Actuarial Studies Seminar Macquarie University

29 July 2009

Aihua Zhang Lecturer, Nottingham University Business School, China

A closed-form solution for the continuous-time consumption model with

endogenous labor income In this paper we study the consumption, labor supply, and portfolio decisions of an in nitely-lived individual who receives a wage rate and income from investment into a risky asset and a risk-free bond. Uncertainty about labor income arises endogenously, because labor supply evolves randomly over time in response to changes in nancial wealth. We derive closed-form solutions for optimal consumption, labor supply and investment strategy. We also obtain approximately log-linear relationships between optimal consumption, labor supply and retirement age, respectively. Moreover, we derive Euler equation under uncertainty of asset returns and derive a similar growth equation for expected optimal labor supply. The effects of risk-aversion coecients on optimal decisions are examined.

A closed-form solution for the continuous-time

consumption model with endogenous labor income

Aihua Zhang∗

Nottingham University Business School, China Campus

July 20, 2009

Abstract

In this paper we study the consumption, labor supply, and portfo-lio decisions of an infinitely-lived individual who receives a wage rateand income from investment into a risky asset and a risk-free bond.Uncertainty about labor income arises endogenously, because laborsupply evolves randomly over time in response to changes in financialwealth. We derive closed-form solutions for optimal consumption, la-bor supply and investment strategy. We also obtain approximatelylog-linear relationships between optimal consumption, labor supplyand retirement age, respectively. Moreover, we derive Euler equationunder uncertainty of asset returns and derive a similar growth equa-tion for expected optimal labor supply. The effects of risk-aversioncoefficients on optimal decisions are examined.

Keywords: Labor supply decisions, portfolio optimization with wage in-come, Euler equation, martingale method

JEL subject classifications: C61; C73; G1; J22

∗Corresponding address: Nottingham University Business School China Campus; Ad-ministration Building, 199 Taikang EastRoad; Ningbo, 315100; China. E-mail: Ai-hua.Zhang@nottingham.edu.cn

1

1 Introduction

In this paper we analyze a continuous-time model of optimal consumption in

which both asset returns and labor income are stochastic. There is a riskless

asset earning an exogenous (and constant) rate of interest. The risky asset

is modeled by a geometric Brownian motion. Labor income is determined by

the interaction of an endogenous labor supply decision with the stochastic

market return on labor supply. The utility function is assumed to be a linear

combination of two CRRA utility functions with respect to consumption and

labor supply, respectively.

Interestingly, our model appears to share some similarities with the model

considered by Bodie, Merton and Samuelson (BMS) (1992) and is even closer

to the more recent work by Bodie et al (2004).1 The similarities include that

all of us study the problem of maximizing expected discounted lifetime util-

ity and consider a utility function of two arguments: consumption and labor

supply/leisure. In BMS (1992), these two arguments are treated as one com-

posite good, which makes the model as if there was only one consumption

good. In our case, we have a more general utility function which is (addi-

tively) separable in consumption and labor supply. Bodie et al (2004) extend

the original BMS’s model by incorporating a habit formation utility. As an-

other similarity, we all assume a complete market, in which wage incomes

are perfectly hedged.

1This model was originally developed independently in my PhD thesis (2007). I wishto take this opportunity to thank two anonymous referees for the comments on an earlyversion of this paper and for making me aware of these two prior works.

1

However, the prior works mentioned above do not have closed forms for

optimal consumption and labor supply except closed forms for optimal port-

folio.2 As one of our main contributions, we obtain closed-form solutions for

consumption and labor supply. This is summarized in Theorem 1. We find

that optimal consumption and labor supply depend on the market deflator

1H(t)

