Information Acquisition and Response in Peer-Effects Networks C. Matthew Leister Monash University
Conference on Economic Networks and Finance LSE, December 11, 2015
Introduction
Setup
Equilibrium
Welfare
Individuals/firms face heterogeneous incentives to acquire and respond to information.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs
Conclusions
Introduction
Setup
Equilibrium
Welfare
Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position
Conclusions
Introduction
Setup
Equilibrium
Welfare
Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position Dual role of information:
Conclusions
Introduction
Setup
Equilibrium
Welfare
Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position Dual role of information: 1. infer the state of the world,
Conclusions
Introduction
Setup
Equilibrium
Welfare
Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position Dual role of information: 1. infer the state of the world, 2. in equilibrium, infer the observations and subsequent actions of neighbors.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Peer-effects networks with incomplete information
ui (x1 , . . . , xN ) = ai + ω + |
X k6=i
{z
σik xk xi −
marginal value to xi
}
1 σii xi2 2 | {z }
O.C. to xi
Introduction
Setup
Equilibrium
Welfare
P ui (x1 , . . . , xN ) = ai + ω + k6=i σik xk xi − 21 σii xi2 A competitive supply chain
up stream firm
+
+
− −
vert. integ. firm
− down stream firm
−
ω : demand for novel product xfirm : production
Conclusions
Introduction
Setup
Equilibrium
Welfare
P ui (x1 , . . . , xN ) = ai + ω + k6=i σik xk xi − 21 σii xi2 Traders with heterogeneous funding constraints
deep pockets
− +
liquidity const.
ω : long term asset value xtrader : market order/inventory
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Basic questions
(1) How does heterogeneity in strategic positioning influence the incentives to acquire information?
Introduction
Setup
Equilibrium
Welfare
Conclusions
Basic questions
(1) How does heterogeneity in strategic positioning influence the incentives to acquire information? (2)
Who over and who under acquires information? Who gains to influence others’ beliefs?
Introduction
Setup
Equilibrium
Welfare
Positive results
EQ
Information response game −−→ value to information. Equilibrium properties: a. game on correlation-adjusted network (second stage), b. negative responses (second stage), c. multiple information acquisition equilibria (first stage).
Conclusions
Introduction
Setup
Equilibrium
Welfare
Welfare results 1. Extent of symmetry among pair-wise peer effects drives direction of two inefficiencies: a. informational externalities (network charact.: in-walks), b. strategic value to information acquisition (network charact.: closed-walks).
2. Symmetric networks (for e.g.) a. “bunching” for moderate peer effects: equilibrium information asymmetries inefficiently low, b. significant strategic substitutes: acquisition of negative responders inefficiently low, c. positive strategic distortion ∝ connectedness in network.
3. “Antisymmetric” networks: inefficiencies reverse.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Policy implications
Transparency-based policy: targeted certification of information investments.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Literature Network games with incomplete information: Calv´ o-Armengol & de Mart´ı (2007,2009), Calv´ o-Armengol, de Mart´ı, Prat (2015), de Mart´ı & Zenou (2015). Coordination games with endogenous information: Novshek & Sonnenschein (1983,1988), Vives (1988,2008), Hauk & Hurkens (2007). Morris & Shin (2002), Hellwig & Veldkamp (2009), Myatt & Wallace (2012,2013), Colombo, Femminis, & Pavan (2014).
Finance: Grossman & Stiglitz (1980), Kyle (1985,1989), Babus & Kondor (2013).
Introduction
Setup
Equilibrium
Welfare
Conclusions
Timeline of the game
each i chooses information quality ei ∈ [0, 1] at cost κi (ei )
each i observes signal θi , then chooses action xi ∈ R
state ω ∈ Ω observed, each i’s ui (x|ω) realized
t=1
t=2
t = 2+
Introduction
Setup
Equilibrium
Welfare
Model primitives: second stage (t = 2) Each i chooses xi ∈ R, yielding i’s payoffs (t = 2): X 1 σik xk xi − xi2 , ui (x|ω) = ω + ρ 2 k6=i
where ω ∈ Ω ⊆ R, σij ∈ R for each i, j, and ρ ∈ [0, 1],
Conclusions
Introduction
Setup
Equilibrium
Welfare
Model primitives: second stage (t = 2) Each i chooses xi ∈ R, yielding i’s payoffs (t = 2): X 1 σik xk xi − xi2 , ui (x|ω) = ω + ρ 2 k6=i
where ω ∈ Ω ⊆ R, σij ∈ R for each i, j, and ρ ∈ [0, 1], i observes signal θi ∈ Θ ⊆ R of quality ei ∈ [0, 1],
Conclusions
Introduction
Setup
Equilibrium
Welfare
Model primitives: second stage (t = 2) Each i chooses xi ∈ R, yielding i’s payoffs (t = 2): X 1 σik xk xi − xi2 , ui (x|ω) = ω + ρ 2 k6=i
where ω ∈ Ω ⊆ R, σij ∈ R for each i, j, and ρ ∈ [0, 1], i observes signal θi ∈ Θ ⊆ R of quality ei ∈ [0, 1], Pure strategy: Xi : Θ × [0, 1] → R.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Model primitives: second stage (t = 2) Each i chooses xi ∈ R, yielding i’s payoffs (t = 2): X 1 σik xk xi − xi2 , ui (x|ω) = ω + ρ 2 k6=i
where ω ∈ Ω ⊆ R, σij ∈ R for each i, j, and ρ ∈ [0, 1], i observes signal θi ∈ Θ ⊆ R of quality ei ∈ [0, 1], Pure strategy: Xi : Θ × [0, 1] → R.
