an optimal job, consumption/leisure, and investment policy

5
Operations Research Letters 42 (2014) 145–149 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl An optimal job, consumption/leisure, and investment policy Gyoocheol Shim a , Yong Hyun Shin b,a Department of Financial Engineering, Ajou University, Suwon 443749, Republic of Korea b Department of Mathematics, Sookmyung Women’s University, Seoul 140742, Republic of Korea article info Article history: Received 3 September 2013 Received in revised form 27 January 2014 Accepted 27 January 2014 Available online 2 February 2014 Keywords: Job choice Consumption Leisure Portfolio selection Martingale method abstract In this paper we investigate an optimal job, consumption, and investment policy of an economic agent in a continuous and infinite time horizon. The agent’s preference is characterized by the Cobb–Douglas utility function whose arguments are consumption and leisure. We use the martingale method to obtain the closed-form solution for the optimal job, consumption, and portfolio policy. We compare the optimal consumption and investment policy with that in the absence of job choice opportunities. © 2014 Elsevier B.V. All rights reserved. 1. Introduction We study an optimal job, consumption, and investment policy of an infinitely-lived economic agent whose preference is characterized by the Cobb–Douglas utility function of consumption and leisure. We consider two kinds of jobs one of which provides higher income but lower leisure than the other. We provide the closed-form solution for the optimal job, consumption, and investment policy by using the martingale and duality approach. We show that there is a threshold wealth level below which the optimally behaving agent chooses the job providing higher income, but above which he chooses the other job providing higher leisure. This is intuitively appealing since leisure is more important than income as the agent’s wealth level gets higher. We show that the agent in our model consumes less (resp. more) when the agent’s wealth is below (resp. above) the threshold level than he would if he did not have such job choice opportunities. We also show that the agent in our model takes more risk than he would without the job choice options. There have been many extensive research studies on continu- ous-time portfolio selection after Merton’s pioneering study (Merton [10,11]). Bodie, Merton, and Samuelson [1] have studied the effect of the labor–leisure choice on portfolio choice of an Corresponding author. E-mail addresses: [email protected] (G. Shim), [email protected] (Y.H. Shin). economic agent who has flexibility in his labor supply, by using the dynamic programming method. However they did not derive the closed-form solution. In this paper we use the martingale method to derive the closed-form solution. Many papers have considered portfolio selection with a retirement option: for example, Choi and Shim [2], Choi, Shim, and Shin [3], Dybvig and Liu [4], Farhi and Panageas [5], Lim and Shin [9], etc. The retirement in these papers is irreversible in that the agent cannot come back to his job after retirement, while the job choices in our model are reversible in that the agent can change the current job at any state and time. The rest of the paper proceeds as follows. Section 2 sets up the optimization problem. Section 3 provides a solution to the problem and Section 4 investigates properties of the optimal policy. 2. The model We consider the continuous-time financial market in an infinite-time horizon. We assume that there are two financial assets in the market: one is a riskless asset and the other is a risky asset. The risk-free interest rate r > 0 is assumed to be a constant and the price S t of the risky asset is governed by the geometric Brownian motion dS t /S t = µdt + σ dB t for t 0, where (B t ) t =0 is a standard Brownian motion on the underlying probability space (, F , P) and the parameters µ and σ > 0 are assumed to be constants. We let {F t } t 0 be the augmentation under P of the natural filtration generated by the standard Brownian motion (B t ) t =0 . Let Θ t denote the job of an economic agent at time t . The job process 2 , (Θ t ) t =0 is F t -adapted. For simplicity, we assume http://dx.doi.org/10.1016/j.orl.2014.01.009 0167-6377/© 2014 Elsevier B.V. All rights reserved.

