DP2016-02

Robust Comparative Statics for
Non-monotone Shocks
in Large Aggregative Games
Carmen CAMACHO
Takashi KAMIHIGASHI
Cagri SAGLAM
February 1, 2016

Robust Comparative Statics for Non-monotone
Shocks in Large Aggregative Games∗
Carmen Camacho†

Takashi Kamihigashi‡Çağrı Sağlam§

February 1, 2016

Abstract
A policy change that involves a redistribution of income or wealthis typically controversial, affecting some people positively but others
negatively. In this paper we extend the “robust comparative statics”
result for large aggregative games established by Acemoglu and Jensen(2010, 49th IEEE Conference on Decision and Control, 3133–3139) topossibly controversial policy changes. In particular, we show thatboth the smallest and the largest equilibrium values of an aggregate
variable increase in response to a policy change to which individuals’
reactions may be mixed but the overall aggregate response is positive.
We provide sufficient conditions for such a policy change in terms of
distributional changes in parameters.
Keywords: Large aggregative games; robust comparative statics; positive shocks; stochastic dominance; mean-preserving spreads∗

Earlier versions of this paper were presented at the “Dynamic Interactions for Economic Theory” Conference in Paris, December 16-17, 2013, and the Novo Tempus andLabex MME-DII Conference on “Time, Uncertainties, and Strategies” in Paris, December
14-15, 2015. We would like to thank participants of these conferences, including Rabah
Amir and Martin Jensen for their helpful comments and suggestions. Financial support
from the Japan Society for the Promotion of Science is gratefully acknowledged.
†
Centre National de la Recherche Scientifique (CNRS) and Université Paris 1 PanthéonSorbonne, France. Email: maria.camacho-perez@univ-paris1.fr‡
Research Institute for Economics and Business Administration (RIEB), Kobe University, Japan. Email: tkamihig@rieb.kobe-u.ac.jp§
Department of Economics, Bilkent University, Turkey. Email: csaglam@bilkent.edu.tr

1

Introduction

Recently, Acemoglu and Jensen (2010, 2015) developed new comparative
statics techniques for large aggregative games, where there are a continuumof individuals interacting with each other only through an aggregate variable(which integrates all individuals’ actions). The surprising insight of theiranalysis is that in such games, one can obtain a “robust comparative statics”
result without considering the interaction between the aggregate variable andindividuals’ actions. In particular, Acemoglu and Jensen (2010) defined apositive shock as a positive parameter change (a set of simultaneous increases
in individuals’ parameters) that positively affects each individual’s action foreach value of the aggregate variable. They showed that both the smallest andthe largest equilibrium values of the aggregate variable increase in response
to a positive shock.
Although positive shocks are common in economic models, many important policy changes in reality tend to be controversial, affecting someindividuals positively but others negatively. For example, a policy changethat involves a redistribution of income necessarily affects some individuals’
income positively but others’ negatively. Such policy changes of practicalimportance cannot be positive shocks.
The purpose of this paper is to show that Acemoglu and Jensen’s (2010,2015) analysis can in fact be extended to such policy changes. Using Acemoglu and Jensen’s (2010) static framework, we consider possibly controversial policy changes by defining an “overall positive shock” to be a parameterchange to which individuals’ reactions may be mixed but the overall aggregate response is positive for each value of the aggregate variable. We showthat both the smallest and the largest equilibrium values of the aggregate
variable increase in response to an overall positive shock. Then we provide
sufficient conditions for an overall positive shock in terms of distributionalchanges in parameters. These conditions enable one to deal with various
policy changes, including ones that involve a redistribution of income.
The concept of overall positive shocks is closely related not only to that of
positive shocks but also to Acemoglu and Jensen’s (2013) concept of “shocks
that hit the aggregator,” which were defined as parameter changes that directly affect the “aggregator” positively along with additional restrictions.
Such parameter changes are not considered in this paper, but they can easily
be incorporated by slightly extending our framework.
This paper is not the first to study comparative statics for distributional
1changes. In a general dynamic stochastic model with a continuum of individuals, Acemoglu and Jensen (2015) considered robust comparative statics forchanges in the stationary distributions of individuals’ idiosyncratic shocks,
but their analysis was restricted to positive shocks in the above sense.1 Jensen(2015) and Nocetti (2015) studied comparative statics for more general distributional changes, but neither of them considered robust comparative statics.
This paper bridges the gap between robust comparative statics and distributional comparative statics.
The rest of the paper is organized as follows. In Section 2 we provide
a simple example of income redistribution and aggregate labor supply to illustrate how Acemoglu and Jensens’s (2010) analysis may fail to apply to apolicy change of practical importance. In Section 3 we present our generalframework along with basic assumptions, and show the existence of a purestrategy Nash equilibrium. In Section 4 we formally define overall positive
shocks. We also introduce a more general definition of “overall monotone
shocks.” We then present our robust comparative statics result. In Section
5 we provide sufficient conditions for an overall monotone shock in termsof distributional changes in parameters based on first-order stochastic dominance and mean-preserving spreads. In Section 6 we apply our results to theexample of income redistribution. In Section 7 we conclude the paper.

