The Theory of Rational Choice

1
Chapter 1

• The Theory of Rational Choice

2
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
  linear5x 3y 192x 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?

3
Answers

• FOC 6x 5 0. x 5/6 is soln. SOC 6 gt 0, so
  we have a minimum.
• x2, y3
• x 5- (25-16)0.5/8 ? x1, x1/4
• E(X) 0.25 0.5 1.5 2.25
• Prob(Y?1.5) 0.25
• First term 2, common ratio 0.5, so sum to
  infinity 2/(1 - 0.5) 4
• Conditional probability 0.5

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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

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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) gt u(b) if and only if the decision-maker
  prefers a to b

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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

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The Theory of Rational Choice

• Preferences ? Ordinal information
• v is another payoff function that represents the
  same preferences as u if
• v(c) gt v(p) ?u(c) gt u(p)
• Any monotonically increasing function of u
  represents the same preferences

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The Theory of Rational Choice

• Example
• u(C)3, u(P)2, u(S)1
• u(C)3 gt u(P)2 gt u(S)1
• f(x)2x
• v(x) f(u(x))
• v(C) f(u(C)) f(3)6, v(P)4, v(S)2
• v(C)6 gt v(P)4 gt v(S)2

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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 AP,C and she always chooses C
• If AP,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)