A closed-form solution to the continuous-time

consumption model with endogenous labor income 1

Aihua Zhang (Chang)

Department of Financial Mathematics, Fraunhofer Institut Techno- und

Wirtschaftsmathematik (ITWM), 67663 Kaiserslautern, Germany

E-mail : aihua.zhang@fraunhofer.itwm.de

August 30, 2007

Abstract

In this paper, we study an intertemporal maximization problem of

an infinitely-lived individual who faces both labor income and asset

return uncertainty. Given the growth rate of wage, the uncertainty of

labor income is caused by the stochastic labor supply which is to be

determined upon the available market information. Closed forms of

consumption, labor supply and portfolio are obtained analytically by

means of the martingale method. The Euler equation under uncer-

tainty is established.

1I am grateful to Professor Ralf Korn and Professor Charles Nolan for their helpfulcomments and suggestions. The financial supports from the Rheinland-Pfalz cluster ofexcellence for ”Dependable Adaptive Systems and Mathematical Modeling” (DASMOD)and from the ”Deutsche Forschungsgemeinschaft” (DFG) are greatly acknowledged withthanks.

1

Keywords: Consumption, Labor Supply, Portfolio, Euler equation, Mar-

tingale

JEL subject classifications: C61; C73; G1; J22

1 Introduction

A closed-form solution to the intertemporal consumption problem, in which

both asset return and labor income uncertainty are considered, is absent.

Much of the existing literature either provides an approximate solution or

relies on restrictive assumptions to obtain analytical results. The former in-

cludes Mason/Wright (2001), Chan/Viceira (2000) and the some references

therein. Chamberlain/Wilson (2000) investigates the asymptotic properties

of the optimal consumption process. Some assumptions are restricted to

the uncertainty of labor income. For example, Toche (2005) assumes that

the uncertainty is about the timing of the income loss. Similarly, Pitch-

ford (1991) takes the form of uncertainty as the timing of the reversal of

an income shock. And some other assumptions are imposed on the utility

function. A quadratic utility can lead to a simple characterization of the

optimal consumption. However, Blanchard and Fischer (1989) pointed out

that quadratic utility is an unattractive description of behavior toward risk

as it implies increasing absolute risk aversion. Moreover, most of the them

treat the labor income as an endowment. The exception is the paper by

Chan/Viceira (2000) which generalizes the work of Bodie/Merton/Samuelson

(1992). As in Bodie/Merton/Samuelson (1992), we include the labor sup-

2

ply to the utility function. Using the Martingale technique (other than the

dynamic programming used by Bodie/Merton/Samuelson (1992)), we de-

rive analytically a closed-form solution to the expected-utility maximization

problem with both asset return and labor income uncertainty. We find that

the uncertainty gives rise to an additional term (corresponding to the market

price of risk) in the Euler equation. This is represented in Eq. (37). The

finding is also supported by the results concluded by Toche (2005) and Ma-

son/Wright (2001). In Toche (2005), the inclusion of an additional term to

the Euler equation is due to the risk of permanent income loss. Mason and

Wright drew the conclusion based on the approximation of the discrete-time

problem.

The rest of the paper is organized as follows. In section 2, we specify

the model which is solved by using the martingale method in section 3. The

closed forms of consumption, labor supply and portfolio are given in Eqs.

(33) and (60) in section 3, where the economic interpretations of the results

are also provided. Section 4 concludes this paper. Frequently used notations

are presented in the Appendix B.

2 Description of the model

It is assumed that an infinitely-lived individual born with no initial wealth

works only when young (that is, before retirement age T ). Given the wage

rate wt with the initial wage w0 and its growth rate a being fixed, he earns

an income of wtLt by supplying an amount of labor Lt at the time t when

3

t ≤ T . The labor income is invested into a risk-free bond offering a gross

return r and a risky asset offering an instantaneous expected gross return µ.2

After the individual retires, his post-retirement consumption is financed by

his savings when young and the capital gain from the investment. His objec-

tive is to maximize his lifetime utility by choosing the optimal consumption,

the amount of work (when young) and portfolio.

2.1 The dynamics of the asset prices

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

by Bt, satisfies

dBt

Bt

= rdt, B0 = 1 (1)

where, r is the nominal interest rate. We will take the price of the risky asset,

denoted by St, as being exogenously given, rather than being endogenously

derived, and it follows a geometric Brownian motion

dSt

St

= 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 and Wt is a Brownian motion

defined on a probability space (Ω,F , P).

2It straightforward to extend it to any number of assets.