Assumption 1 (I − [sij σij ])−1 is well defined for every s ∈ [0, 1]N(N−1) .
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Model primitives: first stage (t = 1)
Each i = 1, . . . , N privately invests in information quality ei ∈ [0, 1]. i’s cost of information quality κi (·) ∈ C 2 satisfies κi (0) , κ′i (0) = 0, with non-decreasing κ′′i (ei ) ≥ 0. Assumption 2 For v0 > 0, there exists an unique ei† ∈ (0, 1) solving v0 ei† = κ′i (ei† ).
Introduction
Setup
Equilibrium
Welfare
Conclusions
Model primitives: first stage (t = 1)
Each i = 1, . . . , N privately invests in information quality ei ∈ [0, 1]. i’s cost of information quality κi (·) ∈ C 2 satisfies κi (0) , κ′i (0) = 0, with non-decreasing κ′′i (ei ) ≥ 0. Assumption 2 For v0 > 0, there exists an unique ei† ∈ (0, 1) solving v0 ei† = κ′i (ei† ). All conditions satisfied for normal state and signals case.
Introduction
Setup
Equilibrium
Welfare
Model primitives: beliefs and expectations Belief: µi (e−i ), density function over e−i ∈ [0, 1]N−1 .
Conclusions
Introduction
Setup
Equilibrium
Welfare
Model primitives: beliefs and expectations Belief: µi (e−i ), density function over e−i ∈ [0, 1]N−1 . Consistency: µi (e−i ) = 1 for t = 1 for given e−i , with µi (e′−i ) = 0 otherwise.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Model primitives: beliefs and expectations Belief: µi (e−i ), density function over e−i ∈ [0, 1]N−1 . Consistency: µi (e−i ) = 1 for t = 1 for given e−i , with µi (e′−i ) = 0 otherwise. E1.
E2.
Ei [ω] = Ei [θi ] = 0, v0 := Ei ω 2 ] = Ei θi2 |ei , Ei [ω|θi , ei ] = ei θi ,
E3. Ei [θj |θi , ei , ej ] = ei ej θi ,
for each ei ∈ [0, 1].
Conclusions
Introduction
Setup
Equilibrium facts
Equilibrium
Welfare
Conclusions
Theorems
1. Multiple IAE e∗ may exist even with a unique IRE β ∗ for each e.
Introduction
Setup
Equilibrium
Welfare
Equilibrium facts
Conclusions
Theorems
1. Multiple IAE e∗ may exist even with a unique IRE β ∗ for each e. 2. Significant strategic substitutes: can have βi∗ < 0.
Introduction
Setup
Equilibrium
Welfare
Equilibrium facts
Conclusions
Theorems
1. Multiple IAE e∗ may exist even with a unique IRE β ∗ for each e. 2. Significant strategic substitutes: can have βi∗ < 0. 3. Significant peer effects required for 1. or 2. to obtain.
Introduction
Setup
Equilibrium
Welfare
Equilibrium facts
Conclusions
Theorems
1. Multiple IAE e∗ may exist even with a unique IRE β ∗ for each e. 2. Significant strategic substitutes: can have βi∗ < 0. 3. Significant peer effects required for 1. or 2. to obtain. Proposition Under Assumptions 1 and 2, there exists a ρ¯ > 0 such that for ρ ∈ [0, ρ¯), a unique IAE e∗ with βi∗ > 0 for all i obtains.
