OSIPP Discussion Paper : DP-2024-E-002

An implementation of
constrained efficient allocations
in hidden information economies
May 24, 2024
Yuya Wakabayashi1
JSPS Research Fellow (DC2), Graduate School of Economics, Osaka University

Abstract: We examine a simple hidden information economy with a single agent, multiple
identical firms, and two goods. The agent’s states are unknown to the firms. The game
unfolds in two stages: in the first stage, firms offer a menu of allocations, and in the second
stage, the agent selects her preferred allocation from the offered allocations. We show that,in almost all cases, the constrained efficient allocation cannot be implemented by asubgame perfect equilibrium in the game.
JEL classification: D47, D82, D86
Keywords: Asymmetric information, Constrained efficiency, Nash implementation

1

Email: u690485a@ecs.osaka-u.ac.jp

An implementation of
constrained eﬃcient allocations
in hidden information economies∗
Yuya Wakabayashi†
May 24, 2024

Abstract
We examine a simple hidden information economy with a single agent, multiple
identical firms, and two goods. The agent’s states are unknown to the firms. The
game unfolds in two stages: in the first stage, firms oﬀer a menu of allocations, andin the second stage, the agent selects her preferred allocation from the oﬀered allocations. We show that, in almost all cases, the constrained eﬃcient allocation cannot be
implemented by a subgame perfect equilibrium in the game.

JEL classification: D47, D82, D86.
Keywords: Asymmetric information, Constrained eﬃciency, Nash implementation.

∗

We are greateful to Takuro Yamashita for helpful comments. We also thank to financial support fromthe Japan Society for the Promotion of Science (23KJ1453).
†
JSPS Research Fellow (DC2), Graduate School of Economics, Osaka University.
Email:
u690485a@ecs.osaka-u.ac.jp1

1

Introduction

In an economy with asymmetric information, it is diﬃcult to achieve an eﬃcient allocation
as an equilibrium because agents can misreport their information. Therefore, much of theliterature focuses on the constrained eﬃcient allocation, the second-best allocation among
incentive compatible allocations, to explore its attainability within an equilibrium framework.
Prescott and Townsend (1984) is one of the pioneering studies to investigate competitive
equilibria in economies with asymmetric information. Their study has inspired numerous researchers to explore various economies with asymmetric information. Rustichini and Siconolfi(2012) consider economies with observable types, where agents’ types are observable but theiraction or state are unobservable. In this situation, they demonstrate that some constrained
eﬃcient allocation cannot be achieved as a competitive equilibrium, that is, there is no pricesupporting it.
In this paper, we focus on a simple hidden information economy, which is a special
case of economies with observable types as discussed by Rustichini and Siconolfi (2012).
Our approach diverges in that we examine the feasibility of implementing a constrained
eﬃcient allocation through a Nash equilibrium in a game structure, rather than a competitive
equilibrium. We demonstrate that, in almost all cases, achieving the constrained eﬃcient
allocation through a subgame perfect equilibrium in a simple game is impossible. This resultaligns with the notion that there exists no price that can support a constrained eﬃcient
allocation in a general equilibrium framework.

1.1

Related literature

We briefly review the related literature and discuss about our contribution. In the studies
about economies with asymmetric information, two main approaches exist for reaching theconstrained eﬃcient allocation. The first is the Walrasian approach, which seeks to identify
prices that support the constrained eﬃcient allocation within a competitive market. The second is the Nash approach, aiming to find a Nash equilibrium that implements the constrained
eﬃcient allocation within a game structure.
A seminal work employing the Walrasian approach is by Prescott and Townsend (1984),
who examine two crucial economies: one with moral hazard and the other with adverse
selection. They demonstrate that the welfare theorems for the constrained eﬃcient allocation
hold in an economy with moral hazard, but may fail in one with adverse selection.
Rothschild and Stiglitz (1976) are pioneers in applying the Nash approach. They showthat a Cournot-Nash equilibrium may not exist in an economy with adverse selection. Conversely, in an economy with moral hazard, Bennardo and Chiappori (2003) demonstrate thatthe constrained eﬃcient allocation can be implemented by a Nash equilibrium in a simple
game.
Given the limitations of the standard approaches in economies with adverse selection,
many researchers have explored additional conditions and solution concepts to achieve theconstrained eﬃcient allocation (Walrasian approach: Bisin and Gottardi 2006; Citanna andSiconolfi 2016; Azevedo and Gottlieb 2017; Nash approach: Netzer and Scheuer 2014; Di2

