There was an interesting problem I encountered in my microeconomics course. Suppose you are risk-averse and you have a friend who has an investment project. There is a \(50 \%\) probability that the project succeeds and if it does, it will be valued at \(x\) kr. Furthermore, let us suppose it is at least double your initial wealth ie. \(x \geq 2 \cdot w_0\) where \(w_0\) is your initial wealth. The price to participate is \(\theta \cdot w_0\) where \(\theta \in [0,1]\) ie. it is some share of your initial wealth. Notably this means that whether it succeeds or not, you still have to pay that price. Should you participate? More precisely, for what \(x\) and \(\theta\) would you accept the gamble? The basic assumption is that you will choose the gamble that yields the highest expected value.

Since we are risk-averse, our preferences will be represented by a strictly concave Von-Neumann-Morgenstern utility function by a well-known theorem. One example of such a utility function is \(u(w) = \ln(w)\). Therefore, the expected value of doing nothing will be … As for the investment project, we can write it as the following gamble:

\[P \left( A=2 \, \middle| \, \dfrac{A^2}{B}>4 \right)\]

and so the expected value for the investment project will be:

\[ M = \begin{bmatrix} \frac{5}{6} & \frac{1}{6} & 0 \\ \frac{5}{6} & 0 & \frac{1}{6} \\ 0 & \frac{5}{6} & \frac{1}{6} \end{bmatrix} \]

How does the expected value ¹ for the investment project depend on \(\theta\) when \(x = 2 \cdot w_0\) or \(x \geq 2 \cdot w_0\)? Furthermore, what value should \(\theta\) be such that you are indifferent between doing nothing and taking the gamble? And what is the optimal ² \(\theta\) (depending on \(x\) and \(w_0\)) that will yield the highest expected value? Ie. what is the ideal scenario in which you would participate despite being risk-averse?

\[ \begin{aligned} V_{\circ} &= \frac{4}{3}\pi r^3 \\ &= 4 \pi \cdot K \end{aligned} \]

This poem by Martin Luther King:

Fleecy locks, and black complexion
Cannot forfeit nature’s claim.
Skin may dinner, but affection
Dwells in black and white the same.
And if I were so tall as to reach the pole,
And to grasp the ocean at a span,
I must be measured by my soul.
The mind is the standard of the man.