Introduction
Setup
Equilibrium
Welfare
Welfare
Conclusions
Introduction
Setup
Equilibrium
Welfare
Welfare
For any e, giving X∗ : νi (X∗ |e) := Ei [ui (X∗ |θi , ei , µ∗i ) |ei , µ∗i ] − κi (ei ) .. . 1 = v0 βi∗2 − κi (ei ) . 2
Conclusions
Introduction
Setup
Equilibrium
Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N
k
Welfare
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N
∂ ∂ei
=
∂νi
P
k νk (X ∗ (X |e)
∂ei
k
∗ |e)
βk∗ ,k6=i
+
X ∂νi (X∗ |e) ∂β ∗ k
k6=i
∂βk∗
∂ei
+
X ∂νk (X∗ |e) ∂β ∗ k
k6=i
∂βk∗
∂ei
.
Introduction
Setup
Equilibrium
Welfare
Conclusions
Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N
∂ ∂ei
P
k
νk (X∗ |e) X βi∗2 ∂ ∗ ′ = v0 − κ (ei ) + v0 βi∗ ei ρσik ek β + ei ∂ei k k6=i | {z } ∗ | = 0 in IAE e f.o.c. {z } k
v0
X k6=i
βk∗
∂ ∗ β . ∂ei k
= 0 in public acquisition eq. epb f.o.c.
|
{z
= 0 in planner’s solution epl f.o.c.
}
Introduction
Setup
Equilibrium
Welfare
Conclusions
Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N
∂ ∂ei
P
k
νk (X∗ |e) X βi∗2 ∂ ∗ ′ ei ρσik ek = v0 − κ (ei ) + v0 βi∗ β + ei ∂ei k k6=i {z } | k
(marginal) strategic value
v0
X k6=i
|
∂ ∗ β . ∂ei k {z }
βk∗
(marginal) externalities
Introduction
Setup
Equilibrium
Welfare
Conclusions
Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N
∂ ∂ei
P
k
νk (X∗ |e) X βi∗2 ∂ ∗ ′ ei ρσik ek − κ (ei ) + v0 βi∗ β + = v0 ei ∂ei k k6=i {z } | k
ξist (X∗ |e)
v0
X k6=i
|
∂ ∗ β . ∂ei k {z }
βk∗
ξiex (X∗ |e)
Introduction
Setup
Equilibrium
Welfare
Conclusions
Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N
∂ ∂ei
P
k
νk (X∗ |e) X βi∗2 ∂ ∗ ′ = v0 ei ρσik ek − κ (ei ) + v0 βi∗ β + ei ∂ei k k6=i | {z k
v0
(marginal) public-value
X k6=i
βk∗
∂ ∗ β . ∂ei k }
Introduction
Setup
Equilibrium
Welfare
Welfare: marginal inefficiencies Theorem (marginal inefficiencies) For information qualities e, consistent beliefs µ and IRE X∗ : βi∗2 ′ 1 Ie ΣIe (I − Ie ΣIe )−1 Ie ΣIe 1i , ei∗ i β∗ ξiex (e, X∗ ) = 2v0 ∗i (β ∗ − βi∗ 1i )′ Ie ΣIe (I − Ie ΣIe )−1 1i . ei ξist (e, X∗ ) = 2v0
Conclusions
Introduction
Setup
Equilibrium
Welfare
Welfare: marginal inefficiencies Theorem (marginal inefficiencies) For information qualities e, consistent beliefs µ and IRE X∗ : βi∗2 ′ 1 Ie ΣIe (I − Ie ΣIe )−1 Ie ΣIe 1i , ei∗ i β∗ ξiex (e, X∗ ) = 2v0 ∗i (β ∗ − βi∗ 1i )′ Ie ΣIe (I − Ie ΣIe )−1 1i . ei ξist (e, X∗ ) = 2v0
ξist (e, X∗ ) ∝ 1′i
∞ X ([ei ej ρσij ]i6=j )τ τ =2
!
1i :
summation of closed walks on [ei ej ρσij ]i6=j beginning and ending on i.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Welfare: marginal inefficiencies Theorem (marginal inefficiencies) For information qualities e, consistent beliefs µ and IRE X∗ : βi∗2 ′ 1 Ie ΣIe (I − Ie ΣIe )−1 Ie ΣIe 1i , ei∗ i β∗ ξiex (e, X∗ ) = 2v0 ∗i (β ∗ − βi∗ 1i )′ Ie ΣIe (I − Ie ΣIe )−1 1i . ei ξist (e, X∗ ) = 2v0
ξiex (e, X∗ ) ∝ (β ∗ − βi∗ 1i )′
∞ X τ =1
([ei ej ρσij ]i6=j )τ
!
