Optimal Investment, Heterogeneous Consumption and Best Time for Retirement
SSubmitted to
Operations Research manuscript (Please, provide the manuscript number!)
Authors are encouraged to submit new papers to INFORMS journals by means ofa style file template, which includes the journal title. However, use of a templatedoes not certify that the paper has been accepted for publication in the named jour-nal. INFORMS journal templates are for the exclusive purpose of submitting to anINFORMS journal and should not be used to distribute the papers in print or onlineor to submit the papers to another publication.
Optimal Investment, Heterogeneous Consumptionand Best Time for Retirement
Zuo Quan Xu
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, [email protected]
Harry Zheng
Department of Mathematics, Imperial College, London SW7 2BZ, UK, [email protected]
This paper studies an optimal investment and consumption problem with heterogeneous consumption ofbasic and luxury goods, together with the choice of time for retirement. The utility for luxury goods isnot necessarily a concave function. The optimal heterogeneous consumption strategies for a class of non-homothetic utility maximizer are shown to consume only basic goods when the wealth is small, to consumebasic goods and make savings when the wealth is intermediate, and to consume small portion in basic goodsand large portion in luxury goods when the wealth is large. The optimal retirement policy is shown to beboth universal, in the sense that all individuals should retire at the same level of marginal utility that isdetermined only by income, labor cost, discount factor as well as market parameters, and not universal, inthe sense that all individuals can achieve the same marginal utility with different utility and wealth. It isalso shown that individuals prefer to retire as time goes by if the marginal labor cost increases faster thanthat of income. The main tools used in analysing the problem are from PDE and stochastic control theoryincluding viscosity solution, variational inequality and dual transformation.
Key words : heterogeneous consumption; non-concave utility; dynamic programming; optimal stopping;variational inequality; dual transformation; free boundary
1. Introduction a r X i v : . [ q -f i n . P M ] A ug u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!)
There has been extensive research in optimal investment and consumption in the literature, seeKaratzas and Shreve (1999), Pham (2009), and references therein for excellent expositions of thetopic. In a complete market model with a fixed investment horizon, the problem can be solvedwith the martingale method and the optimal consumption has a representation in terms of theinverse of the marginal utility of consumption evaluated at the level of the pricing kernel. In anincomplete market model or with optimal stopping time involved, it is no longer possible to use themartingale method to solve the problem as the martingale representation theorem cannot be usedor the optimal stopping time needs to be determined. For a Markov model, the optimal investmentstopping problem may be studied with the stochastic control method. Using the dynamic program-ming principle, one can derive the Hamilton-Jacobi-Bellman (HJB) variational inequality which isin general difficult to solve as it involves the determination of the optimal stopping region and thesolution of a nonlinear partial differential equation (PDE) in the continuation region.One effective method to solve the variational inequality is the dual transformation method.For an optimal investment stopping managers’ decision problem with a mixed power and optionstype non-smooth non-concave utility function, Guan et al. (2017) apply the concavification anddual transformation method to convert the primal variational inequality into a dual variationalinequality, then analyze the properties of its solution and the multiple free boundaries, and finallycharacterize the optimal value function and strategy for the original problem. The key advantageof the dual transformation is that one only needs to solve a linear PDE in the continuation region,which is equivalent to an optimal stopping problem and is relatively easier to solve. For the Black-Scholes model, the dual variational inequality is an American option pricing problem and can besolved numerically with the binomial tree method.Optimal investment and consumption with retirement, coupled with income and labor cost, hasalso been actively studied in the literature. Choi and Shim (2006) discuss lifetime consumption andinvestment problems with retirement, construct the solution of the HJB variational inequality with u and Zheng:
Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) some clever technique, and compare the optimal policies with and without retirement for powerand log utilities. Choi et al. (2006) consider a similar problem with additional leisure componentand its tradeoff with income and use the martingale method to find the optimal solution for theconstant elasticity of substitution utility. Dybvig and Liu (2010) discuss and compare three differentmodels (benchmark, voluntary retirement, and voluntary retirement with no-borrowing constraint)to see their differences in optimal investment and consumption strategies. One common feature ofaforementioned papers is that all models are for lifetime consumption and investment, which makesthe dual problem a perpetual optimal stopping problem with a scalar state variable (discountedpricing kernel) and a single point free boundary that can be determined by the continuity and thesmooth pasting condition. The HJB variational inequality also reduces to a linear second orderordinary differential equation in the continuation region. Yang and Koo (2018) study an optimalconsumption and portfolio selection problem with early retirement option embedded in mandatoryretirement.In the literature of optimal investment and consumption, it is normally assumed that the con-sumption utility is homothetic with a single homogeneous consumption goods and there is nodistinction on types of consumption. In the real world, however, there are many different types ofconsumption, from basic goods everyone needs to luxury goods for the rich. People have differentconsumption behaviors for different types of goods. Hence heterogeneous consumption models arecalled for to study different types of consumption. In the literature of financial economics, thereare some research on different types of consumption and their impact on financial phenomenonand explanatory power for empirical data. For example, Ait-Sahalia et al. 2004 evaluate the riskof holding equity by specifying utility as a non-homothetic function of both luxury and basic con-sumption goods and find that basic consumption overstate the risk aversion whereas for the veryrich, the equity premium is not much a puzzle. Campanale (2018) considers a model with a basicgoods and a luxury goods and finds a substantial reduction in precautionary savings in an equilib-rium heterogeneous agent model compared to a standard homogeneous model. Wachter and Yogo u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) (2010) use the non-homothetic consumption model to predict that the expenditure share for basicgoods declines in total consumption and the variance of consumption growth rises in the level ofconsumption, consistent with empirical findings.
The only literature on optimal heterogeneous consumption with retirement the authors are awareof is a conference presentation, see Koo et al. (2017), which discusses voluntary retirement (opti-mal stopping) and quadratic and HARA utilities for basic and luxury consumption, respectively,together with income and labor cost (disutility). Koo et al. (2017) first solve a post-retirementproblem and then a pre-retirement problem and give some closed form solutions without presentingproofs. Assuming all results are correct in Koo et al. (2017), there is still a number of significantdeficiencies in the modeling. First, there is no mandatory retirement age, which means one canwork as long as one likes, say 100 years old, clearly unrealistic. Second, income and labor cost areconstant over time, which is again unrealistic as both should increase and, as one gets older, themarginal labor cost should be greater than the marginal income to reflect the deteriorating physicalcondition. Third, quadratic function is simple to compute but not suitable as a utility function asit increases and then decreases, but utility function should be a non-decreasing function. Math-ematically, the modeling in Koo et al. (2017) makes the optimal stopping a perpetual Americanoptions pricing problem, which is much easier to solve than a similar finite maturity counterpartas one only needs to determine one number to separate the continuation and exercise regions inthe perpetual case whereas one has to find a curve (free boundary) in the finite maturity case. Wewill address these issues in this paper.A recent paper by Yang and Koo (2018) is possibly the closest work to ours in terms of method-ologies, that is, first to solve a post-retirement problem, then to solve a pre-retirement problemwith dual transformation and variational inequality, finally to characterize the free boundary ofthe continuation and stopping regions. The same methodology has also been used by others, seeGuan et al. (2017) and Ma et al. (2019). The key differences of Yang and Koo (2018) and our u and Zheng:
Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) paper are the following: in Yang and Koo (2018) there is single homogeneous consumption withstrictly concave and infinitely differential utility whereas we have heterogeneous consumption withnon-concave and non-differential utility, which makes our problem more complicated and resultsin considerably different consumption behavior; in Yang and Koo (2018) the income and laborcost are constants whereas our model deals with time-dependent income and labor cost, whichmakes our model more realistic and results in non-monotonic free boundary, in sharp contrast toglobally increasing free boundary of Yang and Koo (2018). Furthermore, Yang and Koo (2018) hasa complete market model whereas we can handle closed convex cone control constraints. This paper considers the dynamic portfolio and consumption problem with two special features.First, there are two types of consumption, namely basic and luxury goods, and the utility forthe luxury goods consumption is not necessarily concave to reflect the phenomenon that one getssatisfaction from luxury consumption only when one can afford to spend sufficiently large amountof wealth on it, otherwise, the utility for luxury consumption is zero. This setup makes the totalutility from consumption an increasing, but not necessarily concave, function. Second, the agenthas an option to choose the time for retirement before a finite mandatory retirement age, dependenton the tradeoff between the income and labor cost if he is working.We show that the heterogeneous consumption problem can be solved by firstly finding the optimaltotal consumption and then the optimal basic goods consumption and the luxury goods consump-tion from a simple local optimization problem (see Example 1). We also show that the optimalheterogeneous consumption strategies are to consume only basic goods when the wealth is small,basic goods and more investment and savings when the wealth is intermediate, and small portionin basic goods and large portion in luxury goods when the wealth is large.For the optimal retirement problem, we separate the problem into a post-retirement problem anda pre-retirement problem. Closed-form optimal strategy is given in the paper for the post-retirementproblem via its dual function. For the pre-retirement problem, we use the dual transformation to u and Zheng:
Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) show that the dual function satisfies a variational inequality where it only involves linear PDE in thecontinuation region. Because closed-form solutions are usually not available for the finite horizonoptimal stopping problem, we provide a numerical method to solve the variational inequality. Ourresults show that the optimal early retirement decision is determined by the marginal utility andthe optimal required wealth depends on the utility. In other words, the optimal retirement policy isnot only universal, in the sense that all individuals should retire at the same level of the marginalutility that is determined by market parameters, discount factor, income and labor cost, but alsonot universal, in the sense that all individuals should retire at different wealth levels due to differentutility preferences. Our findings could have important policy implications for government in makingpension and retirement age decisions. Furthermore, under realistic assumption (Assumption 1), weprove that older individuals (whose age are close to the mandatory retirement age) are more liketo retire as time goes by.Mathematically speaking, the main tools used in analysing the problem are from PDE andstochastic control theory including viscosity solution, variational inequality and dual transforma-tion. In particular, we introduce dual transformation for a nonlinear variational inequality, turningit into a linear one to study. Although the main results of the paper are stated in a completemarket model setup, the same results also hold for constrained portfolio trading strategies in aclosed convex cone, which covers some interesting cases such as no short selling or unavailability ofsome stocks for trading, see Karatzas and Shreve (1999). We emphasize that adding bounded port-folio constraints may lead to the concavification principle invalid and significantly affect economicinsights, see Dai et al. (2019).In summary, this paper shows that for a complicated investment and consumption problem withearly retirement and heterogeneous consumption features, one can decompose the problem into anumber of simpler decision problems, that is, one can decide the optimal allocation of basic andluxury consumption once the total consumption is known, one can determine the optimal stoppingtime and the region of work and retirement with the marginal utility or, equivalently, the dual state u and Zheng:
Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) variable, and one can find the optimal wealth, investment, and total consumption with the dualvariational inequality method. These separation principles may facilitate empirical analysis. Forexample, one may first find the empirical utility for aggregate consumption and empirical marginalutility for early retirement at the master level, then derive individual empirical utilities for basicand luxury consumption and wealth based on specific information at the secondary level, whichsimplifies the decision and empirical procedures.The rest of the paper is organized as follows. In Section 2 we formulate the optimal investmentconsumption model, describe non-homothetic consumption utilities, and illustrate non-concavetotal consumption utility with an example and study the behavior of the optimal consumption.In Section 3 we solve the post-retirement optimal investment and consumption problem. Usingthe concavification and the dual transformation, we show the dual value function is a classicalsolution of a second order linear ordinary differentiable equation, prove a verification theorem, andderive the closed-form dual value function when utilities for basic and luxury goods are powerutilities. In Section 4 we discuss the pre-retirement optimal investment, consumption, and stoppingproblem. We first prove a verification theorem and describe the optimal strategies, then establishthe existence, uniqueness, regularity properties of the solution to the dual variational inequality.In Section 5, we characterize the optimal retirement region with the free boundary independent ofconsumption utilities. We also give a numerical example to show that the free boundary may notbe necessarily increasing for young individuals. Section 6 concludes. Notation
We make use of the following notation throughout this paper: • R n , the n -dimensional real Euclidean space; • M (cid:62) , the transpose of a matrix or vector M ; • (cid:107) M (cid:107) = (cid:113)(cid:80) i,j m ij , the L -norm for a matrix or vector M = ( m ij ).The underlying uncertainty of the financial market is generated by a standard {F t } t (cid:62) -adapted n -dimensional Brownian motion B ( · ) ≡ ( B ( · ) , . . . , B n ( · )) (cid:62) defined on a fixed filtered completeprobability space (Ω , F , P , {F t } t (cid:62) ). u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!)
Given a Hilbert space H with the norm (cid:107) · (cid:107) H , we can define a Banach space L F ([ a, b ]; H ) = ϕ ( · ) : [ a, b ] → H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ( · ) is an {F t } t (cid:62) -adapted, progressively measurableprocess and satisfies (cid:107) ϕ ( · ) (cid:107) F < + ∞ with the norm (cid:107) ϕ ( · ) (cid:107) F = (cid:18) E (cid:20)(cid:90) ba (cid:107) ϕ ( t, ω ) (cid:107) H d t (cid:21)(cid:19) . For two functions f and g , we write f ( x ) (cid:28) g ( x ) if there exists a real constant C > | f ( x ) | (cid:54) Cg ( x ) for all x sufficiently large. Definition 1.
A function f : (0 , + ∞ ) → (0 , + ∞ ) is called power-like decreasing (PLD) if it iscontinuous, strictly decreasing, and satisfies lim x → f ( x ) = + ∞ and f ( x ) (cid:28) x qq − as x → + ∞ forsome 0 < q < Definition 2.
A function f : (0 , + ∞ ) → (0 , + ∞ ) is called power-like increasing (PLI) if it is con-tinuous, strictly increasing, and satisfies lim x → f ( x ) = 0, lim x → + ∞ f ( x ) = + ∞ and f ( x ) (cid:28) x q as x → + ∞ for some 0 < q <
2. Model formulation
We consider an arbitrage-free Black-Scholes financial market with one risk-free bond with theconstant risk-free rate r and n stocks. The dynamics of the stock prices follows the followingstochastic differential equations (SDEs)d S i ( t ) = S i ( t ) (cid:32) b i d t + n (cid:88) j =1 σ ij d B j ( t ) (cid:33) , t (cid:62) , i = 1 , , . . . , n, where b i is the appreciation rates of the stock i and σ ij the volatility coefficients between thestock i and risk source B j . Define the volatility matrix σ = ( σ ij ) and the excess return vector µ = ( b − r, . . . , b n − r ) (cid:62) . The parameters r , µ and σ are constants and σ is nonsingular.Consider a representative individual (“He”). Before the retirement time τ , the individual has anincome I ( t ) at time t , and has no income after retirement. For any time t , before or after retirement, u and Zheng: Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) he consumes c ( t ) on the basic goods and g ( t ) on the luxury goods, also invests π i ( t ) dollars in thestock i . His wealth process, X ( t ), evolves according to (see Karatzas and Shreve (1999))d X ( t ) = ( rX ( t ) + π ( t ) · µ + I ( t ) { t (cid:54) τ } − c ( t ) − g ( t )) d t + π ( t ) · σ d B ( t ) , (1)where π ( t ) := ( π ( t ) , . . . , π n ( t )) is called a portfolio, which is equivalent to holding π i ( t ) /S i ( t ) unitsof stock i at time t for i = 1 , . . . n . In our model bankruptcy is prohibited so the wealth processmust be non-negative all the time. There is also a mandatory retirement age, T , so the individualmust choose a retirement time τ no late than T .The individual aims to find a strategy, which consists of a portfolio π ∗ for stock trading, a non-negative consumption rate c ∗ on the basic goods, a non-negative consumption rate g ∗ on the luxurygoods, and a retirement time τ ∗ no late than T , to maximize the expected utility, namely,sup π,c,g,τ E (cid:20)(cid:90) + ∞ e − ρt u ( c ( t ) , g ( t )) d t − (cid:90) τ e − ρt L ( t ) d t (cid:21) , (2)where u is his utility function on two consumptions, ρ > L ( · ) > Remark 1.
In this paper, we assume the individual will never die. One can also introduce anexogenous random variable to represent the death time into the model. If the death time η takesvalue in [0 , T ] with T > T and has the cumulative distribution function F and is independent ofthe market, then problem (2) becomessup π,c,g,τ E (cid:20)(cid:90) η e − ρt u ( c ( t ) , g ( t )) d t − (cid:90) τ ∧ η e − ρt L ( t ) d t (cid:21) . (3)Taking the conditional expectation over η and using the law of total expectation, we can write (3)equivalently as sup π,c,g,τ E (cid:34)(cid:90) T (1 − F ( t )) e − ρt u ( c ( t ) , g ( t )) d t − (cid:90) τ (1 − F ( t )) e − ρt L ( t ) d t (cid:35) . (4) u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) If η is an exponential variable with parameter λ (mortality rate) and T = ∞ , then (4) reducesto the same infinite horizon problem as (2) with ρ replaced by ρ + λ . If η is not an exponentialvariable, then (1 − F ( t )) e − ρt is a hyperbolic, not exponential, discounting function, which makes(4) a time inconsistent problem that cannot be solved with the dynamic programming principleand the game-theoretic approach would need to be used, which is highly interesting but beyondthe scope of this paper and we leave it for future research. Remark 2.
