graduate school of management and economics microeconomics...

In the Name of God

Sharif University of Technology

Graduate School of Management and Economics

Microeconomics 2 44706 (1394-95 2nd term) - Group 2

Dr. S. Farshad Fatemi

Chapter 13:

Adverse Selection, Signaling, and Screening

In this chapter, we are going to investigate the result of relaxing another assumption of the perfect competitive model:

There is no hidden information between the agents.

Asymmetric information may lead to non-Pareto optimal outcomes.

Microeconomics 2 Dr. F. Fatemi Page 162 Graduate School of Management and Economics – Sharif University of Technology

• Asymmetric Information and Adverse Selection

Akerlof, “The Market for Lemons: Quality Uncertainty and the Market Mechanism”, QJE (1970)

Many identical potential risk neutral employers (firms) competing in a competitive framework

All with the same CRS technology with only

One input: labor

&

One output: the numeraire good (its price equal 1) Microeconomics 2 Dr. F. Fatemi Page 163

Graduate School of Management and Economics – Sharif University of Technology

𝑁 workers with different productivity levels 𝜃 ∈ �𝜃,𝜃� ⊂ ℝ+ with CDF 𝐹(𝜃)

Each worker’s opportunity cost 𝑟(𝜃)

It can be seen as the value of home production or the utility of remaining unemployed

Accepts the contract if the wage offered is greater than or equal 𝑟(𝜃)


• Publicly Observable Types

Given the competitive nature of the market in a competitive equilibrium:

𝑤∗(𝜃) = 𝜃 ∀𝜃

Then each worker accepts the contract if 𝑟(𝜃) ≤ 𝜃

Aggregate surplus:

𝑁� �𝜃𝜃≥𝑟(𝜃)

𝑑𝐹(𝜃) + �𝑟(𝜃)𝜃<𝑟(𝜃)

𝑑𝐹(𝜃)�

It is a Pareto optimal allocation to employ those who have 𝑟(𝜃) ≤ 𝜃.


• Unobservable Types

The wage 𝑤 cannot be a function of worker’s type (productivity)

Then workers accept the contract if 𝑟(𝜃) ≤ 𝑤

If a firm believes that the average productivity of workers who accept employment is 𝜇

Then a firm’s demand for labor is:

𝑧(𝜔) = � 0 if 𝜇 < 𝑤[0,∞) if 𝜇 = 𝑤 ∞ if 𝜇 > 𝑤

Microeconomics 2 Dr. F. Fatemi Page 166


If a firm’s belief about the productivity of workers is correct:

𝜇 = 𝐸𝑥𝑝[𝜃|𝑟(𝜃) ≤ 𝑤]

Definition (MWG 13.B.1): In the competitive labor market model with unobservable worker productivity levels, a competitive equilibrium is a wage rate which satisfies:

𝑤∗ = 𝐸𝑥𝑝[𝜃|𝑟(𝜃) ≤ 𝑤∗]

This definition involves the rational expectation on the firm’s part.


• Pareto Inefficiency

Suppose

𝑟(𝜃) = 𝑟 ∀𝜃

Then at a given wage rate 𝑤

either all workers got employed 𝑤 ≥ 𝑟

or no one got employed 𝑤 < 𝑟

so 𝑤∗ = 𝐸𝑥𝑝[𝜃]

Not Pareto optimal: either too many workers are employed or too few


• Adverse Selection

Adverse selection arises when relatively less productive workers accept the employment at any given wage.