, which is stochastic, implying that labor supply is endogenously stochas-

tic. As shown by the market deflator, when the financial market performs

better (worse), the individual is then allowed to consume more (less) and

work less (more). Similar to Bodie et al (2004), an exogenous retirement

age T is taken into account to separate the individual’s life time into two

distinct periods: working life and retirement. As another contribution, we

find that optimal consumption and labor supply are approximately log-linear

in retirement age and exponentially grow in time. Additionally, optimal la-

bor supply is also log-linear in wage rate. It shows that postponing the

retirement age increases optimal consumption and decreases optimal labor

supply.3 Moreover, We establish the Euler equation under uncertainty of

the financial market, finding that the uncertainty gives rise to an additional

term corresponding to the market price of risk in the Euler equation under

certainty. This is represented in Eq. (28). Finally, by examining the effects

of the individual’s risk-aversion coefficients (γ and η) on optimal consump-

2Bodie et al (2004) do provide explicit solutions for optimal consumption and laborsupply, but the solutions depend on the Shadow price that is to be solved uniquely undersome restrictions made. In this sense, their solutions are not closed, at least, not as closedas ours. But their solutions are for a more general setup.

3This corresponds to the case when consumption and labor supply are gross comple-ments in Bodie et al (2004).

2

tion and labor supply, respectively, we find that (at optimum) the individual

reduces his consumption and boosts his labor supply as he gets more risk

averse. Optimal consumption (labor supply) is smoothened for large values

of the risk-aversion coefficient with respect to consumption γ (with respect

to labor supply η).

The rest of the paper is organized as follows: In section 2, we set up our

model and our main results are discussed in section 3. Technical details are

given in Appendix.

2 Description of the model

We assume that an infinitely-lived individual born with no initial wealth

works only when young (that is, before retirement age T > 04). At each

time t, for t ≤ T , he supplies an amount of labor Lt, receiving a wage

rate wt, so that his wage income in period t is Ltwt. Given an initial wage

w0 > 0, the wage grows at the constant rate a > 05 . The income is invested

into a risk-free bond offering a gross return r and a risky asset offering an

instantaneous expected gross return µ. After the individual retires, his post-

retirement consumption is financed by his savings when young and the capital

gains from the investment. His objective is to maximize his lifetime utility

by choosing an optimal consumption, an optimal amount of labor supply

(during his working life) and an optimal portfolio investment stream.

4The retirement age T is exogenous.5For simplicity, we assume a nonstochastic wage. But the structure of our solution still

applies when wage is perfectly correlated with the risky asset as in Bodie et al (2004) andBMS (1992).

3

2.1 The dynamics of asset prices

In the continuous-time financial market, the price of a risk free bond, denoted

by Bt, satisfies

dBt

Bt

= rdt (1)

where, r > 0 is the nominal interest rate.

The price of the risky asset follows a geometric Brownian motion

dStSt

= µdt+ σdWt, (2)

where µ is the expected nominal return on the risky asset per unit time, σ

(σ 6= 0) is the volatility of the asset price. The ’uncertainty’ of asset price

is generated by a Brownian motion Wt : t ∈ [0,∞) defined on a given

probability space (Ω,F ,P), where Ω is the set of possible states of nature,

F is a σ-algebra of observable events and P is the associated probability

measure. The nature filtration Ft : t ∈ [0,∞), generated by the Brownian

motion Wt : t ∈ [0,∞), captures the flow of information. For simplicity,

the market coefficients r, µ and σ are assumed to be constant.

2.2 The utility function

The instantaneous utility function is defined by

u(Ct, Lt) =C1−γt

1− γ− b L

1+ηt

1 + η(3)

4

where, γ captures the individual’s degree of risk-aversion with respect to

consumption Ct ≥ 0 and we assume that γ > 1 such that utility from con-

sumption is bounded from above,6 η reflects the individual’s disutility from

working and we assume that η > 0 such that disutility from labor supply

Lt ≥ 0 is convex, and the coefficient b is strictly positive, together with the

negative sign, indicating disutility gained from working.

2.3 The wealth process

We assume that the individual stops working after reaching his retirement

age T . In order to capture this fact in his lifetime horizon, we introduce a

dummy variable as follows:

1t =

0 , T ≤ t <∞

1 , 0 ≤ t < T(4)

If we assume that at time t the individual invests a proportion of πt of

his wealth into the risky asset and 1 − πt into the risk-free bond,7 then his

wealth process Xt must satisfy:8

dXt = Xtrdt+ πt[(µ− r)dt+ σdWt] − Ctdt+ wtLt1tdt

X0 = 0 (5)

6It is easy to check from the utility function (3) that zero consumption would incur apenalty of negative infinite utility, ensuring that consumption will always be positive.