2

A Simple Model of Income Redistribution

Consider an economy with a continuum of agents indexed by i ∈ [0, 1]. Agenti solves the following maximization problem:max

ci ,xi ≥0u(ci ) − xis.t. ci = wxi + ei + si ,

(2.1)
(2.2)

where u : R+ → R is strictly increasing, strictly concave, and twice continuously differentiable, w is the wage rate, si is a lump-sum transfer to agent i,and ci , xi , and ei are agent i’s consumption, labor supply, and endowment,
respectively. We assume that ei + si ≥ 0 for all i ∈ [0, 1]. If si < 0, agent
i pays a lump-sum tax of −si . For simplicity, we assume that the upper
1

See Balbus et al. (2015) for monotone comparative statics results on distributional
Bayesian Nash equilibria with strategic complementarities.

2bound on xi is never binding for relevant values of w and is thus not explicitly imposed. This simply means that no agent works 24 hours a day, 7 daysa week. The government has no external revenue and satisfiesZ
si di = 0.
(2.3)
i∈I

Aggregate demand for labor is given by a demand function D(w) such that
D(0) < ∞, D(w) = 0 for some w > 0, and D : [0, w] → R+ is continuousand strictly decreasing. The market-clearing condition isZ
xi di.
(2.4)
D(w) =i∈I

Given (2.3), any change in the profile of si affects some agents’ income
positively but others’ negatively. Hence it cannot be a positive shock in thesense of Acemoglu and Jensen’s (2010). However, one may still ask, how doesa policy change that widens income inequality affect aggregate labor supply
and the wage rate? This may seem a difficult question to answer without
additional assumptions. It turns out that our results help one answer thistype of question in a robust way.

3

Large Aggregative Games

We consider a large aggregative game as defined by Acemoglu and Jensen(2010, Sections II, III). There are a continuum of players indexed by i ∈I ≡ [0, 1]. Player i’s action and action space are denoted by xi and Xi ⊂ R,
respectively. The assumption made in this section are maintained throughout
the paper.
Assumption 3.1. For each i ∈ I, Xi is nonempty and compact. Furthermore, there exists a compact set K ⊂ R such that Xi ⊂ K for all i ∈ I.
Q
Let X = i∈I Xi . Let X be the set of action profiles x ∈ X such thatthe mapping i ∈ I 7→ xi is measurable.2 Let H be a function from K to a
subset Ω of R. We define G : X → Ω, called the aggregator, byZ
G(x) = H
xi di .
(3.1)
i∈I
2

Unless otherwise specified, measurability means Lebesgue measurability.