Suppose

𝑟(𝜃) ≤ 𝜃 ∀𝜃

𝑟′(. ) > 0

It can be shown that in this case w∗ can be found which satisfies

𝑤∗ = 𝐸𝑥𝑝[𝜃|𝑟(𝜃) ≤ 𝑤∗]

The answer is neither necessarily unique nor efficient. Microeconomics 2 Dr. F. Fatemi Page 169


• A Game-Theoretic Approach

Consider the following sequential game:

1) Identical firms (without loss of generality 2 firms) simultaneously announce their wage offers

2) Workers decide whether to accept any of the two offers or remain unemployed

Assume as before:

𝑟(𝜃) ≤ 𝜃 ∀𝜃 and 𝑟′(. ) > 0

𝑓(𝜃) is the PDF associated with 𝜃 and 𝑓(𝜃) > 0 ∀𝜃


We have to find the SPNE of the game:

Proposition (MWG 13.B.1): Let 𝑊∗ denote the set of competitive eq wages and let 𝑤∗ = Max[𝑤:𝑤 ∈ 𝑊∗]

i) If 𝑤∗ > 𝑟�𝜃� and ∃ 𝜀 > 0 such that 𝐸𝑥𝑝[𝜃|𝑟(𝜃) ≤ 𝑤′] > 𝑤′ ∀𝑤′ ∈ [𝑤∗ − 𝜀 ,𝑤∗]

Then there is a unique SPNE of the 2-stage model. In this SPNE, employed workers receive a wage of 𝑤∗; and workers with types in set Θ(𝑤∗) = {𝜃: 𝑟(𝜃) ≤ 𝑤∗} accept employment.

ii) If 𝑤∗ = 𝑟�𝜃�, then there are multiple SPNEs. However, in every pure strategy SPNE each agent’s payoff exactly equals her payoff in the highest-wage competitive eq.


• Constrained Pareto Optimal

We will get back to this after studying principle-agent model.


• Signaling

One obvious solution to the problem of unobserved types, is including the ability to send signals by the informed party.

A second hand car dealer tries to send signals that the car is of good quality;

In a market with low and high ability workers, workers might be able to send signals regarding their ability;

In many countries, the potential tenants for rental houses; provide a letter from the previous landlords;

Recommendation letters for getting admission


A simple model:

Only two type of workers:

0 < 𝜃𝐿 < 𝜃𝐻

0 < 𝜆 = Prob(𝜃 = 𝜃𝐻) < 1

Each worker can obtain some education prior to entering the job market, which has these properties:

Obtaining the education is costly

Education has no effect on worker’s ability (productivity)

Education level is observable Microeconomics 2 Dr. F. Fatemi Page 174


The cost of education level 𝑒 ≥ 0 for type 𝜃:

𝑐(𝑒,𝜃)

Where for ∀𝑒,𝜃:

𝑐(0,𝜃) = 0

𝜕𝑐(𝑒,𝜃)𝜕𝑒

> 0 , 𝜕2𝑐(𝑒,𝜃)𝜕𝑒2

> 0

𝜕𝑐(𝑒,𝜃)𝜕𝜃

< 0 , 𝜕2𝑐(𝑒,𝜃)𝜕𝑒𝜕𝜃

< 0

Worker’s utility if she obtains the education level of 𝑒 and get wage of 𝑤:

𝑢(𝑤, 𝑒|𝜃) = 𝑤 − 𝑐(𝑒,𝜃)


The sequence of the game:

1) Nature determines the type of worker

2) Observing her type; the worker decides about her level of education

3) Observing the worker’s education, but not her type; (two) firms simultaneously make wage offer

4) The worker decides whether to accept any of the offers or remain

unemployed



Start the analysis from the end of the game:

After observing e the firm assigns a probability µ(e) that the worker’s type is 𝜃𝐻

Therefore, a firm’s expected productivity is

𝜇(𝑒)𝜃𝐻 + �1 − 𝜇(𝑒)�𝜃𝐿

In any PBE, the simultaneous game of offering wages is very much like a Bertrand competition setting; in which both firms offer a wage equal to expected productivity.

