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Investment, consumption and best retirement time

Optimal investment, heterogeneous consumption

and the best time for retirement

XU, Zuo Quan

The Hong Kong Polytechnic University

Stochastic Control in Finance, 22 - 26 July 2019, Singapore

Based on joint work with Harry Zheng, Imperial College London

XU Zuo Quan, Hong Kong PolyU, [email protected] Investment, consumption and best retirement time

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Background

Optimal investment and consumption


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Background


PDE/viscosity solution approach

• Merton (1969 RES, 1971 JET, 1975 JFQA): lifetime portfolio

selection

• Fleming & Zariphopoulou (1991 MOR): borrowing constraint

• Zariphopoulou (1994 SICON): πt 6 f (Xt) and Xt > 0

• Vila & Zariphopoulou (1997 JET): borrowing constraint

• Oksendal & Sulem (2002 SICON): (fixed and proportional)

transaction costs

• Xu & Yi (2016 MCRF): ct 6 kXt + `


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Background


Probabilistic/martingale method

• Brennan (1971 JFQA): different borrowing and lending rates

• Cvitanic & Karatzas (1992 AAP, 1993 AAP): portfolio con-

straint

• Bardhan (1994 JEDC): ct > ` and Xt > 0

• Cvitanic & Karatzas (1996 MF): transaction costs

• Karoui, Peng & Quenez (1997 MF): Backward stochastic

differential equations in finance

• Elie & Touzi (2008 FS): Xt > ϑ sups6t Xs


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Background


Monograph

• Yong & Zhou (1999): Stochastic controls: Hamiltonian sys-

tems and HJB equations

• Fleming & Soner (2006): Controlled Markov processes and

viscosity solutions

• Pham (2009): Continuous-time stochastic control and opti-

mization with financial applications

• Karatzas & Shreve (2016): Methods of mathematical finance

(stochastic modelling and applied probability)


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Model

Model formulation


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Model

Features

Model features

• Heterogeneous consumptions: basic goods and luxury goods

• Utility function: two factors, non-concave

• Income: non-monotone, time-dependent, defer retirement

• Labor cost: non-monotone, time-dependent, prevent younger

to retire, encourage older to retire

• Mandatory retirement age

• Mixed controls: portfolio, retirement time, consumptions for

basic goods and luxury goods

• Complete market setup, also hold for convex constrained

trading strategies


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Model

Features

Model features










trading strategies


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Model

Features

Model features










trading strategies


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Model

Features

Model features










trading strategies


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Model

Features

Model features










trading strategies


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Model

Features

Model features










trading strategies


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Model

Features

Model features










trading strategies


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Model

Mathematical formulation

Financial assets

• One bond dS0(t) = rS0(t) dt, t > 0,

S0(0) = s0 > 0.

• n stocksdSi (t) = Si (t)

(bi dt +

n∑j=1

σij dB j(t)

), t > 0,

Si (0) = si > 0.

The parameters r , µ and σ are all constant and σ is nonsingular.


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Model


Wealth process and controls

dX (t) = (rX (t) + π(t) · µ+ I (t)1{t6τ}−c(t)− g(t)) dt

+π(t) · σ dB(t),

X0 = x0.

• I (·): the income process (given)

• π(·): the investment strategy

• τ : the retirement time, no later than the mandatory retire-

ment age T

• c(·): the consumption rate on the basic goods

• g(·): the consumption rate on the luxury goods


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Model


Target

Find a feasible strategy (π, c , g , τ) to maximize

E[∫ +∞

0e−ρtu(c(t), g(t)) dt −

∫ τ

0e−ρtL(t) dt

]

• u(·, ·): the heterogeneous utility function, non-concave for

luxury goods

• ρ: the discount factor

• L(·): the labor cost process (given, deterministic)


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Model

Related work

Non-concave utility maximisation

• Carpenter (2002 JF): Does option compensation increase

managerial risk appetite?

