Excerpt

## Abstract

This research paper will survey models and experiments for decisions involving outcomes of different amounts at different points of time. Experiments have shown that people like to get rewards earlier than later in time. This is what the concept of Intertemporal choice implies. Thus, this paper will consider theoretical concepts including discounted utility, hyperbolic and quasy-hyperbolic models as well as empirical results indicating how people really make such decisions.

Key words: Intertemporal Choice, Discounted Utility, Hyperbolic and Quasy Hyperbolic Models

## Introduction

Intertemporal choice implies decisions in regard to tradeoffs among outcomes at different points in time. From choosing whether to spend one salary immediately or to put it into savings, whether to order candies now that may have health impacts later in life, or whether to postpone an unpleasant task until some future date, are just some of the ingredients of people behaviours engaging in what economics and decision theories describe as Intertemporal choice.

Economists, cognitive scientists, psychologists and many others have realised the importance of being able to find a unique equation that would cover how humans, agents or clients discount the value of rewards as they are delayed further in the future. Economists have analyzed intertemporal decisions using the discounted utility (DU) model for almost a century now. This model assumes that people evaluate enjoyment and pains resulting from a decision in similar ways that people in finance evaluate losses and gains. According to DU model people exponentially discount the value of outcomes according to how delayed they are in time (Berns et al., 2007). Despite that DU model has been used to measure the discount rate associated with many different choices including how much to spend on research and development projects, health, education, clean energy etc., this model has shown to not properly reflect people decisions. Hence, the model of hyperbolic time discounting can be considered as the mostly used observed pattern of behaviour to reflect many of such DU anomalies. In addition, quasi-hyperbolic discounting brings another important perspective to this spectrum.

Discounting thus is central in intertemporal choice, however, what discounting model to employ may be equally important as this represents the preferences of an agent across time. Hence, the precision with which a high number of experiments and econometric results are constructed and interpreted depends on the accuracy of the models used to measure such delay discounting. Additionally, techniques used to elicit subject^{1} discounting rates will have another interesting impact in the final result. This research will first reveal the importance of DU model, its anomalies and the need for hyperbolic and quasy hyperbolic models that have been used to experimentally analyze the differences in discounting different commodities ranging from money, to a variety of consumable goods and projects.

Discounted Utility Model

John Ray further examined the psychological and sociological determinants of intertemporal choice interpreting the joint product of many conflicting motives after Adam Smith called attention to the importance of these choices as important determinants of economic propensity of nations (Loewenstein et al., 2003). Correspondingly, Samuelson went further by bringing up a normative model in his five page publication titled 'Note on Measurement Utility' in 1937 (Camerer et al., 2004). Since then, the theory of discounted utility has been the most widely used framework for analyzing intertemporal choices (Chabris et al., 2006). This theory has been used to describe how decision makers actually behave (positive or descriptive economics), and to prescribe how they should optimally behave (normative economics). Positive economics will be shown later on the paper while explaining some of the normative anomalies.

According to the American Heritage Dictionary the term of discounting means "to reduce in quantity or value". Similarly, in economics the idea of discounting is reduced to the term discounting function that could be expressed in the following form: where D(t) represents the discount function in all the equations used now and later, and nt represents the per-period discount rate, or the discount rate between t and t+1. Hence, assuming that the above constant per-period pure rate of time preference or discount rate nt equals π for all t, we get the Samuelson's DU model^{2}:

Abbildung in dieser Leseprobe nicht enthalten

The Samuelson's normative model is called discounted utility model (DU), which simply is the weighted sum of discounted instant values over a given period of time (Soman et al., 2005). Its initial and main assumption is that the discount factor is constant over time. Let us suppose that an agent believes that the utility derived from gaining one euro falls by five percent from time t to t + 1 , then its utility would be five percent less at t + 2 compared to t + 1, five percent less at t + 3 in relation to t + 2 and so it continues (Musau, 2009). The same assumption leads to exponential discounting that is being used even now in calculating present values in finance. The next assumption in Intertemporal choice assume that diminishing marginal utility has opposing effect to the decision maker if compared to the other property of positive rate of time preference (Frederic et al., 2002). The first means that immediate utility function is concave (having the next additional reward brings less satisfaction to the decision maker compared to the previous one), whereas the next implies that the agent favours sooner rather than later rewards. Positive rate of time preference perceives that delaying a reward, its value is reduced. This has been reflected on the discount function that weakly declines as the delay, τ, increases: D(t) must be monotone decreasing, [Abbildung in dieser Leseprobe nicht enthalten]. However, [Abbildung in dieser Leseprobe nicht enthalten] represents the weight attached in period t for the wellbeing in period t+1 (this is sometimes referred to as felicity or as instantaneous utility). It assumes that having an outcome X, the decision maker would chose the future consumption path [Abbildung in dieser Leseprobe nicht enthalten] over the one in his current position [Abbildung in dieser Leseprobe nicht enthalten] if and only if [Abbildung in dieser Leseprobe nicht enthalten]. In other words, despite that the discount function declines as the delay increases, this assumes that increasing felicity u(Ct+c) in overall would increases decision makers total utility, Ul. Correspondingly, normalizing D(0) to 1, we could combine these assumptions as it follows (Chabris et al., 2006):

Abbildung in dieser Leseprobe nicht enthalten

D(0) means the present tense and no discounting as it means no delay of the outcome. The discounting function D(t) is lower in value than D(0) as the outcome is delayed, represented by higher τ . The discounting function D(t) is often given as discount rate (r) which is the proportional change on value of D(t) over a standard time horizon, or as a discount factor (δ) that is the proportion remaining after delaying an outcome (Read, 2003). In some other psychology texts (Goldin, 2007), the above DU model is also expressed as:

Abbildung in dieser Leseprobe nicht enthalten

Here, U(Xt) is the consequence of option X in period t, and δ is the constant discount factor.

