Optimal Consumption Policies Research Articles

Testing a theory of strategic multi-product choice

This paper tests a theory of strategic multi-product choice (SMPC) using empirical evidence from a large-scale choice experiment, two smaller longitudinal choice experiments, and multi-market panel data. Multi-product choice involves two stages. In the first stage, the consumer chooses a set of substitutable products, where “set” refers to both the variety of alternatives and the quantities of each. In the second stage, the set is consumed. Assuming consumers are strategic, their consumption decisions will consider both the utility of whichever product is selected for consumption and the expected utility (i.e., value) of the set that remains. SMPC therefore requires a dynamic model. We test two such dynamic models in this paper. These models are derived from a basic random utility framework with a stochastic error term for the utility of each product alternative at the moment of consumption. Despite maintaining state variables for the quantity of every alternative, these SMPC dynamic models offer both a value function and optimal consumption policy in closed form. These structures allow us to test for strategic consumption in the second stage and for optimality of the choice sets selected in the first stage. Data from the large-scale choice experiment and the smaller longitudinal choice experiments support strategic consumer decision-making, consistent with SMPC theory. SMPC theory further predicts that the amount of variety consumers select will be higher for lower consumption rates and lower for higher consumption rates. Evidence from panel data of yogurt purchases supports this prediction. While we find that consumption choices are consistent with SMPC theory, they are not consistent with alternative explanations such as variety seeking or diversification bias. Viewed in its entirety, the empirical evidence presented in this paper confirms that both the choice set selected and the way it is consumed are consistent with dynamic models of future preference uncertainty.

Optimal consumption for recursive preferences with local substitution — the case of certainty

We characterize optimal consumption policies in a recursive intertemporal utility framework with local substitution. We establish existence, uniqueness, and a version of the Kuhn–Tucker theorem. The structure of optimal consumption plans is described explicitly for a large class of aggregators. For Epstein-Zin preferences, the solution coincides with the solution of the corresponding time-additive preferences.

Optimal Consumption Under a Habit-Formation Constraint: The Deterministic Case

We formulate and solve a deterministic optimal consumption problem to maximize the discounted constant relative risk aversion utility of an individual’s consumption-to-habit process assuming they only invest in a riskless market and that they are unwilling to consume at a rate below a certain proportion of their consumption habit. Increasing increases the degree of addictiveness of habit formation, with (respectively, ) corresponding to nonaddictive (respectively, completely addictive) model. We derive the optimal consumption policies explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient individuals (or, equivalently, those with more addictive habits) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (or, equivalently, those with less addictive habits) 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 more risk seeking (that is, less risk averse). Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.

Integrating distributed energy resources

The rapid growth of the behind-the-meter (BTM) distributed generation has led to initiatives to reform the net energy metering (NEM) policies to address pressing concerns of rising electricity bills, fairness of cost allocation, and the long-term growth of distributed energy resources. This article presents an analytical framework for the optimal prosumer consumption decision using an inclusive NEM X tariff model that covers existing and proposed NEM tariff designs. The structure of the optimal consumption policy lends itself to near closed-form optimal solutions suitable for practical energy management systems that are responsive to stochastic BTM generation and dynamic pricing. The short and long-run performance of NEM and feed-in tariffs (FiT) are considered under a sequential rate-setting decision process. Also presented are numerical results that characterize social welfare distributions, cross-subsidies, and long-run solar adoption performance for selected NEM and FiT policy designs.

Robust consumption and portfolio choice with derivatives trading

This paper analyzes the optimal consumption and portfolio choice problem for an ambiguity-averse investor with recursive preferences. The investor has access to both the stock and derivatives markets. The stock price process follows a stochastic volatility jump-diffusion model and the investor can have different levels of ambiguity about diffusion and jump risks, respectively. We obtain an exact analytical solution for investors with unit elasticity of intertemporal substitution of consumption and an approximate solution otherwise. We find that the optimal consumption policy is more sensitive to ambiguity aversion with respect to diffusion risks than the jump risk. The optimal exposures to diffusion and jump risks are significantly affected by the corresponding ambiguity aversions in the complete market; however, the optimal stock investment is relatively insensitive to jump misspecification in the incomplete market. We also show that taking into consideration ambiguity aversion to diffusion risks and participating in the derivatives market are essential to reduce the potential welfare loss, while the impact of jump misspecification is marginal.

