Duesenberry's Theory of Consumption: Habit, Learning, and Ratcheting
aa r X i v : . [ ec on . T H ] D ec Duesenberry’s Theory of Consumption: Habit,Learning, and Ratcheting ∗ Kyoung Jin Choi † Junkee Jeon ‡ Hyeng Keun Koo § December 27, 2018
Abstract
This paper investigates the consumption and risk taking decision of an economic agentwith partial irreversibility of consumption decision by formalizing the theory proposed byDuesenberry (1949). The optimal policies exhibit a type of the (s, S) policy: there aretwo wealth thresholds within which consumption stays constant. Consumption increasesor decreases at the thresholds and after the adjustment new thresholds are set. The shareof risky investment in the agent’s total investment is inversely U-shaped within the (s,S) band, which generates time-varying risk aversion that can fluctuate widely over time.This property can explain puzzles and questions on asset pricing and households’ portfoliochoices, e.g., why aggregate consumption is so smooth whereas the high equity premiumis high and the equity return has high volatility, why the risky share is so low whereasthe estimated risk aversion by the micro-level data is small, and whether and when anincrease in wealth has an impact on the risky share. Also, the partial irreversibility modelcan explain both the excess sensitivity and the excess smoothness of consumption.
JEL Classification Codes : D11, E21, G11
Keywords : Duesenberry, consumption, portfolio choice, adjustment costs, time-varyingrisk aversion, habit formation, permanent income hypothesis, excess sensitivity, excesssmoothness ∗ We thank participants of the JAFEE-Columbia-NUS Conference, Tokyo, Japan, the Asian Finance Quan-titative Finance Conference, Guangzhou, China and the Quantitative Methods in Finance Conference 2018 inSydney for helpful and encouraging comments. We thank Philip Dybvig for his inspiring work and discussionswhich provided motivation for this work. † E-mail: [email protected] , Haskayne School of Business, University of Calgary, Canada. ‡ E-mail: [email protected] , Department of Applied Mathematics, Kyung Hee University, Korea. § E-mail: [email protected] , Department of Financial Engineering, Ajou University, Korea. Introduction
This critique is based on a demonstration that two fundamental assumptionsof aggregate demand theory are invalid. These assumptions are (1) that everyindividual’s consumption behaviour is independent of that of every other indi-vidual, and (2) that consumption relations are reversible in time (Duesenberry,1949).This paper analyzes a model of consumption and portfolio choice decision. Buildingon the second part of the quote by Duesenberry (1949) as a critique of the Keynesian con-sumption function, we aim to model the irreversibility of consumption decision. However,we are also aware that each consumption decision is not fully irreversible, but partiallyirreversible . Thus, we model the partial irreversibility by introducing proportional costsfor each consumption adjustment. The partial irreversibility makes consumption a non-smooth function of wealth and change infrequently over time, demonstrating the excesssensitivity and the excess smoothness for moderate shocks. The risky share of the house-hold is U-shaped, which can reconcile several conflicting views in the literature, regardingtime-varying risk aversion and the impact of a wealth change on risky investments.In our model, conditional on consumption never changing, the household has thevon-Neuman Morgenstern preference such as the constant relative risk aversion (CRRA)preference. However, there is a utility cost proportional to the current level of marginalutility whenever the household increases or decreases the consumption level. The former isa learning cost in increasing the consumption level and the latter represents consumptionratcheting. Note that the existence of the utility cost from decreasing consumption isimportant in our model since the model can generate all the properties derived in ourmodel only with the cost of downward adjustment. The optimal polices exhibit a (s, S) type of policy as follows. Suppose the currentconsumption level is c . Then, there are two wealth thresholds cx and c ¯ x , where x and ¯ x are constants, i.e., the current (s, S) band is interval ( cx, c ¯ x ). In this case, consumptionincreases and decreases if and only if the wealth level hits cx and c ¯ x , respectively. Other-wise, consumption stays constant inside the band. Once a boundary is reached, the newconsumption level, c new is set and the next new (s, S) band is updated as ( c new x, c new ¯ x ).The adjustment is made whenever there is a change in consumption. We describe howeach new consumption level is determined in more detail in the main body of the paper. For example, consider a household who lives in Chicago and goes for a vacation within the U.S. once peryear. Suppose the family has a permanent increase in income and decides to visit Europe for a regular vacationfrom this year. Our assumption implies that the household incurs search or learning costs when preparing tovisit a new place. We do not consider a monetary cost of search that should follow every consumption decision.Rather, our model assumes that there is one time utility cost of searching or learning the new consumptionpattern when they increase the consumption level. In addition, the cost of upward adjustment is usually very small in our calibration exercise. We can set itto be zero and generate the same resultf by slightly changing values of other variables. he optimal risky share is U-shaped in wealth for each (s, S) band. The risky assetholdings consist of two components: the myopic component and the hedging component.Noticing the minus sign in this decomposition, the hedging demand takes zero at theboundaries of each (s, S) interval and the maximum value inside the interval. The reasonis to avoid a high utility cost that would be incurred if the household frequently adjustedits consumption. Therefore, the risky share is U-shaped with the maximum at the oneof the boundaries, the minimum somewhere inside the interval. On the other hand,naturally there is heterogeneity among households in utility costs as well as risk aversion.Naturally arises the question of how to infer risk aversion of a certain household. Wedefine RCRRA (revealed coefficient of relative risk aversion) by risk aversion inferred bythe outsider observing the risky asset holdings of a household over time. Since the riskyshare is time-varying, the RCRRA is time-varying. Moreover, the RCRRA is inverseU-shaped inside the (s, S) boundary since the risky share is U-shaped.Having the above properties in mind, our model can explain a number of interestingimplications for the household consumption and risky investment decisions. First, ourmodel can fill a gap between different views on risk aversion in the literature on decisiontheory, structural estimation, behavioral, and asset pricing. Note that most asset pricingmodels use the relative risk aversion coefficient of around 10 or higher for calibration exer-cises to match asset pricing moments (e.g., Bansal et al. (2012), Cocco et al. (2005), andreferences therein ). Along the similar lines, households usually hold 6 −
20% in equity(conditional on participation, up to 40 %), which implies relative risk aversion is at least10 or larger following calibration according to the standard models with widely acceptedmarket parameter values. On the contrary, the estimated individual relative risk aversiontakes values between 0.7 and 2 in decision theory, structural estimation, or behavioralliterature such as Bombardini and Trebbi (2012), Campo et al. (2011), Chetty (2006),Gandelman and Hernndez-Murillo (2013, 2015), Hansen and Singleton (1982), Layard, Mayraz, and Nickel(2008), and Szpiro (1986), among which more recent ones tend to claim that risk aversionis less than 1 or around 1.We show that time-varying risk aversion, i.e., RCRRA in our model is the same asactual risk aversion only when the wealth process hits either one of (s, S) boundaries.The set of these events, however, has measure zero for any stock price sample path.The RCRRA is greater than actual risk aversion for most of times. We note that themaximum value of RCRRA increases with the consumption adjustment costs (RCRRA isalways equal to risk aversion if there is no such cost). We show that the RCRRA takesfairly high values closer to its maximum than actual risk aversion during the times whenthe wealth process stays in the middle range of the (s, S) band and does not have highfluctuation: These are times when the market is neither bullish nor bearish, rather hassmall volatility for a while. This feature is in sharp contrast with that from traditional There are many other asset pricing models to suggest even much higher value than 10. For example,Kandel and Stambaugh (1991) argues γ = 29. abit models since risk aversion in the habit models becomes higher only in downturns. Forexample, Otrok et al. (2002) show that the size of equity premium in the traditional habitmodel is determined by a relatively insignificant amount of high-frequency volatility in USaggregate consumption. Moderate shocks have occurred in the world financial marketsover substantial time periods when the RCRRA takes high values in our model. Therefore,our model can generate the average value of RCRRA consistent with that often used inthe asset pricing literature, while the coefficient of relative risk aversion is close to thatsuggested by the behavioral or experimental literature.Second, but more important is that the U-shaped risky share resulted from our modelhas an interesting implication for the effect of the change of wealth on the risky shareon which the empirical literature looks inclusive. For example, Calvet et al. (2009) andCalvet and Sodini (2014) favor habit, commitment, or DARA (decreasing relative riskaversion) models predicting that the impact is positive, i.e., the risky share increases withan increase in financial wealth. Brunnermeier and Nagel (2008) and Chiappori and Paiella(2011), however, show no relationship (or slightly positive relationship, if any) betweenthe financial (liquid) wealth change and the risky share. We argue that both can happendepending on which time-series is chosen. The U-shape implies that the risky share isdecreasing in wealth in the left side of (s, S) band and increasing in wealth in the rightside of (s, S) band (see Figure 10). The optimal wealth process tends to stay longer inthe increasing (decreasing) regions of each (s, S) interval over time if the market is suchthat good shocks are more frequent than bad shocks (see Figure 3 and its description).Thus, if good shocks occurred more frequently in the stock market during the data period,the relationship tends to be positive. By using this intuition, we simulate four types ofsample paths: (a) bullish, (b) intermediate, (c) bearish, and (d) highly fluctuating. Then,we regress the change of risky share on the change of financial wealth within each sample.For cases (a) and (b), we find a significant positive impact of the wealth increase on therisky share. Moreover, we find no or slightly negative relationship for cases (c) and (d).Thus, our model can reconcile the discrepancy in the empirical literature.Third, the optimal consumption process in our model features two well-documentedempirical regularities: the excess smoothness (Deaton, 1987) and the excess sensitivity(Flavin, 1981). First, consumption tends not to respond to a permanent shock as long asthe shock does not push up or down the wealth level to one of the (s, S) boundaries, whichimplies the excess smoothness. Second, suppose there is a good shock in the permanentincome. While this shock may not increase the current consumption level, it increases theprobability that consumption will increase in the future, by which the excess sensitivityof consumption appears in our model. By the same reason, however, both the excesssmoothness and the excess sensitivity vanish for a large shock that immediately pushesthe wealth process up or down to either of the (s, S) boundaries. In other words, thepartial irreversibility of consumption in our model implies that consumption responds toa large shock consistent with the permanent income hypothesis (Jappelli and Pistaferri(2010)). ourth, we explore asset pricing implications and find that our partial reversibilitymodel has a potential to well the U.S. data well. We follow Constantinides (1990) andMarshall and Parekh (1999), simulate optimal consumption of individuals, and obtain asimulated series of monthly aggregate consumption. We compute several moments suchas the consumption growth rate, the standard deviation of marginal rate of intertemporalsubstitution, and the autocorrelation of consumption growth rate. We find that our modelmatches better the US data than the traditional habit models.Finally, our paper also makes a theoretical contribution to the literature of dynamicoptimization. We first transform the dynamic consumption and portfolio selection prob-lem into a static one, and then derive the dual Lagrangian problem. The advantage ofsolving the dual Lagrangian problem is that we do not need to consider the portfoliochoice and we are left to analyze two sided singular control problem (of consumption).After each adjustment of consumption, the problem is to decide whether to increase or todecrease consumption and if so, how much to change. Then, the original value functionis obtained from the convex duality relation. We characterize the full analytic solution.The optimal portfolio can be derived from the dual value function by the convex dualityand It´o’s lemma. The detail for each step is presented in the appendix. Our paper is motivated from Duesenberry (1949). On one hand, the first part of hiscritique quoted in the beginning of our introduction is closely related to external habitformation and thus has significantly contributed to developing the modern habit modelssuch as Abel (1990), Constantinides (1990), and Campbell and Cochrane (1999). Someof our results resemble those from habit model. For example, our model generates time-varying risk aversion. However, ours are fundamentally different from habit models in thatthe agent in habit models becomes more conservative when the current consumption levelgets closer to the habit stock (e.g., in downturns) while RCRRA in our model becomeshigher in times when the stock market is neither bullish nor bearis, but rather flat andhas low volatility. On the other hand, there is a handful of previous literature that grewout from the second part of the critique such as Dybvig (1995) and Jeon et al. (2018). However, Dybvig (1995) and Jeon et al. (2018) assume that consumption decision is fullyirreversible (other than allowing a predetermined depreciation). Therefore, they are an Note that the second part of critique is well described as follows:At any moment a consumer already has a well-established set of consumption habits... Supposea man suffers a 50 per cent reduction in his income and expects this reduction to be permanent.Immediately after the change he will tend to act in the same way as before... In retrospect he willregret some of his expenditures. In the ensuing periods the same stimuli as before will arise, buteventually he will learn to reject some expenditures and respond by buying cheap substitutes forthe goods formerly purchased (Dusenberry 1949, p. 24). xtreme special case of our model. We better formulate the ratchet effect with partialirreversibility by allowing costly downward adjustment, which generates the U-shape riskyshare and its novel implications that Dybvig (1995) and Jeon et al. (2018) do not.Our model is also related to dynamic consumption and investment models with durableconsumption or consumption commitment such as Grossman and Laroque (1990), Hindy and Huang(1992, 1993), Hindy et al. (1997), Flavin and Nakagawa (2008), and Chetty and Szeidl(2007, 2016). We view our model as complement to the consumption commitment lit-erature. For example, Chetty and Szeidl (2016) show in the commitment model thatthe excess smoothness and excess sensitivity arise for moderate shocks and they vanishfor large shocks. The same result holds from a different channel in our model, namelythrough learning or adjustment costs. In our model there exist two different adjustmentcosts, downward and upward adjustment costs. Thus magnitude of large shocks whichmake immediate adjustment and individuals aggressive in risk taking is in general differ-ent for good shock and for bad shocks. This paves a way for empirical test whether theeffects of large shock on consumption and risk taking are different for these two differentshocks. Furthermore, our model generalizes the models of consumption and portfolio se-lection with durability and local substitution by Hindy and Huang (1992) and Hindy et al.(1997) via the isomorphism discovered by Schroder and Skiadas (2002).The rest of the paper continues as follows. Section 2 describes the model. 3 present theanalysis for the explicit solution. The implications of risky investment and consumptionare provided in Sections 4 and 5, respectively. Section 6 concludes.