In this paper, we assume there is no income or pension after retirement. The storywill be different if one considers pension. For instance, Powell (2014), “For some, making a decisionto stay on the job an extra year or two beyond your planned retirement date will enable youtake advantage of your company’s match in your 401(k) plan, delay triggering Social Securityto maximize monthly payouts later, and accumulate more money in your savings,” says SandraTimmerman, a nationally recognized gerontologist.
Remark 3.
There is no control constraint in (1), that is, π ( t ) ∈ R n for all t . We can show theresults of the paper also hold with control constraints π ( t ) ∈ K for all t , where K is a closed convexcone in R n . There is a unique pricing kernel in the complete market model, but there are infinitelymany of them in the presence of control constraints. With a closed convex cone constraint, onecan choose the so-called minimum pricing kernel with which one can find the replicating portfoliotrading strategy and the model is just like a standard complete market model, see (He and Zhou2011, Section 4), Remarks 9 and 10 below.Constant elasticity of substitution function is usually used to combine multiple consumptiongoods or production inputs into an aggregate quantity in economics. This aggregator functionexhibits constant elasticity of substitution. It also arises as a utility function in consumer theory.Due to different nature of consumptions on basic and luxury goods, in this paper, we restrict ourselfto the following type of non-homothetic utility function u ( c, g ) = U (cid:18) c α α + (( g − a ) + ) β β (cid:19) , u and Zheng: Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) where a >
0, 0 < α < β < U is a differentiable, strictly increasing, unbounded concave function,and a > u ( c, g ) is increasing in c and g , respectively. It is strictly concave in c globally on (0 , ∞ ). It is also strictly concave in g on ( a, ∞ ), but not concave on (0 , ∞ ).We reformulate problem (2) to make it more attractable and study the behavior of optimalconsumption. Define the total utility of consumption at level k (cid:62) u ( k ) = max c, g (cid:62) ,c + g = k u ( c, g ) . Once the best total consumption k ∗ at a time is determined, then the best consumption c ∗ ( k ) on thebasic goods and g ∗ ( k ) on the luxury goods at that time are determined by the above optimizationproblem. Therefore there is no need to consider separately the best consumption on basic and thaton luxury goods; we only need to consider the best total consumption.Clearly, ¯ u ( · ) is a strictly increasing, unbounded function. Because one consume 0 on luxury goodswhen k < a , the total utility function ¯ u ( k ) is equal to u ( k, k → + ¯ u (cid:48) ( k ) = + ∞ . (5)Economically speaking, it motives the individual to consume basic goods even if his wealth is verysmall. Furthermore, it is easily seen that ¯ u ( k ) (cid:28) k β for sufficiently large k , so ¯ u ( · ) satisfies thepower growth condition. Therefore, the function ¯ u ( · ) is of PLI type.Our first important conclusion is Proposition 1.
Any non-homothetic utility maximizer should spend almost all of his wealth onluxury goods as the total consumption increases to infinity.
Although Ait-Sahalia et al. (2004) consider a class of utility that is slightly different from ours, thesame conclusion can be drawn for suitable choice of parameters.We now rewrite problem (2) as followssup τ,k,π E (cid:20)(cid:90) + ∞ e − ρt ¯ u ( k ( t )) d t − (cid:90) τ e − ρt L ( t ) d t (cid:21) , (6) u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) subject to the processd X ( t ) = ( rX ( t ) + π ( t ) · µ + I ( t ) { t (cid:54) τ } − k ( t )) d t + π ( t ) · σ d B ( t ) , (7)with an initial endowment X (0) = x > u ( · ) is not necessarily global concave on [0 , ∞ ), so problem (6) is anon-concave utility maximization problem. Non-concave utility maximization problems in generalare hard to deal with (see, e.g., Bian et al. (2018) for a similar non-concave utility maximizationproblem, and Dai et al. (2019) for the case with trading constraint).We end this section by giving an example of non-homothetic utility function for which the totalutility function as well as the optimal consumptions on basic and luxury goods are explicitly given. Example 1.
Let u ( c, g ) = 2 c / + (( g − a ) + ) / . Then¯ u ( k ) = max c,g (cid:62) ,c + g = k u ( c, g ) = k / , (cid:54) k (cid:54) k ; (cid:114)(cid:113) k − a + − (cid:18)(cid:113) k − a + + 1 (cid:19) , k > k , where k > a is the unique solution for43 (cid:115)(cid:114) k − a + 14 − (cid:18)(cid:114) k − a + 14 + 1 (cid:19) = 2 k / . Given a total consumption k , the optimal consumption pair is( c ∗ ( k ) , g ∗ ( k )) = ( k, , (cid:54) k (cid:54) k ; (cid:16)(cid:113) k − a + − , k + − (cid:113) k − a + (cid:17) , k > k . Figure 1 demonstrates the shapes of ¯ u ( · ) and its the concave envelope (cid:101) u ( · ) (namely, the smallestconcave function dominating ¯ u ( · )). We can see that ¯ u ( · ) is not globally concave. Later, we willshow that the optimal total consumption should never be in the range where ¯ u ( · ) is apart from itsconcave envelope, which is ( k − , k + ) in Figure 1. u and Zheng: Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) (cid:45)(cid:54) (cid:64)(cid:82) (cid:101) u ( k ) (cid:64)(cid:73) ¯ u ( k ) k − k k + k Figure 1
An example of the total utility function ¯ u ( · ) and its concave envelope (cid:101) u ( · ).
3. Post-retirement problem
Now we study problem (6). After retirement, because there is no income, the wealth process (7)reduces to d X ( t ) = ( rX ( t ) + π ( t ) · µ − k ( t )) d t + π ( t ) · σ d B ( t ) , for t > τ . (8)This is a stationary process, so we may define the value function for the post-retirement problem V ( x ) = sup k,π E (cid:20)(cid:90) + ∞ τ e − ρ ( t − τ ) ¯ u ( k ( t )) d t (cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) . (9)Assume ϑ = σ − µ is not equal to the zero vector, and the discount factor ρ is sufficiently large,namely ρ > rβ + β − β ) (cid:107) ϑ (cid:107) . (10)This guarantees that the value function for the post-retirement problem is finite. Without suchassumption, the value function may be infinity. Then the value function V ( · ) is a viscosity solutionof its associated HJB equation sup k,π (cid:8) (cid:107) π · σ (cid:107) V (cid:48)(cid:48) ( x ) + ( rx + π · µ − k ) V (cid:48) ( x ) − ρV ( x ) + ¯ u ( k ) (cid:9) = 0 , x > ,V (0) = 0 , (11) u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) in the class of concave PLI functions. This can be proved by the standard approach, see, e.g., Yongand Zhou (1999). The boundary condition is due to the fact that, given the initial wealth x = 0,the only admissible (thus optimal) strategy is ( π ( · ) , k ( · )) ≡ (0 ,
0) because no bankruptcy is allowed.
Remark 4.
The uniqueness of viscosity solution is in general hard to prove. In this paper, we usea construction and verification method which does not require to prove the uniqueness.Assume the value function V ∈ C . Since V is increasing, V (cid:48) (cid:62)
0. If V (cid:48) vanished at some point,because V is concave, it would attain its global maximum at that point, which contradicts to thefact that V is unbounded. Therefore we conclude that V (cid:48) >
0. This further implies by (11) that V (cid:48)(cid:48) <
0. Here it is critical to assume that the admissible set of trading strategies is a cone. If theadmissible set is merely a bounded set, then the value function would not be concave in general;see more discussions in Dai et al. (2019). We now rewrite the HJB equation (11) as − V (cid:48) ( x ) V (cid:48)(cid:48) ( x ) (cid:107) ϑ (cid:107) + rxV (cid:48) ( x ) − ρV ( x ) + h ( V (cid:48) ( x )) = 0 , x > ,V (0) = 0 , (12)where h ( y ) := sup k (cid:62) (cid:0) ¯ u ( k ) − ky (cid:1) , y > , is a convex PLD function. Remark 5.
One can prove that h ( y ) = sup k (cid:62) (cid:0)(cid:101) u ( k ) − ky (cid:1) , where (cid:101) u is the concave envelope of ¯ u (see Figure 1). If we replace ¯ u in problem (9) by (cid:101) u , thevalue function V does not change, which implies that one should not consume at the level k if¯ u ( k ) < (cid:101) u ( k ); in other words, the optimal total consumption k ∗ ( t ) should satisfy ¯ u ( k ∗ ( t )) ≡ (cid:101) u ( k ∗ ( t ))for all t . In Figure 1, this means one consumes either no more than k − or no less than k + . If theindividual can afford to consume in the middle range (say k ), he would prefer to consume only tothe level k − and use the unconsumed amount k − k − for investment or saving until he can afford u and Zheng: Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) to consume at least to the level k + to buy an expensive luxury goods later, rather than to buya less expensive luxury goods and satisfy his basic needs at present. Our conclusion is that lessexpensive luxury goods would not be considered by consumers. Remark 6.
