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经济代写|行为金融学代写Behavioral Finance代考|Expected utility theory

A fundamental aspect of Kahneman and Tversky’s analysis of prospect theory is their critique of expected utility theory. Concepts of expected value and expected utility were developed by Bernoulli and others from the 18th century onwards but only found their way into mainstream economics in the mid-20th century, most famously in von Neumann and Morgenstern’s (1944) analysis of expected utility theory.

To enable an understanding of Kahneman and Tversky’s critique, a summary of some of the basic principles of EUT are outlined in Box 4.1. The von Neumann and Morgenstern preference axioms include transitivity, completeness, substitution, continuity and invariance. Transitivity means that if $A$ is preferred to $B$ and $B$ is preferred to $C$ then $A$ is preferred to $\mathrm{C}$. Completeness means that in a choice between $\mathrm{A}$ and $\mathrm{B}$, an individual will either prefer A, prefer B or be indifferent between A and B. Substitution implies that if two alternatives are identical then they can be substituted for each other-for example, if an individual is indifferent between two alternatives then they will also be indifferent between the alternatives if these are offered with equal probabilities. Continuity implies that if $A \leq B \leq C$ then $B$ can be expressed as a weighted sum of $A$ and $C$. Invariance implies that the expected utility function can be scaled up without affecting the ordering of preferences.
Overall, these EUT axioms generate a theory in which people have stable, consistent risk preferences. As noted earlier in the context of the St Petersburg Paradox, by including an assumption of risk aversion, the standard utility function becomes concave, as illustrated in Figure 4.1. Expected utility will increase at a decreasing rate and individuals will prefer averages to extremes. Given these axioms, Savage (1954) shows that expected utility is the product of a subjective utility function and a Bayesian subjective probability distribution (Savage 1954; Kahneman and Tversky 1979).

Figure $4.1$ shows an example of a choice between $£ 10$ and $£ 50$. If a person is offered a choice between a $50 \%$ chance of $£ 10$ and a $50 \%$ chance of $£ 50$, then the expected value of this gamble is $£ 30$ but this gives utility at $u_1$ whereas a guaranteed $£ 30$ has a utility of $u_2$ and $u_2>u_1$. In other words, if a person has to be paid a certainty equivalent (the amount which makes them indifferent between the gamble and a guaranteed amount) of $£ 35$ to take a gamble with an expected value of $£ 30$ then they are risk-averse. EUT assumes that people are risk-averse and the more bowed is the utility function, then the higher risk aversion will be. This is captured by the Arrow-Pratt measure of absolute risk aversion (ARA), which captures the curvature of the utility function using the change in marginal utility relative to its level. The coefficient of relative risk aversion (RRA) weights the risk parameter by consumption and further variants of ARA and RRA include constant absolute risk aversion (CARA) and constant relative risk aversion (CRRA). Ultimately, all these measures are similar in embedding stable, measurable risk preferences.

经济代写|行为金融学代写Behavioral Finance代考|Kahneman and Tversky’s critique of EUT

Kahneman and Tversky provide a critique of Savage’s (1954) approach to analysing decisions between uncertain outcomes – the set of risky alternatives that Kahneman and Tversky call “prospects”. They assert that people do not necessarily reason using mathematical/statistical tools and this explains some of the behavioural paradoxes described above. Kahneman and Tversky set out some of the problems with EUT and then devise their own solution in the form of prospect theory – a model which enables us better to understand various anomalies in human decision-making.

In developing prospect theory, Kahneman and Tversky (1979) start with a critique of standard expected utility theory (EUT) and explain how real-world behaviour is better explained by prospect theory. Expected utility theory is both a normative theory capturing how rational people should behave; and a positive/descriptive theory – capturing how people do behave. Many people probably would, in principle at least, prefer not to plan their lives in an illogical, inconsistent way so, as a normative theory, EUT has more merit – even normative issues can be incorporated within it, as seen in models of inequity aversion – explored in Chapter 2. Normatively, if utility functions can be broadened properly to incorporate preferences for non-monetary sources of utility, such as equity, then most people would like to act in a way predicted by EUT.

Some of the mathematics underlying Kahneman and Tversky’s critique of EUT are outlined in the Mathematical Appendix A4.2. According to Kahneman and Tversky, the most profound problem with EUT is its legitimacy as a positive theory. The problem is that people don’t/can’t act as predicted by EUT: EUT lacks predictive power and so its role as a positive descriptive theory is compromised. Kahneman and Tversky argue that EUT does not provide an adequate description of human behaviour in the real world.

In their critique, Kahneman and Tversky focus on the Savage axioms underlying EUT including the expectation axiom (overall utility is the sum of the expected utilities), the asset integration axiom (acceptable prospects are those which integrate with wealth to give a utility greater than the utility of the wealth alone) and the risk aversion axiom, which holds if, and only if, the utility function is concave.