We consider a simple and standard continuous-time financial market. The financial marketconsists of two assets: a riskless asset and a risky asset. We assume that the risk-freerate, the rate of return on the riskless asset, is constant and equal to r . The price S t ofthe risky asset or the market Index evolves as follows: dS t /S t = µdt + σdB t , where µ, σ are constants, µ > r , and B t is a Brownian motion on a standard probabilityspace (Ω , F , P ) endowed with an augmented filtration {F t } t ≥ generated by the Brownianmotion B t .The agent’s wealth process ( X t ) ∞ t =0 evolves according to the following dynamics: dX t = [ rX t + π t ( µ − r ) − c t ] dt + σπ t dB t , X = X > , (1) More precisely, Dybvig (1995) and Jeon et al. (2018) assume that consumption is not allowed to decrease.The model converges to those of Dybvig (1995) and Jeon et al. (2018) if the cost of decreasing consumptiongoes to infinite and the cost of increasing consumption is equal to zero in our model. See Karatzas and Shreve (1998) for details of mathematics and the probability theory. here c t ≥ π t are the consumption rate and the dollar amount invested in the riskyasset, respectively, at time t .For non-negative constants α and β , the agent’s utility function is given by U ≡ E (cid:20)Z ∞ e − δt (cid:0) u ( c t ) dt − αd ( u ( c t )) + − βd ( u ( c t )) − (cid:1)(cid:21) , (2)where δ > u ( · ) is a twice-continuously differentiable,strictly concave, and strictly increasing function. We decompose u ( c t ) = u ( c ) + u ( c t ) + − u ( c t ) − , where u ( c t ) + and u ( c t ) − are non-decreasing processes. This decomposition is well-definedif the consumption process c t is admissible (See Definition 2.1 and equation (4)). (2)implies that the agent instantaneously loses the α or β unit of utility whenever there isone unit increase or decrease in the marginal utility. The example of the first kind is theutility cost of learning how to spend and that of the second is the point made by modelswith “catching up with Joneses” such as Abel (1990), Constantinides (1990), Gali (1994)and Campbell and Cochrane (1999).In this paper, we assume the constant relative risk aversion(CRRA) utility function: u ( c ) = c − γ − γ , γ > , γ = 1 , (3)where γ is the agent’s risk coefficient of relative risk aversion.To define the strategy set, we denote by Π the family of all c´agl´ad, F t -adapted, non-decreasing process with starting at 0 and assume that there exist c + , c − ∈ Π such thatthe agent’s consumption c t can be expressed by c t = c + c + t − c − t , (4)where c is the agent’s initial consumption rate. Then, the agent’s objective (2) is rewrittenas U ≡ E (cid:20)Z ∞ e − δt (cid:0) u ( c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) . (5)We define the set of consumption and risky investment strategies as follows. Definition 2.1.
We call a consumption-portfolio plans ( c + , c − , π ) admissible if(a) A consumption strategy ( c + , c − ) satisfies E (cid:20)Z ∞ e − δt (cid:0) | u ( c t ) | dt + αu ′ ( c t ) dc + t + βu ′ ( c t ) dc − t (cid:1)(cid:21) < ∞ . (6) We denote by Π( c ) the class of all consumption strategies ( c + , c − ) satisfying thecondition (6) .(b) For all t ≥ , π t is measurable process with repsect to F t satisfying Z t π s ds < + ∞ , a.e. (7) c) For all t ≥ , the wealth process is non-negative, i.e., X t ≥ . (8)The following assumption should be satisfied in order to guarantee the existence ofthe special case of α = β = 0, i.e., the classical Merton problem. Assumption 1. K ≡ r + δ − rγ + γ − γ θ > . If the agent instantaneously increases her consumption by a small amount, from c to c + ∆ c between t and t + ∆ t , the intertemporal utility gain from the additional consump-tion is 1 δ ( u ( c + ∆ c ) − u ( c )) ≈ δ u ′ ( c )∆ c. On the other hand, the utility loss from theconsumption increase is αu ′ ( c )∆ c . The gain should be greater than the loss. Otherwise,the agent will never increase consumption in our model. This observation leads to thefollowing assumption. Assumption 2. ≤ δα < . There is no parameter restriction for β ∈ [0 , ∞ ). The problem studied by Dybvig(1995) is an extreme case of ours. Dybvig (1995) consider the case when α = 0 and β → ∞ , which means there is no utility cost of increasing consumption and infinity utilityloss from decreasing consumption (i.e., ratcheting of consumption).Now we state the problem as follows. Problem 1 (Primal Problem (Dynamic Version)) . Given c = c > and X = X > , we consider the following utility maximizationproblem: V ( X, c ) = sup E (cid:20)Z ∞ e − δt (cid:0) u ( c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) (9) where the supremum is taken over all admissible consumption/portfolio plans ( c + , c − , π ) subject to the wealth process (1) . Note that Problem 1 is subject to the dynamics budget constraint (1). In Section3.1, we first transform Problem 1 into a static problem (Problem 2) by the well-knownmethod of linearizing the budget constraint suggested by Karatzas et al. (1987) andCox and Huang (1989). Finally we transform that static problem to a singular controlproblem (Problem 3). Theorem 3.1 shows that the solution to Problem 1 is recoveredfrom the solution to Problem 3 by the duality relationship. We will obtain the solutionto Problem 3 and characterize optimal policies by using it in later sections. Note that theadvantages of dealing with the singular control problem in Problem 3 are described rightbelow the problem statement. Solution Analysis
In order to reformulate Problem 1, first we transform the wealth process satisfying (1)into a static budget constraint. For this purpose we define, for t ≥ θ ≡ µ − rσ , ξ t ≡ e − rt Z t , and Z t ≡ e − θ t − θB t . Let us define an equivalent measure Q by setting d Q d P = Z T , (10)so that the process B Q t = B t + θt is a standard Brownian motion under the measure Q .Then, the wealth process (1) is changed by dX t = [ rX t − c t ] dt + σπ t dB Q t . (11)Applying Fatou’s lemma and Bayes’ rule to e − rt X t , we get the following static budgetconstraint: E (cid:20)Z ∞ ξ t c t dt (cid:21) ≤ X. (12)where X is the initial wealth level, i.e., X = X . Then, we restate Problem 1 as thefollowing problem. Problem 2 (Primal Problem (Static Version)) . Given c = c > and X = X > , we consider the following utility maximizationproblem: V ( X, c ) = sup E (cid:20)Z ∞ e − δt (cid:0) u ( c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) (13) where the supremum is taken over all admissible consumption/portfolio plans ( c + , c − , π ) subject to the static budget constraint (12) . By following Cox and Huang (1989) and Karatzas et al. (1987), it is easy to see thatthe solution to Problem 1 is the same as that to Problem 2. Thus, henceforth the bothproblems are called the primal problem in this paper.Using the static budget constraint (12), we consider the following Lagrangian: L = E (cid:20)Z ∞ e − δt (cid:0) u ( c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) + y (cid:18) X − E (cid:20)Z ∞ ξ t c t dt (cid:21)(cid:19) = E (cid:20)Z ∞ e − δt (cid:0) ( u ( c t ) − ye δt ξ t c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) + yX, (14)where y > y t = ye δt ξ t , t ≥ hich plays the role of the Lagrange multiplier for the budget constraint at time t , and thus( y t ) ∞ t =0 represents the marginal utility(shadow price) of wealth process. We will describehow the marginal utility of wealth process is related to the agent’s optimal consumptionpolicy.Now we introduce the dual problem of Problem 1 or Problem 2: Problem 3 (Dual problem) . J ( y, c ) = sup ( c + ,c − ) ∈ Π( c ) E (cid:20)Z ∞ e − δt (cid:0) ( u ( c t ) − y t c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) = sup ( c + ,c − ) ∈ Π( c ) E (cid:20)Z ∞ e − δt (cid:0) h ( y t , c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) , (15) where h ( y, c ) = u ( c ) − yc. and Π( c ) is the class of all consumption strategies ( c + , c − ) satisfying the condition (6) . Problem 3 is the optimization problem with singular controls over Π( c ). There are twoadvantages in dealing with the dual problem. The first advantage is that we do not needto consider the portfolio choice. This property is, in fact, inherited from the formulationof Problem 2. The second advantage is that now the agent’s problem becomes a singularcontrol problem of deciding either to increase or decrease the level of consumption giventhe current consumption c . Therefore, we can apply a standard method of singular controlproblem developed by Davis and Norman (1990) or Fleming and Soner (2006). The dual value function J ( y, c ) satisfies the following Hamilton-Jacobi-Bellman(HJB)equation:max {L J ( y, c ) + u ( c ) − yc, J c ( y, c ) − αu ′ ( c ) , − J c − βu ′ ( c ) } = 0 , ( y, c ) ∈ R (16)where R ≡ R + × R + and the differential operator L is given by L = θ y ∂ ∂y + ( δ − r ) y ∂∂y − δ. To solve the HJB equation (16), we define the increasing region IR , the decreasing region DR and the non-adjustment region NR as follows: IR = { ( y, c ) ∈ R | J c ( y, c ) = αu ′ ( c ) } , NR = { ( y, c ) ∈ R | − βu ′ ( c ) < J c ( y, c ) < αu ′ ( c ) } , DR = { ( y, c ) ∈ R | J c ( y, c ) = − βu ′ ( c ) } . (17)In what follows in this subsection we describe the explicit form of the dual value functionin each region. First, as shown in Appendix A, the regions IR , NR and DR are rewritten y IR = { ( y, c ) ∈ R | y ≤ u ′ ( c ) b α } , NR = { ( y, c ) ∈ R | u ′ ( c ) b α < y < u ′ ( c ) b β } , DR = { ( y, c ) ∈ R | u ′ ( c ) b β ≤ y } , respectively. See Figure 1 for the graphical representation of each region. It is importantto characterize IR -, NR - and DR -regions in order to understand the optimal strategies,which will be investigated in great detail in Section 3.3. Here, b α and b β are given by b α = (1 − δα ) m − m κ w m − w m − − > b β = (1 + δβ ) m − m w m − κw m − w > κ = 1 − δα δβ . m and m are positive and negative roots of following quadraticequation: θ m + ( δ − r − θ m − δ = 0 . Moreover, w is a unique solution to the equation f ( w ) = 0 in (0 , f ( w ) = ( m − m (1 − w − m )( w m − κ ) − m ( m − w m − w )(1 − κw − m ) . (18)In the following proposition, we provide a solution to Problem 3. Proposition 3.1.
The dual value function J ( y, c ) of Problem 3 is given by J ( y, c ) = D yc (1 − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − + D yc (1 − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − + 1 δ c − γ − γ − ycr , for ( y, c ) ∈ NR ,J (cid:18) y, I ( yb α ) (cid:19) + α (cid:18) u ( c ) − u ( I ( yb α )) (cid:19) , for ( y, c ) ∈ IR ,J (cid:18) y, I ( yb β ) (cid:19) − β (cid:18) u ( c ) − u ( I ( yb β )) (cid:19) , for ( y, c ) ∈ DR , (19) where D = ( α − δ ) m + ( m − b α r m − m , D = ( α − δ ) m + ( m − b α r m − m . Proof.
The proof is given in Appendix A.Finally we summarize the duality relationship between the value function of the primalproblem and the dual value function of Problem 3 in the following theorem.
Theorem 3.1.
For the value function V ( X, c ) of Problem 1 and dual value function J ( y, c ) of Problem 3, the following duality relationship is established: V ( X, c ) = min y> ( J ( y, c ) + yX ) . (20) In addition, there exists a unique solution y ∗ for the minimization problem (20) .Proof. The proof is given in Appendix B. .3 Optimal strategies ✻ dual variable y ✲ marginal utility u ′ ( c ) ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ y = u ′ ( c ) b β ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ y = u ′ ( c ) b α ( DR -region) ( IR -region)( NR -region) ✻❄ ( u ′ ( c ) , y ) ❄✻ Figure 1: DR -region, NR -region, and IR -region Refer Figure 1 for the graphical representation of the optimal consumption behavior.If the initial consumption level c is such that y /u ′ ( c ) lies in the increasing region IR or the decreasing region DR , it jumps immediately to the non-adjustment region NR .Suppose the level of consumption is such that y /u ′ ( c ) lies inside the NR -region. Thelevel of consumption stays constant during the time y t -process lies inside the NR -region.Consumption jumps down if and only if y t process hits u ′ ( c ) b β and it jumps up if andonly if y t hits u ′ ( c ) b α . In this case, the question is how much the consumption leveljumps up or down. Proposition 3.2 explicitly characterizes the amount of the jump ateach time when there is a consumption adjustment. Proposition 3.2.