• Guan, Li, Xu and & Yi (2017 MCRF): A stochastic control

problem and related free boundaries in finance

• Bian, Chen & Xu (2019 SIFIN): Utility maximization under

trading constraints with discontinuous utility


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Model

Related work

Multi-consumption goods

• Breeden (1979 JFE): An intertemporal asset pricing model

with stochastic consumption and investment opportunities

• Madan (1988 JET): Risk measurement in semimartingale

models with multiple consumption goods

• Lakner (1989 PhD thesis): Consumption/investment and equi-

librium in the presence of several commodities

• Ait-Sahalia, Parker and Yogo (2004 JF): Luxury goods and

the equity premium


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Model

Related work

Multi-consumption goods (cont’d)

• Wachter & Yogo (2010 RFS): Why do household portfolio

shares rise in wealth?

• Koo, Roh & Shin (2017 JIA): An optimal consumption, gift,

investment, and voluntary retirement choice problem with

quadratic and HARA utility

• Campanale (2018 B.E.JM): Luxury consumption, precaution-

ary savings and wealth inequality


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Model

Related work

Data from Ait-Sahalia, Parker and Yogo (2004 JF)

Luxury Goods and the Equity Premium 2961

-25

-15

-5

5

15

25

-35 -25 -15 -5 5 15 25 35Excess Returns (%)

Con

sum

ptio

n G

row

th (%

)

PCE nondurables & services Luxury retail sales(A)

-40

-30

-20

-10

0

10

20

30

1961 1966 1971 1976 1981 1986 1991 1996 2001Year

Perc

ent

PCE nondurables & services Luxury retail sales Excess returns

1974

1990

1995

1970

(B)

Figure 1. Response of basic and luxury consumption to stock returns. Panel A is a scatterplot of the growth rate for PCE nondurables and services and sales of luxury retailers againstexcess stock returns (CRSP NYSE-AMEX portfolio over 3-month T-bills). The thin (thick) line isthe least squares regression line for PCE nondurables and services (sales of luxury retailers). PanelB is a time series plot of the growth rate for PCE nondurables and services, the growth rate forsales of luxury retailers, and excess stock returns. All series are normalized to have zero mean andare reported in percent.

is a time series plot of these series. For comparison, we include the growthrate of PCE nondurables and services in both plots. PCE is relatively smoothand almost nonresponsive to excess returns. By contrast, the consumption ofluxuries is both more volatile and more correlated with excess returns. Luxury

Figure: The growth rate for PCE nondurables and services and sales of

luxury retailers against excess stock returns


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Model

Related work

Data from Ait-Sahalia, Parker and Yogo (2004 JF) (Cont’d)

Luxury Goods and the Equity Premium 2961

-25

-15

-5

5

15

25

-35 -25 -15 -5 5 15 25 35Excess Returns (%)

Con

sum

ptio

n G

row

th (%

)

PCE nondurables & services Luxury retail sales(A)

-40

-30

-20

-10

0

10

20

30

1961 1966 1971 1976 1981 1986 1991 1996 2001Year

Perc

ent

PCE nondurables & services Luxury retail sales Excess returns

1974

1990

1995

1970

(B)

Figure 1. Response of basic and luxury consumption to stock returns. Panel A is a scatterplot of the growth rate for PCE nondurables and services and sales of luxury retailers againstexcess stock returns (CRSP NYSE-AMEX portfolio over 3-month T-bills). The thin (thick) line isthe least squares regression line for PCE nondurables and services (sales of luxury retailers). PanelB is a time series plot of the growth rate for PCE nondurables and services, the growth rate forsales of luxury retailers, and excess stock returns. All series are normalized to have zero mean andare reported in percent.

is a time series plot of these series. For comparison, we include the growthrate of PCE nondurables and services in both plots. PCE is relatively smoothand almost nonresponsive to excess returns. By contrast, the consumption ofluxuries is both more volatile and more correlated with excess returns. Luxury

