Discussion Papers In Economics
And Business

Welfare Implications of Mitigating Investment
Uncertainty
Takayuki Ogawa and Jun Sakamoto
Discussion Paper 18-33-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWelfare Implications of Mitigating Investment
Uncertainty
Takayuki Ogawa and Jun Sakamoto
Discussion Paper 18-33-Rev.

December 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWelfare Implications of Mitigating Investment
Uncertainty∗
Takayuki Ogawa†and Jun Sakamoto‡

Abstract
This paper explores the welfare implications of mitigating investment uncertainty inthe context of Easley and O’Hara (2009) [Ambiguity and Nonparticipation: The Role ofRegulation. Review of Financial Studies 22(5), 1817–1843]. While one may expect welfare
gains to be had by encouraging participation in financial markets by ambiguity-averse
investors, we formally show that it hurts other investors and is not Pareto-improving without
appropriate income transfers.
Keywords: Ambiguity, Heterogenous agents, Uncertainty, Welfare eﬀects.
JEL Classification Numbers: D81, G11, G18.

∗ We are indebted to Yuichi Fukuta, Ichiro Gombi, Keiichi Hori, Shinsuke Ikeda, Yoshiyasu Ono, and seminar

participants at Osaka University for their helpful comments.
† Faculty of Economics, Osaka University of Economics, 2-2-8 Osumi, Higashiyodogawa-ku, Osaka 533-8533,
Japan. Tel: +81-6-6328-2431. E-mail: tkogawa@osaka-ue.ac.jp.
‡ Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan.
Tel: +81-6-6850-5847. E-mail: rge010sj@student.econ.osaka-u.ac.jp.

1

Introduction

Knight (1921) and Keynes(1937) emphasize an importance of distinguishing risk and uncertainty(or ambiguity) in situations where the probability law of the economy is unknown to somemarket participants. Decision making with ambiguity aversion is axiomatized by Gilboa andSchmeidler (1989) and Schmeidler (1989) and is applied to various economic issues, prominently
to the analysis of financial markets and asset prices.1 Along this line, we investigate welfare
implications of reducing investment uncertainty, which is usually considered to be beneficial in
the literature, and show that it cannot be Pareto-improving without appropriate income transfers.
The theoretical model employed in this paper bases on Easley and O’Hara (2009), whodevelop a general equilibrium model with two types of investors—i.e., sophisticates (naïfs)who precisely (only imprecisely) know the probability distribution of investment payoﬀs. After demonstrating that the naïf chooses to participate in financial markets as the ambiguity
suﬃciently dissipates, Easley and O’Hara mention that “regulation, particularly regulation ofunlikely events, can moderate the eﬀects of ambiguity, thereby increasing participation in financial markets and generating welfare gains (p. 1818).” Against their perspective, we formallyshow that it at least deteriorates the welfare of sophisticates in their setting.
More accurately, we examine how individual welfare is aﬀected by either increasing theminimum mean of investment payoﬀs perceived by the naïf or decreasing the perceived maximum
variance. Both of them stimulate the naïf’s demand for the risky asset and push up the asset
price, thereby reducing asset holdings of the sophisticate. It corrects a distortion in the asset
price but is undesirable for the sophisticate who has enjoyed the lower asset purchase price. In
other words, the sophisticate with more information loses a benefit arising from miss-pricing.
Therefore, the mitigation of investment uncertainty is harmful to the sophisticate and thus is notdesirable in a Pareto sense. The government needs to appreciately redistribute gains from thenaïfs to the sophisticates.
Several recent studies theoretically explore the welfare eﬀects attributed to various ambiguity.
Most of them report a welfare improvement of decreasing ambiguity within a representative agent
framework. For instance, in a partial equilibrium setting in which an investor is ambiguous abouta prediction model of stock returns, Chen et al. (2014) calibrate the welfare costs caused by
the ambiguity. Ilut and Schneider (2014) estimate confidence shocks about future TFP and findthat they are a major source of business cycle frequency movements.2 Alonso and Prado (2015)
document quite a large benefit of removing all consumption fluctuations when a consumer
confronts ambiguity about the transition probability of stock’s dividend growth rates. As an
exception, Easley et al. (2014) demonstrate the welfare eﬀect to be ambiguous when sometraders face ambiguity about the trading strategies of the others. However, they focus only on
the welfare of the ambiguity-facing traders and do not discuss a spillover eﬀect to the others.
In contrast with the existing literature, we analyze all individuals’ welfare in a heterogeneous
agents model and moreover deals with how to achieve a Pareto improvement.
The rest of the paper is organized as follows. Section 2 presents the structure of themodel. Section 3 analyzes the welfare of each investor to show that a Pareto improvement
cannot be attained only by mitigating investment uncertainty. Section 4 considers two kindsof redistribution policies between the sophisticate and the naïf. It finds that when investment
uncertainty eases, a lump-sum income transfer is useful to accomplish a Pareto improvement
1For theoretical applications, refer to e.g., Dow and Werlang (1992), Epstein and Wang (1994, 1995), Chen and
Epstein (2002), Epstein and Miao (2003), Cao et al. (2005), and Easley and O’Hara (2009). See Anderson et al.
(2009) and Antoniou et al. (2015) for empirical studies and Guidolin and Rinaldi (2013) for a survey.
2See also Cagetti et al. (2002) and Bidder and Smith (2012) for an impact of ambiguity on business cycles.

