Measuring Impatience in Intertemporal Choice.

In general terms, decreasing impatience means decreasing discount rates. This property has been usually referred to as hyperbolic discounting, although there are other discount functions which also exhibit decreasing discount rates. This paper focuses on the measurement of the impatience associated with a discount function with the aim of establishing a methodology to compare this characteristic for two different discount functions. In this way, first we define the patience associated with a discount function in an interval as its corresponding discount factor and consequently we deduce that the impatience at a given moment is the corresponding instantaneous discount rate. Second we compare the degree of impatience of discount functions belonging to the same or different families, by considering the cases in which the functions do or do not intersect.

Introduction

Impatience was already defined in 1960 by [1] as the decrease in the aggregate utility with respect to time. In his work, he stated: “this study started out as an attempt to formulate postulates permitting a sharp definition of impatience, the short term Irving Fisher has introduced for preference for advanced timing of satisfaction” ([1] referred to the 1930 work of Fisher [2]: “The Theory of Interest” (Chapter IV)). This idea of a preference for advancing the timing of future satisfaction has been used in economics since the appearance of Böhm-Bawerk’s work: Positive Theorie des Kapitals [3].

Some authors use the term impulsivity as a synonym of impatience, e.g. [4]. In effect, [5] define impulsivity in intertemporal choice as a “strong preference for small immediate rewards over large delayed ones”. We can find a similar and earlier definition in [6] who defined the impulsiveness (in choices among outcomes of behavior) as “the choice of the less rewarding over more rewarding alternatives”. Observe that impatience has usually been presented in relative terms by comparing the values shown by two intertemporal choices. [7] states that the term impulsivity is often utilized in psychiatric studies on intertemporal choice and cites some examples of impulsive subjects such as smokers, addicts and attention-deficient hyperactivity-disorder patients. The opposite behavior to impulsivity is self-control.

As for the measure of impatience, the rate of discount is commonly taken to indicate the level of impulsivity or impatience in intertemporal choices. [8] offer an interesting review of the empirical research on intertemporal choice and summarize the implicit discount rates from all the studies they reviewed. In the same way, [9] and [10] provide a revision on time-declining discount rates from the observed individual choice, among other approaches.

But, as will be demonstrated in the next section, there are other ways to quantify the impatience. Our main objective in this paper is to develop a measure of the impatience exhibited by the discount function associated with the underlying intertemporal choice. In this case, we will be able to compare the impatience associated with two discount functions.

On this subject, there are many empirical papers trying to compare the degree of impatience of a group of individuals at different points in time (e.g. [11], [12], [13], [14]) or to compare the degree of impatience with different discount functions (e.g. [7], [13], [15], [16], [17], [18]). There is also an alternative measure of discounting: the area under the curve proposed by [19], which allows comparing the impatience between individuals in a model-free way (since it is not tied to any specific theoretical framework). See [20] for a review of this method and reference to several studies in which it has been employed.

Another approach related to this topic is the analysis of the main types of impatience. Thus, when studying the impatience in intertemporal choice, we usually find that it decreases. Following [21], decreasing impatience implies an inverse relationship between the discount rate and the magnitude of the delay and has usually been attributed to hyperbolic discounting. In the same way, [22] treats decreasing impatience as the core property which is parametrically expressed by hyperbolic and quasi-hyperbolic discount functions.

Recently, several studies have included different degrees of impatience and not only decreasing impatience ([23], [24], [25], [26], [27], [28]). [25] report individual evidence of lower discounting for intervals closer to the present than for distant ones, demonstrating concave discounting, which implies increasing impatience.

Additionally, a number of empirical papers have recently appeared which relate impatience with decision-making in games. For example, [29] study the relationship between impatience, risk aversion, and household income. [30] conducts a research on impatience, risk aversion, and working environment. [31] empirically study the patience/impatience of punishers in a multilateral cooperation game. In a similar vein, [32] explore the relationship between impatience and bargaining behavior in the ultimatum game.

In this paper the concept of patience associated with a discount function (F(t)) in an interval [t1, t2] is defined as the value of the discount factor corresponding to F(t) in this interval. Hence, the impatience associated with F(t) in an interval will be calculated as 1 minus the discount factor associated with the given interval, which is the value of the discount corresponding to $1 in this interval. Additionally, we present a procedure to compare the degree of impatience between two discount functions, of the same or different family. In [33] we find a comparison between exponential and hyperbolic discount functions controlling the overall impatience in order to isolate the differences due to self-control problems only. The controlled comparison is made by means of age adjustment which equalizes areas under discount functions.

