Equilibrium consumption and portfolio decisions with stochastic

discount rate and time-varying utility functions

Huiling Wua, Chengguo Wengb, Yan Zengc,∗

aChina Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081,P.R.China

bDepartment of Statistics and Actuarial Science, University of Waterloo, Waterloo N2L 3G1, CanadacLingnan (University) College, Sun Yat-sen University, Guangzhou 510275, P.R. China

Abstract

This paper studies a multi-period investment-consumption optimization problem with astochastic discount rate and a time-varying utility function, which are governed by aMarkov-modulated regime switching model. The investment is dynamically reallocatedbetween one risk-free asset and one risky asset. The problem is time-inconsistent due tothe stochastic discount rate. An analytical equilibrium solution is established by resortingto a game theoretical framework. Numerous sensitivity analysis and numerical examplesare provided to demonstrate the effects of the stochastic discount rate and time-varyingutility coefficients on the decision-maker’s investment-consumption behavior. Our resultsshow that many properties which are satisfied in the classical models do not hold any moredue to either the stochastic discount rate or the time-varying utility function.

Keywords: Nash equilibrium, Stochastic discount rate, Investment-consumption, Regimeswitching.

1. Introduction

The problem of optimal investment-consumption has been one of the time-honored topicsin finance since the seminal work of Samuelson (1969) and Merton (1969, 1971). Variousextensions and applications, in both continuous-time and discrete-time settings, have beendeveloped over the past forty years, including, for example, Richard (1975), Karatzas andShreve (1998), Campbell and Viceira (2001), Chen (2005), Cocco et al. (2005), Cheungand Yang (2007), and Li et al. (2008) among many others. The present paper considersthe problem in a discrete-time setting with many additional components, including regimeswitching models for investment asset prices, stochastic discount rate, time-varying utilityfunction and equilibrium formulation.

In most literature on investment-consumption problems, the discount rate is constantover time, which unfortunately has been strongly questioned with much empirical and ex-perimental evidence. For example, empirical studies on human and animal behavior per-formed by Ainslie (1992) indicate that the discount rate is declining over time, and theexperiments by Harrison et al. (2002) conclude that a constant discount rate only works

∗Corresponding author. Tel.: +86 20 84110516; fax: +86 20 84114823.Email addresses: sunnyling168@hotmail.com (Huiling Wu), c2weng@uwaterloo.ca (Chengguo

Weng), zengy36@mail.sysu.edu.cn (Yan Zeng)

Preprint submitted to OR Spectrum October 16, 2016

for a relatively short period and the discount rate varies significantly in response to thechanges in value of various sociodemographic variables. The discount rate has also beenempirically observed to be age dependent. For instance, according to the studies by Readand Read (2004), the older people discount more than the younger, and the middle agedpeople discount less than either group. For more experimental studies, we refer the readersto Thaler (1981), Loewenstein and Prelec (1992) and Frederick et al. (2002). In fact, thediscount rate reflects the subjective time preference of a decision-maker in the sense that ahigh discount rate implies less patience than a low discount rate in waiting for a consump-tion. In reality, there are often more than one person with possibly distinct time preferencesparticipating in a decision-making process at different stages, and even in the case that asingle decision-maker dominates throughout the whole decision-making process, her tasteoften varies over time, say, in response to the changes in environment. This explains whythe discount rate is assumed to be stochastic and dynamically vary along with certain ex-ogenous process; see for example, Grenadier and Wang (2007), Harris and Laibson (2013)and Pirvu and Zhang (2014). Therefore, for an optimal investment-consumption problem(particularly over a long investment horizon), it is more practical to consider non-constantdiscount rates.

Indeed, the utility-based decision-making problems with non-constant discount rates havereceived considerable attention in recent years. The non-constant discount rates make theproblems time-inconsistent1 and Bellman optimality principle is not applicable for derivingtheir solutions; see Section 2 for more detailed explanation. Referring to Strotz (1956),Grenadier and Wang (2007), Marın-Solano and Navas (2009, 2010) and Zou et al. (2014),there are three possible ways which a decision-maker may follow in dealing with a time-inconsistent problem: (a) A pre-committed decision-maker does not revise her initial strat-egy even if her strategy is time-inconsistent.2 She commits her successors to implementthe strategy devised by herself at the initial time. (b) A naive decision-maker continuouslymodifies her calculated decisions for the future. She solves the associated optimal controlproblem for each time n = 0, 1, . . . , T − 1 and then patches together the optimal decision ateach time. More specifically, if

(dnn, d

nn+1, d

nn+2, . . .

)is an optimal strategy planned at time n

for n = 0, 1, 2, . . ., then the naive decision-maker’s actual policy rule will be (d00, d

11, d

22, . . .).

She does not take into account the fact that her preferences may change in the near futureand that these patched decisions are time-inconsistent. (c) A sophisticated decision-makertakes possible future revisions into account, and looks for an (subgame perfect Nash) equi-librium strategy, which is time-consistent. More detailed discussion can be found in Pollak(1968), Goldman (1980) and Barro (1999).

The equilibrium strategy, as a time-consistent solution to a time-inconsistent problem,has been highly advocated among scholars in recent years; see Kryger and Steffensen (2010),

1A dynamic optimization problem is called a time-inconsistent problem if the Bellman’s optimalityprinciple does not hold (see Bjork and Murgoci (2014), Bjork et al. (2014)). Otherwise, it is a time-consistent problem.

2For a dynamic optimization problem, if the optimal strategy π(n) = (πn, πn+1, . . . , πT−1) at timen(n = 0, 1, . . . , T − 1) is consistent with the optimal strategy π(k) determined at any time k > n, then it iscalled a time-consistent strategy. Otherwise, if there exists some k > n such that the truncated part of π(n),i.e., (πk, πk+1, . . . , πT−1) is not equal to π(k), then the strategy is called a time-inconsistent strategy. Whenthe strategy is time-inconsistent, the strategy π(n) previously decided at time n will not be implemented attime k > n unless some commitment mechanism exists or the decision-maker is self control (Hsiaw (2013)).

2

Wang and Forsyth (2011), Zeng and Li (2011), Wei et al. (2013), Bjork and Murgoci(2014) and Bjork et al. (2014) for mean-variance problems, Ekeland and Lazrak (2006)for the Ramsey consumption-saving problem with a general discount function, Ekelandand Pirvu (2008) for Merton’s investment-consumption problem with a quasi-exponentialdiscount factor and a CRRA utility. There are some studies on equilibrium strategy toinvestment-consumption problems, but most of them are in a continuous-time setting; see,for example, Marın-Solano and Navas (2010), Pirvu and Zhang (2014), Ekeland et al. (2012)and Kronborg and Steffensen (2015).

In contrast, our paper studies the equilibrium strategies to a discrete-time investment-consumption problem with a stochastic discount rate and a time-varying power utility func-tion. Firstly, in view of the studies by Read and Read (2004) and the existing psychologicalreports, we formulate the discount rate to hinge on the age of the decision-maker, whichis equivalent to being time varying. Further, we assume that the discount rate dependson the market state considering the fact that the behaviors including time preference of aninvestor are more or less affected by the financial environment. For example, some peoplemay be more patient and would like to invest their wealth as much as possible in a bullishmarket state because they would like to take the opportunity to accumulate wealth as muchas possible for future consumption. The setting of the state-dependent discount rate is ac-tually consistent with that in Pirvu and Zhang (2014), where a state-dependent exponentialdiscount function is adopted. In this paper, we do not consider the specific modelling of thediscount rate, and instead, we assume a general form ρ(n, ξn), which reflects its dependenceon the market state and the timing. For a decision-maker who is totally not sensitive tomarket state, we can always retrieve the corresponding results by reducing ρ(n, ξn) to ρ(n) asa special case, and further, if the decision maker is sensitive to neither the market state northe timing, some special results can be obtained from our general analysis by substituting aconstant discount rate into. As mentioned above, the non-constant discount rate will lead totime-inconsistency. However, time-inconsistency is not the only issue we want to deal within this paper. We also care about the effects of the time-varying and state-dependent utilitycoefficients on the investment-consumption strategy when the discount rate is reduced toa constant over time. This is motivated by Canakoglu and Ozekici (2009, 2010, 2012). Inorder to study a portfolio selection problem, Canakoglu and Ozekici (2009) consider an ex-ponential function of the form U(i, x) = K(i)−C(i) exp(−x/β), and Canakoglu and Ozekici(2010, 2012) assume a power utility function of the form U(i, x) = K(i) + C(i)(x− β)γ/γ,where i denotes the market state. Interestingly, the derived optimal investment strategiesare independent of the coefficients K(i) and C(i). This motivates us to investigate whetherthese coefficients are also irrelevant to the solutions to an investment-consumption model.In our model, we adopt a power utility of the form Un(i, x) = ϑn(i)xγ/γ with a more generalϑn(i) compared with C(i) in Canakoglu and Ozekici (2010, 2012). It turns out that theutility coefficients ϑn(i) measure the attention degree of consumption. A higher ϑn(i) leadsto a higher consumption proportion at time n, and a higher ϑk(i), k > n for a future periodresults in a less proportion of consumption for the current time n.

In addition to the economic meanings on their own, the stochastic discount rates and thetime-varying utility coefficients enable us to regularize our model and equilibrium solutionsto recover and compare with existing results, including the investment-consumption problemwithout bequests by Samuelson (1969), the investment-consumption model by Cheung andYang (2007) and the investment-only problems by Cheung and Yang (2004) and Canakogluand Ozekici (2010). Moreover, through numerous mathematical analysis and numerical

3

examples, a number of interesting phenomena are observed with respect to the equilibriumstrategy and value function we derived from our model, which have never been observed inthe literature. For example, the “hump saving” phenomenon regarding the expected wealthobserved by Merton (1969) and Samuelson (1969) does not necessarily hold in generalunder our formulation depending on the pattern by which the utility functions vary alongtime; see Proposition 4.4 and Example 4.1. Moreover, the conclusion obtained in theexisting literature with a constant discount rate that the decision-maker will consume alarger proportion of her wealth at a time closer to the end of investment time horizondoes not always hold under our formulation with non-constant discount rates. Finally, theresult that the decision-maker tends to consume more (less) and invest less (more) in a bad(good) investment environment in the existing literature is also not always true under ourformulation with a non-constant discount rate.

As we have previously mentioned, the stochastic discount rate leads to time-inconsistencyin our model. To develop a time-consistent solution, we follow the recent literature andtackle the problem from the perspective of a sophisticated decision-maker. We take theentire problem as a non-cooperative game with one decision-maker at each stage, viewedas the future incarnation of the decision-maker, and develop equilibrium investment andconsumption strategies in a closed form, which is time-consistent. Sensitivity analysis andnumerical examples are provided to demonstrate the effects of stochastic discount rate andtime-varying utility coefficients on the equilibrium strategies.

The rest of the paper proceeds as follows. The proposed discrete-time investment-consumption problem is formulated in Section 2. The equilibrium strategy is derived inSection 3 and some comparisons with the exiting literature are provided in the same sec-tion. Mathematical analysis of the effects on the equilibrium strategy from the time-varyingutility coefficients is given in Section 4, and numerical sensitivity analysis is presented inSection 5. The paper is concluded in Section 6. The proofs of some relevant results arerelegated to the Appendixes.