1but an investment subsidy is not. Section 5 concludes.

2

The Model

The model bases on Easley and O’Hara (2009). A closed economy, which lasts two periods,
is inhabited by two types of investors, j = S, N. There are one safe asset with a zero rate of
return and one risky asset of which stochastic payoﬀ realized in the second period is normally
distributed.3 Only a part of investors (called sophisticates, S) know the true mean and varianceof the stochastic payoﬀ, (v̂, σ̂), whereas the remaining (called naïfs, N) face ambiguity of thepayoﬀ distribution and subjectively possess the possible set of mean payoﬀ { v̄min, · · · , v̄max } andthat of variance { σ̄min, · · · , σ̄max }. It is assumed to satisfyv̂ ∈ (v̄min, v̄max ),σ̂ ∈ (σ̄min, σ̄max ).

(1)

At the beginning of the first period, each investor is endowed with a constant amount of therisky asset, x̄(> 0), and chooses a portfolio, (m j , x j ), where m j and x j denote the demand of theinvestor j(= S, N) for the safe asset and that for the risky asset. Given the current price of therisky asset, p, the budget constraint in the first period isp x̄ = m j + px j .

(2)

In the second period, investors consume their wealth entirely. The realized consumption, c̃ j , isstochastic and given byc̃ j = m j + ṽ x j ,(3)

where ṽ represents the stochastic payoﬀ when holding the risky asset.
All investors share the same preference with respect to c̃ j , in which the absolute risk aversion
is set to be unity. In this case, the utility maximization problem reduces to the mean-varianceapproach (see appendix A). For the sophisticate who knows the true probability distribution, it
ismax US = p x̄ + (v̂ − p)xS −xSσ̂(xS )2 .
2(4)

The first term on the right-hand side represents the value of the initial endowment; the second
term is the excess return of holding the risky asset; and the third term is the disutility fromrisk-taking. Solving this problem yields the sophisticate’s demand function for the risky asset:
xS =v̂ − p.
σ̂

(5)

As axiomatized by Gilboa and Schmeidler (1989), when investors cannot place a uniqueprior on the probability distribution and are ambiguity-averse, they act so as to maximize theminimum expected utility. In the present context, the naïf chooses a portfolio expecting the3The original model of Easley and O’Hara (2009) contains two risky assets, one of which has no uncertainty
over payoﬀs. For the sake of simplicity, we do not introduce it. This is because, as long as the payoﬀs of two risky
assets are independently distributed, the welfare eﬀect of adding one more asset with no uncertainty is negligible.
See Huang et al. (2017) for the case where the payoﬀs of risky assets correlate.