4

2.2 The utility function

The instantaneous utility function is defined by

u(Ct, Lt) =C1−γ

t

1− γ− b

L1+ηt

1 + η(3)

where, γ > 1 3 and η > 0 4. The utility function has two arguments. One

is the consumption per unit time Ct, and the other is the labor supply (or

the amount of work) per unit time Lt. The coefficient b is positive, together

with the negative sign, indicating disutility gained from working. The pa-

rameters γ and η regulate the curvature of the utility function with respect

to consumption and labor supply, respectively.

2.3 The wealth process

The individual works only before age T . In other words, he no longer works

after reaching his retirement age T . In order to capture this fact in the

lifetime horizon, we introduce a dummy variable as follows:

1t =

0 , t > T

1 , t ≤ T(4)

If we assume that the individual with no initial capital invests πt propor-

tion of his wealth into the risky asset at time t, t > 0, and 1 − πt fraction

3So that the utility function with respect to the consumption is bounded4So that the utility function with respect to the labor supply without the negative sign

is convex

5

of his wealth into the risk-free bond, then his wealth process Xt is derived as

follows:5

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

X0 = 0 (5)

It says that the change in wealth must equal the capital gains less infinitesi-

mal consumption plus infinitesimal labor income when t ≤ T and equal the

difference between the capital gains and the infinitesimal consumption when

t > T . The labor income before retirement is equal to the amount of working

Lt times a wage rate wt. The wage rate grows exponentially at a rate of a

with a strictly positive initial wage w0, that is

wt = w0eat (6)

Define the stochastic discount factor Ht by

Ht ≡ e−rt− 12θ2t−θWt (7)

where, θ is the market price of risk defined by

θ ≡ µ− r

σ(8)

Because the financial market under consideration is complete (because the

number of the risky asset is the same as the number of the driving Brownian

5For the derivation of a standard wealth process, we refer to a classic textbook, suchas Korn/Korn (2000) and Shreve (2004).

6

motion) and is absent of arbitrage (because σ 6= 0). As a consequence,

the individual’s current wealth must equal the expected present value of his

future consumption less the expected present value of his labor income. In

other words, the resources for his expected future consumption come from

the current value of his accumulated financial 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, it can be further expressed as

Xt = Et[

∫ ∞

t

Hs

Ht

Csds]− Et[

∫ T

t

Hs

Ht

wsLsds] (10)

In particular, at time zero,6 we have

E0[

∫ ∞

0

HsCsds] = E0[

∫ T

0

HswsLsds] (11)

where E0 is the conditional expectation conditional on the trival information

set F0, which is actually equal to the unconditional expectation E. We will

drop the subscript 0 when it appears in what follows. Intuitively, Eq. (11)

says that the expected discount future consumption when old (i.e. after

retirement age T ) is financed by the expected discounted labor income when

6Note that H0 = 1 and X0 = 0

7

young (i.e. before retirement).

2.4 The maximization problem

We start by defining an admissible set.

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 can be seen from (12) that the wealth before age T is allowed to be-

come negative so long as that the present value of future labor income is

large enough to offset such a negative value (See Karatzas, page 63, for the

argument).

The individual wishes to maximize the expected total discounted utility

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

that

A1 ≡

(Ct, Lt, πt) ∈ A : E[

∫ ∞

0

e−ρtu−(Ct, Lt)dt] < ∞

(13)

with x− ≡ max−x, 0. Namely, his optimization problem is given by

max(Ct,Lt,πt)∈A1

E[

∫ ∞

0

e−ρtu(Ct, Lt)dt] (14)

8

subject to

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

X0 = 0 (15)

where, the discount rate ρ > 0.

3 Solving the optimal problem by the Mar-

tingale approach

It follows from the martingale method that the (dynamic) maximization

problem (14)-(15) is equivalent to the following problem 7

maxCt,Lt

E[

∫ ∞

0

e−ρtu(Ct, Lt)dt] (16)

subject to

E[

∫ ∞

0

HtCtdt] = E[

∫ T

0

HtwtLtdt] (17)

This budget constraint is the same as

E[

∫ ∞

0

Ht(Ct − wtLt1t)dt] = 0 (18)

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

peared from the control variables.

7See for example Cox/Huang (1989), Karatzas (1997) and Korn/Korn (2001)

9

3.1 Optimal consumption and labor supply

We now apply the Lagrangian method to the static problem just described.