By convexity, h is always continuous, but may not be differentiable. In fact, h isdifferentiable if and only if ¯ u is strictly concave (Xu and Yi 2016, Lemma 7.2). In particular, h isnot differentiable when ¯ u is not concave.Instead of finding a viscosity solution for the HJB equation (11), we provide an explicit classicalsolution for it in the following theorem. Theorem 1 (Solution for the post-retirement problem) . Suppose Y evolves according to ageometric Brownian motion d Y ( t ) = Y ( t )(( ρ − r ) d t − ϑ · d B ( t )) , t (cid:62) . (13) Define (cid:98) V ( y ) = E (cid:20)(cid:90) ∞ e − ρt h ( Y ( t )) d t (cid:12)(cid:12)(cid:12)(cid:12) Y (0) = y (cid:21) . (14) Then (cid:98) V ∈ C (0 , ∞ ) is convex, PLD, and the value function V for the post-retirement problem (9) is given by V ( x ) = inf y> ( (cid:98) V ( y ) + xy ) , x > . (15) Moreover, the optimal wealth, the optimal stock trading strategy and the optimal total consumptionat time t after retirement τ are respectively given by X ∗ ( t ) = − ∂ y ˆ V ( Y ( t )) , π ∗ ( t ) = ϑY ( t ) ∂ yy ˆ V ( Y ( t )) , k ∗ ( t ) = − ∂ y h ( Y ( t )) , t (cid:62) τ, (16) where Y ( t ) is the solution of the equation (13) with the initial condition Y (0) = y which is thesolution of the equation ∂ y ˆ V ( y ) + x = 0 with x the initial wealth at time of retirement. u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!)
Remark 7.
The process Y defined in (13) is called the dual state process and the function (cid:98) V in (14) the dual value function . They play pivotal roles in giving explicit representations of theoptimal value V , wealth X ∗ , portfolio π ∗ and total consumption k ∗ for the post-retirement problem(9) via the duality relations (15) and (16), in other words, once one solves the dual problem, onehas solved the primal problem (9). Due to the convexity of the dual value function (cid:98) V , X ∗ ( t ) and Y ( t ) are counter-monotonic, that is, the smaller the Y ( t ), the larger the X ∗ ( t ). We also have aone-fund theorem as the dual state process Y can be interpreted as a fund as follows:d Y ( t ) Y ( t ) = ρ − r − (cid:107) ϑ (cid:107) r d S ( t ) S ( t ) + θ (cid:62) ( σ (cid:62) σ ) − σ (cid:62) (cid:18) d S ( t ) S ( t ) , d S ( t ) S ( t ) , . . . , d S n ( t ) S n ( t ) (cid:19) (cid:62) . where S denotes the money account. From this we see that the optimal stock trading strategyand the optimal total consumption at time t after retirement τ depends on this special fund only. Remark 8.
Since the total consumption utility ¯ u ( · ) is a non-concave function, its concave envelope (cid:101) u ( · ) has s straight line segment in the middle, whose slope corresponds to the marginal utility ofconsumption or some Y in terms of dual variable. One can find a threshold X = − ∂ y ˆ V ( Y ) from(16). If the optimal wealth X ∗ is less than X , then the optimal total consumption k ∗ is kept ata lower level (basic consumption only), whereas if X ∗ is greater than X , then the optimal totalconsumption k ∗ jumps to a higher level (mainly luxury consumption). This jump consumptionphenomenon only happens when there exits a non-concave consumption utility. The gap or theunconsumed part is used for additional investment or savings in the hope of achieving large wealth. Remark 9.
In the presence of closed convex cone control constraint π ( t ) ∈ K , one may define thepositive polar cone of K by (cid:101) K := { v ∈ R n | v · π (cid:62) , ∀ π ∈ K} . Let ˆ v be the minimum solution of the quadratic function (cid:107) σ − ( v + µ ) (cid:107) over v ∈ (cid:101) K . Denote byˆ ϑ = σ − (ˆ v + µ ). Assume ˆ ϑ (cid:54) = 0. If we replace ϑ in (13) by ˆ ϑ , then (cid:98) V ( y ) defined in (14) is thecorresponding dual value function for control constrained problem, see (He and Zhou 2011, Section4). All the results below hold true by replacing all ϑ below by ˆ ϑ . u and Zheng: Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) We end this section by considering an example where the dual value function for the post-retirement problem can be explicitly given.
Example 2.
Ait-Sahalia et al. (2004) consider the following utility function: u ( c, g ) = ( c − c ) − φ − φ + ( g + b ) − ψ − ψ , where the basic consumption c is at least c . This utility is slightly different from the non-homotheticutility. In this case we have (cid:98) V ( y ) = C y − φ + C y + C + ( C y − ψ + C y + C ) { y
5, are explicitly given constants. Although an explicit expressionfor V is not available, we can determine its value via (15) by effective numerical scheme.
4. Pre-retirement problem
Now we go back to problem (6), which may be rewritten as a pre-retirement problemsup τ,k,π E (cid:20)(cid:90) τ e − ρt (¯ u ( k ( t )) − L ( t )) d t + e − ρτ V ( X ( τ )) (cid:21) , (17)where V is given by (15). The last term represents the total utility after retirement. The wealthprocess X before retirement followsd X ( t ) = ( rX ( t ) + π ( t ) · µ + I ( t ) − k ( t )) d t + π ( t ) · σ d B ( t ) , for t (cid:54) τ . (18)We use dynamic programming to solve problem (17). We denote its value function by W .We introduce the following dual variational inequality min (cid:110) − ∂ t (cid:99) W − (cid:107) ϑ (cid:107) y ∂ yy (cid:99) W − ( ρ − r ) y∂ y (cid:99) W + ρ (cid:99) W − yI ( t ) − h ( y ) + L ( t ) , (cid:99) W − (cid:98) V (cid:111) = 0 , (cid:99) W ( T, y ) = (cid:98) V ( y ) , ( t, y ) ∈ S . (19) u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!)
The dual optimal stopping problem corresponding to the dual variational inequality (19) is givenby sup τ E (cid:20)(cid:90) τ e − ρt ( Y ( t ) I ( t ) + h ( Y ( t )) − L ( t )) dt + e − ρτ (cid:98) V ( Y ( τ )) (cid:21) , (20)where Y is the dual process satisfying the SDE (13).We can show there exists a unique (cid:99) W (with nice properties) satisfying the above variationalinequality (19). Denote by W , p,loc ( S ) the local Sobolev space, see Krylov (1980) for the theory ofSobolev space. The following result characterizes the solution for the pre-retirement problem. Theorem 2 (Solution for the pre-retirement problem) . Let (cid:99) W be the solution of (19) in W , p,loc ( S ) . Then (cid:99) W is convex and PLD in y , and the value function W for the pre-retirementproblem (17) is given by W ( t, x ) = inf y> ( (cid:99) W ( t, y ) + xy ) , ( t, x ) ∈ S . Moreover, the optimal wealth, the optimal stock trading strategy and the optimal total consumptionat time t before retirement τ are respectively given by X ∗ ( t ) = − ∂ y (cid:99) W ( t, Y ( t )) , π ∗ ( t ) = ϑY ( t ) ∂ yy (cid:99) W ( t, Y ( t )) , k ∗ ( t ) = − ∂ y h ( Y ( t )) , (cid:54) t (cid:54) τ, where Y ( t ) is the solution of the equation (13) with the initial condition Y (0) = y ∗ and y ∗ is thesolution of the equation ∂ y (cid:99) W (0 , y ) + x = 0 with x the initial wealth at time 0. Remark 10.
In general, there is no C , solution to the variational inequality (19) and we needto work on derivatives in the weak sense. In the presence of a closed convex cone control constraint π ( t ) ∈ K , similar to the case of post-retirement problem, we may replace ϑ in (19) by ˆ ϑ and allresults below, then we get the corresponding dual value function for control constrained problem,see Remark 9 and (He and Zhou 2011, Section 4). u and Zheng: Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!)