2

worst scenario, which is that the minimum (maximum) mean payoﬀ and the maximum variance
are realized when taking a long (short) position in the risky asset:{2
p x̄ + (v̄min − p)xN − σ̄max
if xN ≥ 0,
2 (x N )max UN =
(6)
σ̄
2
max
xN
p x̄ + (v̄max − p)xN − 2 (xN ) if xN ≤ 0.
It implies that the naïf holds the risky asset according tov̄min −p
if p < v̄min , σ̄maxxN = 0if v̄min ≤ p ≤ v̄max , v̄max −p if v̄max < p.
 σ̄max
The equilibrium condition in the risky asset market is(1 − µ)xS + µxN = x̄ (> 0),

(7)

(8)

where µ(∈ (0, 1)) denotes an exogenous ratio of the naïfs in the total population. Since theright-hand side of this equation is positive, the case where all investors take short positions
cannot be equilibrium; that is, from (7), the asset price lies below v̄max in equilibrium. Hence,
there are the following two cases depending on the naïf’s portfolio choice.4
Case 1 (Nonparticipating equilibrium): if v̄min ≤ p ≤ v̄max , the naïfs choose not to participate
in the risky asset market and, from (4) through (8), we havexNN = 0,
UNN = pN x̄,

xSN =

x̄
,
1− µ

pN = v̂ −σ̂USN = pN x̄ + (xSN )2,

σ̂ x̄
,
1− µ

(9)

2
where the superscript N stands for the nonparticipating equilibrium. It is obvious that the
ambiguity parameters (v̄min, v̄max, σ̄max ) have no eﬀect on individuals’ welfare as long as thenaïfs do not participate in the risky asset market. The nonparticipating equilibrium exists if andonly if
Condition 1 (Nonparticipating equilibrium).
v̄min ≤ pN ≤ v̄max ⇐⇒ v̄min ≤ v̂ −σ̂ x̄≤ v̄max .
1− µCase 2 (Participating equilibrium): if p < v̄min , all investors participate in the risky asset
market taking long positions. Using (4) through (8), we obtain the participating equilibrium,indexed by the superscript P:
[
(
)]
1− µ
σ̂ x̄
P
xN =
v̄min − v̂ −,(1 − µ)σ̄max + µσ̂
1− µ()σ̄max x̄µ Pµ
Pv̂ − v̄min += xSN −x ,xS =(1 − µ)σ̄max + µσ̂
µ
1− µ N(1 − µ)σ̄max v̂ + µσ̂ v̄min − σ̂ σ̄max x̄
µσ̂ P
(10)
pP =
= pN +x ,(1 − µ)σ̄max + µσ̂
1− µ Nσ̄max P 2
UNP = pP x̄ +(xN ) ,2σ̂
USP = pP x̄ + (xSP )2,
2
4The equilibrium condition for the safe asset, in zero net supply, is automatically satisfied because Walras’ law
holds.

3

where xSN and pN in the second and third equations are given in (9) and independent from theambiguity parameters. The last two equations indicate that the welfare of each investor, U jP ,is increasing with respect to both the value of the initial endowment, pP x̄, and the risky asset
holdings, x Pj .5 This equilibrium exists if and only if
Condition 2 (Participating equilibrium).
pP < v̄min ⇐⇒ v̂ −σ̂ x̄< v̄min .
1− µNeedless to say, xNP and xSP in (10) have positive values under condition 2 and (1). From thethird and second equations in (10), the participation of the naïfs, xNP > 0, pushes up the risky
asset price, thereby depressing the risky asset demand of the sophisticates.

