Microeconomia 2 - Slides17.pdf

Microeconomía II

Federico Weinschelbaum

UdeSA

Motivation

Traditional Principal Agent Assumes exclusive contracts

Payment made by principal = agent’s consumption

What happens if this is not true? Examples

Coca-Cola’s manager can invest in Pepsi

Agent can sign other contracts

What other people observe?

Observe more than the principal (e¤ort) they can monitor e¤ort

Arnott y Stiglitz (1991) with the family Itoh (1993) with other agents

This paper

Analyzes the robustness of the traditional P-A results to the existence of insurers

Characterizes the equilibria when insurers are present under di¤erent environments

Time Sequences

Sequential O¤ering Sequential Contracting

Quantity of Insurers (One, Finite, Free Entry)

Implementing High E¤ort (2nd Best) remember

two levels of e¤ort {high, low} e¤ort not observable

Cost(implement high)>Cost(implement low)

implement high e¤ort₎Second Best Solutionw(π) (Risky)

EUa(high) =u (Individual Rationality)

What Happens if there is an insurer?

He insures the agent₎ low e¤ort They gain the risk premium

EUa(w2ndBest/high) =EUa(w2ndBest/low)<Ua

h

E(w2ndBest/low)i

The Model

Traditional Principal Agent + Insurers O¤ers are made sequentially

Some General Results

1 The principal is always active (without him nothing can be

implemented)

2 Implementation of low e¤ort, nothing changes.

3 Principal and active Insurers induce the same level of e¤ort

4 The traditional implementation costs are the lower bound costs for

the aggregate contract.(Insurers are not providing anything, if there is a better contract the principal could used before)

5 The traditional implementation costs are the lower bound costs for

Time Sequence (Sequential o¤ering)

Principal o¤ers Insurer 1 o¤ers Insurer 2 o¤ers ...

...

Insurer N o¤ers

Finite Number of insurers

Every Insurer makes zero pro…ts

Implementing Low E¤ort Multiple Equilibria Insurers can participate or not

Principal’s Contract

E(wp_je

l) =traditional

Could be risky

Aggregate Contract

Finite Number of insurers II

Implementing High E¤ort Multiple Equilibria Insurers can participate or not

But Last Insurer Must Participate. Principal’s Contract

1 R _wp_{(eh) =}R _w2ndBest_(eh)

Finite Number of insurers II

Principal’s Contract R

wp(e_h) =R w2ndBest(e_h)

R

w2ndBest(el)>φ[g(el) +u]

R

wp(el)

Departing from 2nd Best, R

w(eh)remains constant

R

w(el) reduces

Characteristics Contract is riskier

Finite Number of insurers III

Aggregate Contract 2ndBest

Last Insurer (only him) take the contract to the Second Best. There is no insurer after him..

Free Entry

Every Insurer makes zero pro…ts (Similarly) Principal’s Contract more expensive

E(wpjeh)>E(w2ndBestjeh)

Aggregate Contract

3rdBest

Third Best

The Problem is

max R W(π)f(π/eh)dπ subject to

R

v(W(π))f(π/eh)dπ g(eh) u (Individual Rationality)

R

v(W(π))f(π/eh)dπ g(eh)

R

v(W(π))f(π/el)dπ g(el)

(Incentive Compatibility) R

v(W(π))f(π/eh)dπ g(eh) v

R

W(π)f(π/el)dπ g(el)

Third Best Results

GNI is Binding, IC is Not If IR is binding

could not proof it is always binding

it is with non decreasing absolute risk aversion.

If IR is binding (we worked with that) GNI can be rewritten

u v(R W(π)f(π/el)dπ) g(el) (Non Insurability)

or

φ(u+g(el))

R

Third Best Results II

3rdBest cost is higher. There is a new constraint that former solution 2ndBest does not ful…lled

Agent utility remains u with higher expected payment. Thus, contract is riskier.

3rdBest contract is increasing inπ under the assumption that the monotone likelihood ratio property holds.

The Model predicts

Free Entry

Results are independent of the sequence

The principal could o¤er third best and nobody enters

Welfare Analysis

Free Entry of Insurers Decreases Welfare

No Insurers (or …nite number),Implements Low E¤ort Nothing Changes

No Insurers (or …nite number), Implements High E¤ort Two Cases Free Entry, Implement High E¤ort Contract is Riskier

Welfare Analysis II

∫ ∫πf(π/eh)dπ−πf(π/el)dπ

[ ]

∫_w3rd_f(_π/_e)_dπ−_φu+_g(_e)

e

e

e

e

∫ ∫πf(π/eh)dπ−πf(π/el)dπ

[ ]

1st _{( / ) ( )}

h l

w fπ e dπ φ− u+g e ∫

∫ ∫πf(π/eh)dπ−πf(π/el)dπ

Figure 1A.Effort implemented in equilibrium: first best.

[ ]

2n d _{( / ) ( )}

h l

w f π e dπ−φ u+g e ∫

e

e

Another Timing (Finite) Sequential Contracting (Agent

proposes)

Principal o¤ers

Agent accepts / rejects Agent o¤ers

Insurer 1 accepts / rejects Agent o¤ers

Insurer 2 accepts / rejects .

..

Agent o¤ers

Another Timing (Finite) Sequential Contracting (Insurer

proposes)

Principal o¤ers

Agent accepts / rejects Insurer 1 o¤ers

Agent accepts / rejects ..

Insurer N o¤ers

Agent accepts / rejects Agent chooses e¤ort level

Extensions

Uncertainty regarding the presence of insurers

Changing informational degree of information of insurers

Quali…cations to the Informativeness Principle

Semi Simultaneous O¤ers

With limited liability Second Best is not implementable even for …nite number.