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Page 1: An optimal job, consumption/leisure, and investment policy

Operations Research Letters 42 (2014) 145–149

Contents lists available at ScienceDirect

Operations Research Letters

journal homepage: www.elsevier.com/locate/orl

An optimal job, consumption/leisure, and investment policyGyoocheol Shim a, Yong Hyun Shin b,∗

a Department of Financial Engineering, Ajou University, Suwon 443749, Republic of Koreab Department of Mathematics, Sookmyung Women’s University, Seoul 140742, Republic of Korea

a r t i c l e i n f o

Article history:Received 3 September 2013Received in revised form27 January 2014Accepted 27 January 2014Available online 2 February 2014

Keywords:Job choiceConsumptionLeisurePortfolio selectionMartingale method

a b s t r a c t

In this paper we investigate an optimal job, consumption, and investment policy of an economic agentin a continuous and infinite time horizon. The agent’s preference is characterized by the Cobb–Douglasutility function whose arguments are consumption and leisure. We use the martingale method to obtainthe closed-form solution for the optimal job, consumption, and portfolio policy. We compare the optimalconsumption and investment policy with that in the absence of job choice opportunities.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

We study an optimal job, consumption, and investmentpolicy of an infinitely-lived economic agent whose preference ischaracterized by the Cobb–Douglas utility function of consumptionand leisure. We consider two kinds of jobs one of which provideshigher income but lower leisure than the other. We providethe closed-form solution for the optimal job, consumption, andinvestment policy by using the martingale and duality approach.We show that there is a threshold wealth level below which theoptimally behaving agent chooses the job providing higher income,but above which he chooses the other job providing higher leisure.This is intuitively appealing since leisure is more important thanincome as the agent’s wealth level gets higher. We show that theagent in our model consumes less (resp. more) when the agent’swealth is below (resp. above) the threshold level than he would ifhe did not have such job choice opportunities. We also show thatthe agent in our model takes more risk than he would without thejob choice options.

There have been many extensive research studies on continu-ous-time portfolio selection after Merton’s pioneering study(Merton [10,11]). Bodie, Merton, and Samuelson [1] have studiedthe effect of the labor–leisure choice on portfolio choice of an

∗ Corresponding author.E-mail addresses: [email protected] (G. Shim), [email protected]

(Y.H. Shin).

http://dx.doi.org/10.1016/j.orl.2014.01.0090167-6377/© 2014 Elsevier B.V. All rights reserved.

economic agentwho has flexibility in his labor supply, by using thedynamic programming method. However they did not derive theclosed-form solution. In this paper we use the martingale methodto derive the closed-form solution. Many papers have consideredportfolio selectionwith a retirement option: for example, Choi andShim [2], Choi, Shim, and Shin [3], Dybvig and Liu [4], Farhi andPanageas [5], Lim and Shin [9], etc. The retirement in these papersis irreversible in that the agent cannot come back to his job afterretirement,while the job choices in ourmodel are reversible in thatthe agent can change the current job at any state and time.

The rest of the paper proceeds as follows. Section 2 sets up theoptimization problem. Section 3 provides a solution to the problemand Section 4 investigates properties of the optimal policy.

2. The model

We consider the continuous-time financial market in aninfinite-time horizon. We assume that there are two financialassets in the market: one is a riskless asset and the other is a riskyasset. The risk-free interest rate r > 0 is assumed to be a constantand the price St of the risky asset is governed by the geometricBrownian motion dSt/St = µdt + σdBt for t ≥ 0, where (Bt)

t=0 isa standard Brownian motion on the underlying probability space(Ω, F , P) and the parameters µ and σ > 0 are assumed tobe constants. We let Ftt≥0 be the augmentation under P ofthe natural filtration generated by the standard Brownian motion(Bt)

t=0.Let Θt denote the job of an economic agent at time t . The job

process 2 , (Θt)∞

t=0 is Ft-adapted. For simplicity, we assume

Page 2: An optimal job, consumption/leisure, and investment policy

146 G. Shim, Y.H. Shin / Operations Research Letters 42 (2014) 145–149

that there are two kinds of jobs, A0 and A1. The agent receivesconstant labor income Yi > 0 and have a leisure rate Li at eachjob Ai, i = 0, 1, where0 ≤ Y0 < Y1 and 0 < L1 < L0.Let ct ≥ 0 and πt denote the consumption rate and the amountof money invested in the risky asset, respectively, at time t . Theconsumption rate process c , (ct)∞t=0 and the portfolio processπ , (πt)

t=0 are Ft-progressively measurable, t0 csds < ∞ for all

t ≥ 0 almost surely (a.s.), and t0 π2

s ds < ∞ for all t ≥ 0 a.s.Thus the agent’s wealth process (Xt)

t=0 with X0 = x evolvesaccording todXt =

rXt + (µ − r)πt − ct + Y01Θt=A0

+ Y11Θt=A1dt + σπtdBt . (2.1)