5. Optimal retirement region
In this section we study the optimal retirement time. Even though the dual problem (20) is muchsimpler than the primal problem (17), one still needs to determine the optimal stopping time τ ,which is a free-boundary problem. We can nevertheless simplify this problem further. Define W ( t, y ) := e − ρt ( (cid:99) W ( t, y ) − (cid:98) V ( y )) , ( t, y ) ∈ S . (21)Then the variational inequality (19) in terms of W becomes min {− ( ∂ t + L ) W − e − ρt ( yI ( t ) − L ( t )) , W } = 0 , ( t, y ) ∈ S ; W ( T, y ) = 0 , (22)where L := (cid:107) ϑ (cid:107) y ∂ yy + ( ρ − r ) y∂ y . The optimal stopping problem corresponding to the dual variational inequality (22) is given bysup τ E (cid:20)(cid:90) τ e − ρt ( Y ( t ) I ( t ) − L ( t )) dt (cid:21) , (23)where Y is the dual process satisfying the SDE (13).We define the retirement region and the working region, respectively, as R = { ( t, y ) ∈ S | W ( t, y ) = 0 } , C = { ( t, y ) ∈ S | W ( t, y ) > } . Let b ( t ) = inf { y > | W ( t, y ) > } , (cid:54) t (cid:54) T, with the convention that inf ∅ = + ∞ . Then we have the following characterization of the optimalstopping region: Proposition 2 (Universal stopping region) . The free boundary b ( · ) is irrelevant to the indi-vidual’s utility function, and R = { ( t, y ) ∈ S | y (cid:54) b ( t ) } , C = { ( t, y ) ∈ S | y > b ( t ) } . Furthermore, a bigger I ( · ) or a smaller L ( · ) leads to a smaller b ( · ) , which implies one defersretirement as the income increases or labor cost drops. u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!)
Proposition 2 tells us that one should retire if his marginal utility is small enough (namely hiswealth is big enough). On the one hand, this result is universal in the sense that all individuals(namely, all choices of u ( · )) choose to retire at the same level of marginal utility. The free boundary b ( · ) does not depend on h ( · ) (or the personal preference u ( · )) and only depends on ρ , I ( · ), L ( · ), andmarket parameters. On the other hand, the result is also not universal in the sense that differentindividuals may choose to retire at different levels of wealth. This is a very important consequenceof our model. For example, two individuals A and B will retire at the same marginal utility, say,1 /
8, but A and B have different total utility functions with u A ( x ) = 4 x / and u B ( x ) = 2 x / . ThenA will retire when the wealth reaches 16 whereas B will not retire unless the wealth is at least 64.Intuitively speaking, everyone should retire earlier if he has arrived at a very good economicsituation. The following result confirms this conjecture and provides an explicit upper bound forthe retirement marginal utility. Proposition 3 (Everybody may retire earlier) . We have b ( t ) (cid:54) L ( t ) I ( t ) for all t ∈ [0 , T ] . Note that this upper bound only depends on the labor cost and income and is independent ofindividual utilities. We may conclude that it is likely a good time to take early retirement whenincome is very high, which provides sensible economic appeal.We propose the following hypothesis on the income and labor cost processes, which is not onlyrealistic but also economically important. It will play an important role in determining the mono-tonicity of the optimal retirement time.
Assumption 1 (Growth condition) . We have L (cid:48) ( t ) L ( t ) (cid:62) ρ (cid:62) I (cid:48) ( t ) I ( t ) for t ∈ [ T − (cid:96), T ] with (cid:96) a positiveconstant (cid:54) T . The economic interpretation of this hypothesis is given as follows. For a young person, his marginallabor cost is decreasing as he is getting more skilled. By contrast, for an older one, his marginallabor cost is increasing as he is becoming aging with less energy and more burdens such as illness,family issue, child care. Therefore, as an individual becomes aging, his marginal labor cost increases u and Zheng:
Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) faster than his marginal income. It will be shown that this hypothesis guarantees the monotonicityof the retirement boundary for t ∈ [ T − (cid:96), T ], but this conclusion is not necessarily true before T − (cid:96) .A numerical example will be given to demonstrate this importance phenomenon.The next graph shows an example of the income process I ( · ) and the labor cost process L ( · )that satisfy Assumption 1. Their shapes before T − (cid:96) can be arbitrary. (cid:45)(cid:54) L ( t ) (cid:64)(cid:73) I ( t ) (cid:64)(cid:82) T − (cid:96) tT Figure 2
An example of the income process I ( · ) and the labor cost process L ( · ) Theorem 3 (Monotonic free boundary) . Assume Assumption 1 holds. Then b ( t ) is increasingfor t ∈ [ T − (cid:96), T ] , with terminal value b ( T − ) := lim t → T b ( t ) = L ( T ) I ( T ) . Economically speaking, older individuals (less than (cid:96) years to the mandatory retirement age) aremore like to retire as time goes by, but for younger individuals, the marginal income may growthfaster than marginal labor cost, so they may not prefer to retire. In particular, if T − (cid:96) = 0, theneveryone is more like to retire as time flies.Figure 3 illustrates the results of Proposition 3 and Theorem 3. u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) (cid:45)(cid:54) L ( T ) I ( T ) • b ( t ) (cid:64)(cid:64)(cid:73) L ( t ) I ( t ) (cid:64)(cid:64)(cid:82) t T − (cid:96) Ty CR Figure 3
An example of the retirement region R and the working region C under Assumption 1. Although, the free boundary b ( · ) is shown in Theorem 3 under Assumption 1 to be monotonicallyincreasing as time approaches to the mandatory retirement age, in the next section, we give anumerical example to show it is not necessarily monotonically increasing globally in general. Remark 11.
Last but not least we discuss how to numerically solve the pre-retirement problem(17) which is well known to be exceedingly difficult as one not only needs to find the free boundarybut also solves a nonlinear PDE in the continuation region. The dual formulation (20) significantlysimplifies the problem as it is an optimal stopping problem of one state variable, which has beenstudied extensively in the literature. Using the smooth pasting condition and the Itˆo-Doob-Meyerdecomposition of the value function, Jacka (1991) and Peskir (2005) show that the free boundary isthe unique solution to a nonlinear integral equation. Since the free boundary is not known a priori,one may reformulate the dual variational inequality (19) as a linear complementarity problemand solve it numerically with the implicit-explicit method of Cont and Voltchkova (2005). Thecrux of our approach is the dual variational inequality (22) which is independent of dual utilityfunctions, so solving one optimal stopping problem (23) is equivalent to solving infinitely manyoptimal stopping problems (20) with a simple connection (21). u and Zheng:
Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) Example 3.
We first construct the functions I ( · ) and L ( · ) to satisfy Assumption 1. Define L ( t ) = a + a t + a t , if t (cid:54) T − (cid:96) ; e Kt , if t > T − (cid:96). To ensure the continuity up to the second order derivative at T − (cid:96) , we let a = e K ( T − (cid:96) ) (cid:0) − K ( T − (cid:96) ) + K ( T − (cid:96) ) (cid:1) ,a = Ke K ( T − (cid:96) ) (1 − K ( T − (cid:96) )) ,a = K e K ( T − (cid:96) ) . Choose 0 < /K < T − (cid:96) , then a < a ( T − (cid:96) ) + a >
0. This ensures L ( · ) is first decreasingand then increasing. Set I ( t ) = Ce K (cid:48) t . To satisfy the growth condition Assumption 1, we require
K > ρ > K (cid:48) .We now choose the following parameters K = 2 , K (cid:48) = 0 . , C = 8 , (cid:96) = 0 . , T = 2 , ρ = 0 . . They satisfy all the requirements above so that Assumption 1 holds. Figure 4 depicts functions I ( · ) and L ( · ). u and Zheng: Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!)
Figure 4
The functions I ( · ) and L ( · ). To find the free boundary, we use the binomial tree method and find the optimal exercise bound-ary with the dynamic programming principle, see Kim et al. (2016). We further choose the followingparameters: µ = 0 . , r = 0 . , σ = 0 . , ∆ t = 0 . , N = 2000 , where N is the tree size and ∆ t is the step size. Figure 5 depicts the optimal retirement boundary b ( · ). We see from the picture that the boundary is decreasing for time t far from maturity T , andincreasing for t (cid:62) T − (cid:96) = 1 .
3. We also see that its terminal value b ( T − ) = 3 . L ( T ) I ( T ) ,which is consistent with Theorem 3. u and Zheng: Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!) Figure 5
The free boundary b ( · ).
6. Conclusions
In this paper we discuss an optimal investment, heterogeneous consumption problem with manda-tory retirement age and options for early retirement. We solve the problem with the dual controlmethod and characterize the free boundary that separates the regions of working and retirement.We show that the optimal heterogeneous consumption strategies are to consume only basic goodswhen the wealth is small, to consume basic goods and make savings when the wealth is interme-diate, and to consume small portion in basic goods and large portion in luxury goods when thewealth is large. We also show that the optimal retirement policy is universal, in the sense that allindividuals should retire at the same level of marginal utility which is determined only by incomeand labor cost, discount factor and market parameters, but independent of individual’s utility. Ourfindings could have important policy implications for government in making pension and retirementage decisions.
Acknowledgment . The authors are grateful to two anonymous reviewers, Associate Editor andArea Editor, whose constructive comments and suggestions have helped to improve the paper ofthe previous version. u and Zheng:
Investment, Heterogeneous Consumption and Retirement Time Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!)
