What is the difference between expected utility and utility of expected outcome?
$U(\boldsymbol x) = \int u(x)dF(x)$ is the expected utility of the outcome
u(\bar x) denotes the utility of the expected value
Define the three risk types
A decision maker exhibits risk loving preferences if
$$
u(\bar x) > U(\boldsymbol x)
$$
A decision maker exhibits risk neutral preferences if
$$
u(\bar x) = U(\boldsymbol x)
$$
A decision maker exhibits risk averse preferences if
$$
u(\bar x) \leq U(\boldsymbol x)
$$
An agent is risk averse if and only if his utility function is….
Concave
What is a fair bet and who would never agree on this?
A risk avert person never takes a fair bet!
Fair bet: Fair bet is when the sure outcome is as good as the bet it self.
A risk neutral individual acceptance of any gamble that is fair or better. A risk averse individual always refuse a fair be.
Example:
If you guess right on heads or tails ($\alpha = .5$ change) you get one dollar. Or you can take sure thing. What is then a fair bet? That is if the sure thing is the expected value of the gamble, e.g., $0.5*1=0.5$. I.e, the sure thing is as good as the bet it selfe, hence it is fair.
What is certainty equivalence?
The certainty equivalent of lottery $F$, denoted by $c(F,u)$ is the amount of money which makes the individual indifferent between the lottery and the certain amount.
What is the insurance premium?
The insurance premium of $F$ denoted by $\pi(F, u)$ is the amount of money the individual is willing to give up in expectations to avoid the risk of $F$, that is
\pi(F,u) = \bar x -c(F,u)
What is SOSD?
Second Order Stocastic Dominance.
This ranking has to do with risk. That is, which lottery is less risky, given that both lotteries has the same expected value.
Definition 1: For any two distributions $F$ and $G$ with the same mean, $F$ SOSD $G$ if for every non-decreasing concave function $u$ we have
$$
\int u(x) dF(x) \geq \int u(x) dG(x)
$$
Definition 2: $G(x)$ is a mean-preserving spread of $F(x)$ if there exists a $H_z(z)$ with mean zero, such that $y \sim G$ is such that y = $x+ y$
Regarding SOSD.
Consider two distributions F and G with the same mean. Then the following statements are equivalent:
- $F$ SOSD $G$
- $G$ is a mean preserving spread of $F$
- $\int^xG(t)dt\geq\int^xF(t)dt$ for all $x$
What are the two ways of measuring risk?
- Absolut risk aversion (Arrow-Pratt coefficient)$r_A(x) := - \frac{u’‘(x)}{u’(x)}$The Arrotw-prat coefficient divides by $u’(x)$ which makes it invariant to any affine transformation.A person that exhibits decreasing absolut risk aversion takes on more risk when they become more wealthy, i.e., they can afford to take chance.
- Relative riskaversion$r_R(x): = xr_A(x)$The relative risk multiplies absolute risk by the amount of money x, hence it makes the risk relative to the amount of money at stake.
What is support and set order?
Support is the intervall for which a CDF goes from 0 → 1. E.g, $F = [0,A]$ is $ \text{supp}F$ if the cumulative probability are = 1 at $A$. The support is the interval in which a PMF or PDF is non-zero.
Set order $\leq_s$ ranks CDF’s according to the lowest value from which a CDF starts to go from 0 → 1.
See pictures in Notion.
