Problem Set 2a: Choice Under Uncertainty and Insurance

1. 考虑 EUT 的公理。他们现实吗？个人是否有这种行为

2. 要求个人评估以下赌博。第一次赌博提供
100 英镑有 40% 的机会和 60% 的机会收到 1000 英镑。第二个相等

3. 个人的效用函数由 𝑈(𝑊) = ln⁡ (𝑊) 给出，其中 𝑊 表示财富。

0.2 的概率。

(i) 个人是否厌恶风险？
(ii) 个人为保险支付的最高保费是多少

(iii) 什么是公平的保险？

4. 考虑两个风险厌恶个体 �⁡ 和 �，他们有两次可微分的 von
Neumann-Morgenstern 效用函数分别为 �(𝑥) 和 �(𝑥)。 �′𝑠⁡实用函数

5. 什么决定了讲座中图二的等值预期财富线的斜率

1. Consider the Axioms of EUT. Are they realistic? Do individuals behave in this way
in practice?

2. An individual is asked evaluate the following gambles. The first gamble provides
£100 with a 40% chance and a 60% chance of receiving £1000. The second an equal
chance of receiving £200 or £1000.
How would an investor with a (natural) log utility function (so U(W) = ln(W)) evaluate
the two gambles above and choose between them? Describe the investors’ attitude
to risk?

3. An individual has a utility function given by 𝑈(𝑊) = ln⁡ (𝑊), where 𝑊 denotes wealth.
They have initial wealth of £200 and face the prospect of a loss of £150 with a
probability of 0.2.

(i) Is the individual risk averse?
(ii) What is the maximum premium the individual will pay for insurance against
this loss?
(iii) What would the fair insurance be?

4. Consider two risk averse individuals, �⁡and �, who have twice differentiable von
Neumann-Morgenstern utility functions �(𝑥) and �(𝑥), respectively. �′𝑠⁡utility function
is given to be �(𝑥) ⁡= ⁡𝑔(�(𝑥)), where 𝑔(. ) is a strictly increasing and strictly concave
function, that is, 𝑔’(. ) ⁡> ⁡0 and 𝑔’’(. ) ⁡< ⁡0. Prove that individual � is more risk averse
than individual � by the Arrow-Pratt measure of Absolute Risk Aversion.

5. What determines the slope of the iso-expected wealth line on Figure II in the Lecture
Notes? What determines the slope of the indifference curve in Figure II?