e.g. Benabou [1993], [1996] and
Durlauf [1996a], [1996b].) In addition, in certain contexts the variables that
we observe (e.g out-of-wedlock births) are consequences of behavior that the
agents - as opposed to the social scientist - are observing and mimicking, and
this behavior may be more naturally modeled as a continuous variable. For
this reason and because it is mathematically simpler we emphasize models
with continuous choices.4
We first examine the conditions under which for each set of exogenous
parameters we obtain a unique equilibrium. A sufficient condition for a
unique equilibrium is that the absolute value of the second derivative of
utility with respect to one’s own action is greater than the absolute value
of the cross partial derivative between one’s own action and the action of
3
4
See e.g. Krugman [1991]
Some results for a model with discrete action space are derived in section 2.4.
2
the peer group. We refer to this condition as the Moderate Social Influence
(MSI) condition. One, perhaps surprising, fact is that uniqueness is guaranteed by this condition with no assumptions about the degree to which
interactions are global or local.
We establish sufficient conditions for the existence of multiple equilibria.
These conditions tend to be satisfied if there exists sufficient non-linearity in
the effect of peer actions on an individual’s own optimal action, and not too
much heterogeneity among agents. Hence when sorting creates homogeneity,
multiple equilibria are more likely.
As mentioned above the main ingredient used to generate multiple equilibria in models of non-market interactions has been strategic complementarities. We show that strategic complementarity is not necessary for multiple
equilibria. We present an example where each agent belongs to one of two
groups. Individuals in each group want to differentiate themselves from the
average individual in the other group in the consumption of a good. No
other social interactions are present. There are multiple equilibria if the
desire to differentiate is strong enough.
Even in cases with multiple equilibria, as long as there is some individual heterogeneity, (generically) there will not be a continuum of equilibria.
Around each equilibrium there is an interval that contains no other equilibria. As the model does not tell us the size of this interval, the empirical
relevance of this finding may be limited. However, it does suggest that empirical approaches to estimating multiple equilibria models should focus on
situations where the different equilibria are sufficiently distinct.
Another result is that, if the MSI condition holds, and strategic complementarity prevails, there is always a well defined social multiplier.5 This
social multiplier can be thought of as the ratio of the response of an individual action to an exogenous parameter (that effects only that person) and
the (per capita) response of the entire peer group to a change in the same
parameter that effects the entire peer group. In the empirical framework
section of the paper, we discuss estimating this multiplier. The presence of
this social multiplier implies that social interactions are generally connected
with unusually high variances of aggregates. In fact, as we will argue later,
it is very difficult to empirically distinguish between models with a unique
equilibrium and a large social multiplier and models with multiple equilibria.
Although the results concerning uniqueness and the social multiplier are
independent of the interaction structure, the same is not true for ergodicity.
However we show that if MSI prevails, shocks are independent and iden5
When the MSI condition fails to hold, the social multiplier becomes unbounded.
3
tically distributed across individuals, and each agent is influenced by the
average action of all other agents, then the equilibrium average action for a
large population is independent of the actual shocks that occur.
We do not discuss the welfare properties of equilibria. Indeed, the presence of heterogeneous individuals in our benchmark model makes it more difficult to rank equilibria. One possibility is to proceed as Brock and Durlauf
[1995] and examine ex-ante welfare. In general, the ranking of equilibria
according to ex-ante welfare typically depends on finer aspects of the interaction structure. We have decided not to focus on these issues here.
In the third section of the paper, we present a linear quadratic version of
the social interaction model which can be used for estimation. We then discuss how different approaches to estimating the extent of social interactions
fit into this framework. There are three primary approaches to estimating
social interactions.
First social interactions are estimated by looking at the covariance between individual outcomes and the average outcome of a peer group. Even
in the best case scenario, ordinary least square coefficients based on these
covariances do not yield consistent coefficient estimators of social interaction parameters. The problem occurs because peer outcomes are partially
determined by the individual’s outcome. Our framework suggests a correction in the univariate case for this problem. This correction will not work if
unobserved individual attributes are correlated with peer attributes. This
correlation can occur either because of omitted community level characteristics or because of differential sorting into neighborhoods. The standard
correction for this mutual interdependence (following Case and Katz, 1991)
is to use an instrumental variables approach which relies on the exogenous
characteristics of peers. This approach will successfully solve problems due
to omitted community attributes, but not if there is sorting into communities. Randomized experiments offer the best chance of identifying peer
effects.
The second empirical approach uses the variance of aggregates. The
third empirical approach uses the logic of the multiplier. In this approach,
the relationship between exogenous variables and outcomes for individuals is
compared with the relationship between exogenous variables and outcomes
for groups. The ratio is a measure of the size of social interactions.
All three of these approaches offer different methods of capturing social
interactions and in many cases, the estimates will be absolutely equivalent.
However, all will suffer when there are omitted variables and sorting. The
empirical hope is that randomization of people with peers (as in Sacerdote
[2000]) will help us to break this sorting. However, this randomization is
4
rare and if we estimate social interactions only in those cases where we have
randomization, then we are likely to have only limited empirical scope for
this type of work.
2
Theoretical Models of Non-Market Interactions
Economics has always been concerned with social interactions. Most often,
economists have written about social interactions that are mediated by the
market. In particular, economists have focused on the interaction that occurs when greater demand of person x for commodity z raises the price of
that commodity and reduces the consumption by person y. This negative
interaction lies at core of our discipline, and when this interaction appears in
our models, it tends to predict ”well-behaved” systems, with unique equilibria, monotonic convergence, etc. Although negative interactions can create
cycles (high demand in one period raises prices the next period) as in the
case of cobweb models, they tend not to create high levels of variability.
However, as economists tried to explain more puzzling phenomena, particularly large variances over time and space, they moved to positive interaction models. The most famous early example of this move is in Keynes,
whose famous ”beauty contest” description of the stock market suggested
the possibility of positive market-mediated interactions. One person’s demand for the stock market could conceivably induce others to also purchase
shares. This type of model has only recently been formalized (e.g. Froot,
Scharfstein and Stein [1992]). Several other authors have focused on different mechanisms where we see market-mediated positive social interactions
(e.g. Murphy, Shleifer and Vishny [1989]). These models create the possibility of multiple equilibria or high levels of variability for a given set of
fundamentals.
Our interest is fundamentally in non-market interactions. The literature
on these interactions has paralleled the rise in market-mediated positive interaction models and has many similarities. Schelling [1971,1972] pioneered
the study of non-market interactions in economics. Following in Schelling’s
tradition, economists have examined the role of non-market interactions in
a myriad of contexts.
Many of the recent papers on non-market interactions use random field
models, also known as interactive particle systems, imported from statistical physics. In these models one typically postulates individual’s interdependence and analyzes the macro behavior that emerges. Typical questions
concern the existence and multiplicity of macro phases that are consistent
5
with the postulated individual behavior. Follmer [1974] was the first paper
in economics to use this framework. He modeled an economy with locally
dependent preferences, and examined when randomness in individual preferences will affect the aggregate, even as the number of agents grows to infinity.
Brock [1993] and Blume [1993] recognized the connection of a class of interactive particle models to the economic literature on discrete choice. Brock
and Durlauf [1995] develops many results on discrete choice in the presence
of social interactions. Other models inspired in statistical physics start with
a more explicit dynamic description on how agents choices at a point in time
are made, conditional on other agents’ previous choices, and discuss the evolution of the macro behavior over time. Bak, Chen, Scheinkman and Woodford [1993] (see also Scheinkman and Woodford [1995]) study the impact
of independent sectoral shocks on aggregate fluctuations with a “sandpile”
model that exhibits self-organized criticality and show that independent
sectoral shocks may affect average behavior, even as the numbers of sectors
grow large. Durlauf [1993] constructs a model based on local technological
interactions to examine the possibility of productivity traps, in which low
productivity techniques are used, because other producers are also using low
productivity processes. Glaeser, Sacerdote and Scheinkman [1996] use what
is referred to as the voter model in the literature on interacting particle
systems.6 to analyze the distribution of crime across American cities. Topa
[1997] examines the spatial distribution of unemployment with the aid of
contact processes.7
A related literature, that we do not discuss here, studies social learning.8
In these models agents typically learn from observing other agents and base
decisions on the observed decisions of others.