The present value of the future labor income stream is Yi/r forΘt = Ai where i = 0, 1. Since Y1/r > Y0/r and the job stateprocess 2 is chosen endogenously by the agent, we let X0 = x >−Y1/r and the agent faces the following wealth constraint:

Xt ≥ −Y1

r, for all t ≥ 0 a.s. (2.2)

We call a triple of control (2, c, π) satisfying the above conditionsincluding (2.2) with X0 = x > −Y1/r admissible at x. Let A(x) bethe set of all admissible policies.

We assume that the agent has the Cobb–Douglas utility functionu(ct , lt), as in Farhi and Panageas [5]:

u(ct , lt) ,1α

cαt l

1−αt1−γ

1 − γ, 0 < α < 1 and 0 < γ = 1, (2.3)

where γ is the agent’s coefficient of relative risk aversion, α is aconstant, and lt is the leisure rate at time t . Let γ1 , 1− α(1− γ ),and then 0 < γ1 = 1 and the Cobb–Douglas utility function u(·, ·)in (2.3) can be rewritten as

u(ct , lt) = lγ1−γt

c1−γ1t

1 − γ1.

Remark 2.1. If γ > 1, then γ > γ1 > 1, γ11−γ1

< 0 and Lγ1−γ

γ10

− Lγ1−γ

γ11 < 0. If 0 < γ < 1, then 0 < γ < γ1 < 1, γ1

1−γ1> 0 and

Lγ1−γ

γ10 − L

γ1−γγ1

1 > 0. Thus the following inequality always holds:

γ1

1 − γ1

L

γ1−γγ1

0 − Lγ1−γ

γ11

> 0.

Problem 2.1. The agent’s optimization problem is tomaximize theexpected utility

J(x; 2, c, π) = E

0e−ρt

Lγ1−γ

0c1−γ1t

1 − γ11Θt=A0

+ Lγ1−γ

1c1−γ1t

1 − γ11Θt=A1

dt

,

over (2, c, π) ∈ A(x), where ρ > 0 is a subjective discount factor.

Thus the value function V (x) is given byV (x) = sup

(2,c,π)∈A(x)J(x; 2, c, π).

Assumption 2.1. We assume, as in Farhi and Panageas [5], that

K1 , r +ρ − r

γ1+

γ1 − 12γ 2

1θ2 > 0,

where θ , (µ − r)/σ , called the market price of risk.

3. The solution to the optimization problem

We denote the state price density by Ht :

Ht , e−

r+ 1

2 θ2t−θBt

.

For any fixed T ∈ [0, ∞), we denote the equivalent martingalemeasure byPT :PT (A) = E

e−

12 θ2T−θBT 1A

, for A ∈ FT .

By theGirsanov theorem, thenewprocessBt = Bt+θ t is a standardBrownian motion for t ∈ [0, T ] under the measurePT . As shownin Proposition 7.4 in Section 1.7 of Karatzas and Shreve [7], thereexists a unique probability measureP onF∞ which agrees withPT

on FT , for T ∈ [0, ∞), andBt is a standard Brownian motion fort ∈ [0, ∞) underP. Thus Eq. (2.1) can be rewritten as

dXt =rXt − ct + Y01Θt=A0 + Y11Θt=A1

dt + σπtdBt . (3.1)

By (2.2) and (3.1), we derive, similarly to Lim and Shin [9], thefollowing budget constraint:

E

0

ct − Y01Θt=A0 − Y11Θt=A1

Htdt

≤ x.