1i :
summation of walks on [ei ej ρσij ]i6=j beginning with j and ending on i, weighted by βj and aggregate over j 6= i.
Introduction
Setup
Equilibrium
Welfare
Conclusions
Example: three-player symmetric network, common κ ei .80
2
+⊗
.75
-.1 .1
1
b
.70
+
.65
⊗
+⊗
e†
b
b
eipl +eipb ei∗ ⊗
.60
-.1 3
b
.55 1
2
3
i
Introduction
Setup
Equilibrium
Welfare
Conclusions
Example: three-player symmetric network, common κ ei .80
2
.75
⊗
⊗
+
+
b
e†
b
-1/3 .70
.1
1
.65 eipl +eipb ei∗ ⊗
-1/3 3
.60 .55
+
b
b
1 e1pl = 0
2
3
i
Introduction
Setup
Equilibrium
Welfare
Conclusions
Example: three-player symmetric network, common κ ei
⊗
+
+
b
.80
2
⊗
b
.75
e†
-1/3 .70
1/3
1
.65 eipl +eipb ei∗ ⊗
-1/3 3
.60
b
.55 1
2
e1pl = e1pb = e1∗ = 0
3
i
Introduction
Setup
Equilibrium
Welfare
Conclusions
Example: two-player antisymmetric network, common κ
ei .75
b
+ ⊗
e† ⊗
1/3 1
2
.65
-1/3
b
+
eipl +eipb ei∗ ⊗
b
.55 1
2
i
Introduction
Setup
Equilibrium
Welfare
Welfare and policy design
Conclusions
Introduction
Setup
Equilibrium
Welfare
Welfare and the neutral player symmetric networks ξipl (II ) ei†
(III )
(I )
βi∗
antisymmetric networks ξipl (VI )
(IV )
(V )
ei†
βi∗
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Market efficiency in liquidity crises
conclusion
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
N = 8 traders comprise non-trivial share of market.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
N = 8 traders comprise non-trivial share of market. xi : i’sPinventory/market order (e.g. Kyle (1985)); x¯ := 8i=1 xi .
Conclusions
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
N = 8 traders comprise non-trivial share of market. xi : i’sPinventory/market order (e.g. Kyle (1985)); x¯ := 8i=1 xi . t = 2 market price φ(¯ x ) = A + B x¯, B > 0.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
N = 8 traders comprise non-trivial share of market. xi : i’sPinventory/market order (e.g. Kyle (1985)); x¯ := 8i=1 xi . t = 2 market price φ(¯ x ) = A + B x¯, B > 0. ω: risky asset’s long term value.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
N = 8 traders comprise non-trivial share of market. xi : i’sPinventory/market order (e.g. Kyle (1985)); x¯ := 8i=1 xi . t = 2 market price φ(¯ x ) = A + B x¯, B > 0. ω: risky asset’s long term value. t = 2 payoffs: ui (x|ω) = (ω + pi φ(¯ x )) xi − xi2 X xk xi − (1 − pi B) xi2 . = ω + pi A + pi B k6=i
Conclusions
Introduction
Setup
Equilibrium
Market efficiency in liquidity crises
Liquidity flush market: pi < 0 for each unconstrained i.
Welfare
Conclusions
Introduction
Setup
Equilibrium
Market efficiency in liquidity crises
Liquidity flush market: pi < 0 for each unconstrained i. Market crowding in information acquisition.
Welfare
Conclusions
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
Liquidity flush market: pi < 0 for each unconstrained i. Market crowding in information acquisition. Traders set ei∗ , βi∗ < e † (region (II)): over-acquire; over exertion in informationally inefficient markets.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
Liquidity crises: Liquidity spirals `a la Brunnermeier and Pedersen (2009) → upward sloping demand.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
Liquidity crises: Liquidity spirals `a la Brunnermeier and Pedersen (2009) → upward sloping demand. pi > 0 for liquidity-constrained trader i.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises Market structure: pi B < 0 pi B > 0
− +
+
− −
constrained traders
unconstrained traders
Conclusions
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
Liquidity crisis paradigm shift: Constrained traders set ei∗ , βi∗ > e † 1. Flush market: antisymmetric relationships → over-acquire. 2. Crisis: symmetric relationships → under-acquire.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Market efficiency in liquidity crises
ei
antisymmetric peer effects
symmetric peer effects
1.0
pl ecnst.