7Here there is no restriction on short selling, so πt is allowed to become both positiveand negative.

8For a standard wealth process, we refer to classic financial mathematics textbooks,such as Korn/Korn (2000) and Shreve (2004).

5

This says that the change in wealth must equal capital gains less (infinites-

imal) consumption plus (infinitesimal) labor income during working life or

equal to the difference between capital gains and (infinitesimal) consumption

during retirement. Labor income is equal to the amount of labor supply Lt

times a wage rate wt and wage grows exponentially at a constant rate of a

with a strictly positive initial wage w0,9 that is

wt = w0eat (6)

Define the stochastic discount factor (also known as state price density)

Ht by

Ht ≡ e−rt−12θ2t−θWt (7)

where, θ is the market price of risk defined by

θ ≡ µ− rσ

(8)

It is clear that the resulting financial market is complete and free of

arbitrage. As a consequence, the individual’s current wealth must equal the

present value of his future consumption less the present value of his future

labor income on average. In other words, the resources for his expected

future consumption come from the current value of his accumulated financial

9Note that although we assume a non-stochastic wage rate, labor supply evolves en-dogenously in response to stochastic shocks to asset returns, so labor income wtLt is stillstochastic. We should emphasize that, when wage is stochastic and perfectly correlatedwith the risky asset, our results still hold.

6

wealth through investment plus the expected present value of his future labor

income if he is still working. Formally, the wealth process Xt must satisfy

that

Xt = Et

[∫ ∞t

Hs

Ht

Csds

]− Et

[∫ ∞t

Hs

Ht

wsLs1sds

](9)

where, Et is the conditional expectation given the information up to and

including time t which is denoted by Ft and Ft ⊆ F . By the definition of

the dummy variable 1t in (4), Eq. (9) can be further expressed as

Xt = Et

[∫ ∞t

Hs

Ht

Csds

]− Et

[∫ T

t

Hs

Ht

wsLsds

](10)

In particular, at time zero,10 we have

E[∫ ∞

0

HsCsds

]= E

[∫ T

0

HswsLsds

]. (11)

Eq. (11) implies that expected life time consumption is expected to be fi-

nanced by future labor incomes at origin. This is because that there is no

initial wealth to be invested in the financial market, and the future financial

wealth comes from investing labor incomes into the market.

2.4 The maximization problem

We start by defining admissibility, which is equivalent to the non-negative

constraint on expected life-time consumption.

10Note that H0 = 1 and X0 = 0

7

Definition 2.1. A consumption-labor supply-portfolio process set (Ct, Lt, πt)

is said to be admissible if

Xt + Et

[∫ ∞t

Hs

Ht

wsLs1sds

]≥ 0, for all t, (12)

with probability one. The resulting class of admissible sets is denoted by A.

It follows from (9) that the inequality of (12) is equivalent to

Et

[∫ ∞t

Hs

Ht

Csds

]≥ 0, for all t, almost surely. (13)

It can be seen from (12) that the wealth before retirement age T is al-

lowed to become negative so long as that the present value of future labor

income is large enough to offset such a negative value (See Karatzas (1997),

page 63, for the same argument)11.

The individual wishes to maximize the expected total discounted utility

by choosing an optimal consumption-labor supply-portfolio set over the class

A1 ≡

(Ct, Lt, πt) ∈ A : E[∫ ∞

0

e−ρtu−(Ct, Lt)dt

]<∞

, (14)

where, u− ≡ max−u, 0.11However, this assumption is not always realistic, in particular when liquidity is low

and restrictions are made on the amount of borrowing against future expected income.A way to include liquidity features into the model has been shown, for example, by ElKaroui and Jeanblanc (1998), Bodie (2004) and used by Zhang and Ewald (2009) to solvea related problem.