The optimal consumption c ∗ t for t ≥ is given by c ∗ t = c + c ∗ , + t − c ∗ , − t , where y ∗ t = y ∗ e δt ξ t and y ∗ is the unique solution to the minimization problem (20) and c ∗ , + t = max ( , − c + sup s ∈ [0 ,t ) (cid:18) c ∗ , − s + I ( y ∗ s b α ) (cid:19)) ,c ∗ , − t = max ( , c + sup s ∈ [0 ,t ) (cid:18) c ∗ , + s − I ( y ∗ s b β ) (cid:19)) . (21) Proof.
The proof is given in Appendix C.Processes c ∗ , + t and c ∗ , − t are non-decreasing regulators such that y ∗ t /u ′ ( c ∗ t ) lies insidethe NR -region. c ∗ , + t and c ∗ , − t stay mostly constant. Consumption increases whenever
10 20 30 40 50 60 70 8000.511.522.533.5 (a) y t u ′ ( c ∗ t ) (b) c ∗ t (c) c ∗ , + (d) c ∗ , − Figure 2: Simulation of optimal consumption, y/u ′ ( c ∗ ), c ∗ , + and c ∗ , − with α = 5, β = 10.The other parameter values are as follows: ρ = 0 . , µ = 0 . , r = 0 . , σ = 0 . , γ =2 , X = 50, c = 1 and T = 80. c ∗ , + t increases, but consumption decreases whenever c ∗ , − t increases. More precisely, asseen in Figure 1, the process c ∗ , + t stays constant within the non-adjustment region NR and increases if and only if y ∗ t /u ′ ( c t ) hits the free boundary b α . Similarly, the process c ∗ , − t also stays constant within the non-adjustment region NR . c ∗ , − t increases if and only if y ∗ t hits the free boundary b β . Figure 2 plots sample paths of c ∗ , + t and c ∗ , − t together withthose of y t u ′ ( c ∗ t ) and c ∗ t . See Figure 2 for simulated paths of y t u ′ ( c ∗ t ) , c ∗ , + t , c ∗ , − t , and c ∗ t .Proposition 3.2 describes the consumption path according to the ratio of the shadowprice of wealth to the marginal utility process. Then, how is the consumption processrelated to the agent’s wealth process? The following theorem provides the answer to thisquestion. Theorem 3.2.
Pick an arbitrary time s ≥ and let c ∗ s = c be the optimal consumptionat s for some constant c > . For t ≥ s , the optimal consumption is fixed as c ∗ t = c during the time in which y t is inside the NR region (by Proposition 3.2). In this case, he optimal wealth process X ∗ t follows X ∗ t = cr − c D m (1 − γ + γm ) b α (cid:18) y ∗ t c − γ b α (cid:19) m − + D m (1 − γ + γm ) b α (cid:18) y ∗ t c − γ b α (cid:19) m − ! . (22) In addition, there exist two positive numbers x and ¯ x such that c ∗ t = c for t ≥ s if andonly if x < X ∗ t c < ¯ x or cx < X ∗ t < c ¯ x. (23) The explicit forms of x and ¯ x are given in the proof.Proof. The proof is given in Appendix D.Suppose c ∗ s = c is the optimal consumption level at a certain time s as in Theorem3.2. Equation (22) explicitly describes the optimal wealth process during the times whenthe shadow value of wealth, y t -process stays inside the NR -region for t ≥ s .The consumption level increases according to the first equation in (21) if and only if y t hits the lower threshold u ′ ( c s ) b α or touches the IR -region. The latter part of Theorem 3.2tells that this condition is equivalent to the case when X ∗ t hits the upper threshold c ¯ X .Conversely, the consumption level decreases according to the second equation in (21) ifand only if y t hits the upper threshold u ′ ( c s ) b β or touches the DR -region. This conditionis equivalent to the case when X ∗ t hits the lower threshold cx . A B initial wealth X ( c = c )wealth X t ✛ ✲ c x c ¯ x X t ( c = c d ) ✛✲ c d xA d B d c d ¯ x X t ( c = c u ) ✛ ✲ c u x c u ¯ xA u B uX t ( c = c dd ) ✛✲ c dd xA dd B dd c dd ¯ x X t ( c = c du ) ✛ ✲ c du xA du B du c du ¯ x X t ( c = c ud ) ✛ ✲ c ud x c ud ¯ xA ud B ud X t ( c = c uu ) ✛ ✲ c uu x c uu ¯ xA uu B uu ✑✑✑✑✑✑✑✑✑✑✑✰ X t hits c x anddecreases to c d x (bad shock)with c d < c . ✡✡✡✡✡✡✡✢ X t hits c d x anddecreases to c dd x (bad shock) ❏❏❏❏❏❏❏❫ with c dd < c d . X t hits c d ¯ x andincreases to c du ¯ x (good shock) ❏❏❏❏❏❏❏❫ with c du > c d ✄✄✄✄✄✄✎ X t hits c u x anddecreases to c ud x (bad shock) ❏❏❏❏❏❏❏❫ with c ud < c d ❩❩❩❩❩❩❩❩❩⑦ X t hits c u ¯ x andincreases to c uu ¯ x (good shock) ❏❏❏❏❏❏❏❫ with c uu > c u ❩❩❩❩❩❩❩❩❩❩⑦ X t hits c ¯ x andincreases to c u ¯ x (good shock)with c u > c Figure 3: A discrete example of wealth and consumption.
The optimal consumption policy looks like a (s, S) policy over wealth. Note that thetwo threshold levels x and ¯ x depend on the market parameter values and ( α, β ). Equation(23) implies that for a given consumption c at a certain point of time, the thresholds are x and c ¯ x . Once the agent’s wealth reaches either one of two thresholds of wealth, thenew consumption level c ′ is determined by (21). Then, the two new thresholds levels areset as c ′ x and c ′ ¯ x .Figure 3 describes a discrete-time version of the wealth and consumption movement.Assume that current wealth is X and consumption is c . Initially the wealth processlies in interval ( A, B ). Suppose X t hits c ¯ x (point B ) after several good shocks. Then,a new consumption level is set as c u and the new (s, S) band is interval ( A u , B u ). Atthis instant, the wealth level is at point B u . If a positive shock arrives again at point B u ,a new consumption level is immediately set as c uu and the new (s, S) band is interval( A uu , B uu ). If a negative shock arrives at point B u , however, X t moves inside ( A u , B u )and the consumption level stays at c u until X t hits either c u x (point A u ) or c u ¯ x (point B u ).On the contrary, let us go back to the time when the wealth process lies in interval( A, B ). Suppose X t hits c x (point A ) after several bad shocks. Then, a new consumptionlevel is set as c d and the new (s, S) band is interval ( A d , B d ). At this instant, the wealthlevel is at point A d . If a negative shock arrives again at point A d , a new consumptionlevel is immediately set as c dd and the new (s, S) band is interval ( A dd , B dd ). If a positiveshock arrives at at point A d , however, X t moves inside ( A d , B d ) and the consumptionlevel stays at c u until X t hits either c d x (point A d ) or c d ¯ x (point B d ). (a) X ∗ t (b) c ∗ t (c) π ∗ t Figure 4: Simulation of optimal consumption, wealth and portfolio with α = 5, β = 10. Theother parameter values are as follows: ρ = 0 . µ = 0 . r = 0 . σ = 0 . γ = 2, X = 50, C = 1 and T = 80. Now let us turn to the optimal risky investment, which is given in Proposition 3.3below. The explicit form of the optimal portfolio in is easily obtained by applying Ito’slemma to the optimal wealth process in (22) and drawing out the corresponding term.Figure 4 shows a simulated path of X ∗ t and plots its corresponding consumption and riskyportfolio. Notice from the figure that the number of portfolio rebalances is far more thanthe number of consumption adjustment over time. In other words, the household keepsrebalancing risky stock holdings within the (s, S) band while consumption is adjustedonly at the boundary. roposition 3.3. Let c ∗ s = c be the optimal consumption level at time s for some constant c . The optimal portfolio π ∗ t is given by π ∗ t = θσ c D m ( m − − γ + γm ) b α (cid:18) y ∗ t c − γ b α (cid:19) m − + D m ( m − − γ + γm ) b α (cid:18) y ∗ t c − γ b α (cid:19) m − ! (24) for t ≥ s before the next consumption adjustment happens.Proof. The proof is given in Appendix E.Based on the classical portfolio selection results, we can easily see that the optimalrisky portfolio in Equation (24) in Proposition 3.3 (a) will be decomposed as follows: π ∗ t = µ − rγσ X ∗ t − (cid:18) µ − rγσ X ∗ t − π ∗ t (cid:19) or π ∗ t X ∗ t = µ − rγσ − (cid:18) µ − rγσ − π ∗ t X ∗ t (cid:19) . (25)The first term is a myopic term and the remaining term is a hedging term. Note theminus sign in front of the second term. Intuitively, the second term in (25) is the demandthat crows out the myopic demand, in order to maintain the current consumption levelsince frequent changes of the consumption level incur high utility cost. In this sense, thesecond term itself is positive. As seen in Panel (b) of Figure 5, the risky share is U-shapedand the ratio of the hedging demand takes the largest value at the certain point insidethe (s, S) band at which the total risky share takes the minimum value, which generatesinteresting implications. We will discuss the implications of the U-shaped risky share overtime in Section 4 (e).
25 30 35 40 45 50 55 60 65 701015202530354045 (a) saving & portfolio
25 30 35 40 45 50 55 60 65 700.250.30.350.40.450.50.550.60.650.70.75 (b) savingwealth & portoliowealth
25 30 35 40 45 50 55 60 65 701.522.533.544.5 (c) RCRRA
Figure 5: Parameter value are as follows : δ = 0 . , r = 0 . , µ = 0 . , σ = 0 . , γ = 2 , X =50 , c = 1 , α = 5 , β = 100. In this case, x = 23 .
93 and ¯ x = 72 .
82. In particular, ˆ X = 42 .
70 inpanel (c). The maximum RCRRA value is 4.1712.
While decomposition (25) is intuitive, putting (24) into (25) is complicated and thusit is not much informative. In order to better understand risky investment pattern, wedefine the revealed coefficient of relative risk aversion (RCRRA) as follows:RCRRA( t ) ≡ µ − rσ X ∗ t π ∗ t . (26)The RCRRA is the level of relative risk aversion inferred from the agent’s portfolio allo-cation at time t by outsiders who does not know the agent’s actual risk aversion. If the gent is unconstrained ( α = β = 0, i.e., the Merton case), the RCRRA is always the sameas the agent’s true relative risk aversion γ . However, in general RCRRA is time-varying.In fact, it has maximum and minimum values as shown in Theorem 3.3. The next the-orem provides the properties of RCRRA and confirms the above intuition regarding thehedging demand. Theorem 3.3. (a) RCRRA ( t ) ≥ γ for all t ≥ .(b) Pick any s ≥ . Consider the wealth process X ∗ t ∈ ( cx, c ¯ x ) for t ≥ s , where c ∗ s = c is the optimal consumption level at time s for some constant c . Then, RCRRA ( t ) approaches γ as X ∗ t approaches either cx or c ¯ x . Moreover, there exist ˆ X ∈ ( cx, c ¯ x ) such that RCRRA attains the maximum at X t = ˆ X . RCRRA decreases in wealthfor X ∗ t ∈ ( cx, ˆ X ) and increases in wealth for X ∗ t ∈ ( ˆ X, c ¯ x ) .Proof. The proof is given in Appendix F.Panle (c) of Figure 5 plots a typical RCRRA as a function of wealth. RCRRA ishump-shaped, which is opposite to the U-shaped risky share in Panel (b). It attainsthe minimum value γ at the two ends of interval [ cx, c ¯ x ] and the maximum value atˆ X ∈ ( cx, c ¯ x ). As described in Theorem 3.3(b), RCRRA increases in wealth when thewealth level is smaller than ˆ X and it decreases in wealth when the wealth level is greaterthan ˆ X . This section investigate the properties of the optimal risky investment policy obtained inSection 3.3 in detail.
A puzzle in the classical dynamic portfolio selection literature is that while householdsusually hold 6 −
20% in equity (conditional on participation, up to 40%), standard modelspredicts much higher values. For example, see the first row of Table 1 that is the Mertoncase with CRRA preference with about 7% risk premium and 25% volatility. The riskyshare is fairly high such as 124% and 75% when γ = 0 . .