Optimal Investment and Consumption under a Habit-Formation Constraint

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 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.

Optimal asset allocation, consumption and retirement time with the variation in habitual persistence

In this paper, we study the individual's optimal asset allocation, consumption and retirement time under habitual persistence. To depict the phenomenon that the individual feels equally satisfied with a lower habitual level and is more reluctant to change the habitual level after retirement, we assume that both the level and the sensitivity of the habitual consumption decline at the time of retirement. We establish the concise form of the habitual evolutions, and obtain the optimal retirement time and consumption policy based on martingale and duality methods. The optimal consumption experiences a sharp decline at retirement, but the excess consumption raises because of the reduced sensitivity of the habitual level. This result is consistent with the evidence observed in the “retirement consumption puzzle”. Particularly, the optimal retirement and consumption policies are balanced between the wealth effect and the habitual effect. Larger wealth increases consumption, and larger growth inertia (sensitivity) of the habitual level decreases consumption and brings forward the retirement time.

Optimal Consumption Under a Habit-Formation Constraint

We propose a new optimal consumption model in which the degree of addictiveness of habit formation is directly controlled through a constraint on admissible consumption. In particular, we assume that the individual is unwilling to consume at a rate below a certain proportion 0<α≤1 of her consumption habit, which is the exponentially-weighted average of past consumption rates. α=1 prohibits the habit process to decrease and corresponds to the completely addictive model. α=0 makes the habit-formation constraint moot and corresponds to the non-addictive model. 0<α<1 leads to partially addictive models, with the level of addictiveness increasing with α. In contrast to the existing habit-formation literature, our constraint cannot be incorporated in the objective function through infinite marginal utility, and alters the analysis of the corresponding control problem. 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 individual's consumption-to-habit process subject to the said habit-formation constraint. We derive the optimal consumption policies explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient individuals (or, equivalently, those with more addictive habits) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (or, equivalently, those with less addictive habits) 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 more risk seeking (that is, less risk averse). Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.

Optimal contribution rate of PAYGO pension

ABSTRACT In this paper, we study the optimal contribution rate of pay-as-you-go (PAYGO) pension under a Nash equilibrium between the participants and the government. Given the fixed contribution rate, the participants of different cohorts choose optimal consumption and asset allocation policies to achieve their objectives. Using the variational method, we derive the closed-form solution. The value function of the participants is monotonous with respect to the PAYGO contribution rate under constant population growth rate. As such, we modify the ‘Samuelson-Aaron’ criterion that the preference between PAYGO and fully funded pensions depends not only on the demography, investment and salary parameters but also on the age of the cohort. Moreover, we establish the critical age to separate the preference and the ex-post admissible scope for the contribution rate adjustment. As a central planner, the government is fully aware of the participants' optimal feedback functions. The objective of the government is to maximize the overall utility of the participants weighted by each cohort's population. For a shrinking population, the fully funded scheme is more preferable and only the elderly cohorts prefer PAYGO scheme. However, taking into account the larger political power of the elderly cohorts in the objective function, the optimal contribution rate raises slightly.

A Stochastic Control Model of Investment and Consumption with Applications to Financial Economics

This study considers a stochastic control model in which an economic unit has productive capital and liabilities in the form of debt. The worth of capital changes over time through investment and random Brownian fluctuations in the unit price of capital. Income from production is also subject to the random Brownian fluctuations. The existence of the solutions to the associated Hamilton Jacobi Bellman equation for this model is established and the optimal policies are characterized. The optimal advertising rate as a function of the market share, the optimal consumption rate and the fraction of the wealth invested in stock at any time are obtained. The worth of the capital and the optimal consumption policy are derived for the stochastic optimal investment consumption model associated with the Hamilton Jacobi Bellman equation. Analysis and numerical simulations are then presented.