The objective of this paper is interesting for the following reasons:

First of all, fitting the preferences exhibited by an individual or a group of individuals to a well-known discount function has an important advantage. In effect, the questionnaires used in intertemporal choice include a limited number of pairs of amounts and delays. Nevertheless, a discount function fitted to data from respondents allows us to analyze the preferences between any pair of monetary rewards.

Inevitably, most researches on this issue show the discount functions which, in each case, better fit the data. In effect, [34] estimate the parameters of the main intertemporal choice models: exponential, simple hyperbolic, quasi-hyperbolic, and q-exponential. Subsequently, they compare the impatience shown by two groups by simply comparing the discount rates of the corresponding discount functions. Obviously, this is not an accurate procedure because some discount functions are biparametric and so it should require a comparison of both parameters defining the function. Moreover, this is a simplification because it would be interesting to compare the impatience in a certain time interval where, among other circumstances, the relative position of the impatience levels can change. Even the use of the q-exponential discount function assumes working with an exponential, a hyperbolic or a generalized hyperbolic discount function, depending on the concrete values of q and k. Therefore, the most important thing is to obtain the discount function which better fits the collected data, and then it is likely that the subsequent comparison can involve discount functions belonging to different families. Even the comparison between two discount functions belonging to the same family (for instance, two hyperbolic discount functions) is also noteworthy because they usually exhibit different parameters.

Several researches have considered the impatience shown by individuals of different nationalities, genders or socio-economic levels. The comparison of the discount functions involved in these studies is important in order to design, for example, a market segmentation strategy according to the former criteria.

[35] state that the intertemporal impatience can be applied to the acquisition of material objects instead of money. This makes the issue of impatience very interesting in marketing and consumer behavior. They point out some culture-related differences between western and eastern participants in the empirical study conducted by them: the former valued immediate consumption more than the latter. In the same way, [14] experimentally compared intertemporal choices for monetary gains and losses by American and Japanese subjects, demonstrating that Westerners are more impulsive and time-inconsistent than Easterners. [36] also recognize the accuracy of discounting to explain impatience in marketing.

Finally, [37] have found that gender and autobiographical memory can have an effect on delay discounting: there is a significant difference between men and women because, in the case of higher memory scores, the former showed less impatience when discounting future rewards. In the experimental analysis, they used the standard hyperbolic and the quasi-hyperbolic models.

It is therefore apparent that the comparison of discount functions will be of interest to segment a market depending on the impatience exhibited by individuals who are classified by different criteria (geographical, gender, culture, etc.).

This paper is organized as follows. After this introduction, in Section 2 we will formally define the impatience (impulsivity) ranging from the discount corresponding to $1 in an interval [t1, t2] (a two-parameter function, referred to as impatience-arc) to the instantaneous discount rate at an instant t (a one-parameter function, referred to as instantaneous impatience). The value of the instantaneous rate at t = 0 (a constant) can also be taken into account. Obviously, any simplification in the measurement of impatience will result in a reduction in the amount of information thus obtained. Therefore, in Section 3, we will compare the impatience associated with two discount functions, considering two cases: when the functions do not intersect and the functions do intersect. In Section 4, all the obtained results will be applied to well-known families of discount functions. Finally, Section 5 summarizes and concludes.

Defining impatience (impulsivity) in intertemporal choice

In economics and other social sciences it is common practice to try to simplify the complexity of the models describing the behavior corresponding to a group of people. This is the case of discount functions in the framework of intertemporal choice within the field of finance. In effect, a (dynamic) intertemporal choice can be described by a two-variable discount function ([38]), that is, a continuous function such that where F(d, t) represents the value at d (delay) of a $1 reward available at instant d + t. In order to make financial sense, this function must satisfy the following conditions:

F(d, 0) = 1,

F(d, t) > 0, and

For every d, F(d, t) is strictly decreasing with respect to t.

A discount function is said to be

With bounded domain if, for every , there exists an instant , depending on d, such that F(d, t) = 0.

With unbounded domain if, for every and , one has F(d, t) > 0. Within this group, a discount function can be:

Regular if , for every .

Singular if , for every .