2. Model setup

2.1. Market model and wealth process

Suppose that the return processes of the investment assets are governed by a Markov-modulated regime switching model, where the financial market state is assumed to switchamong a finite set of distinct regimes {1, . . . , L} with its dynamics being fully dictated bya time-homogeneous Markov chain {ξn, n = 0, 1, · · · } with a transition matrix Q. Here, therandom variable ξn represents the market state over period n, i.e., time interval [n, n + 1)for n = 0, 1, . . .. In the specific calibration of a regime switching financial model, two orthree states are commonly assumed for the Markov chain. They respectively represent abullish state and a bearish state in a model with two states, and a third one, if exists, istypically interpreted as an intermediate (or normal) state. The regime switching frameworkfor modeling econometric series offers a transparent and intuitive way to capture marketbehavior through different economic conditions. It has been widely used in econometricssince the pioneering work of Hamilton (1989). For its applications in dynamic portfolioselection and portfolio insurance, we refer the readers to Canakoglu and Ozekici (2009,2010, 2012), Wu and Li (2012), Weng (2013, 2014) and references therein.

Consider an investor who chooses to invest in a risk-free asset and a risky asset tradablein the market. Denote the gross returns of the two assets over period [n, n+ 1), given ξn =

4

i ∈ {1, . . . , L}, respectively by rf (i) (for the risk-free asset) and Rn(i) (for the risky asset),where rf (i) is a positive constant, {R0(i), R1(i), . . . , RT−1(i)} are non-negative identicallydistributed random variables, and Rn(i) is independent of Rm(j) for n 6= m and i, j ∈{1, . . . , L}. In addition, it is reasonable to assume that E[Rn(i)] > rf (i) > 1 for all n and i ∈{1, · · · , L}. For stochastic optimization problems, a Markov-modulated regime switchingmodel often retains the same level of tractability as in a setting with a stationary assetprocess. That is, given the market state i, the returns of the risky assets at different timeperiods are independent identically distributed (i.i.d.) Although most research papers aboutportfolio optimization with regime switching have adopted this assumption, for example,Cheung and Yang (2004), Cheung and Yang (2007) and Li et al. (2008), actually the i.i.d.assumption does not hold in some real-world situations. It should be noted that, we adoptthe assumption that the risky returns at different time periods are identically distributedjust for notation convenience. No essential difficulty occurs even if this assumption is notadopted. We just need to change the notation of the investment strategy α(i) in Lemma 3.1and other related notations to be time-dependent. However, if we do not assume that therisky returns at different periods are independently distributed, then our problem becomesa decision-making problem with serial correlation. The serial correlation together with therandom discount rate will give rise to much more technical difficulties to obtain the closed-form results and analyze their properties. Therefore, we take a simple assumption aboutthe random returns of the risky asset to highlight the effects of the stochastic discount rateand time-varying utility coefficients. It should be noted that, however, the unconditionalreturns {R0(ξ0), R1(ξ1), . . . , RT−1(ξT−1)} in our model are not independently distributed.Instead, it leads to a nonstationary time series with a nice structure for calibration andoptimization. In the conclusion section, we point out the limitations of our paper andpresent some related research topics worthy of further study.

We assume that the investor has an initial wealth w0 and plans her investment and con-sumption for T consecutive periods. The investor may also be referred to as the “decision-maker” throughout the paper. For n = 0, 1, . . . , T − 1, let cn be the consumption amountby the investor at time n, and immediately after the consumption, a proportion αn of theremaining wealth is invested in the risky asset and the remaining proportion 1 − αn inthe risk-free asset over period [n, n + 1). Then πn := (cn, αn) represents an investment-consumption strategy of the investor over period n, and π(n) := (πn, πn+1, . . . , πT−1) isa strategy throughout periods n, n + 1, · · · , T − 1. Obviously, the wealth process W π

n

(n = 0, 1, . . . , T ) recursively evolves as follows:

W πn+1 = (W π

n − cn) [(1− αn)rf (ξn) + αnRn(ξn)] , n = 0, 1, . . . , T − 1. (1)

Throughout the paper, the investor is assumed to be prohibited from consuming more thanher wealth over each period and borrowing is not allowed, so that cn and αn are respectivelyconstrained by 0 ≤ cn ≤ W π

n and 0 ≤ αn ≤ 1 for n = 0, 1, . . . , T − 1.

Remark 2.1. The wealth process (1) is non-negative since both rf (ξn) and Rn(ξn) are non-negative and the investment-consumption strategy is subject to constraints cn ∈ [0,Wn] andαn ∈ [0, 1], n = 0, 1, . . . , T − 1.

2.2. Discount rate and performance function

We assume that the discount rate is dynamically varying according to the market state ξnand the age of the investor. Specifically, the discount rate at time n is given by ρ(y+n, ξn)

5

for certain function ρ(·, ·), n = 1, . . . , T , where y denotes the age of the decision-maker atthe time of decision making so that she is aged at y + n up to time n. For presentationconvenience, simplified notation ρ(n, ξn) := ρ(y + n, ξn) will be used hereafter as long asthe initial age of the decision-maker is irrelevant to the analysis. Accordingly, we define theperformance function as follows:

Jn(i, wn; π(n)) = En,i,wn

[T−1∑k=n

Uk (ck)

(1 + ρ(n, i))k−n+

UT (W πT )

(1 + ρ(n, i))T−n

], n = 0, 1, . . . , T−1, (2)

where En,i,wn [·] = E[·|ξn = i,W πn = wn], and similar to Canakoglu and Ozekici (2009, 2010,

2012), for k = 0, . . . , T , the utility function is given by

Uk(x) = ϑk(ξk)xγ

γwith γ < 1, γ 6= 0 and ϑk(ξk) > 0. (3)

Next, we make some comments on the utility function (3) and its interplay with thenon-constant discount rate. Considering that the sense of satisfaction from consumptionand wealth is realized at times n, n + 1, . . . , T , we assume that the utilities, i.e., the eval-uation criterions of these feelings from consumption or terminal wealth, depend on thecorresponding time periods and market states in the future. For example, the utility fromthe consumption realized at time k is set to depend on time k and the corresponding mar-ket state ξk, i.e., Uk(ck) = ϑk(ξk)(ck)

γ/γ. The discount rate, however, measures the currentvalue of future utilities, and hence, the discount rate at time n should instead depend onthe subjective information at time n. When it moves to time n + 1, the market state willtransfer into a state j with a transition probability Pr(ξn+1 = j|ξn = i), j ∈ {1, 2, . . . , L},and then the discount rate is changed to ρ(n+ 1, j) accordingly. There are two reasons forus not to combine the coefficient ϑk(ξk) into the discount rate ρ(n, i) in (2). First, they ob-viously bear clear economic meanings by their owns. Second, the existence of the stochasticdiscount rate ρ(n, i) gives rise to the issue of time-inconsistency of the problem which willbe seen shortly, whereas ϑk(ξk) does not.

2.3. Pre-committed solution and time inconsistency

The specific formulation for the decision making problem we study in this paper is givenin the next subsection. It involves optimizing the performance function as defined in (2). Inthis subsection, we investigate the time inconsistency issue attributed to the non-constantdiscount rate, and we start from the classical investment-consumption problem at time n:

ICn (ρ(n, ξn)) : V (n)n (ξn, wn) = max

π(n)Jn (ξn, wn; π(n)) , n = 0, 1, . . . , T − 1, (4)

where π(n) = (πn, πn+1, . . . , πT−1) and πn ∈ [0, wn] × [0, 1]. Obviously, the problemICn(ρ(n, ξn)) aims to achieve the maximal expected utility from inter-temporal consumptionand terminal wealth based on the information up to time n, such as the discount rate. Astime moves from n to n+ 1, the decision-maker faces problem ICn+1(ρ(n+ 1, ξn+1)), wherethe discount rate is adapted from ρ(n, ·) to ρ(n + 1, ·) and accordingly, the performancefunction changes to

Jn+1(j, wn+1; π(n+ 1)) = En+1,j,wn+1

[T−1∑k=n+1

Uk (ck)

(1 + ρ(n+ 1, j))k−n−1+

UT (W πT )

(1 + ρ(n+ 1, j))T−n−1

],

6

where j denotes the market state realized at time n + 1. Due to the dynamic adjustmentof the time preferences, the optimal strategy denoted by π(n+ 1) = (πn+1, πn+2, . . . , πT−1)for the problem over a shorter time horizon

ICn+1 (ρ(n+ 1, ξn+1)) : maxπ(n+1)

Jn+1 (ξn+1, wn+1; π(n+ 1)) ,

is not necessarily consistent with the optimal strategy π∗(n) =(π∗n, π

∗n+1, . . . , π

∗T−1

)de-

rived at time n which solves problem ICn (ρ(n, ξn)). In other words, the optimal solutionπ∗(n + 1) :=

(π∗n+1, . . . , π

∗T−1

)truncated from π∗(n) does not necessarily solve problem

ICn+1 (ρ(n+ 1, ξn+1)) which the decision-maker faces at time n+ 1.Proposition 2.1 below gives the explicit solution to problem ICn(ρ(n, ξn)) for a fixed n.

It is worth emphasizing that the optimal investment strategy α(n) := (α(n)n , . . . , α

(n)T−1) and

consumption strategy c(n) := (c(n)n , . . . , c

(n)T−1) presented there only optimize the particular

performance function Jn which represents the satisfaction of the decision-maker at time n.The solution (α(n), c(n)) does not involve the optimization of the decision-maker’s satisfactionat any other time k > n in any sense. This solution requires the collaboration and pre-commitment from the future decision making to exclusively optimize the objective at timen and thus, it is called a pre-committed solution.

Proposition 2.1. For k = n, n+1, . . . , T −1, the optimal investment-consumption strategyof problem ICn(ρ(n, ξn)) is given by

α(n)k (ξk) =

arg max

α(n)k ∈[0,1]

Ek,ξk

[α

(n)k Rk(ξk) + (1− α(n)

k )rf (ξk)]γ, if 0 < γ < 1,

arg minα(n)k ∈[0,1]

Ek,ξk

[α

(n)k Rk(ξk) + (1− α(n)

k )rf (ξk)]γ, if γ < 0,

(5)

c(n)k (ξk, wk) =

wk

1 +(A

(n)k (ξk)

) 11−γ

. (6)

The value function V(n)k (ξk, wk) is given by

V(n)k (ξk, wk) =

(wk)γ

γϑk(ξk)

(1 +

(A

(n)k (ξk)

) 11−γ)1−γ

, k = n, n+ 1, . . . , T − 1, (7)

where

A(n)T (i) = 0, i = 1, 2, . . . , L,

A(n)k (ξk) =

Y (ξk)

(1 + ρ(n, ξn))ϑk(ξk)Ek,ξk

[(1 +

(A

(n)k+1(ξk+1)

) 11−γ)1−γ

ϑk+1(ξk+1)

], (8)

k = n, n+ 1, . . . , T − 1.

Proof. We omit the proof because the optimal strategy and the value function can be derivedin the same way as a special case of our forthcoming equilibrium results; see Remark 3.1and Special case 1 presented right after Theorem 3.1 in Section 3.

