DP2017-34

Robust Comparative Statics for
Non-Monotone Shocks in Large
Aggregative Games
Carmen CAMACHO
Takashi KAMIHIGASHIÇağrı SAĞLAM
December 15,2017

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

Takashi Kamihigashi‡Çağrı Sağlam§

December 4, 2017

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 as well as Editor Xavier Vives and three anonymous referees fortheir helpful comments and suggestions. Financial support from the Japan Society for the
Promotion of Science (KAKENHI No. 15H05729) 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. Rather surprisingly, in such games, one can obtain a “robust comparative statics” result without considering the interaction between the aggregate
variable and individuals’ actions. In particular, Acemoglu and Jensen (2010)
defined a positive shock as a positive parameter change that positively affects each individual’s action for each value of the aggregate variable. Thenthey showed that both the smallest and the largest equilibrium values of theaggregate 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.1 These conditions enable one to deal with various
policy changes, including ones that involve a redistribution of income.
This paper is not the first to study comparative statics for distributionalchanges. 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,
1

The concept of overall positive shocks is related not only to that of positive shocks but
also to Acemoglu and Jensen’s (2013) concept of “shocks that hit the aggregator,” whichwere defined as parameter changes that directly affect the “aggregator” positively alongwith additional restrictions. Such parameter changes are not considered in this paper, butthey can easily be incorporated by slightly extending our framework.

1but their analysis was restricted to positive shocks in the above sense. Jensen(2016) and Nocetti (2016) 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 in large aggregative games.2
Before showing our robust comparative statics results, we establish theexistence of the smallest and the largest equilibrium values of the aggregate
variable. This result is closely related to the literature on the existence ofa Nash equilibrium for games with a continuum of players. The seminal
result in this literature is Schmeidler’s (1973) existence theorem. Mas-Colell(1984) reformulated Schmeidler’s model and equilibrium concept in terms of
distributions rather than measurable functions, offering an elegant approachto the existence problem. In this paper, while we use measurable functions
to obtain our existence result, we consider distributions to develop sufficientconditions for robust comparative statics. Mas-Colell’s (1984) distributional
approach was extended by Jovanovic and Rosenthal (1988) to sequential
games.
Rath (1992) provided a simple proof of Schmeidler’s (1973) existence
theorem, which was extended by Balder (1995). Although the existence ofan equilibrium in this paper follows from one of his results, the existenceof the smallest and the largest equilibrium values of the aggregate variable
does not directly follow from the existence results available in the literature,including more recent results (e.g., Khan et al., 1997; Khan and Sun, 2002;
Carmona and Podczeck, 2009). The existence of extremal equilibria wereshown by Vives (1990), Van Zandt and Vives (2007), and Balbus et al. (2015)
for different settings.
The rest of the paper is organized as follows. In Section 2 we provide
a simple motivating example of income redistribution and aggregate labor
supply. In Section 3 we present our general framework along with basic assumptions, and show the existence of the smallest and the largest equilibrium
values of the aggregate variable. In Section 4 we formally define overall positive shocks. We also introduce a more general definition of “overall monotone
shocks.” We then present our general robust comparative statics result. In
Section 5 we provide sufficient conditions for an overall monotone shock interms of distributional changes in parameters based on first-order stochastic
2

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

2

dominance and mean-preserving spreads. In Section 6 we apply our results
to the example of income redistribution.

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 upperbound 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
D(w) =
xi di.
(2.4)
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 (2010). However, one may still ask, for example, how does a policy change that widens income inequality affect aggregate
labor supply and the wage rate?
This question cannot be answered using standard methods such as theimplicit function theorem if the policy change in question is a discrete jumpfrom one policy to another. If one insists on applying the implicit function
3

theorem, then one needs to introduce a policy parameter that affects income
distribution in a differentiable way, and find a set of equations that characterize aggregate labor supply and the wage rate. Even then, one typicallyneeds to assume the existence of a unique equilibrium and the assumptionsof the implicit function theorem.
It turns out that, using our results, one can answer the above and otherquestions in a “robust” way without introducing these extra assumptions.