Regular discount functions are the most usual valuation financial tools. Nevertheless, and as indicated at the beginning of this Section, this discounting model can be simplified by using a function F(t) independent of delay d. More specifically, a one-variable discount function F(t) ([38] and [39]) is a continuous real function such that defined within an interval [0, t0) (t0 can even be +∞), where F(t) represents the value at 0 of a $1 reward available at instant t, satisfying the following conditions:

F(0) = 1,

F(t) > 0, and

F(t) is strictly decreasing.

The following theorem of representation provides the relationship between the preferences existing in a scenario of intertemporal choice and its associated discount function.

Theorem 1. A discount function F(t) gives rise to the total preorder ≽ defined by satisfying the following conditions:

If t1 ≤ t2, then (C, t1) ≽ (C, t2), and

If C1 ≥ C2, then (C1, t) ≽ (C2, t).

Reciprocally, every total preorder ≽ satisfying conditions (i) and (ii) defines a discount function.

Theorem 1 shows that, in intertemporal choice, an agent can indistinctly use a discount function or a total preorder. In this way, the concept of impatience has been mainly treated with a total preorder. For example, [40] propose the following choice: “$10 in a year or $15 in a year and a week”. In this way, they state that: “If an individual A prefers the first option ($10 in a year) while B prefers the second option ($15 in a year and a week), it is said that A is more impulsive than B because A prefers a smaller, but more immediate reward, whereas B prefers to wait a longer time interval to receive a greater reward”. Nevertheless, our aim here is to define the concept of impatience by using discount functions. In effect, given a one-variable discount function F(t), the patience associated with F(t) in an interval [t1, t2] (t1 < t2) is defined as the value of the discount factor f(t1, t2) corresponding to this interval, viz: where is the instantaneous discount rate of F(t) at instant x. Obviously, the inequality 0 < f(t1, t2) < 1 holds. Observe that the greater the discount factor, the less sloped is the discount function in the interval [t1, t2]. In this case, people are willing to wait for a long time to receive a future amount because they have to renounce a small part of their money. On the other hand, the impatience associated with F(t) in the interval [t1, t2] (t1 < t2) is defined as the value of the discount D(t1, t2) corresponding to this interval, viz: which lies in the interval [0, 1].

Some comments:

It is logical that the impatience can be measured by the amount of money that the agent is willing to lose in exchange for anticipating the availability of a $1 reward.

Any function with the same monotonicity as f(t1, t2) (resp. D(t1, t2)) can be used as a measure of patience (resp. impatience). For example, is a measure of the impatience. Consequently, for an infinitesimal interval (t, t + dt), the measure of the impatience is given by δ(t).

The term impulsivity is used on most occasions as a synonym of impatience, but we prefer its use for intervals of the type [0, t] or, from an infinitesimal point of view, δ(0).

[40] use the term “self-control” as the opposite of impulsivity and therefore as a synonym of patience.

Comparing the impatience represented by two discount functions

Most empirical studies on intertemporal choice present a set of data based on the preferences of outcomes shown by a group of individuals. The analysis of the impatience exhibited by the group is very difficult to realize because individual members of the group will show a wide variety of preferences with regard to amounts and time delays. Therefore, it is preferable to fit the resulting data to a discount function belonging to any of the noteworthy families of discount functions, viz, linear, hyperbolic, generalized hyperbolic, exponentiated hyperbolic, or exponential. The necessary adjustment can be made by using the q-exponential discount function (see [38] and [41]) since it includes the majority of the aforementioned functions as particular cases ([42]). Once a discount function is obtained which represents all the information coming from the individual questionnaires, it is easier to obtain the instantaneous impatience and the impatience-arc, that is to say, the impatience corresponding to a time interval. To do this, we can make use of all the tools of mathematical analysis. Moreover, the comparison between the impatience shown by two groups of people is more accurate and more easily understood, and the results can be used in designing and implementing future strategies.

Case in which the two functions do not intersect

Let F1(t) and F2(t) be two discount functions. Assume that the ratio is increasing. This implies that, for every t > 0, and so F1(t) < F2(t). Let us recall that the patience is measured by the discount factor defined by Eq (1). As is increasing, for every t1 and t2 such that t1 < t2, , from where . Therefore, and so

In particular, for every t and h > 0, or equivalently

Therefore, if F(t) is differentiable, then that is to say

The converse implication is also true, whereby we can enunciate the following result.