The results in Proposition 2.1 hold for a general decision making time n = 0, 1, . . . , T−1,and equation (8) indicates that the optimal strategy for problem ICn (ρ(n, ξn)) depends on

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the discount rate ρ(n, ξn) at time n. As time goes by, if there exists an m ≥ n+ 1 such thatρ(m, ξm) 6= ρ(n, ξn), then the optimal consumption amounts for the problem ICm(ρ(m, ξm))become

c(m)k (ξk, wk) =

wk

1 +(A

(m)k (ξk)

) 11−γ

, k = m,m+ 1, . . . , T − 1,

where

A(m)T (i) = 0, i = 1, 2, . . . , L,

A(m)k (ξk) =

Y (ξk)

(1 + ρ(m, ξm))ϑk(ξk)Ek,ξk

[(1 +

(A

(m)k+1(ξk+1)

) 11−γ)1−γ

ϑk+1(ξk+1)

], (9)

k = m,m+ 1, . . . , T − 1.

Because ρ(m, ξm) 6= ρ(n, ξn), we obtain c(m)k (ξk, wk) 6= c

(n)k (ξk, wk) for k = m,m+1, . . . , T−1.

This means that the globally optimal strategy at time n does not solve problem ICm (ρ(m, ξm))at a future time m. In other words, due to the stochastic discount rate, the pre-committedsolutions as obtained in Proposition 2.1 are time inconsistent.

2.4. Problem formulation

Time-consistency is a basic requirement of dynamic decision-making. Consequently, awise decision-maker will take possible future revisions into account, give up the globallyoptimal strategy at time n and turn to a subgame Nash equilibrium strategy. To thisend, following the idea from Bjork and Murgoci (2014) and the relevant references therein,we view the decision-making process as a non-cooperative game with one distinct decision-maker, referred to as the decision-maker n, over period n, for n = 0, 1, . . . , T −1 and assumethat the decision-maker n only decides the optimal policy at time n to maximize her overallsatisfaction she feels at time n, on the premise that she has determined her strategies toexercise in the future to optimize her satisfaction in the future over each possible scenarios.More precisely, the (subgame perfect Nash) equilibrium strategy is defined below.

Definition 2.1. Let π be a fixed feasible control law and define

π(n) = (πn, πn+1, . . . , πT−1) , n = 0, 1, . . . , T − 1,

with a strategy πn ∈ Ψ(wn) := [0, wn]× [0, 1]. If for any n = 0, 1, . . . , T −1, i ∈ {1, 2, . . . , L}and wn ≥ 0, we have

maxπn∈Ψ(wn)

Jn (i, wn; π(n)) = Jn(i, wn; π(n)),

then π is said to be a subgame perfect Nash equilibrium strategy (or simply equilibriumstrategy) to problem (2). In addition, if an equilibrium π exists, the equilibrium valuefunction is defined as Vn(i, wn) = Jn(i, wn; π(n)).

In view of Definition 2.1, it is appealing to resort to backward recursion for an equilibriumsolution:

(a) The equilibrium strategy πT−1 = (cT−1, αT−1) is obtained by letting the decision-makerT − 1 optimize the objective function

JT−1(i, wT−1; πT−1) = ET−1,i,wT−1

[UT−1 (cT−1) +

UT(W

πT−1

T

)1 + ρ(T − 1, i)

].

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(b) At time T − 2, considering that the decision-maker T − 1 might not follow the strategymade at time T −2, the decision-maker T −2 takes possible future revisions into accountand maximizes her objective function JT−2, given the knowledge that the decision-makerT − 1 will use the strategy πT−1. Therefore, πT−2 = (cT−2, αT−2) is the optimal controlthat optimizes the function JT−2 (i, wT−2; (πT−2, πT−1)).

(c) Generally, at time n, the decision πn = (cn, αn) of the decision-maker n is determined bymaximizing Jn given that the forthcoming decision-makers n+ 1, n+ 2, . . ., T − 1 havechosen the strategy π(n+ 1) = (πn+1, πn+2, . . . , πT−1), i.e.,

Vn(i, wn) = maxπn∈Ψ(wn)

Jn (i, wn; (πn, πn+1, . . . , πT−1)) , n = 0, 1, . . . , T − 1, (10)

πn = arg maxπn∈Ψ(wn)

Jn (i, wn; (πn, πn+1, . . . , πT−1)) , n = 0, 1, . . . , T − 1.

The above backward recursion algorithm always implies that an optimal solution is asubgame-perfect Nash-equilibrium of the specific game. It also indicates that, when thediscount rate is reduced to a constant, an equilibrium strategy is also reduced to the classicalglobally optimal strategy. By adopting the subgame perfect Nash equilibrium (equilibriumfor short hereafter) strategy, the basic requirement of time-consistency for the strategyhas been satisfied. It should be emphasized that the term of time-consistency, used inmany existing papers to describe the equilibrium strategy, means the persistence of thedecision-maker to the strategy as time evolves but not the globally optimality of the strategy.According to the above backward recursion, every decision-maker at the time chain hasno impulse to deviate from the equilibrium strategy because this strategy is optimallydetermined by herself given the best decisions she will make in the future.

To sum up, the optimization problem that this paper aims to solve is given by :

NICn (ρ(n, i)) : maxπn∈Ψ(wn)

Jn (i, wn; (πn, πn+1, . . . , πT−1)) ,

s.t. πk solves NICk (ρ(k, ξk)) , k = n+ 1, . . . , T − 1,(11)

with terminal period problem given as

NICT−1 (ρ(T − 1, ξT−1)) : maxπT−1∈Ψ(wT−1)

ET−1,ξT−1,wT−1

[UT−1 (cT−1) +

UT(W

πT−1

T

)1 + ρ(T − 1, ξT−1)

].

From (11), the maximizer πn seems to be a function of the wealth wn. Thus, it appearsto be impractical to solve the problem since one would have to compute πn for each pos-sible value of wn. Actually, by the forthcoming Lemmas 3.1 and 3.2, we can separate theproblem into two: an investment problem and a consumption problem, because these twolemmas imply that there is no link between the investment and consumption decisions. Inour derivation, we first obtain the investment strategy αn and then study the consumptionstrategy cn by substituting the obtained αn into the consumption problem. Such a separa-bility between the consumption and investment indeed has been observed in a number ofexisting literature, for example, Merton (1969, 1971), Samuelson (1969), Richard (1975),Chen (2005), Cheung and Yang (2007), Ekeland and Pirvu (2008), Li et al. (2008).

9

Remark 2.2. The time-varying coefficients ϑk(ξk) in the power utility function Uk(x) =ϑk(ξk)

xγ

γcan be viewed as regularization parameters for the investment-consumption model.

When the discount rate is a constant, taking certain special values for these coefficientswill lead to particular models in the existing literature. For example, if we set ϑ0 = ϑ1 =· · · = ϑT−1 = 0, our formulation reduces to be an investment-only problem, which has beenstudied by Canakoglu and Ozekici (2010); furthermore, if ϑ0 = ϑ1 = · · · = ϑT−1 = 0 andϑT = 1, our formulation becomes a model which has been studied by Cheung and Yang(2004). Moreover, if ϑT (·) = 0 is set, the problem becomes an investment-consumptionproblem without bequests, for which a similar case with an identical consumption utilityfunction for each period has been studied by Samuelson (1969). Furthermore, if the utilityfunctions are deterministic and identical over time, the investment-consumption model (11)boils down to the one in Cheung and Yang (2007) without absorbing state.

3. Equilibrium strategy and equilibrium value function

To obtain an equilibrium strategy, we start from establishing a recursive formula forequilibrium value functions Vn defined in (10). From (2),

Jn(i, wn; π)

=En,i,wn

(1 + ρ(n+ 1, ξn+1))T−n−1

(1 + ρ(n, i))T−nJn+1(ξn+1,W

πnn+1; π) + Un (cn)

+T−1∑k=n+1

(1 + ρ(n, i))T−k − (1 + ρ(n+ 1, ξn+1))T−k

(1 + ρ(n, i))T−nUk (ck)

. (12)

Equation (10) means Vn+1(i, wn+1) = Jn+1 (i, wn+1; π(n+ 1)), and hence (12) implies thefollowing recursive formula:

Vn(i, wn) = maxπn∈Ψ(wn)

Jn (i, wn; (πn, πn+1, . . . , πT−1))

= maxcn,αn∈Ψ(wn)

En,i,wn

(1 + ρ(n+ 1, ξn+1))T−n−1

(1 + ρ(n, i))T−nVn+1(ξn+1,W

πnn+1) + Un (cn)

+T−1∑k=n+1

(1 + ρ(n, i))T−k − (1 + ρ(n+ 1, ξn+1))T−k

(1 + ρ(n, i))T−nUk (ck)

(13)

with n = 0, 1, . . . , T − 1 and terminal condition

VT (i, wT ) = ϑT (i)(wT )γ

γ. (14)

Remark 3.1. The recursion for Vn in (13) differs from that in the classical models, wherethe discount rates are constant and the value functions Vn depend on the first two terms onthe right hand side of (13) only. As shown in (13), Vn for our model additionally dependson the inter-temporal consumption utility Uk(ck), k = n+ 1, . . . , T − 1, which substantiallyincreases the technical difficulty in solving the problem. In the special case where the discountrate is constant over time, the coefficient of Uk(ck) is equal to 0 and the recursion reduces tobe exactly the same as in the classical models. Therefore, it is anticipated that the solutions

10

obtained based on recursion (13) can be used to recover those to the classical models, i.e.,the pre-committed solution given in Proposition 2.1. See Remark 3.3 for comments onthe connections between the equilibrium solution and the pre-committed solution for somespecific models.

To develop an equilibrium solution to our problem (11), some technical lemmas andnotations are necessary as given below.

Lemma 3.1. There exists a unique α(i) such that

α(i) = arg maxα∈[0,1]

En,i [αRn(i) + (1− α)rf (i)]γ

for 0 < γ < 1, and

α(i) = arg minα∈[0,1]

En,i [αRn(i) + (1− α)rf (i)]γ

for γ < 0, where En,i[·] = E[·|ξn = i].

Proof. See Proposition 1 in Cheung and Yang (2004).

Lemma 3.2. Given w > 0 and a function f > 0, if 0 < γ < 1, then

sup0≤c≤w,α

{cγ

γ+

(w − c)γ

γf(α)

}=

1

γsup

0≤c≤w

{cγ + (w − c)γ

[supαf(α)

]}; (15)

else if γ < 0, then

sup0≤c≤w,α

{cγ

γ+

(w − c)γ

γf(α)

}=

1

γinf

0≤c≤w

{cγ + (w − c)γ

[infαf(α)

]}. (16)

Proof. See Appendix A.

