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|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。

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