Theorem 2. Let F1(t) and F2(t) be two discount functions. The following three statements are equivalent:

The ratio is increasing.

The impatience represented by F1(t) is greater than the impatience represented by F2(t), that is to say, f1(t1, t2) < f2(t1, t2), for every t1 and t2 such that t1 < t2.

If F1(t) and F2(t) are differentiable, δ1(t) > δ2(t), for every t.

Example 1. Let and be two hyperbolic discount functions where F1(t) < F2(t) (so i1 > i2). is increasing since

According to Theorem 2, must be greater than . In effect,

Fig 1 shows that it is not easy to graphically observe that the ratio F2(t) (shown in red) to F1(t) (in blue) is increasing.

Fig 1

Hyperbolic discount functions of Example 1.

For this reason we are going to formulate the following

Corollary 1. Let F1(t) and F2(t) be two discount functions such that F2(t) − F1(t) is increasing. In this case, any of the three equivalent conditions of Theorem 2 is satisfied. For a proof, see Appendix.

Example 2. Let be a regular hyperbolic discount function of parameter i1 and be a singular hyperbolic discount function of parameters i1 and i2 (so necessarily i1 > i2). Fig 2 shows that the difference F2(t) − F1(t) is increasing.

Fig 2

Hyperbolic discount functions of Example 2.

In effect,

According to Corollary 1, must be greater than , which can easily be verified. Then the impatience represented by F1(t) is greater than the impatience represented by F2(t). Fig 2 shows the general situation described by Corollary 1. Finally, the results obtained in Theorem 2 and Corollary 1 can be summarized in Fig 3.

Fig 3

Summary of the results in Theorem 2 and Corollary 1.

Let us now consider a third situation. Let us suppose that the ratio reaches a local maximum at instant t0. A possible graphic representation is depicted in Fig 4.

Fig 4

Discount functions of Example 3 and their ratio.

By Theorem 2, for intervals [t1, t2] included in [0, t0] (t1, t2 < t0), the impatience represented by F1(t) is greater than the one represented by F2(t). After instant t0, the opposite situation occurs, that is, the impatience represented by F1(t) is less than that represented by F2(t), but this situation can change because t0 is a local maximum and so there exists the possibility of another local extreme. For example, if F2(t) is singular and F1(t) is regular, there will exist a neighborhood of infinity where is increasing.

Example 3. Let be a hyperbolic discount function of parameter i and , i > 0. Obviously, F1(t) < F2(t) and reaches a maximum at (see Fig 4 where i = 0.10). In accordance with the previous paragraph, is greater than in the interval , and contrarily δ2(t) is greater than δ1(t) in .

We can now formulate the following statement.

Theorem 3. Let F1(t) and F2(t) be two discount functions such that F1(t) < F2(t). If F2(t) − F1(t) reaches a local maximum at , then the factor reaches a local maximum at a later instant t0 (eventually, t0 can be +∞).

Example 4. Observe that, for the discount functions of Example 1 with i1 = 0.05 and i2 = 0.10, F2(t) − F1(t) reaches its local maximum at and t0 = +∞, as predicted by Theorem 3.

Table 1 schematically represents the result obtained in Theorem 3. For the sake of simplicity, we will suppose that both F2(t) − F1(t) and reach a unique local maximum.

Table 1

Several implications arising from the relationship between the local maxima of F2(t) − F1(t) and .

Intervals
(0,t0′)	(t0′,+∞)
F2(t) − F1(t) increasing	F2(t) − F1(t) decreasing
⇓   Intervals   ⇑
(0, t0)	(t0,+∞)
F_2(t)F_1(t) increasing	F_2(t)F_1(t) decreasing
δ2(t) < δ1(t)	δ2(t) > δ1(t)
f1(t1, t2) < f2(t1, t2)	f1(t1, t2) > f2(t1, t2)

Although t0 is the instant which separates the intervals where δ1(t) > δ2(t) and δ1(t) < δ2(t), there are some intervals [t1, t2], where t1 < t0 < t2, such that f1(t1, t2) < f2(t1, t2). In effect, given t1 < t0, this instant t2 must satisfy:

The maximum value of t2 must satisfy the following equation:

Example 5. Let F1(t) and F2(t) be the discount functions of Example 3. Taking t1 = 7, we have to solve the following equation (see Fig 5): for which the solution is t2 = 14.367.

