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This paper
provides a closed form distribution for the probability of intertemporal
indifference between a certain quantity of a commodity now, *Q*(0) = *q*^{0} , and some future quantity *Q*(*T*) =* q* at time t = T assuming a discount weight, *w*(*T*) ∈ (0,1).

This paper combines two psychological posits, through Leibniz’s Principle of sufficient reason, to provide a probabilistic theory for intertemporal indifference. Leibniz’s principle, often associated with ex nihilo nihil fit^{1}, has been influential in the thinking of several philosophers and was recognised as one of the four laws of thought in the 18^{th} century. The principle states that the occurrence/existence/truthfulness of an event/entity/ statement implies the occurrence/existence/truthfulness of a sufficient explanation. In this paper, we will repeatedly make use of this principle, especially in condensing two psychological theories: the theory that we generally prefer to consume now than to consume in the future and the theory that our perception of time does not need to coincide with the “hourglass” passage of time. Through Leibniz’s principle, we aim to achieve face validation of a probabilistic theory for intertemporal indifference.

Patterns of intertemporal indifferences have captured the interest of psychologists and economists for several decades. Procrastination, addiction and willingness to save are a few observed behaviours involving time trade- offs [

A very general discount weight function is of the form,

Prelec’s

Empirical investigations on time preferences have, however, rarely converged [

• An individual is expected to value a future quantity compared to a current quantity, q^{0}, by placing a discount weight,

With an expectation in place, we, next, seek to identify the source of variability. Among the different senses of the human being, time perception is a sense that is increasingly gaining the interest of psychologists. A key issue in modern neuroscience is the association that an individual’s perception bears with her neural events [

Strictly speaking, the present is an infinitely small-time interval separating the past and the future. However, it is argued that the perceived present, or the so-called specious present, is an interval which needs not be infinitely small—a concept first observed in 1882 [

The assumption that an individual’s perception of a future interval is a sequence of random specious presents coupled with the assumption that the individual is expected to place a weight on future consumption are sufficient assumptions to formulate a probabilistic distribution for intertemporal indifference—a direct consequence of Leibniz’s principle. Given that perceived passage of time is not always equal to hour-glass passage of time and that individuals generally prefer to consume now than in the future, then from Leibniz’s Principle of sufficient reason, the existence of an intertemporal indifference probability distribution is implied.

The assignment of probability measures is done by deriving a maximum entropy distribution for the probability that an individual/animal is indifferent between some quantity

With Equation (1) being the unconditional expectation of

Next, it is important to note that an individual or animal will only express a time preference provided she perceives time to pass (or provided the interval of time that she perceives is bigger than her specious present). Since the specious presents are random time durations, we can assume that a given fixed indifference amount at a specific time point has a random duration being equal to the individual’s specious present at that time. Furthermore, at the instant that the next specious present begins, the indifference amount increases continuously with certainty. Thus, we ensure that an individual’s intertemporal indifference path is a continuous function of time. As a result the process adheres to the typical characterisations of diffusion processes; usually restricted by a jump constraint on their probability measure,

Equation (3) ensures that the indifference path is time-continuous. For example, an individual being given an amount q^{0} now will always be indifferent to the same amount, q^{0}, in any future interval which lies in her specious present. It is at the instant the new specious present begins that the individual’s indifference path exits the state q^{0} moving, at all times, along a cádlág path^{2}.

We now wish to assign probability measures such that the surprisal or hidden information is maximized. Surprisal maximization is the standard principle in devising probability distributions. According to the principle of maximum-entropy, if we have a partial knowledge about a random variable (whether it is discrete or continuous, it’s range, mean, etcetera), we first obtain a family of probability distributions that are all consistent with our information and then, we select, from that family, the single distribution whose uncertainty is the greatest. Most commonly used distributions are MaxEnt^{3} given a current state of knowledge. For example, if we know nothing about a system of continuous random variables except it’s range, we get the uniform distribution, or except it’s positive mean, we get the exponential distribution, or except it’s mean and standard deviation, we get the normal distribution, and so on.