8

The optimization problem is then given by

max(Ct,Lt,πt)∈A1

E[∫ ∞

0

e−ρtu(Ct, Lt)dt

](15)

subject to

dXt = Xtrdt+ πt[(µ− r)dt+ σdWt] − Ctdt+ wtLt1tdt

X0 = 0 (16)

where, the discount rate satisfies ρ > 0.12

3 Optimal policies

Following the Martingale method we conclude that the (dynamic) maximiza-

tion problem (15)-(16) is equivalent to the following problem13

maxCt,Lt

E[∫ ∞

0

e−ρtu(Ct, Lt)dt

](17)

subject to

E[∫ ∞

0

HtCtdt

]= E

[∫ T

0

HtwtLtdt

]. (18)

12This is a sufficient condition for the transversality condition to hold, i.e.

lims→∞

∫ s

0

e−ρtu(Ct, Lt)dt = 0.

13See for example Cox/Huang (1989), Karatzas (1997) and Korn/Korn (2001)

9

This budget constraint is the same as

E[∫ ∞

0

Ht(Ct − wtLt1t)dt]

= 0. (19)

We see that, in the maximization problem above, the portfolio πt has disap-

peared from the control variables. We solve the problem (17)-(18) (or (19))

for optimal consumption C∗t and labor supply L∗t first and then recover the

optimal portfolio π∗t from the original budget constraint (16).14 The details

of the computation are given in Appendix. We only subtract the main results

below.

Theorem 1. Consider the problem (15)-(16) and the utility function (3).

The corresponding optimal consumption, labor supply and portfolio, for all

t ∈ [0,∞), are given by

C∗t = Aη

γ+η

T e−ργtH− 1γ

t , (20)

L∗t = A− γγ+η

T eρηtH

1η

t

(wtb

) 1η1t, (21)

π∗t =1

γ

µ− rσ2

+

(1

γ+

1

η

)µ− rσ2

1

β

(1− e−β(T−t)) wtL∗t

X∗t1t, (22)

and the corresponding optimal wealth X∗t satisfies

X∗t =1

αC∗t −

1

β(1− e−β(T−t))wtL

∗t1t, (23)

14The superscript * denotes the corresponding optimal quantity throughout the paper.

10

provided that ρ 6= (η + 1)(r − a− θ2

2η).15 Here

AT ≡ αb−1ηw

η+1η

0

1

β(1− e−βT ), (24)

with

α ≡ γ − 1

γ

(r +

θ2

2γ

)+ρ

γ

β ≡ η + 1

η

(r − a− θ2

2η

)− ρ

η(25)

The optimal portfolio in (22), during working life, consists of two parts:

one is Merton’s classical portfolio rule for the model with one consumption

good, the other is the correction term which is proportional to the ratio of

current labor income to current financial wealth exclusive of future labor in-

comes.16 So labor flexibility promotes greater risk-taking in financial invest-

ment as found in BMS (1992). By inspection of Eq. (22), we can also draw

the following conclusion as common with BMS: as the individual approaches

retirement age T , he tends to exhibit more conservative investment.17

However, what distinguish our work from those of BMS and Bodie et al

are the closed forms of optimal consumption and labor supply. We will now

devote our efforts to study the solutions of consumption and labor supply

and their implications in economics. Eqs (20)-(21) show that both of con-

15This is to ensure that β 6= 0. Note also that α 6= 0, since γ > 1, r > 0 and ρ > 0.16Retirement portfolio only has the first term. This is of no surprise, as labor supply

stops and there is only one consumption good during retirement.17To see this, note that 1

β

(1− e−β(T−t)) is positive and increasing in T −t for t ∈ [0, T ).

When the expected return µ of the risky asset is below the risk-free rate r, the individualis actually selling his share of the risky asset.

