1. The Theory of Rational Choice
Chapter 1
The Theory of Rational Choice

2. Practice problems
Consider the problem: min 3x2 - 5x What is the FOC, how do we find the solution, how do we know this is a minimum?
We have a system of 2 simultaneous equations linear: 5x + 3y = 19 2x + 6y = 22 What are x and y?
g(x) = 4x2 - 5x What value(s) of x set g(x) = 0?
A random variable X takes the value of 1 with probability ¼ , 2 with probability ¼ , and 3 with probability ½ . What is E(X)?
A random variable Y is uniformly distributed between 0 and 2. What is the probability that a particular draw from this distribution is greater than or equal to 1.5?
What is the sum of …. ?
Event A occurs with probability If event A occurs, event B may also occur. If the unconditional probability that B occurs is 0.2, what is the conditional probability that B occurs, given that A has occurred?

3. Answers
FOC 6x – 5 = 0. x = 5/6 is soln. SOC: 6 > 0, so we have a minimum.
x=2, y=3
x = [5+- (25-16)^0.5]/8 → x=1, x=1/4
E(X) = = 2.25
Prob(Y1.5) = 0.25
First term = 2, common ratio = 0.5, so sum to infinity = 2/( ) = 4
Conditional probability = 0.5

4. The Theory of Rational Choice
A rational decision-maker chooses the best action according to her preferences, among the actions available to her.
Set of available actions
Preferences
Complete
Consistent (transitive)
Rational ≠ Selfish

5. The Theory of Rational Choice
Payoff function: associates a number with each action in such a way that actions with higher number are preferred.
For a and b in some set A
u(a) > u(b) if and only if the decision-maker prefers a to b

6. The Theory of Rational Choice
Example:
A = {Coke, Pepsi, Sprite} = {C, P, S}
Decision-maker prefers C to P and P to S
u(C)=3, u(P)=2, u(S)=1 Or
u(C)=10, u(P)=0, u(S)=-2

7. The Theory of Rational Choice
Preferences  Ordinal information
v is another payoff function that represents the same preferences as u if
v(c) > v(p) u(c) > u(p)
Any monotonically increasing function of u represents the same preferences

8. The Theory of Rational Choice
Example:
u(C)=3, u(P)=2, u(S)=1
u(C)=3 > u(P)=2 > u(S)=1
f(x)=2*x
v(x) = f(u(x))
v(C) = f(u(C)) = f(3)=6, v(P)=4, v(S)=2
v(C)=6 > v(P)=4 > v(S)=2

9. The Theory of Rational Choice
The action chosen by a decision-maker is at least as good, according to her preferences, as every other available action.
Decisions should not be affected by irrelevant alternatives.
Example:
If A={P,C} and she always chooses C
If A’={P,C,S} and she chooses P
Inconsistent with the Theory of Rational Choice
To be consistent she must choose C or S
- See the Weak Axiom of Revealed Preference (WARP)