Signaling Economics 302 - Microeconomic Theory II: Strategic Behavior Shih En Lu

Economics 302 - Microeconomic Theory II: Strategic Behavior
Shih En Lu
Simon Fraser University
(with thanks to Anke Kessler)
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Most Important Things to Learn Today
Understand the economic intuition of signaling.
Know how to …nd signaling equilibria.
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Quick Review of Probabilities
Let P (X ) be the probability of an event X occurring.
Let P (XY ) be the probability both events X and Y occurring.
Let P (X jY ) be the probability of event X occurring given that event
Y has occurred.
Example: X = Canucks win; Y = Your friend is happy
What is P (X jY ) in terms of P (X ), P (Y ) and P (XY )?
What is P (X ) in terms of P (Y ), P (X jY ) and P (X jnot Y )?
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Sometimes, part of the informed side of a market wants to reveal
Example: smart job applicants.
How to do so credibly?
Idea: certain tasks are easier for some people than others.
Example: going to school is easier for smarter (in an academic sense)
If such tasks are worthwhile for some people, but not for others, they
can provide information about people’s type.
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Education as a Signal
This model is due to Spence (1973).
Two types of workers: skilled (s = 1) and unskilled (s = 0).
Fraction p of the worker pool is skilled.
A …rm …nds it optimal to pay workers their expected skill level:
w = E [s ].
Education (e = 1 if educated, e = 0 if not) does NOT enhance skills.
Skilled workers have utility w ce and unskilled ones have utility
w ke: education imposes cost c on skilled workers and k > c on
unskilled ones.
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Equilibrium in a Signaling Game
Each type of the informed side (here skilled and unskilled workers)
must act optimally.
The uninformed side has beliefs µ: probabilities on the types of the
informed side, conditional on the observables. It acts optimally based
on those beliefs.
Example: "Educated workers are skilled with probability 0.7, and
non-educated workers are skilled with probability 0.2." This can be
denoted as µ(s = 1je = 1) = 0.7, µ(s = 1je = 0) = 0.2.
Beliefs must be consistent with what the informed side is doing.
Example: If fraction q of the skilled and fraction q 0 of the unskilled
get educated, what must the …rms’beliefs be?
Note: beliefs after unexpected behaviour can be anything. So if
nobody gets educated, then the …rm is free to have any belief if,
hypothetically, it sees an educated person.
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Pooling Equilibrium
Let w0 be the wage of uneducated workers, w1 be the wage of
educated workers.
In a pooling equilibrium, all types do the same thing.
For example, nobody gets educated, and everyone is paid w0 = p.
This is only sustainable if skilled workers don’t …nd it worthwhile to
be educated. So we must have w1 c p.
So any pro…le of the following form is a pooling equilibrium: "Nobody
gets educated. w0 = µ(s = 1je = 0) = p,
w1 = µ(s = 1je = 1) 2 [0, c + p ]."
Exercise (to do at home): …nd all equilibria where everyone gets
educated. When do such equilibria exist?
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Separating Equilibrium
In a separating equilibrium, each type does something di¤erent.
Here, it must be that the skilled get educated, and the unskilled
don’t. (Why can’t it be the opposite?)
As a result, w0 = µ(s = 1je = 0) = 0 and w1 = µ(s = 1je = 1) = 1.
For this equilibrium to exist, it must be that:
the skilled …nd an education worthwhile: c 1; and
the unskilled don’t …nd an education worthwhile: k 1.
Thus, the signaling device must be not too costly for some and costly
enough for others.
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Semi-Separating (or Hybrid) Equilibrium
In a semi-separating equilibrium, one type plays a mixed strategy.
For example, fraction q of skilled workers get an education.
Thus, skilled workers must be indi¤erent between e = 1 and e = 0,
so c = w1 w0 .
Hence, unskilled workers will strictly prefer being uneducated, so
w1 = µ(s = 1je = 1) = 1.
Thus, w0 = 1 c. Also, we must have
p (1 q )
w0 = µ(s = 1je = 0) = 1 pq .
Thus, 1
p (1 q )
1 pq ,
so q =
p +c 1
cp .
Since q 2 (0, 1), this equilibrium exists if and only if c 2 (1
p, 1).
Exercise (to do at home): …nd all equilibria where the unskilled type
plays a mixed strategy.
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Signaling Summary
Credible mechanism for (potentially) revealing information.
Three categories of equilibria: pooling (no information is transmitted),
separating (type is always revealed) and semi-separating.
Whether (and how many) equilibria exist within each category
depends on model parameters.
Everywhere in life: gifts, how you dress, idiosyncratic application
requirements, etc.
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