Becker and Murphy (2001) presents a particularly far ranging analysis
of the social interactions in economics. This volume extends Becker’s (1991)
earlier analysis of restaurant pricing when there are social interactions in
the demand for particular restaurants. This work is particularly important
because of its combination of social interactions with maximizing behavior
and classic price theory. In the same vein, the work of Pesendorfer [1995]
on cycles in the fashion industry examines how a monopolist would exploit
the presence of non-market interactions.
In the remainder of this section we develop a model of non-market in6
e.g. Ligget [1985]
e.g. Ligget [1985]
8
e.g. Arthur [1989], Banerjee [1992], Bickhchandani, Hirshleifer and Welch [1992],
Ellison[1993], Ellison and Fudenberg [1994], Gul and Lundholm [1995], Kirman [1993] and
Young [1993].
7
6
teractions that will hopefully serve to clarify the mechanisms by which nonmarket interactions affect macro behavior. The model is self-consciously
written in a fashion that makes it close to traditional economic models,
though it can accommodate versions of the models that were inspired by
statistical physics. We consider only a static model, although we also examine some ad hoc dynamics. On the other hand, our framework encompasses
several of the examples in the literature, and is written so that special cases
can be used to discuss issues in the empirical evaluation of non-market interactions (see Section 3).9 In the spirit of the work of Pesendorfer [1995]
or Becker and Murphy [2001] we explicitly allow for the presence of prices,
so that the model can be used to study the interplay between market and
non market interactions.
We next write down the details of the model. In 2.2 we describe some
examples in the literature that fit into our framework. We present results
for the case of continuous actions in 2.3. These results concern sufficient
conditions for uniqueness or multiplicity of equilibria, the presence of a social
multiplier, the stability of equilibria, and ergodicity. Models with discrete
actions and large populations are treated in 2.4. At the end of this section
we give a short discussion of endogenous peer group formation.
2.1
A model of non-market interactions
We identify each agent with an integer i = 1, . . . , n. Associated with each
agent i are his peer, or reference, groups Pik , k = 1, . . . K, each one a subset
of {1, . . . , n} that does not contain i. We allow for multiple reference groups
(K > 1) in order to accommodate some of the examples that have appeared
in the literature,( Example 5 below.) Later we will discuss the case where
the agent chooses his reference groups, but in most of what follows the Pik ’s
are fixed. Each agent is subject to a “taste shock” θi , a random variable with
support on a set Θ. Finally, each agent chooses an action a ∈ A. Typically
the set A will be a finite set, the discrete choice case, or an interval of the
real line, the continuous choice case. The utility of agent i depends on the
action chosen by him, ai , and the actions chosen by all other agents in his
peer group. More precisely we assume that:
U i = U i (ai , A1i , . . . , AK
i , θi , p), where
(1)
9
In particular, our framework is related to the pioneering work of Cooper and
John[1988] on coordination failures. They study a particular version of our framework
where each agent interacts with the average agent in the economy, and focus on models
where there is no heterogeneity across agents.
7
Aki =
n
X
k
γij
aj ,
(2)
j=1
n
k ≥ 0, γ k = 0 if j 6∈ P k ,
k
with γij
j=1 γij = 1, and p ∈ Π is a vector of
ij
i
parameters.
In other words, the utility enjoyed by agent i depends on his own chosen action, on an weighted average of the actions chosen by agents in his
reference groups, on his taste shock, and on a set of parameters. We allow
the utility function to vary across agents because in some cases we want to
identify variations in the parameters θi explicitly with variations on income
or other parameters of the problem. We also assume that the maximization
problem depends only on the agent being able to observe the relevant average action of peers. In some applications we will allow for Pik = ∅, for some
of the agents. In this case we may set Aki to be an arbitrary element of A,
and U i to be independent of Aki . In many examples each agent will have a
unique reference group. In this case we drop the superscript k and write Pi
instead of Pi1 etc...
Typically p will be interpreted as the exogenous (per-unit) price of taking
action.10 In addition we will think of θi = (yi , ζi ), where yi represents income
and ζi a shock to taste. In this case,
P
1
K
U i (ai , A1i , . . . , AK
i , θi , p) = V (ai , Ai , . . . , Ai , ζi , yi − pai ).
(3)
An equilibrium is defined in a straightforward way. For given vectors θ =
(θ1 , . . . , θn ) ∈ Θn and p, an equilibrium for (θ, p) is a vector a = (a1 , . . . , an )
such that, for each i,
ai ∈ argmax U i (ai ,
n
X
1
γij
aj , . . . ,
j=1
n
X
K
γij
aj , θi , p),
(4)
j=1
This definition of equilibrium requires that, when making a decision,
agent i observes Aki - the summary statistics of other agents’ actions that
affect his utility. As usual we can interpret this equilibrium as a steady
state of a dynamical system in which at each point in time, agents make
a choice based on the previous choices of other agents, though additional
assumptions, such as those in Proposition 4 below are needed to guarantee
that the dynamical system will converge to such a steady state.
10
One can also extend the notion of equilibrium to allow for an endogenous p.
8
2.2
Some examples
Example 1 The discrete choice model of Brock and Durlauf [1995].11
Here the set A = {−1, 1}, and Θ = R. Each agent has a single reference
group, all other agents, and the weights γij ≡ 1/(n − 1). This choice of
reference group and weights is commonly referred to as global interactions.
Let
1 − ai
U i (ai , Ai , θ) = hai − J(Ai − ai )2 +
θi ,
(5)
2
where h ∈ R, J > 0. The θi ’s are assumed to be independently and identically
distributed with
1
Prob (θi ≤ z) =
,
1 + exp(−νz)
for some ν > 0. h measures the preference of the average agent for one of the
actions, J the desire for conformity, and θi is a shock to the utility of taking
the action ai = −1. Brock and Durlauf also consider generalized versions of
this model where the γij ’s vary, thus allowing each agent to have a distinct
peer group.
Example 2 Glaeser and Scheinkman [2001].
The utility functions are:
1−β 2 β
ai − (ai − Ai )2 + (θi − p)ai .
2
2
Here 0 ≤ β ≤ 1 measures the taste for conformity. In this case,
U i (ai , Ai , θi , p) = −
ai = [βAi + θi − p].
(6)
Notice that when p = 0, β = 1 and the A0i s are the average action of all other
agents, this is a version of the Brock-Durlauf model with continuous actions.
P
Unfortunately, this case is very special. Equilibria only exist if i θi = 0,
and in this case a continuum of equilibria would exist. The model is, as we
will show below, much better behaved when β < 1.
In Glaeser and Scheinkman [2001] the objective was to admit both local
and global interactions in the same model to try to distinguish empirically
between them. This was done by allowing for two reference groups, and
setting Pi1 = {1, . . . , n} − i, A1i the average action of all other agents, Pi2 =
{i − 1} if i > 1, P12 = {n}, and writing:
U i (ai , A1i , ai−1 , θi , p) = −
11
β2
1 − β1 − β2 2 β1
ai − (ai −A1i )2 − (ai −ai−1 )2 +(θi −p)ai .
2
2
2
A related example is in Aoki[1995]
9
Example 3 The class of models of strategic complementarity discussed in Cooper and John [1988].