asakos and Koufopoulos 2018; Dosis 2018, 2019, 2022). This contrasts with the relative
success in economies with moral hazard.
However, Jerez (2005) points out a flaw in Prescott and Twonsend’s (1986) framework,
which imposes incentive compatible constraints solely on consumers, rather than firms. He
sees this restriction as a potential conceptual issue, questioning how these constraints are
enforced in a decentralized economy. He argues for viewing the incentive compatible constraints as restrictions on the set of allocations that firms can oﬀer to consumers. Thus,
investigating the attainability of the constrained eﬃcient allocation in economies with moral
hazard attached with this natural restriction is also necessary.
In situations where incentive compatible constraints are placed on firms, Rustichini andSiconolfi (2012) take the Walrasian approach. They demonstrate that competitive equilibrium may not exist and the Second Welfare Theorem may fail in an economy with observabletypes, including one with moral hazard.
The contribution of our paper is to take the Nash approach in a special case of economies
with observable types. We show that for almost all cases, we cannot achieve a constrained eﬃcient allocation as a Nash equilibrium in a simple game. Our impossibility result corresponds
to that there is no price that support a constrained eﬃcient allocation.1

2

Model

We consider a simple hidden information economy. There exist one agent, m (≥ 2) identicalfirms, and two objects. Let J = {1, . . . , m} be the set of firms, and L = {1, 2} be the setof objects. The agent has two states, g(= good) and b(= bad). Let S = {g, b} be the set ofstates. Each state s ∈ S occurs with the probability qs ∈ (0, 1) such that qg + qb = 1.
Each firm contracts with the agent before the agent knows her state. A (state-dependent)
allocation is denoted by a vector x = (x1g , x1b , x2g , x2b ) ∈ R4 , where xls is the net amount of
the good l ∈ L when the agent’s state is s ∈ S. The agent consumes allocations among theconsumptions set denoted by X ⊆ R4 with 0 ∈ X.
The firms know the agent’s preference, but they cannot observe the state of the agent.
The agent has a utility function u over X and the firms have a profit function π over X. We
assume that the agent and the firms share the same parameters (ηg1 , ηb1 , ηg2 , ηb2 ) ∈ (0, 1)4 suchthat ηg1 + ηg2 = 1 and ηb1 + ηb2 = 1. The parameter ηsl is the weight for the object l ∈ L when
the state is s ∈ S. By using these parameters, the agent’s utility of x ∈ X is denoted byu(x) = qg (ηg1 vg1 (x1g ) + ηg2 vg2 (x2g )) + qb (ηb1 vb1 (x1b ) + ηb2 vb2 (x2b )),= qg ηg1 vg1 (x1g ) + qb ηb1 vb1 (x1b ) + qg ηg2 vg2 (x2g ) + qb ηb2 vb2 (x2b ),where vsl : R → R is a continuous, strictly concave, strictly increasing function such thatvsl (0) = 0. The firms’ profit of x ∈ X is denoted by
π(x) = −{qg (ηg1 x1g + ηg2 x2g ) + qb (ηb1 x1b + ηb2 x2b )},= −(qg ηg1 x1g + qb ηb1 x1b + qg ηg2 x2g + qb ηb2 x2b ).

1

See Section 4 for more detailed discussion.

3

One of important economies in our model is an insurance economy. We consider a car
insurance. Assume ηg1 < ηb1 (or equivalently ηg2 > ηb2 ). Here, object 1 represents the amountof money the agent can obtain when he/she is involved in a traﬃc accident, while object 2
denotes the amount when he/she is not. In this context, ηs1 represents the probability of anagent getting into a traﬃc accident when their state is s. Additionally, ηg1 < ηb1 indicatesthat if the agent’s state is good, he/she can decrease the probability of being involved in a
traﬃc accident. Thus this situation reflects a car insurance economy.