5, respectively, while thesevalues fall into the reasonable range of risk aversion estimated by the recent literature(see the first paragraph of Section 4.2).The optimal risky share in our case is not constant, but is U-shaped (or V-shaped)(see Panel (b) in Figure 5 or Panels (b) and (d) in Figure 6). The maximum value of theratio of risky asset holdings is the same as that of the Merton case only when the wealthprocess hits the ( s, S ) boundary. But, it is a measure zero event in a sample path. Inmost of times, the risky share is strictly smaller as seen in Figure 5(b). As we will discuss = 0 . γ = 1 . γ = 3 . γ = 6 γ = 10( α, β ) = (0 ,
0) 124% 75% 32% 19% 11%( α, β ) = (5 ,
10) 100% – 124% 55% – 75% 24% – 32% 11% – 19% 9% – 11%( α, β ) = (25 , α, β ) = (49 , Table 1: The risky share for different values of ( α, β ) ′ s : the first row corresponds to thestandard Merton case. The other parameters values are δ = 0 . , r = 0 . , µ = 0 . σ = 0 . in Section 4.4, the ratio of risky asset holdings tends to be close to the minimum levelsin times when there are moderate shocks and the wealth fluctuation is also moderate(equivalently, the RCRRA tends to be high for those times). For example, a householdwith γ = 1 . α, β ) = (49 , There is a vast literature on measuring risk aversion. The most widely accepted values arebetween 0.7 and 2. More recent literature tends to argue that risk aversion is close to 1or even less than 1 (For example, see Chetty (2006), Layard, Mayraz, and Nickel (2008),Bombardini and Trebbi (2012), and Gandelman and Hernndez-Murillo (2015)).
Actual Risk aversion γ = 0 . γ = 1 . γ = 1 . γ = 1 . γ = 1 . α , β ) (40,10000) (40,6000) (40,4500) (40,3200) (40, 2200)( α , β ) (45,5000) (45,3000) (45,2200) (45,1500) (45, 1100)( α , β ) (49,1000) (49,500) (49,400) (49,260) (49, 180)Table 2: Actual risk aversion and the calibrated ( α, β ) ′ s to generate RCRRA with maximum13: The other parameters values are δ = 0 . , r = 0 . , µ = 0 . , σ = 0 . , X = 50, and c = 1. However, most of asset pricing literature takes the level of risk aversion as about 10 orhigher for the calibration analysis (e.g., Bansal et al. (2012), and Cocco et al. (2005)). If e consider the investment aspect of these asset pricing models, it is related to the pointmade in Section 4.1 in the sense that γ = 10 results in the risky share consistent withthe empirical observation for households’ stock holdings (see the column for γ = 10 when( α, β ) = (0 ,
0) in Table 1).In summary, the estimated results directly tell that risk aversion is small around unitywhile the calibration exercise in asset pricing indicates that the plausible level of riskaversion should be much higher. Our model can reconcile the gap in these two lines ofliterature. Table 2 shows our numerical exercises. It calibrates ( α, β ) values that generate13 as the maximum RCRRA for each risk aversion around unity with other widely acceptedmarket parameter values given in the table.Note that the conventional habit model can also generate time-varying risk aversion.However, it requires to define a internal or external habit stock process, which are usuallyad hoc. Our model can perform a precise calibration analysis that can fit the maximumand minimum values of risk aversion.
20 30 40 50 60 70 801.41.51.61.71.81.922.12.22.3 =10=15=20 (a) α = 5
20 30 40 50 60 70 800.450.50.550.60.650.70.750.8 =10=15=20 (b) α = 5 =50=100=150 (c) α = 49 =50=100=150 (d) α = 49 Figure 6: Panels (a) and (c) plot RCRRA curves for shares for different α and β values.Panels (b) and (d) plot the corresponding risky share, respectively. Other parameter valuesare as follows: δ = 0 . , r = 0 . , µ = 0 . , σ = 0 . , c = 1 and γ = 1 . α , β ) σ = 0 . σ = 0 . σ = 0 . σ = 0 . δ = 0 . , r =0 . , µ = 0 . , γ = 0 . , X = 50, and c = 1. In Figure 5(c), the actual risk aversion is γ = 3 and the maximum value of RCRRA isabout 5.32. This difference does not seem large. However, the difference between theactual level of risk aversion and the RCRRA value can be dramatically large. Figures 7and 8 plot a sample path of RCRRA( t ). While the actual risk aversion level is γ = 0 . cx nor c ¯ x (See the left panel in each figure), which means that the consumption pathstays constant in Figures 7 and 8.The maximum value of RCRRA increases in α and β , respectively (see Table 3). Ahigh α or β implies a high cost when the agent changes the consumption level. Therefore,the intertemporal hedging deman become large (with negative sign) to crow out the myopicdemand (see Eq. (25) and explanation below the equation), which makes the RCRRAlevel higher. In addition, a high risk, i.e., a high volatility σ increase the maximum levelof RCRRA by the same token. Figure 7: RCRRA can be very large while the actual risk aversion is very low ( γ = 0 . δ =0 . , r = 0 . , µ = 0 . , σ = 0 . , T = 80 , X = 64 . , c = 1 , α = 49 , β = 1000 . Themaximum RCRRA value is 37.46. Note that β is more important since α cannot exceed δ because of Assumption 2.
20 40 60 8020406080100120140160180 0 20 40 60 800510152025303540
Figure 8: RCRRA can be very large while the actual risk aversion is very low ( γ = 0 . δ =0 . , r = 0 . , µ = 0 . , σ = 0 . , T = 80 , X = 64 . , c = 1 , α = 49 , β = 1000 . Themaximum RCRRA value is 37.46.
We know that RCRRA can be very large while the given risk aversion is very small.Then, when does it happen? In habit models, it happens in the downturns, in particular,when the current consumption level is very close to the habit stock. The mechanism toinduce the difference between the actual risk aversion and the RCRRA in our model isvery different from that of habit models.Notice in our model that RCRRA tends to be small during the times when there areconsecutive large shocks and thus in these times there are high fluctuations in wealth. Forexample, those times are between t = 45 and t = 60 in Figure 7 and between t = 20 and t = 50 in Figure 8. On the other hand, it tends to be high during the time when thereare little or modest fluctuations in wealth due to moderately alternating good and badshocks. For example, those times are before t = 45 in Figure 7 and after t = 50 in Figure8. There are other examples in the Appendix.In other words, high cost for consumption adjustment amplifies risk aversion oversmall or moderate shocks (particularly during the times when the wealth level stays inthe mid range of (s, S) band), making the household look very conservative. However,the household may look very aggressive (if her actual risk aversion is very low), showingRCRRA close to her actual risk aversion, during the time when there are large shocks.Notably, if the household has a long sequence of moderate good shocks, then the riskaversion gradually decreases (equivalently the household gradually increases risky assetholdings). This pattern looks like that the household gains more confidence due to thesuccess in the stock market.In this sense, our utility cost model can generate substantial risk aversion in times ofmoderate risk events and great risk-taking in times of large risk events. This result is uite consistent with the puzzle reported in the behavioral literature. (a) bull market (b) intermediate (c) bear market (d) highly fluctuating Figure 9: Four different cases of stock price evolution: Parameter value are as follows : δ = 0 . , r = 0 . , µ = 0 . , σ = 0 . , γ = 1 . c = 1. Here we investigate how the risky share changes relative to an increase in wealth. Notethat models with habit, commitment, or DRRA (decreasing relative risk aversion) oftenpredict that the relationship is positive while standard models with CRRA preferencepredicts no relationship. The empirical literature, however, seems rather inconclusive. Forexample, Calvet et al. (2009) and Calvet and Sodini (2014) favor the habit or commitmentor DRRA models, and Chiappori and Paiella (2011) and Brunnermeier and Nagel (2008)favor the classical CRRA model and show even slightly negative relationship. We shedon light on the debate in the literature. We argue that whether the change in wealthhas positive, negative, or no impact on the risky share depends on what kind of a samplepath (time-series of the stock market data) researchers use although the data generatingprocess is fixed.Figure 9 shows the four types of sample paths we consider. Panel (a) is a typicallong-term bull market, , Panel (b) is intermediate, Panel (c) is a typical bear market, andPanel (d) is the highly-fluctuating market over time. We will show that the impact of anwealth change on the risky share is different for each sample path. To do so, we generatethe population with random ( α, β ) ′ s and simulate their wealth and risky share over time.Basically our estimation analysis follows that of Brunnermeier and Nagel (2008). Con-sider the following equation ∆ k log π t X t = ρ ∆ k log X t , (27)where ∆ k denotes a k -period(year) first-difference operator, ∆ k y t ≡ y t − y t − k . Below webriefly explain how we regress Eq. (27). (Step 1) Generate initial consumption/wealth distributions of N individuals. • We divide the interval [0 , T ] into 12 × T subintervals with end points t j = 1 , , ..., × T . (Here, we assume that T is a positive integer) We set equal initial consumption, c = 1 for each individual and generate eachindividual’s initial consumption x randomly according to a uniform distributionover ( cx, c ¯ x ). • For each i, i = 1 , , ...N and given pairs ( m α , v α ), ( m β , v β ), we generate log-normallydistributed random variables α i and β i whose the mean and variance are ( m α , v α )and ( m β , v β ), respectively. • Generate a 12 × T random vector ω that follows a standard normal distribution.Using this vector ω , we generate the process of the risky asset returns ∆ S t /S t , andthe dual process y ∗ t in Proposition 3.2 for all the N individuals. By Proposition 3.2,we can simulate the optimal wealth and portfolio processes of N individuals. (Step 2) Compute changes in the ratio of risk asset holdings and changes in wealth. • Let ( X , P ) , ( X , P ) , ..., ( X N , P N ) be the simulated wealth/portfolio processes for N individuals in our utility cost model obtained in (Step 1) . (Note that X i and P i are (12 × T + 1) random vectors for i = 1 , , ..., N ). • For given T and k , there are ( T − k + 1) numbers of ∆ k .For i = 1 , , ..., N and j = 1 , , ..., ( T − k + 1) DR ( i, j ) = ∆ k log P ij + k X ij + k = log P i (12 × ( j + k −
1) + 1) X i (12 × ( j + k −
1) + 1) − log P i (12 × ( j −
1) + 1) X i (12 × ( j −
1) + 1) ,X ( i, j ) = ∆ k log X ij + k = log X i (12 × ( j + k −
1) + 1) − log X i (12 × ( j −
1) + 1) . • We regress Eq. (27) with
OLS using the simulate results DR and X . ( m α , v α ) ( m β , v β ) bull markets intermediate bear markets highly fluctuating(5, 5 ) (50, 20 ) 0 . ∗∗∗ − . ∗∗∗ − . ∗∗∗ (10, 5 ) (100, 20 ) 0 . ∗∗∗ . ∗∗∗ − . ∗∗∗ − . ∗∗∗ (10, 5 ) (150, 50 ) 0 . ∗∗∗ . ∗∗∗ -0.0478 (0.01) -0.0271 (0.26)(15, 10 ) (150, 50 ) 0 . ∗∗∗ . ∗∗∗ -0.0387 (0.04) -0.0341 (0.17)(15, 10 ) (200, 50 ) 0 . ∗∗∗ . ∗∗∗ -0.0164 (0.43) -0.0379 (0.14) N = 1500, T = 5, k = 2. Table 4: The regression coefficients for markets (a), (b), (c), and (d) in Figure 9: We generatethe distribution of households with α and β using the log-normal distributions with mean m α , m β and variance v α , v β respectively. The values in the parentheses are p-values when thep-value is greater than 1%. ∗ ∗ ∗ means the p-value smaller than 1%. The other parametersvalues are δ = 0 . , r = 0 . , µ = 0 . , γ = 1 .
5, and c = 1. The results are summarized in Table 4. The regression results show the positive impactof the wealth increase on the risky share when the market is generally going up (Panel The log-normal distribution implies that there are households having fairly large values of α ’s althoughtheir density in the population is very small. We drop those household whose α values violates Assumption 2. a)) or intermediate with moderate up-and-downs. There is no wealth impact on the riskyshare or slightly negative (if any) when the market is generally going down (Panel (c)) orhas a huge volatility (Panel (d)). The intuition behind these results originates from the decreasing region increasing region Figure 10: The decreasing region and increasing region of the risky share in the (s, S) band. ratio of risky asset holding dynamics and wealth process. The risky share is hump-shapedin each (s, S) band, which implies that there are two regions with the band: the oneis the decreasing region in which the risky share decreases with wealth and the other isthe increasing region in which the risky share increases with wealth (see Figure 10). Thedecreasing region is in the left part of the (s, S) band and the increasing region is in theright part of the (s, S) band. In general, the increasing region longer than the decreasingregion since the risk premium is positive. In addition, recall Figure 3 and its descriptionbelow Theorem 3.2. In Figure 3, if there are consecutive good shocks, the wealth processtends to move in the following way: X → B → B u → B uu . During the times of thisjourney, the wealth process will stay longer in the increasing region of each (s, S) bandsince there are more good shocks than bad shocks in size and amount. On the other hand,if there are consecutive bad shocks, the wealth process tends to move in the other way: X → A → A d → A dd . During the times of this journey, the wealth process will staylonger in the decreasing region of each (s, S) band since there are more bad shocks thangood shocks.Having the above property in mind, if the market is doing well in the long-run as inPanel (a) of Figure 9), over time the wealth process is more likely to stay in the increasingregion of each (s, S) band. Thus, the risky share tends to increase with wealth duringthese times. On the other hand, if the market is doing poorly for a while as in Panel (c)of Figure 9), the wealth process is more likely to stay in the decreasing region of each (s,S) interval. Thus, the risky share decreases with wealth during these times.The intermediate case (Panel (b) of Figure 3) tends to be closer to the case of the bull arket and the highly fluctuating period (Panel (d)) of 3) tends to be closer to the case ofthe bear market. It is because, as mentioned above, the right half part of the (s, S) band(increasing region) is generally longer than left half part of the (s, S) band (decreasingregion). Here we provide two implications of optimal consumption. Our model can explain theexcess sensitivity and excess smoothness puzzles. In addition, our model can generatethe reasonable asset pricing moments such as auto-correlation coefficient of consumption,which is often pointed out as a weakness of habit models.