3

Assumption 3.2. The set Ω is compact and convex, and H : K → Ω iscontinuous.3
Each player i’s payoff depends on his own action xi ∈ Xi , the entire action
profile x ∈ X through the aggregate G(x), and a parameter ti specific toplayer i. In other words, player i’s payoff takes the form πi (xi , G(x), ti ). Any
additional information on xj and tj with j 6= i is irrelevant to his decisionmaking.
Let
Q Ti be the underlying space for ti for each i ∈ I; i.e., ti ∈ Ti . LetT ⊂ i∈I Ti . We regard T as a set of well-behaved parameter profiles; forexample, T can be a set of measurable functions from I to R. We only
consider parameter profiles t in T .
Assumption 3.3. For each i ∈ I, player i’s payoff function πi maps each(k, Q, τ ) ∈ K × Ω × Ti into R.4 Furthermore, for each t ∈ T , πi (·, ·, ti ) is
continuous on K × Ω, and for each (k, Q) ∈ K × Ω, πi (k, Q, ti ) is measurable
in i ∈ I.
The game here is aggregative in the sense that each player’s payoff isaffected by other players’ actions only through the aggregate G(x). Accordingly, each player i’s best response correspondence depends on other players’
actions only through Q = G(x). Let Ri (Q, ti ) denote player i’s best response
correspondence:
Ri (Q, ti ) = arg max πi (xi , Q, ti ).

(3.2)

xi ∈Xi

The following technical assumption ensures that given any Q ∈ Ω, we canfind a measurable action profile x ∈ X such that xi ∈ Ri (Q, ti ) for all i ∈ I.5
Assumption 3.4. For each open subset U of K, the set {i ∈ I : Xi ∩ U 6= ∅}is measurable.
Throughout the paper, we restrict attention to pure-strategy Nash equilibria, which we simply call equilibria. To be more precise, given t ∈ T , anequilibrium of this game is an action profile x ∈ X such that xi ∈ Ri (G(x), ti )3

Given the assumptions on H and K, the properties of Ω here can be assumed without
loss of generality.
4
If πi is initially defined only on Xi × Ω × Ti , then this means that πi can be extended
to K × Ω × Ti in such a way as to satisfy Assumption 3.3.
5
See the paragraph below (A.1) for details.

4

for all i ∈ I. Given t ∈ T , we define an equilibrium aggregate as Q(t) ∈ Ωsuch that Q(t) = G(x) for some equilibrium x. We define Q(t) and Q(t) as
the smallest and largest equilibrium aggregates, respectively, provided thatthey exist. The following result shows that an equilibrium as well as thesmallest and largest equilibrium aggregates exist.
Theorem 3.1. For any t ∈ T , an equilibrium exists. Furthermore, the setof equilibrium aggregates is compact (and nonempty). Thus the smallest andlargest equilibrium aggregates Q(t) and Q(t) exist.
Proof. See Appendix A.
Following Acemoglu and Jensen (2010, Theorem 1), we prove the above
result using Kakutani’s fixed point theorem and Aumann’s (1965, 1976) results on the integral of a correspondence. Our result differs from Acemoglu
and Jensen’s in that we assume a continuum of player types rather than afinite number of player types.6
The existence of an equilibrium can alternatively be shown by using Theorem 3.4.1 in Balder (1995) under slightly different assumptions. There areother general existence results for games with a continuum of players in theliterature (e.g., Khan et all. 1997; Carmona and Podczeck, 2009). However,these results do not directly apply here since they assume a common strategy space for all players and do not explicitly establish the existence of thesmallest and largest equilibrium aggregates.

4

Overall Monotone Shocks

By a parameter change, we mean a change in t ∈ T from one profile toanother. Given t, t ∈ T , the parameter change from t to t means the change
in t from t to t. The following definitions take t, t ∈ T as given.
Definition 4.1 (Acemoglu and Jensen, 2010). The parameter change fromt to t is a positive shock if (a) T is equipped with a partial order ≺, (b) G(·)
is an increasing function,7 (c) t ≺ t, and (d) for each Q ∈ Ω and i ∈ I, thefollowing properties hold:6

Acemoglu and Jensen (2015) allow for a continuum of player types by using the Pettis
integral in (3.1).
7
In this paper, “increasing” means “nondecreasing,” and “decreasing” means “nonincreasing.”5(i) For each xi ∈ Ri (Q, ti ) there exists xi ∈ Ri (Q, ti ) such that xi ≤ xi .
(ii) For each y i ∈ Ri (Q, ti ) there exists y i ∈ Ri (Q, ti ) such that y i ≤ y i .
For comparison purposes, Acemoglu and Jensen’s (2010) key assumptions
are included in the above definition. We introduce additional definitions.
Definition 4.2. The parameter change from t to t is a negative shock if theparameter change from t to t is a positive shock. A parameter change is amonotone shock if it is a positive shock or a negative shock.
Acemoglu and Jensen (2010, Theorem 2) show that if the parameterchange from t to t is a positive shock, then the following inequalities hold:Q(t) ≤ Q(t),Q(t) ≤ Q(t).