2.1

Incentive compatibility and constrained eﬃciency

Let φ : {g, b} → {g, b} be a manipulation which the agent takes, and Φ be the set of
all manipulations. Given an allocation x ∈ X and a manipulation φ ∈ Φ, we denote amanipulated utility byuφ (x) = qg ηg1 vg1 (x1φ(g) ) + qb ηb1 vb1 (x1φ(b) ) + qg ηg2 vg2 (x2φ(g) ) + qb ηb2 vb2 (x2φ(b) ).
Similarly, denote a manipulated profit by
πφ (x) = qg ηg1 x1φ(g) + qb ηb1 x1φ(b) + qg ηg2 x2φ(g) + qb ηb2 x2φ(b) .
Let φI ∈ Φ be the truth-telling manipulation, that is, φI (g) = g and φI (b) = b. Let φG ∈ Φ
be a manipulation such that she always reports the good state, that is, φG (g) = φG (b) = g.
We similarly define φB ∈ Φ by φB (g) = φB (b) = b. Let φE be a manipulation such that she
exchanges her states to report, that is, φE (g) = b and φE (b) = g. We can rewrite the set ofmanipulations by Φ = {φI , φG , φB , φE }.
An allocation x ∈ X is feasible if qg x1g + qb x1b ≤ 0 and qg x2g + qb x2b ≤ 0. Let X F ⊆ X
be the set of all feasible allocations. An allocation x ∈ X is incentive compatible if foreach φ ∈ Φ, u(x) ≥ uφ (x). Let X IC ⊆ X be the set of incentive compatible allocations. We
denote the set of all feasible and incentive compatible allocations by X F IC = X F ∩ X IC .
Note that, since 0 is feasible and incentive compatible, X F IC is non-empty.
An allocation x ∈ X is constrained eﬃcient if (i) x ∈ X F IC , and (ii) there is nox′ ∈ X F IC such that u(x′ ) > u(x). Let X CE ⊆ X be the set of constrained eﬃcient
allocations.

2.2

Subgame perfect implementability

We consider the following game form:
Stage 1: Each firm j ∈ J oﬀers a set of allocations! Dj ⊆ X.
Stage 2: The agent chooses an allocation x ∈ j∈J Dj and a manipulation φ ∈ Φ.

Let D = 2X be the firms’ strategy space. Let f : D → X × Φ be a strategy of the agent, andF be the set of all the agent’s strategies. Given D ∈ D and f ∈ F, let xf (D) and φf (D) be
an allocation and a manipulation obtained by the strategies.

4Given D ∈ D, let 1D : X → {0, 1} be a indicator function such that for each x ∈ X,
1D (x) = 1 if x ∈ D and 1D (x) = 0 if x ∈/ D. A profile of strategies ((Dj )j∈J , f ) is a subgame
perfect equilibrium if(i) for each D ∈ D,
(xf (D), φf (D)) ∈ arg max uφ (x),(x,φ)∈D×Φ(ii) for each j ∈ J and Dj′ ∈ D,
1Dj′ (xf (D′ ))
1Dj (xf (D))
f""
π f (x (D)) ≥π f ′ (xf (D′ )),f (D)) φ (D)f (D ′ )) φ (D )′ (xf (D ′ )) +
1
(x
1
1
(x
D
D
D
k
k
k∈J
k̸=j
j

!
where D = ∪k∈J Dk and D′ = Dj′ ∪ ( k̸=j Dk ). The first condition is about the agent’s
optimality, and the second condition is about the firms’ optimality.
An allocation x ∈ X is subgame perfect implementable if there exists a subgame
perfect
equilibrium ((Dj )j∈J , f ) ∈ Dm × F such that (xf (D), φf (D)) = (x, φI ), where D =!
j∈J Dj .

3

ResultOur main result states that if the parameters are not symmetric, then any non-zero constrained eﬃcient allocation is not subgame perfect implementable.
Theorem 1. Assume that ηg1 ̸= ηb1 or ηg2 ̸= ηb2 . Then, for each x ∈ X CE \ {0}, x is not
subgame perfect implementable.
In the following, we undertake the proof of Theorem 1. Initially, we present some preliminary results. Subsequently, we analyze certain properties of the constrained eﬃcient
allocation. Finally, we proceed to establish our main theorem.

