Download Version for International Economic Review

Version for International Economic Review .... after first revisions.\
-> for every single result find theory paper that shows it. For example, for spillover, show
jealousy result of Dupor et al 1998 that it increases equilibrium consumption. Similar
results are from conformity. The best is to discuss results in terms of. "The results explain
that .... has positive effect, that .... has this effect, that ... has this effect. Even though the
authors call the interactions effects different names, a careful examination and side by
side comparisons of utility reveals that all discuss the same social interactions process".
-> put in appendix a matrix representation of the effects, but make sure it is not over
1page! Referee said it was cleaner.
-> update references -for example Glaeser is now published.
-> you can also check comments from the defense.
created 5-24-03
[INTUITION!!! - take from paper02, see stuff right after equation 3 when I discuss the
intuition of the result - what really happens.]
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A Partial Equilibrium Analysis in the
Markets with Social Interactions
Andrew Grodner
Department of Economics at East Carolina University,
A423 Brewster Building, East Carolina University, Greenville, N.C. 27858, [email protected]
December 2004
Abstract
This paper analyzes a demand side of a two-good economy where the demand of one
good is affected by social interactions between consumers. We focus on two types of
interactions: a spillover effect in the form of positive externality from other consumers'
choices, and a conformity effect representing a need for making similar choices as other
people. We show that a positive spillover effect shifts the demand curve for the good with
interactions upwards, and the conformity effect makes the demand curve to pivot around
the average market demand to make the slope flatter.
JEL: D11
Keywords: consumer demand, social interactions, spillover, conformity.
* We would like to thank Richard Ericson, Peter Wilcoxen, Lester Zeager, and Buhong
Zheng for their helpful comments and suggestions, and Kristina Lambraight for help in
preparing the manuscript.
1. Introduction
Economic approach to human behavior that assumes a single decision maker is
sometimes criticized as an example of excessive abstraction leading to un-informative
behavioral implications and inaccurate predictions. One reason economic researchers
have avoided models with interdependent agents is complexity and no generally accepted
theoretical framework for examining interdependent economic agents (Manski 2000,
Moffitt 2001, Durlauf and Young 2001). Our research builds on Brock and Durlauf
(2001b) to present a simple model to analyze the demand part of the market. Comparative
statics provide a clear and intuitive predictions that can be successfully used in many
applications involving social interactions. The paper focuses on the market with two
goods and examines the qualitative, and in some cases, quantitative effects of two types
of interactions in utility: spillover from others' decisions and conformity with others'
choices.
Social interactions are of much policy relevance. If the social reference group's mean
value affects the individual ??? incomes ??? outcomes then social interactions affect the
taxation programs or policies directed toward improving the well-being of the
unemployed (Blomquist 1993). When there is a substantial amount of interdependence,
then an optimal policy design needs to consider a social multiplier effect because
individuals will react to the actions of others (Becker and Murphy 2000, Glaeser,
Sacerdote, and Scheinkman 2002). If workers care about their relative positions in the
income distribution then regulatory policy that does not disturb relative incomes receives
too low a benefit in conventional cost-benefit calculations (Frank and Sunstein 2001,
Kniesner and Viscusi 2003). Therefore ignoring social interactions can overstate the
welfare of the society (Soetevent and Kooreman 2002) but there may be conditions
where well-being in society can be improved (UPDATE Kooreman and Schoonbeek
2004).
The obvious interest in social interactions has generated an extensive literature with
enormous variety of forms for interdependence (ex. spillovers, peer group effects,
fairness, conformity, neighborhood effects, externalities, social norms, herding, identity,
social capital, and contagion, etc.). The modelling approaches range from overlapping
generations considerations, evolutionary game theory, to Bayesian learning as the feature
(see Durlauf .... Handbook or Urban for a comprehensive review). Still, most theoretical
exercises involving household economic interactions share one common feature: the
utility of the individual is somehow affected by either utility or choices made by
members of the individual's reference group, which is comprised of persons with whom
the individual somehow interacts or relates.
Studies of demand analysis that introduce social interactions (LES) .... other papers on
interactions in demand - Kooreman, and other stuff.... ... In particular The analysis of
demand with social interactions was
[PUT NEW LITERATURE HERE AND ONLY FOCUS ON DEMAND AND SOCIAL
INTERACTIONS. Recognizing the importance of interdependence, it has long been
hypothesized that a person's utility depends not only on his or her own choices but also
on the actions or utility of others with whom a person interacts socially or economically
(Becker 1974). Economic theory formally considered many forms of interactions:
spillovers (Roback 1982), peer group effects (de Bartolome 1990), fairness (Rabin 1993),
conformity (Bernheim 1994), neighborhood effects (Durlauf 1996), externalities
(Chamley 1999), social norms (Lindbeck, Nyberg, and Weibull 1999), herding (Smith and
Sorensen 2000), social capital (Glaeser, Laibson, and Sacerdote 2000, Becker and
Murphy 2000), identity (Akerlof and Kranton 2000), and contagion (Rigobon 2001). The
basic theoretical setup generally differs across applications, ranging from overlapping
generations considerations to Bayesian learning as the feature. All theoretical exercises
involving household economic interactions share one common feature: the utility of the
individual is somehow affected by either utility or choices made by members of the
individual's reference group, which is comprised of persons with whom the individual
somehow interacts or relates.]
The present paper generalizes the results for a particular class of models with social
interactions. It include Linear Expensditure often used in lit (cit). We follow the
approach introduced by Brock and Durlauf (2001a,b), and we use the concepts of the
total utility and the social utility that represents interactions. The contribution of the
research is the demonstration of how interactions affect the market demands for a general
utility function and presenting comparative statics results for various forms of
interdependence. We analyze spillover (externality from other consumers' behavior) and
conformity (there is a penalty for people behaving different from the norm), where social
interactions are directly embedded into the utility function in the form of the social
utility. Generally, positive spillover effect shifts the demand curve upwards and the
conformity pivots the demand curve around the expected market demand to make the
demand more inelastic. Moreover, when interactions are endogenous and the consumer
directly affects the expected market demand, the spillover effect is more pronounced but
the observed effect of conformity diminishes.
[On the other hand, when interaThe paper also [the model show importance .... .. for
researchers working on the demand and estimating demand equations ... ] Contribution:
clear distinction between exogenous and endogenous, effect on the good without
interactions, intuitive graphic representation.]
REFERENCES TO CITE ... probably only are enough + whatever is the most recent.
!!!!!!! Cite it: A Model of Demand with Interactions among Consumers
AU: Cowan, Robin; Cowan, William; Swann, Peter
SO: International Journal of Industrial Organization, vol. 15, no. 6, October 1997, pp.
711-32
Life-Cycle Consumption under Social Interactions
AU: Binder, Michael; Pesaran, M Hashem
SO: Journal of Economic Dynamics and Control, vol. 25, no. 1-2, January 2001, pp. 3583
Binder, Michael; Pesaran, M Hashem. 1998. Decision Making in the Presence of
Heterogeneous Information and Social Interactions, International Economic Review, vol.
39, no. 4, November 1998, pp. 1027-52.
!!! and add model for LES!!! - the classic for consumption that actually has been tested.
Kateyn work and the following papers.
2003 Conspicuous Consumption and Social Segmentation
Theory January 5 1
38010
Journal of Public Economic
Spillover effects? Use Social Utility
Running to Keep in the Same Place: Consumer Choice as a Game of Status
Ed Hopkins, Tatiana Kornienko. The American Economic Review. Nashville: Sep
2004.Vol.94, Iss. 4; pg. 1085, 23 pgs
+ Conspicuous Consumption and Pareto Improvements in Journal of Public Economic
THeory.
Kooreman P., Schoonbeek L., 2004, Characterizing Pareto improvements in an
interdependent demand system, Journal of Public Economic Theory, August, vol. 6, no. 3,
pp. 427 - 443.
Botond Köszegi, Matthew Rabin. 2006. A Model of Reference-Dependent Preferences,
Quarterly Journal of Economics November 2006, Vol. 121, No. 4: 1133-1165.
We develop a model of reference-dependent preferences and loss aversion where
"gain–loss utility" is derived from standard "consumption utility" and the reference point
is determined endogenously by the economic environment. We assume that a person's
reference point is her rational expectations held in the recent past about outcomes, which
are determined in a personal equilibrium by the requirement that they must be consistent
with optimal behavior given expectations. In deterministic environments, choices
maximize consumption utility, but gain–loss utility influences behavior when there is
uncertainty. Applying the model to consumer behavior, we show that willingness to pay
for a good is increasing in the expected probability of purchase and in the expected
prices conditional on purchase. In within-day labor-supply decisions, a worker is less
likely to continue work if income earned thus far is unexpectedly high, but more likely to
show up as well as continue work if expected income is high.
*__________ applications to willingness to buy and labor supply
Impotance - multiplier - cite Gleaser
My innovation seems to be to show effects in demand curve directly and to show what
happens to the non-interactions good. These papers research single good economy. (?) I
also distinqusih spillover and conformity, though one of the papers does something very
similar.
Conformity in labor supply - can people work the same hours worked and that is why
supply curves are flat?
2. Model
Using the specification suggested by Brock and Durlauf (2001a,b), suppose we have the
total utility function
V x, y;  ,  x   V ux, y , S  x;  x ,  
st. p x x  p y y  M
where x and y are actions/choices made by an individual with the corresponding
x, y is the private utility associated with a choice bundle 
x, y,
prices p x and p y , u
x is the conditional probability measure of choices (expectation/belief) that a person
places on the choices of others in the reference group 1 at the time of decision making,
S
x; x ,  is the social utility associated with the choice of individual and his or her
expectation of the choices of others,  is the parameter indicating the importance of
x; x , 0 ), and M
social utility in total utility (assume that for  0 we have S
is the total resource available to the individual. The individual maximizes the total utility
2. The approach has been used with much
1 subject to the budget constraint 
success and the recent example includes Hopkins and Kornienko (2004).