3

WelfareIn this section we analyze the welfare eﬀects of reducing uncertainty in the risky asset market,
interpreted as either increasing the minimum mean payoﬀ v̄min or decreasing the maximum
variance σ̄max .
Easley and O’Hara (2009) suggest how to mitigate investment uncertainty through policy
interventions such as investor education and financial regulations. Education for uninformednaïfs obviously helps lead their wrong beliefs to the true payoﬀ distribution. Regulations
designed to make the worst case perceived by the naïfs impossible play the same role. For
example, a government guarantee against risky investment raises the perceived minimum mean
payoﬀ, whereas the Investment Act, which forces mutual funds to diversify investment, decreases
the perceived maximum variance. These interventions intend to improve welfare by inducing
the naïfs to participate in the risky asset market. In this section, however, we demonstrate thatthey hurt the sophisticates without appropriate income transfers.
Compare conditions 1 and 2 to understand that an increase in v̄min changes the equilibriumfrom case 1 to case 2. From (10) in which condition 2 holds, the resulting participation in therisky asset market by the naïfs (xNP > 0) raises the asset price (pP > pN ), thereby lowering therisky asset holdings of the sophisticates (xSP < xSN ). Since both the endowment value p x̄ and theasset holdings xN increase, the naïf’s welfare apparently improves:
UNP − UNN = (pP − pN ) x̄ +σ̄max P 2
(xN ) > 0,2(11)which comes from the fourth equations in (9) and (10). On the sophisticate’s welfare, we canshow that the beneficial eﬀect of a rise in p x̄ is dominated by the harmful eﬀect of a decrease inxS :USP − USN = −µ2 σ̂xNP {(1 − µ)[(v̂ − v̄min ) + 2(σ̄max − σ̂) x̄] + σ̂ x̄}2(1 − µ)2 [(1 − µ)σ̄max + µσ̂]

< 0,

(12)

of which sign is determined by (1). Therefore, an increase in v̄min enhances the welfare of thenaïf but worsens that of the sophisticate.
5Payoﬀs of the risky asset are not only more profitable but also more volatile than those of the safe asset. With aunitary absolute risk aversion, the former benefit dominates the latter disutility from risk-taking. Thus, the welfare
is increasing with respect to risky asset holdings, x jP .

4

To evaluate the aggregate eﬀect, we define the social welfare as follows:
UW ≡ (1 − µ)US + µUN .

(13)

Using (11) and (12) implies that an increase in v̄min improves the social welfare:
UWP − UWN =

[
(
)]
µxNP
σ̂ x̄
v̄min − v̂ −
> 0.
2
1− µ

(14)

See appendix B for the closed-form solution of (11) and the derivations of (12) and (14).
We next examine an eﬀect in the participating equilibrium by totally diﬀerentiating (10) and(14) with respect to either v̄min or σ̄max :
dxNP
dxSP
dUNP
dUSP
dUWP
dpP
> 0,
< 0,
> 0,
> 0,
< 0,
> 0,
dv̄mindv̄mindv̄mindv̄mindv̄mindv̄min
dxNP
dxSP
dUNP
dUSP
dUWP
dpP
< 0,
> 0,
< 0,
< 0,
> 0,
< 0.
dσ̄maxdσ̄maxdσ̄maxdσ̄maxdσ̄maxdσ̄maxSee appendix B for the analytical details. The implication is the same as the previous argument—i.e., as the naïfs’ asset demand expands, the asset price rises and the sophisticates’ asset holdings
decreases. The welfare eﬀect is negative for the sophisticate and positive for the naïfs.
The above results are summarized as follows:
Proposition 1. On the risky asset with uncertainty, both an increase in the minimum of possible
mean payoﬀ v̄min and a decrease in the maximum variance σ̄max raise the social welfare. However,it cannot be Pareto-improving—i.e., it is beneficial to the naïf and harmful to the sophisticate.
Against the perspective of Easley and O’Hara (2009), with no further policy intervention, themitigation in investment uncertainty is not desirable in a Pareto sense.

4

Redistribution Policy

Proposition 1 states that mitigating investment uncertainty has a positive impact on naïf’s
welfare larger than a negative impact on sophisticate’s welfare in the absolute value, so thatthe social welfare improves. This section examines whether policy makers can achieve a
Pareto improvement by implementing income transfers from the improved naïfs to the damaged
sophisticates simultaneously when investment uncertainty eases.