The optimal consumption level is infrequently adjusted in our model (see, for example,Figures 4 and 5). The excess smoothness appears since consumption in our model doesnot immediately respond to a permanent shock with a moderate level when the wealthlevel after the shock does not reach the (s, S) boundary. In other words, the household isless likely to change a consumption level with response to a small unexpected permanentshock (income), while they do increase or decrease the ratio of risky asset holdings for apositive (negative) shock.The excess sensitivity also arises in our model by a similar reason. After a moderategood shock, the wealth level becomes higher than before. Although consumption does notimmediately respond, it is more likely to increase later since the probability of increasingconsumption in the next period becomes higher after the good shock. Therefore, ourmodel can explain the excess sensitivity puzzle.However, all the above arguments do not really work if the size of shock is large enough.For a large good shock, the wealth level immediately reaches the upper threshold, whichmakes consumption increase immediately. By the same token, the consumption leveldecreases immediately after a large bad shock. So, the consumption moves in the largeshock events, following the permanent income hypothesis.
Here we provide how the model can generate consumption data and try to match severalasset pricing moments. By using the time series of aggregate consumption we compute themean and standard deviation of consumption growth rates, the intertemporal marginalrate of substitution(IMRS) and the theoretical equity premium(EP). Based on our con-sumption model, we simulate optimal consumption processes for N = 100 individualsfollowing three steps: (Step 1) Generate initial consumption/wealth distributions of N individuals. [∆ c ] σ (∆ c ) EP Std of IMRS AC1(∆ c )Data 0.0192 0.0212 – – 0.4600Our Model 0.0181 0.0236 0.0052 0.0775 0.4900Merton with temporalaggregation 0.0200 0.0858 0.0526 0.2996 0.1677Table 5: Mean, standard deviation, autocorrelation of the consumption growth rate, theoret-ical equity premium, and standard deviation of the IMRS. The parameter values as follows: δ = 0 . , r = 0 . , µ = 0 . , σ = 0 . , γ = 3 .
5, and α = 5 , β = 10. The data in thefirst row is from Bansal et al. (2012), sampled on an annual frequency with the period from1930 to 2008. • We divide the interval [0 ,
79] into 2 × ×
79 subintervals with end points t j =1 , .., × × • Similar to Marshall and Parekh (1999), we set equal initial consumption, c = 1for each individual and generate each individual’s initial consumption x randomlyaccording to a uniform distribution over ( cx, c ¯ x ). • Generate a 2 × ×
79 random vector ω that follows a standard normal distribution.Using this vector ω , we generate the process of the risky asset returns ∆ S t /S t , andthe dual process y ∗ t in Proposition 3.2 for all the N individuals. By Proposition 3.2,we can simulate the optimal consumption processes of N individuals. (Step 2) Aggregation of Consumption • Let C , C , ..., C N be the simulated consumption processes for N individuals in ourutility cost model obtained in (Step 1) . • (Cross sectional aggregation) The cross sectionally aggregated consumption process CA of C , C , ..., C N is defined as follows:For j = 1 , , ..., × × CA ( t j ) = 1 N N X i =1 c i ( t j ) . • (Temporal aggregation) We temporally aggregate the cross sectionally aggregatedseries CA to create monthly CA ∗ (that is, ∆ t = 1 /
12) as follows:For j = 1 , , ..., × CA ∗ ( i ) = X j =1 CA ( t × ( i − j ) . (Step 3) Compute the consumption growth rate, IMRS, and theoretical EP of aggregatedconsumption CA ∗ γ α β E [∆ c ] σ (∆ c ) EP Std of IMRS AC1(∆ c )0.015 3.5 5 10 0.0181 0.0236 0.0052 0.0775 0.490015 0.0182 0.0232 0.0050 0.0758 0.488820 0.0183 0.0229 0.0049 0.0747 0.48810.015 3.5 0 20 0.0182 0.0233 0.0050 0.0762 0.48895 0.0183 0.0229 0.0049 0.0747 0.488110 0.0184 0.0226 0.0048 0.0736 0.48690.015 3.5 30 100 0.0191 0.0217 0.0044 0.0699 0.482550 100 0.0192 0.0215 0.0044 0.0693 0.480850 1000 0.0192 0.0215 0.0044 0.0691 0.48020.010 3.5 5 10 0.0194 0.0242 0.0054 0.0791 0.48880.015 0.0181 0.0236 0.0052 0.0775 0.49000.050 0.0093 0.0204 0.0039 0.0683 0.49530.015 0.9 5 10 0.0738 0.0963 0.0052 0.0775 0.48891.5 0.0431 0.0563 0.0052 0.0775 0.48983.5 0.0181 0.0236 0.0052 0.0775 0.490010 0.0063 0.0082 0.0052 0.0775 0.4900Table 6: Mean, standard deviation, autocorrelation of the consumption growth rate, theoret-ical equity premium, and standard deviation of the IMRS. The parameter values as follows: r = 0 . , µ = 0 . σ = 0 . • We use the simulated returns on the risky asset r ( t j ) = 1 , , ..., × • We derive the following time-series i = 1 , , ..., × − CG ( i ) = CA ∗ ( i + 1) − CA ∗ ( i ) CA ∗ ( i ) . (IMRS) I ( i ) = e − δ ∆ t (cid:18) CA ∗ ( i + 1) CA ∗ ( i ) (cid:19) − γ . (EP) EP ( i ) = − cov (cid:0) e − δ ∆ t ( CA ∗ ( i + 1) /CA ∗ ( i )) − γ , ( r ( i + 1) /r ( i )) (cid:1) E [ e − δ ∆ t ( CA ∗ ( i + 1) /CA ∗ ( i )) − γ ] . Using these time-series, we obtain desired statistics(the mean and the standardderivation of consumption growth, the IMRS, the theoretical EP and the auto-correlation of aggregated consumption CA ∗ ).Since each time-series depends on the random vector ω , repeat (Step 1)–(Step 3) ata well with reasonable values for the market and preference parameters. In particular,the auto-correlation simulated by our model is well matched with the data, which is notpossible with the standard Merton case. It is surprising that a fairly small values of ( α, β )can generate the autocorrelation value close to its historical data. Table 6 confirms thisresult. We model the partial irreversibility of consumption decision motivated by Duesenberry(1949). In order to do so, we introduce the adjustment cost of consumption. We show thatthe consumption partial irreversibility model can generate a number of novel implications.Some of our results are similar to those derived from habit formation or consumptioncommitment models. However, the mechanism to generate time-varying risk aversion orthe excess sensitivity and excess smoothness consumption is very different from that fromthese literature. In this sense, we view that our model as complement to the existingliterature. In addition to these results, we find that the consumption adjustment cost canreconcile the gap between the asset pricing literature and the literature on estimating riskaversion. Also, we shed light on the debate on how the wealth change has impact on thehouseholds risky share.We believe that the (partial) irreversibility is a very realistic and important aspect ofconsumption decision. While we model it by introducing the adjustment cost, we admitthat there would be other ways to model it. We hope that our model contributes tobuilding other works toward this direction. One of important future works will be buildinga general equilibrium model with the partial irreversibility of consumption decision thatcan investigate further asset pricing implications (e.g., Choi et al. (2018)). This type ofextension will help to provide the microfoundation for the relative income hypothesis byDuesenberry (1949).
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Economics Letters , 20(1),19–21. ppendixA Derivation of the dual value function J ( y, c ) In this section, we will derive a solution to Problem 3 by solving the HJB equation (16).The following theorem guarantees that the solution to the HJB equation (16) is the solutionto the dual problem.
Theorem A.1 (Verification Theorem) .
1. Suppose that the HJB equation (16) has a twice continuously differentiable solution J ( y, c ) : R → R satisfying the following conditions:(1) For any admissible consumption strategy ( c + , c − ) , the process defined by Z t e − δs ( − θ ) y s J y ( y s , c s ) dB s , t ≥ , is a martingale.(2) For any admissible consumption strategy ( c + , c − ) , lim inf t →∞ E [ e − δt J ( y t , c t )] ≥ . Then, for initial condition ( y , c ) ∈ R and any admissible consumption strategy ( c + , c − ) , J ( y , c ) ≥ E (cid:20)Z ∞ e − δt (cid:0) h ( y t , c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) .
2. Given any initial condition ( y, c ) ∈ R , suppose that there exist an admissible consumptionstrategy ( c ∗ , + , c ∗ , − ) such that, if c ∗ is the associated consumption process, then ( y t , c ∗ t ) ∈ { ( y, c ) ∈ R : L J ( y, c ) + h ( y, c ) = 0 } , Lebesgue-a.e., P -a.s., Z t e − δs (cid:0) J c ( y s , c ∗ s ) − αu ′ ( c ∗ s ) (cid:1) dc ∗ , + s = 0 , for all t ≥ , P − a.s., Z t e − δs (cid:0) − J c ( y s , c ∗ s ) − βu ′ ( c ∗ s ) (cid:1) dc ∗ , − s = 0 , for all t ≥ , P − a.s, (28) and lim t →∞ E h e − δt J ( y t , c ∗ t ) i = 0 (Transversality condition) . Then, J ( y, c ) is the dual value function for Problem 3 and ( c ∗ , + , c ∗ , − ) is the optimal consumptionstrategy.Proof. (Proof of 1.) For given consumption process { c t } ∞ t =0 , define a process M ct = Z t e − δs (cid:0) ( u ( c s ) − y s c s ) ds ) − αu ′ ( c s ) dc + t − βu ′ ( c s ) dc − t (cid:1) + e − δt J ( y t , c t ) . (29)By the generalized It´o’s lemma (See Harrison (1985)), dM ct = e − δt ( u ( c t ) − y t c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t + (cid:16) e − δt dJ ( y t , c t ) − e − δt δJ ( y t , c t ) dt (cid:17) = e − δt (cid:18) θ y t J yy ( y t , c t ) + ( δ − r ) y t J y ( y t , c t ) − δJ ( y t , c t ) + u ( c t ) − y t c t (cid:19) dt + e − δt ( J c ( y t , c t ) − αu ′ ( c t )) d ( c + t ) c + e − δt ( − J c ( y t , c t ) − βu ′ ( c t )) d ( c − t ) c + e − δt ( J ( y t , c t ) − J ( y t , c t − ) − αu ′ ( c t )∆ c t ) { ∆ c t > } + e − δt ( J ( y t , c t ) − J ( y t , c t − ) + βu ′ ( c t )∆ c t ) { ∆ c t < } − θe − δt y t J y ( y t , c t ) dB t (30) here ( c + ) c and ( c − ) c are the continuous parts of c + and c − , respectively.Hence, for any fixed T > M cT − M ct = Z Tt e − δs (cid:18) θ y s J yy ( y s , c s ) + ( δ − r ) y s J y ( y s , c s ) − δJ + u ( c s ) − y s c s (cid:19) ds | {z } ( A ) + Z Tt (cid:0) J c ( y s , c s ) − αu ′ ( c s ) (cid:1) d ( c + s ) c + Z Tt (cid:0) − J c ( y s , c s ) − βu ′ ( c s ) (cid:1) d ( c − s ) c | {z } ( B ) + X t ≤ s ≤ T e − δs (cid:0) J ( y s , c s ) − J ( y s , c s − ) − αu ′ ( c s )∆ c s (cid:1) { ∆ c s > } | {z } ( C ) + X t ≤ s ≤ T e − δs (cid:0) J ( y s , c s ) − J ( y s , c s − ) + βu ′ ( c s )∆ c s (cid:1) { ∆ c s < } | {z } ( D ) + Z Tt ( − θ ) e − δs y s J y ( y s , c s ) dB s | {z } ( E ) . (31)Since max {L J + u ( c ) − yc, J c ( y, c ) − αu ′ ( c ) , − J c ( y, c ) − βu ′ ( c ) } = 0 , we deduce that ( A ) ≤ B ) ≤ . Moreover,( C ) = X t ≤ s ≤ T e − δs Z c s c s − ∆ c s (cid:0) J c ( y s , c s ) − αu ′ ( c ) (cid:1) dc · { ∆ c s > } ≤ , ( D ) = X t ≤ s ≤ T e − δs Z | ∆ c s | (cid:0) − J c ( y s , c s − | ∆ c s | + c ) − βu ′ ( c ) (cid:1) dc · { ∆ c s < } ≤ , (32)and by assumption, E [( E )] = 0.Thus, we can conclude that E t [ M cT − M ct ] ≤ , and { M ct } t ≥ is a super-martingale.This implies that E [ M cT ] ≤ J ( y , c ) and J ( y , c ) ≥ E (cid:20)Z T e − δs (cid:0) ( u ( c s ) − y s c s ) ds ) − αu ′ ( c s ) dc + t − βu ′ ( c s ) dc − t (cid:1)(cid:21) + e − δT J ( y T , c T ) . (33)By assumption lim inf T →∞ E [ e − δT J ( y T , c T )] ≥ . and Fatou’s lemma, we deduce that J ( y , c ) ≥ E (cid:20)Z ∞ e − δs (cid:0) ( u ( c s ) − y s c s ) ds ) − αu ′ ( c s ) dc + t − βu ′ ( c s ) dc − t (cid:1)(cid:21) . (34)The relation (34) holds for any admissible consumption strategy ( c + , c − ), we obtain J ( y , c ) ≥ sup ( c + ,c − ) ∈ Π( c ) E (cid:20)Z ∞ e − δt (cid:0) h ( y t , c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) . (Proof of 2.) y assumption, we can show that in (31) E [( A )] = E [( B )] = E [( C )] = E [( D )] = E [( E )] = 0 for the process M c ∗ t . This implies that { M c ∗ t } t ≥ is a martingale and J ( y , c ) = E (cid:20)Z T e − δs (cid:0) ( u ( c ∗ ) − y s c ∗ ) ds − αu ′ ( c ∗ ) dc ∗ , + t − βu ′ ( c ∗ ) dc ∗ , − t (cid:1)(cid:21) + e − δT J ( y T , c T ) . (35)The transversality condition leads to J ( y , c ) = E (cid:20)Z ∞ e − δs (cid:0) ( u ( c ∗ ) − y s c ∗ ) ds − αu ′ ( c ∗ ) dc ∗ , + t − βu ′ ( c ∗ ) dc ∗ , − t (cid:1)(cid:21) . (36)Thus, J ( y , c ) = sup ( c + ,c − ) ∈ Π( c ) E (cid:20)Z ∞ e − δs (cid:0) ( u ( c ) − y s c ) ds − αu ′ ( c ) dc + t − βu ′ ( c ) dc − t (cid:1)(cid:21) and the consumption strategy ( c ∗ , + , c ∗ , − ) attains the maximum. Hence ( c ∗ , + , c ∗ , − ) is the optimal.Now, we will obtain the analytic characterization of the dual value function by using the thevariational inequality (16).As Dai and Yi (2009), we consider the double obstacle problem arising from variational in-equality (16) as follows: L w ( y, c ) + u ′ ( c ) − y ≥ , for w ( y, c ) = αu ′ ( c ) , L w ( y, c ) + u ′ ( c ) − y ≤ , for w ( y, c ) = − βu ′ ( c ) , L w ( y, c ) + u ′ ( c ) − y = 0 , for − βu ′ ( c ) < w ( y, c ) < αu ′ ( c ) , (37)Consider the following substitution: w ( y, c ) = u ′ ( c ) H ( z ) and z = yu ′ ( c ) . Then, the double obstacle problem (37) can be changed by L H ( z ) + 1 − z ≥ , for H ( z ) = α, L H ( z ) + 1 − z ≤ , for H ( z ) = − β, L H ( z ) + 1 − z = 0 , for − β < H ( z ) < α, (38)The following proposition provides the exact solution of the double obstacle problem (38). Proposition A.1.