References
Ait-Sahalia Y, Parker JA, and Yogo M (2004) Luxury goods and the equity premium,
Journal of Finance
SIAM Journal on Financial Mathematics
B.E. Journal ofMacroeconomics,
18, issue 1.Choi KJ and Shim G (2006) Disutility, optimal retirement, and portfolio selection,
Mathematical Finance
Mathematical Finance
SIAM Journal on Numerical Analysis https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3422276
Dybvig PH and Liu H (2010) Lifetime consumption and investment: retirement and constrained borrowing,
Journal of Economic Theory
John Wiley Sons , New York.Guan CH, Li X, Xu ZQ, and Yi FH (2017) A stochastic control problem and related free boundaries infinance,
Mathematical Control and Related Fields
Mathematical Finance
Mathematical Finance
Methods of Mathematical Finance , Springer-Verlag, New York.Kim YS, Stoyanov S, Rachev S, and Fabozzi F (2016) Multi-purpose binomial model: Fitting all momentsto the underlying geometric Brownian motion,
Economics Letters u and Zheng:
Investment, Heterogeneous Consumption and Retirement Time
Article submitted to
Operations Research ; manuscript no. (Please, provide the manuscript number!)
The third International Conference on Engineeringand Computational Mathematics, 31 May - 2 June, 2017 , , page 95.Krylov N (1980) Controlled Diffusion Processes , Springer-Verlag, New York.Ma JT, Xing J, and Zheng H (2019) Global closed-form approximation of free boundary for optimal invest-ment stopping problems,
SIAM Journal on Control and Optimization
Journal of Computational Mathematics and Optimization
Mathematical Finance
Continuous-time Stochastic Control and Optimization with Financial Applications , Berlin,Springer.Powell R (2014) Pre-retirement checklist: 10 tasks to complete,
USA TODAY , .Wachter JA, and Yogo M (2010) Why do household portfolio shares rise in wealth? The Review of FinancialStudies
Math-ematical Control and Related Fields
Mathematics of Operations Research
Stochastic Controls: Hamilton Systems and HJB equation , New York,Springer. -companion to
Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time ec1
Proofs of Statements
EC.1. Proof of Proposition 1
For any non-homothetic utility function u ( c, g ) = U (cid:18) c α α + (( g − a ) + ) β β (cid:19) , we need to show lim k →∞ g ∗ ( k ) k = 1 . Suppose this were not true. Then, because 0 (cid:54) g ∗ ( k ) (cid:54) k , thereexit a constant 0 < (cid:15) < k n such that g ∗ ( k n ) < (1 − (cid:15) ) k n for all n .Thus, ¯ u ( k n ) = u ( c ∗ ( k n ) , g ∗ ( k n )) (cid:54) u ( k n , g ∗ ( k n )) (cid:54) U (cid:18) k αn α + (((1 − (cid:15) ) k n − a ) + ) β β (cid:19) . On there other hand, since β > α , we have(((1 − (cid:15)/ k n − a ) + ) β β > k αn α + (((1 − (cid:15) ) k n − a ) + ) β β for sufficiently large k n , which leads to u (cid:0) (cid:15)k n / , (1 − (cid:15)/ k n (cid:1) > U (cid:18) (((1 − (cid:15)/ k n − a ) + ) β β (cid:19) > U (cid:18) k αn α + (((1 − (cid:15) ) k n − a ) + ) β β (cid:19) (cid:62) ¯ u ( k n ) . This clearly contradicts the definition of ¯ u . EC.2. Example 1
Let f ( c ) = 2 c / + 43 (( k − c − a ) + ) / = c / + ( k − c − a ) / , (cid:54) c (cid:54) k − a ;2 c / , c (cid:62) k − a. • When 0 (cid:54) k (cid:54) a , we have ¯ u ( k ) = max (cid:54) c (cid:54) k f ( c ) = max (cid:54) c (cid:54) k c / = 2 k / , and the optimal pair is c ∗ ( k ) = k, g ∗ ( k ) = 0 . c2 e-companion to Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time • When k > a , let c ∗ = (cid:114) k − a + 14 − . Then 0 (cid:54) c ∗ < k − a and f (cid:48) ( c ∗ ) = 0. Because f (cid:48)(cid:48) (cid:54) f is concave on [0 , k − a ]. Hencemax (cid:54) c (cid:54) k − a f ( c ) = f ( c ∗ ) = 2( c ∗ ) / + 43 ( c ∗ ) / = 43 (cid:115)(cid:114) k − a + 14 − (cid:18)(cid:114) k − a + 14 + 1 (cid:19) . Consequently, ¯ u ( k ) = max (cid:54) c (cid:54) k f ( c ) = max (cid:110) sup (cid:54) c
On the function V : Choose the policy π ( t ) = 0 and k ( t ) = rX ( t ) / t (cid:62) τ , then the wealthprocess follows d X ( t ) = rX ( t ) / t, for t > τ . -companion to Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time ec3
It follows that X ( t ) = X ( τ ) e r ( t − τ ) / and k ( t ) = rX ( t ) / rX ( τ ) e r ( t − τ ) / / t (cid:62) τ . Hence, V ( x ) (cid:62) E (cid:20)(cid:90) + ∞ τ e − ρ ( t − τ ) ¯ u ( rX ( τ ) e r ( t − τ ) / /
2) d t (cid:12)(cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) = (cid:90) + ∞ e − ρt ¯ u ( rxe rt/ /
2) d t. Because ¯ u is unbounded, the value function V is unbounded. On the function h : The convexity of h follows from its definition and this implies its continuityon (0 , ∞ ). The estimate h ( y ) (cid:28) y pp − as y → + ∞ follows from the fact that ¯ u ( k ) (cid:28) k p ; and h ( y ) → + ∞ as y → + follows from (5). Obviously h is decreasing. If it were not strictly decreasing, then h would attain a local (positive) minimum at some point, which would also be the global minimumby its convexity, but this contradicts the estimate h ( y ) (cid:28) y pp − → y → + ∞ . Proof of Theorem 1:
Without loss of generality, we may assume the retirement time τ = 0.By (10) and the upper bound h ( y ) (cid:28) y pp − , we have (cid:98) V ( y ) (cid:28) y pp − . It is not hard to verify the claimthat (cid:98) V ∈ C (0 , ∞ ) is convex, PLD and satisfies12 (cid:107) ϑ (cid:107) y (cid:98) V (cid:48)(cid:48) ( y ) + ( ρ − r ) y (cid:98) V (cid:48) ( y ) − ρ (cid:98) V ( y ) + h ( y ) = 0 , (EC.1)and lim N → + ∞ E (cid:104) e − ρN (cid:98) V ( Y ( N )) (cid:12)(cid:12) Y (0) = y (cid:105) = 0for any y >
0. Define ϕ ( x ) = inf y> ( (cid:98) V ( y ) + xy ) , x > , then we have the dual relationship (cid:98) V ( y ) = sup x> ( ϕ ( x ) − xy ). Moreover, ϕ ( x ) = (cid:98) V ( y ∗ ) + xy ∗ with y ∗ being the solution of the euqation (cid:98) V (cid:48) ( y ) + x = 0, which leads to ϕ ∈ C (0 , ∞ ) by the implicitfunction theorem and ϕ (cid:48) ( x ) = y ∗ and (cid:98) V (cid:48)(cid:48) ( y ) ∂ x y ∗ + 1 = 0. In particular, ϕ (cid:48)(cid:48) ( x ) = ∂ x y ∗ <
0. So ϕ is astrictly concave PLI solution for the HJB equation (12). Alsolim N → + ∞ E (cid:2) e − ρN ϕ ( X ( N )) (cid:12)(cid:12) X (0) = x (cid:3) = 0for any x > c4 e-companion to Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time
