Education as a Signaling Device and Investment in

Human Capital

Topic 3Part 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 = {s1, s2}: success/failure

S = {s1, s2,...,sn-1,sn}: number of n units sold

Event

• An event is a collection of those states “si” 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, A1 or A2) occurs is measured using the concept of probability

• P(Ai) expresses the probability that the event Ai occurs

• We assume that

Ai = S

P ( Ai) = 1

0 P (Ai) 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 Bc 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” = Bc

• The Bayes rule can be stated as

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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|Bc) = 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(Bc) = 0.95

Bayes’ Rule

• Using Bayes’ Rule we get

50.)95)(.05(.)05)(.95(.

)05)(.95(.

)()|()()|(

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