The Lagrangian function is written as

L(λ; Ct, Lt) = E[

∫ ∞

0

e−ρtu(Ct, Lt)dt] + λ

(0− E[

∫ ∞

0

Ht(Ct − wtLt1t)dt]

)(19)

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

∂u

∂Ct

= λeρtHt

∂u

∂Lt

= −λeρtHtwt1t (20)

From the utility function given in (3), we know that

∂u

∂Ct

= C−γt

∂u

∂Lt

= −bLηt (21)

Substituting them back into the first order conditions (20) leads to 8

C∗t = λ−

1γ e−

ργ

tH− 1

γ

t

L∗t = λ1η e

ρηtH

1η

t

(wt

b

) 1η1t (22)

8The superscript * denotes the corresponding optimal quantity throughout the paper.

10

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

tuting C∗t and L∗t into the budget constraint (17), we get

λ−1γ E[

∫ ∞

0

e−ργ

tHγ−1

γ

t dt] = λ1η b−

1η E[

∫ T

0

eρηt(Htwt)

η+1η dt]

According to the Fubini theorem, we can interchange the oder of the expec-

tation and integration as

λ−1γ

∫ ∞

0

e−ργ

tE[Hγ−1

γ

t ]dt = λ1η b−

1η

∫ T

0

eρηtE[(Htwt)

η+1η ]dt (23)

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

Hγ−1

γ

t = e−γ−1

γ(r+ θ2

2)t− γ−1

γθWt

= e−γ−1

γθWt− 1

2( γ−1

γ)2θ2t · e−

γ−1γ

(r+ θ2

2γ)t (24)

Noting that e−γ−1

γθWt− 1

2( γ−1

γ)2θ2t is a martingale and thus has expectation of

unit,9 we obtain

E[Hγ−1

γ

t ] = e−yt, with y ≡ γ−1γ

(r + θ2

2γ). (25)

Similarly, we can get that

E[(Htwt)η+1

η ] = wη+1

η

0 e−zt, with z ≡ η+1η

(r − a− θ2

2η) (26)

9exWt− 12 x2t for a fixed x is also referred to an exponential martingale. See for example

Shreve (2004).

11

The substitution of E[Hγ−1

γ

t ] and E[(Htwt)η+1

η ] from (23) gives us that

λ−1γ

∫ ∞

0

e−ργ

te−ytdt = λ1η b−

1η w

η+1η

0

∫ T

0

eρηte−ztdt (27)

Making the following denotations

y ≡ y +ρ

γ

z ≡ z − ρ

η(28)

we can rewrite (27) as

λ−1γ

∫ ∞

0

e−ytdt = λ1η b−

1η w

η+1η

0

∫ T

0

e−ztdt (29)

A simple calculation leads to10

λ−1γ1

y= λ

1η b−

1η w

η+1η

0

1

z(1− e−zT ) (30)

provided that ρ 6= (η+1)(r−a− θ2

2η).11 Multiplying both side by λ−

1η y results

in12

λ−γ+ηγη = AT , with AT ≡ yb−

1η w

η+1η

01z(1− e−zT ) (31)

10γ > 1, ρ > 0 and r > 0, so y > 011This condition is to ensure that z 6= 0.12The subscript T indicates that A depends on T

12

So we have

λ−1γ = A

ηγ+η

T

λ1η = A

− γγ+η

T

(32)

Replacing λ−1γ and λ

1η in (22) by (32), we obtain the optimal consumption

and optimal labor supply as follows

C∗t = A

ηγ+η

T e−ργ

tH− 1

γ

t ,

L∗t = A− γ

γ+η

T eρηtH

1η

t

(wt

b

) 1η1t (33)

3.2 Economic interpretation and the Euler equation

In the last subsection, we have obtained the optimal consumption and la-

bor supply policies in (33). By inspecting the Eq. (33), it is clear that the

individual will work more when his wage rate wt rises but work less when

the relative weight of the disutility from working b is bigger. With constant

risk aversions (i.e. γ and η are held fixed), the individual will work more

and consume less when the rate of time preference ρ becomes larger. More-

over, we can see from (33) that C∗t is decreasing in Ht and L∗t is increasing

in Ht. If we refer the stochastic discount factor Ht to the market deflater,

these phenomena can then be interpreted as that the individual is allowed

to consume more if the market as a whole performs well but has to work

harder if the market develops bad as he has to compensate for the market

13

bad performance.

