The Theory of Rational Choice Chapter 1 The Theory of Rational Choice
Practice problems Consider the problem: min 3x2 - 5x + 3. 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 + 1 . 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 2 + 1 + 0.5 + 0.25 + …. ? Event A occurs with probability 0.4. 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?
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) = 0.25 + 0.5 + 1.5 = 2.25 Prob(Y1.5) = 0.25 First term = 2, common ratio = 0.5, so sum to infinity = 2/(1 - 0.5) = 4 Conditional probability = 0.5
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
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
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
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
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
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)