About delay aversion

Abstract

In this paper, we study the behaviour of decision makers who show preferences for advancing the timing of future satisfaction. We give two definitions that are representative of this kind of attitude and investigate their implications in (an intertemporal version of) three popular models used in decision theory: the Expected Utility, the Choquet Expected Utility and the MaxMin Expected Utility models. The first definition reveals interesting links with the theory studying the impossibility of aggregating infinite streams of income, while keeping both strong monotonicity and equality among all generations. Our second definition turns out to be a behavioural characterization of what Irving Fisher called impatience. Finally, we make a connection with the notion of domination of one stream of income over another, for all interest rates.

Introduction

“One today is worth two tomorrows”.¹

As an individual’s attitude towards time is crucial in almost all economic problems, it is not surprising that the economic literature studying time preferences is extremely rich and dates back to the 19th century with Böhm-Bawerk’s “The Positive Theory of Capital” (von Böm-Bawerk, 1923). Our research follows the stream of papers initiated by the seminal works of Koopmans (1960) and Diamond (1965), as we focus on a Decision Maker (DM), or Social Planner, who has preferences over positive, bounded sequences. In Section  3, sequences are treated interchangeably as infinite streams of income or consumption. By contrast, in Section  4, we think about them just as streams of income. This difference in interpretation will be clarified below when the main concepts are introduced.

Our main concern is the study of preferences for advancing the time of future satisfaction. This general idea of having an inclination for immediate utility over delayed utility has been given different names in the literature, mostly used as synonyms. We would like to stress straight away, to avoid confusion, that when we are dealing with the concepts of (long-term or short-term) delay aversion, impatience and myopia, we are in fact referring to precise behavioural definitions. All these notions will be formally defined in the main body of the paper, starting from the preferences of the DM over infinite, bounded sequences.

Since the seminal paper of Samuelson (1937), the use of the (exponential) discounting model has been the paradigm for describing such intertemporal tastes. In this model, a stream of income $(x_0,x_1,\dots )$ is evaluated by the utility function:

$$U(x_0,x_1,\dots )=\sum _{t=0}^{\infty }{\delta }^{t}u\left(x_t\right)\text{.}$$

The attitude towards the future is represented by the discount factor $\delta \in (0,1)$. We think that this approach is not totally satisfactory. For instance, it does not allow the concepts of impatience, myopia and delay aversion to be distinguished.

We therefore depart from the traditional analysis and we work with three popular, theoretical models of decision making, adapted to our framework: the Expected Utility model, the Choquet Expected Utility model and the MaxMin Expected Utility model. The idea that these models are suitable to study time preferences can already be found in Marinacci (1998) and Chateauneuf and Ventura (2013). In  Marinacci (1998) the author links the MaxMin Expected Utility model with the concept of patience, whereas in  Chateauneuf and Ventura (2013) the authors make a study of the Choquet Expected Utility model and its connection with impatience and myopia. Even if these models are mostly used to analyse decisions under uncertainty in the literature, we want to stress that in our framework no uncertainty is involved. These models are used as flexible tools in order to generalize the notion of the weights that a DM uses in order to evaluate different points in time.

The main contribution of the paper is represented by the introduction of two novel definitions that represent preferences for advancing the time of future satisfaction and their mathematical representation in the three models cited above. In the rest of the introduction we describe these two notions and we briefly present our main findings.

Recent work by Benoît and Ok (2007) describes and characterizes situations when one DM is more delay averse (in some precise sense) than another. Starting from their paper, we define the concept of long-term delay aversion, which is compatible with their work. Suppose that an agent has to choose between two extra payments of, say, $1000 and $10,000. The $1000 are paid within a month whereas the $10,000 will be paid much later. We believe that if the second and bigger payment is made sufficiently far in the future, then the agent will choose the first one. More formally, let us consider a DM who is supposed to receive two additional amounts of income or consumption good, $a$ and $b$, with $a\le b$, delivered respectively in periods $n_0$ and $n$ with $n_0<n$. Then she will be long-term delay averse if she prefers $a$ over $b$ provided that $n$ is sufficiently big. The use of the adjective “long-term” underlines the fact that $n$, the period of time in which the bigger extra amount is given, can be a very large number. We want to emphasize that there is no uncertainty regarding the date at which the payment $b$ is made. The preference for the sooner payment $a$ derives purely from the fact that the DM has to wait too much, according to her intertemporal tastes, in order to obtain $b$.