For a Lagrange multiplier λ > 0, a dual value function isV (λ) + λx

= sup(2,c,π)∈A(x)

E

0e−ρt

Lγ1−γ

0c1−γ1t

1 − γ11Θt=A0

+ Lγ1−γ

1c1−γ1t

1 − γ11Θt=A1

dt

− λ

0

ct − Y01Θt=A0 − Y11Θt=A1

Htdt

+ λx

= E

0e−ρt u0(zt)10<zt≤z +u1(zt)1zt>z

dt

+ λx, (3.2)

if the job and consumption strategy (Θλt , cλ

t ) is given by

Θλt =

A0, if 0 < zt ≤ z,A1, if zt > z,

cλt =

Lγ1−γ

γ10 (zt)

−1γ1 , if 0 < zt ≤ z,

Lγ1−γ

γ11 (zt)

−1γ1 , if zt > z,

where

ui(z) = supc≥0

Lγ1−γ

ic1−γ1

1 − γ1− cz

+ Yiz

= Lγ1−γ

γ1i

γ1

1 − γ1z−

1−γ1γ1 + Yiz, i = 0, 1,

zt , λeρtHt = λeρ−r− 1

2 θ2t−θBt

, (3.3)

and z is the solution to the algebraic equationu0(z) =u1(z):

z =

γ1

1−γ1

L

γ1−γγ1

0 − Lγ1−γ

γ11

Y1 − Y0

γ1

> 0, (3.4)

which is positive by Remark 2.1. Similarly to Proposition 6.5 inKaratzas andWang [8], the Lagrangemultiplierλ is chosen in (3.12)so that

E

0

cλt − Y010<zt≤z − Y11zt>z

Htdt

= x (3.5)

Page 3: An optimal job, consumption/leisure, and investment policy

G. Shim, Y.H. Shin / Operations Research Letters 42 (2014) 145–149 147

is satisfied. From the equality (3.5), we see that there exists areplicating portfolio π such that (2, c, π) is admissible at x (seeTheorem 9.4 in Section 3.9 of Karatzas and Shreve [7]).

ThusV (λ) can be rewritten as

V (λ) = E

0e−ρt

L

γ1−γγ1

0γ1

1 − γ1z−

1−γ1γ1

t + Y0zt

10<zt≤z

+

L

γ1−γγ1

1γ1

1 − γ1z−

1−γ1γ1

t + Y1zt

1zt>z

dt

.

Itô’s formula to the process zt in (3.3) implies the stochasticdifferential equation (SDE) dzt/zt = (ρ − r)dt − θdBt , z0 = λ.Now we define the function

φ(t, z) , Ezt=z

te−ρs

L

γ1−γγ1

0γ1

1 − γ1z−

1−γ1γ1

s

+ Y0zs

10<zs≤z +

L

γ1−γγ1

1γ1

1 − γ1z−

1−γ1γ1

s

+ Y1zs

1zs>z

ds

, (3.6)

then, by Feynman–Kac formula, the functionφ(t, z) =

φ0(t, z), for 0 < z ≤ z,φ1(t, z), for z > z, in (3.6) is the solution to the

following partial differential equations (PDEs)

Lφ0 + e−ρtL

γ1−γγ1

0γ1

1 − γ1z−

1−γ1γ1 + Y0z

= 0,

for 0 < z ≤ z,

Lφ1 + e−ρtL

γ1−γγ1

1γ1

1 − γ1z−

1−γ1γ1 + Y1z

= 0,

for z > z,

(3.7)

with the boundary conditions φ0(t, z) = φ1(t, z) and∂φ0∂z (t, z) =

∂φ1∂z (t, z), where the partial differential operator is defined by

L ,∂

∂t+ (ρ − r)z

∂z+

12θ2z2

∂2

∂z2.

Remark 3.1. For later use, we consider the quadratic equation

f (n) ,12θ2n2

+

ρ − r −

12θ2n − ρ

=12θ2(n − n+)(n − n−) = 0,

where two roots are n+ > 1 and n− < 0.

Remark 3.2. Note that

n− < −1 − γ1

γ1< n+ (3.8)

since f (−(1 − γ1)/γ1) = −K1 < 0.

