Prof A Zimper's article in Journal of Economic Theory

The point of departure for this article are two empirical observations. First, people tend to underestimate their survival chances when they are young whereas they overestimate these chances when they are old. Second, people save too little when they are young whereas they save too much when they are old. Intuitively, these age-dependent biases between subjective survival beliefs and objective survival probabilities look like a good candidate for explaining the observed age-dependent savings patterns. After all, people who misperceive a low (high) survival likelihood should tend to save less (more) for the future than predicted by models that assume rational expectations for which survival beliefs and objective survival probabilities coincide. The above article aims to rigorously formalize this intuition within a model of life-cycle savings decisions.

The main conceptual challenge was the construction of a plausible decision-theoretic model of survival beliefs formation that can replicate the stylized facts about age-dependent survival biases. Recall that the standard literature on survival belief formation either assumes rational expectations, where subjective beliefs are given as objective probabilities, or it assumes Bayesian learning, where subjective survival beliefs converge to their objective counterparts as the individual observes more and more data over the life-cycle. Neither rational expectations nor (standard) Bayesian learning are consistent with the observed survival belief biases. In particular, the observation that older people increasingly overestimate their survival chances is at odds with a standard Bayesian learning model according to which an older age (i.e., more observed data) should result in convergence to objective probabilities. Based on non-additive probabilities about the data-generating process, we construct a model of Bayesian learning under ambiguity. In contrast to the convergence behavior of standard Bayesian learning, our learning model allows for persistent, or even increasing, survival belief biases. Intuitively, under ambiguity the decision maker does not ‘trust’ the observed data anymore whereby such ‘doubts’ might be driven by psychological factors. Importantly, our model generates the stylized biases observed in the data with a minimum of additional parameters compared to the standard model.

Formally, the survival beliefs that are generated through our learning model are given as neo-additive capacities, i.e., non-additive probability measures that stand for a mixture between standard additive beliefs and additional decision weights attached to (i) the best (maximal survival) and (ii) to the worst (immediate death) possible event. To describe the life-cycle savings behavior of a representative agent, we plug these neo-additive survival beliefs into an otherwise standard life-cycle model (i.e., we assume exponential time-discounting and an additive time-separable utility function). Because of the non-additivity of survival beliefs our agent is described as a Choquet expected utility rather than a standard expected utility decision maker (the Choquet integral is the standard mathematical technique of integration with respect to non-additive measures).

As a secondary challenge we had to address the fact that the resulting life-cycle model with ambiguous survival beliefs is dynamically inconsistent, i.e., the future decision maker wants to deviate from a savings plan that is optimal from the ex-ante perspective of the same decision maker. We characterize the resulting savings behavior for the two benchmark cases of naïve versus sophisticated dynamically decision makers. Whereas naive agents are not aware of their dynamic inconsistency, sophisticated agents fully understand their deviating future preferences. As a consequence, sophisticated agents play a game against their future selfs which is solved through backward induction.

Finally, we bring our theoretical model to the data whereby we use survival belief data from the Health and Retirement study. Our quantitative analysis shows that agents with calibrated neo-additive survival beliefs (i) save less than originally planned, (ii) exhibit undersaving at younger ages, and (iii) hold larger amounts of assets in old age than their rational expectations counterparts who correctly assess their survival chances. Our neo-additive life-cycle model can therefore simultaneously accommodate three important empirical findings on household saving behavior. The model performs thereby better for naïve than for sophisticated agents.