انتخاب های مصرف پرتفوی بهینه و برنامه ریزی بازنشستگی

We examine consumption and investment decisions in a life-cycle model with habit formation, stochastic opportunity set, stochastic wages and labor supply flexibility. Retirement is taken into account by specifying an age at which labor earnings stop, but consumption spending continues. Explicit solutions are obtained for optimal consumption, labor supply and the financing portfolio. We examine the structure and determinants of the optimal portfolio. We also study the effects of the retirement date and of habits on optimal decisions. Finally, we conduct a preliminary analysis to assess the effects of a liquidity constraint on optimal consumption–leisure choices.

Quantitative models for retirement planning have, in recent years, become the subject of intense practical interest. The reason is the rapid growth of self-directed retirement accounts, such as 401k, 403b, IRA, Keogh, etc. A defining feature of these accounts is that people must decide how much to contribute and how to allocate their accumulations among the choices offered. Investment advisory services have responded by providing interactive retirement planning “tools” which embody quantitative models for computing desired savings and optimal asset allocations. At the core of these planning tools is the single-period mean–variance portfolio selection model developed by Markowitz (1952).1 Because this model ignores so many factors that are relevant to real-world investment decisions, critics have questioned the usefulness of planning calculators.2 Fortunately, over the past three decades, scholars in economics and finance have produced lifetime optimization models that capture many of the important features of reality, such as differing time horizons, family status, human-capital endowments, and habit formation. This paper builds on these contributions by suggesting further aspects that would improve the quantitative models that should be used in retirement planning. For this purpose we study consumption, labor/leisure and portfolio choices in a life-cycle model with labor income flexibility and habit formation. Our contribution can best be understood relative to Bodie et al. (BMS) (1992) who were first to endogenize the labor/leisure decision in an intertemporal consumption–portfolio choice model à la Samuelson (1969) and Merton 1969 and Merton 1971. BMS demonstrated that human capital has a critical impact on optimal policies. Among other items they showed (a) that investment behavior, typically, becomes more conservative as retirement approaches, (b) that labor flexibility smooths consumption behavior and (c) that it promotes greater risk-taking in financial investments. This paper extends their analysis by incorporating (i) habit formation, (ii) two distinct periods during the life-cycle, i.e. an accumulation period and a retirement period and (iii) a more general financial market with multiple assets and stochastic coefficients.3 It also presents a preliminary analysis of the impact of a liquidity constraint. In this context we provide explicit solutions for, or detailed characterizations of, the optimal policies. One of our contributions is to derive the optimal portfolio in closed form. Another one is to show how properties of the optimal policies, such as (a)–(c), are amended in this general setting. Kenc (1999) also extends the ideas of BMS by including (i) freedom to retire, (ii) variations in labor productivity over the life-cycle and (iii) nontradedness of human capital and liquidity constraints. Each of these aspects, however, is considered in isolation.4 Moreover, his analysis assumes a constant opportunity set and focuses on time separable preferences with Cobb–Douglas utility. Our model gives results for more general preferences, a stochastic opportunity set and a wage rate process with random coefficients. It also provides the detailed structure of the optimal portfolio in this general environment. Additional related literature includes Merton (1983) who takes the individual's life span to be random and Sundaresan and Zapatero (1997) who investigate the optimal date of retirement if the individual enjoys an accumulation period before receiving a one time retirement payment. These papers assume either defined contribution or defined benefits plans; they also assume that pensions are paid by firms and not by individuals. Our analysis emphasizes the importance of labor flexibility, habits, liquidity constraints and the presence of two distinct periods during an individual's life-cycle. Our general setting identifies new features of optimal consumption, labor/leisure and portfolio policies. As in the prior literature on habits we find that current consumption depends on past consumption rates. With labor flexibility it will also depend on the current wage rate as long as the utility function is nonadditive in consumption and leisure. More importantly though, we find that past wages also have an impact when the utility is nonadditive. This effect arises because the individual seeks to sustain historical consumption rates which are influenced by past wages. This dependence on past wages shows up even during retirement. The optimal investment policy reveals a complexity of effects at play. To provide a flavor of the results let us focus on the optimal portfolio during the accumulation period of the life-cycle. This policy decomposes into two parts, one which finances consumption during the accumulation period View the MathML source and the other one which finances retirement consumption View the MathML source. The accumulation period portfolio View the MathML source reflects, in particular, the individual's ability to purchase leisure at a cost which fluctuates over time. This aspect alone motivates two hedging terms which complement more traditional