0.96
bc bc bc
bc
bc
bc
bc
bc bc
bc bc
bc
bc
bc bc
∗ ecnst.
e†
0.92 bc
0.88
0
1
2
3
4
5
6
7
8
# cnst.
Introduction
Setup
Equilibrium
Welfare
Market efficiency in liquidity crises
Liquidity crisis paradigm shift: Constrained traders set ei∗ , βi∗ > e † 1. Flush market: antisymmetric relationships → over-acquire. 2. Crisis: symmetric relationships → under-acquire.
Unconstrained traders set ei∗ , βi∗ < e † 1. Flush market: symmetric relationships → over-acquire. 2. Crisis: antisymmetric relationships → under-acquire. 3. Extreme crisis: few unconstrained traders set ei∗ , βi∗ < 0.
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Market efficiency in liquidity crises ei
symmetric peer effects
antisymmetric peer effects
1.0 bc
bc bc
0.8
e†
bc bc
bc bc
bc bc
pl eunc.
bc
bc
∗ eunc. bc
4
5
0.6
negative response bc
0.4 0.2 0.0
0
1
2
3
bc
bc
6
7
8
# cnst.
Introduction
Setup
Equilibrium
Welfare
Conclusions
Policy suggestion in liquidity crises
Constrained traders impose symmetric, positive informational externalities on each other: under acquire, with positive strategic values...
Introduction
Setup
Equilibrium
Welfare
Conclusions
Policy suggestion in liquidity crises
Constrained traders impose symmetric, positive informational externalities on each other: under acquire, with positive strategic values... Couple stress-tests with certification of information investments of constrained traders.
Introduction
Setup
Equilibrium
Welfare
Conclusions
Conclusions 1. Introduce problem of costly information acquisition into new context: general network of peer effects. 2. Symmetric networks: a. Equilibrium information inefficiently symmetric. b. Players moving against their information do so too little. c. Strategic values to information are positive.
3. Direction of welfare and strategic motives determined by network “position” and extent of symmetry in relationships: direction of inefficiencies reverse in antisymmetric networks. 4. Information externalities and “position”: βi∗ w.r.t. ei† and origin, Strategic values and “position”: connectedness.
Introduction
Setup
Equilibrium
Welfare
Conclusions
Conclusions II
1. Liquidity crisis paradigm shift: over acquisition of information in liquid markets, under acquisition in constrained markets. 2. Unconstrained “shorters” in crisis: inefficient. 3. Transparency-based policy intervention: stress test with information investment certification.
Introduction
Setup
Equilibrium
Welfare
Equilibrium characterization
Conclusions
Introduction
Setup
Equilibrium
Welfare
Equilibrium characterization Theorem (t = 2 information-response equilibrium (IRE)) Under Assumption 1, for any e and consistent µ there exists a unique linear IRE of the form: X∗ = [Xi∗ (θi |ei )] = [βi∗ θi ] , P where each βi∗ solves βi∗ = ei + k6=i ei ek ρσik βk∗ : β ∗ := (I − [ei ej ρσij ]i6=j )−1 e ∞ X ([ei ej ρσij ]i6=j )τ e. = τ =0
Conclusions
Introduction
Setup
Equilibrium
Welfare
Conclusions
Equilibrium characterization Theorem (t = 2 information-response equilibrium (IRE)) Under Assumption 1, for any e and consistent µ there exists a unique linear IRE of the form: X∗ = [Xi∗ (θi |ei )] = [βi∗ θi ] , P where each βi∗ solves βi∗ = ei + k6=i ei ek ρσik βk∗ : β ∗ := (I − [ei ej ρσij ]i6=j )−1 e ∞ X ([ei ej ρσij ]i6=j )τ e. = τ =0
βi∗ : i’s “informational centrality” (weighted Bonacich centrality).
Introduction
Setup
Equilibrium
Welfare
Equilibrium characterization
Conclusions
Back
Theorem (t = 1 information-acquisition equilibrium (IAE)) Under Assumption 1, for IRE X∗ and consistent beliefs µ there exists a (generically unique∗ ) IAE e∗ . For any such IAE, and ∀ i with ei∗ ∈ (0, 1): β ∗2 v0 i∗ = κ′i (ei∗ ) . ei
Introduction
Setup
Equilibrium
Welfare
Conclusions
Equilibrium characterization
Back
Theorem (t = 1 information-acquisition equilibrium (IAE)) Under Assumption 1, for IRE X∗ and consistent beliefs µ there exists a (generically unique∗ ) IAE e∗ . For any such IAE, and ∀ i with ei∗ ∈ (0, 1): β ∗2 v0 i∗ = κ′i (ei∗ ) . ei
ei∗ 2 -.1 .1
.80 1
.75 .70
b
b
2
3
b
-.1 3
.65
1
i