Figure: The growth rate for PCE nondurables and services, the growth

rate for sales of luxury retailers, and excess stock returns


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Model

Related work

Mixed control with PDE

• Choi & Shim (2006 MF): Disutility, optimal retirement, and

portfolio selection

• Choi, Shim & Shin (2008 MF): Optimal portfolio, consumption-

leisure and retirement choice problem with CES utility

• Lim & Shin (2008 QF): Optimal investment, consumption

and retirement decision with disutility and borrowing con-

straints

• Guan, Li, Xu & Yi (2017 MCRF): A stochastic control prob-

lem and related free boundaries in finance


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Model

Related work

Mixed control with RBSDE

• Buckdahn & Li (2011 AMAS): Stochastic differential games

with reflection and related obstacle problems for Isaacs equa-

tions

• Karatzas & Wang (2000 SICON): Utility maximization with

discretionary stopping

• Hamadene & Lepeltier (2000 SPTA): Reflected BSDEs and

mixed game problem


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Model

Solution

Methods

• Combine heterogeneous consumptions to a single total con-

sumption

• Turn the non-concave utility into a concave utility

• Post-retirement problem: stationary life-time problem, ex-

plicit solution

• Pre-retirement problem: a nonlinear variational inequality

• Dual method: turn nonlinear variational inequalities into lin-

ear ones


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Model

Solution

Dual approach in probability

• Bismut (1973 JMAA): Conjugate convex functions in optimal

stochastic control

• Karatzas, Lehoczky, Shreve & Xu (1991 SICON): Martingale

and duality methods for utility maximization in an incomplete

market

• Shreve & Xu (1992 AAP): A duality method for optimal con-

sumption and investment under short-selling prohibition. I.

general market coefficients; and II. constant market coeffi-

cients

• Cvitanic & Karatzas (1992 AAP): Convex duality in con-

strained portfolio optimization


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Model

Solution

Dual approach in probability/PDE

• Hugonnier & Kramkov (2004 AAP): Optimal investment with

random endowments in incomplete markets

• Hugonnier, Kramkov & Schachermayer (2005 MF): On utility-

based pricing of contingent claims in incomplete markets

• Xu & Yi (2016 MCRF): An optimal consumption-investment

model with constraint on consumption

• Guan, Li, Xu and & Yi (2017 MCRF): A stochastic control

problem and related free boundaries in finance


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Model

Solution

Optimal stopping

• Shiryaev (1978): Optimal stopping rules

• Barndorff-Nielsen & Shiryaev (2010): Change of time and

change of measure

• Dai & Xu (2011 MF): Optimal redeeming strategy of stock

loans with finite maturity

• Xu & Zhou (2013 AAP): Optimal stopping under probability

distortion

• Xu & Yi (2019 MOR): Optimal redeeming strategy of stock

loans under drift uncertainty


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Model reformulation

Model reformulation


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Model reformulation

Overall utility

Definition

The overall utility is

u(k) = supc, g>0,c+g=k

u(c, g).

Assume it satisfies the Inada conditions with power growth rate

• limk→+∞ u(k) = +∞

• limk→0+ u′(k) = +∞

• u(k)� kp with 0 < p < 1


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Model reformulation

Overall utility

Example 1: u(c , g ) = u1(c) + u2(g )

• If u(c , g) = u1(c) + u2(g) is increasing and strictly concave

in both c and g , and

limc→0+

u′1(c) = limg→0+

u′2(g) = +∞.

Then u(·) is globally concave.


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Model reformulation

Overall utility

Example 2: u(c , g ) =√c +

√(g − a)+

• If u(c , g) =√c +

√(g − a)+ for some a > 0. Then

u(k) = supc,g>0,c+g=k

u(c , g) =

√k, 0 6 k 6 2a;√

2k − 2a, k > 2a.

• The concave envelope of u(·) is given by

u(k) =

√k, 0 6 k < a;

12√a

(k + a), a 6 k 6 3a;√

2k − 2a, k > 3a.


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Model reformulation

Overall utility


√(g − a)+

• If u(c , g) =√c +

√(g − a)+ for some a > 0. Then

u(k) = supc,g>0,c+g=k

u(c , g) =

√k, 0 6 k 6 2a;√

2k − 2a, k > 2a.

• The concave envelope of u(·) is given by

u(k) =

√k, 0 6 k < a;

12√a

(k + a), a 6 k 6 3a;√

2k − 2a, k > 3a.


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Model reformulation

Overall utility


√(g − a)+ (cont’d)

-

6

0

u(k)

a 2a 3a k

u(k)

Figure: Non-concave u(·) and its concave envelope u(·).