The variational inequality (38) has a unique C -solution, which is H ( z ) = α, for z ≤ b α ,D (cid:18) zb α (cid:19) m + D (cid:18) zb α (cid:19) m + 1 δ − zr , for b α < z < b β , − β, for z ≥ b β , (39) where D = ( α − δ ) m + ( m − b α r m − m , D = ( α − δ ) m + ( m − b α r m − m , and m , m are positive and negative roots of following quadratic equation: θ m + ( δ − r − θ m − δ = 0 . and b α , b β are defined as b α = (1 − δα ) m − m κ w m − w m − − , b β = (1 + δβ ) m − m w m − κw m − w . ith κ = 1 − δα δβ .Here, w is the unique root to the equation f ( w ) = 0 in (0 , , where f ( w ) = ( m − m (1 − w − m )( w m − κ ) − m ( m − w m − w )(1 − κw − m ) . (40) Also, H ′ ( b α ) = H ′ ( b β ) = 0 , H ′ ( z ) < for z ∈ ( b α , b β ) and H ′ ( z ) attains minimum at b m ∈ ( b α , b β ) defined by b m = b α · (cid:18) − D m ( m − D m ( m − (cid:19) m − m . Proof.
The uniqueness of the solution is guaranteed due to the maximum principle of the partialdifferential equation theory(see Lieberman (1996)).Now, we will prove the remain part of proposition in the following steps. (Step 1)
We first consider the following free boundary problem: L H + 1 − z = 0 , b α < z < b β ,H ( b α ) = α, H ′ ( b α ) = 0 ,H ( b β ) = − β, H ′ ( b β ) = 0 . (41)Then we can extend the solution H onto R + by H ( z ) = α if z ∈ (0 , b α ) and H ( z ) = − β if z ∈ ( b β , ∞ ) . (42)Next, we show that H ( z ) is the solution to variational inequality (38). We can let the generalsolution for (41) in the form of H ( z ) = D (cid:18) zb α (cid:19) m + D (cid:18) zb β (cid:19) m + 1 δ − zr . From the smooth-pasting condition H ( b α ) = α and H ′ ( b α ) = 0, H ( b α ) = D + D + 1 δ − b α r = α,H ′ ( b α ) = m D + m D − r = 0 . (43)Therefore, D and D are given by D = ( α − δ ) m + ( m − b α r m − m , D = ( α − δ ) m + ( m − b α r m − m . (44)Similarly, H ( b β ) = D (cid:18) b β b α (cid:19) m + D (cid:18) b β b α (cid:19) m + 1 δ − b β r = − β,H ′ ( b β ) = m D b α (cid:18) b β b α (cid:19) m − + m D b α (cid:18) b β b α (cid:19) m − − r = 0 , (45)and D = − ( β + δ ) m + ( m − b β r m − m (cid:18) b α b β (cid:19) m , D = − ( β + δ ) m + ( m − b β r m − m (cid:18) b α b β (cid:19) m . (46)From (44) and (46),( α − δ ) m + ( m − b α r = (cid:18) − ( β + 1 δ ) m + ( m − b β r (cid:19) · (cid:18) b α b β (cid:19) m , ( α − δ ) m + ( m − b α r = (cid:18) − ( β + 1 δ ) m + ( m − b β r (cid:19) · (cid:18) b α b β (cid:19) m . (47) et us define w = b α b β . From (47), m m ( α − δ ) + ( β + δ ) w m ( α − δ ) + ( β + δ ) w m = m − m − w m − ww m − w . (48)From (48), we define f ( w ) as (18). (Step 2) f ( w ) has a unique solution w ∈ (0 ,
1) and w ∈ (0 , κ ).For w ∈ ( κ, f ( w ) = w − m (( m − m ( w m − w )( w m − κ ) − m ( m − w m − w )( w m − κ ))
1) with w ∈ (0 , κ ). (Step 3) The two free boundaries b α and b β are uniquely determined. Moreover, 0 < b α < (1 − δα ) and (1 + δβ ) < b β < ∞ .From (47), b α = (1 − δα ) m − m κ w m − w m − − b β = (1 + δβ ) m − m w m − κw m − w . Thus, b α and b β are uniquely determined.Let us temporarily denote f ( w ) = w m − m w + κm − κ. Since f ′ ( w ) = m ( w m − − > f (0) > , we deduce that f ( w ) > , κ ) . This leads to m − m w m − κw m − w > b β > (1 + δβ ).Let us also temporarily denote f ( w ) = ( m − w m − κm w m − + κ. Since f ′ ( w ) = ( m − m w m − ( w − κ ) < f ( κ ) = κ − κ m > , we deduce that f ( w ) > , κ ) . This leads to m − m κ w m − w m − − < nd b α < (1 − δα ). (Step 4) In z ∈ ( b α , b β ), H ′ ( z ) < b m .First, we will show that D > D < . Since D = ( α − δ ) m + ( m − b α r m − m , D = ( α − δ ) m + ( m − b α r m − m ,D > ⇐⇒ ( α − δ ) m + ( m − b α r < . ⇐⇒ b α > − m m − rδ (1 − δα ) . ⇐⇒ (1 − δα ) m − m κ w m − w m − − > − m m − rδ (1 − δα ) . ⇐⇒ κ w m − w m − − > . ⇐⇒ κw m − > w m . (55)Similarly, we can deduce that D < H ′ ( b α ) = H ′ ( b β ) = 0 and H ′′ ( z ) = D m ( m − b α (cid:18) zb α (cid:19) m − + D m b α (cid:18) zb α (cid:19) m − . Since H ′′ ( b m ) = 0, it is enough to show that b α < b m < b β . By the definition of b m , b α < b m < b β ⇐⇒ < (cid:18) − D m ( m − D m ( m − (cid:19) m − m < x . Since D = ( α − δ ) m + ( m − b α r m − m , D = ( α − δ ) m + ( m − b α r m − m , we can easily check that1 < (cid:18) − D m ( m − D m ( m − (cid:19) m − m ⇐⇒ b α < (1 − δα ) . Also, we know that D = − ( β + δ ) m + ( m − b β r m − m (cid:18) b α b β (cid:19) m , D = − ( β + δ ) m + ( m − b β r m − m (cid:18) b α b β (cid:19) m . This implies that (cid:18) − D m ( m − D m ( m − (cid:19) m − m < x ⇐⇒ b β > (1 + δβ ) . Thus, we deduce that b α < b m < b β . and H ′′ ( z ) < , on ( b α , b m ) and H ′′ ( z ) > , on ( b m , b β ) . Hence, H ′ ( z ) attains minimum at z = b m and H ′ ( z ) < b α , b β ). (Step 4) H ( z ) satisfies the variational inequality (38). For z ∈ ( b α , b β ). it is clear that L H + 1 − z = 0 . (56)Since H ( b α ) = α, H ( b β ) = − β and H ′ ( z ) is strictly decreasing function on ( b α , b β ), − β < H ( z ) < α on ( b α , b β ) . • For z ≤ b α , H ( z ) = α and L H + 1 − z = 1 − δα − z ≥ . • For z ≥ b β , H ( z ) = − β and L H + 1 − z = 1 + δβ − z ≤ . From (Step 1) ∼ (Step 4) , we have proved the desired result.By Proposition A.1, w ( y, c ) given by w ( y, c ) = αu ′ ( c ) , for yu ′ ( c ) ≤ b α ,D u ′ ( c ) (cid:18) yu ′ ( c ) b α (cid:19) m + D u ′ ( c ) (cid:18) yu ′ ( c ) b α (cid:19) m + u ′ ( c ) δ − yr , for b α < yu ′ ( c ) < b β , − βu ′ ( c ) , for yu ′ ( c ) ≥ b β , (57)is a solution of the double obstacle problem (37).Using the w ( y, c ) in the equation (57), we construct the dual value function J ( y, c ) as follows:(i) For b α u ′ ( c ) < y < b β u ′ ( c ), J ( y, c ) = Z c u ′ ( x ) D (cid:18) yu ′ ( x ) b α (cid:19) m dx − Z ∞ c u ′ ( x ) D (cid:18) yu ′ ( x ) b α (cid:19) m dx + u ( c ) δ − ycr (58)(ii) For b α u ′ ( c ) ≥ y , J ( y, c ) = J (cid:18) y, I ( yb α ) (cid:19) + α (cid:18) u ( c ) − u ( I ( yb α )) (cid:19) . (59)(iii) For b β u ′ ( c ) ≤ y , J ( y, c ) = J (cid:18) y, I ( yb β ) (cid:19) − β (cid:18) u ( c ) − u ( I ( yb β )) (cid:19) . (60)where the function I ( · ) : R + → R + is defined by I ( y ) ≡ ( u ′ ) − ( y ) = y − γ . Remark A.1.
It is easy to check that m > and m < . and for u ( c ) = c − γ − γ with γ ( = 1) > , m < − − γγ < m . Thus, the two integrals in (58) are well-defined and J ( y, c ) is J ( y, c ) = D yc (1 − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − + D yc (1 − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − + 1 δ c − γ − γ − ycr . emark A.2. For J ( y, c ) defined in (58) , (59) and (60) , we can easily confirm that J c ( y, c ) = w ( y, c ) . Proposition A.2.
For the function J ( y, c ) defined in (58) , (59) and (60) , the following state-ments are true:1. J ( y, c ) is a twice continuously differentiable and satisfies the HJB equaton (16) . Moreover,the regions IR , NR and DR are represented by IR = { ( y, c ) ∈ R | y ≤ u ′ ( c ) b α } , NR = { ( y, c ) ∈ R | u ′ ( c ) b α < y < u ′ ( c ) b β } , DR = { ( y, c ) ∈ R | u ′ ( c ) b β ≤ y } , respectively.2. For any admissible consumption strategy ( c + , c − ) , Z t ( − θ ) y s J y ( y s , c s ) ds, ∀ t ≥ is a martingale.3. For any admissible consumption strategy ( c + , c − ) , lim t →∞ e − δt E [ J ( y t , c t )] = 0 . Proof. (Proof of 1.)
