actuarial studies seminar macquarie university 29 july ...€¦ · considered by bodie, merton and...
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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: [email protected]
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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