经济代写|行为金融学代写Behavioral Finance代考|FINM3407

经济代写|行为金融学代写Behavioral Finance代考|Expected utility theory

卡尼曼和特沃斯基分析前景理论的一个基本方面是他们对预期效用理论的批判。预期价值和预期效用的概念是伯努利和其他人从 18 世纪开始发展起来的,但直到 20 世纪中叶才进入主流经济学,最著名的是冯·诺依曼和摩根斯坦 (1944) 对预期效用理论的分析。

为了理解 Kahneman 和 Tversky 的批评,方框 4.1 概述了 EUT 的一些基本原则。冯诺依曼和摩根斯坦偏好公理包括传递性、完整性、替代性、连续性和不变性。传递性意味着如果一个优先于乙和乙优先于C然后一个优先于C. 完整性意味着在一个和乙,一个人要么偏爱 A,偏爱 B,要么在 A 和 B 之间漠不关心。替代意味着如果两个选择相同,那么它们可以相互替代——例如,如果一个人在两个选择之间无差异,那么他们也会如果这些选项以相等的概率提供,则对这些选项无动于衷。连续性意味着如果一个≤乙≤C然后乙可以表示为的加权和一个和C. 不变性意味着可以在不影响偏好排序的情况下扩大预期效用函数。
总体而言,这些 EUT 公理产生了一种理论,其中人们具有稳定、一致的风险偏好。正如前面在圣彼得堡悖论的背景下所指出的,通过包含风险厌恶的假设,标准效用函数变得凹形,如图 4.1 所示。预期效用将以递减的速度增加,个人更喜欢平均值而不是极端值。鉴于这些公理,Savage (1954) 表明预期效用是主观效用函数和贝叶斯主观概率分布的乘积(Savage 1954;Kahneman 和 Tversky 1979)。

数字4.1显示了一个选择的例子££10和££50. 如果一个人被提供一个选择50%的机会££10和一个50%的机会££50,则本次赌博的期望值为££30但这提供了实用性在1而保证££30有一个效用在2和在2>在1. 换句话说,如果必须向某人支付确定性等值(使他们在赌博和保证金额之间无动于衷的金额)££35以期望值进行赌博££30那么他们是风险厌恶的。EUT 假设人们厌恶风险,效用函数越低,风险厌恶程度越高。这可以通过绝对风险厌恶 (ARA) 的 Arrow-Pratt 度量来捕捉,该度量使用边际效用相对于其水平的变化来捕捉效用函数的曲率。相对风险厌恶系数 (RRA) 通过消费对风险参数进行加权,ARA 和 RRA 的进一步变体包括恒定绝对风险厌恶 (CARA) 和恒定相对风险厌恶 (CRRA)。最终,所有这些措施在嵌入稳定、可衡量的风险偏好方面都是相似的。

经济代写|行为金融学代写Behavioral Finance代考|Kahneman and Tversky’s critique of EUT

Kahneman 和 Tversky 批评了 Savage (1954) 分析不确定结果之间的决策的方法——Kahneman 和 Tversky 称之为“前景”的一组风险选择。他们断言人们不一定使用数学/统计工具进行推理,这解释了上面描述的一些行为悖论。Kahneman 和 Tversky 列出了 EUT 的一些问题,然后以前景理论的形式设计了他们自己的解决方案——这个模型使我们能够更好地理解人类决策中的各种异常情况。

在发展前景理论时,Kahneman 和 Tversky (1979) 从对标准预期效用理论 (EUT) 的批判开始,并解释了前景理论如何更好地解释现实世界的行为。期望效用理论既是一种规范理论,它反映了理性的人应该如何行事;和一个积极/描述性的理论——捕捉人们的行为方式。至少在原则上,许多人可能更愿意不以不合逻辑、不一致的方式来计划他们的生活,因此,作为一种规范性理论,EUT 具有更多优点——甚至规范性问题也可以纳入其中,正如在不公平厌恶模型中所看到的那样– 在第 2 章中进行了探讨。从规范上讲,如果效用函数可以适当地扩展以包含对非货币效用来源(如股权)的偏好,那么大多数人都希望以 EUT 预测的方式行事。

数学附录 A4.2 概述了卡尼曼和特沃斯基对 EUT 的批评所依据的一些数学。根据 Kahneman 和 Tversky 的说法,EUT 最深刻的问题是其作为实证理论的合法性。问题是人们不能/不能按照 EUT 的预测行事:EUT 缺乏预测能力,因此它作为积极描述理论的作用受到损害。Kahneman 和 Tversky 认为 EUT 没有提供对现实世界中人类行为的充分描述。

在他们的批评中,Kahneman 和 Tversky 关注 EUT 背后的野蛮公理,包括​​期望公理(整体效用是预期效用的总和)、资产整合公理(可接受的前景是那些与财富整合以提供大于仅财富的效用)和风险厌恶公理,当且仅当效用函数是凹的时,该公理才成立。

经济代写|行为金融学代写Behavioral Finance代考

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