Assumption 2.1 is equivalent to n− < −(1 − γ1)/γ1 as shownin Remark 3.2, which again is equivalent to γ1 > 1/(1 − n−). Thatis, Assumption 2.1 is that γ1 is not too small so that the marginalutility ∂u(c, l)/∂c = lγ1−γ c−γ1 of consumption decreases not tooslowly as c goes to infinity for a given l. This assumption ensuresa finite value function (for example, see the paragraph below thecondition (2.6) in Karatzas et al. [6]).

Remark 3.3. If we define a function g1(x) for n− < x < n+ asfollows:

g1(x) , −f (x)

x − n−

= −12θ2(x − n+) > 0,

then g1(x) is a decreasing function for n− < x < n+. Thus we have

g1

1 − γ1

γ1

> g1(1) > 0 ⇒ −

K11−γ1γ1

+ n−

>r

1 − n−

> 0

1−γ1γ1

+ n−

K1+

1 − n−

r> 0. (3.9)

Also if we define a function g2(x) for n− < x < n+ as follows:

g2(x) , −f (x)

x − n+

= −12θ2(x − n−) < 0,

then g2(x) is also a decreasing function for n− < x < n+. Thus wehave

g2(1) < g2

1 − γ1

γ1

< 0 ⇒

r1 − n+

< −K1

1−γ1γ1

+ n+

< 0

1−γ1γ1

+ n+

K1+

1 − n+

r> 0. (3.10)

Proposition 3.1. Let

v(z) =

C1zn+ + L

γ1−γγ1

0γ1

(1 − γ1)K1z−

1−γ1γ1 +

Y0

rz,

for 0 < z ≤ z,

D2zn− + Lγ1−γ

γ11

γ1

(1 − γ1)K1z−

1−γ1γ1 +

Y1

rz, for z > z,

where

C1 =

n−γ1+1−γ1γ1K1

+1−n−

r

(n+ − n−)zn+−1(Y1 − Y0) > 0 and

D2 =

n+γ1+1−γ1γ1K1

+1−n+

r

(n+ − n−)zn−−1(Y1 − Y0) > 0,

and then φ(t, z) = e−ρtv(z) is a solution to the PDEs (3.7) with theboundary conditions.

Proof. For 0 < z ≤ z, we have the PDE

Lφ + e−ρtL

γ1−γγ1

0γ1

1 − γ1z−

1−γ1γ1 + Y0z

= 0.

If we conjecture a trial solution of the form φ(t, z) = e−ρtv(z),then we derive the ordinary differential equation (ODE) withrespect to z

12θ2z2v′′(z) + (ρ − r)zv′(z) − ρv(z)

+ Lγ1−γ

γ10

γ1

1 − γ1z−

1−γ1γ1 + Y0z = 0. (3.11)

So the solution of the ODE (3.11) is

v(z) = C1zn+ + Lγ1−γ

γ10

γ1

(1 − γ1)K1z−

1−γ1γ1 +

Y0

rz.

Similarly, for z > z, we obtain the solution

v(z) = D2zn− + Lγ1−γ

γ11

γ1

(1 − γ1)K1z−

1−γ1γ1 +

Y1

rz.

Page 4: An optimal job, consumption/leisure, and investment policy

148 G. Shim, Y.H. Shin / Operations Research Letters 42 (2014) 145–149

Now we use the smooth-pasting condition at z = z to deter-mine the constants C1 and D2 as follows:

C1 =

Lγ1−γ

γ10 −L

γ1−γγ1

1

(n−γ1+1−γ1)

(1−γ1)K1z−

1γ1 +

(1−n−)(Y1−Y0)r

(n+ − n−)zn+−1

=

n−γ1+1−γ1γ1K1

+1−n−

r

(n+ − n−)zn+−1(Y1 − Y0),

and

D2 =

Lγ1−γ

γ10 −L

γ1−γγ1

1

(n+γ1+1−γ1)

(1−γ1)K1z−

1γ1 +

(1−n+)(Y1−Y0)r

(n+ − n−)zn−−1

=

n+γ1+1−γ1γ1K1

+1−n+

r

(n+ − n−)zn−−1(Y1 − Y0).

From (3.9) and (3.10), we see that C1 > 0 and D2 > 0, respec-tively.