Note that α(i) defined in Lemma 3.1 is irrelevant to n since R0(i), R1(i), . . . , and RT−1(i)are identically distributed as assumed. Thus, if we define

Y (i) = E {[(1− α(i))rf (i) + α(i)Rn(i)]γ} , i = 1, . . . , L, (17)

then Y (i) is also irrelevant to time n and merely depends on the given market state i. Toproceed, for n = 0, 1, · · · , T − 1, and i = 1, . . . , L, we further define

HT (i) = 0, (18)

AT (i) = 0, (19)

An(i) =Hn(i)Y (i)

ϑn(i), (20)

and

Hn(i) = En,i

T∑

k=n+1

k−1∏m=n+1

((Am(ξm))

11−γ

1+(Am(ξm))1

1−γ

)γY (ξm)

(1 + ρ(n, i))k−n· ϑk(ξk)(

1 + (Ak(ξk))1

1−γ

)γ , (21)

11

where, by convention, a productm∏k=n

Θ(k) with m < n is set to be equal to 1 for any sequence

{Θ(k), k = 0, 1, . . .}. It follows from (20) and the forthcoming equation (23) that An(i) is akey indicator measuring the attention degree of risk investment. A larger An(i) results inmore wealth invested in the risky asset.

Remark 3.2. From (20) and (21), it is easy to verify that Hn(i) > 0 and An(i) > 0 forn = 0, 1, . . . , T − 1 and i = 1, . . . , L. This fact is used in the proof of Theorem 3.1.

Theorem 3.1. One equilibrium strategy {(αn, cn), n = 0, 1, . . . , T − 1} for problem (11) isgiven as follows:

αn(i) =

arg maxαn∈[0,1]

E [αnRn(i) + (1− αn)rf (i)]γ , if 0 < γ < 1,

arg minαn∈[0,1]

E [αnRn(i) + (1− αn)rf (i)]γ , if γ < 0,

(22)

cn(i, wn) =wn

1 +(Y (i)Hn(i)ϑn(i)

) 11−γ

=wn

1 + (An(i))1

1−γ. (23)

And the equilibrium value function Vn is given by

Vn(i, wn) =(wn)γ

γϑn(i)

(1 + (An(i))

11−γ

)1−γ, n = 0, 1, . . . , T − 1. (24)

Proof. See Appendix B.

Next we show that Theorem 3.1 can be used to retrieve the pre-committed solution whichwe have previously obtained in Proposition 2.1. We also show how Theorem 3.1 can be usedto recover those solutions derived in Samuelson (1969) and Cheung and Yang (2004) for thecorresponding models. The analysis is conducted as three special cases as follows.

Special case 1. In this special case, we fix a time n ∈ {0, 1, . . . , T − 1} and assumethat the discount rates for the future periods are all the same as ρ(n, ξn), the discountrate at time n. We show that the equilibrium solution given in Theorem 3.1 reduces tothe pre-committed solution obtained in Proposition 2.1. Indeed, according to (21), for anyk ≥ n, we have

Hk(ξk) =Ek,ξk

T∑

l=k+1

l−1∏m=k+1

((Am(ξm))

11−γ

1+(Am(ξm))1

1−γ

)γY (ξm)

(1 + ρ(n, ξn))l−kϑl(ξl)(

1 + (Al(ξl))1

1−γ

)γ

=1

1 + ρ(n, ξn)Ek,ξk

ϑk+1(ξk+1)(1 + (Ak+1(ξk+1))

11−γ

)γ +

((Ak+1(ξk+1))

11−γ

1 + (Ak+1(ξk+1))1

1−γ

)γ

Y (ξk+1)

×T∑

l=k+2

l−1∏m=k+2

((Am(ξm))

11−γ

1+(Am(ξm))1

1−γ

)γY (ξm)

(1 + ρ(n, ξn))l−k−1

ϑl(ξl)(1 + (Al(ξl))

11−γ

)γ

12

=1

1 + ρ(n, ξn)Ek,ξk

ϑk+1(ξk+1)(

1 + (Ak+1(ξk+1))1

1−γ

)γ+

((Ak+1(ξk+1))

11−γ

1 + (Ak+1(ξk+1))1

1−γ

)γ

Y (ξk+1)Hk+1(ξk+1)

=1

1 + ρ(n, ξn)Ek,ξk

ϑk+1(ξk+1)(

1 + (Ak+1(ξk+1))1

1−γ

)γ+

((Ak+1(ξk+1))

11−γ

1 + (Ak+1(ξk+1))1

1−γ

)γ

Y (ξk+1)Ak+1(ξk+1)ϑk+1(ξk+1)

Y (ξk+1)

=

1

1 + ρ(n, ξn)Ek,ξk

[(1 + (Ak+1(ξk+1))

11−γ

)1−γϑk+1(ξk+1)

],

where the third equality follows from the assumption that ρ(k, ξk) = ρ(n, ξn) for k > n.Consequently,

AT (i) =0, i = 1, 2, . . . , L,

Ak(ξk) =Y (ξk)

(1 + ρ(n, ξn))ϑk(ξk)Ek,ξk

[(1 + (Ak+1(ξk+1))

11−γ

)1−γϑk+1(ξk+1)

],

k = n, n+ 1, . . . , T − 1,

which is the same as equation (8) in Proposition 2.1. Therefore, the equilibrium strategyreduces to the pre-committed solution given in Proposition 2.1 in this case.

Special case 2. By Theorem 3.1, for all n = 0, 1, . . . , T − 1, we have

limϑn(i)→0

ϑn(i)(

1 + (An(i))1

1−γ

)1−γ= Hn(i)Y (i),

and

Hn(i) = En,i

T−1∏

m=n+1

Y (ξm)ϑT (ξT )

(1 + ρ(n, i))T−n

.This implies that for all i ∈ {1, 2, . . . , L}, n = 0, 1, · · · , T − 1, Vn(i, wn) = (wn)γ

γHn(i)Y (i)

when ϑn(i) = 0. Furthermore, if ρ(n, i) = 0 and ϑT (i) = 1, for all i ∈ {1, 2, . . . , L}, thenHn(i) simplifies into

Y (i)Hn(i) = Y (i)En,i

[T−1∏

m=n+1

Y (ξm)

], n = 0, 1, . . . , T − 1, (25)

where Y (i) = E [(1− α(i))rf (i) + α(i)Rn(i)]γ , i = 1, . . . , L, is obviously equivalent to Q(1)i

in Cheung and Yang (2004). By (25), we obtain Y (i)HT−1(i) = Y (i) = Q(1)i . For n < T −1,

13

according to (25), Y (i)Hn(i) can be rewritten as

Y (i)Hn(i) =Y (i)L∑j=1

Q(i, j)Y (j)En+1,j

(T−1∏

m=n+2

Y (ξm)

)

=Y (i)L∑j=1

Q(i, j) (Y (j)Hn+1(j)) ,

where Q(i, j) = Pr (ξn+1 = j|ξn = i). In conclusion, when ϑn(i) = 0, ϑT (i) = 1 and ρ(n, i) =0 for n ∈ {0, 1, . . . , T−1} and i ∈ {1, 2, . . . , L}, our value function reduces to that in Cheungand Yang (2004). For the strategy, according to (23), we have

limϑn(i)→0

cn(i, wn) = 0

while αn(i) is also the same as α∗i in Cheung and Yang (2004).

Special case 3. When there is only one market state in the financial market, ϑn(i) = 1,it is equivalently to put in our model ϑT (i) = 0 and ρ(n, i) = ρ for n ∈ {0, 1, . . . , T − 1}and i ∈ {1, 2, . . . , L}. Firstly, it follows from (21) that HT−1 = 0 with a consequence ofcT−1(wT−1) = wT−1. Secondly, referring to (F.1) in Appendix F, An in (20) can be simplifiedinto

An =Y

1 + ρ

(1 + (An+1)

11−γ

)1−γ.

Moreover, denote by φn the consumption proportion at time n, and one can apply (23) toobtain

φn =1

1 + (An)1

1−γ=

1

1 +

(Y

1+ρ

(1 + (An+1)

11−γ

)1−γ) 1

1−γ

=1

1 +(

Y1+ρ

) 11−γ(

1 + (An+1)1

1−γ

) =a1

a1 + 1 + (An+1)1

1−γ,

where a1 =(

1+ρY

)1/(1−γ). In the meanwhile,

a1φn+1

1 + a1φn+1

=a1

1 + (An+1)1

1−γ

1

1 + a1

1+(An+1)1

1−γ

=a1

a1 + 1 + (An+1)1

1−γ.

Therefore, we obtain

φn =a1φn+1

1 + a1φn+1

,

which is consistent with the recursion formulas in Samuelson (1969) if ignoring the differenceof the notations. In addition, we notice that

w∗ = arg maxw

∫ ∞0

[(1− w)(1 + r) + wZ]γ dP (Z)

is actually equal to αn in our paper under the assumption that there is only one market state,where P (Z) is the distribution function of the risky return and 1 + r is the deterministicreturn of the risk-free asset.

14

Remark 3.3. If one looks at the details of the two papers Cheung and Yang (2004) andSamuelson (1969), one will find that both α∗i and w∗ have no essential difference from αngiven in equation (22) though the models behind are different from each other. This is dueto the fact that, as in our model, a power utility function is adopted in both papers. Whenthere is no constraint imposed to link the consumption and the investment strategies, itfollows from Lemma 3.2 that αn is independent of the wealth level, the discount rate, as wellas the utility coefficients. As a consequence, no matter what the underlying model is, αnhas the same general expression as α∗i in Cheung and Yang (2004) and w∗ in Samuelson(1969). Nonetheless, the economic meanings of αn in the present paper differ from bothα∗i and w∗. While both α∗i and w∗ mean the investment proportion, i.e., the ratio of thecapital amount invested in risky asset to the total wealth available, αn is the ratio of theinvestment wealth to the wealth after consumption. In our investment-consumption model,the investment proportion is actually ηn defined in (26). More discussion on the propertiesof the investment proportion ηn will be given in the next section, and one can see shortlythat the investment proportion ηn in our model carries certain different properties comparedwith α∗i and w∗.

4. The effects of ϑn on equilibrium strategy

In this section, some analytical results will be established to demonstrate how the utilitycoefficients affect the decision-maker’s consumption-investment behavior and the dynam-ics of the wealth process. To this end, we assume that the discount rate is a constantover time to highlight the effects of the utility coefficients on the investment-consumptionstrategy. Under this assumption, our investment-consumption strategy reduces to be thepre-committed solution given in Proposition 2.1. However, as we have previously mentionedin Section 1, time-inconsistency is not the only aspect we are concerned with. We are also in-terested in how the utility coefficients can affect the investment-consumption strategy. Theother reason to consider a constant discount rate in this section is as follows. In Section1, we pointed out that the introduction of K(i) and C(i) in Canakoglu and Ozekici (2009,2010, 2012) has no influence on the optimal strategy and this motivates us to investigatewhether these coefficients are also irrelevant to the optimal solutions for an investment-consumption optimization problem. Hence, we need to maintain a constant discount rateso that our results are comparable with those in Canakoglu and Ozekici (2009, 2010, 2012).

4.1. Effects on consumption-investment behavior

Let ηn(i) and φn(i), respectively, denote the proportions of total wealth invested in therisky asset and consumed at time n under the equilibrium strategy. Note that ηn(i) andφn(i) do not necessarily sum to 1 because there may be some investment in the risk-freeasset. Most subsequent analysis in this section will be conducted via these quantities ηn(i)and φn(i), which, according to Theorem 3.1, admits the following expressions:

ηn(i) =(wn − cn(i))αn(i)

wn=

(An(i))1

1−γ

1 + (An(i))1

1−γαn(i), (26)

n = 0, 1, . . . , T − 1,

φn(i) =cn(i)

wn=

1

1 + (An(i))1

1−γ, n = 0, 1, . . . , T − 1. (27)

15

For presentation convenience, ηn(i) and φn(i) will be respectively referred to as “investmentproportion” and “consumption proportion”. Hereafter, by “increasing” and “decreasing” fora function, we mean “non-decreasing” and “non-increasing” respectively. For a multivariatefunction f(x1, . . . , xL), by saying “f is increasing (decreasing) in (x1, . . . , xL)”, we mean thatf is increasing (decreasing) in each argument xi.