 is strictly quasi-concave function (to guarantee satisfaction for the
We assume that u
utility maximization conditions) with u x 0, u xx 0, u y 0, u yy 0 (subscripts denote
partial derivative) which implies that both x and y are goods with increasing but
diminishing marginal utility. We also assume the positive sign on the derivative Vu 0
, however, the derivative V S has an uncertain sign and depends on the form of
interactions.
We consider two forms of social interactions: positive spillover and conformity. Positive
spillover is characterized by the positive contribution to the total utility, VS 0 (for
example increased human capital of the members of the reference group increases
everybody's human capital). The conformity is associated with the negative contribution
to the utility because there is a disutility for being different, VS 0 (for example
usually alcohol drinkers expect their colleagues to drink as well and the outcasts may not
be accepted). Note that the analysis can easily be extended to negative spillover by
assuming VS 0 , or non-conformity by taking VS 0 .
Spillover is generally defined as an externality effect relative to the behavior of the
reference group. For example, in the labor supply setting the spillover effect can be
viewed as a positive externality generated by the labor supplied in the reference group,
where a higher mean of hours worked in the reference group decreases the individual's
disutility from working. A way to interpret the spillover effect in the labor market is that
one feels less pain from working if one knows that others also work. Therefore, the forms
x; x , x x all could be
x; x , x x , S s2 
x; x , x x 2 , S s3 
like S s1 
called a spillover effect. However, even though all functions are strictly increasing the
impact of the spillover effect on the individual behavior may be different depending on
the specific form. For example, we show that the derivative S xx always affects the slope
s2
s3
of the demand curve. Because S s1
xx 0 , S xx 0 , S xx 0 , the effect of the spillover
on the demand for good x is different depending on the assumed form of interactions.
Full results of the derivatives for selected functional forms of interactions are presented
in table 1A.
Conformity in behavior and attitudes is one of the fundamental building blocks that
historically contributed to the emergence of the field of social psychology (Sherif 1935).
The general idea is that individuals tend to conform to broadly defined social norms and
the magnitude of the response depends on the cohesiveness, group size, and social
support. For example, in the labor supply setting we can think that the person feels
penalized when working a different amount of hours than what is typical in the reference
x; x , 1/|x x | where the
group. Therefore, one can model conformity as S c 
individual is rewarded for behaving according to the norm. However, the form of social
utility in S c is difficult to work with analytically, and we would need to impose at least
a restriction on x x . Thus, without the loss of generality (the signs on the critical
derivatives are the same), we assume conformity as a quadratic loss of utility:

S c1 
x; x , 
x x 2 , S c2 
x; x , 12
x x 4 ,


2
S c3 
x; x , 
x 2 2x 2 (notice that we put the negative sign on S to indicate

4
the fact that VS 0 ). Again, depending on the form of conformity the effect of
interactions on the demands for x and y may differ in a non-trivial way (note that
c2
c3
S c1
xx 0 , S xx 0 , S xx 0 ). Complete results for the derivatives are presented in table
1A.
3. Results
We now turn to the comparative statics results and examine the impact of different forms
of interactions on the general utility problem stated in (1) and (2). The results are derived
by setting up a constrained maximization problem, taking the total differential of the first
order conditions, and solving for the endogenous differentials, dx , dy , and d (where
 is the marginal utility of income M ), in terms of the exogenous differentials dp x ,
d endogenous
dp y , dM , d , and dx . We analyze the partials d exogenous
to examine the
behavior of the endogenous changes around the equilibrium due to the change in one of
the exogenous factors while holding other exogenous variables constant (differentials are
set to zero). See derivations in the appendix for details.
3.1. Exogenous interactions
In equation (1) interactions are represented by the expectation of the demand for good x
by a particular consumer, x . In a perfect world or in a small community the individual
may be able to observe other people's demands for good x and make sensible inferences
about the expected demand by way of computing the sample mean, median, or mode.
However, there are often cases where the market is so large that the individual has no
means to infer about others' behavior and may resort to using existing norms. For
example, one norm may be that a full-time status for a working person means that an
individual is working 40 hours per week. Sometimes such norm can even be enforced by
the law.
When the interactions are exogenous the variable x is exogenous and thus taking the
x; x ,  becomes:
total differential of S
dS S xx dx S xx dx S xd (3)
We can see that the difference between non-interactions case and any case with
interdependence is exhibited by the presence of the partials S xx , S xx , and S x .
3.1.1. Demand for good with interactions, x
The change of the demand for good x around equilibrium, approximated by dx , is
dx 
p y u xy p x u yy V u dM p x p y yp y V u u xy yp x V u u yy dp y xp x V u u yy xp y V u u xy p 2y dp x
detH
2
2
p y V S S xdp y V S S xx dx

detH
where the matrix H is the Hessian from the maximization of (1) subject to (2). The
determinant of H is
detH 2p x p y V u u xy p 2x V u u yy p 2y V u u xx 
p 2y V S S xx 0. (5)


 is concave and V
 is without interactions, the
Notice that if the function u
u

detH
0

concavity of
guarantees
(determinant of the bordered Hessian).
2
However, with interactions present we still need to determine the sign of p y V S S xx .
The results of comparative statics that summarize the effects of variables that affect the
demand curve are:
substitution effect part
income effect part
p 2y
xp y V u u xy p x V u u yy 
change in price : dx 
(6)
2
dp x
2p x p y V u u xy p x V u u yy p 2y V u u xx p 2y V S S xx 
p 2y V S S x
change in magnitude of interactions : dx 
d
detH
2
p
x  y V S S xx
change in average market demand : 
detH

x
Equation (6) represents the effect of price on the demand for x which can be easily
decomposed into income and substitution effects. Both components from the Slutky
equation are affected because interactions enter the denominator through the term
p 2y VS S xx ; it is true for all further cases below.

Figure 1.1 presents how exogenous positive spillover affects the demand curve for good
(7)
(8)
(4)
x . All forms of the spillover cause the demand curve to shift outwards because
S x, S xx 0 (see table 1A and use (7) with VS 0 ). However, the functional form for
social interactions has a profound effect on how exactly the demand curve shifts. For S s1
the shift is parallel, for S s2 the effect is larger for higher levels of x , and for S s3 the
effect is smaller for higher levels of x . Not only the level of shift differs but also the
S s2
S s3
slope changes in a non-trivial way. Because S s1
xx 0,
xx 0,
xx 0 the first
demand curve has the same slope as the baseline, second demand curve is still a straight
line with steeper slope, but third demand curve is non-linear (see table 1A and use (9)
with VS 0 ). The effect of increase in average demand, x , is qualitivatively the
same as change in importance of social interactions,  (see table 1A for S xx and use
(8) with VS 0 ).
Even though qualitatively all spillover effects have the same impact on the demand curve,
quantitative implications are dramatically different. Each new demand curve have
different elasticity and potential policy implications can vary greatly. For example when
a researcher needs to calculate the deadweight loss associated with the fiscal policy such
as taxation, the results differ various demand curves. Notice that for spillover 1 the
deadweight loss is the same as for the baseline case. But for spillover 2, the deadweight
loss would higher, and for spillover 3 the deadweight loss would lower than the baseline
case of no interactions.
The effect of conformity in the utility function is summarized by figure 1.2. For all forms
of interactions the demand curve pivots around the point where x x because at that
level of x we have S x 0 . Moreover, the demand curve always moves rightward
around the average demand, x (see table 1A for S x and use (7) with VS 0 for
different levels of x ). However, the slope of new demand can be uniformly flatter for
S c1 , or become flatter as x changes ( S c2 ), or the slope can change from flatter to
steeper as in S c3 (see table 1A for S xx and use (6) with VS 0 for different levels of
x ). The effect of increase in average demand, h , is also not uniform though all
demand curves shift rightwards (see table 1A for S xx and use (8) with VS 0 ).
The intuition of the result for conformity is that because there is a penalty for being
different from the norm, there is a natural tendency for consumers to behave the same.
Therefore, the demand curve is less elastic. However, non-linearity of the conformity
effect creates of break-even points where consumers change behavior from being less
responsive to the change in price, to be more sensitive to price changes. Some of the
behavior resembles the Loss Aversion hypothesis.
[->> the intuition for spillover: on graph, same effect as externatility - shift rightward
but shift different, .... depends on marginal social utility, meaning, how social individual
is. Can you actually label kinds of social individuals by their social utility function? So
what would be linear spillover, quadratic, square? What is the marginal utility of social
utility? How would you label those guys?]
[->> Conclude paper by saying that the obvious extension is to put together conformity
and spillover and show that together they can probably model any type of shift/move.
That shows the generality of the approach for demand modeling. Not only that, it has
really nice interpretation. It is possible that some spillover / conformity effects
counteract.]
The analysis not only stresses the need for modelling non-linear social interactions, but
also underlines the fact that modeling interdepence by (1) and (2) is suprisingly flexible
and can accomodates many realistic cases.
3.1.2. Demand for good without interactions, y
We also analyze how interactions present in good x affect second good that does not
have interactions, y . The change of the demand for good y around equilibrium,
approximated by dy , is
dy 
p x V u u xy p y V u u xx dM p x p y xp x V u u xy xp y V u u xx dp x yp x V u u xy p 2x yp y V u u xx dp y
detH
p y V S S xx dM xp y V S S xx dp x yp y V S S xx dp y p x p y V S S xdp x p y V S S xx dx