Again the reference group of agent i is Pi = {1, . . . , n} − i. The set A is
1 P
an interval on the line and Ai = n−1
aj6=i . There is no heterogeneity and
i
the utility of each agent is U = U (ai , Ai ). Cooper and John[1988] examine
symmetric equilibria. The classic production externality example fits in this
framework. Each agent chooses an effort ai and the resulting output is
f (ai , ā). Each agent consumes his per-capita output and has a utility function
u(ci , ai ). Write
U (ai , Ai ) = u(f (ai ,
(n − 1)Ai + ai
), ai ).
n
Example 4 A simple version of the model of Diamond [1982] on
trading externalities.
Each agents draws an ei which is his cost of production of an unit of
the good. The ei ’s are distributed independently across agents and with a
distribution H and density h > 0, with support on a (possibly infinite) interval [0, d]. After a period in which the agent decides to produce or not,
he is matched at random with a single other agent, and if they have both
produced, they exchange the goods and each enjoys utility u > 0. Otherwise,
if the agent has produced, he obtains utility θi ≥ 0 from the consumption of
his own good. If the agent has not produced he obtains utility 0. We assume
that all agents use a cut-off policy, a level xi such that the agent produces,
if and only if ei ≤ xi . We set
ai = H(xi ),
the probability that agent i will produce. Here the reference group is again
all j 6= i, and
P
j6=i aj
Ai = E(aj |j 6= i) ≡
n−1
Hence, if he uses policy ai an agent has an expected utility that equals
U i (ai , Ai , θi ) =
Z
H −1 (ai )
[uAi + θi (1 − Ai ) − e]h(e)de.
0
Optimality requires that xi = min{uAi + θi (1 − Ai ), d}.
Suppose first that θi ≡ 0. A symmetric equilibrium (ai ≡ a) will exist
whenever there is a solution to the equation
a = H(ua).
10
(7)
If H is the uniform distribution in [0, u] then every a ∈ [0, 1] is a symmetric
equilibrium, As we will show in Proposition 2 this situation is very special.
For a fixed H, for almost every vector θ = (θ1 , . . . , θn ), (interior) equilibria
are isolated.
Example 5 A matching example that requires multiple reference
groups (Pesendorfer[1995]).
In a simple version, there are two groups, leaders (L) and followers (F ),
with nL and nF members respectively. An individual can use one of two
kinds of clothes. Buying the first one (a = 0) is free, buying the second
(a = 1) costs p. Agents are matched randomly to other agents using the
same clothes. Suppose the utility agent i , who is of type t ∈ {L, F } and is
matched to an agent of type t0 , is Vi (t, t0 , a, p, θi ) = u(t, t0 ) − ap + θi a, where
θi is a parameter that shifts the preferences for the second kind of clothes.
Assume that:
u(L, L) − u(L, F ) > p > u(F, L) − u(F, F ) > 0,
(8)
where we have abused notation by writing u(L, L) instead of u(t, t0 ) with
t ∈ L and t0 ∈ L etc...In this example, each agent has two reference groups.
If i ∈ L then Pi1 = L − {i} and Pi2 = F . On the other hand, if i ∈ F then
Pi1 = L and Pi2 = F − {i}.
2.3
Equilibria with continuous actions
In this subsection we derive results concerning the existence, number of
equilibria, stability and ergodicity of a basic continuous action model. We
try not to rely on a specific structure of reference groups or to assume a
specific weighting for each reference group. We assume that A is a (possibly
unbounded) interval in the real line, that each U i is at least twice continuously differentiable, and that the second partial derivative with respect an
i < 0.12 Each agent i has a single reference group P .
agent’s own action U11
i
The choice of a single peer group for each agent and of a scalar action is not
crucial, but it substantially simplifies the notation.
We also assume that the optimal choices are interior and hence, since
i 6∈ Pi , the first order condition may be written as:
U1i (ai , Ai , θi , p) = 0
(9)
12
i
As usual this inequality can be weakened by assuming that U11
≤ 0, and that at the
optimal choice strict inequality holds.
11
i < 0, then a = g i (A , θ , p) is well defined and,
Since U11
i
i i
g1i (Ai , θi , p) = −
i (a , A , θ , p)
U12
i
i i
i (a , A , θ , p) .
U11
i
i i
(10)
We will write G(a, θ, p) for the function defined in Rn × Θn × Π given by:
G(a, θ, p) = (g 1 (A1 , θ1 , p), . . . , g n (An , θn , p)).
Recall that for a given vectors θ = (θ1 , . . . , θn ) ∈ Θn and p, an equilibrium for (θ, p) is a vector a(θ, p) = (a1 (θ, p), . . . , an (θ, p)) such that, for each
i,
ai (θ, p) = g i (Ai (a(θ, p)), θi , p).
(11)
Proposition 1 gives conditions for the existence of an equilibrium.
Proposition 1 Given a pair (θ, p) ∈ Θn × Π, suppose that I is a closed
bounded interval such that, for each i, g i (Ai , θi , p) ∈ I, whenever Ai ∈ I.
Then there exists at least one equilibrium a(θ, p) ∈ I n . In particular, an
◦
equilibrium exists if there exists an m ∈ R, with [−m, m] ⊂A, and such that,
for any i and Ai ∈ [−m, m], U1i (−m, Ai , θi , p) ≥ 0 and U1i (m, Ai , θi , p) ≤ 0.
Proof: If a ∈ I n , since Ai is a convex combination of the entries of a,
Ai ∈ I. Since g i (Ai , θi , p) ∈ I, whenever Ai ∈ I, the (continuous) function G(·, θ, p) maps I n into I n , and therefore must have at least one fixed
point. The second part of the proposition follows since U11 < 0 implies that
g i (Ai , θi , p) ∈ [−m, m], whenever Ai ∈ [−m, m]. QED
Proposition 1 gives us sufficient conditions for the existence of an equilibrium for a given (θ, p). The typical model however describes a process
for generating the θi0 s in the cross section. In this case not all pairs (θ, p)
are equally interesting. The process generating the θi ’s will impose a distribution on the vector θ, and we need only to check the assumptions of
Proposition 1 on a set of θ’s that has probability 1. For a fixed p, we define
an invariant interval I as any interval such that there exists a set Λ ⊂ Θn
with Prob (Λ) = 1, such that for each i, and for all θ ∈ Λ, g i (Ai , θi , p) ∈ I,
whenever Ai ∈ I. If multiple disjoint compact invariant intervals exist,
multiple equilibria prevail with probability one.
It is relatively straightforward to construct models with multiple equilibria that are perturbations of models without heterogeneity.13 Suppose
13
A model without heterogeneity is one where all utility functions U i and shocks θi are
identical. We choose the normalization θi ≡ 0. We will consider perturbations in which
the utility functions are still uniform across agents, but the θi can differ across agents.
12
that Θ is an interval containing 0 and that g(A, θ) is a smooth function that
is increasing in both coordinates. The assumption that g is increasing in
θ is only a normalization. In contrast, the assumption that g is increasing
in A is equivalent to U12 > 0, i.e. an increase in the average action by the
members of his reference group, increases the marginal utility of an agent’s
own action. This assumption was called strategic complementarity in Bulow, Geanokoplos and Kemplerer [1985]. Let x be a stable fixed point of
g(·, 0) i.e. g(x, 0) = 0 and g1 (x, 0) < 1. If the interval Θ is small enough,
there exists an invariant interval containing x. In particular, if a model without heterogeneity has multiple stable equilibria, the model with small noise,
that is where θi ∈ Θ, Θ a small interval, will also have multiple equilibria.
The condition on invariance must hold for almost all θ ∈ Θ. In particular if
we have multiple disjoint invariant intervals and we shrink Θ, we must still
have multiple disjoint invariant intervals. On the other hand if we expand
Θ, we may loose a particular invariant interval and multiple equilibria are
no longer assured. An implication of this reasoning is that when individuals
are sorted into groups according to their θ’s, and agents do not interact
across groups, then multiple equilibria are more likely to prevail. In section
2.5 we discuss a model where agents sort on their θ’s.