Proposition 4.1. Given a market state i ∈ {1, . . . , L}, the consumption proportion φn(i)is strictly increasing along with ϑn(i) while the investment proportion ηn(i) is strictly de-creasing with ϑn(i).

Proof. See Appendix C.

From Proposition 4.1, the effects of the utility coefficients ϑn(i) on the decision-maker’sconsumption-investment behavior are two-fold. First, a financial market state i correspond-ing to a larger coefficient ϑn(i) will encourage the decision-maker to consume more and investless. This reflects how the decision-maker adjusts her investment-consumption behavior inresponse to the changes in financial market state. Second, Proposition 4.1 also hints thedifference between two decision-makers with different utility coefficients. If ϑn(i) is largerfor one decision-maker than the other in each of all the possible market states i = 1, . . . , L,the one with a larger coefficients ϑn(i) will consume more and invest less than the otherregardless of what a state the financial market stays in.

Proposition 4.2. Let ϑT = (ϑT (1), . . . , ϑT (L)) and assume that the discount rates areconstant. Then, for each n = 0, 1, . . . , T −1, the consumption proportion φn(i) is decreasingin ϑT and the investment proportion ηn(i) is increasing in ϑT . Furthermore, when all theelements in the transition matrix Q are positive, the monotonic property of φn(i) and ηn(i)with respect to ϑT is strict.

Proof. See Appendix D.

By Proposition 4.2, the coefficient ϑT exerts an opposite effect on the decision-maker’sconsumption-investment behavior in contrast to ϑn(·) for n ≤ T − 1. While a large ϑn(·)stimulates more consumption and less investment, a large ϑT (·) encourages more investmentand less consumption at time n.

Proposition 4.3. Let ϑk = (ϑk(1), . . . , ϑk(L)) for k = 0, 1, . . . , T , and assume that thediscount rates are constant. Then, for each k = n + 1, n + 2, . . . , T − 1, the consumptionproportion φn(i) is decreasing in ϑk and the investment proportion ηn(i) is increasing in ϑk.Furthermore, when all the elements in the transition matrix Q are positive, the monotonicproperty of φn(i) and ηn(i) with respect to ϑk is strict.

Proof. See Appendix E.

The results in Proposition 4.3 are consistent to the economic meanings of ϑn(·) as co-efficients in the utility functions. Sitting at time n, a decision-maker with a larger ϑk fork > n means that she has more utility on a future consumption and therefore she wouldlike to reduce her consumption proportion and increase her investment proportion at timen so as to enhance the chances to consume more in the future.

Propositions 4.1-4.3 confirm that the utility coefficients measure the attention degree ofthe consumption. The higher the attention degree is, the higher the consumption proportion

16

is. At the same time, the higher the attention degree for a future period, the less proportionof the consumption for the current period. Moreover, when ϑn(·) ≡ 0 for n = 0, 1, . . . , T −1and ϑT (·) ≡ 1 are set, the consumption amount given in equation (23) reduces to be zeroand the investment proportion coincides with the solution to the model analyzed in Cheungand Yang (2004).

4.2. Effects on wealth process

In this section, we explore some properties about the wealth process for ϑT (·) ≡ 0, i.e.,no bequest taken into account in decision-making. We will prove that the “hump saving”phenomenon for the expected wealth observed by Merton (1969) and Samuelson (1969)still holds for our model under some specific conditions. Moreover, we will show via anumerical example that the “hump saving” phenomenon does not necessarily hold whenthe decision-maker has a time-varying consumption utility function.

To proceed, we note from (1) and (23) that the wealth under the strategy given inTheorem 3.1 satisfies

Wn+1 = (Wn − cn) [(1− αn)rf (i) + αnRn(i)] ,

=Wn(An(ξn))

11−γ

1 + (An(ξn))1

1−γ[(1− αn)rf (i) + αnRn(i)] , n = 0, 1, . . . , T. (28)

Repeatedly applying the above recursion yields

WT = w0

T−1∏k=0

(Ak(ξk))1

1−γ

1 + (Ak(ξk))1

1−γ[(1− αk)rf (ξk) + αkRk(ξk)].

With ϑT (·) ≡ 0, it follows from (20) and (21) that

HT−1(i) = ET−1,i

[ϑT (ξT )

1 + ρ(T − 1, i)

]= 0 and AT−1(i) = 0, i = 1, . . . , L,

and thus, the terminal wealth WT is zero. This result accords with our intuition well. Thedecision-maker will not keep any wealth at terminal time since a zero utility is assigned tothe terminal wealth.

To study the conditions for the expected wealth process to increase, we follow Merton(1969) and define the following expected rate of growth in wealth over period [n, n+ 1):

Λn(i) =En,i (Wn+1)−Wn

Wn

, n = 0, 1, . . . , T − 1.

It follows from (28) that

Λn(i) =En,i (Wn+1)−Wn

Wn

=(An(i))

11−γ

1 + (An(i))1

1−γZ(i)− 1, (29)

where Z(i) := E [(1− αn)rf (i) + αnRn(i)] is the expected gross return of the investment

portfolio in the i-th market state. By (29), Λn(i) > 0 if and only if Z(i)−1 > 1/(An(i))1

1−γ .

17

Therefore, when Z(i)− 1 > 1/(An(i)1

1−γ , we have

En,i (Wn+1)−Wn

Wn

> 0 and

cn (i, wn) =wn

1 + (An(i))1

1−γ<

(An(i))1

1−γ

1 + (An(i))1

1−γZ(i)wn = En,i (Wn+1) .

Note that Z(i) − 1 is the expected return rate on the equilibrium portfolio, and by (23),

1/(An(i))1

1−γ measures the ratio of consumption amount to investment amount. Thus, theabove analysis means that, if the expected return rate of investment is larger than the ratioof consumption to investment, the expected wealth will increase and the decision-maker attime n will not consume more than the wealth she expects to have at time n+ 1.

Merton (1969) and Samuelson (1969) discover that the expected wealth is decreasingover time or has the shape of “hump saving” for the consumption-investment problemwith identical utility coefficients. Next, we will show that the “hump saving” phenomenonhappens in our model under specific conditions. To this end, we consider a consumption-investment with a fixed financial market state. Under this assumption, we denote ϑn(i) =ϑn, ρ(n, i) = ρ(n) and Λn(i) = Λn for simplification.

Proposition 4.4. If the discount rate ρ(n) is a constant over time and

ϑ0

ϑ1

≤ ϑ1

ϑ2

≤ · · · ≤ ϑT−3

ϑT−2

≤ ϑT−2

ϑT−1

, ϑT = 0, (30)

then we have Λn+1 ≤ Λn.

Proof. See Appendix F.

Proposition 4.4 indicates that the expected rate of growth of wealth is decreasing overtime when the ratio of the utility coefficients between the neighboring periods is also de-clining over time. Under this circumstance, a similar argument from Merton (1969) andSamuelson (1969) can be used to justify the “hump saving” phenomenon for the discrete-time model with proper time-varying utility coefficients. When the expected rate of growthin wealth at the initial time Λ0 < 0, the expected wealth will be decreasing over the wholedecision-making time horizon. If Λ0 > 0 instead, there must exist a time, say t, such thatthe expected wealth is increasing before time t and decreasing thereafter. The existence ofsuch a time t is guaranteed by the fact that the terminal wealth must be zero, which wehave proved previously. The condition (30) indicates that the “hump saving” phenomenonholds in a more general situation. When ϑn = 1, n = 0, 1, . . . , T − 1, ϑT = 0, (30) holdstrue and our result is reduced to that in Samuelson (1969) with identical utility coefficients.Another example is ϑn = a+ n+ 1, n = 0, 1, . . . , T − 1, ϑT = 0 where a ≥ 0, which can alsolead to the phenomenon of “hump saving”.

However, when (30) does not hold, the “hump saving” phenomenon is not necessarilytrue as demonstrated in the following example.

Example 4.1. Consider our discrete-time investment-consumption model with T = 5 pe-riods. For simplicity, assume that the market state is fixed on certain regime where thelog-return of the risky asset is subject to a normal distribution with mean µ = 0.2 andstandard deviation σ = 0.1. Let the discount rate ρ = 0.6, the risk aversion coefficient

18

Time

0 1 2 3 4 5E

xp

ecte

d W

ea

lth

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 1: The expected wealth process

γ = 0.5, and the return of the risk-free asset rf = 1.035. Further assume the followingutility coefficients for each period:

(ϑ0, ϑ1, ϑ2, ϑ3, ϑ4, ϑ5) = (5, 4, 1, 4, 5, 0).

With initial wealth w0 = 1, we calculate the expected wealth under the equilibrium strategydeveloped in Theorem 3.1 for each period, and illustrate the results in Figure 1. Obviously,the expected wealth does not comply with the “hump saving” phenomenon. Instead, it is firstdecreasing up to time 2, increasing for one period and then finally declining all the way tothe terminal time. The decrease of the expected wealth over time horizon [0, 2] results fromlarger consumption coefficients assumed while the increase of the expected wealth in [2, 3]results from lower attention degree of the consumption.

5. Numerical sensitivity analysis

In this section, we numerically study how the stochastic discount rate affects the equilib-rium strategy and the equilibrium value function. The interplay of stochastic discount rateand time-varying utility coefficients can be very complicated as one can see from the ex-pressions of the equilibrium strategies and the equilibrium value function given in Theorem3.1, and it is impossible for us to consider all the possible patterns of the equilibrium strat-egy and equilibrium value function resulted from all the possible models for discount rateand utility coefficients. We are interested in showing some phenomena which either contra-dict the classical conclusions or have not been observed in the existing investment and/orinvestment-consumption models. To this end, we consider the special case where all theutility coefficients are constant and equal to 1. Such a hypothesis is not from any empiricalsupports, and instead, it is imposed simply for the convenience of numerical illustration.

5.1. The closer to terminal time, the larger consumption proportion?

In the case of constant discount rate, Cheung and Yang (2007) conclude that the decision-maker consumes a larger proportion of her wealth as she gets closer to the terminal of thetime horizon. In this subsection, we will numerically show that it is not always the caseunder a non-constant discount rate. To this end, we consider a simple case where themarket state is fixed on a single regime throughout the whole investment time horizon andthe discount rate is time-deterministic so that we can write ρ(n) := ρ(n; i). Further assume

19

that the decision-maker has an initial age of 35, and consider an investment period of 10years with semi-annual rebalances so that there are T = 20 periods in the model. Furtherassume that the gross return per period is 1.015 for the risk-free asset and subject to alog-normal distribution with mean µ = 0.04 and standard deviation σ = 0.2 for the riskyasset. Let the risk aversion coefficient γ = 0.5.