11

sumption and labor supply are affected by the stochastic discount factor Ht

and therefore can be predicted by the market performance: when the market

performs better (worse), that is, when the market price of risk θ is higher

(lower) (so that Ht is smaller (bigger)), the individual is allowed to consume

more (less) and work less (more). Due to the uncertainty of financial market,

both optimal consumption and labor supply are stochastic. It is thus more

convenient to study their economical behaviors in expectation. We do so in

the sequel.

3.1 Optimal consumption and the Euler equation un-

der uncertainty

Taking expectation on both hand sides of Eq. (20), we get the expected

optimal consumption as

C∗t ≡ E[C∗t ] = Aη

γ+η

T e−ργtE[H− 1γ

t

]= A

ηγ+η

T e−ργtE[eθγWt− θ2

2γ2t · e

1γ(r+ γ+1

2γθ2)t

]= A

ηγ+η

T e1γ(r−ρ+ γ+1

2γθ2)t (26)

where, the last equality is obtained by noticing that the process

eθγWt− θ2

2γ2t

for t ∈ [0,∞)

is a Martingale and thus has expectation of one.

12

In logs, we have the following linear approximation:18

ln(C∗t ) ≈ αc +η

γ + ηln(T ) + gct, (27)

where,

αc =η

γ + η

[ln(α)− 1

ηln(b) +

η + 1

ηln(w0)

]

and gc is the growth rate of expected optimal consumption and is given by

gc =1

γ

(r − ρ+

γ + 1

2γθ2

)(28)

Figure 1:

18To get this approximation, note that ln(

1β (1− e−βT )

)≈ ln(T ), for small βT .

13

Figure 2:

Eq. (27) implies that the expected optimal consumption is approximately

log-linear in retirement age and that one percent change of retirement age

will result in ηγ+η

percent change of optimal consumption. Figure 1 (Figure

2) plots the expected (log) optimal consumption against (log) retirement age

T of the individual at age 30. Here, we choose the same values as in BMS for

the parameters: µ = 0.09, r = 0.03, σ = 0.35, b = 0.5, ρ = 0.06 and the initial

wage w0 = 60, 000$ per year; Unlike in BMS, we set the growth rate of wage

at a = 0.01, γ = η = 3 and let retirement age T change from 50 to 68. From

Figure 2, we see that the slop of the fitted line is 0.5, which is exactly equal

to ηγ+η

= 33+3

, confirming the approximation in (27).

Eq. (28) is the Euler equation of our intertemporal maximization problem

14

(under uncertainty).19 It is interesting that the risk-aversion coefficient η

with respect to labor supply does not affect the Euler equation. This is

presumably because of the separability in the utility of consumption and

labor supply.20

From (28), it is easy to see that the growth rate of the expected consump-

tion is strictly positive when ρ < r+ γ+12γθ2, strictly negative if ρ > r+ γ+1

2γθ2

and constant if ρ = r + γ+12γθ2. Intuitively, as the discount rate captures

the consumer’s preference over time, a smaller discount rate implies that

the consumer is more patient and therefore prefers less consumption today

than tomorrow (that is, consumption is rising). Similarly, he will be less pa-

tient if ρ is larger, in particular, when the discount rate exceeds the critical

value r + γ+12γθ2, he will prefer to consume more earlier than later (that is,

consumption is falling).

The positive term θ2 in (28) captures the uncertainty of the financial

market, indicating that a risky financial market induces the consumer to

shift consumption more frequently. When other things are equal, a higher

market price of risk θ leads to a steeper slope of the expected consumption.

In the case that the asset does not pay a risk premium, i.e. when µ = r, all

the wealth will be optimally invested into the risk-free bond to secure a fixed

income. The Euler equation (under uncertainty) will then coincide with the

19The Euler equation are standard in models without labor supply. See the recent paperby Luo, Smith and Zou (2009), who derive the Euler equation for a CARA utility functionand a Ornstein-Uhlenbeck process for wage income. Toche (2005) and Marson and Wright(2001) also find a similar structure of the Euler equation under uncertainty. In Toche(2005), the inclusion of an additional term to the Euler equation is due to the risk ofpermanent income loss while, in Mason/Wright (2001), the conclusion is drawn based onthe approximation of a discrete-time problem.