3

Large Aggregative Games

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 assumptions made in this section are maintained throughout the paper.
Assumption 3.1. For each i ∈ I, Xi is nonempty and compact. Thereexists a compact convex 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.3 Let H be a function from K to a
subset Ω of R. We define G : X → Ω, called the aggregator, byZ
xi di .
(3.1)
G(x) = Hi∈I

Assumption 3.2. The set Ω ⊂ R is compact and convex, and H : K → Ωis continuous.4
Given x ∈ X and i ∈ I, player i’s payoff takes the form πi (xi , G(x), ti ),where ti is player Q
i’s parameter. Let Ti be the underlying space for ti ; i.e.,ti ∈ Ti . Let T ⊂ i∈I Ti . We regard T as a set of well-behaved parameter
profiles; for example, T can be a set of measurable functions from I to R.
We only consider parameter profiles t in T .
3Unless otherwise specified, measurability means Lebesgue measurability.
Given the assumptions on H and K, the properties of Ω here can be assumed without
loss of generality.
44Assumption 3.3. For each i ∈ I, player i’s payoff function πi maps each(k, Q, τ ) ∈ K × Ω × Ti into R.5 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 since each player’s payoff is affected by otherplayers’ actions only through the aggregate G(x). Accordingly, each player
i’s best response correspondence depends only on Q = G(x) and ti :Ri (Q, ti ) = arg max πi (xi , Q, ti ).

(3.2)

xi ∈Xi

The following assumption ensures that given any Q ∈ Ω, one can find ameasurable action profile x ∈ X such that xi ∈ Ri (Q, ti ) for all i ∈ I.
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 the game is an action profile x ∈ X such that xi ∈ Ri (G(x), ti )for all i ∈ I. We define an equilibrium aggregate as Q ∈ Ω such that Q = G(x)for some equilibrium x ∈ X . The following is a useful observation.
Remark 3.1. Given t ∈ T , Q ∈ Ω is an equilibrium aggregate if and only if
Q ∈ G(Q, t), where
G(Q, t) = {G(x) : x ∈ X , ∀i ∈ I, xi ∈ Ri (Q, ti )} .

(3.3)

For t ∈ T , define Q(t) and Q(t) as the smallest and largest equilibrium
aggregates, respectively, provided that they exist.
Theorem 3.1. For any t ∈ T , the set of equilibrium aggregates is nonempty
and compact. Therefore, both Q(t) and Q(t) exist.
Proof. See Appendix A.1.
Our primary concern here is not the existence of an equilibrium but that ofQ(t) and Q(t). Although the existence of an equilibrium for our model follows
5

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

from Theorem 3.4.1 in Balder (1995) under more general assumptions,6 thecompactness of the set of equilibrium aggregates does not directly follow
from his result or other existence results in the literature, as mentioned in
the introduction.
Theorem 3.1 differs from Theorem 1 in Acemoglu and Jensen (2010) inthat we assume a continuum of player types rather than a finite number ofplayer types.7 But our proof follows the basic strategy of their proof.

4

Overall Monotone Shocks

By a parameter change, we mean a change in t ∈ T from one parameter
profile to another. We fix t, t ∈ T in Sections 4 and 5.
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) H(·)
is an increasing function,8 (c) t ≺ t, and (d) for each Q ∈ Ω and i ∈ I, thefollowing properties hold:(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).

6

(4.1)

In particular, the continuity requirement in Assumption 3.3 can be relaxed as follows:for each t ∈ T , πi (·, ·, ti ) is upper semicontinuous on K × Ω, and πi (k, ·, ti ) is continuouson Ω for each k ∈ K. Furthermore, the aggregator G can be a multidimensional function
in a specific way; see Balder (1995, Assumption 3.4.2).
7
Acemoglu and Jensen (2015) allow for a continuum of player types, which can be acontinuum of random variables, by using the Pettis integral in (3.1).
8
In this paper, “increasing” means “nondecreasing,” and “decreasing” means “nonincreasing.”

6

The following definitions allow us to show that the above inequalities holdfor a substantially larger class of parameter changes.
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 is an overall positive shock. We areready to state our general result on robust comparative statics:
Theorem 4.1. 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.
Proof. See Appendix A.2.
The proof of this result is similar to that of Theorem 2 in Acemoglu andJensen (2010). The latter result is immediate from Theorem 4.1 under ourassumptions since a positive shock is an overall positive shock. The dynamic
version of their result established by Acemoglu and Jensen (2015, Theorem5) can also be extended to overall monotone shocks in a similar way.

5

Sufficient Conditions

In this section we provide sufficient conditions for overall monotone shocks byassuming that players differ only in their parameters ti . To be more specific,
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 ,7

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 Camacho et al., 2016, Lemma A.1). 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.
For the rest of this section, we take t, t ∈ T as given.

5.1

First-Order Stochastic Dominance

Given two 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 function φ : Rn → R, where Rn is equippedwith the usual partial order ≤. As is well known (e.g., Müller and Stoyan,
2002, Section 1), 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. Suppose that Ft stochastically dominates Ft , and that bothR(Q, τ ) and R(Q, τ ) are increasing (resp. decreasing) Borel functions of τ ∈T for each Q ∈ Ω. Then the parameter change from t to t is an overall
positive (resp. negative) shock.
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).
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
R(Q, z)dFt (z) = H
R(Q, ti )di ∈ G(Q, t),
(5.7)
≤H
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 nota 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).9 Any of those conditions can
9These results originate from games with strategic complementarities, which were popularized by Vives (1990) and Milgrom and Roberts (1990). Other related studies include
Roy and Sabarwal (2008), Van Zandt and Vives (2007), and Balbus et al. (2015).