detH
Comparative statics results are summarized with the derivatives:
substitution effect part
income effect part
p 2x
yp x V u u xy p y V u u xx p y V S S xx 
(10)
2p x p y V u u xy p 2x V u u yy p 2y V u u xx p 2y V S S xx 
p x p y V S S x
dy
change in magnitude of interactions :

(11)
d
detH
p x p y V S S xx
dy
change in average market demand :

(12)
dh
detH
change in price :
dy

dp y
Interactions affect every derivative through the denominator but the interdependence also
p y VS S xx . The result stems from the
influences the income effect through the term 
fact that because the budget constraint is binding any change to good x due to change in
price for good y will change available income for good y .
When there is a positive spillover effect in good x the demand for good y shifts
leftwards for all forms of spillover (see table 1A for S x and use (11)). By the same
token, increase in average demand increases the demand for y , even though the change
differs for various forms of spillover (see table 1A for S xx and use (12)). The slope,
however, can only be established for S s1 - it is the same as the baseline case and the
xx 0 ; the slope for x did not
demand curve shifts parallel leftwards (because S s1
change either). Two other cases of spillover for which S xx 0 have uncertain change in
p 2y VS S xx  in (10) has both flattening and steepening effect.
the slope because term 
Notice that the change in the demand for x due to interactions does not have a parallel
effect on the demand for y : increase in the demand for x does not automatically mean
decrease in the demand for y due to the change in p y . The intuition of the result comes
from the fact that we need to know whether x and y are substitutes or complements.
In the case of conformity we can say that when the average demand for x increases, x
, the demand curve for y will move to the left and the change may create non-linearities
of the demand curve (see table 1A and use S xx ). On average consumers will be induced
to buy more of good x and therefore they will spend less money on good y .
However, it is not clear what happens to the entire demand curve for y because the
derivative S xx affects both the nominator and the denominator of (10) and the derivative
S x has an uncertain sign (see table the 1A). For example, if we focus on the the simplest
case, S c1 , we know that S x x x 0 , and thus dy/d p x p y VS S x/ detH 0 .
Therefore, conditional on the level of x we can say that for individuals with x h
the demand for good y is higher 3 . For x h the demand for good y is lower
x/
 0 ). But we still can not determine how
because more of good x is consumed ( 
the demand for y changes on the range of x because there is no reference point like
x on the demand schedule for good y .
A graphical illustration can help. If we can think of an extreme case when the demand for
good x becomes perfectly inelastic due to the conformity, then the demand for good y
becomes M x /p y , where M x M p x x and x represents a constant demand for
good x . Depending on whether goods x and y are complements or substitutes the
change to the demand curve for good y is different.
We present the effect of the interactions in good x on the demand curve for good y in
figure 2. If without the presence of social interactions goods x and y are substitutes (

x/
p y 0 , and on figure 2 the offer curve is downward sloping), after introduction of
the extreme conformity in x the resulting demand curve for good y is less elastic than
the demand curve for good y . Intuitively we can think that with the presence of the
substitute the demand for y was more elastic because consumers could demand more of
x to substitute for the higher price of y . When the demand for x is fixed consumer
cannot substited y with x .
On the other hand, when without the interdepence the goods x and y are complements
x/
p y 0 , and on the figure 2 the offer curve is upward sloping), the demand for
(
good y with extreme conformity in x becomes more elastic relative to the case of no
interactions. In other words, when there were no social interactions in x , both x and y
were relatively tightly connected by being complements. However, when x is fixed and
certain part of the income is spent on good y , mechanically the demand for good y
becomes more elastic.
We can see that the effect of interactions in good x on the demand for good y is
dependent on the relationship between the two goods. When we make certain
assumptions about the two goods in some cases we can make the inference, but as we
demonstrated, the analysis becomes more involved and due to the complications the
results may be less useful and intuitive. Nevertheless, we show that interactions in only
one good affect the behavior of the goods in the entire market as long as the goods are in
the same expenditure bundle for the consumers.
3.2. Endogenous interactions
Until now we analyzed exogenous interactions, meaning the consumer takes the
expectation of the market demand x as given. We turn to the case where the individual
not only is able to correctly observe the expected market demand but also affects it when
he or she makes a choice about the consumption of x . Therefore, we suppose that x is
x. Thus, dx becomes 
endogenous and it is a function of x , as 
x dx (where

x 
x/
x ), and when we take the total differential of S
x; x ,  we get:
dS S xx dx S xx 
S xd (12)
x dx 
Due to the complexity of endogenous interactions we only examine the simplest
x; x , xx 
x and
interactions forms of spillover and conformity, S s1 
2

c1
S 
x; x , 2 
x x 
x respectively. However, most likely the conclusions of
the following sections are applicable to other forms of interdependence.
3.2.1. Demand for good with interactions, x
The change of the demand for good x around the equilibrium, approximated by dx , is
dx 
p y u xy p x u yy V u dM p x p y yp y V u u xy yp x V u u yy dp y xp x V u u yy xp y V u u xy p 2y dp x
detH
2
p y V S S xd

detH
where the matrix H is the Hessian from the maximization of (1) subject to (2):
2
detH 2p x p y V u u xy p 2x V u u yy p 2y V u u xx 
p 2y V S S xx 
x p y V S S xx 0 (14)
Notice that in endogenous, relative to exogenous interactions, the term associated with
x disappears because it is affected directly by the variable x . Also, the interactions
2
enter into detH through two terms. The term p y VS S xx represents the effect of
2