20For the same reason, γ does not affect the growth equation (32) of labor supply.

15

well-known Euler equation for the case of certainty, which is 21

gc =r − ργ

. (29)

When there is no uncertainty, the growth of consumption is strictly decreas-

ing in γ or strictly increasing in the elasticity of substitution between con-

sumptions 1γ: when γ is smaller (larger), the less (more) marginal utility

changes as consumption changes, the more (less) the individual is willing to

substitute consumption between periods.

When the uncertainty of asset returns exists, i.e., when θ > 0, the effects

of γ on consumption are twofold: (i) it captures the individual’s williness

of substitution between consumptions under certainty, so consumption de-

creases as γ increases; (ii) it also governs the individual’s risk aversion toward

the uncertainty of financial market: with small risk aversion γ, he will in-

vest a high proportion of wealth into the risky asset (since he is not very risk

averse), causing high fluctuation with financial wealth and consequently high

frequency in adjusting consumption, therefore we see a steep consumption

pattern for small values of γ as shown in the figures below;22 when the indi-

vidual is very risk averse (i.e., for large γ), he will invest a relatively small

proportion of wealth into the asset, so his financial wealth will be relatively

stable, implying a relatively smooth consumption stream (see Figures 3-4 for

the change of consumption when γ becomes large.)

21See e.g. Romer (2006) for more detailed discussions of the Euler equation when thereis no uncertainty.

22Figure 3 shows the case of a decreasing consumption stream, while Figure 4 presentsthe case of an increasing consumption stream.

16

Figure 3:

Figure 4:

17

3.2 Optimal labor supply

Similarly, the expectation of optimal labor supply for 0 ≤ t < T is computed

as

L∗t ≡ E[L∗t ] = A− γγ+η

T

(wtb

) 1ηeρηtE[H

1η

t

]= A

− γγ+η

T

(wtb

) 1ηe−

1η(r−ρ+ η−1

2ηθ2)t. (30)

In logs, we have the following linear approximation for the expected op-

timal labor supply:

ln(L∗t ) ≈ αl −γ

γ + ηln(T ) + glt+

1

ηln(wt), (31)

where

αl = − γ

γ + η

[ln(α) +

1

γln(b) +

η + 1

ηln(w0)

]

and gl is the growth rate of expected optimal labor supply and is given by

gl = −1

η

(r − ρ+

η − 1

2ηθ2

). (32)

When θ = 0, it becomes

gl = −r − ρη

. (33)

Eq. (31) implies that the expected optimal labor supply is approximately

log-linear in retirement age and that increasing the retirement age by one

18

percent will decrease the expected optimal labor supply by γγ+η

percent.

Figure 5 (Figure 6) shows the relationship between the expected (log) optimal

labor supply and (log) retirement age T of the individual at age 30. In Figure

6, the slop of −0.5 of the fitted line confirms the approximation in (31) for

γ = η = 3 and other parameters shown at the bottom of the Figure.

Figure 5:

It can be concluded from (32) that the expected labor supply is constant

when ρ = r + η−12ηθ2, and it is strictly decreasing in time when ρ < r +

η−12ηθ2 while strictly increasing when ρ > r + η−1

2ηθ2. Similar to the optimal

consumption, optimal labor supply is relatively smooth for large values of the

risk aversion η. This can be seen in Figure 7-8. Figure 7 shows the case that

the expected optimal labor supply is increasing in t, while Figure 8 shows

the case that it increasing in t for very small η but decreasing in t for large

η.

19

Figure 6:

Figure 7:

20

Figure 8:

Acknowledgments

This paper has been presented at the Scottish Economic Society annual Con-

ference (2008) held in Perth, UK and the Econometric Society Australasian

Meeting in 2009 in Canberra. I am grateful to my supervisor Professor Ralf

Korn and Professor Charles Nolan for their helpful comments and sugges-

tions for the original draft. I would also like to thank one of the anonymous

referees for the applause of the value of the contribution of this paper and

for his kind suggestions, dealing with the writing of an early version of this

paper, which have improved the exposition of this version.