For instance, an individual with an yearly discount factor of 91% would be indifferent between an option with a utility of 91 Euros today and an option with a utility of 100 next year, because 91 is 91% of $100. Here, δ is the constant discount factor of 91, such that δ = 1/(1 +π), or relating it to the example δ = 1 / (1 +0.09), where π is referred to as the discount rate of 9%. As the following equation demonstrates, larger discount rates (π) are associated with smaller discount factors (δ) - that is, a decision maker whose discount rate is thirty percent will care less about the future than a decision maker with a discount rate of only five percent. Stated differently, decision makers with small discount rates are more patient than decision makers with large discount rates (Goldin, 2007).

According to Camerer and his colleagues (2004) there have been three important additional assumptions or properties of DU mentioned, utility independence, consumption utility and stationary utility. Such properties are sometimes included under the concept of dynamic consistency and self control (Laibson, 2001). Utility independence assumes that the utility of a choice is equal to the sum of its utility in each time period. Based on this, the utility is distributed on the same manner through years depending on the discounting dictation that punishes the utility experienced in the future (Camerer, 2004). Thereafter, this model allows us to calculate the overall utility of an option by multiplying each utility by a discount factor (δ). Another practical example would be to consider that "Student A" has three options that give to her three different utilities, to go to some friend party would give a utility of 200, to solve some homework a utility of only 30, and to watch a movie 100 utils. Time is the constrain leaving only two options, (1) to solve her homework first and later go to the party, or (2) to first watch a movie and then do the homework while not having time left for the party. The discount rate (π) between present (t) and later (t+1) is supposed to be 0.5. Thus, "homework now" option has an overall utility of doing the homework now plus the discounted utility of going to the party later, which is 30 + (0.67 * 200) = 164. Similarly, the utility of the "movie now" option plus the discounted utility of homework later is equal to 100 + (0.67 * 30) = 120. The Intertemporal choice shall follow the result form the discounted utility, which implies that "Student A" should do her homework now so that she can go to the party later. However, if the student is faced with the same options on the days to come over and over again, the DU consumption independence would imply that the student should do her homework first and go to the party later all the time in the future over and over again. This assumption implies that the well-being in period t is independent of his or her consumption in any other time period. This is to say that the outcome's utility is completely unaffected by other outcome experienced in past or future periods. Additionally, the stationary immediate utility derived from this model highlights a similar assumption that one's cardinal utility function υ(Γί+τ) or generated well being stays unchanged during the whole range of different time horizons (Loewenstein et al., 2003). The final assumption is independence of discounting from consumption representing that the discount function or rate of time preference is constant regardless the kind of consumption nature that is in question (Musau, 2009). The form of consumption meant to be discounted consistently could be anything from money, leisure or basic groceries.

Although instances like those above could give some understanding of why DU has been so straightforward method for modeling intertemporal behavior and decisions, its assumption seem to fail recognising the habit formation and preference or taste changes of decision makers over time (Lowensetine et al, 2004). Despite that it still has helped to better realize people's willingness to trade off between current and future satisfaction including patience or impatience, summarized by a single number - the discount rate, scientists have continued to test if people behave correspondingly to this model opening this way the next phase of experimental analysis.

Eliciting methods and DU Anomalies^{3}

First of all, experimental analyses have used different elicitation methods on how to elicit discounting. Methods of eliciting have their important role in better understanding how scientist have really found anomalies of DU and compared it with later developed intertemporal choice models. Numerous

**[...]**

^{1} Subjects have the meaning of people, agents, decision makers or participants in the experiments. Thus, as these terms may be used interchangeably, in this paper they all imply the same meaning.

^{2} The absolute year is represented by t, while the relativistic year in the future is denoted by τ (relativistic because they depend on the year - t in which the person starts delaying - τ). Thus the difference between them is that the date is denoted by t and the delay by τ (Rasmusen, 2008). http://www.rasmusen.org/special/hyperbolic- rasmusen.pdf

^{3} DU anomalies refer to the deviation of subjects' behavior from the normative equation of DU. However, this does not imply in any way that people behave incorrectly. Life expectancy or death, other natures of risks associated with waiting for rewards, marginal rate of substitution, and many other different psychological, political and soci- economic factors cause behaviors that we do not intend to further explain in this paper.

- Quote paper
- PhD candidate Ilir Hajdini (Author), 2012, Intertemporal Choices in Management Decision Making, Munich, GRIN Verlag, https://www.grin.com/document/202566

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