asymmetric information - applicationsshreyas/eco/lecture17.pdf · in settings of asymmetric...
TRANSCRIPT
Adverse selection and signaling
How can we characterise market equilibria
in settings of asymmetric information?
Examples:
1. When a firm hires a worker, the firm may
know less about the worker’s innate ability
than the worker herself;
2. In the used-car market, a prospective
seller may have much better information about
her car’s quality than a prospective buyer.
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3. When an individual buys health insur-
ance, he may know more about his propen-
sity to contract a serious disease than the
insurance company does.
In these cases, market equilibria may often
fail to be Pareto optimal.
Moreover, this problem may be further com-
pounded by adverse selection.
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Adverse selection arises when an informed
individual’s trading decisions depend on her
privately-held information in a manner that
adversely affects uninformed market partic-
ipants.
User-car example: individual more likely to
sell her car when she knows it is not very
good.
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Akerlof’s labour-market (‘lemons’) model
Many identical potential firms that can hire
workers;
Each produces identical output using a CRS
technology;
Labour is the only input;
Firms are risk-neutral, seek to maximise ex-
pected profits and act as price takers;
Price of output is 1 (in terms of numeraire)
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Workers differ in their productivity, θ (num-
ber of units they can produce);
[θ, θ] ⊂ R - set of possible worker productiv-
ity levels, 0 ≤ θ ≤ θ <∞;
Proportion of workers with productivity of θ
or less given by F (θ). We assume F(.) is
non-degenrate;
Total number (measure) of workers is N.
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Workers seek to maximise amount they can
earn from their labour (in terms of numeraire);
A worker of type θ can earn r(θ) on her own
(opportunity cost of working) ⇒ she will ac-
cept employment at a form iff her wage is
at least r(θ)
What is the CE of this model when workers’
productivity levels are publicly observable?
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There is a distinct equilibrium wage w∗(θ)
for each type θ
Given competitive, CRS nature of firms, w∗(θ) =
θ for all θ.
Set of workers accepting employment in a
firm is {θ : r(θ) ≤ θ}
This CE is Pareto optimal.
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What is the CE when worker productivity
levels are not observable by firms?
Wage rate is now independent of worker
type, so single wage rate w for all workers.
Set of workers willing to accept employment
at wage rate w is: Θ(w) = {θ : r(θ) ≤ w}
Suppose firm believes that average produc-
tivity of workers who accept employment is
µ.
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What is demand for labour as a function of
w?
z(w) =
0, µ < w
[0,∞), µ = w∞, µ > w
If worker types in set Θ∗ are accepting em-
ployment offers in a CE, and if firms’ beliefs
about productivity of potential employees
correctly reflect the average productivity of
workers hired in this equilibrium, then we
must have µ = E[θ/θ ∈ Θ∗]
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Thus, demand for labour must equal sup-
ply in an equilibrium with a positive level of
employment iff w = E[θ/θ ∈ Θ∗]
Definition: In a competitive labour market
model with unobservable worker productiv-
ity levels, a CE is a wage rate w∗ and a set
Θ∗ of worker types who accept employment
such that
Θ∗ = {θ : r(θ) ≤ w∗}
w∗ = E[θ/θ ∈ Θ∗]
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Typically, a CE as defined above will not be
Pareto optimal - i.e., there will be an inef-
ficient allocation of workers between firms
and home production.
Consider the case where r(θ) = θ - every
worker is equally productive at home.
Suppose F (r) ∈ (0,1) - there are some work-
ers with θ > r and some with θ < r. Pareto
optimal allocation will have those with θ ≥ r
accepting employment at a firm and those
with θ < r not doing so.
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In a CE, set of workers willing to accept
employment at a given wage Θ∗(w) is either
[θ, θ] (if w ≥ r) or ∅ (if w < r).
Thus E[θ/θ ∈ Θ(w)] = E[θ] for all w and so,
equilibrium wage rate is w∗ = E[θ].
If E[θ] ≥ r, all workers accept employment
at a firm; if E[θ] < w, no one does. Which of
these equilibria will arise depends on fraction
of high and low productivity workers.
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Signaling - Spence model
Two types of workers with productivities θH
and θL respectively, with θH > θL > 0 - pri-
vate information;
λ = Pr(θ = θH) ∈ (0,1)
Before entering job-market, worker can get
some education - amount of education a
worker receives is observable.
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Assume education has no effect on worker
productivity!
Cost of obtaining education level e for a type
θ worker (monetary/psychic cost) given by
twice continuously differentiable function c(e, θ)
Assume c(0, θ) = 0; ce(e, θ) > 0; cee(e, θ) >
0; cθ(e, θ) < 0 ∀e > 0; ceθ(e, θ) < 0 - both
cost and MC of education are lower for high-
ability workers
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Workers’ utility u(w, e/θ) = w − c(e, θ)
r(θ) - opportunity cost of working, or value
of outside option. For simplicity, we assume
r(θH) = r(θL) = 0
Implication: in the absence of ability to sig-
nal, unique equilibrium has all workers em-
ployed at firms at wage w∗ = E[θ], and is
Pareto efficient.
Our analysis of signaling here therefore em-
phasises potential inefficiencies of signaling.
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A set of strategies and a belied function
µ(e) ∈ [0,1] giving the firms’ common prob-
ability assessment that the worker is of high-
ability after observing education level e is a
weak PBE if:
(i) The worker’s strategy is optimal given
the firms’ strategies;
(ii) Belief function µ(e) is derived from the
worker’s strategy using Baye’s rule, where
possible;
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(iii) Firms’ wage offers following each choice
e constitute a NE of the simultaneous-move
wage offer game in which the probability
that the worker is of high-ability is µ(e).
We begin our analysis at the end of the
game.
Suppose after seeing some education level
e, firms attach probability of µ(e) that the
worker is of type θH.
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Then, expected productivity of worker is µ(e)θH+
(1− µ(e)θL
In a simultaneous-move wage offer game,
the firms’ pure strategy NE wage offers equal
workers’ expected productivity.
Thus, in any pure-strategy PBE, we must
have both firms offering same wage which
is exactly equal to expected productivity.
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Knowing this, what is the worker’s strategy
- choice of education level contingent on her
type?
Workers’ preferences over (wage,education)
pairs - single crossing property.
Arises because worker’s MRS between wages
and education at any given (w,e) pair is
(dwde )u = ce(e, θ) which is decreasing in θ
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w(e) - equilibrium wage offer that results for
each education level.
In any PBE, w(e) = µ(e)θH +(1−µ(e)θL for
the equilibrium belief function µ(e), hence
w(e) ∈ [θL, θH]
Separating equilibrium: Let e∗(θ) be worker’s
equilibrium education choice as a function
of her type, and let w∗(e) be the firms’ equi-
librium wage offer as a function of workers’
education level.
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Lemma: In any separating PBE, w∗(e(θH)) =
θH and w∗(e(θL)) = θL; each worker type re-
ceives wage equal to her productivity level.
Lemma: In any separating PBE, e∗(θL) =
0; a low-ability worker chooses to get no
education.
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