35 36 37 38 39 40 41 42 43 440.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Age

Consum

ption p

roport

ion

(a) With constant discount rate

35 36 37 38 39 40 41 42 43 440.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Age

Consum

ption p

roport

ion

(b) With time-varying discount rate

Figure 2: Consumption proportion at each age

(a) We first experiment with ρ(n) = 0.2 for n = 0, 1, . . . , 19, and obtain the consumptionproportion as shown in Figure 2(a), which is obviously increasing over time, an observationconsistent to the conclusion by Cheung and Yang (2007).

(b) Then we consider the time-varying discount rates as follows:

ρ(0 : 9) =(0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15)′,

ρ(10) =0.1, ρ(11 : 19) = (0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6)′.

The above discount rates are specified to be consistent to the findings by Read and Read(2004) that a decision-maker in middle age tends to discount least resulting from more secu-rity and less uncertainty at this age, while the younger and older incarnation discounts moredue to uncertainty early in life and declining capacity late in life. The resulting consump-tion proportions are demonstrated in Figure 2(b), which obviously shows a substantiallydifferent shape from the preceding experiment with a constant discount rate. The consump-tion proportion in this case does not increase all the way with age n. Instead, it shows a“U”-shape, meaning that the decision-maker will consume a larger proportion of her wealthover earlier periods, and then cuts down the consumption proportion for a while until shedecides to rise it up again over the last periods.

As put by Read and Read (2004), “the young people don’t yet know if their world isrisky or safe, they are better off acting as if there is no tomorrow”. Consequently, when thedecision-maker is relatively young, she is unable to resist her impulses for seconds and thenconsume more of her wealth in her decision. When she gets older, she has more feelings ofsecurity and becomes more willing to wait for consumption, so the consumption proportionis declining as she gets older. In this experiment, we may interpret the results as follows.The decision-maker may achieve the maximum value of the health, work experience andreproductive opportunities up to age 40 (i.e., time 10), and consequently consumes theleast over the period around the time. Moreover, aging is accompanied by a degeneration of

20

health and work opportunities, and thus she begins to gain more pleasure from immediateconsumption.

5.2. The better market state, the higher investment proportion and lower consumption pro-portion?

Assume that there are three possible financial market regimes representing a “bullish”,“normal” and “bearish” state respectively, and the log-return of the risky asset has a normaldistribution with mean µi and standard deviation σi for each state as given below:

µ1 = −0.0159, µ2 = 0.0013, µ3 = 0.0027,

andσ1 = 0.0609, σ2 = 0.0254, σ3 = 0.0141.

The transition matrix is as follows

Q =

0.8239 0.1761 0.00000.0154 0.9743 0.01030.0000 0.0069 0.9931

.

The above parameters for the risky asset are calibrated from 1245 weekly closing datapoints of the Standard and Poor’s 500 index from January 01, 1990 throughout November10, 2013. With the above parameter values of µi and σi (i = 1, 2, 3), the first regime mightbe explained as a bearish market state, the third regime should be a bullish state and thesecond one might be explained as the normal state. The gross weekly return of the risk-freeasset is a constant rf = 1.0000126. Furthermore, we assume that one year has 52 weeksand T = 52×3 weeks, i.e., 3-year investment time horizon. Let the risk reversion coefficientγ = 0.5.

In this subsection, we investigate whether a better market environment will stimulatethe decision-maker to invest more and consume less. The answer is negative, and the resultdepends on the specific value of the discount rate as we demonstrate in the following twoexperiments.

Table 1: Discount rate

ρ state 1 state 2 state 3n = 0 : 51 0.60 0.55 0.50n = 52 : 103 0.45 0.40 0.35n = 104 : 155 0.30 0.25 0.20

(a) Assume that the investment-consumption strategy is adjusted weekly and the decision-maker is 25 years old at time 0 so that she is 28 years old up to the end of decision making.For simplicity, we further assume that the discount rates are constant within each year butvarying from one year to the next as specified in Table 1. The values in Table 1 implicitlyassume that the decision-maker has less patience to wait for the consumption in a badmarket state (with a larger discount rate) than in a good market state (with a smaller dis-count rate). The resulting investment and consumption proportions are reported in Figure3 for ages of 25, 26 and 27, corresponding to times 0, 53 and 105 (in weeks) respectively.Figure 3 indicates that the decision-maker tends to consume more and invest less in a bad

21

investment environment, and consume less and invest more in a good investment setting.This observation is consistent with the conclusions obtained by Cheung and Yang (2004,2007) for classical models with a constant discount rate.

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

State

Investm

ent P

roport

ion

Age=25

Age=26

Age=27

(a) Investment proportion

1 2 3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

State

Consum

ption P

roport

ion

Age=25

Age=26

Age=27

(b) Consumption proportion

Figure 3: Investment-consumption proportion: lower discount rates in better market states

Table 2: Discount rate

ρ state 1 state 2 state 3n = 0 : 51 0.50 0.55 0.60n = 52 : 103 0.35 0.40 0.45n = 104 : 155 0.20 0.25 0.30

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

State

Investm

ent P

roport

ion

Age=25

Age=26

Age=27

(a) Investment proportion

1 2 30.3

0.35

0.4

0.45

0.5

0.55

0.6

State

Consum

ption P

roport

ion

Age=25

Age=26

Age=27

(b) Consumption proportion

Figure 4: Investment-consumption proportion: higher discount rates in better market states

(b) Next, we inherit all the numerical settings from preceding experiment but alter thediscount rates to those as given in Table 2, where the relation of the discount rate to theinvestment environment is contrary to that in Table 1 for the preceding experiment. WithTable 2, the decision-maker is assumed to have more patience to wait for the pleasure ofconsumption in a bad investment environment than in a good one. The resulting investment

22

and consumption proportions are demonstrated in Figure 4, and the consumption proportionshows a completely different relation to the market state, which indeed has never beenobserved in a classical investment-consumption model with a constant discount rate. Thus,the non-constancy of the discount rate plays an important role in the investor’s consumptionbehavior. The results in Figure 4 may be interpreted as follows. When a better marketstate is associated with a higher discount rate, a favorable investment environment does notseem appealing enough for the investor to invest more because she feels pessimistic aboutthe future incomes. Therefore, she wants to enjoy the current pleasure of life and thenconsumes a larger proportion of her wealth.

0.3 0.35 0.4 0.45 0.50.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Discount rate at state 1

Consum

ption P

roport

ion a

t sta

te 1

n=29

(a) ρ2 and ρ3 are fixed

0.25 0.3 0.35 0.4 0.450.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

Discount rate at state 2

Consum

ption P

roport

ion a

t sta

te 2

n=29

(b) ρ1 and ρ3 are fixed

0.2 0.25 0.3 0.35 0.4

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

Discount rate at state 3

Consum

ption P

roport

ion a

t sta

te 3

n=29

(c) ρ1 and ρ2 are fixed

Figure 5: Consumption proportion when discount rate increases

5.3. The higher discount rate, the larger consumption proportion?

In this subsection, we apply the same parameter values for the investment assets as cali-brated with the three-state regime switching model in the preceding subsection. Moreover,let rf = 1.0000126 and γ = 0.5. We further assume that the investor is 27 years old attime 0 and the planning time horizon is one year with weekly adjustments so that T = 52.We shall study the change pattern of the consumption proportion as we vary the discountrate. For simplicity, the discount rate is assumed to depend on the market state only andwe write ρi := ρ(n, i). We consider the following three scenarios:

(1) fix ρ2 = 0.25 and ρ3 = 0.20, and vary ρ1 from 0.30 to 0.5 in a step size of 0.05;23

(2) fix ρ1 = 0.30 and ρ3 = 0.20, and vary ρ2 from 0.25 to 0.45 in a step size of 0.05;

(3) fix ρ1 = 0.30 and ρ2 = 0.25, and vary ρ3 from 0.20 to 0.40 in a step size of 0.05.

To investigate how the consumption proportion responds to the changes in discount rate,we select φ29(i), the one at time n = 29 among the 52 time periods, as a representative.We note that the consumption proportion at the other time periods present similar results.The values of φ29(i) versus to ρi for i = 1, 2, 3 are shown in plots (a), (b) and (c) of Figure 5respectively, which clearly demonstrate that the consumption proportion φ29(i) is increasingalong with the discount rate ρi, i = 1, 2, 3. Such a conclusion accords to our intuition well,since the decision-maker with a higher discount rate exhibits less patience and thereforewill consume more and invest less.

Nevertheless, the consumption proportion φn(i) is not always increasing with ρj for i 6= j.Figures 6(a) and 6(b) are the results of two among a variety of combinations of i, j ∈ {1, 2, 3}which we have experimented with. Figure 6(a) implies that the consumption proportion atstate 1 is not increasing all the way with ρ3. We also note that the consumption proportionat the last period, corresponding to n = 51, in Figure 6(b) remains unchanged regardlessof what a value is assigned for ρ3. In this example, n = 51 corresponds to the last period,and the decision in the last period will not be affected by the information from the previousperiods except the wealth w51. Since the market is assumed to be in state 1 when weconsider the value of φ51(1), the discount rate ρ3 for market state 3 does not have any effecton the decision in the last period.

0.2 0.25 0.3 0.35 0.40.4086

0.4086

0.4086

0.4086

0.4086

0.4086

0.4086

0.4086

0.4087

0.4087

0.4087

Discount rate at state 3

Consum

ption P

roport

ion a

t sta

te 1

n=29

(a) ρ1 and ρ2 are fixed: n = 29

0.2 0.25 0.3 0.35 0.40.625

0.6255

0.626

0.6265

0.627

0.6275

0.628

0.6285

0.629

0.6295

0.63

Discount rate at state 3

Consum

ption P

roport

ion a

t sta

te 1

n=51

(b) ρ1 and ρ2 are fixed: n = 51

Figure 6: Consumption proportion when discount rate increases

5.4. How does the state-dependent discount rate affect the equilibrium value function?

In this subsection, we aim to numerically compare our equilibrium value function withthe optimal value function under a constant discount rate, and demonstrate the effects of thestochastic discount rate on the equilibrium value function. For the sake of convenience, weassume that the discount rates merely depend on the market state, and independent of theage of the decision-maker. We inherit all the numerical settings from preceding subsectionexcept the discount rates. In addition, suppose that the initial wealth w0 = 1 and theinitial market state i0 = 2. Denote by SDR the vector of state-dependent discount ratescorresponding to each of the three market states, and EV the equilibrium value function at

24

time 0 in our model. Let CDR and OV, respectively, denote the constant discount rate andthe optimal value function at time 0 in the model with a constant discount rate as obtainedby Cheung and Yang (2007).

Table 3: Comparison of the value functions

SDR CDR EV OV OV-EV(0.30,0.25,0.20) 0.25 3.3371 3.3377 0.0006(0.35,0.25,0.20) 0.25 3.3366 3.3377 0.0011(0.40,0.25,0.20) 0.25 0.3359 3.3377 0.0018(0.45,0.25,0.20) 0.25 0.3350 3.3377 0.0027(0.25,0.25,0.25) 0.25 3.3377 3.3377 0

When the initial information in both decision-making processes are the same, the optimalvalues are reported in Table 3, from which we have the following observations.