(4.1)

In this section we show that these inequalities hold for a substantially larger
class of parameter changes. To this end, for Q ∈ Ω and t ∈ T , we defineG(Q, t) = {G(x) : x ∈ X , ∀i ∈ I, xi ∈ Ri (Q, ti )} .

(4.2)

The following definitions, which do not require T to be partially ordered andG(·) to be increasing, play a central role in our comparative statics results.
Definition 4.3. The parameter change from t to t is an overall positive shock
if for each Q ∈ Ω the following properties hold:(i) For each q ∈ G(Q, t) there exists q ∈ G(Q, t) such that q ≤ q.
(ii) For each r ∈ G(Q, t) there exists r ∈ G(Q, t) such that r ≤ r.
Definition 4.4. The parameter change from t to t is an overall negative
shock if the parameter change from t to t is an overall positive shock. A
parameter change is an overall monotone shock if it is an overall positive
shock or an overall negative shock.
It is easy to see that a positive shock (which requires G(·) to be increasing)is an overall positive shock. We are ready to state our result on robust
comparative statics:
Theorem 4.1. Let t, t ∈ T . Suppose that the parameter change from t to t
is an overall positive shock. Then both inequalities in (4.1) hold. The reserve
inequalities hold if the parameter change is an overall negative shock.
6

Proof. See Appendix B.
The proof of this result closely follows that of Theorem 2 in Acemoglu
and Jensen (2010). The latter result is immediate from Theorem 4.1 under
our assumptions since a positive shock is an overall positive shock.8

5Sufficient Conditions

In this section we provide sufficient conditions for overall monotone shocks
by assuming that players differ only in their parameters ti . This assumption
by itself is innocuous since it can be ensured by redefining ti as (ti , i). Morespecifically, we assume the following for the rest of the paper.
Assumption 5.1. There exists a Borel-measurable convex set T ⊂ Rn
(equipped with the usual partial order) with n ∈ N such that Ti ⊂ T for
all i ∈ I. There exists a convex-valued correspondence X : T → 2T suchthat Xi = X (ti ) for all i ∈ I and ti ∈ Ti . Moreover, there exists a function
π : K × Ω × T → R such that∀i ∈ I, ∀(k, Q, τ ) ∈ K × Ω × T ,πi (k, Q, τ ) = π(k, Q, τ ).

(5.1)

This assumption implies that player i’s best response correspondence
Ri (Q, τ ) does not directly depend on i; we denote this correspondence by
R(Q, τ ). For (Q, τ ) ∈ Ω × T , we defineR(Q, τ ) = min R(Q, τ ),R(Q, τ ) = max R(Q, τ ).

(5.2)

Both R(Q, τ ) and R(Q, τ ) are well-defined since R(Q, τ ) is a compact set foreach (Q, τ ) ∈ (Ω, T ) (see Lemma A.1). To consider distributional changes int ∈ T , we assume the following for the rest of the paper.
Assumption 5.2. T is a set of measurable functions from I to T , andH : K → Ω is an increasing function.
For any t ∈ T , let Ft : Rn → I denote the distribution function of t:Z
Ft (z) =1{ti ≤ z}di,
(5.3)
i∈I

where 1{·} is the indicator function; i.e., 1{ti ≤ z} = 1 if ti ≤ z, and = 0
otherwise. Note that Ft (z) is the proportion of players i ∈ I with ti ≤ z.
8

Acemoglu and Jensen (2015, Theorem 5) establish a dynamic version of their result
based on positive shocks. Their dynamic result can also be extended to overall monotone
shocks in a similar way.