As we also see from (33), both of the optimal consumption and labor

supply are stochastic and depend on the market prices through the market

deflater Ht. It is thus more convenient to study their growths in terms of

expectation. Similar to (24), we have

H− 1

γ

t = e1γ(r+ θ2

2)t+ θ

γWt

= eθγ

Wt− θ2

2γ2 t · e1γ(r+ γ+1

2γθ2)t (34)

The expectation of eθγ

Wt− θ2

2γ2 tequals one, so

E[H− 1

γ

t ] = e1γ(r+ γ+1

2γθ2)t (35)

And consequently, the expected optimal consumption is given by

C∗t ≡ E[C∗

t ] = Aη

γ+η

T e−ργ

tE[H− 1

γ

t ]

= Aη

γ+η

T e1γ(r−ρ+ γ+1

2γθ2)t (36)

The growth rate of the expected optimal consumption then equals

1

C∗t

d(C∗t )

dt=

1

γ(r − ρ +

γ + 1

2γθ2) (37)

This is referred to the Euler equation for the intertemporal maximization

above (under uncertainty). The positive term θ2 captures the uncertainty of

the financial market. A risky financial market induces the consumer to shift

14

consumption over time. It can be seen that the growth rate is decreasing in γ

or increasing in the elasticity of substitution between consumptions 1γ: when

γ is smaller, the less marginal utility changes as consumption changes, the

more the individual is willing to substitute consumption between periods.

When the difference r − ρ is fixed, a higher market price of risk θ leads to a

steeper slope of the expected consumption, thus a more prudent behavior. If

the risk premium µ−r equals nothing, then the market price of risk becomes

zero and therefore all the wealth will be optimally invested into the risk-free

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

well-known Euler equation for the case of certainty and becomes.

1

C∗t

d(C∗t )

dt=

r − ρ

γ(38)

It states that, when the nominal interest rate exceeds the discount rate, the

expected consumption of the individual with a constant relative risk aversion

(with respect to the consumption) is rising and falling if the reverse holds.

From (37), it is easy to see that the growth rate of the expected consump-

tion is strictly positive when ρ < r + γ+12γ

θ2, strictly negative if ρ > r + γ+12γ

θ2

and constant if ρ = r + γ+12γ

θ2. 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 is more willing to shift consumption

between different periods (that is, consumption is rising). Similarly, he will

be less patient if ρ is larger, in particular, when the discount rate exceed the

critical value r + γ+12γ

θ2, he will prefer to consume more earlier than later

15

(that is, consumption is falling).

Parallel to the analysis of the optimal consumption, we have

H1η

t = e−1η(r+ θ2

2)t− θ

ηWt

= e− θ

ηWt− θ2

2η2 t · e−1η(r+ η−1

2ηθ2)t (39)

So the expectation is equal to

E[H1η

t ] = e−1η(r+ η−1

2ηθ2)t (40)

The expected optimal labor supply for t ≤ T is then computed as

L∗t ≡ E[L∗t ] = A− γ

γ+η

T

(wt

b

) 1ηe

ρηtE[H

1η

t ]

= A− γ

γ+η

T

(wt

b

) 1ηe−

1η(r−ρ+ η−1

2ηθ2)t (41)

The growth rate of the expected labor supply equals

1

L∗t

d(L∗t )

dt= −1

η(r − ρ +

η − 1

2ηθ2) (42)

When θ = 0, it becomes

1

L∗t

d(L∗t )

dt= −r − ρ

η(43)

It can be included from (40) that the expected labor supply is constant when

ρ = r + η−12η

θ2. And it is strictly decreasing in time when ρ < r + η−12η

θ2 while

16

strictly increasing when ρ > r + η−12η

θ2.

Using the first order conditions (20) and the marginal utility functions

(21), we can obtain the tradeoff between consumption and labor supply

bLηt

C−γt

= wt1t (44)

This implies that labor supply when t ≤ T is decreasing in consumption

(due to the diminishing marginal utility with respect to consumption) and

increasing in the wage rate.

3.3 Optimal wealth and portfolio rule

It is well-known that the optimal portfolio for the consumption-portfolio

problem with CRRA utility is constant over time and given by13

π∗t =1

γ

µ− r

σ2(45)

Since there is no labor supply after retirement, the intertemporal consumption-

labor supply-portfolio problem for t > T collapses to the intertemporal consumption-

portfolio problem starting from the time point T with a constant relative risk

aversion (CRRA) utility.14 That is to say, the optimal portfolio rule for our

problem when t > T is given by (45). We now focus on the case when t ≤ T .