From (3.6) and Proposition 3.1, we haveV (λ) = φ(0, λ) = v(λ).Thus we use the Legendre transform inverse formula to obtain thevalue function V (·) as follows:

V (x) = infλ>0

(v(λ) + λx) , (3.12)

for any x ∈ (−Y1/r, ∞).

Theorem 3.1. The value function V (·) is given by

V (x) =

C1(λ0)n+ + L

γ1−γγ1

0γ1

(1 − γ1)K1(λ0)

−1−γ1γ1

+

x +

Y0

r

(λ0), for x ≥ x,

D2(λ1)n− + L

γ1−γγ1

1γ1

(1 − γ1)K1(λ1)

−1−γ1γ1

+

x +

Y1

r

(λ1), for − Y1/r < x < x,

where λ0 and λ1 are determined from the following algebraicequations

x = −n+C1(λ0)n+−1

+ Lγ1−γ

γ10

1K1

(λ0)−

1γ1 −

Y0

r(3.13)

and

x = −n−D2(λ1)n−−1

+ Lγ1−γ

γ11

1K1

(λ1)−

1γ1 −

Y1

r, (3.14)

respectively.

Remark 3.4. If we substitute z in (3.4) for λ0 and λ1 into (3.13) and(3.14), respectively, then we can define the wealth level x as

x =

−n+n−γ1−n++n+γ1γ1K1

+n+n−−n+

r

n+ − n−

+L

γ1−γγ1

0

Lγ1−γ

γ10 − L

γ1−γγ1

1

1 − γ1

γ1K1

(Y1 − Y0) −Y0

r

=

−n+n−γ1−n−+n−γ1γ1K1

+n+n−−n−

r

n+ − n−

+L

γ1−γγ1

1

Lγ1−γ

γ10 − L

γ1−γγ1

1

1 − γ1

γ1K1

(Y1 − Y0) −Y1

r.

Theorem 3.2. The optimal policy to Problem 2.1 is (Θ∗, c∗, π∗) suchthat

Θ∗

t =

A1, if − Y1/r < Xt < x,A0, if Xt ≥ x,

c∗

t =

L

γ1−γγ1

1

zλ1t

−1γ1

, if − Y1/r < Xt < x,

Lγ1−γ

γ10

zλ0t

−1γ1

, if Xt ≥ x,

and

π∗

t =

θ

σ

n−(n− − 1)D2

zλ1t

n−−1+ L

γ1−γγ1

11

γ1K1

zλ1t

−1γ1

,

if −Y1/r < Xt < x,

θ

σ

n+(n+ − 1)C1

zλ0t

n+−1+ L

γ1−γγ1

01

γ1K1

zλ0t

−1γ1

,

if Xt ≥ x,

where zλ0t and zλ1

t are determined from the algebraic equations

Xt = −n+C1

zλ0t

n+−1+ L

γ1−γγ1

01K1

zλ0t

−1γ1

−Y0

r(3.15)

and

Xt = −n−D2

zλ1t

n−−1+ L

γ1−γγ1

11K1

zλ1t

−1γ1

−Y1

r, (3.16)

respectively.

Remark 3.5. It can easily be shown that dXt/dzλ0t < 0 and

dXt/dzλ1t < 0 so that Xt in Theorem 3.2 is a decreasing function

of zλ0t and zλ1

t , for Xt ≥ x and for −Y1/r < Xt < x, respectively.

4. The properties of the solution

We compare the optimal consumption and investment ruleswith those without the job choice options. If the agent’s job werepermanently Ai ∈ A0, A1 without job-switching opportunity,then the optimal consumption/investment strategy, say (cMi , πMi),under the wealth constraint Xt ≥ −Yi/r with X0 = x ≥ −Yi/r ,would be

cMit = L

γ1−γγ1

i

zMit

−1γ1

= K1

Xt +

Yi

r

,

πMit =

θ

σγ1L

γ1−γγ1

i1K1

zMit

−1γ1

σγ1

Xt +

Yi

r

,

where

zMit =

Lγ1−γ

i

K γ11

Xt +

Yi

r

−γ1

, i = 0, 1, (4.1)

Page 5: An optimal job, consumption/leisure, and investment policy

G. Shim, Y.H. Shin / Operations Research Letters 42 (2014) 145–149 149

which can be proved in the sameway as inMerton [10] or Karatzaset al. [6].