7

5.1

First-Order Stochastic Dominance

Given a pair of distributions F and F , F is said to (first-order) stochasticallydominate F ifZ
Z
φ(z) dF (z) ≤φ(z) dF (z)(5.4)

for any increasing bounded Borel-measurable function φ : Rn → R, where
Rn is equipped with the usual partial order ≤. It is well known (e.g., Müller
and Stoyan, 2002, Section 1) that in case n = 1, F stochastically dominates
F if and only if∀z ∈ R,F (z) ≥ F (z).

(5.5)

The following result provides a sufficient condition for an overall monotone shock based on stochastic dominance.
Theorem 5.1. Let t, t ∈ T . Suppose that Ft stochastically dominates Ft .
Suppose that both R(Q, τ ) and R(Q, τ ) are increasing (resp. decreasing) Borelmeasurable functions of τ ∈ T for each Q ∈ Ω. Then the parameter change
from t to t is an overall positive (resp. negative) shock.
Proof. We only consider the increasing case; the decreasing case Ris symmetric.
Let q ∈ G(Q, t). Then there exists x ∈ X such that q = H( i∈I xi di) andxi ∈ R(Q, ti ) for all i ∈ I. Since xi ≤ R(Q, ti ) for all i ∈ I by (5.2), and since
H is an increasing function by Assumption 5.2, we have

Z
Z

q≤H
R(Q, ti )di = H
R(Q, z)dFt (z)
(5.6)
i∈I

Z
Z

≤H
R(Q, z)dFt (z) = H
R(Q, ti )di ∈ G(Q, t),
(5.7)
i∈I

where the inequality in (5.7) holds since Ft stochastically dominates Ft andR(Q, ·) is an increasing function. It follows that condition (i) of Definition 4.3
holds. By a similar argument, condition (ii) also holds. Hence the parameterchange from t to t is an overall positive shock.
If the parameter change from t to t is a positive shock, then it is easy tosee from (5.3) and (5.5) that Ft stochastically dominates Ft . However, there
are many other ways in which Ft stochastically dominates Ft . Figure 1 showsa simple example. In this example, the parameter change from t to t is not
81.0
Cumulative Prob.

Parameter Value

1.0
0.8
0.6

ti

0.4

ti

0.2

0.8
0.6
0.4

F-t

0.2

F-t

0.0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Parameter Valuei

Figure 1: The parameter change from t to t is not a monotone shock (leftpanel), but Ft stochastically dominates Ft (right panel).
a monotone shock, but Ft stochastically dominates Ft by (5.5). Thus theparameter change here is an overall positive shock by Theorem 5.1 if bothR(Q, τ ) and R(Q, τ ) are increasing in τ .
There are well known sufficient conditions for both R(Q, τ ) and R(Q, τ )to be increasing or decreasing; see Milgrom and Shannon (1994, Theorem 4),
Topkis (1998, Theorem 2.8.3), Vives (1999, P. 35), Amir (2005, Theorems
1, 2), and Roy and Sabarwal (2010, Theorem 2). Any of the conditions canbe combined with Theorem 5.1 to replace the assumption that both R(Q, τ )and R(Q, τ ) are increasing or decreasing. Here we state a simple result using
the well-known conditions shown in Amir (2005, Lemma 1, Theorems 1, 2).
Corollary 5.1. Let t, t ∈ T . Suppose that Ft stochastically dominates Ft .
Suppose that T ⊂ R and the upper and lower boundaries of X (τ ) are increasing (resp. decreasing) functions of τ ∈ T . Suppose further that for eachQ ∈ Ω, π(k, Q, τ ) is twice continuously differentiable in (k, τ ) ∈ K × Tand ∂ 2 π(k, Q, τ )/∂k∂τ ≥ 0 (resp. ≤ 0) for all (k, τ ) ∈ K × T . Then theparameter change from t to t is an overall positive (resp. negative) shock.

5.2

Mean-Preserving Spreads

Following Acemoglu and Jensen (2015), we say that Ft is a mean-preserving
spread of Ft if (5.4) holds for any Borel-measurable convex function φ : T →R.9 Rothschild and Stiglitz (1970, p. 231) and Machina and Pratt (1997,9

Our approach differs from that of Acemoglu and Jensen (2015) in that while they
consider positive shocks induced by applying a mean-preserving spread to the stationary
distribution of each player’s idiosyncratic shock, we consider non-monotone shocks induced
by applying a mean-preserving spread to the entire distribution of parameters.

