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Suppose an investor has a preference represented by the relation $\succ$ for which there is a von-Neumann Morgenstern representation with the utility function $u$: $$u(x)=\begin{cases} x & x\ge 0 \\ \lambda x &x<0 \end{cases}$$ Determine all values $\lambda$ for which $\mu \succ 0 \succ \upsilon$, where $\upsilon=\frac 12 \delta_{\{-100\}}+\frac 12 \delta_{\{150\}}$ and $\mu=\frac 12 \delta_{\{-100\}}+\frac 12 \delta_{\{200\}}$.

My try: $$\mu \succ 0 \succ \upsilon$$ $$\frac 12 \delta_{\{-100\}}+\frac 12 \delta_{\{200\}} \succ 0 \succ \frac 12 \delta_{\{-100\}}+\frac 12 \delta_{\{150\}}$$ We subtract $\frac 12 \delta_{\{-100\}}$: $$\frac 12 \delta_{\{200\}} \succ -\frac 12 \delta_{\{-100\}} \succ \frac 12 \delta_{\{150\}}$$ We substitute the function $u$: $$200 \succ 100\lambda \succ 150$$ $$2 \succ \lambda \succ 1,5$$

My asks:
First of all, I don't know if it's allowed to perform such transformations - I treated the relation $\succ$ as a regular $>$. If this is correct, what theorems should I rely on to make the task accurate? If the way is incorrect, how else can I approach the task?

dsk62
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1 Answers1

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I am not familiar with the notation you have used, but I have assumed $v$ is a lottery with equal probabilities of wealth $-100$ and $150$ and $\mu$ is a lottery with equal probability of wealth $-100$ and $200$.

To answer the question, we just have to find the expected utilities of zero wealth for sure and of the two lotteries and compare them.

The expected utility of a wealth of zero for sure is zero.

For the lottery $v$ the expected utility is

$$ \frac{1}{2}u(-100)+\frac{1}{2} u(150)=\frac{1}{2}\lambda(-100)+\frac{1}{2}150 $$

and for lottery $\mu$ the expected utility is

$$ \frac{1}{2}u(-100)+\frac{1}{2} u(200)=\frac{1}{2}\lambda(-100)+\frac{1}{2}200. $$

Clearly it is always the case that lottery $\mu$ is preferred to lottery $v$.

The expected utility from lottery $v$ is less than zero if

$$150<\lambda (100) \iff \lambda>1.5$$

The expected utility from lottery $\mu$ is larger than zero if

$$200>\lambda (100) \iff \lambda<2$$

So the condition $\mu\succ 0\succ v$ is true when $\lambda\in(1.5,2)$.

In your working, this line does not really make sense:

$$\frac 12 \delta_{\{200\}} \succ -\frac 12 \delta_{\{-100\}} \succ \frac 12 \delta_{\{150\}}$$

This is because the preference relation is defined only over lotteries, and $\frac 12 \delta_{\{200\}}$ is not a lottery.

However, mathematically it leads you to the right answer because your working could be made correct by writing:

$$\mu \succ 0 \succ \upsilon$$ $$\iff$$ $$\frac 12 \delta_{\{-100\}}+\frac 12 \delta_{\{200\}} \succ 0 \succ \frac 12 \delta_{\{-100\}}+\frac 12 \delta_{\{150\}}$$ $$\iff$$ $$\frac 12 u(-100)+\frac 12 u(200)> u(0) > \frac 12 u(-100)+\frac 12 u(150)$$ We subtract $\frac12 u(-100)$: $$\frac 12 u(200)> \frac12 u(-100) > \frac 12 u(150)$$ We substitute the function $u$ (and multiply by $2$): $$200>100\lambda>150$$ $$\iff$$ $$2> \lambda> 1.5$$

smcc
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