3.1

Preliminaries

The first result shows the necessary and suﬃcient condition for incentive compatibility.
Lemma 1. For each x ∈ X, x ∈ X IC if and only if u(x) ≥ uφG (x) and u(x) ≥ uφB (x).
Proof. Let x ∈ X. If x ∈ X IC , then u(x) ≥ uφG (x) and u(x) ≥ uφB (x) are obvious. Thus,
assume u(x) ≥ uφG (x) and u(x) ≥ uφB (x), and show u(x) ≥ uφE (x). By definition,uφG (x) + uφB (x)=(qg ηg1 vg1 (x1g ) + qb ηb1 vb1 (x1g ) + qg ηg2 vg2 (x2g ) + qb ηb2 vb2 (x2g ))+ (qg ηg1 vg1 (x1b ) + qb ηb1 vb1 (x1b ) + qg ηg2 vg2 (x2b ) + qb ηb2 vb2 (x2b ))= u(x) + uφE (x).
5Hence, by u(x) ≥ uφG (x) and u(x) ≥ uφB (x),u(x) + u(x) ≥ uφG (x) + uφB (x) = u(x) + uφE (x),which implies u(x) ≥ uφE (x). Therefore, x ∈ X CE .
Given x ∈ X, by a simple calculation,u(x) ≥ uφG (x)⇔qg ηg1 vg1 (x1g ) + qb ηb1 vb1 (x1b ) + qg ηg2 vg2 (x2g ) + qb ηb2 vb2 (x2b )≥ qg ηg1 vg1 (x1g ) + qb ηb1 vb1 (x1g ) + qg ηg2 vg2 (x2g ) + qb ηb2 vb2 (x2g )⇔ qb ηb1 vb1 (x1b ) + qb ηb2 vb2 (x2b ) ≥ qb ηb1 vb1 (x1g ) + qb ηb2 vb2 (x2g )⇔ ηb1 (vb1 (x1b ) − vb1 (x1g )) ≥ ηb2 (vb2 (x2g ) − vb2 (x2b )).

In the same way, we can show that u(x) ≥ uφB (x) if and only if ηg2 (vg2 (x2g ) − vg2 (x2b )) ≥
ηg1 (vg1 (x1b ) − vg1 (x1g )). Hence, we get the following result from Lemma 1.
Corollary 1. For each x ∈ X, x ∈ X IC if and only if(i) ηb1 (vb1 (x1b ) − vb1 (x1g )) ≥ ηb2 (vb2 (x2g ) − vb2 (x2b )),(ii) ηg2 (vg2 (x2g ) − vg2 (x2b )) ≥ ηg1 (vg1 (x1b ) − vg1 (x1g )).

The next result is about the relation between the amounts of the object 1 and object 2.
Lemma 2. For each x ∈ X IC , x1b ≥ x1g if and only if x2g ≥ x2b .
Proof. Let x ∈ X IC . If x1b ≥ x1g , then since vg1 is increasing, vg1 (x1b ) − vg1 (x1g ) ≥ 0, and so by
Corollary 1 (ii),ηg2 (vg2 (x2g ) − vg2 (x2b )) ≥ ηg1 (vg1 (x1b ) − vg1 (x1g )) ≥ 0.

Since vg2 is increasing, by the above inequality, x2g ≥ x2b . Similarly, if x2g ≥ x2b , then by
Corollary 1 (i), x1b ≥ x1g .

3.2

Properties

The first result states that for any non-zero constrained eﬃcient allocation, at least one
incentive compatible constraint holds with strict inequality.
Proposition 1. For each x ∈ X CE \ {0}, u(x) > uφG (x) or u(x) > uφB (x).
Proof. Let x ∈ X CE \ {0}. Suppose that u(x) = uφG (x) and u(x) = uφB (x). We showu(x) < 0 = u(0). This is a contradiction to constrained eﬃciency because 0 is feasible andincentive compatible.
Without loss of generality, assume x1b ≥ x1g . The converse case can be demonstrated
similarly. If x1b = x1g , then by Lemma 2, x2g = x2b , and so by feasibility, x1g = x1b = x2g = x2b = 0.
However, this contradicts x ̸= 0. Hence, x1b > x1g . By Lemma 2, x2g > x2b .
6