4.1

Lump-sun transfers

A lump-sum transfer from the naïfs to the sophisticates is introduced into the previous model.
The budget equations are then rewritten as
p x̄ + s0j = m j + px j ,c̃ j = m j + s1j + ṽ x j ,where stj (t = 0, 1. j = S, N) denotes a subsidy-cum-tax and, at each point in time, satisfies thebudget constraint of the government,(1 − µ)sSt + µstN = 0
5

for t = 0, 1.

Taking account of them, we can represent individual utility byÛS ≡ US + (sS0 + sS1 ),ÛN ≡ UN −1− µ 0(sS + sS1 ),µwhere US and UN are given in (4) and (6). It indicates that the lump-sum transfer does not alter
the demand functions for the risky asset (5) and (7), the equilibrium values (xN , xS, p) in (9) and(10), and conditions 1 and 2.
Suppose that the economy is at first in the nonparticipating equilibrium with null transfer andthat the government implements the transfer as soon as an increase in v̄min shifts the equilibriumfrom case 1 to case 2. The social welfare is unaﬀected by such a transfer and given by (14):ÛWP − ÛWN ≡ (1 − µ)(ÛSP − ÛSN ) + µ(ÛNP − ÛNN )= (1 − µ)(USP − USN ) + µ(UNP − UNN ) > 0.

(15)

Keeping the social welfare unchanged, a moderate transfer compensates the sophisticate’s welfare
loss. A Pareto improvement is achieved if ÛSP − ÛSN > 0 and ÛNP − ÛNN > 0, or equivalently,)(− USP − USN < sS0 + sS1 <)µ ( PUN − UNN ,1− µwhere UNP − UNN > 0 and USP − USN < 0 are shown in (11) and (12).6 Note that such a Paretoimproving transfer always exists because the last inequality in (15) is satisfied.
The result is summarized in the following proposition:
Proposition 2. When investment uncertainty eases, the government can achieve a Pareto improvement by simultaneously implementing a lump-sum income transfer from the naïfs to thesophisticates.

4.2

Investment subsidies

One may propose an investment subsidy as an alternative instrument for Pareto-improving
redistribution because the sophisticates hold the risky asset more than the naïfs do. However, it
is shown not to work.
Let us rewrite the individual budget equations as(p − τ 0 ) x̄ − z0 = m j + (p − τ 0 )x j ,c̃ j = m j + (ṽ + τ 1 )x j − z1,

where τ t and zt (t = 0, 1) are an investment subsidy rate and a lump-sum tax, both of which
are not discriminated on the basis of investor’s type j(= S, N). In this case, individual utility
reduces toσ̂US = (p − τ 0 ) x̄ − z0 − z1 + (v̂ − p + τ 0 + τ 1 )xS − (xS )2,
2
{
0
0
1
0
1
2
(p − τ ) x̄ − z − z + (v̄min − p + τ + τ )xN − σ̄max
2 (x N )UN =2(p − τ 0 ) x̄ − z 0 − z1 + (v̄max − p + τ 0 + τ 1 )xN − σ̄max
2 (x N )6One can obtain a closed-form condition by using (A.5) and (A.6) in appendix B.

6

if xN ≥ 0,
if xN ≤ 0.

For a given asset price p, the investment subsidy stimulates investment demand sincev̂ − p + τ 0 + τ 1,σ̂v̄min −p+τ 0 +τ 1

if p < v̄min + τ 0 + τ 1 ,

σ̄max


xN = 0if v̄min + τ 0 + τ 1 ≤ p ≤ v̄max + τ 0 + τ 1 ,
 v̄max −p+τ0 +τ1 if v̄
01
max + τ + τ < p.