Optimal Portfolio and Consumption Policies with Stochastic Volatility

Optimal asset allocation and consumption policies have been important issues in finance in the past decades. We study these issues under constant relative risk aversion (CRRA) utility functions in a general setting: stochastic volatility, incomplete markets and finite investment horizons. So far, numerical computation has been the main method for obtaining solutions in this general setting. We present a closed-form approximate solution for this dynamic optimization problem. We show that our theoretical predictions are in good agreement with numerical results and our approximation error is even smaller than the parameter-estimation errors in underlying dynamics.

Present-Bias and the Value of Sophistication

This paper develops a dynamic wealth management model for risk-averse investors displaying present-bias in the form of hyperbolic discounting. The investor chooses an optimal consumption policy and allocates her funds between a risk-free asset, a traded liquid asset, and a non-traded illiquid asset. We characterize these policies for both sophisticated and naive present-biased investors. There are three results. First, sophisticated investors over-consume more than their naive counterparts if and only if their coefficient of relative risk-aversion is smaller than one. As a result, sophistication is welfare reducing (increasing) when risk-aversion is low (high). Second, increasing asset illiquidity always benefits the sophisticated investor more than the naive investor. Thus, the welfare gap between sophisticated and naive investors is increasing in the proxy for asset illiquidity. Finally, present-biased investors accumulate a larger share of their wealth in the non-traded illiquid asset than in the traded risky stock compared to the neoclassical exponential discounter investor. As a consequence, from the perspective of present-biased investors, the equity premium puzzle (1985) and the private equity puzzle (2002) are two sides of the same coin.

Optimal power consumption analysis of a load-dependent server activation policy for a data service center

This paper analyzes a data service center with optimal power consumption policy by using an M/M/s/∞ queueing system with queue-dependent heterogeneous and unreliable servers. This load-dependent server activation policy is that the servers deploying in a data service center were divided into fundamental and reserved servers. The fundamental servers are always available and reserved servers are switched off initially. The decision of switching the reserved server on or off is depending on the load level of system. Using the matrix-analytical approach, we derive the formulae for computing the rate matrix and obtain the stationary probabilities. Except for investigating the effects of system parameters on the various system characteristics, we also propose a mathematical model to determine the optimal threshold values of switching on/off reserved servers so as minimize the power consumption on a given expected waiting time.

Optimal consumption analysis for a stochastic growth model with technological shocks

AbstractAn optimal consumption problem of maximizing the expected discounted value of utility is discussed for an economic growth model with the random technological shocks. Applying the technique of dynamic programming principle, we derive the Hamilton‐Jacobi‐Bellman equation corresponding to the optimization problem and prove that the value function is a unique viscosity solution to the equation. Moreover, the optimal consumption policy is given in a feedback form under weak assumptions.

Asset management with endogenous withdrawals under a drawdown constraint

Asset preservation is a major concern for foundations that are hostile to large wealth downfalls. Implications for optimal consumption and investment policies are explored in a dynamic setting where wealth is restrained from falling below a fraction of its all-time high. Risky investment regulates wealth growth and mitigates the ratchet effect of the constraint, and may decrease as wealth approaches its maximum. The correspondence found between habit formation over consumption and wealth ratcheting provides a rational explanation for the proliferation of capital protection oriented funds.