We show that ηg1 vg1 (x1g )+ηg2 vg2 (x2g ) < 0 and ηb1 vb1 (x1b )+ηb2 vb2 (x2b ) < 0. These two inequalities
imply u(x) < 0. Since we can prove each inequality in the same way, we only show ηg1 vg1 (x1g )+ηg2 vg2 (x2g ) < 0. By x1g < x1b and feasibility, qg x1g + qb x1g < qg x1g + qb x1b ≤ 0, and so x1g < 0.
Similarly, by x2b < x2g and feasibility, x2b < 0.
Case 1: x1g < x1b ≤ 0.
By Corollary 1 (ii),ηg2 vg2 (x2g ) < ηg2 (vg2 (x2g ) − vg2 (x2b ))
0>x2b

=

u(x)=uφB (x)ηg1 (vg1 (x1b ) − vg1 (x1g )) ≤ −ηg1 vg1 (x1g ),x1b ≤0and so ηg1 vg1 (x1g ) + ηg2 vg2 (x2g ) < 0.
Case 2: x1g < 0 < x1b .
If x2g ≤ 0, then by x1g < 0, ηg1 vg1 (x1g ) + ηg2 vg2 (x2g ) < 0. Hence, we assume x2g > 0. By x2b < 0,x2b < 0 < x2g . By strict concavity,
−x1g
x1b
1 1
·
v
(x
)
+
· v 1 (x1 ) < vg1
x1b − x1g g g
x1b − x1g g b

#

−x1g
x1b
1
·
x
+
· x1
x1b − x1g g x1b − x1g b$= vg1 (0) = 0.

(1)

By (1),− vg1 (x1g ) +
⇔
⇔

#

−x1g
x1b
1 1
·
v
(x
)
+
· v 1 (x1 )x1b − x1g g g
x1b − x1g g b

x1g
−x1g
1 1
·
v
(x
)
+
· v 1 (x1 ) < −vg1 (x1g )x1b − x1g g g
x1b − x1g g b$< −vg1 (x1g ) + 0−x1g· (vg1 (x1b ) − vg1 (x1g )) < −vg1 (x1g )x1b − x1g

−x1g
⇔ 1
· ηg1 (vg1 (x1b ) − vg1 (x1g )) < −ηg1 vg1 (x1g )1xb − xg(2)

In the same way, by x2b < 0 < x2g and strict concavity, we can showηg2 vg2 (x2g ) <x2g· ηg2 (vg2 (x2g ) − vg2 (x2b )).
22
xg − xb(3)

By feasibility,x2g
qb x2g − qb x2b
qb x1b − qb x1g
−x1g
≤
=
q
=
≤
.
b
x2g − x2b qg x2g +qb x2b ≤0 x2g − x2bx1b − x1g qg x1g +qb x1b ≤0 x1b − x1g(4)

By u(x) = uφB (x) and Corollary 1 (ii),ηg2 (vg2 (x2g ) − vg2 (x2b )) = ηg1 (vg1 (x1b ) − vg1 (x1g )).
7(5)

Hence,
x2g
−x1g
2 2 2
2 2 2
2 2
ηg vg (xg ) < 2· ηg (vg (xg ) − vg (xb )) ≤ 1· ηg1 (vg1 (x1b ) − vg1 (x1g )) < −ηg1 vg1 (x1g ),21(3) xg − xb(2)(4,5) xb − xgand so ηg1 vg1 (x1g ) + ηg2 vg2 (x2g ) < 0.
Similarly, we can show ηb1 vb1 (x1b ) + ηb2 vb2 (x2b ) < 0, and so u(x) < 0.
The next result states that the feasibility constraints are binding for any constrained
eﬃcient allocation.
Proposition 2. For each x ∈ X CE , qg x1g + qb x1b = 0 and qg x2g + qb x2b = 0.
Proof. Let x ∈ X CE . If x = 0, then the result is obvious, and so we assume x ∈ X CE \ {0}.
By x ∈ X CE \ {0} and Proposition 1, u(x) > uφG (x) or u(x) > uφB (x). We only show
qg x1g + qb x1b = 0 since we can prove qg x2g + qb x2b = 0 in the same way. To show the result,
suppose qg x1g + qb x1b < 0.
Without loss of generality, assume u(x) > uφG (x). We can derive a contradiction in thesame way for the case u(x) > uφB (x). By u(x) > uφG (x) and Corollary 1 (i), ηb1 (vb1 (x1b ) −vb1 (x1g )) > ηb2 (vb2 (x2g ) − vb2 (x2b )). Let ε > 0 be such that qg (x1g + ε) + qb x1b < 0 andηb1 (vb1 (x1b ) − vb1 (x1g )) > ηb1 (vb1 (x1b ) − vb1 (x1g + ε)) > ηb2 (vb2 (x2g ) − vb2 (x2b )).