σ̄max
xS =

The asset demand expansion in turn raises the equilibrium price, pN and pP , proportionately:
pN = v̂ −σ̂ x̄+ τ 0 + τ 1,
1− µ

pP = p N +

µσ̂ P
x ,
1− µ N

where xNP in the second equation is given by the first equation in (10). Substituting theseequilibrium prices into the asset demand functions gives the equilibrium asset holdings of each
investor. Consequently, whether the government implements the investment subsidy or not, theresulting demand for the risky asset remains the same as the first and second equations in (9)and (10). Conditions 1 and 2 are also unaﬀected. In sum, the investment subsidy just pushes up
the asset price.
We shall turn to the welfare analysis. The government finances the investment subsidy by
imposing lump-sum taxation in each period:
τ 0 x̄ + z0 = (1 − µ)τ 0 xS + µτ 0 xN ,z1 = (1 − µ)τ 1 xS + µτ 1 xN .
In equilibrium in which (8) holds, they are
z0 = 0,z1 = τ 1 x̄.

Applying these equations, the equilibrium demand for the risky asset and the equilibrium asset
price to utility, we can find that the investment subsidy is neutral to individual’s welfare as wellas the social welfare.
The result is summarized as follows:
Proposition 3. An investment subsidy financed by lump-sum taxation is neutral to individuals’
welfare as well as the social welfare. Thus, it is not an eﬀective policy instrument to achieve a
Pareto improvement.

5

Conclusion

Recently, numerous theoretical and empirical studies report considerable impacts of uncertainty
on financial markets and asset prices. They usually presume that reducing uncertainty overinvestment payoﬀs is desirable since it corrects distorted asset prices. In this paper, we analyze
individuals’ welfare to show that mitigating the uncertainty cannot be Pareto-improving byhurting some investors while it enhances the social welfare. We also show that appropriate
income transfers are needed to attain a Pareto improvement. A moderate lump-sum transferfrom the improved naïfs to the damaged sophisticates is helpful to carry out the purpose. On theother hand, an investment subsidy financed by lump-sum taxation is not an eﬀective instrumentfor Pareto-improving redistribution.
7

While the simple model used in this paper achieves a main part of our aim, there are somedirections to be extended for further research. First, we can obtain implications on economic
growth and poverty traps by developing a dynamic model with capital accumulation such as
Fukuda (2008). Second, it is important to endogenize a population ratio of sophisticates andnaïfs.7 Third, to assess quantitative eﬀects, we have to built a larger scale model and estimate
structural parameters including absolute risk aversion and the probability distribution perceived
by naïfs.

Appendix A: Derivations of (4) and (6)This appendix demonstrates that the utility maximization problems are represented by the meanvariance approach, (4) and (6). Consider the following constant absolute risk aversion utility
function, u(·):[][]
E j u(c̃ j ) = E j − exp(−c̃ j )for j = S, N.
Note that in the presence of uncertainty the expectation operator takes a diﬀerent form between
the sophisticate and the naïf. Since the stochastic process of c̃ j follows the normal distribution,
it reduces to
)]
[ (
[
]
1
(A.1)
E j u(c̃ j ) = − exp − E j [c̃ j ] − Var j [c̃ j ] .
2
From (3), the subjective expected mean and variance of c̃ j are respectively
E j [c̃ j ] = m j + E j [ṽ]x j= p x̄ + (E j [ṽ] − p)x j ,Var j [c̃ j ] = Var j [ṽ](x j )2,(A.2)

where the second equality in the first equation comes from (2).
The sophisticate knows the true probability distribution of investment payoﬀs, so that wehave
ES [ṽ] = v̂,VarS [ṽ] = σ̂.

(A.3)

The naïf does not know the true distribution and considers the worst case among the possible
set of mean payoﬀ and variance to avoid ambiguity (see Gilboa and Schmeidler 1989):{v̄min if xN ≥ 0,
EN [ṽ] =VarN [ṽ] = σ̄max .
(A.4)
v̄max if xN ≤ 0,
The first equation implies that the worst case of possible mean payoﬀ depends on a long andshort position. Substituting (A.2) into (A.1) and applying (A.3) and (A.4) to the result give (4)and (6) in the text.

7Mele and Sangiorgi (2015) analyze an incentive to reduce investment uncertainty by costly acquiring information on the payoﬀ distribution and find that the price swings occur.