21

Appendix

A. Proof of Theorem 1:

Proof. We apply the Lagrangian formalism to the problem of (17) and (19).

The Lagrangian is written as

L(λ;Ct, Lt) = E[∫ ∞

0

e−ρtu(Ct, Lt)dt

]+ λ

(0− E

[∫ ∞0

Ht(Ct − wtLt1t)dt])

(34)

where, λ is the Lagrangian multiplier. The first order conditions are

∂u

∂Ct= λeρtHt

∂u

∂Lt= −λeρtHtwt1t. (35)

From (3) we conclude that23

∂u

∂Ct= C−γt

∂u

∂Lt= −bLηt . (36)

23Note that the first order conditions imply the tradeoff between consumption and laborsupply:

bLηtC−γt

= wt1t.

22

Substituting back into the first order conditions in (35) leads to

C∗t = λ−1γ e−

ργtH− 1γ

t

L∗t = λ1η e

ρηtH

1η

t

(wtb

) 1η

1t. (37)

The multiplier λ can then be obtained from the budget constraint: Sub-

stituting C∗t and L∗t into the budget constraint of (18), we get

λ−1γE[∫ ∞

0

e−ργtH

γ−1γ

t dt

]= λ

1η b−

1ηE[∫ T

0

eρηt(Htwt)

η+1η dt

].

As both integrals above are finite, we can apply the Fubini theorem, which

allows us to interchange the order of expectation and integration to obtain

λ−1γ

∫ ∞0

e−ργtE[H

γ−1γ

t

]dt = λ

1η b−

1η

∫ T

0

eρηtE[(Htwt)

η+1η

]dt. (38)

From the definition of Ht in Eq. (7), we have that

Hγ−1γ

t = e−γ−1γ

(r+ θ2

2)t− γ−1

γθWt

= e−γ−1γθWt− 1

2( γ−1γ

)2θ2t · e−γ−1γ

(r+ θ2

2γ)t. (39)

Noting that e−γ−1γθWt− 1

2( γ−1γ

)2θ2t is a martingale and thus has expectation of

one, we obtain

E[Hγ−1γ

t ] = e−yt, with y ≡ γ−1γ

(r + θ2

2γ). (40)

23

Similarly, we can get that

E[(Htwt)η+1η ] = w

η+1η

0 e−zt, with z ≡ η+1η

(r − a− θ2

2η). (41)

The substitution of E[Hγ−1γ

t ] and E[(Htwt)η+1η ] from (38) gives us that

λ−1γ

∫ ∞0

e−ργte−ytdt = λ

1η b−

1ηw

η+1η

0

∫ T

0

eρηte−ztdt. (42)

We use the following notation:

α ≡ y +ρ

γ

β ≡ z − ρ

η(43)

and rewrite Eq. (42) as

λ−1γ

∫ ∞0

e−αtdt = λ1η b−

1ηw

η+1η

0

∫ T

0

e−βtdt. (44)

A simple calculation leads to24

λ−1γ

1

α= λ

1η b−

1ηw

η+1η

0

1

β(1− e−βT ) (45)

provided that ρ 6= (η + 1)(r − a − θ2

2η).25 Multiplying both sides by λ−

1ηα

results in26

λ−γ+ηγη = AT , with AT ≡ αb−

1ηw

η+1η

01β(1− e−βT ) (46)

24γ > 1, ρ > 0 and r > 0, so α > 025This condition is to ensure that β 6= 0.26The subscript T indicates that A depends on T

24

and therefore

λ−1γ = A

ηγ+η

T

λ1η = A

− γγ+η

T . (47)

Replacing λ−1γ and λ

1η in (37) using (47) then gives the optimal consumption

and optimal labor supply as in (20) and (21), respectively.

Next, we show the optimal portfolio and the corresponding optimal wealth.

