## D91 Intertemporal Consumer Choice; Life Cycle Models and Saving

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#### Document Type

- Working Paper (18)
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#### Keywords

- Financial Literacy (3)
- Labor income risk (3)
- Portfolio choice (3)
- stochastic differential utility (3)
- FBSDE (2)
- Health shocks (2)
- Stochastic mortality risk (2)
- asset pricing (2)
- consumption-portfolio choice (2)
- incomplete markets (2)

- Asset pricing under rational learning about rare disasters : [Version 28 Juli 2011] (2011)
- This paper proposes a new approach for modeling investor fear after rare disasters. The key element is to take into account that investors’ information about fundamentals driving rare downward jumps in the dividend process is not perfect. Bayesian learning implies that beliefs about the likelihood of rare disasters drop to a much more pessimistic level once a disaster has occurred. Such a shift in beliefs can trigger massive declines in price-dividend ratios. Pessimistic beliefs persist for some time. Thus, belief dynamics are a source of apparent excess volatility relative to a rational expectations benchmark. Due to the low frequency of disasters, even an infinitely-lived investor will remain uncertain about the exact probability. Our analysis is conducted in continuous time and offers closed-form solutions for asset prices. We distinguish between rational and adaptive Bayesian learning. Rational learners account for the possibility of future changes in beliefs in determining their demand for risky assets, while adaptive learners take beliefs as given. Thus, risky assets tend to be lower-valued and price-dividend ratios vary less under adaptive versus rational learning for identical priors. Keywords: beliefs, Bayesian learning, controlled diffusions and jump processes, learning about jumps, adaptive learning, rational learning. JEL classification: D83, G11, C11, D91, E21, D81, C61

- Optimal gradual annuitization : quantifying the costs of switching to annuities (2007)
- We compute the optimal dynamic asset allocation policy for a retiree with Epstein-Zin utility. The retiree can decide how much he consumes and how much he invests in stocks, bonds, and annuities. Pricing the annuities we account for asymmetric mortality beliefs and administration expenses. We show that the retiree does not purchase annuities only once but rather several times during retirement (gradual annuitization). We analyze the case in which the retiree is restricted to buy annuities only once and has to perform a (complete or partial) switching strategy. This restriction reduces both the utility and the demand for annuities.

- Planning and financial literacy : how do women fare? (2008)
- Many older US households have done little or no planning for retirement, and there is a substantial population that seems to undersave for retirement. Of particular concern is the relative position of older women, who are more vulnerable to old-age poverty due to their longer longevity. This paper uses data from a special module we devised on planning and financial literacy in the 2004 Health and Retirement Study. It shows that women display much lower levels of financial literacy than the older population as a whole. In addition, women who are less financially literate are also less likely to plan for retirement and be successful planners. These findings have important implications for policy and for programs aimed at fostering financial security at older ages.

- The reaction of consumer spending and debt to tax rebates – evidence from consumer credit data (2008)
- We use a new panel dataset of credit card accounts to analyze how consumer responded to the 2001 Federal income tax rebates. We estimate the monthly response of credit card payments, spending, and debt, exploiting the unique, randomized timing of the rebate disbursement. We find that, on average, consumers initially saved some of the rebate, by increasing their credit card payments and thereby paying down debt. But soon afterwards their spending increased, counter to the canonical Permanent-Income model. Spending rose most for consumers who were initially most likely to be liquidity constrained, whereas debt declined most (so saving rose most) for unconstrained consumers. More generally, the results suggest that there can be important dynamics in consumers’ response to “lumpy” increases in income like tax rebates, working in part through balance sheet (liquidity) mechanisms.

- Life insurance demand under health shock risk : [Version: 7 February 2014] (2014)
- This paper studies the life cycle consumption-investment-insurance problem of a family. The wage earner faces the risk of a health shock that significantly increases his probability of dying. The family can buy term life insurance with realistic features. In particular, the available contracts are long term so that decisions are sticky and can only be revised at significant costs. Furthermore, a revision is only possible as long as the insured person is healthy. A second important and realistic feature of our model is that the labor income of the wage earner is unspanned. We document that the combination of unspanned labor income and the stickiness of insurance decisions reduces the insurance demand significantly. This is because an income shock induces the need to reduce the insurance coverage, since premia become less affordable. Since such a reduction is costly and families anticipate these potential costs, they buy less protection at all ages. In particular, young families stay away from life insurance markets altogether.

- Consumption habits and humps : [Version 23 June 2013] (2013)
- We show that the optimal consumption of an individual over the life cycle can have the hump shape (inverted U-shape) observed empirically if the preferences of the individual exhibit internal habit formation. In the absence of habit formation, an impatient individual would prefer a decreasing consumption path over life. However, because of habit formation, a high initial consumption would lead to high required consumption in the future. To cover the future required consumption, wealth is set aside, but the necessary amount decreases with age which allows consumption to increase in the early part of life. At some age, the impatience outweighs the habit concerns so that consumption starts to decrease. We derive the optimal consumption strategy in closed form, deduce sufficient conditions for the presence of a consumption hump, and characterize the age at which the hump occurs. Numerical examples illustrate our findings. We show that our model calibrates well to U.S. consumption data from the Consumer Expenditure Survey.

- Stochastic differential utility as the continuous-time limit of recursive utility (2013)
- We establish a convergence theorem that shows that discrete-time recursive utility, as developed by Kreps and Porteus (1978), converges to stochastic differential utility, as introduced by Dufffie and Epstein (1992), in the continuous-time limit of vanishing grid size.

- Consumption and wage humps in a life-cycle model with education : [version 11 june 2014] (2014)
- he observed hump-shaped life-cycle pattern in individuals' consumption cannot be explained by the classical consumption-savings model. We explicitly solve a model with utility of both consumption and leisure and with educational decisions affecting future wages. We show optimal consumption is hump shaped and determine the peak age. The hump results from consumption and leisure being substitutes and from the implicit price of leisure being decreasing over time; more leisure means less education, which lowers future wages, and the present value of foregone wages decreases with age. Consumption is hump shaped whether the wage is hump shaped or increasing over life.

- Asset pricing and consumption-portfolio choice with recursive utility and unspanned risk : [version 4 august 2014] (2014)
- We study consumption-portfolio and asset pricing frameworks with recursive preferences and unspanned risk. We show that in both cases, portfolio choice and asset pricing, the value function of the investor/ representative agent can be characterized by a specific semilinear partial differential equation. To date, the solution to this equation has mostly been approximated by Campbell-Shiller techniques, without addressing general issues of existence and uniqueness. We develop a novel approach that rigorously constructs the solution by a fixed point argument. We prove that under regularity conditions a solution exists and establish a fast and accurate numerical method to solve consumption-portfolio and asset pricing problems with recursive preferences and unspanned risk. Our setting is not restricted to affine asset price dynamics. Numerical examples illustrate our approach.

- Consumption-investment problems with stochastic mortality risk : [version 3 march 2014] (2014)
- I numerically solve realistically calibrated life cycle consumption-investment problems in continuous time featuring stochastic mortality risk driven by jumps, unspanned labor income as well as short-sale and liquidity constraints and a simple insurance. I compare models with deterministic and stochastic hazard rate of death to a model without mortality risk. Mortality risk has only minor effects on the optimal controls early in the life cycle but it becomes crucial in later years. A diffusive component in the hazard rate of death has no significant impact, whereas a jump component is desired by the agent and influences optimal controls and wealth evolution. The insurance is used to ensure optimal bequest such that there is no accidental bequest. In the absence of the insurance, the biggest part of bequest is accidental.