Education as a Signaling Device and Investment in Human Capital Topic 3 Part I

Outline Tools –Probability Theory –Game Theory Games of Incomplete Information Perfect Bayesian Equilibrium A Model of Education as a Signaling Device of the Productivity of the Worker (Spence, 1974) Education as Human Capital Accumulation (Becker, 1962) Empirical Evidence

Probability: Basic Operations and Bayes’ Rule We need to use probabilities in deriving the Perfect Bayesian Equilibrium Then, we will review basic probability operations and the Bayes’ Rule

Sample Space Let “S” denote a set (collection) of all possible states of the environment known as the sample space A typical state is denoted as “s” Examples S = {s 1, s 2 }: success/failure S = {s 1, s 2,...,s n-1,s n }: number of n units sold

Event An event is a collection of those states “s i ” that result in the occurrence of the event An event can imply that one state occurs or that multiple states occur

Probability The likelihood that an uncertain event (or set of events, for example, A 1 or A 2 ) occurs is measured using the concept of probability P(A i ) expresses the probability that the event Ai occurs We assume that  A i = S P (  A i ) = 1 0  P (A i )  1, for any i

Addition Rules The probability that event “A or event B” occurs is denoted by P(A  B) If the events are mutually exclusive (events are disjoint subsets of S, so that A  B=  ), then the probability of A or B is simply the sum of the two probabilities P(A  B) = P(A) + P(B) If the events are not mutually exclusive (events are not disjoint, so that A  B ‡  ), we use the modified addition rule P(A  B) = P(A) + P(B) – P(A  B)

Multiplication Rules The probability that “event A and event B occur” is denoted by P(A  B) Multiplication rule applies if A and B are independent events. A and B are independent events if P(A) does not depend on whether B occurs or not, and P(B) does not depend on whether A occurs or not. Then, P(A  B)= P(A)*P(B) We apply the modified multiplication rule when A and B are not independent events. Then, P(A  B) = P(A)*P(B/A) where, P(B/A) is the conditional probability of B given that A has already occurred

Bayes’ Rule Bayes’ Rule (or Bayes’ Theorem) is used to revise probabilities when additional information becomes available Example: We want to assess the likelihood that individual X is a drug user given that he tests positive –Initial information: 5% of the population are drug users –New information: individual X tests positive. The test is only 95% effective (the test will be positive on a drug user 95% of the time, and will be negative on a non-drug user 95% of the time)

Bayes’ Rule Let A be the event “individual X tests positive in the drug test”. Let B be the event “individual X is a drug user”. Let B c be the complementary event “individual X is not a drug user” We need to find P(B|A), the probability that “individual X is a drug user given that the test is positive”. We assume S consists of “B” and “not B” = B c The Bayes rule can be stated as

Bayes’ Rule Information given Test effective 95%. Then, –Probability that the test results positive given that the individual is a drug addict = P(A|B) = 0.95 (test correct) –Probability that the test results positive given that the individual is not a drug addict = P(A|B c ) = 0.05 (test wrong) 5% of population are drug users: –Probability of being a drug addict = P(B) = 0.05 –Probability of not being a drug addict = P(B c ) = 0.95

Bayes’ Rule Using Bayes’ Rule we get