9

Theorem 3) show that in case n = 1, Ft is a mean-preserving spread of Ft if
ZZ
Ft (z)dz = Ft (z)dz,(5.8)
and if there exists z̃ ∈ R such that(
≤ 0 if z ≤ z̃,Ft (z) − Ft (z)≥ 0 if z > z̃.

(5.9)

The following result provides a sufficient condition for an overall monotone shock based on mean-preserving spreads.
Theorem 5.2. Let t, t ∈ T . Suppose that Ft is a mean-preserving spreadof Ft . Suppose that both R(Q, τ ) and R(Q, τ ) are Borel-measurable convex(resp. concave) functions of τ ∈ T for each Q ∈ Ω. Then the parameterchange from t to t is an overall positive (resp. negative) shock.
Proof. The proof is essentially the same as that of Theorem 5.1 except thatthe inequality in (5.7) holds since Ft is a mean-preserving spread of Ft andR(Q, τ ) is convex in τ .
Figure 2 shows a simple example of a mean-preserving spread. As can beseen in the left panel, the parameter change from t to t is not a monotone
shock. However, it is a mean-preserving spread by (5.8) and (5.9), as can beseen in the right panel. Thus the parameter change here is an overall positive
shock by Theorem 5.2 if both R(Q, τ ) and R(Q, τ ) are convex in τ ∈ T .
Sufficient conditions for R(Q, τ ) or R(Q, τ ) to be convex or concave areestablished by Jensen (2015). Here we state a simple result using his conditions (Lemmas 1, 3, Theorem 2, Corollary 2).
Corollary 5.2. Let t, t ∈ T . Suppose that Ft is a mean-preserving spread of
Ft . Suppose that the upper and lower boundaries of X (τ ) are convex (resp.
concave), continuous functions of τ ∈ T . For each (Q, τ ) ∈ Ω × T , supposethat π(k, Q, τ ) is strictly quasi-concave and continuously differentiable in k ∈K, and that R(Q, τ ) < max X (τ ) (resp. R(Q, τ ) > min X (τ )). For eachQ ∈ Ω, suppose that ∂π(k, Q, τ )/∂k is quasi-convex (resp. quasi-concave) in(k, τ ) ∈ K × T . Then the parameter change from t to t is an overall positive(resp. negative) shock.

10

1.0
Cumulative Prob.

Parameter Value

1.0
0.8
0.6

ti

0.4

ti

0.2

0.8
0.6
0.4

F-t

0.2

F-t

0.0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Parameter Valuei

Figure 2: The parameter change from t to t is not a monotone shock (leftpanel), but Ft is a mean-preserving spread of Ft (right panel).

6

Applications

Recall the model of Section 2. Let ti = ei + si for i ∈ I. The first-ordercondition for the maximization problem (2.1)-(2.2) is written as(≤ 1 if xi = 0,0u (wxi + ti )w(6.1)= 1 if xi > 0.
Let x(w, ti ) denote the solution for xi as a function of w and tRi .
To see that this model is a large aggregative game, let Q = i∈I x(w, ti )di.
Then (2.4) implies that w = D−1 (Q). Let τ > 0 and T = [0, τ ]. The model
here is a special case of the game in Section 5 with
π(k, Q, τ ) = u(D−1 (Q)k + τ ) − k,X (τ ) = K = Ω = [0, k],(6.2)

where k is a constant satisfying k > max(w,τ )∈[0,w]×T x(w, τ ).
First suppose that si = 0 and ti = ei for all i ∈ I. Let ti = ei andti = ei be as in Figure 1. Then the parameter change from t to t is not a
monotone shock. However, it is straightforward to verify the conditions ofCorollary 5.1 to conclude that the parameter change is an overall negative
shock. Hence the smallest and largest equilibrium values of aggregate labor
supply decrease in response to this parameter change, which implies that thesmallest and largest equilibrium values of the wage rate increase.
Now suppose that ei = e and ti = si for all i ∈ I for some e > 0.
Let ti = e + si and ti = e + si be as in Figure 2. Then Ft is a meanpreserving spread of Ft . The parameter change from t to t widens income
inequality, and is not a monotone shock. However, it is straightforward toverify the conditions of Corollary 5.2 to conclude that the parameter change isan overall positive shock. Hence the smallest and largest equilibrium values
11