Fig 5

Intersection of the two instantaneous discount rates.

Finally, this reasoning can be continued by considering the following local extreme of (in this case, a local minimum), and so on.

Case in which the two functions intersect

For the sake of simplicity, in this Subsection, we will assume that functions F1(t) and F2(t) only intersect at an instant t1. In this case, we will distinguish between the following two subcases:

F1(t) and F2(t) are secant. This situation does not affect the results obtained in Theorems 2 and 3.

F1(t) and F2(t) are tangent. In this case, reaches a local extreme at this point and so we can apply Theorem 3. More specifically, reaches a local minimum at t1 (see Fig 6) and so, by Theorem 3, δ1(t) is less than δ2(t) on the left of t1, and contrarily δ1(t) is greater than δ2(t) on the right of t1. But observe also that reaches a local maximum at t0. Thus, the global situation can be summarized in Table 2.

Fig 6

Intersection of the two discount functions: case of tangency.

Table 2

Patience / impatience according to different intervals.

Intervals	(0, t0)	(t0, t1)	(t1,+∞)
F_2(t)F_1(t)	↗	↘	↗
Greater impatience	F1(t)	F2(t)	F1(t)
Greater patience	F2(t)	F1(t)	F2(t)

An application to well-known discount functions

In experimental analysis, it is usual to fit the available data from several groups of individuals to discount functions belonging to the same family. It is therefore necessary to compare the impatience represented by two discount functions coming from the same general family.

Comparison of two generalized hyperbolic discount functions

These functions are the well-known q-exponential discount functions introduced by [41]. Let F1(t) and F2(t) be two generalized hyperbolic discount functions: and where i1 > i2. Let us calculate the first derivative of :

We are going to assume that s1 ≠ s2. Otherwise, the comparison between F1(t) and F2(t) would be the same as two hyperbolic discount functions. Making this derivative equal to zero, we obtain:

If s1 > s2, then t0 < 0 and is increasing in . Thus, by Theorem 2(iii), δ1(t) > δ2(t) and so the impatience represented by F1(t) is greater than the impatience represented by F2(t).

If s1 < s2, we can consider two subcases:

, in which case t0 > 0 is a local maximum of . Thus, by Theorem 2, δ1(t) > δ2(t) in (0, t0) and δ1(t) < δ2(t) in (t0,+∞) and therefore, according to Theorem 2, the impatience represented by F1(t) is greater than the impatience represented by F2(t) in the interval (0, t0) and less in the interval (t0,+∞).

, in which case t0 < 0 and is increasing in . Thus, again by Theorem 2, δ1(t) > δ2(t) and so the impatience represented by F1(t) is greater than the impatience represented by F2(t).

In order to compare the impatience of several well-known discount functions, in Table 3, we have considered the linear, hyperbolic, generalized hyperbolic, and exponential discounting both in the column on the left and on the upper row. Each cell of this table has been divided into three parts. We have represented the cases in which two discount functions F1(t) and F2(t) (F1(t) < F2(t)) satisfy the three equivalent conditions of Theorem 2. In this case, the first part of the cell shows the relationships to be satisfied by the parameters of F1(t) and F2(t) in order to satisfy Theorem 2. On the other hand, we have represented in bold those cases where F1(t) and F2(t) do not satisfy the conditions of Theorem 2. In theses cases, the first level of the cell exhibits the relationships between the parameters of F1(t) and F2(t) so that F1(t) < F2(t) in a neighborhood of zero; the second level of the cell includes the maximum t0 of ; and, finally, the third level contains the maximum of F2(t) − F1(t). The relative position of these time instants was discussed in Theorem 3.

Table 3

Cases of application of Theorem 2 or Theorem 3 (in bold), where F1 < F2.