In this literature, strategic complementarity is the usual way to deliver
the existence of multiple equilibria. The next example shows that in contrast to the results of Cooper and John[1988], in our model, because we
consider a richer structure of reference groups, strategic complementarity is
not necessary for multiple equilibria.
Example 6 This is an example to show that in contrast to the case of purely
global interactions, strategic complementarity is not a necessary condition
for multiple equilibria. There are two sets of agents {S1 } and {S2 }, and n
agents in each set. For agents of a given set, the reference group consists of
all the agents of the other set. If i ∈ Sk ,
Ai =
1 X
aj ,
n j∈S
`
` 6= k. There are two goods and the relative price is normalized to one. Each
agent has an initial income of one unit and his objective is to maximize:
U i (ai , Ai ) = log ai + log(1 − ai ) +
λ
(ai − Ai )2 .
2
(12)
Only the first good exhibits social interactions, and agents of each set want
i < 0.
to differentiate from the agents of the other set. Provided λ < 8, U11
13
However there is no strategic complementarity - an increase in the action
of others, (weakly) decreases the marginal utility of an agent’s own action.
We will look for equilibria with ai constant within each set. An equilibrium
of this type is described by a pair x, y of actions for each set of agents. In
equilibrium we must have:
1 − 2x + λx(1 − x)(x − y) = 0,
(13)
1 − 2y + λy(1 − y)(y − x) = 0.
(14)
Clearly x = y = 1/2 is always an equilibrium. It is the unique equilibrium
that is symmetric across groups. Provided λ < 4 the Jacobian associated
with equations (13) and (14) is positive, which is compatible with uniqueness
even if we consider asymmetric equilibria. However whenever λ > 4 the
Jacobian becomes negative and other equilibria must appear. For instance if
λ = 4.04040404, x = .55 and y = .45 is an equilibrium, and consequently
so is x = .45 and y = .55. Hence at least three equilibria obtain, without
strategic complementarity.
Proposition 1 gives existence conditions that are independent of the
0 s. Also, the existence
structure of the reference groups and of the weights γij
of multiple invariant intervals is independent of the structure of interactions
embedded in the Pi ’s and γij ’s, and is simply a result of the choice of an
individual’s action, given the “average action” of his reference group, the
distribution of his taste shock, and the value of the exogenous parameter p.
In some social interaction models, such as the Diamond search model,
(Example 4 above) there may exist a continuum of equilibria. The next
proposition shows that these situations are exceptional.
Proposition 2 Suppose Θ is an open subset of Rk and that there exists
∂U i
a coordinate j such that j1 6= 0, that is θij has an effect in the marginal
∂θi
utility of the action. Then, for each fixed p, except for a subset of Θn of
Lebesgue measure zero, the equilibria are isolated. In particular if the θij ’s
are independently distributed with marginals that have a density with respect
to the Lebesgue measure then, for each fixed p, except for a subset of Θn of
zero probability, the equilibria are isolated
Proof: For any p, consider the map F (a, θ) = a − G(a, θ, p). The matrix
of partial derivatives of F with respect to θj is a diagonal matrix with entry
∂U i
dii 6= 0, since j1 6= 0. Hence for each fixed p, DF has rank n and it is a
∂θi
consequence of Sard’s theorem,(see e.g. Mas-Colell page 320) that except
14
perhaps for a subset of Θn of Lebesgue measure zero F1 has rank n. The
implicit function theorem yields the result. QED
Consider again the search model discussed in Example 4. Suppose that
u ≤ d and that the each θi is in an open interval contained in (0, d). Then
at any interior equilibrium the assumptions of the Proposition are satisfied.
This justifies our earlier claim that the continuum of equilibria that exists
when θi ≡ 0 is exceptional. In the model discussed on Example 2, if p =
0, β = 1, and the reference group of each agent is made up by all other
P
agents (with equal weights), then if θi 6= 0, there are no equilibria, while
P
if
θi = 0, there is a continuum. Again, the continuum of equilibria is
exceptional. However if β < 1 there is a unique equilibrium for any vector
θ. This situation is less discontinuous than it seems. In equilibrium,
P
P
ai
1
θi
=
.
n
1−β n
P
Hence, if we fix
θi and drive β to one, the average action becomes unbounded.
Although Proposition 2 is stated using the θi0 s as parameters, it is also
true that isolated equilibria become generic if there is heterogeneity across
individuals’ utility functions.
One occasionally proclaimed virtue of social interaction models is that
they create the possibility that multiple equilibria might exist. Proposition
1 gives us sufficient conditions for there to be multiple equilibria in social
interactions models. One way to insure uniqueness is this context is to place
a bound on the effect of social interactions.
We will say that Moderate Social Influence (henceforth MSI) prevails, if
the marginal utility of an agent’s own action is more affected (in absolute
value) by a change on his own action than by a change in the average action
of his peers. More precisely we say that MSI prevails if
U i (a , A , θ , p) 12 i i i
< 1.
i
U11 (ai , Ai , θi , p) (15)
From equation (10) the MSI condition implies:
|g1i (Ai , θi , p)| < 1.
(16)
This last condition is, in fact, weaker than inequality (15), since it is equivalent to inequality (15) when ai is optimal, given (Ai , θi , p). We use only
inequality (16), and therefore we will refer to this term as the MSI condition.
15
The next proposition shows that if the MSI condition holds, there will
be a unique equilibrium.14
Proposition 3 If for a fixed (θ, p), MSI holds (that is inequality (16) is
verified for all i,) then there exists at most one equilibrium a(θ, p).
Proof: Let F (a, θ, p) = a − G(a, θ, p). The matrix of partial derivatives with
respect to a, that we denote by F1 (a, θ, p), has diagonal elements equal to 1
and, using equation (10), off diagonal elements dij = −g1i (Ai , θi , p)γij . Also,
for each i,
X
j6=i
|dij | = |g1i (Ai , θi , p)|
X
γij = |g1i (Ai , θi , p)| < 1.
j6=i
Hence F1 (a, θ, p) is a matrix with a positive dominant diagonal, and as a
consequence, for each (θ, p), F (a, θ, p) = 0 has a unique solution (Mckenzie
[1960], or Gale and Nikaido [1963]). QED
To guarantee that uniqueness always prevail, MSI should hold for all
(θ, p) ∈ Θn × Π. The assumption in Proposition 3 is independent of the
structure of interactions embedded in the Pi ’s and the γij ’s. An example
where MSI is satisfied is when U (ai , Ai , θi , p) = u(ai , θi , p) + w(ai − Ai , p),
where u11 < 0, and, for each p, w(·, p) is concave.
If in addition to MSI we assume strategic complementarity (U12 > 0), we
can derive stronger results. Suppose p has a component, say p1 , such that
each g i has a positive partial derivative with respect to p1 . In equilibrium,
we have
∂a
∂g 1
∂g n
= (F1 )−1 (a, θ, p)( 1 , . . . , 1 )0 .
(17)
1
∂p
∂p
∂p
Since F1 has a dominant diagonal that is equal to one, we may use the
Neumann expansion to write:
(F1 )−1 = I + (I − F1 ) + (I − F1 )2 + . . . .
(18)
Recall that all diagonal elements of (I − F1 ) are zero and that the offdiagonal elements are −dij = g1i (Ai , θi , p)γij > 0. Hence each of the terms
in this infinite series is a matrix with non-negative entries, and
∂a
∂g 1
∂g n 0
=
(I
+
H)(
,
.
.
.