13See for example Merton (1971), Karatzas (1997) and Korn/Korn (2001).14As the horizon is infinite, it does not matter for the optimal investment strategy

whether it starts at the time zero or starts at a positive time point T .

17

Clearly, the optimally invested wealth X∗t , t ≤ T , satisfies Eq. (10) at

the optimum, that is

X∗t = Et[

∫ ∞

t

Hs

Ht

C∗s ds]− Et[

∫ T

t

Hs

Ht

wsL∗sds] (46)

We compute the first term on the right-hand side of Eq. (46) below. The

second term can 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 expectation15

Et[

∫ ∞

t

Hs

Ht

C∗s ds] = C∗

t Et[

∫ ∞

t

HsC∗s

HtC∗t

ds] (47)

Substituting the optimal consumption obtained in (32) gives us

Et[

∫ ∞

t

Hs

Ht

C∗s ds] = 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−ργ

sE[Hγ−1

γs ]ds

= C∗t

∫ ∞

0

e−ργ

se−ysds

= C∗t

∫ ∞

0

e−ysds

= C∗t

1

y(48)

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

15The reason C∗t is Ft measurable is simply because C∗

t is a function of Ht which is Ft

measurable. The fact that C∗t can be taken out from the conditional expectation is due

to the property of ’Taking out what is known’ of conditional expectation.

18

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 (25) to get the fourth equality. Similarly,

we can get that

Et[

∫ T

t

Hs

Ht

wsL∗sds] =

1

z(1− e−z(T−t))wtL

∗t (49)

So we now have obtained that

X∗t =

1

yC∗

t −1

z(1− e−z(T−t))wtL

∗t (50)

Rearranging it, we get

C∗t = yX∗

t +y

z(1− e−z(T−t))wtL

∗t (51)

Noting that both y and yz(1−e−z(T−t)) are strictly positive (so long as z 6= 0),

we can say that the consumption before retirement, at the optimum, is both

proportional to the financial wealth and proportional to the labor income.

Discounting the optimally invested wealth in (50) by the stochastic dis-

count factor Ht and then taking differentiation, we can get16

d(HtX∗t ) = −Ht

(1

yC∗

t

γ − 1

γ− 1

z(1− e−z(T−t))wtL

∗t

η + 1

η

)θdWt

−HtC∗t dt + HtwtL

∗t dt (52)

16The details of the derivation is given in Appendix A

19

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)

From the definition of the dummy variable, we have 1t = 0 when t > T and

1t = 1 when t ≤ T . So the last equation becomes

d(HtXt) = HtXt(πtσ − θ)dWt −HtCtdt + HtwtLtdt, for t ≤ T (55)

This also holds at the optimum as

d(HtX∗t ) = HtX

∗t (π∗t σ − θ)dWt −HtC

∗t dt + HtwtL

∗t dt (56)

A Comparison of (52) with (56) gives us the optimal portfolio rule for the

case when t ≤ T

π∗t =

(1−

1yC∗

tγ−1

γ− 1

z(1− e−z(T−t))wtL

∗t

η+1η

X∗t

)θ

σ(57)

20

with

θ =µ− r

σ(58)

and C∗t , L∗t satisfy (33). By comparing the numerator in the parenthesis in

(57) with X∗t in (50), it is trivial to conclude that

π∗t → 0, when γ →∞ and η →∞ (59)

In words, when the investor is extremely risk-averse, he will invest almost all

of his wealth in the risk-free bond for the reason of safety.

The optimal portfolio (57) can be further written as

π∗t =1

γ

µ− r

σ2+ (

1

γ+

1

η)µ− r

σ2

1

z(1− e−z(T−t))

wtL∗t

X∗t

(60)

So the share of the wealth optimally invested into the risky asset is made up

of two parts:

• the classical Merton’s portfolio rule for the consumption-portfolio problem

1γ

µ−rσ2 plus

• the correction term which is proportional to the optimal labor income rel-

ative to the optimally invested wealth, ( 1γ

+ 1η)µ−r

σ21z(1− e−z(T−t))

wtL∗tX∗

t.

21

4 Conclusion

It has been found that, at the optimum, a smaller discount rate implies lower

expected labor supply before retirement and higher expected consumption

during retirement. When the discount rate is below some critical values (for

example, when ρ < minr+ γ+12γ

θ2, r+ η−12η

θ2), the expected labor supply will

drop during the working period and reach its minimum at the retirement age;

while the consumption is expected to grow with no limit. When the discount

rate exceeds these critical values, the expected labor supply will increase

and reaches its maximum at the retirement age; but the consumption is

expected to decrease and converge to zero. We have derived that when labor

income supplements total wealth, the classical Merton’s portfolio rule needs

to be adjusted by adding an additional share of the total wealth which is

proportional to the labor income.

