Does consumer behavior exhibit time inconsistency? This is an essential, yet difficult question to answer. Since the pioneering contribution of Samuelson (1937), it has become a standard assumption in dynamic economics models that consumers have an exponential time discount function, {1, ?, ?2, ...}, which implies that consumer behavior is time consistent. A significant body of evidence in experimental psychology and economics literature, however, suggests that consumers discount the future hyperbolically, not exponentially. The essential feature of hyperbolic discounting is that consumers are time inconsistent. In the last decade, a particular kind of hyperbolic discounting, the quasi-hyperbolic discount function, {1, ??,??2, ...}, has been widely studied due to its analytic simplicity.1 Many researchers have applied this discount function to explain various economic anomalies, such as procrastination, retirement, addiction and credit card borrowing.2 This paper also adopts this formulation, which shall be simply referred to as hyperbolic discounting in later discussion.

The recent use of hyperbolic discounting has been criticized for lack of convincing empirical evidence.3 An ideal test is to compare consumers’ long-run plans with their later actions, which will be consistent for exponential consumers but inconsistent for hyperbolic consumers. In the real world, it is difficult to track long-run plans or later actions — especially long-run plans. This paper examines time inconsistency using a large-scale randomized experiment in the credit card market, with which we have a unique opportunity to conduct a reasonably good test. In the experiment, 600,000 consumers were each randomly assigned to one of six different groups, denoted as Market Cells A to F, which were mailed six different credit card offers. The six offers had different introductory interest rates and different durations: Market Cell A (4.9% for 6 months), B (5.9% for 6 months), C (6.9% for 6 months), D (7.9% for 6 months), E (6.9% for 9 months) and F (7.9% for 12 months). All other characteristics of the solicitations were identical across the six market cells. Consumer responses and subsequent usage of respondents for 24 months were observed.

One advantage of this experiment is that the 600,000 subjects do not change their behavior due to their participation in the experiment; indeed, they do not even know that they are part of an experiment. A second advantage of this experiment is that consumer long-run plans can be inferred from their actions. Consumer plans are identified from their responses to different offers, such as A (4.9% for 6 months) and F (7.9% for 12 months). For example, if the consumers who receive the short introductory offer (A) are more likely to accept the credit card than those who receive the longer introductory offer (F) — and, given the randomized experimental treatment, the two groups may be viewed as identical — it implies that the consumers expect their credit card debt to be short-lived. For purposes of inferring experimental subjects’ long-run plans, actions speak much louder than words. A third advantage of this experiment is that the number of experimental subjects (600,000 consumers solicited, and more than 5,000 consumers accepting the solicitation) is quite large, ensuring that the inferences drawn will be precise. Combining the subjects’ inferred plans with their later actions, we have a unique opportunity to test for time consistency.

There are two phenomena in this dataset suggestive of time inconsistency. First, significantly more consumers in Market Cell A are found to accept their offers than in Market Cell F. This ex ante preference becomes puzzling after observing that respondents, ex post, keep on borrowing on this card well after introductory periods. We will show in a later section that respondents in Market Cell A would pay less interest if their cards were repriced as offer F. Why do not all their counterparts in Market Cell F accept the F offer? We term the first puzzle as “rank reversal.” Second, consumer switching behavior is not consistent over time. The majority of respondents (60%) stay with this card after the introductory period, and their debts remain at the same level as when they accepted this card. Given the same debt level, it should be worthwhile to switch a second time since it was optimal to accept this offer before. Obviously, there would be no puzzle if respondents did not receive new low-rate solicitations from other credit card issuers after the end of the introductory period. However, the number of solicitations averaged at least three per qualified household per month during the sample period. A typical solicitation from the observed issuer (and other credit card issuers contemporaneously) included a 5.9% introductory interest rate for 6 months. 96% of the respondents remain credit-worthy after 6 months, which will be discussed in greater detail in section 3.

At least two explanations are possible for consumer behavior that on the surface appears to be time inconsistent. First, consumers may behave in a time inconsistent fashion because they have hyperbolic time preferences. Hyperbolic consumers have a much higher discount rate in the short run than in the long run. Therefore their credit card choice, which is largely determined by shortrun benefit, may not be optimal from the long-run perspective. Second, consumers are subject to random shocks, the ex post realizations of which may generate divergences between consumers’ initial plans and later actions, even if their preferences are time consistent.

In this paper, we examine the validity of both hypotheses. To build up a basic intuition, we analyze a multi-period credit card choice model without uncertainty. The simple model shows that exponential consumers will never exhibit “rank reversal”. Exponential agents always prefer an offer requiring less interest payment. This is due to their time consistency, which makes their short-run choice (credit card choice) also optimal from the long-run perspective (later interest payment). However, “rank reversal” is possible for hyperbolic consumers. There are two kinds of hyperbolic preferences which have been widely studied in the literature: sophisticated and naive. Our studies show that both versions are able to explain “rank reversal”, even though the underlying economic stories are different. A sophisticated hyperbolic consumer who recognizes her time inconsistency problem would like to precommit to avoid overspending in the future. Accepting a shorter introductory offer, rather than a longer one, serves as a commitment device, even though she would pay less interest if she accepted the longer offer. A naive hyperbolic consumer, however, trades a longer offer for a shorter one because she underestimates the amount she will borrow in the future. This underestimation is due to the fact that she naively believes that her future selves will be as patient as she desires now.

To explore the possibility of explaining behavior with random shocks, we develop a dynamic model which incorporates three important random processes. First, consumer income has both persistent and transitory shocks. Second, receiving new introductory offers is probabilistic. Third, accepting a new offer causes the consumer to incur a random switching cost. A realistic dynamic model is required because some researchers argue that exponential discounting can explain anomalies if “even a small degree of” uncertainty is incorporated,For example Fernandez-Villaverde (2002). which we show is not necessarily the case here.

We find that an exponential model still cannot reconcile respondents’ continued borrowing and preference for the shorter offer A, even with random shocks. The intuition for the failure is that the behavioral discrepancy observed is not for some individuals but for a large group of consumers. An individual exponential consumer may, ex ante, accept an offer that proves, ex post, to be a bad deal based on the realized random shocks. However, a relatively large group of exponential consumers should prefer the offer that on average provides the lowest interest payment. Hyperbolic time preferences are also incorporated into the dynamic model, from which we estimate time preference parameters with a reasonable degree of precision.

Estimation results show that the second puzzle can only be explained by the stochastic nature of switching costs, which are traditionally assumed to be constant for an individual. Our random switching cost appears to be a more realistic treatment, because it captures either fluctuations in free time or fluctuations due to subjective, psychological factors that strongly affect realized switching costs. Under this interpretation, respondents in this experiment accept the offers due to their low realized switching costs at the time of solicitation. However, their mean switching costs are much higher, which can be partially inferred from the low response rate (1%). This high mean will keep the majority of respondents from switching a second time after the introductory period.

The paper is organized as follows. There have been many empirical studies in support of hyperbolic discounting, both from laboratory experiments and field studies, which will be discussed in detail in the following section. In Section 3, the experiment is introduced and the two puzzles are elaborated. Section 4 rigorously defines what we call “rank reversal” and proves that it is impossible in an exponential model with certainty. A simple 3-period model illustrates that “rank reversal” is possible for hyperbolic agents. The dynamic model with uncertainty, which accommodates both exponential and hyperbolic time preferences, is presented in Section 5. The estimation strategy and results are discussed in Section 6. Section 7 concludes.

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