Clearly, the optimally invested wealth X∗t , satisfies Eq. (9) at the optimum,

that is

X∗t = Et[

∫ ∞t

Hs

Ht

C∗sds]− Et[

∫ ∞t

Hs

Ht

wsL∗s1sds] (48)

We compute the first term on the right-hand side of Eq. (48) below. The

second term can then be computed in a similar manner. Multiplying and

dividing the integrand of the first term by C∗t and noting that C∗t is Ft

measurable and therefore can be taken out from the conditional expectation27

Et[

∫ ∞t

Hs

Ht

C∗sds] = C∗t Et[

∫ ∞t

HsC∗s

HtC∗tds] (49)

27The reason C∗t is Ft measurable is simply because C∗t is a function of Ht which is Ftmeasurable. The fact that C∗t can be taken out from the conditional expectation is dueto the property of ’Taking out what is known’ of conditional expectation.

25

Substituting the optimal consumption obtained in (20) gives us that

Et

[∫ ∞t

Hs

Ht

C∗sds

]= C∗t Et

[∫ ∞t

e−ργ(s−t)

(Hs

Ht

) γ−1γ

ds

]

= C∗t E

[∫ ∞t

e−ργ(s−t)

(Hs

Ht

) γ−1γ

ds

]

= C∗t

∫ ∞0

e−ργsE[H

γ−1γ

s

]ds

= C∗t

∫ ∞0

e−ργse−ysds

= C∗t

∫ ∞0

e−αsds

= C∗t1

α(50)

where, the conditional expectation is replaced by the unconditional expecta-

tion (the second equality) since the increment of a Brownian motion Ws−Wt

is independent of Ft for s ≥ t. The third equality is obtained by relabeling

s − t as s for the reason that Ws − Wts≥t is again a Brownian motion.

We have used the result obtained in Eq. (40) to get the fourth equality.

Similarly, we can get that

Et

[∫ ∞t

Hs

Ht

wsL∗s1sds

]=

1β(1− e−β(T−t))wtL

∗t , t ∈ [0, T )

0, t ∈ [T,∞)

So we can write the optimal wealth as

X∗t =1

αC∗t −

1

β(1− e−β(T−t))wtL

∗t1t, for all t ∈ [0,∞). (51)

Discounting the optimal wealth by the stochastic discount factor Ht and

26

then taking differentials, we can get

d(HtX∗t ) = −Ht

(1

αC∗tγ − 1

γ− 1

β(1− e−β(T−t))wtL

∗t

η + 1

η

)θdWt

−HtC∗t dt+HtwtL

∗tdt. (52)

On the other hand, we know, by applying Ito’s lemma to the stochastic

discount factor Ht, that

dHt = −Ht(rdt+ θdWt) (53)

and, by further applying the stochastic product rule to HtXt, that

d(HtXt) = HtdXt +XtdHt + dHtdXt

= HtXt(πtσ − θ)dWt −HtCtdt+HtwtLt1tdt. (54)

This also holds at the optimum as

d(HtX∗t ) = HtX

∗t (π∗t σ − θ)dWt −HtC

∗t dt+HtwtL

∗t1tdt. (55)

A comparison of Eq. (56) with Eq. (53) gives us the optimal portfolio rule

in (22).

27

B. Notations

T : retirement age

r: (constant) nominal interest rate

µ: drift term of the stock price

σ: volatility of the stock price

Wt: Brownian motion

Ct: consumption per unit time

Lt: labor supply per unit time

b: weight of the dis-utility from working added to the instantaneous utility

γ: relative risk aversion w.r.t. consumption

η: relative risk aversion w.r.t. labor supply

wt: wage rate

a: growth rate of wage

ρ: discount rate

πt: share of portfolio invested into the risky asset at time t

X: financial wealth process

θ = µ−rσ

(market price of risk)

Ht = e−(r+ θ2

2)t−θWt (stochastic discount factor)

1Ht

: market deflator

y ≡ γ−1γ

(r + θ2

2γ)

z ≡ η+1η

(r − a− θ2

2η)

α ≡ y + ργ

β ≡ z − ρη

AT ≡ αb−1ηw

η+1η

01β(1− e−βT )

28

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31