	Discount function F2(t)
Discount function F1(t)	Linear F2(t) = 1 − d2t	Hyperbolic F_2(t)=11+i_2t	Generalized hyperbolic F_2(t)=1(1+i_2t)^s_2	Exponential F_2(t)=1(1+i_2)^t
Linear F1(t) = 1 − d1t	d1 > d2	d1 > i2	d1 > s2i2	d1 > ln(1 + i2)
	–	–	–	–
	–	–	–	–
Hyperbolic F_1(t)=11+i_1t	i1 > d2	i1 > i2	i1 > s2i2 and s2 < 1	i1 > ln(1 + i2)
	t_0=i_1-d_22i_1d_2	–	–	t_0=i_1-ln(1+i_2)i_1ln(1+i_2)
	t0′=(i_1/d_2)^1/2-1i_1	–	–	(1+i_1t0′)^2(1+i_2)^t0′=i_1ln(1+i_2)
Generalized Hyperbolic F_1(t)=1(1+i_1t)^s_1	s1 i1 > d2	s1i1 > i2 and s1 > 1	s1i1 > s2i2 and s1 > s2	s1 i1 > ln(1 + i2)
	t_0=s_1i_1-d_2i_1d_2(s_1+1)	–	–	t_0=s_1i_1-ln(1+i_2)i_1ln(1+i_2)
	t0′=(s_1i_1/d_2)^1/(s_1+1)-1i_1	–	–	(1+i_1t0′)^s_1+1(1+i_2)^t0′=s_1i_1ln(1+i_2)
Exponential F_1(t)=1(1+i_1)^t	ln(1 + i1) > d2	ln(1 + i1) > i2	ln(1 + i1) > s2i2	i1 > i2
	t_0=ln(1+i_1)-d_2d_2ln(1+i_1)	–	–	–
	t0′=ln ln(1+i_1)-lnd_2ln(1+i_1)	–	–	–

Conclusion

The term impatience was introduced by [2] in 1930 to refer to the preference for advanced timing of future satisfaction. More recently the concept of decreasing impatience has been applied to those situations in which discount rates are decreasing. Usually this property has also been labeled as hyperbolic discounting, although there are other discount functions involving decreasing discount rates. In this paper, we have focused on measuring the degree of impatience of discount functions in both intervals and instants.

In effect, in experimental research into impatience in intertemporal choice, the data from questionnaires are usually fitted to discount functions from different families of functions. Leaving aside the problem of whether this fitting is good, once the experimental discount functions corresponding to two groups of people have been obtained, there arises the problem of comparing the impatience exhibited by each of them.

At first glance, the faster the function decreases, the higher is the degree of impatience. That is, if F2(t) − F1(t) is increasing, the impatience shown by F1(t) is higher than the impatience shown by F2(t). But this graphic criterion only represents a condition sufficient to compare degrees of impatience. Nevertheless, it is convenient to state a necessary and sufficient condition for the impatience of F1(t) to be higher than the impatience of F2(t); this condition could be that the ratio F2(t)/F1(t) is increasing. In Theorem 2, that allows us to compare the impatience associated with two discount functions, we present two conditions equivalent to the former. Unfortunately, it is difficult to observe this property graphically in most cases (unless we consider the difference ln F2(t) − ln F1(t)). In other cases, however, the monotonicity of F2(t)/F1(t) changes, necessitating the calculation of its maximum and minimum extreme values (Theorem 3). In most cases, the comparison of the impatience will be made using two discount functions belonging to the same family. Therefore, the problem of determining the local extremes of F2(t)/F1(t) can be solved explicitly or, at least, their existence must be demonstrated.

The main contributions of this paper are Theorems 2 and 3. In Table 3 we compare the impatience shown by pairs of discount functions belonging to the most important families of temporal discounting (linear, hyperbolic, generalized hyperbolic and exponential discount functions). Thus, a restriction in red represents a condition sine qua non for a pair of discount functions in order to satisfy Theorem 2. Nevertheless, there are other pairs of discount functions not satisfying the conditions of Theorem 2. In this case, we have deduced (as shown in bold) the expressions of t0 and (the equation to be satisfied by) which allows us to check the statement in Theorem 3.

These Theorems allow us to compare the impatience shown by two individuals or two groups of people, once their preferences have been fitted to a suitable discount function belonging to a well-known family of functions. Finally, this methodology can be applied to two-variable (amount and time) discount functions when some anomalies in intertemporal choice (for example, delay or magnitude effect) are taken into account.

Appendix

Proof of Corollary 1. As F2(t) − F1(t) is increasing and F1(t) is decreasing, then is increasing. Therefore, (and consequently ) is increasing which is condition (i) of Theorem 3.

Proof of Theorem 3. As F2(t) − F1(t) reaches a local maximum at , then from where

As F2(t) > F1(t), and and are negative (remember that F(t) is decreasing), one has

Hence and therefore the factor is increasing at , leading to a local maximum at an instant (eventually t0 could be +∞).