Difference between \(\theta\) and \(\rho\)

Think of theta as a person who really values equality, and rho as a person who really values the present. Thus, it is clear that 
This will lead us to the definitions: 

Utilitarianism and Rawlsian Social Welfare Function:

The Utilitarian calculation:
\(U = \frac{C_1 ^{1 - \theta}}{1 - \theta} + ... + \frac{C_n ^{1 - \theta}}{1 - \theta} \), where \(\theta \leq 0 \) if \(\theta = 0\), not utilitarianism
The Rawlsian calculation:
\(U = min(c_1, c_2, c_3,..., c_n)\), notice, it's a Leontief 
Example
Person 1: \(\theta = \frac{1}{2}\) with a function of \(\sqrt C_1 + \sqrt C_2\)
vs. 
Person 2: \(\theta = 2\) with a function of \(- \frac{1}{C_1} - \frac{1}{C_2}\)
Choice between:
Person 1 will choose Bundle 1, since \(11 > 6.5\)
Person 2 will choose Bundle 2, since \(-0.4 > -1.01\). This represents the Rawlsian persepctive and the preference for consumption smoothing.
* Equity-premium puzzle: Why don't people invest in stocks? They have a relatively low risk and high return. Economists have used this to estimate that \(\theta = 25\)... signifying that people really, really don't like waiting. This is unsatisfactory.

Uncertainty and Risk

The probability of event of A is defined as follows:
Event \(A \subseteq E\) where \(E\) is all events
\(P(A) + P(B) + P(C) + ... + P(N) = 1\), if all events, \(A, B, ... N\) are disjoint and \(A \cup B ... \cup N = E\)
Classic Bernoulli Trial (most probability theory is built upon this)
Distribution: \(P(E_1) = P\) and \(P(E_2) = 1 - P\)
The expectation of a random variable X:
\(E(X) = P(E_1) X_1 + P(E_2)X_2 ... + P(E_n) X_n\)
Square root of variance is equal to the standard deviation, or, the absolute average between 2 random variables. In the Bernoulli, \(\sqrt p(1 - p)\)

Detours

Betting market: horserace (subjective event that people choose through reason) vs. Roulette wheel (objective event with essential randomness)
Subjective probability: 
(3, 3)*    (0, 1)
(1, 0)      (1, 1)*
Here, the top left is the payoff dominant equilibrium. The bottom right is the risk dominant equilibrium. 

Von Neumann - Morgenstern

This is the workhorse of modern economics for decisions under uncertainty. Fun fact: Von Neumann was smarter than Einstein per people who knew them both. Rationality is preferences over lotteries, which are descriptions of probabilities paired with a numerical payoff (\(X, Y\)). 
For VNM lotteries, a person's preferences over lotteries must be:
  1. Complete (have an opinion about each one)
  2. Transitive (can compare different bundles)
  3. Continuous (if \(L_1 > L_2 > L_3\), there exists some mean between the two extremes that makes you indifferent toward the second option: \(-L_1 - (1 - p)L_3 \approx L_2\))
  4. Independent (eliminating framing effects, so if you add the same thing to both sides you still have the same preferences)
General example:
\(L_1: p_1, X_1\) and \( (1 - p_1), Y_1\)
\(L_2 : p_2, X_2\) and \((1 - p_2), Y_2\)
Particular example:
Lottery one:
\(p_1 = 0.6, 1,000,000\)
\((1 - p_1)= 0.4, 200,000\)
Lottery two:
\(p_2 = 0.9, 4,000,000\)
\((1- p_2)= 0.1, 100,000\)
Which will be chosen? Sum up the individual's preferences over risky outcomes like this:
\(V = p_1 \cdot u(X_1) + (1 - p_1) \cdot u(Y_1)...\)
Insurance Example:
max \(p \ln (W - I) + (1 - p) \ln (W - I - L + PI)\)
Where W is wealth, I is insurance, L is the loss in a bad state, and P is payoff of insurance in a bad state. The goal is to equalize marginal utility across all states (so a variant of high theta preferences).
F.O.C.
\(\frac{p}{W - I} + \frac{1 - p}{W - I - L + PI} = 0\)
If insurance markets are competitive, the total premia (money invested in insurance, \(\mathbb{P}\)) must equal the total payoff to insured in the bad state. Such that
\(I = (1 - p) \mathbb{P}\)