0.8

x(w,ti )

0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

ti

Figure 3: Individual labor supply as a function of ti with u(c) = c0.7 /0.7 andw = 0.9.
of aggregate labor supply increase in response to this parameter change,
which implies that the smallest and largest equilibrium values of the wage
rate decrease.
We can confirm the above results by solving (6.1) for xi = x(w, ti ):(max {[u0−1 (1/w) − ti ]/w, 0} if w > 0,x(w, ti ) =(6.3)
0
if w = 0.
This function is decreasing, piecewise linear, and convex in ti ; see Figure 3.
Hence the above comparative statics results directly follow from Theorems
5.1 and 5.2.

7

Concluding Comments

Acemoglu and Jensen (2010) established that the smallest and largest equilibrium aggregates of a large aggregative game both increase in response toa positive shock, which is a positive parameter change that positively affects
each player’s action for each value of the aggregate variable. In this paper wehave extended their result to an overall positive shock, to which individuals’
reactions may be mixed but the overall aggregate response is positive foreach value of the aggregate variable. We have provided sufficient conditions
for an overall positive shock in terms of distributional changes in parameters
based on stochastic dominance and mean-preserving spreads.
Although we have considered only one-dimensional distributions in our
figures and example, Theorems 5.1 and 5.2 do not require one-dimensional
distributions. Therefore more interesting or complex policy changes can beanalyzed using our results.
12Appendix A

Proof of Theorem 3.1

Since t is fixed here, we suppress the dependence of πi , Ri , and G on t and ti
throughout the proof. For i ∈ I, define µ(i, Q) = Ri (Q)(= Ri (Q, t)).
Lemma A.1. For any Q ∈ Ω, the correspondence µ(·, Q) from i ∈ I to
Ri (Q) ⊂ K has nonempty compact values, and admits a measurable selection.
Proof. Fix Q ∈ Ω. We show this lemma by applying the measurable maximum theorem (Aliprantis and Border 2006, p. 605) to player i’s maximization
problem with i ∈ I taken as a parameter:max πi (xi , Q).

xi ∈Xi(A.1)

Let φ denote the correspondence i ∈ I 7→ Xi ⊂ K. By Assumption 3.1,
φ has nonempty compact values. Note that the set {i ∈ I : Xi ∩ U 6= ∅} inAssumption 3.4 is the lower inverse of U under φ; see Aliprantis and Border(2006, p. 557). Thus Assumption 3.4 means that φ is weakly measurable;
see Aliprantis and Border (2006, p. 592). Assumption 3.3 means that the
mapping (i, k) ∈ I × K 7→ πi (k, Q) ∈ R is a Carathéodory function.
It follows that the measurable maximum theorem applies to (A.1); thus
the correspondence i 7→ µ(i, Q) has nonempty compact values, and admits ameasurable selection.
Lemma A.2. For each Q ∈ Ω, the set G(Q) is nonempty and convex, where
G is defined in (4.2).
Proof. Fix Q ∈ Ω. Since µ(·, Q) admits a measurable selection by Lemma
A.1, G(Q) is nonempty. To see that G(Q) is convex, note from Aumann(1965, Theorem 1) that the setZ
xi di : x ∈ X , ∀i ∈ I, xi ∈ µ(i, Q)
(A.2)
i∈I

is convex. The image of this convex set under H is convex since H is continuous and real-valued.10 Recalling (4.2) we see that G(Q) is convex.
Lemma A.3. The correspondence G(·) has compact values and a closed (infact, compact) graph.
10

The image may not be convex if the range of H is not one-dimensional.

13

Proof. Fix i ∈ I. Note that Xi does not depend on Q; thus the correspondence Q 7→ Xi is continuous in a trivial way. By Assumption 3.3, πi (k, Q)
is continuous in (k, Q) ∈ Xi × Ω. Hence by the Berge maximum theorem(Aliprantis and Border, 2006, p. 570) and the closed graph theorem (Aliprantis and Border, 2006, p. 561), the correspondence µ(i, ·) has a closed graph.
In other words,
F (i) ≡ {(k, Q) ∈ Xi × Ω : k ∈ µ(i, Q)} is closed.