Appendix

A. The derivation of Eq. (52)

Multiplying Eq. (52) by Ht and then substituting C∗t and L∗t obtained in Eq.

(33) gives us

HtX∗t =

1

yHtC

∗t −

1

z(1− e−z(T−t))HtwL∗t

=1

yA

ηγ+η

T e−ργ

tHγ−1

γ

t − 1

z(1− e−z(T−t))A

− γγ+η

T b−1η w

η+1η

t eρηtH

η+1η

t

(61)

22

Taking the differentiation

d(HtX∗t )

=1

yA

ηγ+η

T d

(e−

ργ

tHγ−1

γ

t

)− 1

zA− γ

γ+η

T b−1η w

η+1η

0 d

(e( ρ

η+a η+1

η)t(1− e−z(T−t))H

η+1η

t

)(62)

By applying the Ito’s lemma and then using Eq. (53), we can get

d

(H

γ−1γ

t

)= −H

γ−1γ

t (γ − 1

γθdWt + ydt) (63)

and

d

(H

η+1η

t

)= −H

η+1η

t (η + 1

ηθdWt + (z + a

η + 1

η)dt) (64)

Applying the product rule to e−ργ

tHγ−1

γ

t and e( ρη+a η+1

η)t(1− e−z(T−t))H

η+1η

t ,

respectively, and using (28), (63) and (64), we get

d

(e−

ργ

tHγ−1

γ

t

)= H

γ−1γ

t d(e−

ργ

t)

+ e−ργ

td

(H

γ−1γ

t

)= −ρ

γe−

ργ

tHγ−1

γ

t dt− e−ργ

tHγ−1

γ

t (γ − 1

γθdWt + ydt)

= −e−ργ

tHγ−1

γ

t (γ − 1

γθdWt + ydt) (65)

23

and

d

(e( ρ

η+a η+1

η)t(1− e−z(T−t))H

η+1η

t

)= d

((e( ρ

η+a η+1

η)t − e−zT e(z+a η+1

η)t)

Hη+1

η

t

)= H

η+1η

t d(e( ρ

η+a η+1

η)t − e−zT e(z+a η+1

η)t)

+(e( ρ

η+a η+1

η)t − e−zT e(z+a η+1

η)t)

d(Hη+1

η

t )

= Hη+1

η

t

((ρ

η+ a

η + 1

η)e( ρ

η+a η+1

η)t − (z + a

η + 1

η)e−zT e(z+a η+1

η)t

)dt

−(e( ρ

η+a η+1

η)t − e−zT e(z+a η+1

η)t)

Hη+1

η

t

(η + 1

ηθdWt + (z + a

η + 1

η)dt

)= −e( ρ

η+a η+1

η)t(1− e−z(T−t))H

η+1η

t

η + 1

ηθdWt − ze( ρ

η+a η+1

η)tH

η+1η

t dt (66)

Substituting (65) and (66) back to (62) and then collecting terms and using

(6), (28) and (33) results in

d(HtX∗t )

= −1

yA

ηγ+η

T e−ργ

tHγ−1

γ

t (γ − 1

γθdWt + ydt)

+1

zA− γ

γ+η

T b−1η w

η+1η

0

(e( ρ

η+a η+1

η)t(1− e−z(T−t))H

η+1η

t

η + 1

ηθdWt + ze( ρ

η+a η+1

η)tH

η+1η

t dt

)= −Ht

(1

yC∗

t

γ − 1

γ− 1

z(1− e−z(T−t))wtL

∗t

η + 1

η

)θdWt −HtC

∗t dt + HtwL∗t dt

(67)

B. Notations

T : retirement age

r: (constant) nominal interest rate

µ: drift term of the stock price

σ: volatility of the stock price

24

Wt: Brownian motion

Ct: consumption per unit time

Lt: amount of work

b: weight of the disutility 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

a: growth rate of wage

ρ: discount rate

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

Xt: wealth process

θ = µ−rσ

(market price of risk)

Ht = e−(r+ θ2

2)t−θWt (stochastic discount factor or market deflater)

y ≡ γ−1γ

(r + θ2

2γ)

z ≡ η+1η

(r − a− θ2

2η)

y ≡ y + ργ

z ≡ z − ρη

AT ≡ yb−1η w

η+1η

01z(1− e−zT )

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