Microeconomics 2 Dr. F. Fatemi 7 Graduate School of Management and Economics – Sharif University of Technology

In the signaling games; two types of equilibrium can be considered:

Separating Eq.: Different types of the sender send different signals, then are distinguishable by the receiver (eg. in our example low ability and high ability workers acquire different levels of education)

Pooling Eq.: Different types of the sender send the same signal, then it is impossible for the receiver to distinguish them from each other (eg. in our example both types of workers acquire the same level of education)

For a more rigorous study of these models:

Laffont & Martimort, The Theory of Incentives, the Principal-Agent Model (2002) Microeconomics 2 Dr. F. Fatemi Page 178


In any separating eq. You should check two conditions for each type of worker:

Participation Constraint: Worker has incentive to participate in the labor market at the given level of wage for his type

Incentive Compatibility: Worker has no incentive to pretend that she is the other from the other type (mimic other type’s behavior)


In any separating PBE each type of worker receives her productivity level; that is:

𝑤∗�𝑒∗(𝜃𝐿)� = 𝜃𝐿 and 𝑤∗�𝑒∗(𝜃𝐻)� = 𝜃𝐻

In any separating PBE the low-ability worker chooses “no education”; that is:

𝑒∗(𝜃𝐿) = 0

Comparing to the case of no signaling, in any separating PBE:

The low-ability worker is worse off

The high-ability worker might be better off or worse off


It is possible that the signalling game has a pooling PBE;

In this case both types of workers get the same level of education, and should receive the same wage:

𝑒∗(𝜃𝐿) = 𝑒∗(𝜃𝐻) = �̂�

𝑤∗(�̂�) = 𝜆𝜃𝐻 + (1 − 𝜆)𝜃𝐿

It is trivial that a pooling PBE is weakly dominated by the no-signaling outcome.


• Screening

Another solution to the problem of unobserved types is to distinguish the types through offering them different contracts.

The similar setting to the signaling model:

Only two type of workers:

0 < 𝜃𝐿 < 𝜃𝐻

0 < 𝜆 = Prob(𝜃 = 𝜃𝐻) < 1

Suppose again:

𝑟(𝜃𝐻) = 𝑟(𝜃𝐿) = 0 Microeconomics 2 Dr. F. Fatemi Page 182


Firm can ask each worker to do a different task level 𝑡:

Tasks are costly to worker

For simplicity and make it comparable to signaling setting suppose that the higher task levels add nothing to the output of workers (productivity)

The cost of task level 𝑡 ≥ 0 for type 𝜃:

𝑐(𝑡,𝜃)

𝑢(𝑤, 𝑡|𝜃) = 𝑤 − 𝑐(𝑡,𝜃)

Where for ∀𝑒,𝜃:

𝑐(0,𝜃) = 0

𝜕𝑐(𝑡,𝜃)𝜕𝑡

> 0 , 𝜕2𝑐(𝑡,𝜃)𝜕𝑡2

> 0

𝜕𝑐(𝑡,𝜃)𝜕𝜃

< 0 , 𝜕2𝑐(𝑡,𝜃)𝜕𝑡𝜕𝜃

< 0 Microeconomics 2 Dr. F. Fatemi Page 183


The sequence of the game:

1) Two firms simultaneously announce a finite number of contracts (𝑤, 𝑡)

2) The worker decides whether to accept any of the offers or remain

unemployed

If the worker is indifferent between two offers chooses the one with the lower t



If the type of workers were observable in any SPBE, firms make zero profit and:

(𝑤𝑖∗, 𝑡𝑖∗) = (𝜃𝑖 , 0) 𝑖 = 𝐿,𝐻

If types are unobservable:

In any Eq. firms make zero profit

No pooling PBE exists



In any Separating Eq,

Each type of worker receives her productivity:

Low-ability workers accept contract (𝜃𝐿 , 0)

High-ability workers accept contract (𝜃𝐻, �̂�𝐻) which low-ability worker is indifferent between this and her own contact:

𝜃𝐻 − 𝑐(�̂�𝐻,𝜃𝐿) = 𝜃𝐿 − 𝑐(0,𝜃𝐿)



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