(a) The last column of Table 3 indicates that the optimal value function is always largerthan the equilibrium value function. This observation can be explained as follows. Theinvestment-consumption problem with a constant discount rate is globally optimal andthen maximizes the objective function at time 0. Nevertheless, the equilibrium strategy isnot a generally optimal strategy at time 0, and the equilibrium value must be no largerthan the optimal value. From a game theoretical point of view, the non-cooperation ofeach player in the decision process destroys the welfare of the whole game, and it leads toa smaller equilibrium value relative to the optimal value.

(b) Comparing all the rows of Table 3 for different sets of discount rates, we can concludethat, the more the state-dependent discount rate deviates from the constant discount rate,the larger distance between the equilibrium value function and the optimal value function.This is reasonable since a larger discount rate results in a smaller discounted expected utilityof the inter-temporal consumption and terminal wealth, and therefore, lager components ofthe discount rate vector lead to a smaller equilibrium value function.

(c) The last row of Table 3 confirms our previous findings again that our equilibriumvalue function can be reduced to the optimal value function when the discount rates are allconstant.

5.5. Summary of main findings from the section

In this section, by numerically studying the effects of the non-constant discount rate,we find some conclusions in the case with constant discount rate do not always hold. InSubsection 5.1, we assume that the market state is fixated on a single regime and thediscount rates show a “U” shape over time referring to Read and Read (2004). Under thisassumption, the decision-maker will not consume a larger proportion of her wealth at a timecloser to the end of time horizon, as demonstrated by Cheung and Yang (2007). In contrast,the consumption proportions are declining over the earlier periods and are increasing overthe last periods. That is, it also shows a “U” shape as the discount rates. In Subsection5.2, the conclusion “A better market state results in a lower consumption proportion anda higher investment proportion” might not hold. If a better market state does not seemappealing enough for a pessimistic decision-maker, a larger proportion of wealth will beconsumed even in a better market state. In Subsection 5.3, when discount rate is assumedto depend on the market state only and denote ρi := ρ(n, i) for convenience, we find that

25

the consumption proportion at market state i will be increasing along with ρi. However, theconsumption proportion at market state i is not increasing all the way with ρj(j 6= i). InSubsection 5.4, by numerically comparing our equilibrium value function with the optimalvalue function under a constant discount rate, we show that our equilibrium value functionis less than the optimal value function given the same initial information. Furthermore, themore the time-varying discount rate deviates from the constant discount rate, the largerdistance between the equilibrium value function and the optimal value function.

6. Conclusion

Recent years have seen an upsurge of interests in the effects of random behaviors of thedecision-maker on the resulting decision strategies. To reflect the stochastic feature of thehuman being’s decision behavior, this paper adopts time-varying power utility functions andstochastic discount rates in an investment-consumption model. The discount rates dependon the age of the decision-maker in addition to the financial market state. The non-constantnature of the discount rates leads our model time-inconsistent in the sense that Bellman’soptimality principle is not applicable. That is, the strategy optimally determined at time nmight not be optimal for a future time k > n, and consequently, the forthcoming decisionmakers might deviate from the strategy made at time n. However, time consistency is a basicrequirement of dynamic decision-making. To tackle such a time-inconsistent problem, wefollow the idea from Bjork and Murgoci (2014) and the relevant references therein, and thenderive a subgame Nash equilibrium strategy for the investment-consumption problem. Weview the decision-making process as a non-cooperative game, and assume that the decision-maker at each time n only decides on the optimal policy given that her successors havedetermined their policies. By adopting the Nash equilibrium strategy, the basic requirementof time-consistency is satisfied in the sense that the decision-makers will stick to the Nashequilibrium strategy throughout the decision making process.

Referring to the definition of subgame Nash equilibrium strategy, we obtained the equilib-rium investment-consumption strategy in a closed form. The obtained equilibrium strategyis hinged on the coefficients of the utility functions. To be more precise, the coefficient ofthe utility function ϑk(k = 0, 1, . . . , T − 1) measures the attention degree of the consump-tion at time k and ϑT measures the attention degree of the terminal wealth. The largerthe ϑn+1, ϑn+2, . . . , ϑT are, the lower the consumption proportion at time n is. Moreover,a larger ϑn results in a higher consumption proportion at time n. As for the effects of thediscount rates, through numerical analysis, we found that many well-known conclusions re-garding a classical investment-consumption model, where discount rate is constant, may beno longer legitimate for our derived equilibrium strategy. For example, the conclusions that“The closer to the terminal time, the larger the consumption proportion” and “A bettermarket state leads to a higher investment proportion and a lower consumption propor-tion” might not hold in the case with non-constant discount rate. The graphical shapes ofthe investment-consumption strategy depend on the time preference of the decision-makerduring the decision making.

Many existing psychological experiments indicate the discount rate varies throughoutpeople’s life cycle. This paper adopts a general form of discount rate and assume that itis stochastically varying according to the development of financial market state and theage of the decision-maker. Given the predictive values of the discount rates, a decision-maker can use our results to produce an equilibrium strategy which she has no impulse

26

to deviate from even in the future. Furthermore, the decision-maker can also obtain theinvestment-consumption strategy which is fitting her own utility preference by adjustingproperly ϑ0, ϑ1, . . . , ϑT . In other words, our theoretical results together with the psycholog-ical results can potentially help to sort out an executable strategy tailored for the decisionmaker. This paper has adopted very general assumptions regarding the problem formu-lation, but there are still some limitations to improve in future research: (i) there is noconstraint that links the investment and consumption decisions, which is a more practi-cal topic worthy of further study; (ii) in order to obtain the explicit expressions for theinvestment-consumption strategy, we assume that the returns of the risky assets over twoperiods are conditionally independent given the market states. Thus, it is worth our furtherinvestigation to establish equilibrium strategy under other serial dependence structures forthe asset returns than the regime switching model; (iii) we do not have any real data re-garding the discount rates from any existing psychological reports, so numerical analysis isconducted only for certain hypothetical settings.

Acknowledgements

Wu acknowledges the funding support from the National Natural Science Foundationof China (Nos. 11301562, 11671411), Beijing Social Science Foundation (No. 15JGB049)and the Program for Innovation Research in Central University of Finance and Economics.Weng thanks funding support from the Natural Sciences and Engineering Research Councilof Canada (RGPIN-2016-04001), and Society of Actuaries Centers of Actuarial ExcellenceResearch Grant. Zeng acknowledges the funding support from the National Natural ScienceFoundation of China (Nos. 71201173, 71571195), Fok Ying Tung Education Foundation forYoung Teachers in the Higher Education Institutions of China (No. 151081), GuangdongNatural Science Funds for Distinguished Young Scholar (No. 2015A030306040), GuangdongNatural Science for Research Team (2014A030312003) and Science and Technology PlanningProject of Guangdong Province (2016A070705024).

Appendix A. The proof of Lemma 3.2

Proof. We only give the proof of (16) for γ < 0, as (15) can be proved in a similar way.When γ < 0,

sup0≤c≤w,α

{cγ

γ+

(w − c)γ

γf(α)

}=

1

γinf

0≤c≤w,α{cγ + (w − c)γf(α)} .

Consider the following maximization problem:

inf0≤c≤w

{cγ + (w − c)γH} ,

where H is a positive constant. Since

d2

dc2[cγ + (w − c)γH] = γ(γ − 1)

(cγ−2 + (w − c)γ−2H

)> 0,

the minimal value of cγ + (w − c)γH is achieved at

c =w

1 +H1

1−γ. (A.1)

27

As a consequence,

sup0≤c≤w

{cγ

γ+

(w − c)γ

γH

}=

1

γinf

0≤c≤w{cγ + (w − c)γH} =

1

γwγ[1 +H1/(1−γ)

]1−γ, (A.2)

which is a decreasing function of H, and thus we have (16).

Appendix B. Proof of Theorem 3.1

Proof. The proof will be shown only for 0 < γ < 1, as the results for γ < 0 can be provedin a parallel way. Throughout the proof, we always keep the constraints 0 ≤ cn ≤ wn and0 ≤ αn ≤ 1 for all n = 0, 1, . . . , T −1. The proof will be achieved by an induction argument.To proceed, let

$k =

[(Ak(ξk))

11−γ

1 + (Ak(ξk))1

1−γ

]γ, k = 0, 1, . . . , T − 1,

and writeζn(α, i) = [(1− α)rf (i) + αRn(i)]γ .

Then, according to Lemma 3.1, there exits a unique solution α(i) such that

α(i) = arg maxα∈[0,1]

E [ζn(α, i)] ,

and according to (17), Y (i) = E [ζn(α(i), i)]. Note that both α(i) and Y (i) depend on themarket regime i only and is independent of time n. We shall also use shorthand

ζn = ζn(α(ξn), ξn), n = 0, 1, . . . , T − 1.

When n = T − 1, it follows from (13) and (21) that

VT−1(i, wT−1) = supπT−1

ET−1,i,wT−1

[ϑT−1(i)(cT−1)γ

γ+ϑT (ξT )

(W

πT−1

T

)γ/γ

1 + ρ(T − 1, i)

]

= supcT−1,αT−1

ϑT−1(i)(cT−1)γ

γ

+HT−1(i)(wT−1 − cT−1)γ

γE [ζT−1(αT−1, i)]

(B.1)

where, obviously E [ζT−1(αT−1, i)] > 0. Consequently, applying Lemmas 3.1 and 3.2 to(B.1), we obtain

VT−1(i, wT−1) = supcT−1

[ϑT−1(i)(cT−1)γ

γ+HT−1(i)Y (i)

(wT−1 − cT−1)γ

γ

]and

cT−1(i, wT−1) =wT−1

1 +(Y (i)HT−1(i)

ϑT−1(i)

) 11−γ

=wT−1

1 + (AT−1(i))1

1−γ.

28

Substituting cT−1(i, wT−1) into VT−1 yields

VT−1(i, wT−1) =(wT−1)γ

γϑT−1(i)

(1 + (AT−1(i))

11−γ

)1−γ.

This implies that (22)-(24) hold for n = T − 1. Generally, we assume that (22)-(24) holdfor T − 1, T − 2, . . . , n+ 1, and we shall prove that they hold for n as well. From the givenassumptions, we obtain

Wπn,πn+1,...,πk−1

k = W πnn+1

k−1∏m=n+1

(Am(ξm))1

1−γ

1 + (Am(ξm))1

1−γ((1− αm)rf (ξm) + αmRm(ξm))

for k = T − 1, T − 2, . . . , n+ 1. This recursive equation implies(W

πn,πn+1,...,πk−1

k

)γγ

=

(W πnn+1

)γγ

k−1∏m=n+1

($mζm

), (B.2)

which along with (23) implies

En,i,wn [Uk(ck)] = En,i,wn

(W

(πn,πn+1,...,πk−1)k

)γγ

ϑk(ξk)((Ak(ξk))

11−γ + 1

)γ

= En,i,wn

(W πnn+1

)γγ

ϑk(ξk)k−1∏

m=n+1

($mζm

)(

(Ak(ξk))1

1−γ + 1)γ

= En,i,wn

Eξn+1,...,ξk

(W πnn+1

)γγ

ϑk(ξk)k−1∏

m=n+1

($mζm

)(

(Ak(ξk))1

1−γ + 1)γ

.