(A.3)

Let G be the graph of the correspondence G(·):
G = {(Q, S) ∈ Ω × Ω : S ∈ G(Q)}.

(A.4)

To verify that G is closed, it suffices to show that G contains the limit ofany sequence {(Qj , S j )}j∈N in G that converges in Ω2 . For this purpose, let{(Qj , S j )}j∈N be a sequence in G that converges to some (Q∗ , S ∗ ) ∈ Ω2 .
For each j ∈ N we have S j ∈ G(Qj ); thusR there exists a measurable selection xj ∈ X of µ(·, Qj ) such that S j = H( i∈I xji di). Taking a subsequenceRof {S j }, we can assume that ξ j ≡ i∈I xji di converges to some ξ ∗ ∈ K as
j ↑ ∞. Since H is continuous by Assumption 3.2, it follows that S ∗ = H(ξ ∗ ).
Recalling that xj is a selection of µ(·, Qj ) for all j ∈ N, we see from
(A.3) that (xji , Qj ) ∈ F (i) for all i ∈ I and j ∈ N. Since F (i) is closed, any
convergent subsequence of {(xji , Qj )}j∈N converges in F (i). Since Qj → Q∗
as j ↑ ∞, it follows that any limit point yi of {xji }j∈N satisfies (yi , Q∗ ) ∈ F (i);i.e., yi ∈ µ(i, Q∗ ). In addition, |xji | ≤ max{|k| : k ∈ K} for all i ∈ I andj ∈ N. Hence by Aumann (1976,
there exists x∗ ∈RX such that
R Lemma),∗∗∗∗xi ∈ µ(i, Q ) for all i ∈ I and i∈I xi di = ξ . Thus S ∗ = H( i∈I x∗i di), andS ∗ ∈ G(Q∗ ); i.e., (Q∗ , S ∗ ) ∈ G . It follows that G is closed.
Since G ⊂ Ω × Ω, which is compact, it follows that G is compact. Hence
G(·) has compact values.
Now by Kakutani’s fixed point theorem (Aliprantis and Border, 2006, p.
583), Assumption 3.2, and Lemmas A.2 and A.3, the set of fixed points ofthe correspondence G(·) is nonempty and compact. Let Q ∈ Ω be a fixed
point. Then
there exists a measurable selection x ∈ X of µ(·, Q) such thatR
Q = H( i∈I xi di); i.e., x is an equilibrium. Hence an equilibrium exists.
The preceding argument shows that any fixed point of G(·) is an equilibrium aggregate. Since the set of fixed points of G(·) is nonempty andcompact, it follows that the set of equilibrium aggregates is also nonempty
and compact; thus the smallest and largest equilibrium aggregates exist.
14Appendix B

Proof of Theorem 4.1

Since the correspondence Q 7→ G(Q, t) has compact values by Lemma A.3,
the minimum and the maximum of G(Q, t) exist for each Q ∈ Ω. DefineG(Q, t) = min G(Q, t) G(Q, t) = max G(Q, t). Since G(·, t) has convex values
by Lemma A.2, we have∀Q ∈ Ω,

G(Q, t) = [G(Q, t), G(Q, t)].

(B.1)

By Lemma A.3, G(·, t) has a compact graph. Hence it is easy to see that
G(·, t) is “continuous but for upward jumps”; see Milgrom and Roberts (1994,
p. 447). To conclude both inequalities in (4.1) from Milgrom and Roberts(1994, Corollary 2), it remains to show that for all Q ∈ Ω we haveG(Q, t) ≤ G(Q, t),G(Q, t) ≤ G(Q, t).

(B.2)

To see the first inequality in (B.2), let r = G(Q, t). By Definition 4.3(ii),there exists r ∈ G(Q, t) with r ≤ r. Since G(Q, t) ≤ r, the desired inequality
follows. The second inequality in (B.2) can be verified in a similar way.

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