exogenous interactions whereas the term p y VS S xx x represents endogenous
interactions, or the feedback effet. Therefore, the effect of the interdependence on the
slopes for x and y is more complicated than in the case when the interactions are
exogenous because there is always an intial change which is followed by the endogenous
change.
Changes to the demand for good x around the equilibrium computed from the
comparative statics are:
(13)
substitution effect part
income effect part
p 2y
xp y V u u xy p x V u u yy 
change in price : dx 
(15)
2
2
2
dp x
2p x p y V u u xy p x V u u yy p y V u u xx p 2y V S S xx 
x p y V S S xx 
2
change in magnitude of : dx  p y V S S x
d
detH
interactions
x and
The complication in endogenous interactions is hindged upon the derivatives 


x which represent the nature of the impact that the consumer has on the market
demand. It is reasonable to assume that 0 
x 1 , meaning the consumer increases
expected average demand by increasing individual demand. We can call it local

x cannot be easily
interactions (see Glaeser et al, 2002, for a discussion). However, 
determined. It is possible that even when some consumers tend to have very high or vey
low demands after a treshold their behavior may be ignored by the market because they
may be treated as outliers. On the other hand, individuals who have very good reputation
may trigger a much stronger response from the market when their demand is extreme (ex.
respectable experts in the finantial markets).
x the demand with endogenous spillover will be higher than
For a general function x 
s1,endogenous
S s1,exogenous

x
x x (use (16)). We
that with the exogenous case because S x

cannot say that the new demand will be shifted by x x because the denominator in (15)
also changes. Even though the move can be decomposed into the exogenous change and
the feedback change, the uncertain feedback effect does not let us determine the slope of
a new demand and say whether the shift is parallel or non-linear.
For further insight we analyze the case where the changes to the demand are global. One
such case is when one individual is so influencial that their behavior changes the entire
demand. Another case is when the changes to the individual demand happen for
n
everybody by the same amount, say  x . For example, if x   i1 x i /n and for
every individual we have x i  x , the average market demand will change by



x
 x xi   xx 1 . As a result, we can now easily sign
x  x . Therefore 
x



x 1 and x 0 and futher analyze what happens to the demand for x .
The change in demand due to endogenous spillover spillover is presented in figure 3.1.
As we have shown above the exogenous effect first shifts the demand curve outwards.
The endogenous effect will further increase the demand but we can also say that the
feedback effect will make the shift bigger with higher demand levels to make the demand
curve more elastic (use table 2A and (16), and notice that both social interactions terms
decrease the denominator in (15)). The result is very intuitive: the global change affects
every consumer through the feedback effect but those with higher demands will
experience higher impact on their utility and thus they will respond more to the
exogenous shock. The process is similar to the macroeconomic multiplier.
c1,endogenous

1 
S c1,exogenous
x
x
For the endogenous conformity we can see that S x
.


1 x 1 and the effect of
Local interactions with 0 x 1 imply 0 
(16)
endogenous interactions on the demand for x is qualitatively the same as for the
exogenous conformity (pivot around x ), but the magnitude is smaller. Increased effect
of the individual on the market makes the demand more elastic than in the case of
exogenous conformity, but still less elastic than in the baseline case.
When interdependence becomes global, meaning 
x 1 , the feedback effect
completely counterweights the effect of exogenous interactions and the endogenous
conformity case is observationally equivalent to the baseline case. The intuition of the
result is that because each consumer now affects the market demand, when anyone
changes their demand it triggers the response from the entire market that puts back
pressure on the individual. We present the result in figure 3.2 by showing that first with
the exogenous conformity the demand curve for x becomes less elastic, and then with
endogenous conformity the demand for x approaches the baseline case.
Note that "no change" endogenous conformity result (demand curve does not change
from the baseline case) is experienced in all three forms of conformity. It not only
underlines the complexity of the models with social interactions, but also the difficulty
with the identification of the exact effects of interactions on individual choices and the
sources of observed effects. Especially because endogenous spillover exagerated the
effect of exogenous spillover, whereas endogenous conformity diminished the effect of
exogenous conformity. [PUT IN CONCLUSION? How about the fact that in fact in
conformity the individual see oneself in the middle of exogenous interactions curve?]
[Because all the derivatives are zero (see table 2A), remarkably, there is no change from
the baseline case. Therefore, with global change the feedback effect completely
outweighs the effect of exogenous effect which made the demand curve less elastic. The
intuition of the result is that each consumer now affects the market demand and when
anyone changes their demand it triggers the response from the entire market that puts
0, 1 the demad with
x  
back pressure on the individual person. Moreover, for 
endogenous conformity is between the baseline demand and the demand with exogenous
conformity.]
[For example, if the person consumed less than the average demand before interactions
were introduced, the consumer will have higher demand with exogenous conformity. But
when the interactions are endogenous, the consumer now affects the average demand
directly and pushes it down.
0, 1 the demad is between the baseline demand and the demand
x  
Notice that for 
with exogenous conformity.
Suppose we have a consumer with higher demand than average demand. When we have
exogenous conformity the consumer will be induced to consume less in order to reduce
the negative effect of social interactions. But with endogeneity the individual demand
decrease the average demand. Therefore, consumers who had higher demand suddenly
find themselves that they have lower demand and they increase their demand. .....
What is the perceived average? it may be that each consumer is on the social interactions
demand curve but when he is at the center.
So each customer may see themselves as if the are in the middle of the conformity curve
and they do not move. I guess it is about the response of the rest of the group.
Therefore, the decrease in individual demand will decrease the average demand and the
consumer will be induced to decrease the demand even more. However, notice that by
lowering the demand everyone on the other side is affected???
the the fact but and it counterweights the effect of conformity that makes the demand
curve less elastic. We can also see it by noticing that endogeneity in conformity