,
),
∂p1
∂p1
∂p1
14
(19)
Cooper and John [1988] had already remarked that this condition is sufficient for
uniqueness in the context of their model
16
where H is a matrix with non-negative elements. The non-negativity of
the matrix H, means that there is a social multiplier (as in Becker and
Murphy [2001])15 . An increase in p1 , holding all aj ’s j 6= i constant, leads
to a change
∂g i (Ai , θi , p) 1
dai =
dp ,
∂p1
while, in equilibrium, that change equals


∂g i (Ai , θi , p) X
∂g j (Aj , θj , p)  1

+
H
dp .
ij
∂p1
∂p1
j
The effect of a change in p1 on the average
P
Ā ≡
i ai
n
is, in turn:


1 X (∂g i (Ai , θi , p) X
∂g j (Aj , θj , p)  1
dĀ = 
+
H
dp
ij
n i
∂p1
∂p1
i,j
This same multiplier also impacts the effect of the shocks θi . Differences
in the sample realizations of the θi ’s are amplified through the social multiplier effect.
∂g
The size of the social multiplier depends on the value of g1i ≡ ∂A
. If
i
these numbers are bounded away from one, one can bound the social multiplier. However, as these numbers approach unity the social multiplier effect
gets arbitrarily large. In this case, two populations with slightly distinct
realizations of the θi ’s could exhibit very different average values of the actions. In the presence of unobserved heterogeneity it may be impossible to
distinguish between a large multiplier (that is g1 is near unity) and multiple
equilibria.
Propositions (1) and (3) give us conditions for multiplicity or uniqueness.
At this level of generality it is impossible to refine these conditions. It is
easy to construct examples where g1 > 1 in some range, but still only one
equilibrium exits.
One common way to introduce ad-hoc dynamics in social interaction
models is to simply assume that in period t each agent chooses his action
15
Cooper and John[1988] define a similar multiplier by considering symmetric equilibria
of a game.
17
based on the choices of the agents in his reference group at time t−1.16 Such
processes are not guaranteed to converge, but the next proposition shows
that when MSI prevails convergence occurs.
Let at (θ, p, a0 ) be the solution to the difference equation
at+1 = G(at , θ, p),
with initial value a0 .
Proposition 4 If for a fixed (θ, p), |g1i (·, θi , p)| < 1, for all i, then
lim at (θ, p, a0 ) = a(θ, p).
t→∞
Proof: For any matrix M , let kM k = maxi j |Mij |, be the matrix
t
norm. Then, maxi |at+1
i −ai (θ, p)| ≤ supy kG1 (y, θ, p)k (maxi |ai −ai (θ, p)|) ≤
t
t
maxi |ai − ai (θ, p)|. Hence the vectors a stay in a bounded set B and, by
assumption, supy∈B kG1 (y, θ, p)k < 1. Hence limt→∞ at (θ, p, a0 ) = a(θ, p).
QED
One intriguing feature of social interaction models is that in some of
these models, individual shocks can determine aggregate outcomes for large
groups. In contrast to the results presented earlier, which are independent of
the particular interaction structure, ergodicity depends on a more detailed
description of the interactions. For instance, consider the model in Example
2 above with p = 0, the θi0 s i.i.d., P1 = ∅ and Pi = {1} for each i > 1. That
is, agent 1 is a “leader” that is followed by everyone. Then a1 = θ1 and
ai = θi + βa1 . Hence the average action, even as n → ∞ depends on the realization of θ1 , even though the assumption of Proposition 3 holds. Our next
proposition shows that when MSI holds, shocks are i.i.d., and individuals’
utility functions depend only on their own actions and the average action of
their peer group, then, under mild technical conditions, the average action of
a large population is independent of the particular realization of the shocks.
P
Proposition 5 Suppose that
1. θi is identically and independently distributed.
2. U i (and hence g i ) is independent of i.
3. Pi = {1, . . . , i − 1, i + 1, . . . , n}.
16
In social interaction models, ad hoc dynamics is frequently used to select among
equilibria as in Young [1993, 1998]or Blume and Durlauf[1999].
18
4. γi,j ≡
1
n−1 .
5. A is bounded
6. MSI holds uniformly, that is:
sup |g1 (Ai , θi , p)| < 1
Ai ,θi
Let an (θ, p) denote the equilibrium when n agents are present and agent i
receives shock θi . Then there exists an Ā(p) such that with probability one,
lim
n→∞
n
X
ani (θ, p)
n
i=1
= Ā(p)
(20)
an (θ)
Proof: We omit the argument p from the proof. Let An (θ) = ni=1 i n . The
boundedness of A insures that there are convergent subsequences Ank (θ).
Suppose the limit of one such convergent subsequence is A(θ). Notice that
|Ani k (θ) − Ank (θ)| ≤ b/nk , for some constant b. Hence, for any > 0, we can
find K such that if k ≥ K,
P
|
nk nk
X
ai (θ)
i=1
nk
−
nk
X
g(A(θ), θi )
nk
i=1
nk
X
g(Ani k , θi )
| = |
i=1
nk
−
≤ nk
X
g(A(θ), θi )
i=1
nk
|
(21)
Furthermore, since the θi are i.i.d. and g1 is uniformly bounded, there exists
a set of probability one, that can be chosen independent of A, such that,
n
X
g(A, θi )
i=1
n
→
Z
g(A, y)dF (y),
Θ
where F is the distribution of each θi . Hence, given any > 0, if k is
sufficiently large,
nk
|A (θ) −
Z
g(A(θ), y)dF (y)| ≤ ,
Θ
or
Z
A(θ) =
g(A(θ), y)dF (y).
Θ
in the hypothesis of the proposition guarantees that g(·, θi ) is a contraction and, as a consequence, this last equation has at most one solution, Ā.
19
In particular, all convergent subsequences of the bounded sequence An (θ)
converge to Ā and hence, An (θ) → Ā.Q.E.D.
The assumptions in the proposition are sufficient, but not necessary, for
ergodicity. In general, models in which shocks are i.i.d. and interactions are
local tend to display ergodic behavior.
2.4
“Mean field” models with large populations and discrete
actions
In this subsection we will examine models with discrete action spaces (actually two possible actions), in which the utility function of the agents depends
on their own action and the average action taken by the population. Much
of our framework and results are inspired by the treatment by Brock and
Durlauf [1995] of Example 1 described above. The action space of individuals is {0, 1}. As in Brock and Durlauf we will assume that
U i = U (ai , A, p) + (1 − ai )θi ,
that is the shock θi is the extra utility an agent obtains from taking action
0. We will assume that U (ai , ·, ·) is smooth and that the θi0 s are i.i.d. with a
c.d.f. F with continuous density f. Agents do not internalize the effect that
their action has on the average action.
We also assume strategic complementarity, which in this context we take
to be: U2 (1, A, p) − U2 (0, A, p) > 0, that is, an increase in the average action
increases the difference in utility between action 1 and action 0.
Given A, agent i will take action 1, if and only if, θi ≤ U (1, A, p) −
U (0, A, p). In a large population a fraction F (U (1, A, p) − U (0, A, p)) will
take action 1, the remainder will take action 0.
A mean-field equilibrium, thereafter MFE, is an average action Ā such
that
F (U (1, Ā, p) − U (0, Ā, p)) − Ā = 0.
(22)
This definition of an MFE is exactly as in the Brock and Durlauf treatment of Example 1 above. The next proposition correspond to their results
concerning equilibria in that example.
Proposition 6 An MFE always exists. If 0 < Ā < 1 is an equilibrium
where
f (U (1, Ā, p) − U (0, Ā, p))[U2 (1, Ā, p) − U2 (0, Ā, p)] > 1,
(23)
then there are also at least two other MFE’s, one on each side of Ā. On
the other hand, if at every MFE, f (U (1, Ā, p) − U (0, Ā, p))[U2 (1, Ā, p) −
U2 (0, Ā, p)] < 1, there exists a single MFE.
