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Register a new userA consumption-investment model with state-dependent lower bound constraint on consumption
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This paper considers a life-time consumption-investment problem under the Black-Scholes framework, where the investors consumption rate is subject to a lower bound constraint that linearly depends on the investors wealth. Due to the state-dependent control constraint, the standard stochastic control theory cannot be directly applied to our problem. We overcome this obstacle by examining an equivalent problem that does not impose state-dependent control constraint. It is shown that the value function is a third-order continuously differentiable function by using differential equation approaches. The feedback form optimal consumption and investment strategies are given. According to our findings, if the investor is more concerned with long-term consumption than short-term consumption, then she should, regardless of her financial condition, always consume as few as possible; otherwise, her optimal consumption strategy is state-dependent: consuming optimally when her financial condition is good, and consuming at the lowest possible rate when her financial situation is bad.
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A continuous-time consumption-investment model with constraint is considered for a small investor whose decisions are the consumption rate and the allocation of wealth to a risk-free and a risky asset with logarithmic Brownian motion fluctuations. The consumption rate is subject to an upper bound constraint which linearly depends on the investors wealth and bankruptcy is prohibited. The investors objective is to maximize total expected discounted utility of consumption over an infinite trading horizon. It is shown that the value function is (second order) smooth everywhere but a unique possibility of (known) exception point and the optimal consumption-investment strategy is provided in a closed feedback form of wealth, which in contrast to the existing work does not involve the value function. According to this model, an investor should take the same optimal investment strategy as in Mertons model regardless his financial situation. By contrast, the optimal consumption strategy does depend on the investors financial situation: he should use a similar consumption strategy as in Mertons model when he is in a bad situation, and consume as much as possible when he is in a good situation.
This paper studies an optimal investment and consumption problem with heterogeneous consumption of basic and luxury goods, together with the choice of time for retirement. The utility for luxury goods is not necessarily a concave function. The optimal heterogeneous consumption strategies for a class of non-homothetic utility maximizer are shown to consume only basic goods when the wealth is small, to consume basic goods and make savings when the wealth is intermediate, and to consume small portion in basic goods and large portion in luxury goods when the wealth is large. The optimal retirement policy is shown to be both universal, in the sense that all individuals should retire at the same level of marginal utility that is determined only by income, labor cost, discount factor as well as market parameters, and not universal, in the sense that all individuals can achieve the same marginal utility with different utility and wealth. It is also shown that individuals prefer to retire as time goes by if the marginal labor cost increases faster than that of income. The main tools used in analysing the problem are from PDE and stochastic control theory including viscosity solution, variational inequality and dual transformation.
We propose a new optimal consumption model in which the degree of addictiveness of habit formation is directly controlled through a consumption constraint. In particular, we assume that the individual is unwilling to consume at a rate below a certain proportion $0<alphale1$ of her consumption habit, which is the exponentially-weighted average of past consumption rates. $alpha=1$ prohibits the habit process to decrease and corresponds to the completely addictive model. $alpha=0$ makes the habit-formation constraint moot and corresponds to the non-addictive model. $0<alpha<1$ leads to partially addictive models, with the level of addictiveness increasing with $alpha$. In contrast to the existing habit-formation literature, our constraint cannot be incorporated in the objective function through infinite marginal utility. Assuming that the individual invests in a risk-free market, we formulate and solve an infinite-horizon, deterministic control problem to maximize the discounted CRRA utility of the consumption-to-habit process subject to the habit-formation constraint. Optimal consumption policies are derived explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals consume at the minimum rate if their wealth-to-habit ratio is below a threshold, and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Additionally, we prove that the optimal consumption path is hump-shaped if the initial wealth-to-habit ratio is either: (1) larger than a high threshold; or (2) below a low threshold and the agent is less risk averse. Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.
We extend the result of our earlier study [Angoshtari, Bayraktar, and Young; Optimal consumption under a habit-formation constraint, available at: arXiv:2012.02277, (2020)] to a market setup that includes a risky asset whose price process is a geometric Brownian motion. We formulate an infinite-horizon optimal investment and consumption problem, in which an individual forms a habit based on the exponentially weighted average of her past consumption rate, and in which she invests in a Black-Scholes market. The novelty of our model is in specifying habit formation through a constraint rather than the common approach via the objective function. Specifically, the individual is constrained to consume at a rate higher than a certain proportion $alpha$ of her consumption habit. Our habit-formation model allows for both addictive ($alpha=1$) and nonaddictive ($0<alpha<1$) habits. The optimal investment and consumption policies are derived explicitly in terms of the solution of a system of differential equations with free boundaries, which is analyzed in detail. If the wealth-to-habit ratio is below (resp. above) a critical level $x^*$, the individual consumes at (resp. above) the minimum rate and invests more (resp. less) aggressively in the risky asset. Numerical results show that the addictive habit formation requires significantly more wealth to support the same consumption rate compared to a moderately nonaddictive habit. Furthermore, an individual with a more addictive habit invests less in the risky asset compared to an individual with a less addictive habit but with the same wealth-to-habit ratio and risk aversion, which provides an explanation for the equity-premium puzzle.
We investigate the general structure of optimal investment and consumption with small proportional transaction costs. For a safe asset and a risky asset with general continuous dynamics, traded with random and time-varying but small transaction costs, we derive simple formal asymptotics for the optimal policy and welfare. These reveal the roles of the investors preferences as well as the market and cost dynamics, and also lead to a fully dynamic model for the implied trading volume. In frictionless models that can be solved in closed form, explicit formulas for the leading-order corrections due to small transaction costs are obtained.
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