1 
x , with x 0 giving the exogenous effect and
diminishes its effect by factor 
with 
x 1 giving maximum endogenous effect. As the effect of individual demand on
average demand increases, that is 
x approaches one, clear when we consider that
endogeneity in general make the response to any change more pronounced, and
thereforein conformity is that the initial effect of social interactions made people have
higher negative utility effect
With weaker feedback effect, that is 
x 1 , consumers respond less.
What about derivative"

x

x
?
-- >>> Interesting but NOT TRUE!!! delete - The intuition behind the result is that
endogeneity in general makes the demand curve more elastic, be it spillover or
conformity. Therefore, when exogenous conformity makes consumers less responsive
because they are punished for having their demand different from the norm, the feedback
effect counterweights the effect. The idea is that by consumers now affecting the average
demand when they change their individual demand
The shift is the same as in the case of exogenous interactions because S x is the same
and two effects at the denominator must cancel out? Where is the feedback effect? How
do you decompose it?
x 1 . meaning that there is a global change in the
. assume a global change that 
demands in the economy that shifts everyone's demand for good x by the same amount
2
2
, the derivatives become S xx 2 0 , and S x 2 0 . Thus both p y VS S xx 0
2

and p y V S S xx x 0 make the slope for the demand of good x more negative and
more elastic (because positive denominator becomes smaller).
As an illustration suppose that For example suppose that x 
we have x i  x , we also have x  x . Therefore

x

x
n
xi
 i
1

n
x
x i
, and then if i
  xx 1 .


2
0 ,
xx 
x 
Notice that in the simplest spillover case we have S xx 


S xx  
  x x 0 , and S x 
x 0 .
x   x
xx 
2

 x 
1 
x  0 ,
x  x 
Also, in the simplest conformity case we have S xx  

S xx 
x 10 , and

 x x 1

x  


x
In a market with numerous consumers a change of one consumer's demand may be
negligable and the result for exogenous interactions would apply. However, larger
number of consumers adjusting their demand will significantly affect average market
demand, and in the extreme case we can think of a scenario where every consumer
changes his or her demand due the change in some exogenous factor. The global change
x 1 and everyone's demand for x shifts by the
in the demand would mean that 
2
same amount.
In the case of positive spillover with the global change the effect of endogenous
interactions will exagerate the effect of exogenous effect. Figure 3.1 presents the changes
where the first effect is represented by the paralell shift of the exogenous interactions,
and futher shift is caused by the feedback effect. which also results in increased slope
(use table 2A). in the demand curve when there are exogenous social interactions and
then when the the interactions are endogenous. The demand curve shifts further outwards
because
3.2.2. Demand for good without interactions, y
The good not affected by the social interactions changes around the equilibrium with:
dy 
p x V u u xy p y V u u xx dM p x p y xp x V u u xy xp y V u u xx dp x yp x V u u xy p 2x yp y V u u xx dp y
detH

dp x yp y V S S xx yp y V S S xx 
dp y p x p
p y V S S xx p y V S S xx x dM xp y V S S xx xp y V S S xx 
x
x

detH
The effect of each exogenous change will be affected by social interactions in good x
because of the omipresence of the social interactions terms. The derivatives indicating the
slope of the demand curve for y and the magnitude of social interactions are:
substitution effect part
change in price :
income effect part
p 2x
yp x V u u xy p y V u u xx p y V S S xx p y V S S xx 
dy
x

(18)
2
2
2

2
dp y
2p x p y V u u xy p x V u u yy p y V u u xx p y V S S xx x p y V S S xx 
change in magnitude of : dy  p x p y V S S x
d
detH
interactions
Focusing only on the case of global change in x , meaning 
x 1 , we can only
determine that endogenous spillover will decrease demand for y with increased
magnitude of interactions in good x (see table 2A and use (19)). The result is very
intuitive because with more money spent on good x the consumer has less resources to
spend on y . However, we cannot determine what happens to the slope for y .
In conformity for good x the demand for good y is also very difficult to determine
x 1 the response of y to the price
except for the case of global change in x . When 
is the same as before social interactions. Further results are possible with more explicity
utility function and the use of numerical simulations.
4. Conclusion
We analyze social interactions in the partial market equilibrium study with two goods.
Each individual's the choice of one good is affected by the choices of other consumers in
the market, whereas the other good's demand is not influenced by social interactions.
Interdependence in one good are imbedded directly in the utility function and affects the
demands for both goods: good with interactions directly, and good without interactions
indirectly. We analyze cases of spillover where individuals experience positive
externality from other individuals' demands in the market, and conformity where
individuals experience penalty for having demands different from other individuals in the
market. We perform comparative statics on the demand side of the market where we
select several functional forms for the part of the utility function representing
interdependence. The results of our study are useful for policy changes and welfare
analysis because even though the qualitative effects to the demand curves are relatively
clear, the quantitative outcomes may have profound consequences on the correct
measurement of the deadweight loss or behavioral effects of taxation.
Generally, a positive spillover effect shifts the demand curve upwards, however, specific
functional forms for the social utility make the slope for the demand curve to change in a
non-trivial way. The demand can become both more or less elastic. Therefore the
magnitude of interactions is important because in some cases part of the population in the
market (say, low-demand consumers) may be much strongly affected by the interactions
then the rest of the consumers in the economy. When the conformity is present the
(19)
demand curve pivots around the expected market demand and the demand curve becomes
less elastic. However, selection of the specific functional form can exaggerate or diminish
the general changes to the demand curve.
We also show that interactions in one good significantly affect the demand for the good
that does not have interactions. The effect is difficult to determine and hinges upon the
relationship between both goods as being complements or substitutes. Therefore, we can
not analytically provide answers as to how non-interactions good is affected by the
presence of the interactions type good without some prior knowledge about the
relationship between two commodities. Numerical analysis with carefully selected
functional forms and properly calibrated parameters may potentially aid in more
comprehensive study of the market changes when the consumers' demands are
interdependent. However, we can still say that non-interactions good will be impacted by
the interdependence as long as the budget constraint is binding and the goods are related.
Finally, we show that the partial market analysis with the present endogenous social
interactions is more difficult because one needs to determine exactly how an individual
affects the expectation of the market demand. In the case of spillover the endogeneity
mimics the effects derived for exogenous interactions but the magnitudes are
exaggerated. The conformity case, however, represents the situation where the
endogeneity bring the demand curve back to where it was before interdependence was
introduced.
Also, we acknowledge that there are many other possible forms for interactions