First, with reference to the construction of the dual value function J ( y, c ), we will show that J is a continuously differentiable if we prove that J y , J yy , and J cc are continuous along the freeboundaries c = I ( yb α ) and c = I ( yb β ).Then, we can compute J y ( y, c ) = J y ( y, I ( yb α )) + (cid:18) J c ( y, I ( yb α )) − αu ′ ( I ( yb α )) (cid:19) ddy (cid:18) I ( yb α ) (cid:19) for y ≤ u ′ ( c ) b α = J y ( y, I ( yb α )) , (61)and J y ( y, c ) = J y ( y, I ( yb β )) − (cid:18) − J c ( y, I ( yb β ) − βu ′ ( I ( yb α )) (cid:19) ddy (cid:18) I ( yb β ) (cid:19) for y ≥ u ′ ( c ) b β = J y ( y, I ( yb β )) . (62)Thus, J y ( y, c ) is continuous along the free boundaries.Similarly, we can obtain J yy ( y, c ) = J yy ( y, I ( yb α )) , for y ≤ u ′ ( c ) b α ,J yy ( y, c ) = J yy ( y, I ( yb β )) , for y ≤ u ′ ( c ) b β , (63)and hence J yy is continuous along the free boundaries.We know that J c ( y, c ) = u ′ ( c ) H ( y/u ′ ( c )) and H ( z ) is C -function. Thus, it is clear that J cc ( y, c ) is continuous function and we conclude that J ( y, c ) is C -function.Next, we will show that J ( y, c ) satisfies the HJB-equation (16). • The region NR :Since J c ( y, c ) = u ′ ( c ) H ( y/u ′ ( c )), NR = { ( y, c ) ∈ R | − βu ′ ( c ) < J c ( y, c ) < αu ′ ( c ) } = { ( y, c ) ∈ R | − β < H ( yu ′ ( c ) ) < α } = { ( y, c ) ∈ R | b α < yu ′ ( c ) < b β } . lso, we can easily confirm that L J + u ( c ) − yc = 0 . • The region IR :We deduce that IR = { ( y, c ) ∈ R | J c ( y, c ) = αu ′ ( c ) } = { ( y, c ) ∈ R | yu ′ ( c ) ≤ b α } . Clearly, − J c ( y, c ) − βu ′ ( c ) = − ( α + β ) u ′ ( c ) < . Since J y ( y, c ) = J y ( y, I ( yb α )) and J yy ( y, c ) = J yy ( y, I ( yb α )) on IR , L J ( y, c ) + u ( c ) − yc = (cid:18) L J ( y, I ( yb α )) + u ( I ( yb α )) − yI ( yb α ) (cid:19)| {z } =0 + δJ ( y, I ( yb α )) − δJ ( y, c ) + u ( c ) − yc − ( u ( I ( yb α )) − yI ( yb α ))= Z I ( ybα ) c (cid:0) δJ c ( y, η ) − ( u ′ ( η ) − y ) (cid:1) dη = Z I ( ybα ) c u ′ ( η ) (cid:18) yu ′ ( η ) − (1 − δα ) (cid:19) dη ≤ (cid:18) ∵ yu ′ ( η ) < b α < − δα on IR (cid:19) . (64) • The region DR :Similarly, DR = { ( y, c ) ∈ R | J c ( y, c ) = βu ′ ( c ) } = { ( y, c ) ∈ R | yu ′ ( c ) ≥ b β } . and J c ( y, c ) − αu ′ ( c ) = − ( α + β ) u ′ ( c ) < , Since J y ( y, c ) = J y ( y, I ( yb β )) and J yy ( y, c ) = J yy ( y, I ( yb β )) on IR , L J ( y, c ) + u ( c ) − yc = (cid:18) L J ( y, I ( yb β )) + u ( I ( yb β )) − yI ( yb β ) (cid:19)| {z } =0 + δJ ( y, I ( yb α )) − δJ ( y, c ) + u ( c ) − yc − ( u ( I ( yb α )) − yI ( yb α ))= − Z cI ( ybβ ) (cid:0) δJ c ( y, η ) − ( u ′ ( η ) − y ) (cid:1) dη = − Z cI ( ybβ ) u ′ ( η ) (cid:18) yu ′ ( η ) − (1 + δβ ) (cid:19) dη ≤ (cid:18) ∵ yu ′ ( η ) > b β > δβ on DR (cid:19) . (65)Thus, J ( y, c ) satisfies the HJB-equationmax {L J + u ( c ) − yc, J c − αu ′ ( c ) , − J c − βu ′ ( c ) } = 0 . (Proof of 2.) Let N t = Z t e − δs ( − θy s ) J y ( y s , c s ) dB s . o show the process N t is a martingale, it is suffice to prove that E (cid:20)Z t (cid:16) e − δs ( − θy s ) J y ( y s , c s ) (cid:17) dt (cid:21) < ∞ , for ∀ t ≥ . (see Chapter 3 in Oksendal (2005))First, we consider the case when ( y t , c t ) ∈ NR . Then, I ( y t b α ) < c t < I ( y t b β ) or b α < (cid:18) y t u ′ ( c t ) (cid:19) < b β . Since yJ y ( y, c ) = D m yc (1 − γ + γm ) b α (cid:18) yu ′ ( c ) b α (cid:19) m − + D m yc (1 − γ + γm ) b α (cid:18) yu ′ ( c ) b α (cid:19) m − − ycr , there exist constants K , K > | y t J y ( y t , c t ) | ≤ K y t c t ≤ K ( y t ) − − γγ . (66)When ( y t , c t ) ∈ IR , we know that J y ( y t , c t ) = J y ( y t , I ( y t b α )) , if ( y t , c t ) ∈ IR . In this case, yJ y ( y, I ( yb α )) = D m yI ( yb α )(1 − γ + γm ) b α + D m yI ( yb α )(1 − γ + γm ) b α − yI ( yb α ) r , Thus, there exist constants K , K > | y t J y ( y t , c t ) | ≤ K y t I ( y t b α ) ≤ K ( y t ) − − γγ . (67)Similarly, when ( y t , c t ) ∈ DR , there exist constants K , K > | y t J y ( y t , c t ) | ≤ K y t I ( y t b β ) ≤ K ( y t ) − − γγ . (68)By (66), (67) and (68), for any ( y t , c t ) ∈ R , | y t J y ( y t , c t ) | ¯ K ( y t ) − − γγ . for some constant K > E (cid:20)Z t (cid:16) e − δs ( − θy s ) J y ( y s , c s ) (cid:17) dt (cid:21) ≤ K E (cid:20) e − δt ( y − − γγ t ) (cid:21) = K Z t e − ( K − θ ( γ − γ ) ) s ds < ∞ . (69)This implies that K t is a martingale for t ≥ (Proof of 3.) If ( y t , c t ) ∈ NR , then b α < (cid:18) y t u ′ ( c t ) (cid:19) < b β . Since, | J ( y, c ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D yc (1 − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − + D yc (1 − γ + γm ) b β (cid:18) yc − γ b β (cid:19) m − + 1 δ c − γ − γ − ycr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (70)there exists a constant K > | J ( y t , c t ) | ≤ K ( y t ) − − γγ . (71) hen ( y t , c t ) ∈ IR , we know that J ( y t , c t ) = J (cid:18) y t , I ( y t b α ) (cid:19) + α (cid:18) u ( c t ) − u ( I ( y t b α )) (cid:19) . (72)Thus, there exits a constant K such that | J ( y t , c t ) | ≤ K ( y t ) − − γγ + α | u ( c t ) | . (73)Similarly, for ( y t , c t ) ∈ DR , there exists a constant K such that | J ( y t , c t ) | ≤ K ( y t ) − − γγ + β | u ( c t ) | . (74)By (71), (73) and (74), for any ( y t , c t ) ∈ R , there exist a constant K > | J ( y t , c t ) | ≤ K (cid:16) ( y t ) − − γγ + | u ( c t ) | (cid:17) . (75)By the admissibility of the consumption strategy, E (cid:20)Z ∞ e − δt | u ( c t ) | dt (cid:21) < + ∞ . Since { c t } ∞ t =0 is a finite variation process, its sample paths can have at most countable disconti-nuities. Hence, applying Fubini’s theorem, we deduce that Z ∞ E h e − δt | u ( c t ) | i dt = E (cid:20)Z ∞ e − δt | u ( c t ) | dt (cid:21) < + ∞ . and lim t →∞ E h e − δt | u ( c t ) | i = 0 . From (75),lim t →∞ E h e − δt | J ( y t , c t ) | i ≤ K (cid:16) lim t →∞ E h e − δt ( y t ) − − γγ i + lim t →∞ E h e − δt | u ( c t ) | dt i(cid:17) = 0 . (76)Thus, we can conclude that for any admissible consumption strategy ( c + , c − ) and its associatedconsumption process c , lim t →∞ E h e − δt J ( y t , c t ) i = 0 . B Proof of Theorem 3.1
We will show that the duality relationship in the following steps: (Step 1)
First, we will prove that the dual value function J ( y, c ) is strictly convex in y :By direct computation, y ∂ J∂y = c D m ( m − − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − + D m ( m − − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − ! . (77)Since D > , D < , − γ + γm > , and 1 − γ + γm < , we deuce that ∂ J∂y > y > , and thus J ( y, c ) is strictly convex in y . et us denote the Lagrangian L defined in (14) by L ( y, c ) for the Lagrangian multiplier y and consumption profile c . (Step 2) We will show that there exist a unique solution y ∗ > y ∗ and the optimalconsumption { c ∗ t } ∞ t =0 maximize the Lagrangian L ( y, c ).From Proposition 3.1, we deduce that ∂J∂y ( y, c ) = cr − D m c (1 − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − + D m c (1 − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − ! , for ( y, c ) ∈ NR ,∂J∂y (cid:18) y, I ( yb α ) (cid:19) , for ( y, c ) ∈ IR ,∂J∂y (cid:18) y, I ( yb β ) (cid:19) , for ( y, c ) ∈ DR . (78)For a sufficiently small y >
0, ( y, c ) ∈ IR , and for a sufficiently large y >
0, ( y, c ) ∈ DR . This implies that lim y → ∂J∂y ( y, c ) = lim y → ∂J∂y (cid:18) y, I ( yb α ) (cid:19) = + ∞ , lim y →∞ ∂J∂y ( y, c ) = lim y →∞ ∂J∂y (cid:18) y, I ( yb β ) (cid:19) = 0 . (79)Since J ( y, c ) is strictly convex in y , for given X >
0, there exists a unique y ∗ such that X = − ∂J∂y ( y ∗ , c ) . (80)Thus, there exist optimal consumption strategy ( c ∗ , + , c ∗ , − ) such that J ( y ∗ , c ) = E (cid:20)Z ∞ e − δt (cid:0) h ( y ∗ t , c ∗ t ) dt − αu ′ ( c ∗ t ) dc ∗ , + t − βu ′ ( c ∗ t ) dc ∗ , − t (cid:1)(cid:21) , (81)where y ∗ t = y ∗ e βt H t , ( c ∗ , + , c ∗ , − ) ∈ Π( c ) and c ∗ t = c + c ∗ , + t − c ∗ , − t .This means that since the Lagrangian is concave, y ∗ and c ∗ are maximizers of the Lagrangian. (Step 3) c ∗ satisfies the budget constraint with equality.Define y + h = y ∗ + h and y − h = y ∗ − h with y ∗ ≥ h > t ). Then, L ( c ∗ , y ± h ) ≥ L ( c ∗ , y ∗ ) . Since c ∗ , y ∗ maximizers of the Lagrangian L , we havelim sup h ↓ L ( c ∗ , y + h ) − L ( c ∗ , y ∗ ) h ≤ , lim inf h ↑ L ( c ∗ , y − h ) − L ( c ∗ , y ∗ ) h ≤ , and thus we deduce ± (cid:18) X − E (cid:20)Z ∞ H t c ∗ t dt (cid:21)(cid:19) ≤ . his leads to X = E (cid:20)Z ∞ H t c ∗ t dt (cid:21) This implies that c ∗ satisfies the budget constraint with equality. (Step 4) c ∗ is optimal consumption.Let ( c + , c − ) ∈ Π( c ) be a feasible consumption strategy, i.e., it is admissible and satisfies thebudget constraint.Since c satisfies the budget constraint, E (cid:20)Z ∞ e − δt (cid:0) u ( c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) ≤ E (cid:20)Z ∞ e − δt (cid:0) u ( c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) + y ∗ (cid:18) X − E (cid:20)Z ∞ H t c t dt (cid:21)(cid:19) (82)where y ∗ is defined in (Step 2) .Since y ∗ and c ∗ maximize the Lagrangian L and c ∗ satisfies the budget constraint withequality, E (cid:20)Z ∞ e − δt (cid:0) u ( c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) ≤ E (cid:20)Z ∞ e − δt (cid:0) u ( c t ) dt − αu ′ ( c t ) dc + t − βu ′ ( c t ) dc − t (cid:1)(cid:21) + y ∗ (cid:18) X − E (cid:20)Z ∞ H t c t dt (cid:21)(cid:19) ≤ E (cid:20)Z ∞ e − δt (cid:0) u ( c ∗ t ) dt − αu ′ ( c ∗ t ) dc ∗ , + t − βu ′ ( c ∗ t ) dc ∗ , − t (cid:1)(cid:21) + y ∗ (cid:18) X − E (cid:20)Z ∞ H t c ∗ t dt (cid:21)(cid:19) = E (cid:20)Z ∞ e − δt (cid:0) u ( c ∗ t ) dt − αu ′ ( c ∗ t ) dc ∗ , + t − βu ′ ( c ∗ t ) dc ∗ , − t (cid:1)(cid:21) . (83)Therefore, ( c ∗ t ) ∞ t =0 is optimal. Step 5)