Proposition 4.1. We havec∗

t > cM0t , if Xt ≥ max(−Y0/r, x)

c∗

t < cM1t , if − Y1/r < Xt < x.

Proof. For Xt ≥ max(−Y0/r, x), substituting zM0t in (4.1) for zλ0

tinto Xt in (3.15), and then we obtain

Xt |zλ0t =z

M0t

= −n+C1

zM0t

n+−1+ L

γ1−γγ1

01K1

zM0t

−1γ1

−Y0

r

= −n+C1

Lγ1−γ

0

K γ11

Xt +

Y0

r

−γ1n+−1

+ Xt +Y0

r−

Y0

r< Xt ,

where the inequality is obtained from the fact C1 > 0 in Proposi-tion 3.1. SinceXt is a decreasing functionwith respect to zt , we have

zM0t > zλ0

t , (4.2)

and consequently we obtain

cM0t = L

γ1−γγ1

0

zM0t

−1γ1

< Lγ1−γ

γ10

zλ0t

−1γ1

= c∗

t .

For −Y1/r < Xt < x, substituting zM1t in (4.1) for zλ1

t into Xt in(3.16), and then we also obtain

Xt |zλ1t =z

M1t

= −n−D2

zM1t

n−−1+ L

γ1−γγ1

11K1

zM1t

−1γ1

−Y1

r

= −n−D2

Lγ1−γ

1

K γ11

Xt +

Y1

r

−γ1n−−1

+ Xt +Y1

r−

Y1

r> Xt ,

where the inequality is obtained from the facts n− < 0 and D2 > 0in Proposition 3.1. Since Xt is a decreasing function with respect tozt , we have

zM1t < zλ1

t ,

and consequently we obtain

cM1t = L

γ1−γγ1

1

zM1t

−1γ1

> Lγ1−γ

γ11

zλ1t

−1γ1

= c∗

t .

Proposition 4.2. We haveπ∗

t > πM0t , if Xt ≥ max(−Y0/r, x)

π∗

t > πM1t , if − Y1/r < Xt < x.

Proof. For Xt ≥ max(−Y0/r, x), we obtain

π∗

t =θ

σ

n+(n+ − 1)C1

zλ0t

n+−1+ L

γ1−γγ1

01

γ1K1

zλ0t

−1γ1

σγ1L

γ1−γγ1

01K1

zλ0t

−1γ1

σγ1L

γ1−γγ1

01K1

zM0t

−1γ1

= πM0t ,

where the first inequality is obtained from the fact n+(n+−1)C1 >0 and the second inequality is obtained from the inequality (4.2).

For −Y1/r < Xt < x, we obtain

π∗

t =θ

σ

n−(n− − 1)D2

zλ1t

n−−1+ L

γ1−γγ1

11

γ1K1

zλ1t

−1γ1

σγ1

γ1n−(n− − 1)D2

zλ1t

n−−1+ L

γ1−γγ1

11K1

zλ1t

−1γ1

σγ1

−n−D2

zλ1t

n−−1+ L

γ1−γγ1

11K1

zλ1t

−1γ1

+ n−

1 + γ1(n− − 1)

D2

zλ1t

n−−1

σγ1

−n−D2

zλ1t

n−−1+ L

γ1−γγ1

11K1

zλ1t

−1γ1

=

θ

σγ1

Xt +

Y1

r

= π

M1t ,

where the inequality comes from (3.8) and the fourth equality from(3.16).

Acknowledgments

We are grateful to an anonymous referee and an area editor,J.S. Keppo for their helpful comments and suggestions. We thankHyeng Keun Koo for his kind advice. The work of Prof. Shimwas supported by the WCU (World Class University) programthrough the National Research Foundation of Korea funded by theMinistry of Education, Science and Technology (R31-2009-000-20007-0) and thework of Prof. Shinwas supported by Basic ScienceResearch Program through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (Grant No. NRF-2013R1A1A2058027).

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