 besides spillover and conformity variations; the
represented by the social utility S
analytical representations may vary and interactions may operate through different
channels: budget constraint, parameters, or else. However, we believe that the spillover
and conformity may exhaust most of the real life interactions problems and the examples
demonstrate the flexibility of proposed modelling stratety. The results are intuitive and
the graphical representation aids understanding.
[Note that the spillover interactions are often modeled in urban economics in
agglomeration studies, human capital studies as knowledge spillovers, and many other
applications. On the other hand, conformity reflects modeling strategy for the analysis of
social norms and can be crudely said to be the foundation of the Social Psychology.]
Endnotes
1
The reference group is any set of individuals in the population (including the entire
population) to which the individual refers when making a demand decision. For example
members of the reference group may be neighbors, family members, or co-workers
depending on the phenomena studied.
CHECK
!!!2
For example suppose that x 
n
xi
 i
1
n

x

x
, and then if i we have

 x xi   xx 1 .
x i  x , we also have x  x . Therefore

x
3
0 , and thus the individual consumes less of
Notice that for x h we have 

good x , there is more resources for good y , and thus the individual consumes more of
y because the budget constraint is binding and the prices and incomes do not change.
REFERENCES
Akerlof, George A., and Rachael E. Kranton. 2000. Economics and Identity. Quarterly
Journal of Economics 115(3)(August): 715--754.
Angrist, Joshua D. and Kevin Lang. 2002. How Important are Classroom Peer Effects?
Evidence from Boston's Metco Program. NBER Working Paper No. 9263. Cambridge,
MA: National Bureau of Economic Research.
Aronsson, Thomas, Sö ren Blomquist, and Hans Sacklé n. 1999. Identifying
Intertemporal Behaviour in An Empirical Model of Labour Supply. Journal of Applied
Econometrics 14(6)(November-December): 607--626.
de Bartolome, Charles A.M. 1990. Equilibrium and Inefficiency in a Community Model
with Peer Groups Effects. Journal of Political Economy 98(1)(February): 110--133.
Becker, Gary S. 1974. A Theory of Social Interactions. Journal of Political Economy
82(6)(November-December): 1063--1093.
Becker, Gary S., and Kevin M. Murphy. 2000. Social Economics: Market Behavior in a
Social Environment. Cambridge, MA: Harvard University Press.
Benabou, Roland. 1996a. Equity and Efficiency in Human Capital Investment: The Local
Connection. Review of Economic Studies 63(2)(April): 237--264.
Benabou, Roland. 1996b. Heterogeneity, Stratification, and Growth: Macroeconomic
Implications of Community Structure and School Finance. American Economic Review
86(3)(June): 584--609.
Bernheim, Douglas B. 1994. A Theory of Conformity. Journal of Political Economy
102(5)(October): 841--877.
Blomquist, N. Soren. 1993. Interdependent Behavior and the Effect of Taxes. Journal of
Public Economics 51(2)(June): 211--218.
Brock, William A., and Steven N. Durlauf. 2001a. Interactions-Based Models. In J.J.
Heckman and E. Leamer (eds.), Handbook of Econometrics, Volume 5. Amsterdam, The
Netherlands: Elsevier Science B.V., pp. 3297--3280.
Brock William A., and Steven N. Durlauf. 2001b. Discrete Choice with Social
Interactions. Review of Economic Studies 68(2)(April): 235--260.
Case, Anne C., and Lawrence F. Katz. 1991. The Company You Keep: The Effects of
Family and Neighborhood on Disadvantaged Youths. NBER Working Paper No. 3705.
Cambridge, MA: National Bureau of Economic Research.
Chamley, Christophe. 1999. Coordinating Regime Switches. Quarterly Journal of
Economics 114(3)(August): 869--905.
Corcoran, Mary, Roger Gordon, Deborah Laren, and Gary Solon. 1992. The Association
Between Men's Economic Status and Their Family and Community Origins. Journal of
Human Resources 27(4)(Fall): 575--601.
Crane, Jonathan. 1991. The Empiric Theory of Ghettos and Neighborhood Effects on
Dropping Out and Teenage Childbearing. American Journal of Sociology 96(5)(March):
1226--1259.
Datcher, Linda. 1982. Effects of Community and Family Background on Achievement.
Review of Economics and Statistics 64(1)(February): 32--41.
Duflo, Esther, and Emmanuel Saez. 2002. The Role of Information and Social
Interactions in Retirement Plan Decisions: Evidence from a Randomized Experiment.
Working Paper 48, Center for Labor Economics: University of California, Berkeley.
Durlauf, Steven N. 1996. A Theory of Persistent Income Inequality. Journal of Economic
Growth 1(1)(March): 75--93.
Durlauf, Steven N. and H. Peyton Young. 2001. The New Social Economics. In S.
Durlauf and H.P. Young (eds.), Social Dynamics. Washington D.C.: MIT and Brookings
Institutions Press, pp.1--14.
Evans, William N., Wallace E. Oates, and Robert M. Schwab. 1992. Measuring Peer
Group Effects: A Study of Teenage Behavior. Journal of Political Economy
100(5)(October): 966--991.
Frank, Robert H. 1985. The Demand for Unobservable and Other Nonpositional Goods.
The American Economic Review. 75(1)(March): 101--116.
Frank, Robert H., and Cass R. Sunstein. 2001. Cost Benefit Analysis and Relative
Position. The University of Chicago Law Review 68(2)(Spring): 1--40.
Gaertner, Wulf, 1974, A Dynamic Model of Interdependent Consumer Behavior,
Zeitschrift fur Nationalokonomie, Vol. 34, No. 3-4, pp. 327-44.
Glaeser, Edward L., David Laibson, and Bruce Sacerdote. 2000. The Economic
Approach to Social Capital. NBER Working Paper No. 7728. Cambridge MA: National
Bureau of Economic Research.
Glaeser, Edward L., Bruce I. Sacerdote, and Jose A. Scheinkman. 1996. Crime and Social