For any admissible stopping time τ and strategy ( k ( · ) , π ( · )), let θ n,N = inf { t (cid:62) τ : X ( t ) (cid:62) X ( τ ) + n or | π ( t ) | (cid:62) n } ∧ ( τ + N ) . Then by Itˆo’s lemma and (11),d( e − ρt ϕ ( X ( t )))= e − ρt ( − ρϕ ( X ( t )) + (cid:107) π · σ (cid:107) ϕ (cid:48)(cid:48) ( X ( t )) + ( rX ( t ) + π ( t ) · µ − k ( t )) ϕ (cid:48) ( X ( t ))) d t + e − ρt ϕ (cid:48) ( X ( t )) π ( t ) · σ d B ( t ) (cid:54) − e − ρt ¯ u ( k ( t )) d t + e − ρt ϕ (cid:48) ( X ( t )) π ( t ) · σ d B ( t ) . (EC.2)Hence, ϕ ( X ( τ ))) (cid:62) e − ρ ( θ n,N − τ ) ϕ ( X ( θ n,N )) + (cid:90) θ n,N τ e − ρ ( t − τ ) ¯ u ( k ( t )) d t (EC.3) − (cid:90) θ n,N τ e − ρt ϕ (cid:48) ( X ( t )) π ( t ) · σ d B ( t ) . Taking the conditional expectation on both sides, we have by the positivity of ϕ and the martingaleproperty of Itˆo’s integral that ϕ ( x ) (cid:62) E (cid:20)(cid:90) θ n,N τ e − ρ ( t − τ ) ¯ u ( k ( t )) d t (cid:12)(cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) . Letting n → ∞ and N → ∞ in above, by Fatou’s lemma we obtain ϕ ( x ) (cid:62) E (cid:20)(cid:90) ∞ τ e − ρ ( t − τ ) ¯ u ( k ( t )) d t (cid:12)(cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) . Maximizing the right hand side over ( k ( · ) , π ( · )), we get ϕ ( x ) (cid:62) V ( x ).On the other hand, we use feedback controls π ∗ ( t ) = − ϕ (cid:48) ( X ( t )) ϕ (cid:48)(cid:48) ( X ( t )) ϑ, k ∗ ( t ) = arg max k (cid:62) (˜ u ( k ) − kϕ (cid:48) ( X ( t )))in the wealth process (8). The corresponding inequalities in (EC.2) and (EC.3) become identities,so ϕ ( x ) = E (cid:20) e − ρ ( θ n,N − τ ) ϕ ( X ( θ n,N )) (cid:12)(cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) + E (cid:20)(cid:90) θ n,N τ e − ρ ( t − τ ) ¯ u ( k ( t )) d t (cid:12)(cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) . -companion to Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time ec5
Letting n → ∞ in above, by the dominated convergent theorem and the stationary property of theprocess (8), we obtain ϕ ( x ) = E (cid:20) e − ρN ϕ ( X ( τ + N )) (cid:12)(cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) + E (cid:20)(cid:90) τ + Nτ e − ρ ( t − τ ) ¯ u ( k ( t )) d t (cid:12)(cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) = E (cid:20) e − ρN ϕ ( X ( N )) (cid:12)(cid:12)(cid:12)(cid:12) X (0) = x (cid:21) + E (cid:20)(cid:90) τ + Nτ e − ρ ( t − τ ) ¯ u ( k ( t )) d t (cid:12)(cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) . It follows by letting N → ∞ that ϕ ( x ) = E (cid:20)(cid:90) ∞ τ e − ρ ( t − τ ) ¯ u ( k ( t )) d t (cid:12)(cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21) (cid:54) V ( x ) . On Example 2:
Let u ( c ) = ( c − c ) − φ − φ , u ( g ) = ( g + b ) − ψ − ψ . Then u ( c, g ) = u ( c ) + u ( g ). The dual functions of u and u are respectively given by (cid:98) u ( y ) = sup c (cid:62) c (cid:0) u ( c ) − cy (cid:1) = φ − φ y − φ − c y, and (cid:98) u ( y ) = sup g (cid:62) (cid:0) u ( g ) − gy (cid:1) = ψ − ψ y − ψ + b y, y < b − ψ ; − ψ b − ψ , y (cid:62) b − ψ . Therefore, h ( y ) = sup k (cid:62) c sup c (cid:62) c , g (cid:62) ,c + g = k (cid:0) u ( c, g ) − ky (cid:1) = sup c (cid:62) c ,g (cid:62) (cid:0) u ( c ) + u ( g ) − ( c + g ) y (cid:1) = sup c (cid:62) c (cid:0) u ( c ) − cy (cid:1) + sup g (cid:62) (cid:0) u ( g ) − gy (cid:1) = (cid:98) u ( y ) + (cid:98) u ( y )= φ − φ y − φ − c y + − ψ b − ψ + (cid:0) ψ − ψ y − ψ + b y − − ψ b − ψ (cid:1) { y
Investment, Heterogeneous Consumption and Retirement Time which satisfies (cid:98) V ( y ) (cid:28) y pp − . The parameters can be easily determined to be C = − φ − φ φ (cid:107) ϑ (cid:107) (1 − φ ) − ρφ − r ( φ − φ ) , C = − c r , C = ρ (1 − ψ ) b − ψ ,C = − ψ − ψ ψ (cid:107) ϑ (cid:107) (1 − ψ ) − ρψ − r ( ψ − ψ ) , C = b r , C = − ρ (1 − ψ ) b − ψ . Since h ∈ C (0 , ∞ ), by (EC.1), we see that (cid:98) V ∈ C (0 , ∞ ). One can show that (cid:98) V satisfies therequirements. EC.4. Pre-retirement problem
On the function (cid:99) W : The convexity and PLD property in y variable follows immediately fromits definition. By standard penalty method, we can show (cid:99) W is a solution for problem (19), which isconvex and PLD for each fixed t ∈ [0 , T ). Moreover, (cid:99) W , ∂ y (cid:99) W are continuous in S , ∂ t (cid:99) W , ∂ yy (cid:99) W arebounded in any bounded subdomain of S ; the free boundary, defined by the boundary of { (cid:99) W = (cid:98) V } ,is Lipschitz in both time and space variable. For the proof of the Lipschitz continuity of the freeboundary, we refer to (Nystrom 2007, Theorem 16 and Remark 2). The uniqueness of the solutionfollows from the following comparison principle, which can be proved by penalty method (also seeFriedman (1982)). Theorem EC.1 (Comparison principle) . Let u i ( t, y ) , i = 1 , , be the solutions of the followingvariational inequalities min (cid:8) − ( ∂ t + M ) u i − f i ( t, y ) , u i − g i ( t, y ) (cid:9) = 0 , ( t, y ) ∈ S ,u i ( T, y ) = h i ( y ) , where M is a linear elliptic operator on y . If f (cid:62) f , g (cid:62) g , h (cid:62) h , and | u ( t, y ) | + | u ( t, y ) | (cid:54) Ce Cy in S , for some C > , then u ( t, y ) (cid:62) u ( t, y ) , ( t, y ) ∈ S . In fact (cid:99) W is the value function for the following optimal stopping problem (cid:99) W ( t, y ) = sup t (cid:54) τ (cid:54) T E (cid:20)(cid:90) τt e − ρ ( s − t ) ( I ( s ) Y ( s ) + h ( Y ( s )) − L ( s )) d s + e − ρ ( τ − t ) (cid:98) V ( Y ( τ )) (cid:12)(cid:12)(cid:12)(cid:12) Y ( t ) = y (cid:21) , -companion to Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time ec7 where the underlying process Y ( · ) follows (13). Proof of Theorem 2:
Define ζ ( t, x ) = inf y> ( (cid:99) W ( t, y ) + xy ) , ( t, x ) ∈ S , which is clearly non-decreasing in x . One can also show that ∂ xx ζ < ϕ (cid:48)(cid:48) ( x ) < min (cid:26) − sup k,π { ( ∂ t + L ) ζ − ρζ + ¯ u ( k ) − L ( t ) } , ζ − V (cid:27) = 0 ,ζ ( T, x ) = V ( x ) , ( t, x ) ∈ S := [0 , T ) × (0 , ∞ ) . (EC.4)where L = (cid:107) π · σ (cid:107) ∂ xx + ( rx + π · µ + I ( t ) − k ) ∂ x , which can be reformulated as min (cid:110) − ∂ t ζ + ∂ x ζ ∂ xx ζ (cid:107) ϑ (cid:107) − ( rx + I ( t )) ∂ x ζ − h ( ∂ x ζ ) + ρζ + L ( t ) , ζ − V (cid:111) = 0 ,ζ ( T, x ) = V ( x ) , ( t, x ) ∈ S . (EC.5)Let y = y ( t, x ) = ∂ x ζ ( t, x ) for ( t, x ) ∈ S . Then ∂ x y = ∂ xx ζ ( t, x ) and (cid:99) W ( t, y ) = ζ ( t, x ) − xy. (EC.6)Differentiating the last equation in x gives ∂ y (cid:99) W ( t, y ) ∂ xx ζ ( t, x ) = ∂ x ζ ( t, x ) − y − x∂ xx ζ ( t, x ) = − x∂ xx ζ ( t, x ) , that is, ∂ y (cid:99) W ( t, y ) = − x, (EC.7)Differentiating it in x again gives ∂ yy (cid:99) W ( t, y ) ∂ xx ζ ( t, x ) = − . c8 e-companion to Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time