The characterizations of long-term delay aversion for the models considered yield interesting features. First, as long-term delay aversion proves to be a very weak notion, it allows a separation between tastes and evaluation of time to be made. In fact we proved that long-term delay aversion depends only upon the “weights” that the DM attaches to periods (or subsets of periods) of time, as long as she has a strictly increasing and continuous utility function.

Let us now turn to a social planner who has preferences over flows of income in which each period represents the wealth of one generation. The usual discounting model implies very demanding notions of intergenerational inequality, analysed in Chateauneuf and Ventura (2013), called impatience and myopia. Impatience states that an increase in wealth for a finite number of generations, with all the future generations receiving zero income afterwards, is preferred to the original income stream as soon as there are enough generations which are better off. Myopia represents the following notion: suppose that one stream of income is strictly preferred to another. Let us further assume that a fixed, arbitrarily-large amount of extra income is added to the second stream for all but a finite number of generations. Then myopia says that the preference order is not reversed whenever this increase in wealth happens to start for a generation sufficiently far into the future.

One of the main reasons that inspired us to define the concept of long-term delay aversion is exactly the aim of proposing a notion that is weaker than the two described above. A good feature of the Choquet, MaxMin and Expected Utility models is that they allow to represent preferences exhibiting long-term delay aversion without neither myopia nor impatience. Such a property is not shared by the discounting model. We show this in Proposition 3.6 and Example 3.1 in Section  3.3.

Finally, and more interestingly, our definition is linked to the literature that studies the contrast between intergenerational inequalities and Pareto optimality. Basu and Mitra (2003) have shown that there is no aggregating function which satisfies strong monotonicity and equality among generations. In fact, we prove that for the Expected Utility and the MaxMin Expected Utility models, long-term delay aversion is equivalent to the strong monotonicity of preferences, while for the Choquet Expected Utility model strong monotonicity is a necessary condition. Since long-term delay aversion is clearly incompatible with treating all generations equally, this result provides an insight about why strong monotonicity of preferences and equality among generations are contradictory.

The other definition that we propose comes from a straightforward observation. An agent should show preferences for advancing time of future satisfaction if, when dealing with an extra payment that can be made in two consecutive periods, she always chooses the one made at the earlier date. We call this notion short-term delay aversion. The word “short-term” is used to underline precisely the fact that we are considering two consecutive dates.

Short-term delay aversion is a demanding notion in terms of preferences of the DM. In discussing this concept, we have in mind that preferences are defined only with respect to streams of income, and not of consumption. While it is not plausible that a DM with an uneven distribution of consumption over time would always prefer to consume at earlier dates, we believe that, when facing two extra monetary payments, an agent should invariably choose the one made at the earlier date. As expected, short-term delay aversion implies both properties of the weights that the DM attaches to periods of time and of the (marginal) utility function. If we focus on the constant marginal utility of wealth, then short-term delay aversion becomes equivalent to discounting (attaching decreasing weights to periods of time).

Short-term delay aversion turns out to be a behavioural counterpart to the definition of preferences for advancing the time of future satisfaction given by Fisher (1930). We call Fisher’s definition F-impatience. According to Fisher, an individual is F-impatient if she has a marginal rate of intertemporal substitution that is always greater than 1 (see footnote 18 p. 82 of  Benoît and Ok, 2007). This is exactly our characterization of short-term delay aversion for the Expected Utility model.

Linked to the short-term delay aversion, we propose the concept of temporal domination, which was studied recently in a paper by Foster and Mitra (2003) who are interested in characterizing when one cash flow dominates another at all interest rates. We say that one stream of income temporally dominates another if the sum of the first $k$ cash flows of the former is always higher than the sum of the first $k$ cash flows of the latter. We find that a DM is short-term delay averse if and only if whenever one sequence temporally dominates another then she prefers the first to the second. Since accordance of preferences with temporal domination is a natural requirement, its equivalence with short-term delay aversion provides an additional justification to this latter notion. Finally, we show that one infinite cash flow temporally dominates another if and only if all the discounters with constant marginal utility prefer the former to the latter.

The rest of the paper is organized as follows. Section  2 gathers some preliminary notions. In Section  3, we define long-term delay aversion and provide the characterizations. The next section presents the results on short-term delay aversion and temporal domination. Section  5 contains some concluding remarks. The proofs which are not in the main body of the paper are given in the Appendix.