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Model reformulation

Model reformulation

New formulation

• The new wealth process followsdX (t) = (rX (t) + π(t) · µ+ I (t)1{t6τ}−k(t)) dt

+π(t) · σ dB(t),

X0 = x0,

• k(·) = c(·) + g(·): the total consumption process

• The new target is

supτ,k,π

E[∫ +∞

0e−ρt u(k(t)) dt −

∫ τ

0e−ρtL(t) dt

]• u: the overall utility, non-concave in general


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Post-retirement problem



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Problem formulation

Value function

• The wealth process after retirement t > τ follows

dX (t) = (rX (t) + π(t) · µ− k(t)) dt + π(t) · σ dB(t).

• Define the value function for the post-retirement problem

V (x) = supk,π

E[∫ +∞

τe−ρ(t−τ)u(k(t)) dt

∣∣∣ X (τ) = x

]. (1)

• Difficulty: non-concave utility

• Approach: dual method


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Solution

Dual utility

• Define

h(y) = supk>0

(u(k)− ky

), y > 0.

• Then

h(y) = supk>0

(u(k)− ky

),

u(k) = infy>0

(h(y) + ky).


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Solution


√(g − a)+ (cont’d)

• h(y) = 14y + ( 1

4y − ay)+.

• The supreme is attained at

(c∗(y), g∗(y)) =

( 14y2 , a + 1

4y2 ), 0 < y < 12√a

;

( 14y2 , 0), y > 1

2√a.

• Either c∗(y) + g∗(y) 6 a or > 3a.

• Never optimal to consume a < c∗(y) + g∗(y) < 3a.

• Either g∗(y) = 0 or g∗(y) > 2a.

• Never optimal to consume 0 < g∗(y) < 2a for luxury goods.


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Solution

Data from Ait-Sahalia, Parker and Yogo (2004 JF) (Cont’d)2966 The Journal of Finance

Figure 2. Consumption of basic and luxury goods under nonhomothetic utility. Thefigure plots the consumption of basic and luxury goods as a function of total expenditure.

two conditional Euler equations,

Et

!β(Ct+1 − a)−φ

(Ct − a)−φ

"Rt+1 − R f

t+1

#$= 0, (9)

Et

!β(Lt+1 + b)−ψ

(Lt + b)−ψ

Pt

Pt+1

"Rt+1 − R f

t+1

#$= 0. (10)

The law of iterated expectations implies the unconditional versions of theseequations.

The focus of the previous literature is on the unconditional version of equa-tion (5), or if one takes the view that luxuries are not contained in NIPA non-durables consumption, of equation (9). We instead focus on the estimation andtesting of equation (10). Equation (10) provides a test of whether the consump-tion Euler equation holds for wealthy households.

Our choice of utility function implies that the relevant curvature parame-ter that determines a household’s attitude toward risk depends on the levelof its total expenditures X.3 Consider the Arrow–Pratt definition of relativerisk aversion γ (X) = −Xu′′(X)/u′(X). The coefficient of relative risk aversionwith respect to gambles over C is γC(C ) = φC/(C − a), which falls with C andasymptotically approaches φ. Hence, for households with sufficiently low lev-els of X that only consume C, γ (X) = γC(C ), so φ is the curvature parameterthat controls risk aversion. Risk aversion with respect to gambles over L is

3 Risk aversion that varies with wealth is an inherent feature of any nonhomothetic intra-periodutility function. There is no utility function that admits nonhomothetic Engel curves and deliv-ers constant relative risk aversion (see Stiglitz (1969), Hanoch (1977), and the discussion of theelasticity of intertemporal substitution in Browning and Crossley (2000)).

Figure: Consumption of basic and luxury goods under nonhomothetic

utility


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Solution

Dual value function

• Define

V(y) = E[∫ ∞

0e−ρth(Y (t)) dt

∣∣∣∣ Y (0) = y

], (2)

where

dY (t) = Y (t)((ρ− r) dt + ϑ · dB(t)).

• Define the concave conjugate function of V by

V(x) = infy>0

(V(y) + xy), x > 0. (3)


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Solution

Verification theorem for the post-retirement problem

Theorem 1

The function V is the same as the value function V of the post-

retirement problem.