Proof of duality-relationship in (20)Since c ∗ is optimal consumption, for y >
0, we deduce V ( X, c ) = E (cid:20)Z ∞ e − δt (cid:0) u ( c ∗ t ) dt − αu ′ ( c ∗ t ) dc ∗ , + t − βu ′ ( c ∗ t ) dc ∗ , − t (cid:1)(cid:21) = E (cid:20)Z ∞ e − δt (cid:0) u ( c ∗ t ) dt − αu ′ ( c ∗ t ) dc ∗ , + t − βu ′ ( c ∗ t ) dc ∗ , − t (cid:1)(cid:21) + y (cid:18) X − E (cid:20)Z ∞ H t c ∗ t dt (cid:21)(cid:19) ≤ sup ( c + ,c − ) ∈ Π( c ) E (cid:20)Z ∞ e − δt u ( c t ) dt (cid:21) + y (cid:18) X − E (cid:20)Z ∞ H t c t dt (cid:21)(cid:19) = J ( y, c ) + yX, where ( c t ) ∞ t = is the optimal consumption process for Problem 3 for y >
0. This implies that V ( X, c ) ≤ inf y> (cid:16) J ( y, c ) + yX (cid:17) . However, we know that V ( X, c ) = E (cid:20)Z ∞ e − δt (cid:0) u ( c ∗ t ) dt − αu ′ ( c ∗ t ) dc ∗ , + t − βu ′ ( c ∗ t ) dc ∗ , − t (cid:1)(cid:21) = J ( y ∗ , c ) + y ∗ X. Thus, V ( X, c ) = min y> (cid:16) J ( y, c ) + yX (cid:17) . This completes the proof. Proof of Proposition 3.2
First, we show that the optimal consumption strategy ( c ∗ , + , c ∗ , − ) given in (21) is admissible. Wecan see that for optimal consumption strategy ( c ∗ , + , c ∗ , − ) and its associated consumption c ∗ , b α ≤ y t ( c ∗ t ) − γ ≤ b β for ∀ t ≥ . (84)Then, there exist constants K , K > | u ( c ∗ t ) | ≤ K ( y t ) − − γγ and y t c ∗ t ≤ K y − − γγ t , (85)for all t ≥ E (cid:20)Z ∞ e − δt | u ( c ∗ t ) | dt (cid:21) ≤ K E (cid:20)Z ∞ e − δt ( y t ) − − γγ dt (cid:21) < + ∞ . Since ( c ∗ , + t , c ∗ , − t ) is the optimal strategy, it is clear that E (cid:20)Z ∞ e − δt (cid:0) αu ′ ( c ∗ t ) dc ∗ , + t + βu ′ ( c ∗ t ) dc ∗ , − t (cid:1)(cid:21) < + ∞ . Above two inequalities imply ( c ∗ , + t , c ∗ , − t ) is admissible consumption strategy.Moreover, by (84) and (85), there exists a constant K > | J ( y t , c ∗ t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D y t c ∗ t (1 − γ + γm ) b α (cid:18) y t ( c ∗ t ) − γ b α (cid:19) m − + D y t c ∗ t (1 − γ + γm ) b β (cid:18) y t ( c ∗ t ) − γ b β (cid:19) m − + 1 δ ( c ∗ t ) − γ − γ − y t c ∗ t r (cid:12)(cid:12)(cid:12)(cid:12) ≤ K ( y t ) − − γγ (86)This implies lim t →∞ E h e − δt | J ( y t , c ∗ t ) | i ≤ K lim t →∞ EE h e − δt ( y t ) − − γγ i = K lim t →∞ e − Kt = 0 . (87)From the construction of the optimal consumption strategy, it is easy to check that theconsumption strategy ( c ∗ , + , c ∗ , − ) given in (21) satisfies the following assumption in TheoremA.1: ( y t , c ∗ t ) ∈ { ( y, c ) ∈ R : L J ( y, c ) + h ( y, c ) = 0 } , Lebesgue-a.e., P -a.s., Z t e − δs (cid:0) J c ( y s , c ∗ s ) − αu ′ ( c ∗ s ) (cid:1) dc ∗ , + s = 0 , for all t ≥ , P − a.s., Z t e − δs (cid:0) − J c ( y s , c ∗ s ) − βu ′ ( c ∗ s ) (cid:1) dc ∗ , − s = 0 , for all t ≥ , P − a.s. (88) D Proof of Theorem 3.2
From Theorem 3.1, we know that there exists a unique solution y ∗ for the minimization problem(20). The first-order condition implies that X = − ∂J∂y ( y ∗ , c )= cr − D m c (1 − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − + D m c (1 − γ + γm ) b α (cid:18) yc − γ b α (cid:19) m − ! . (89) ince Problem 3 is time-consistent, y ∗ s = y ∗ e δs H s is the minimizer for the duality relationshipstarting at s ≥
0. Thus, for optimal wealth X ∗ s at time s , we have X ∗ s = c ∗ s r − c ∗ s D m (1 − γ + γm ) b α (cid:18) y ∗ s ( c ∗ s ) − γ b α (cid:19) m − + D m (1 − γ + γm ) b α (cid:18) y ∗ s ( c ∗ s ) − γ b α (cid:19) m − ! . (90)During the time in which y ∗ t is inside the NR , the optimal consumption c ∗ is constant, i.e., c ∗ t = c ∗ s and thus the optimal wealth X ∗ t for t ≥ s is given by X ∗ t = c ∗ s r − c ∗ s D m (1 − γ + γm ) b α (cid:18) y ∗ t ( c ∗ s ) − γ b α (cid:19) m − + D m (1 − γ + γm ) b α (cid:18) y ∗ t ( c ∗ s ) − γ b α (cid:19) m − ! . (91)By Proposition 3.2, we know that the agent does not increase or decrease his/her consumptionin the region b α < y ∗ t ( c ∗ t ) − γ < b β . Let us define x , ¯ x as follows: x = X ( b β ) , ¯ x = X ( b α ) , where X ( y ) is X ( y ) = 1 r − D m (1 − γ + γm ) b α (cid:18) yb α (cid:19) m − + D m (1 − γ + γm ) b α (cid:18) yb α (cid:19) m − ! . Since ∂ X ∂y ( y ) = − D m ( m − − γ + γm ) b α (cid:18) yb α (cid:19) m − + D m ( m − − γ + γm ) b α (cid:18) yb α (cid:19) m − ! and D > , D < , − γ + γm > , and 1 − γ + γm < X ( y ) is strictly increasing functionof y .This means that the consumption stays for t ≥ s constant if and only if c ∗ s x < X ∗ t < c ∗ s ¯ x. This completes the proof.
E Proof of Proposition 3.3
Proof of (a).By applying the generalized It´o’s lemma(see Harrison (1985)) to the optimal wealth process X ∗ t , dX ∗ t = − ∂ J∂y ( y ∗ t , c ∗ t ) dy ∗ t − ∂ J∂y ( y ∗ t , c ∗ t )( dy ∗ t ) − ∂ J∂y∂c ( y ∗ t , c ∗ t ) dc ∗ , + t + ∂ J∂y∂c ( y ∗ t , c ∗ t ) dc ∗ , − t . (92)If ( y ∗ t , c ∗ t ) ∈ NR , the agent does not adjust his/her consumption. This means that dc ∗ , + t = dc ∗ , − t = 0 and thus ∂ J∂y∂c ( y ∗ t , c ∗ t ) dc ∗ , + t = ∂ J∂y∂c ( y ∗ t , c ∗ t ) dc ∗ , − t = 0 . If ( y ∗ t , c ∗ t ) ∈ IR , the agent should increase his/her consumption. This implies that ∂J∂c ( y ∗ t , c ∗ t ) = αu ′ ( c ∗ t ) , dc ∗ , − t = 0 , nd hence ∂ J∂y∂c ( y ∗ t , c ∗ t ) dc ∗ , + t = ∂ J∂y∂c ( y ∗ t , c ∗ t ) dc ∗ , − t = 0 . Similarly, we also obtain ∂ J∂y∂c ( y ∗ t , c ∗ t ) dc ∗ , + t = ∂ J∂y∂c ( y ∗ t , c ∗ t ) dc ∗ , − t = 0 . when ( y ∗ t , c ∗ t ) ∈ NR .Therefore, by comparing the equation (92) with the wealth dynamics (1) and using the factthat dy ∗ t = ( δ − r ) y ∗ t dt − θy ∗ t dB t , we deduce the optimal portfolio policy π ∗ t as follows: π ∗ t = θσ y ∗ t ∂ J∂y ( y ∗ t , c ∗ t ) , (93)and π ∗ t = θσ c ∗ s D m ( m − − γ + γm ) b α (cid:18) y ∗ t ( c ∗ t ) − γ b α (cid:19) m − + D m ( m − − γ + γm ) b α (cid:18) y ∗ t ( c ∗ t ) − γ b α (cid:19) m − ! . (94)Proof of (b). F Proof of Theorem 3.3
For c ∗ s x < X ∗ t < c ∗ s ¯ x , for t ≥ s , the consumption stays constant. Thus, for simplicity, we canassume c ∗ t = 1.By Theorem 3.2 and Proposition 3.3, we deduce that X ∗ t ( y ∗ t ) − γσθ π ∗ t ( y ∗ t ) = − H ′ ( y ∗ t ) , (95)where H ( · ) is defined in Proposition A.1.Then, G ( y ∗ t ) , γσσ π ∗ t ( y ∗ t ) X ∗ t ( y ∗ t ) = 1 + H ′ ( y ∗ t ) X ∗ t ( y ∗ t ) . (96)Since H ′ ( b α ) = H ′ ( b β ) = 0, G ( b α ) = G ( b β ) = 1 . We will show that there exists a unique ˆ b ∈ ( b α , b β ) such that G ( · ) is a strictly decreasingfunction on ( b α , ˆ b ) and strictly decreasing function on (ˆ b, b β ). G ′ ( y ) = H ′′ ( y ) X ( y ) − H ′ ( y ) X ′ ( y )( X ( y )) . (97)(For convenience of notation we drop the time subscript t and the optimal subscript ∗ .)Let us define the numerator of G ′ ( y ) as ¯ G ( y ), i.e.,¯ G ( y ) = H ′′ ( y ) X ( y ) − H ′ ( y ) X ′ ( y ) . (98)By the proof in Proposition A.1, we know that H ′′ ( b α ) < , H ′′ ( b β ) > . Thus, ¯ G ( b α ) < G ( b β ) > . Since X ( y ) = 1 r − D m (1 − γ + γm ) b α (cid:18) yb α (cid:19) m − + D m (1 − γ + γm ) b α (cid:18) yb α (cid:19) m − ,H ′ ( y ) = D m b α (cid:18) yb α (cid:19) m − + D m b α (cid:18) yb β (cid:19) m − − r , e have X ′ ( y ) = − D m ( m − − γ + γm ) b α (cid:18) yb α (cid:19) m − + D m ( m − − γ + γm ) b α (cid:18) yb α (cid:19) m − ,H ′ ( y ) = D m ( m − b α (cid:18) yb α (cid:19) m − + D m ( m − b α (cid:18) yb β (cid:19) m − . Hence,¯ G ( y ) = γr D m ( m − (1 − γ + γm ) b α (cid:18) yb α (cid:19) m − + D m ( m − (1 − γ + γm ) b α (cid:18) yb α (cid:19) m − ! − γD D m m ( m − m ) (1 − γ + γm )(1 − γ + γm ) b α (cid:18) yb α (cid:19) m + m − = γy m − b α D m ( m − r (1 − γ + γm ) (cid:18) yb α (cid:19) m − m − D D m m ( m − m ) (1 − γ + γm )(1 − γ + γm ) b α (cid:18) yb α (cid:19) m − + D m ( m − r (1 − γ + γm ) ! , γy m − b α G ( y ) . We know that m > , m < , D > , D < , − γ + γm > − γ + γm < , thus, G ( y ) is a strictly increasing function of y .Since ¯ G ( b α ) < , ¯ G ( b β ) >
0, we deduce that G ( b α ) < , G ( b β ) > . Thus, there exists a unique ˆ b ∈ ( b α , b β ) such that G (ˆ b ) = 0 . This implies that G ′ ( y ) < , for y ∈ ( b α , ˆ b ) and G ′ ( y ) > , for y ∈ (ˆ b, b β ) . To sum up, we conclude that RCRRA is a strictly increasing for X ∈ ( cx, b X ) and a strictlydecreasing for X ∈ ( b X, c ¯ x ). Moreover, RCRRA approaches γ when X approaches cx or c ¯ x .(Here, ˆ X = c X (ˆ b ) and ˆ b ∈ ( b α , b β ) is a unique solution of the following algebraic equation: G ( y ) = D m ( m − r (1 − γ + γm ) (cid:18) yb α (cid:19) m − m − D D m m ( m − m ) (1 − γ + γm )(1 − γ + γm ) b α (cid:18) yb α (cid:19) m − + D m ( m − r (1 − γ + γm ) . (99)(99)