20
Proof: H(A) = F (U (1, A, p) − U (0, A, p)) − A satisfies H(0) ≥ 0, and
H(1) ≤ 0 and is continuous. If inequality (23) holds, then H(Ā) = 0 and,
H 0 (Ā) > 0. QED.
The first term on the left hand side of inequality (23) is the density
of agents that are indifferent between the two actions, when the average
action is Ā. The second term is the marginal impact of the average action
on the preference for action 1 over action zero, which, by our assumption
of strategic complementarity, is always > 0. This second term corresponds
exactly to the intensity of social influence that played a pivoting role in
determining the uniqueness of equilibrium in the model with a continuum
of actions.
If there is a unique equilibrium,17 then the social multiplier will equal:
∂ Ā
f (U (1, Ā, p) − U (0, Ā, p))[U3 (1, Ā, p) − U3 (0, Ā, p)]
=
.
∂p
1 − f (U (1, Ā, p) − U (0, Ā, p))[U2 (1, Ā, p) − U2 (0, Ā, p)]
(24)
The numerator in this expression is exactly the average change in action,
when p changes, and agents consider that the average action remains constant. The denominator is, if uniqueness prevails, positive.
As we emphasized in the model with continuous actions, there is a continuity in the multiplier effect. As the parameters of the model (U and F )
approach the region of multiple equilibria, the effect of a change in p on the
equilibrium average action approaches infinity.
In many examples the distribution F satisfies:
1. Symmetry: (f (z) = h(|z|))
2. Monotonicity (h is decreasing.)
If in addition the model is unbiased (U (1, 1/2, p) = U (0, 1/2, p)), then A =
1/2 is an MFE. The fulfillment of inequality (23) now depends on the value of
f (0). This illustrates the role of homogeneity of the population in producing
multiple equilibria. If we consider a parameterized family of models in which
the random variable θi = σxi where σ > 0, then f σ (0) = σ1 f 1 (0). As σ → 0
(σ → ∞) inequality (23) must hold (resp. must reverse). In particular if
the population is homogeneous enough, multiple equilibria must prevail in
the unbiased case.
These reasoning can be extended to biased models, if we assume that
[U2 (1, ·, p) − U2 (0, ·, p)] is bounded and bounded away from zero, and that
17
In here and in what follows we require strict uniqueness that is, the left hand side of
inequality (23) is less than one.
21
the density f 1 is continuous and positive.18 For, in this case, for σ large,
sup{f σ (U (1, A, p) − U (0, A, p))[U2 (1, A, p) − U2 (0, A, p)]} < 1.
(25)
A
Hence equilibrium will be unique, if the population displays sufficient heterogeneity. On the other hand as σ → 0, inequality (25) is reversed and
multiple equilibria appear.
We can derive more detailed properties if we assume,that in addition to
the symmetry and monotonicity properties of f, that U22 (1, A, p)−U22 (0, A, p) ≤
0, that is the average action A has a diminishing marginal impact on the
preference for the high action. In that case, it is easy to show that there are
at most three equilibria.
2.5
Choice of peer group
The mathematical structure and the empirical description of peer or reference groups varies from model to model. In several models (e.g Gabszewicz
and Thisse [1996],Benabou [1993], Glaeser, Sacerdote and Scheinkman [1996]
or Mobius [1999]) the reference group is formed by geographical neighbors.
To obtain more precise results one must further specify the mathematical
structure of the peer group relationship - typically assuming either that all
fellow members of a given geographical unit form a reference group or that
each agent’s reference group is formed by a set of near-neighbors. Mobius
[1999] shows that, in the context that generalizes Schelling’s [1972] tipping
model, the persistence of segregation depends on the particular form of the
near-neighbor relationship. Glaeser, Sacerdote and Scheinkman [1996] show
that the variance of crime-rates across neighborhoods or cities would be a
function of the form of the near-neighbor relationship.
Kirman [1983], Kirman, Oddou and Weber [1986], and Ioannides [1990]
use random graph theory to treat the peer group relationship as random.
This approach is particularly useful in deriving properties of the probable
peer groups as a function of the original probability of connections. Another
literature deals with individual incentives for the formation of networks (e.g.
Boorman [1975], Jackson and Wolinsky [1996], Bala and Goyal [2000])19
18
An example that satisfies these conditions is the model of Brock and Durlauf described
in Example 1 above. Brock and Durlauf use a slightly different state space, but once the
proper translations are made, U2 (1, A, p) − U2 (0, A, p) = kJ for a positive constant k and
0 < f 1 (z) ≤ ν.
19
A related problem is the formation of coalition in games e.g. Myerson [1991].
22
One way to model peer group choice is to consider a set of neighborP
hoods indexed by ` = 1, . . . , m each with n` slots with ` n` ≥ n.20 Every
agent chooses a neighborhood to join after the realization of the θi0 s. To join
neighborhood P ` one must pay q` . The peer group of agent i, if he joins
neighborhood `, consists of all other agents j that joined ` with γij = γij 0
for all peers j and j0. We will denote by A` , the average action taken by all
agents in neighborhood `. Our equilibrium notion, in this case will parallel
Tiebout’s equilibrium (see e.g. Bewley [1981].)
For given vectors θ = (θ1 , . . . , θn ) ∈ Θn and p, an equilibrium will be a set
of prices (q1 , . . . , qm ), an assignment of agents to neighborhoods, and a vector
of actions a = (a1 , . . . , an ), that is an equilibrium given the peer groups
implied by the assignment, such that, if agent i is assigned to neighborhood
`, there is no neighborhood `0 such that
0
sup U i (ai , A` , θi , p) − q`0 > sup U i (ai , Ai , θi , p) − q`
ai
(26)
ai
In other words, in an equilibrium with endogenous peer groups we add the
additional restriction that no agent prefers to move.
To examine the structure of the peer groups that arise in equilibrium
we assume, for simplicity, that the U i ’s are independent of i, that is that
all heterogeneity is represented in the θi ’s. If an individual with a higher θ
gains more utility from an increase of the average action than an individual
with a lower θ, then segregation obtains in equilibrium. More precisely if Θ
is an interval [to , to ] of the line, and
V (A, θ, p) ≡ sup U (ai , A, θ, p),
ai
satisfies:
V (A, θ, p) − V (A0 , θ, p) > V (A, θ0 , p) − V (A0 , θ0 , p)
whenever, A > A0 and θ > θ0 , there exist points t0 = t0 < t1 , . . . , < tm = t0
such that, agent i chooses neighborhood ` if and only if θi ∈ [t`−1 , t` ]. (e.g.
Benabou [1993], Glaeser and Scheinkman [2001]). Though other equilibria
exist, these are the only “stable” ones.
20
This treatment of peer group formation is used in Benabou [1993] and Glaeser and
Scheinkman[20021]. However, in several cases, peer groups have no explicit fees for entry.
Mailath, Samuelson and Shaked [1996] examine the formation of peer groups when agents
are matched to others from the same peer group.
23
3
Empirical approaches to social interactions
The theoretical models of social interaction models discussed above are,
we believe, helpful in understanding a wide variety of important empirical
regularities. In principle, large differences in outcomes between seemingly
homogeneous populations, radical shifts in aggregate patterns of behavior,
and spatial concentration and segregation can be understood through social
interaction models. But these models are not only helpful in understanding
stylized facts, they can also serve as the basis for more rigorous empirical
work. In this section, we outline the empirical approaches that can be and
have been used to actually measure the magnitude of social interactions.
For simplicity, in this empirical section we focus on the linear-quadratic
version of the model discussed in Example 2. Our decision to focus on
the linear-quadratic model means that we ignore some of the more important questions in social interactions. For example, the case for place-based
support to impoverished areas often hinges upon a presumption that social
interactions have a concave effect on outcome. Thus, if impoverished neighborhoods can be improved slightly by an exogenous program, then the social
impact of this program (the social multiplier of the program) will be greater
than if the program had been enacted in a more advantaged neighborhood.