Consequently, with the given infinitesimal mean, Equation (8.2), we first devise a MaxEnt distribution for a sufficiently small time interval, h, and thereafter we shall project that distribution through time given the con-

straint that, for sufficiently small h,

Note that the infinitesimal distribution, at any time, would be very much dependent on the weight function, ^{4} an exit from a preference amount q in the interval

Differentiating ^{ }is a probability measure and

It is interesting to remark that the use of the Dirac measure above assumes unit jumps and hence Equations (4) and (5) provide a good analogy for countable indifference amounts, such as euros^{5}. Furthermore, let us assume that we observe the individual for an instant, or, say, for a sufficiently small time interval^{6}. At that instant, either the individual would have perceived time to pass or not and, consequently, either there exists an exit from a preference amount or not; wherefore we obtained Equations (4) and (5) which formed the infinitesimal distribution. Equation (1) however requires that

The general Chapman-Kolmogorov equation is given by

In our case, since we have only two possible outcomes in the interval h, no growth or an infinitesimal growth as specified in Equation (2), our Chapman-Kolmogorov is then approximately:

giving

Consider the backward difference operator,

Now, since

where ^{7}. The Chapman-Kolmogorov equation can thus be written as

Now, taking

Equation (8) can be readily solved through standard methods [

surface

longer have a partial differential equation. We can thus let

where c is a constant independent of q and t. Writing

Next, with

Substituting Equations (11) and (10) into Equation (9) gives

Next, we need to eliminate

from which, we get

Again, substitution of

resulting in

Integrability on the probability space,

and putting

It’s not unimportant to note that the probability distribution is not a distribution for time preferences but rather a distribution for intertemporal indifferences. To be more precise, we find a closed-form distribution for the prob- ability that an individual will actually be indifferent between some quantity of

An important assumption that we made is that we have assumed that the animal/individual lives forever implying that a choice can always be made. In case of investigations involving mortality, the latter can be rectified by taking into account mortality tables or parametric mortality laws and conditioning the probability of a choice on the probability that the choice exists. That specification is beyond this paper since it is more useful for us to assume that the individual lives forever as the latter assumption appears fairly well-founded for society. It is thus necessary that we discuss the assumptions under which the distribution is valid for society.

We will first assume that society lives forever^{8} and second, we will assume that society possesses a societal sense of time as with the “average” individuals comprising it. Individuals with Parkinson’s disease, attention deficit hyperactivity disorder (ADHD) and schizophrenia have a marked difference in time perception compared to an “average” individual [

Consequently, even the “average” individual is bound to experience a marked alteration in time perception if drugs, such as psilocybin [^{9}, become common usage. Although the latter appears farfetched, an economic scenario entailing a general alteration in the “average” individual’s time perception appears reasonably realistic. Neurobiologists and psychologists often make the following associations: opioid = pleasure, dopamine = happiness, serotonin deficit = depression, oxytocin = love, nucleus accumbens = reward or amygdala = fear, etcetera [^{10}.

As such, we assume some ethical^{11} weights among the preferences of individuals comprising a temporal society so that we can assume that society, as a whole, also experiences the passage of time as a sequence of societal specious presents. Then a society will be indifferent to a single fixed quantity for the duration of its specious present, before feeling the need for some extra amounts, with a social rate of time preference, srtp, when it perceives the present to have become past. Moreover, we can also assume non-subjective rates.

Although, our considerations, thus far, have only concerned a subjective rate of time preference, in the case where rates are not subjective, such as the rate on government bonds, without loss of generality, the probability of intertemporal indifference for a society with an external social discount rate, SDR, can be equivalently seen to follow the same distributional laws as given in Equation (8) since the main source of variability is assumed to be in the perception of time. To be specific, the expectation is taken as given and it is only the occurrence times of events, identified by zeroes or ones, which are random and assigned probability measures through surprisal maximization. So, society’s probability of intertemporal indifference,