Self-Sustainability of Energy Harvesting Systems: Concept, Analysis, and Design

Ambient energy harvesting is touted as a low cost solution to prolong the life of low-powered devices, reduce the carbon footprint, and make the system self-sustainable. Most research to date has focused either on the physical aspects of the energy conversion process or on the optimal consumption policies of the harvested energy at the system level. However, although intuitively understood, to the best of our knowledge, the idea of self-sustainability is yet to be systematically studied as a performance metric. In this paper, we provide a mathematical definition of the concept of self-sustainability of an energy harvesting system, based on the complementary idea of eventual energy outage, rather than the usual energy outage. In particular, we analyze a harvest-store-consume system with infinite battery capacity, stochastic energy arrivals, and fixed energy consumption rate. Using the random walk theory, we identify the necessary condition for the system to be self-sustainable. General formulas are given for the self-sustainability probability in the form of integral equations. Since these integral equations are difficult to solve analytically, an exponential upper bound for eventual energy outage is given using martingales. This bound guarantees that the eventual energy outage can be made arbitrarily small simply by increasing the initial battery energy. We then give an asymptotic formula; and for the special case when the energy arrival follows a Poisson process, we are also able to find an exact formula for the eventual energy outage probability. We also show that the harvest-store-consume system is mathematically equivalent to a GI / ${G}$ /1 queueing system, which allows us to easily find the energy outage probability, in case the necessary condition for self-sustainability is violated. Monte-Carlo simulations verify our analysis.

Robust Consumption and Portfolio Policies When Asset Prices Can Jump

We study the consumption-portfolio allocation problem in continuous time when asset prices follow Levy processes and the investor is concerned about potential model misspecification. We derive optimal consumption and portfolio policies that are robust to uncertainty about the hard-to-estimate drift rate, jump intensity and jump size parameters. We also provide a semi-closed form formula for the detection-error probability and compare various portfolio holding strategies, including robust and non-robust policies. Our quantitative analysis shows that ignoring uncertainty leads to significant wealth loss for the investor.

Optimal consumption and portfolio selection problems under loss aversion with downside consumption constraints

This paper investigates continuous-time optimal portfolio and consumption problems under loss aversion in an infinite horizon. The investor’s goal is to choose optimal portfolio and consumption policies to maximize total discounted S-shaped utility from consumption. The consumption rate process is subject to a downside constraint. The optimal consumption and portfolio policies are obtained through the martingale method and replication technique. Numerical results indicate the differences between the loss averse investor and the constant relative risk averse (CRRA) investor on the optimal consumption and portfolio policies: the loss averse investor likes consuming more money but exposing less to risk than that of the CRRA investor, and the optimal wealth, as a function of state price density, drops faster for the CRRA investor than that for the loss averse investor.

Parameter Uncertainty: The Missing Piece of the Liquidity Premium Puzzle?

I analyze a dynamic investment problem with stochastic transaction cost and parameter uncertainty. I solve the problem numerically and obtain the optimal consumption and investment policy and the least-favorable transaction cost process. Using reasonable parameter values, I confirm the liquidity premium puzzle, i.e., the representative agent model (without robustness) produces a liquidity premium which is by a magnitude lower than the empirically observed value. I show that my model with robust investors generates an additional liquidity premium component of 0.05%-0.10% (depending on the level of robustness) for the first 1% proportional transaction cost, and thus it provides a partial explaination to the liquidity premium puzzle. Additionally, I provide a novel non-recursive representation of discrete-time robust dynamic asset allocation problems with transaction cost, and I develop a numerical technique to efficiently solve such investment problems.

OPTIMAL PORTFOLIO AND CONSUMPTION MODELS UNDER LOSS AVERSION IN INFINITE TIME HORIZON

This paper investigates continuous-time optimal portfolio and consumption problems under loss aversion in an infinite time horizon. The investor's goal is to choose the optimal portfolio and consumption policies to maximize total discounted S-shaped utility from consumption. The problems are solved under two different situations respectively for the reference level: exogenous or endogenous. For the case of exogenous reference level, which is independent of the consumption policy, the optimal consumption policy and wealth process are obtained through the martingale method and replicating technique. For the case of endogenous reference level, which is related to the past actual consumption, the optimization problem with stochastic reference level is first transformed into an equivalent optimization problem with zero reference point, the corresponding relationship between them is proved, and then the relevant optimal consumption policy and wealth process are also obtained. When the investment opportunity sets are constants, the closed-form solutions of the portfolio and consumption policies are derived under two different situations respectively.