The case for desegregation also tends to hinge on concavity of social interactions. Classic desegregation might involve switching low human capital
people from a disadvantaged neighborhood and high human capital people
from a successful neighborhood. This switch will be socially advantageous
if moving the low human capital people damages the skilled area less than
moving the high human capital people helps the less skilled area. This will
occur when social interactions operate in a concave manner.
As important as the concavity or convexity of social interactions have
been, most of the work in this area has focused on estimating linear effects.21
To highlight certain issues that arise in the empirical analysis, we make
many simplifying assumptions that help us focus on the relevant problems.22
We will use the linear model in Example (2). We assume we can observe
data on C, equally sized23 , groups. All interactions occur within a group.
21
Crane (1991) is a notable exception. He searches for non-linearities across a rich range
of variables and finds some evidence for concavity in the social interactions involved in outof-wedlock births. Reagon, Weinberg and Yankow (2000) similarly explore non-linearities
in research on work behavior and find evidence for concavity.
22
A recent survey of the econometrics of a class of interaction based binary choice
models, and a review of the empirical literature, can be found in Brock and Durlauf [to
appear].
23
The assumption of equally sized groups is made only to save on notation.
24
Rewriting equation (6) for the optimal action, to absorb p in the θi0 we
have:
ai = βAi + θi .
(27)
We will examine here a simple form of global interactions. If agent i
belongs to group `,
1 X
Ai =
aj ,
n − 1 j6=i
where the sum is over the agents j in group `, and n is the size of a group.
We will also assume that θi = λ` + εi , where the ε0i s are assumed to be i.i.d.,
with mean zero λ` is a place specific variable (perhaps price) that affects
everyone in the group, while εi is an idiosyncratic shock that is assumed to
be independent across people.
The average action within a group is:
P
i ai
n
P
λ`
i εi
=
+
.
1 − β n(1 − β)
(28)
The optimal action of agent i is then:
P
β j6=i εj
λ`
(n − 1 − βn + 2β)εi
ai =
+
+
.
1−β
(n − 1 + β)(1 − β)
(n − 1 + β)(1 − β)
(29)
The variance of actions on the whole population, is
σλ2
Var(ai ) =
+σε2 1 +
(1 − β)2
β
1−β
2
3(n − 1) − 2β(n − 2) − β 2
(n − 1 + β)2
!
. (30)
σ2
2
λ
As n → ∞ this converges to (1−β)
2 + σε . In this case, and in the cases that
are to follow, even moderate levels of n (n = 30+) yield results that are
quite close to the asymptotic result. For example, if n = 40, and β ≤ .5,
then the bias is at most −.05σε2 . Higher values of β are associated with
more severe negative biases, but when n = 100, a value of β = .75 (which
we think of as being quite high) is associated with a bias of only −.135σε2 .
3.1
Variances Across Space
The simplest, although hardly the most common, method of measuring the
size of social interactions is to use the variance of a group average. The
intuition of this approach stems from early work on social interactions and
multiple equilibria (see, e.g. Schelling [1978], or Becker [1989], or Sah [1991]).
25
These papers all use different social interaction models to generate multiple
equilibria for a single set of parameter values.
While multiple equilibria are often used as an informal devise to explain
large cross-sectional volatity, in fact this multiplicity is not needed. What
produces high variation is that social interactions are associated with large
differences across time and space that cannot be fully justified by fundamentals. Glaeser, Sacerdote and Scheinkman [1996] use this intuition to create
a model where social interactions are associated with a high degree of variance across space without multiple equilibria. Empirically it is difficult to
separate out extremely high variances from multiple equilibria, but Glaeser
and Scheinkman [2001] argue that for many variables high variance models
with a single equilibrium are a more parsimonious mean of describing the
data.
Suppose we obtain m ≤ n observations of members of a group. The sum
of the observed actions, normalized by dividing by the square root of the
number of observations, will have variance:
P
Var
a
√i i
m
=
2
mσλ2
σε2
2 β (n − 2) − 2β(n − 1)
+
+
(n
−
m)σ
.
ε
(1 − β)2 (1 − β)2
(1 − β)2 (n − 1 + β)2
(31)
nσ 2
2
σε
λ
When m = n, (31) reduces to: (1−β)
2 + (1−β)2 , which is similar to the
variance formula in Glaeser, Sacerdote and Scheinkman [1996] or Glaeser
and Scheinkman[2001]. Thus, if m = n and σλ2 = 0, as n → ∞ the ratio
of the variance of this normalized aggregate to the variance of individual
1
actions converges to (1−β)
2 . Alternatively, if m is fixed, then as n grows
large, the aggregate variance converges to
mσλ2
+ σε2 ,
(1 − β)2
and the ratio of the aggregate variance to the individual variance (when
σλ2 = 0) converges to one.
The practicality of this approach hinges on the extent to which σλ2 is
either close to zero or known.24 As discussed above, λ` may be non-zero either because of correlation of background factors or because there are place
specific characteristics that jointly determine the outcomes of neighbors. In
some cases, researchers may know that neighbors are randomly assigned
24
In principle we could use variations in n across groups, and the fact that when m = n
the variance of the aggregates is an affine function of m to try to separately estimate σλ
and σε .
26
and that omitted place specific factors are likely to be small. For example,
Sacerdote [2000] looks at the case of Dartmouth freshman year roommates
who are randomly assigned to one another. He finds significant evidence
for social interaction effects. In other contexts (see Glaeser, Sacerdote and
Scheinkman [1996]) there may be methods of putting an upper bound on σλ2
which allows the variance methodology to work. Our work found extremely
high aggregate variances that seem hard to reconcile with no social interactions for reasonable levels of σλ2 . In particular, we estimated high levels
of social interactions for petty crimes and crimes of the young. We found
lower levels of social interactions for more serious crimes.
3.2
Regressing Individual Outcomes on Group Averages
The most common methodology for estimating the size of social interactions
is to regress an individual outcome on the group average. Crane [1991],
discussed above is an early example of this approach. Case and Katz [1991]
is another early paper implementing this methodology (and pioneering the
instrumental variables approach discussed below). Since these papers, there
has been a torrent of later work using this approach and it is the standard
method of trying to measure social interactions.
We will illustrate the approach considering a univariate regression where
an individual outcome is regressed on the average outcome in that individual’s peer group (not including himself). In almost all cases, researchers
control for other characteristics of the subjects, but these controls would
add little but complication to the formulae. The univariate ordinary least
squares coefficient for a regression of an individual action on the action of
his peer is
P
Cov ai , j6=i aj /(m − 1)
P
.
(32)
Var
j6=i aj /(m − 1)
√
The denominator is a transformation of (31), where m-1 replaces m :
!
P
Var
j6=i aj
=
m−1
σλ2
+
(1 − β)2
mσε2
(33)
[(n − 1 + β) − β(n − m)]2 + β 2 m(n − m)
.
(m − 1)2 (1 − β)2 (n − 1 + β)2
The numerator is:
P
Cov ai ,
j6=i aj
m−1
!
=
σλ2
(2n − 2 − βn + 2β)
+ βσε2
.
2
(1 − β)
(1 − β)2 (n − 1 + β)2
27
(34)
When σλ = 0 then the coefficient reduces to:
Coeff =
(m − 1)2
2β(n − 1) − β 2 (n − 2)
.
m
(n − 1 + β)2 − (n − m)[2β(n − 1) − β 2 (n − 2)]
(35)
When m = n:
Coeff = 2β
(n − 1)2
(n − 1)2
− β2
.
n(n − 1 + β)
(n − 1 + β)2
(36)
Hence as n → ∞, the coefficient converges to 2β − β 2 . Importantly, because
of the reflection across individuals, the regression of an individual outcome
on a group average cannot be thought of as a consistent estimate of β.