Differentiating (EC.6) in t gives ∂ t (cid:99) W ( t, y ) + ∂ y (cid:99) W ( t, y ) ζ tx ( t, x ) = ∂ t ζ ( t, x ) − xζ tx ( t, x ) , ∂ t (cid:99) W ( t, y ) = ∂ t ζ ( t, x ) , in view of (EC.7). Simple calculation shows that (cid:16) − ∂ t ζ + ∂ x ζ ∂ xx ζ (cid:107) ϑ (cid:107) − ( rx + I ( t )) ∂ x ζ − h ( ∂ x ζ ) + ρζ + L ( t ) (cid:17)(cid:12)(cid:12)(cid:12) ( t,x ) = (cid:16) − ∂ t (cid:99) W − (cid:107) ϑ (cid:107) y ∂ yy (cid:99) W − ( ρ − r ) y∂ y (cid:99) W + ρ (cid:99) W − yI ( t ) − h ( y ) + L ( t ) (cid:17)(cid:12)(cid:12)(cid:12) ( t,y ) , (EC.8)which is nonnegative by (19). Using (19) again, (cid:99) W (cid:62) (cid:98) V on S , so ζ (cid:62) V by the dual expressions. Toprove (EC.5), it is only left to show (cid:16) − ∂ t ζ + ∂ x ζ ∂ xx ζ (cid:107) ϑ (cid:107) − ( rx + I ( t )) ∂ x ζ − h ( ∂ x ζ ) + ρζ + L ( t ) (cid:17) (cid:0) ζ − V (cid:1)(cid:12)(cid:12)(cid:12) ( t,x ) = 0 . (EC.9)There are two cases: • If (cid:99) W ( t, y ) > (cid:98) V ( y ), then by (19) and (EC.8), we have (cid:16) − ∂ t ζ + ∂ x ζ ∂ xx ζ (cid:107) ϑ (cid:107) − ( rx + I ( t )) ∂ x ζ − h ( ∂ x ζ ) + ρζ + L ( t ) (cid:17) (cid:12)(cid:12)(cid:12) ( t,x ) = 0 . So (EC.9) holds. • If (cid:99) W ( t, y ) = (cid:98) V ( y ), then since (cid:99) W (cid:62) (cid:98) V on S by (19), we see that the function (cid:99) W − (cid:98) V attaintsits minimum value 0 at ( t, y ). Hence the first order condition gives (cid:98) V (cid:48) ( y ) = ∂ y (cid:99) W ( t, y ) = − x, by (EC.7). This means y is a stationary point of the convex function z (cid:55)→ (cid:98) V ( z ) + xz , so (cid:98) V ( y ) + xy = min z ( (cid:98) V ( z ) + xz ) = V ( x ) , by (15). Together with (EC.6), we obtain ζ ( t, x ) = (cid:99) W ( t, y ) + xy = (cid:98) V ( y ) + xy = V ( x ) , and thus (EC.9) holds. -companion to Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time ec9
We now show that ζ = W . It suffices to prove that ζ (0 , x ) = W (0 , x ) for any x >
0. For anystopping time τ (cid:54) T and strategy ( k ( · ) , π ( · )), let θ n = inf { t (cid:62) X ( t ) (cid:62) x + n or | π ( t ) | (cid:62) n } ∧ τ. By Itˆo’s lemma and (EC.4)d( e − ρt ζ ( t, X ( t ))) = e − ρt ( ∂ t ζ + L ζ − ρζ ) d t + e − ρt ∂ x ζπ ( t ) · σ d B ( t ) (cid:54) − e − ρt (¯ u ( k ( t )) − L ( t )) d t + e − ρt ∂ x ζπ ( t ) · σ d B ( t ) . (EC.10)Taking integral on both sides and rearranging terms, we have ζ (0 , x ) (cid:62) e − ρθ n ζ ( θ n , X ( θ n )) + (cid:90) θ n e − ρt (¯ u ( k ( t )) − L ( t )) d t − (cid:90) θ n e − ρt ∂ x ζπ ( t ) · σ d B ( t ) , (EC.11)which leads to, by taking expectation, ζ (0 , x ) (cid:62) E (cid:2) e − ρθ n ζ ( θ n , X ( θ n )) (cid:3) + E (cid:20)(cid:90) θ n e − ρt (¯ u ( k ( t )) − L ( t )) d t (cid:21) . (EC.12)Let n go to ∞ in above, by Fatou’s lemma and (EC.4), we have ζ (0 , x ) (cid:62) E (cid:2) e − ρτ ζ ( τ, X ( τ )) (cid:3) + E (cid:20)(cid:90) τ e − ρt (¯ u ( k ( t )) − L ( t )) d t (cid:21) (cid:62) E (cid:2) e − ρτ V ( X ( τ )) (cid:3) + E (cid:20)(cid:90) τ e − ρt (¯ u ( k ( t )) − L ( t )) d t (cid:21) . By arbitrariness of τ and strategy ( k ( · ) , π ( · )), we obtain ζ (0 , x ) (cid:62) W (0 , x ).Use the feedback controls π ∗ ( t ) = − ∂ x ζ ( t,X ( t )) ∂ xx ζ ( t,X ( t )) ϑ, k ∗ ( t ) = arg max k (cid:62) (˜ u ( k ) − k∂ x ζ ( t, X ( t ))) , in (18). Let τ ∗ = inf { t (cid:62) ζ ( t, X ( t )) = V ( X ( t )) } , then τ ∗ (cid:54) T by (EC.4). Furthermore the inequalities in (EC.10), (EC.11), and (EC.12) becomeidentities under the above controls. Let n go to ∞ in (EC.12), by the Dominated ConvergenceTheorem, ζ (0 , x ) = E (cid:104) e − ρτ ∗ ζ ( τ ∗ , X ( τ ∗ )) (cid:105) + E (cid:34)(cid:90) τ ∗ e − ρt (¯ u ( k ∗ ( t )) − L ( t )) d t (cid:35) = E (cid:104) e − ρτ ∗ V ( X ( τ ∗ )) (cid:105) + E (cid:34)(cid:90) τ ∗ e − ρt (¯ u ( k ∗ ( t )) − L ( t )) d t (cid:35) (cid:54) W (0 , x ) . c10 e-companion to Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time
Therefore, we conclude that ζ is equal to the value function W . Moreover, ( τ ∗ , π ∗ ( · ) , k ∗ ( · )) definedabove is an optimal control. EC.5. Optimal retirement region
Proof of Proposition 2:
By (22), W is independent of h ( · ), so the stopping region is irrelevantto the individual’s utility function. We next show that the function W is increasing in y for eachfixed t ∈ [0 , T ). For any fixed κ >
1, and define W κ ( t, y ) = W ( t, κy ) for ( t, y ) ∈ S . Then min {− ( ∂ t + L ) W κ − e − ρt ( κyI ( t ) − L ( t )) , W κ } = 0 , ( t, y ) ∈ S ; W κ ( T, y ) = 0 . Applying Theorem EC.1, we see that W κ ( t, y ) (cid:62) W ( t, y ). Because κ > W is increasing in y , which ensures the existence of the free boundary b ( t ). Furthermore,Theorem EC.1 implies that a bigger I ( · ) or a smaller L ( · ) leads to a smaller b ( · ). Proof of Proposition 3: In { ( t, y ) ∈ S | y < b ( t ) } ⊂ R , we have W = 0, so the variationalinequality (22) yields − ( ∂ t + L ) W − e − ρt ( yI ( t ) − L ( t )) = e − ρt ( L ( t ) − yI ( t )) (cid:62) , namely, L ( t ) (cid:62) yI ( t ). Hence the claim follows. Proof of Theorem 3:
For any small ε >
0, set W ε ( t, y ) = W ( t − ε, y ) for ( t, y ) ∈ [( T − (cid:96) ) ∨ ε, T ] × (0 , ∞ ). Then min (cid:8) − ( ∂ t + L ) W ε − e − ρ ( t − ε ) ( yI ( t − ε ) − L ( t − ε )) , W ε (cid:9) = 0 , ( t, y ) ∈ [( T − (cid:96) ) ∨ ε, T ] × (0 , ∞ ); W ε ( T, y ) (cid:62) . Assumption 1 implies e − ρ ( t − ε ) ( yI ( t − ε ) − L ( t − ε )) (cid:62) e − ρt ( yI ( t ) − L ( t )) , ( t, y ) ∈ [( T − (cid:96) ) ∨ ε, T ] × (0 , ∞ ) . By Theorem EC.1, we see that W ε ( t, y ) (cid:62) W ( t, y ), which implies W is decreasing in t and thus b ( t )is increasing. -companion to Xu and Zheng:
Investment, Heterogeneous Consumption and Retirement Time ec11
Now suppose b ( T − ) < L ( T ) I ( T ) (see Figure EC.1). (cid:45) L ( T ) I ( T ) • b ( t ) L ( t ) I ( t ) b ( T − ) • tT RC Figure EC.1
When b ( T − ) < L ( T ) I ( T ) . For ( t, y ) ∈ (cid:110)(cid:2) T, T − ε (cid:1) × (cid:16) b ( T − ) , L ( T ) I ( T ) (cid:17)(cid:111) ∩ C , we have − ( ∂ t + L ) W − e − ρt ( yI ( t ) − L ( t )) = 0 , W ( t, y ) > , W ( T, y ) = 0 , hence, − ∂ t W (cid:12)(cid:12) ( T,y ) = e − ρT ( yI ( T ) − L ( T )) < , contradicting that W is decreasing in tt