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Solution

Example 3: u(c , g ) =(c−a)1−φ+

1−φ + (g+b)1−ψ

1−ψ (Ait-Sahalia et al.)

• The utility function is

u(c , g) =(c − a)1−φ+

1− φ+

(g + b)1−ψ

1− ψ.

• Then

h(y) = φ1−φy

1− 1φ − ay + 1

1−ψb1−ψ

+( ψ1−ψy

1− 1ψ + by − 1

1−ψb1−ψ)1{y<b−ψ} .

• The dual value function is

V(y) = C1y1− 1

φ +C2y +C3 + (C4y1− 1

ψ +C5y +C6)1{y<b−ψ} .


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Pre-retirement problem



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Problem formulation

Value function

• The wealth process before retirement t 6 τ follows

dX (t) = (rX (t) + π(t) · µ+ I (t)− k(t)) dt + π(t) · σ dB(t).

• The pre-retirement problem is

supk,π,τ

E[∫ τ

0e−ρt(u(k(t))− L(t)) dt + e−ρτV (X (τ))

].

• Difficulty: non-concave utility, mixed controls

• Approach: dual method


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Solution

Verification theorem for the pre-retirement problem

Theorem 2

If w is a classical solution of the variational inequality (VI)min

{− sup

k,π{(∂t + L)W − ρW + u(k)− L(t)} ,W − V

}= 0,

W (T , x) = V (x), (t, x) ∈ S := [0,T )× (0,∞);(4)

where

L = 12‖π · σ‖

2∂xx + (rx + π · µ+ I (t)− k)∂x .

Then w is the value function of the pre-retirement problem.


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Solution

Dual value function

Consider the following dual variational inequalitymin

{− ∂tW − 1

2‖ϑ‖2y2∂yyW − (ρ− r)y∂yW + ρW

−yI (t)− h(y) + L(t), W − V}

= 0,

W(T , y) = V(y). (t, y) ∈ S;

(5)


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Solution

Dual value function

Related optimal stopping problem

W(t, y) = supt6τ6T

E{∫ τ

te−ρ(s−t)

(I (s)Y (s)+h(Y (s))−L(s)

)ds

+ e−ρ(τ−t)V(Y (τ))∣∣∣ Y (t) = y

},

where the underlying process Y (·) follows a GBM

dY (t) = Y (t)((ρ− r) dt + ϑ · dB(t)).


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Solution

Dual value function: Existence

Theorem 3 (Existence)

The problem (5) has a solution W, which is convex and decreas-

ing in y . Moreover, W, ∂yW are continuous in S, ∂tW, ∂yyWare bounded in any bounded subdomain of S; the free boundary,

defined by the boundary of {W = V}, is Lipschitz in both time

and space variable.

Idea to prove: The existence of the solution can be proved by

standard penalty method. For the proof of the Lipschitz continuity

of the free boundary, we refer to Nystrom (2007).


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Solution

Dual value function: Uniqueness

Theorem 4 (Comparison principle)

Let ui (t, y), i = 1, 2, be the solutions of the following VIsmin{− (∂t +M)ui − fi (t, y), ui − gi (t, y)

}= 0,

ui (T , y) = hi (y), (t, y) ∈ S,

where M is a linear elliptic operator on y . If f1 > f2, g1 > g2,

h1 > h2, and |u1(t, y)|+|u2(t, y)| 6 CeCy2

in S, for some C > 0,

then

u1(t, y) > u2(t, y), (t, y) ∈ S.


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Solution

Value function of the pre-retirement problem

As a consequence, we have

Corollary 5

The dual variational inequality (5) has a unique solution W.

Theorem 6

Let

W(t, x) = infy>0

(W(t, y) + xy), (t, x) ∈ S.

Then W is the value function of the pre-retirement problem.


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Solution

Value function of the pre-retirement problem

As a consequence, we have

Corollary 5

The dual variational inequality (5) has a unique solution W.

Theorem 6

Let

W(t, x) = infy>0

(W(t, y) + xy), (t, x) ∈ S.

Then W is the value function of the pre-retirement problem.