However, under some conditions (m = n, large, σλ2 = 0), the ordinary least
squares coefficient does have an interpretation as a simple function of β.
Again, the primary complication with this methodology is the presence of
correlated error terms across individuals. Some of this problem is corrected
by controlling for observable individual characteristics. Indeed, the strength
of this approach relative to the variance approach is that it is possible to
control for observable individual attributes. However, in most cases, the
unobservable characteristics are likely to be at least as important as the
observable ones and are likely to have strong correlations across individuals
within a given locale. Again, this correlation may also be the result of
place-specific factors that affect all members of the community.
One approach to this problem is the use of randomized experiments that
allocate persons into different neighborhoods. The Gautreaux experiment
was an early example of a program that used government money to move
people across neighborhoods. Unfortunately, the rules used to allocate people across neighborhoods are sufficiently opaque that it is hard to believe
that this program really randomized neighborhoods.
The Moving to Opportunity experiment contains more explicit randomization. In that experiment, funded by the department of Housing and
Urban Development, individuals from high poverty areas were selected into
three groups: a control group and two different treatment groups. Both
treatment groups were given money for housing which they used to move
into low poverty areas. By comparing the treatment and control groups,
Katz, Kling and Liebman [2001] are able to estimate the effects of neighborhood poverty without fear that the sorting of people into neighborhoods
is contaminating their results. Unfortunately, they can’t tell whether their
effects are the results of peers or other neighborhood attributes. As such,
this work is currently the apex of work on neighborhood effects but it cannot
28
really tell us about the contribution of peers vs. other place based factors.
Sacerdote [2000] also uses a randomized experiment. He is able to compare
people who are living in the same building but who have different randomly
assigned roommates. This work is therefore a somewhat cleaner test of peer
effects.
Before randomized experiments became available, the most accepted approach for dealing with cases where σλ2 6= 0 was to use peer group background characteristics as instruments for peer group outcomes. Case and
Katz (1991) pioneered this approach and under some circumstances it yields
valid estimates of β. To illustrate this approach, we assume that there is a
parameter (x) which can be observed for all people and which is part of the
individual error term, i.e. i = γxi + µi . Thus, the error term can be decomposed into a term that is idiosyncratic and unobservable and a term that is
directly observable. Under the assumptions that both components of i are
orthogonal to λ` and to each other, using the formula for an instrumental
variables estimator we find that:
Cov ai ,
Cov
P
P
j6=i aj /(m
j6=i xj /(m
− 1),
− 1)
P
j6=i xj /(m
=
− 1)
β
n−1
β + (1 − β) m−1
(37)
When m = n, this reduces to β. Thus, in principle, the instrumental
variables estimator can yield consistent estimates of the social interaction
term of interest.
However, as Manski (1993) stresses, the assumptions needed for this
methodology may be untenable. First, the sorting of individuals across
communities may mean that Cov (xi , µj ) 6= 0 for two individuals i and j
living in the same community. For example, individuals who live in high
education communities may have omitted characteristics that are unusual.
Equation (37) is no longer valid in that case, and in general the instrumental variables estimator will overstate social interactions when there is
sorting of this kind. Second, sorting may also mean that Cov (xi , λ` ) 6= 0.
Communities with people who have high schooling levels, for example, may
also have better public high schools or other important community level
characteristics.
Third, the background characteristic of individual j may directly influence the outcome of person i, as well as influencing this outcome through
the outcome of individual j. Many researchers consider this problem to be
less important, because it only occurs when there is some level of social interaction (i.e. the background characteristic of person j influencing person
29
i). While this point is to some extent correct, it is also true that even a small
amount of direct influence of xj on ai can lead to wildly inflated estimates
of β , when the basic correlation of xj and aj is low. (Indeed, when this
correlation is low, sorting can also lead to extremely high estimates of social
interaction.) Because of this problem, instrumental variables estimates can
often be less accurate than ordinary least squares estimates and need to be
considered quite carefully, especially when the instruments are weak.
3.3
Social Multipliers
A final approach to measuring social interactions is discussed in Glaeser and
Scheinkman [2001] and Glaeser, Laibson and Sacerdote [2000], but to our
knowledge has never been really utilized. This approach is derived from a
lengthier literature on social multipliers where these multipliers are discussed
in theory, but not in practice (see Schelling [1978].) The basic idea is that
when social interactions exist, the impact of an exogenous increase in a
variable can be quite high if this increase impacts everyone simultaneously.
The effect of the increase includes not only the direct effect on individual
outcomes, but also the indirect effect that works through peer influence.
Thus, the impact on aggregate outcomes of an increase in an aggregate
variable may be much higher than the impact on an individual outcome of
an increase in an individual variable.
This idea has been used to explain how the pill may have had an extremely large effect on the amount of female education (see Goldin and
Katz [2000]). Goldin and Katz [2000] argue that there is a positive complementarity across women who delay marriage that occurs because when
one women decides to delay marriage, her prospective spouse remains in the
marriage market longer and is also available to marry other women. Thus,
one women’s delaying marriage may increase the incentives for other women
to delay marriage and this can create a social multiplier. Berman [2000]
discusses social multipliers and how they might explain how government
programs appear to have massive effects on labor practices among orthodox Jews in Israel. In principle, social multipliers might explain phenomena
such as the fact that there is a much stronger connection between out-ofwedlock births and crime at the aggregate level than at the individual level
(see Glaeser and Sacerdote [1999]).
In this section, we detail how social multipliers can be used in practice
to estimate the size of social interactions. Again, we assume that the individual disturbance term can be decomposed into i = γxi + µi , and that
m = n. When we estimate the micro regression of individual outcomes on
30
characteristic x, when x is orthogonal to all other error terms, the estimated
coefficient is:
Individual Coeff = γ
(1 − β)n + (2β − 1)
.
(1 − β)n − (1 − β)2
(38)
This expression approaches γ as n becomes large, and for even quite modest
levels of n (n=20), this expression will be quite close to γ.
Our assumption that the xi terms are orthogonal to the ui terms is
probably violated in many cases. The best justification for this assumption
is expediency– interpretation of estimated coefficients becomes quite difficult when the assumption is violated. One approach, if the assumption is
clearly untenable, is to use place-specific fixed effects in the estimation. This
will eliminate some of the correlation between individual characteristics on
unobserved heterogeneity.
An ordinary least squares regression of aggregate outcomes on aggregate
x variables leads to quite a different expression. Again, assuming that the
xi terms are orthogonal to both the λ` and µi terms, then the coefficient
γ
from the aggregate regression is 1−β
.
The ratio of the individual to the aggregate coefficient is therefore:
Ratio =
(1 − β)n + 2β − 1
n−1+β
(39)
As n grows large, this term converges to 1 − β, which provides us with yet
another means of estimating the degree of social interactions. Again, this
estimate hinges critically on the orthogonality of the error terms, which generally means an absence of sorting. It also requires (as did the instrumental
variables estimators above), the assumption that the background characteristics of peers have no direct effect on outcomes.
3.4
Reconciling the Three Approaches
While we have put forward the three approaches as distinct ways to measure
social interactions, in fact they are identical in some cases. In general, the
micro-regression approach of regressing individual outcomes on peer outcomes (either instrumented or not) requires the most data. The primary
advantage of this approach is that it creates the best opportunity to control
for background characteristics. The variance approach is the least data intensive, as it generally only requires an aggregate and an individual variance.
In the case of a binary variable, it requires only an aggregate variance. Of
course, as Glaeser, Sacerdote and Scheinkman [1996] illustrate, this crude
31
measure can be improved upon with more information. The social multiplier
approach lies in the middle. This approach is closest to the instrumental
variable approach using micro data.
32
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