(6)

By Corollary 1 (ii),ηg2 (vg2 (x2g ) − vg2 (x2b )) ≥ ηg1 (vg1 (x1b ) − vg1 (x1g )) > ηg1 (vg1 (x1b ) − vg1 (x1g + ε))(7)Hence, by (6), (7), and Corollary 1, (x1g +ε, x1b , x2g , x2b ) is incentive compatible. However, sinceu(x1g + ε, x1b , x2g , x2b ) > u(x) and (x1g + ε, x1b , x2g , x2b ) is feasible, this contradicts constrained
eﬃciency.

3.3

Proof of Theorem 1

Finally, we show Theorem 1.
Proof of Theorem 1. Without loss of generality, assume ηg1 > ηb1 . By ηg1 + ηg2 = 1 andηb1 + ηb2 = 1, ηg2 < ηb2 . Let x ∈ X CE \ {0}. If x1g = x1b , then by Lemma 2 and Proposition 2,
x1g = x1b = x2g = x2b = 0. However, this contradicts x ∈ X CE \ {0}. Hence, x1g ̸= x1b .
Case 1: x1g > x1b .
By Lemma 2, x2g < x2b . By Proposition 2, x1g > 0 > x1b and x2g < 0 < x2b . By ηg1 > ηb1 ,x1g > 0 > x1b and Proposition 2,
qg ηg1 x1g + qb ηb1 x1b > qg ηb1 x1g + qb ηb1 x1b = 0.

(8)

By ηg2 < ηb2 , x2g < 0 < x2b and Proposition 2,
qg ηg2 x2g + qb ηb2 x2b > qg ηg2 x2g + qb ηg2 x2b = 0.
8(9)By (8) and (9), π(x) < 0. If a firm j oﬀers {0}, then j’s profit becomes zero. Hence, x is not
subgame perfect implementable.
Case 2: x1g < x1b .
By Lemma 2, x2g > x2b . By Proposition 2, x1g < 0 < x1b and x2g > 0 > x2b . By ηg1 > ηb1 ,1
xg < 0 < x1b and Proposition 2,
qg ηg1 x1g + qb ηb1 x1b < qg ηg1 x1g + qb ηg1 x1b = 0.

(10)

By ηg2 < ηb2 , x2g > 0 > x2b and Proposition 2,
qg ηg2 x2g + qb ηb2 x2b < qg ηb2 x2g + qb ηb2 x2b = 0.

(11)

By (10) and (11), π(x) > 0. By x ∈ X CE \ {0} and Proposition 1, u(x) > uφG (x) oru(x) > uφB (x). Without loss of generality, assume u(x) > uφG (x). By Corollary 1 (i),ηb1 (vb1 (x1b )−vb1 (x1g )) > ηb2 (vb2 (x2g )−vb2 (x2b )). Let ε > 0 be such that π(x) > π(x1g +ε, x1b , x2g , x2b ) >1π(x) andmηb1 (vb1 (x1b ) − vb1 (x1g )) > ηb1 (vb1 (x1b ) − vb1 (x1g + ε)) > ηb2 (vb2 (x2g ) − vb2 (x2b )).

(12)

By Corollary 1 (ii),ηg2 (vg2 (x2g ) − vg2 (x2b )) ≥ ηg1 (vg1 (x1b ) − vg1 (x1g )) > ηg1 (vg1 (x1b ) − vg1 (x1g + ε)).

(13)

Let y = (x1g +ε, x1b , x2g , x2b ). By (12), (13) and Corollary 1, u(y) > min{uφG (y), uφB (y), uφE (y)}.
If firm j oﬀers {y}, then by u(y) > u(x) and u(y) > min{uφG (y), uφB (y), uφE (x)}, the agent
chooses y and takes φI . Hence, by π(y) > m1 (x), x is not subgame perfect implementable. !