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Optimal retirement region



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Free boundary problem

Variational inequality

• Define

W(t, y) := e−ρt(W(t, y)− V(y)), (t, y) ∈ S.

• Thenmin {−(∂t + L)W− e−ρt(yI (t)− L(t)), W} = 0,

W(T , y) = 0, (t, y) ∈ S;(6)

where

L := 12‖ϑ‖

2y2∂yy + (ρ− r)y∂y .


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Retirement region and working region

• Define the retirement region

R = {(t, y) ∈ S |W(t, y) = 0},

and the working region

C = {(t, y) ∈ S |W(t, y) > 0}.

• Then

R = {(t, y) ∈ S | y 6 b(t)},

C = {(t, y) ∈ S | y > b(t)},

where the free boundary b(t) = inf{y > 0 |W(t, y) > 0}.


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Properties of the free boundary

• We have b(t) 6 L(t)I (t) for all t ∈ [0,T ].

• Because W is independent of u(·), the free boundary b(·) is

irrelevant to the individual’s utility function! It is universal.


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Properties of the free boundary

• We have b(t) 6 L(t)I (t) for all t ∈ [0,T ].

• Because W is independent of u(·), the free boundary b(·) is

irrelevant to the individual’s utility function! It is universal.


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Hypothesis on growth condition

Hypothesis 1

We have L′(t)L(t) > ρ > I ′(t)

I (t) for t ∈ [T − `,T ] with ` a positive

constant 6 T .

• For a young person, his marginal labor cost is decreasing as

he gets more skilled.

• For an older one, his marginal labor cost is increasing as he

becomes ageing with less energy and more burdens such as

illness, family issue, child care.


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Hypothesis on growth condition

Hypothesis 1

We have L′(t)L(t) > ρ > I ′(t)

I (t) for t ∈ [T − `,T ] with ` a positive

constant 6 T .

• For a young person, his marginal labor cost is decreasing as

he gets more skilled.

• For an older one, his marginal labor cost is increasing as he

becomes ageing with less energy and more burdens such as

illness, family issue, child care.


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Hypothesis: income process, labor cost process

-

6

0

L(t)

T − ` tT

I (t)


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Monotonicity of the free boundary

Theorem 7

Assume Hypothesis 1 holds. Then b(t) is increasing for t ∈ [T −`,T ] with the terminal value

b(T−) := limt→T

b(t) =L(T )

I (T ).


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Monotonicity of the free boundary

-

6 L(T )I (T )•

b(t)� y = L(t)I (t)

�

y0

T

T − `

t

R C

Figure: The two regions R and C under Hypothesis 1

.


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A numerical example

• Define

L(t) =

a0 + a1t + 12a2t

2, if t 6 T − `;

eKt , if t > T − `,

where

a0 = eK(T−`) (1− K (T − `) + 12K

2(T − `)2),

a1 = KeK(T−`) (1− K (T − `)) ,

a2 = K 2eK(T−`).


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A numerical example (cont’d)

• Choose 1/K < T − ` so that L(·) is first decreasing and then

increasing.

• Set I (t) = CeK′t .

• Choose the following parameters

K = 2, K ′ = 0.4, C = 8, ` = 0.7, T = 2, ρ = 0.5.

They satisfy all the requirements and Hypothesis 1.


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Figure: The functions L(·) and I (·).


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Figure: The non-monotone free boundary b(·).


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Conclusion

Optimal consumption

• Consume only basic goods when the wealth is small

• Consume basic goods and make savings when the wealth is

intermediate

• Consume small portion in basic goods and large portion in

luxury goods when the wealth is large


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Conclusion

Optimal consumption



intermediate




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Conclusion

Optimal consumption



intermediate




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Conclusion

Optimal retirement time

• Prefer to work for young people

• Prefer to retire near mandatory retirement age

• Not universal: different wealth levels for individuals with dif-

ferent preferences

• Universal: same level marginal consumption utilities for dif-

ferent individuals, determined only by market parameters and

income process and labor cost process


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Conclusion





ferent preferences





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Conclusion





ferent preferences





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Conclusion





ferent preferences





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Conclusion

Thank you for your attention!


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Optimal investment, heterogeneous consumption and the best … › ... › 2019 › qfinance ›...

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