Managerial economics guide 6

Risk aversion, risk neutral, risk lover

If an individual is presented with a lottery and a chance to get the mean of the lottery for sure, the individual's preference indicates their risk profile:

• Risk averse: If he prefers to get the mean of the lottery for sure, then he is said to be risk averse.
• Risk neutral: If he's indifferent between getting the mean of the lottery for sure and facing the lottery itself, then he is said to be risk neutral.
• Risk lover: If he prefers the lottery itself to getting the mean for sure, then he is said to be a risk lover.

Two distinct states of nature

We restrict ourselves to two distinct states of nature (two markets) X₁ and X₂. X₁ represents a simple random variable.

On the fraction, we can track u(X₁) and u(X₂) by reading along the curve. The mean is a point on the horizontal axis: (\( \overline{x} \) = \( \frac{x_1 + x_2}{2} \)).

If p₁=p₂=1/2, then the mean will be the midpoint of the AB segment, \( \overline{x} = \overline{x} \).

The utility of the mean (or the instantaneous utility of the mean) can be found on the function. It is also the expected utility of the lottery payoffs.

The expected utility of the lottery will be somewhere on the segment of the arc, in between u(x₁) and u(x₂).

Proof

The convex combination of the two points are points in the segment joining these two points. Suppose we call the 1^st point A and the 2^nd point B, the convex combination of A and B is: (1 - λ)A + λB

The convex combination will lie along the arc connecting A & B. Their coordinates can be expressed as:

(λx₁ + (1 - λ)x₂, (1 - λ)u(A) + λu(B))

p represents coordinates between 0 and 1, these coordinates can also be interpreted as probability. Probabilities are normally ranged from 0 to 1 and sum up to 1, which is exactly what you need for making a convex combination: coordinates which are positive whose sum equals to 1.

Therefore, we can write the coordinates as follows:

u([λx₁ + (1 - λ)x₂]) = λu(x₁) + (1 - λ)u(x₂)

The expected utility will lie on the arc that’s joining the two points. The corresponding point on the function is the utility of the mean.

Certainty equivalent of the lottery

Another important concept is the certainty equivalent of the lottery. In a word, it’s the utility of something that you get for sure.