9be combined with Theorem 5.1. Here we state a simple result based on Amir
(2005, Lemma 1, Theorems 1, 2).
Corollary 5.1. Assume the following: (i) Ft stochastically dominates Ft ; (ii)T ⊂ R; (iii) the upper and lower boundaries of X (τ ) are increasing (resp.
decreasing) functions of τ ∈ T ; and (iv) for each Q ∈ Ω, π(k, Q, τ ) is twice
continuously differentiable in (k, τ ) ∈ K × T and ∂ 2 π(k, Q, τ )/∂k∂τ ≥ 0(resp. ≤ 0) for all (k, τ ) ∈ K × T . Then the parameter change from t to t isan 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 convex function φ : T → R.10
Rothschild and Stiglitz (1970, p. 231) and Machina and Pratt (1997, Theorem3) 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. Suppose that Ft is a mean-preserving spread of Ft , and thatboth R(Q, τ ) and R(Q, τ ) are Borel convex (resp. concave) functions of τ ∈ T
for each Q ∈ Ω. Then the parameter change 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 .
10

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.

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).
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 (2016). The following result is based on Jensen (2016,
Lemmas 1, 2, Theorem 2, Corollary 2).
Corollary 5.2. Assume the following: (i) Ft is a mean-preserving spread of
Ft ; (ii) the upper and lower boundaries of X (τ ) are convex (resp. concave)
continuous functions of τ ∈ T ; (iii) for each (Q, τ ) ∈ Ω × T , π(k, Q, τ ) isstrictly quasi-concave and continuously differentiable in k ∈ K; (iv) R(Q, τ ) <max X (τ ) (resp. R(Q, τ ) > min X (τ )); and (v) for each Q ∈ Ω, ∂π(k, Q, τ )/∂kis quasi-convex (resp. quasi-concave) in (k, τ ) ∈ K × T . Then the parameterchange from t to t is an overall positive (resp. negative) shock.

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,
u0 (wxi + ti )w(6.1)= 1 if xi > 0.
of w and ti . Let Q =RLet x(w, ti ) denote the solution for xi as a function
−1
x(w,
t
)di.
Then
(2.4)
implies
that
w
=
D
(Q).
Let τ > 0 and T =
i
i∈I
11

[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 mean-preserving
spread of Ft . Thus the parameter change from t to t widens income inequality,
and is not a monotone shock. However, it is straightforward to verify theconditions of Corollary 5.2 to conclude that the parameter change is an overall
positive shock. Hence the smallest and largest equilibrium values 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.
The above comparative statics results can also be confirmed 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.
Thus the above results directly follow from Theorems 5.1 and 5.2 under (6.3).

Appendix A
A.1

Proofs

Proof of Theorem 3.1

Fix t ∈ T . The existence of an equilibrium follows from Balder (1995, Theorem 3.4.1); thus the set of equilibrium aggregates is nonempty. Recalling
Remark 3.1, it remains to verify that the set of fixed points of G(·, t) is compact. The following result is shown in Camacho et al. (2016, Lemma A.3).
Lemma A.1. The correspondence Q 7→ G(Q, t) has a compact graph.
12

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.
By this result and Lemma 17.51 in Aliprantis and Border (2006), the setof fixed points of G(·, t) is compact, as desired.

A.2

Proof of Theorem 4.1

The following result is shown in Camacho et al. (2016, Lemma A.2).
Lemma A.2. The correspondence Q 7→ G(Q, t) has nonempty convex values.
Let t ∈ T and Q ∈ Ω. Let G(Q, t) = min G(Q, t) and G(Q, t) =max G(Q, t). Both exist by Lemma A.1, and G(Q, t) = [G(Q, t), G(Q, t)]
by Lemma A.2. This together with Lemma A.1 implies that G(·, t) is “continuous but for upward jumps” in the sense of Milgrom and Roberts (1994,
p. 447). Suppose that the parameter change from t to t is an overall positive
shock. Then Definition 4.3 implies that G(Q, t) ≤ G(Q, t) and G(Q, t) ≤G(Q, t). Thus both inequalities in (4.1) follow from Milgrom and Roberts(1994, Corollary 2). If the parameter change is an overall negative shock,then the reverse inequalities hold similarly.

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