8

Appendix B: Proof of Proposition 1
This appendix provides a mathematical proof of proposition 1. It is obvious the sign of (11) tobe ensured by the third equation in (10). If needed, we can present a closed-form solution by
substituting the third and first equations in (10) to eliminate pP and xNP from (11):UNP − UNN =

xNP
µ

{

[
(
)] }
µ2 σ̂ x̄µ(1 − µ)σ̄max
σ̂ x̄+v̄min − v̂ −,1 − µ 2[(1 − µ)σ̄max + µσ̂]
1− µ

(A.5)

which is positive under condition 2.
To obtain (12), we subtract the fifth equation in (9) from the fifth equation in (10) andeliminate pP and xSP from the result by using the third and second equations in (10):]σ̂ [ P 2
USP − USN = (pP − pN ) x̄ +(xS ) − (xSN )2
2
[
]
P
µσ̂xN
µ
N
P
=
x̄ − xS +x .
1− µ2(1 − µ) N
Eliminating xSN and xNP from the bracket in the second equality by substituting the second
equation in (9) and the first equation in (10) gives (12) in the text, or equivalently,
USP − USN = −

{ 2
[
(
)] }
xNP
µ2 σ̂
µ σ̂ x̄σ̂ x̄−v̄min − v̂ −.
1 − µ 1 − µ 2[(1 − µ)σ̄max + µσ̂]
1− µ

(A.6)

We now derive (14) in the text. From (13), we have))((
UWP − UWN = (1 − µ) USP − USN + µ UNP − UNN .
Applying (A.5) and (A.6) to this equation yields (14) in the text.
Let us turn to comparative statics in the participating equilibrium. Remember that xSN and
pN in the second and third equations in (10) are independent from v̄min and σ̄max . The total
diﬀerentiation of (10) and (13) with respect to v̄min generates
dxNP
1− µ
=
> 0,
dv̄min (1 − µ)σ̄max + µσ̂
dxSPµ dxNP=−< 0,
dv̄min1 − µ dv̄min
dpPµσ̂ dxNP=> 0,
dv̄min 1 − µ dv̄min
dUNP
dx P
dpP
= x̄
+ σ̄max xNP N > 0,dv̄mindv̄mindv̄minPdUSµ2 σ̂[(v̂ − v̄min ) + (σ̄max − σ̂) x̄]} dxNP=−
< 0,
dv̄min(1 − µ)[(1 − µ)σ̄max + µσ̂] dv̄mindUWP= µxNP > 0,dv̄min9where the sign of the fifth equation is determined by (1). Similarly, we totally diﬀerentiate (10)and (13) with respect to σ̄max to have
[
(
)]
dxNP
(1 − µ)2σ̂ x̄=−v̄min − v̂ −< 0,dσ̄max1− µ[(1 − µ)σ̄max + µσ̂]2
dxSPµ dxNP=−> 0,dσ̄max
1 − µ dσ̄max
dpPµσ̂ dxNP=< 0,
dσ̄max 1 − µ dσ̄max
dxNP
dUNP
1− µ
=
A
< 0,
dσ̄max 2(1 − µ)[(1 − µ)σ̄max + µσ̂] dσ̄max
dUSPµ2 σ̂[(v̂ − v̄min ) + (σ̄max − σ̂) x̄]} dxNP=−
> 0,
dσ̄max(1 − µ)[(1 − µ)σ̄max + µσ̂] dσ̄max
dUSPµ(xNP )2=−< 0,
dσ̄max
2
where

[

(

)]
σ̂ x̄
A ≡(1 − µ)σ̄max v̄min − v̂ −
1− µ
[
]
σ̂ x̄
+ µσ̂ (v̂ − v̄min ) + 2(σ̄max − σ̂) x̄ +
> 0.
1− µ

Under condition 2, the first equation is to be negative. Condition 2 and (1) ensure a positive signof A and thus determine the sign of the fourth equation. The fifth equation has a positive value
due to (1).

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