Asymmetric information -

applications

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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Pooling equilibrium: In a pooling equilib-

rium, both types of workers choose same

level of education, e∗(θL) = e∗(θH) = e∗.

Since firms’ beliefs must be correctly derives

from the equilibrium strategies and Baye’s

rule when possible, we must have w∗(e∗) =

λθH + (1− λ)θL = E[θ].

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