Given the market states ξn+1, ξn+2, . . . , ξk, these variables $m(m = n+ 1, . . . , k− 1), ϑk(ξk)and Ak(ξk) are not random any more. In other words, these variables are all measurable withrespect to the filtration generated by the sequence of random variables ξn+1, ξn+2, . . . , ξk.Therefore, further noticing the formula Y (ξm) = Em,ξm(ζm) and the conditionally indepen-dent assumption of {R0(·), R1(·), . . . , RT−1(·)} given the market states, we obtain

En,i,wn [Uk(ck)] = En,i,wn

(W πnn+1

)γγ

ϑk(ξk)k−1∏

m=n+1

($mY (ξm))((Ak(ξk))

11−γ + 1

)γ . (B.3)

Substituting (B.2) and (B.3) into (13) implies

Vn(i, wn) := maxcn,αn

[Un (cn) + I1(n, i, wn)− I2(n, i, wn) +

(W πnn+1)γ

γHn(i)

], (B.4)

29

where

I1(n, i, wn) = En,i,wn

(W πn

n+1)γ

γ

(1 + ρ(n+ 1, ξn+1))T−n−1

(1 + ρ(n, i))T−nϑn+1(ξn+1)

×(

1 + (An+1(ξn+1))1

1−γ

)1−γ

,and

I2(n, i, wn) = En,i,wn

(W πn

n+1)γ

γ

T∑k=n+1

(1 + ρ(n+ 1, ξn+1))T−k

(1 + ρ(n, i))T−n

× ϑk(ξk)(1 + (Ak(ξk))

11−γ

)γ k−1∏m=n+1

$mY (ξm)

.

According to the expressions of I1(n, i, wn) and I2(n, i, wn) and by further calculation,we have

I1(n, i, wn)− I2(n, i, wn)

= En,i,wn

(W πn

n+1)γ

γ

(1 + ρ(n+ 1, ξn+1))T−n−1

(1 + ρ(n, i))T−n

× ϑn+1(ξn+1)(An+1(ξn+1))

11−γ(

1 + (An+1(ξn+1))1

1−γ

)γ

−En,i,wn

(W πn

n+1)γ

γ

T∑k=n+2

(1 + ρ(n+ 1, ξn+1))T−k

(1 + ρ(n, i))T−n

× ϑk(ξk)(1 + (Ak(ξk))

11−γ

)γ k−1∏m=n+1

$mY (ξm)

=: 41(n, i, wn)−42(n, i, wn).

Using (21) and the fact that Ak(ξk) = Hk(ξk)Y (ξk)ϑk(ξk)

, we obtain

41(n, i, wn)

= En,i,wn

(W πn

n+1)γ

γ

(1 + ρ(n+ 1, ξn+1))T−n−1

(1 + ρ(n, i))T−n

×Hn+1(ξn+1)Y (ξn+1)

((An+1(ξn+1))

11−γ

1 + (An+1(ξn+1))1

1−γ

)γ

= En,i,wn

(W πnn+1)γ

γ

(1 + ρ(n+ 1, ξn+1))T−n−1

(1 + ρ(n, i))T−nY (ξn+1)$n+1

× En+1,ξn+1

T∑

k=n+2

1

(1 + ρ(n+ 1, ξn+1))k−n−1

× ϑk(ξk)(1 + (Ak(ξk))

11−γ

)γ k−1∏m=n+2

$mY (ξm)

30

= En,i,wn

(W πn

n+1)γ

γ

T∑k=n+2

(1 + ρ(n+ 1, ξn+1))T−k

(1 + ρ(n, i))T−n

× ϑk(ξk)(1 + (Ak(ξk))

11−γ

)γ k−1∏m=n+1

$mY (ξm)

= 42(n, i, wn),

which leads to I1(n, i, wn)− I2(n, i, wn) = 0. Consequently, (B.4) is equivalent to

Vn(i, wn) = maxcn,αn

[Un (cn) +

(W πnn+1)γ

γHn(i)

]= max

cn,αn

[ϑn(i)

(cn)γ

γ+

(wn − cn)γE(ζn(i))

γHn(i)

]. (B.5)

Since E(ζn(i))Hn(i) > 0 and is a function of αn, we can apply Lemma 3.2 to obtain

Vn(i, wn) = supcn

[ϑn(i)

(cn)γ

γ+

(wn − cn)γY (i)

γHn(i)

],

where the supremum of the value function is attained at

cn (i, wn) =wn

1 +(Y (i)Hn(i)ϑn(i)

) 11−γ

=wn

1 + (An(i))1

1−γ.

Substituting cn(i, wn) into Vn(i, wn) and rearranging yield

Vn(i, wn) =(wn)γ

γϑn(i)

(1 + (An(i))

11−γ

)1−γ.

This implies that the desired results also hold for n, and thus, by induction principle, theproof is complete.

Appendix C. Proof of Proposition 4.1

Proof. It follows from (21) that Hn(i) does not depend on ϑn(i), which along with (23)immediately yields that the equilibrium consumption proportion φn(i) is strictly increasingin ϑn(i). Further note that ηn(i) = (1− φn(i))αn(i) and αn(i) is irrelevant to ϑn(i). Thus,ηn(i) is strictly decreasing with ϑn(i).

Appendix D. Proof of Proposition 4.2

Proof. When the discount rates are a constant, (21) can be written as

Hn(i) =1

1 + ρEn,i

ϑn+1(ξn+1)

(1 +

(Hn+1(ξn+1)Y (ξn+1)

ϑn+1(ξn+1)

) 11−γ)1−γ

. (D.1)

By (18), HT (ξT ) = 0 and thus (D.1) implies

HT−1(i) = ET−1,i

[ϑT (ξT )

1 + ρ

]=

L∑j=1

Q(i, j)ϑT (j)

1 + ρ,

31

which is obviously increasing in ϑT for each i = 1, . . . , L. Consequently, it follows from (20)and (27) that φT−1(i) is decreasing in ϑT for each i = 1, . . . , L.

For n = T − 2, (D.1) implies

HT−2(i) =1

1 + ρ

L∑j=1

Q(i, j)ϑT−1(j)

(1 +

(HT−1(j)Y (j)

ϑT−1(j)

) 11−γ)1−γ

. (D.2)

Thus, for each i = 1, . . . , L, HT−2(i) is also increasing in ϑT as a property inherited fromHT−1(i), and consequently, φT−2(i) is decreasing in ϑT due to (20) and (27).

In general, we assume that Hn+1(i) is increasing in ϑT for each i = 1, . . . , L, and from(D.1) we can obtain a similar equation for Hn(·) in terms of Hn+1(·) similar to (D.2) toconclude that Hn(i) is indeed increasing in ϑT . Then, by (20) and (27), φn(i) is decreasingin ϑT , and therefore, by the induction principle, we complete the proof w.r.t. the monotonicproperty of φn in the general case.

The results for ηn(i) follow straightforward from the equation ηn(i) = (1 − φn(i))αn(i)and the fact that αn(i) is irrelevant to ϑn(i). The proof for results with all elements of Qbeing positive can be obtained in a complete parallel way.

Appendix E. Proof of Proposition 4.3

Proof. When the discount rates are constant, (21) can be equivalently written as equation(D.1), from which it follows for k = n, . . . , T − 1 that

Hk(i) =1

1 + ρ

L∑j=1

Q(i, j)ϑk+1(j)

(1 +

(Hk+1(j)Y (j)

ϑk+1(j)

) 11−γ)1−γ

, (E.1)

where Hk+1(j) does not depend on ϑk+1. Therefore, it is straightforward to verify that, forany i, j = 1, . . . , L and k = n, . . . , T − 1,

∂

∂ϑk+1(j)Hk(i) =

Q(i, j)

1 + ρ

(1 +

(Hk+1(j)Y (j)

ϑk+1(j)

) 11−γ)−γ

≥ 0, (E.2)

which implies that Hk(i) is increasing in ϑk+1. On the other hand, by (20) and (27), φn(i)admits the following expression

φn(i) =1

1 +(Hn(i)Y (i)ϑn(i)

) 11−γ

, (E.3)

and hence it is decreasing in ϑn+1. Moreover, (E.1) and (E.3) together imply that φn(i)depends on ϑn+2 only via Hn+1(·). Since (E.2) indicates that Hn+1(j) is increasing in ϑn+2,φn(i) is also decreasing in ϑn+2. Repeatedly using recursion (E.1) and the positiveness ofthe derivative in (E.2), we can conclude by the induction principle that the equilibriumconsumption proportion φn(i) is decreasing in ϑk for each k = n+ 1, n+ 2, . . . , T − 1.

The increasing property of ηn(i) with respect to ϑk for each k = n+ 1, n+ 2, . . . , T − 1follows trivially from the equation ηn(i) = (1 − φn(i))αn(i) and the fact that αn(i) isirrelevant to ϑn(i). The results with all elements Q being positive can be obtained in acomplete parallel way.

32

Appendix F. Proof of Proposition 4.4

Proof. When the assumptions in Proposition 4.4 hold, by (18) and (21), Hn can be simplifiedas

Hn =T∑

k=n+1

k−1∏m=n+1

((Am)

11−γ

1+(Am)1

1−γ

)γY

(1 + ρ)k−nϑk(

1 + (Ak)1

1−γ

)γ=

1

1 + ρ

ϑn+1(1 + (An+1)

11−γ

)γ+1

1 + ρ

((An+1)

11−γ

1 + (An+1)1

1−γ

)γ

Y Hn+1,

where Y = E [((1− αn)rf + αnRn)γ] > 0. According to (20), substituting Y Hn = ϑnAninto the above equation yields

ϑnAnY

=ϑn+1

1 + ρ

1(1 + (An+1)

11−γ

)γ+ϑn+1

1 + ρ

((An+1)

11−γ

1 + (An+1)1

1−γ

)γ

An+1

=ϑn+1

1 + ρ

(1 + (An+1)

11−γ

)1−γ.

Therefore, we have

An =ϑn+1Y

ϑn(1 + ρ)

(1 + (An+1)

11−γ

)1−γ, n = 0, 1, . . . , T − 1, (F.1)

with AT = 0. Next, under the assumptions in Proposition 4.4, we show that An > An+1 (n =0, 1, . . . , T − 1) by using induction principle. To this end, by (20), we first note that

AT−1 = 0, and AT−2 =ϑT−1Y

ϑT−2(1 + ρ)≥ AT−1 = 0.

In general, with the assumption An+1 ≥ An+2, we obtain

An =ϑn+1Y

ϑn(1 + ρ)

(1 + (An+1)

11−γ

)1−γ

≥ ϑn+1Y

ϑn(1 + ρ)

(1 + (An+2)

11−γ

)1−γ

=(ϑn+1)2

ϑnϑn+2

ϑn+2Y

ϑn+1(1 + ρ)

(1 + (An+2)

11−γ

)1−γ=

(ϑn+1)2

ϑnϑn+2

An+1.

If ϑnϑn+1≤ ϑn+1

ϑn+2, n = 0, 1, . . . , T − 3, then An ≥ An+1. Consequently, by (29), we obtain

Λn+1 − Λn = Z

[1

1 + (An)1

1−γ− 1

1 + (An+1)1

1−γ

]≤ 0.

33

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