4

Discussion

In this section, we explore the relationship between our result and Rustichini and Siconolfi’s(2012) result, hereafter referred to as H & S. To accomplish this, we begin by defining a competitive equilibrium. Let σ ∈ ∆(X) be a lottery on X, and let p ∈ RX be a price" vector over
X. A pair "(σ, p) is a competitive equilibrium if (i) σ ∈ arg maxσ′ ∈∆(X) x∈X u(x)σ ′ (x)subject to x∈X p(x)σ ′ (x) ≤ 0, (ii) σ ∈ arg maxσ′ ∈∆(X F IC ) p(x)σ ′ (x), and (iii) σ ∈ ∆(X F ).
The first condition is about utility maximization, the second is about profit maximization,and the third is feasibility condition.
Given x ∈ X, let δx ∈ ∆(X) be the degenerate lottery such that δx (x) = 1 and δx (x′ ) = 0
for x′ ̸= x. To be consistent"with H & S’s model, given a equilibrium price p, the profit
function must satisfy π(x) = x′ ∈X p(x′ )δx (x′ ) = p(x) for all x ∈ X.
R & H establish the existence of competitive equilibria through two steps. First, they
identify a price vector that supports the constrained eﬃcient allocation, i.e., the allocation
maximizes agent’s utility at the price. Next, they verify whether the constrained eﬃcient
allocation maximizes the firm’s profit under the obtained price.
9

R & H show the necessary and suﬃcient condition for a price vector to support the constrained eﬃcient allocation, termed the price supportability condition. If the constrained
eﬃcient allocation is degenerate for x, then the price supportability condition says thatp(x) = 0 and for each x′ ̸= x, p(x′ ) ≥ 0. As we have shown in Theorem 1, if parametersηsl are asymmetric, then it follows that π(x) ̸= 0, and hence p(x) ̸= 0. Thus, the result ofTheorem 1 is attributed to the absence of price supportability.
Finally, we illustrate that even when the price supportability condition is satisfied and the
profit is maximized, indicating the existence of a competitive equilibrium, the constrained
eﬃcient allocation is not subgame perfect implemtnable. This distinction arises from the factthat firms only oﬀer feasible allocations in R & S’s model, whereas they can oﬀer non-feasible
allocations in ours.
Example 1. Let qg = qb = 0.5 and ηg1 = ηb1 = ηg2 = ηb2 = 0.5. Let u be such that for eachx ∈ X,
u(x) = x1g (2 − x1g ) + 3x1b (2 − x1b ) + 6x2g (2 − x2g ) + 2x2b (2 − x2b ).

Then, x∗ = (−0.5, 0.5, 0.5, −0.5) is the unique constrained eﬃcient allocation. By π(x∗ ) = 0,
the price supportability condition is satisfied. Moreover, for each x ∈ X F IC , feasibility
conditions, qg x1g + qb x1b ≤ 0 and qg x2g + qb x2b ≤ 0, and symmetric parameters, ηg1 = ηb1 andηg2 = ηb2 , imply the nonpositive profit π(x) ≤ 0. Thus, x∗ maximizes the profit over X F IC ,and so (x∗ , p∗ ) is a competitive equilibrium, where p∗ (x) = π(x) for x ∈ X.
However, x∗ is not subgame perfect implementable. To see this, let x̃ = (−0.64, 0.5, 0.6, −0.5).
Then,u(x̃) = −1.6896 + 2.25 + 5.04 − 2.5 = 3.1004,
uφG (x̃) = −1.6896 − 5.0688 + 5.04 + 1.68 = −0.0384,
uφB (x̃) = 0.75 + 2.25 − 7.5 − 2.5 = −7,π(x̃) = −0.25(−0.64 + 0.5 + 0.6 − 0.5) = 0.01.

Hence, by u(x̃) = 3.1004 > 3 = u(x∗ ), u(x̃) > min{uφG (x̃), uφB (x̃), uφE (x̃)}, and π(x̃) =0.01 > 0 = π(x∗ ), x∗ is not subgame perfect implementable. This is because we assume thatthe firms can oﬀer non-feasible allocations in the game.

5

Conclusion

In this study, we examine a simple hidden information economy. We demonstrate that when
the parameters of a utility function exhibit asymmetry, any non-zero constrained eﬃcient
allocation is not subgame perfect implementable. We leave the question of subgame perfect
implementability open in the case of symmetric parameters.

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