Interactions. Quarterly Journal of Economics 111(2)(March): 507--548.
Glaeser, Edward L., Bruce I. Sacerdote, and Jose A. Scheinkman. 2002. The Social
Multiplier, Working Paper 9153, National Bureau of Economic Research: Cambridge,
MA.
Hopkins Ed, Kornienko Tatiana. 2004. "Running to Keep in the Same Place: Consumer
Choice as a Game of Status." The American Economic Review. 94(4)(September): 10851108.
Kapteyn Arie, Sara van de Geer, Huib van de Stadt, and Tom Wansbeek, 1997,
Interdependent Preferences: An Econometric Analysis, Journal of Applied Econometrics,
November-December, Vol. 12, No. 6, pp. 665-86.
Katz, Lawrence F., Jeffrey Kling, and Jeffrey B. Liebman. 2001. Moving to Opportunity
in Boston: Early Results of a Randomized Mobility Experiment. Quarterly Journal of
Economics 116(2)(May): 607--654.
Kelly, Morgan, and Cormac Ó Grá da. 2000. Market Contagion: Evidence from the
Panics of 1854 and 1857. American Economic Review 90(5)(December): 1110--1124.
Kniesner, Thomas J., and W. Kip Viscusi. 2003. A Cost-Benefit Analysis: Why Relative
Economic Position Does Not Matter. Yale Journal on Regulation 20(1)(Winter): 1--26.
UPDATE Kooreman P. and Schoonbeek L., Characterizing Pareto improvements in an
interdependent demand system, Journal of Public Economic Theory, forthcoming.
Kremer, Michael. 1997. How Much Does Sorting Increase Inequality? Quarterly Journal
of Economics 112(1)(February): 115--139.
Lindbeck, Assar, Sten Nyberg, and Jorgen W. Weibull. 1999. Social Norms and
Economic Incentives in the Welfare State. Quarterly Journal of Economics
64(1)(February): 1--35.
Manski, Charles F. 1993. Identification of Endogenous Social Effects. Review of
Economic Studies 60(204)(July): 531--542.
Manski, Charles F. 2000. Economic Analysis of Social Interactions. Journal of Economic
Perspectives 14(3)(Summer): 115--136.
Moffitt, Robert A. 2001. Policy Interventions Low-level Equilibria and Social
Interactions. In S. Durlauf and H.P. Young (eds.), Social Dynamics. Washington D.C.:
MIT and Brookings Institutions Press, pp.45--82.
Pollak Robert A., 1976, Interdependent Preferences, American Economic Review, June,
Vol. 66, No. 3, pp. 309-320.
Pollak Robert A., and Terence J. Wales. 1992. Demand System Specification and
Estimation. New York: Oxford University Press.
Rabin, Matthew. 1993. Incorporating Fairness Into Game Theory and Economics.
American Economic Review 83(5)(December): 1281--1302.
Rigobon, Robert. 2001. Contagion: How to Measure it? NBER Working Paper No. 8118.
Cambridge MA: National Bureau of Economic Research.
Roback, Jennifer. 1982. Wages, Rents, and the Quality of Life. Journal of Political
Economy 90(6)(December): 1257--1278.
Rosenbaum, James E. 1991. Black Pioneers-Do Their Moves to the Suburbs Increase
Economic Opportunity for Mothers and Children? Housing Policy Debate 2(4): 1179-1213.
Sacerdote, Bruce. 2001. Peer Effects with Random Assignment: Results for Dartmouth
Roommates. Quarterly Journal of Economics 116(2)(May): 681--704
Sherif, M. 1935. A Study of Some Social Factors in Perception. Archives of Psychology
27(187): 1--60 .
Smith, Lones, and Peter Sorensen. 2000. Pathological Outcomes of Observational
Learning. Econometrica 68(2)(March): 371--398.
Soetevent, Adriaan and Peter Kooreman 2002. Simulation of a Linear Expenditure
System with Social Interactions. Working paper, Faculty of Economics, University of
Groningen, January 28.
Table 1. Results of the comparative statics for different interactions functional forms
when interactions are exogenous relative to the case of no interactions.
Partials
Interactions
Effect
None-baseline
Spillover
αμxx
αμxx2
αμx?x
dx/dpx
dx/dα
dx/dμx
dy/dpy
dy/dα
dy/dμx
-
0
0
-
0
0
--+
+
++
+-
+
++
+-
-?
-?
--+
--+
Conformity
-+
+/+
-?
?
?
α(x-μx)2/2
- ++
+ +/- ++/+-?
?
?
α(x-μx)4/12
- -/- +
+-/-++
-?
?
?
α(x2-μx2)2/4
Notation:
All symbols are relative to baseline case.
0
– derivative is zero.
?
– uncertain / can not determine.
– negative derivative
-– negative derivative and the change is more than the baseline case
-+
– negative derivative and the change is less than the baseline case
+
– positive derivative
++
– positive derivative and the change is more than the baseline case
+– positive derivative and the change is less than the baseline case
/
– lists various cases when the derivative is changing signs.
Figure 1.1. Demonstration of the effect of the spillover interactions on the demand for good x
with different functional forms.
Price of good x
Spillover 3 –
with Sxx<0
Baseline - no
interactions
Spillover 1
– with Sxx=0
Spillover 2 –
with Sxx>0
px
x
Quantity of good x
Figure 1.2. Demonstration of the effect of the conformity interactions on the demand for good x
with different functional forms.
Conformity 1
Conformity 2
Price of good x
Conformity 3
px
Baseline no interactions
(x/sqrt(3))
x
Quantity of good x
Figure 2. Demonstration of the effect of the conformity in good x on the demand for good y when
there is an extreme conformity in good x (consumer consumes fixed amount of good x); graphs
represent relationship between offer curves and demand curves (for good y).
no
interactions substitutes
extreme
conformity no interactions
- complements
y
y
x
x
py
Price of good x
Figure 3.1. Demonstration of the effect of the global endogenous and exogenous spillover
interactions on the demand for good x.
Spillover endogenous
px
Spillover exogenous
Baseline - no
interactions
x
Quantity of good x
Figure 3.2. Demonstration of the effect of the global endogenous and exogenous conformity
interactions on the demand for good x.
Price of good x
Conformity
endogenous
with ’x<1
Conformity
exogenous
px
Baseline no interactions
and
Conformity
endogenous with
’x=1
x
Quantity of good x