demand components. The retirement planning portfolio View the MathML source is comprised of a mean–variance component and several intertemporal hedging demands. One of these is motivated by the stochastic nature of the wage rate. This is especially surprising because the retirement portfolio is designed for the sole purpose of financing consumption during retirement, a phase of the life-cycle during which the individual does not work. The reasons for this effect can be traced to the history dependence implied by habits and the nonseparability of consumption and labor decisions. Our model can be used to analyze the impact of a variety of factors on the consumption and labor choices. We focus more specifically on two questions: the impact of a regulatory decision postponing retirement and the effect of habits relative to a model with time separable preferences. Our investigations show the complexity of these effects. Take, for instance, the case of a regulation postponing retirement. When consumption and leisure are gross complements we find that the individual will work less during his/her working life and consume more except during the newly added work period. Investment behavior during retirement could become more aggressive. This is the case in a setting with a single risky asset, a deterministic opportunity set and under the additional assumption of decreasing absolute risk aversion. Failure of one or more of these assumptions can, however, lead to different behavior. The inability to borrow against future labor income is a fundamental aspect of life-cycle decisions. It imposes a state-by-state constraint on choices since liquid wealth (the value of consumption net of labor income) must be nonnegative at all times. Studies of this constraint, in the context of the standard model, can be found in He and Pages (1993), El Karoui and Jeanblanc-Picque (1998) and Detemple and Serrat (2003). Here, we conduct a preliminary analysis of its impact in our more general setting with labor flexibility and habit formation. We find that the liquidity constraint affects the optimal policies through several channels. As in the standard model with a single consumption good, it reduces the cost of consumption at times at which the liquidity constraint binds. Ceteris paribus this promotes a contemporaneous increase in consumption, thereby improving liquidity. With labor flexibility the constraint simultaneously reduces the cost of leisure. Taken in isolation this effect promotes a decrease in contemporaneous labor (increase in leisure) which will also improve liquidity. In practice both effects materialize simultaneously at times of a liquidity crunch and will combine to ensure that total expenditures on consumption and leisure are increased so as to maintain a nonnegative liquidity. As in the standard model, these adjustments are permanent: the liquidity multiplier is a nondecreasing process which will reduce the cost of consumption and leisure at all times following the liquidity crunch. In addition, we also find an effect related to habit formation. Even if the constraint is inactive at a particular date, the possibility of it binding in the future reduces the immediate cost of consumption (but not the cost of leisure). This follows because the marginal cost of consumption reflects the marginal disutility associated with a variation in future standards of living, which is affected by the possibility of a binding constraint in the future. A positive probability of a future liquidity crunch will then reduce the current cost of consumption (taking the current liquidity multiplier as given) and promote an immediate increase in consumption. With habits individual consumption anticipates on future events, in particular those associated with liquidity shortages. Section 2 presents the model. In Section 3 we describe the consumption and labor supply policies. Section 4 presents the optimal portfolio. The impact of a change in the mandatory retirement date is studied in Section 5. Section 6 focuses on the effects of habits. Section 7 introduces a liquidity constraint. Conclusions follow. Proofs are collected in the appendices.

In this paper we have explored the effects of habit formation, labor flexibility and retirement on the life-cycle consumption–labor pattern of an individual. The generality of our setting has led us to identify new effects generated by the combination of some of these factors. For instance we have shown that utilities that exhibit habit formation and consumption–leisure complementarities induce an impact of past wages on the consumption of retirees. Likewise, complementarity between consumption and leisure implies that the portfolio financing accumulation consumption will contain a hedging demand against fluctuations in wages. These new insights complement earlier results of Bodie et al. (1992) and illustrate the importance of accounting for phenomena such as habits and consumption–leisure complementarities for the purpose of formulating life-cycle investment plans. Some of the results in our paper can be extended to a more general class of preferences with strictly positive marginal utility of leisure when idle (i.e. l=1). This generalization, which can be found in Bodie et al. (2000), is of importance for the purpose of developing parametric examples that admit more explicit solutions. The preliminary analysis of liquidity constraints that we provided is also worth developing further. Our consumption–leisure formulas have documented the impact of the constraint through multiple channels, and structural effects on the optimal portfolio have been sketched. A detailed analysis of the magnitudes and the directions of these effects would be of interest. For this purpose it would also be useful to focus on parametric examples which admit more detailed characterizations