Teoria das organizações e contratos

Teoria das organizacoes e contratos

Chapter 7: The Adverse Selection Problem with a Continuum ofTypes

Mestrado Profissional em Economia

3o trimestre 2015

EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 1 / 25

Continuum of types

We consider a risk-neutral principal

The profit function is Π(e) when the agent’s effort is e

Risk-averse (or neutral) agent

U(w, e|k) = u(w)− kv(e)

Effort e ∈ R+

The disutility parameter k belongs to [k, k]


Information

Only the agent knows the exact value of k

The principal has ex-ante beliefs about the distribution of theagent’s type

These beliefs are represented by a cumulative distribution functionF : [k, k]→ [0, 1] with density f(·) > 0

F (k) =

∫ k

kf(x)dx


Contracts

The contract menu is a set of two functions

{e(·), w(·)}

where e(·) and w(·) are functions of the agent’s type k

Since the principal is risk-neutral, there is no need to allow wagesto be contingent on the results

The incentive compatibility condition is

∀k ∈ [k, k], k ∈ argmax{U(w(k′), e(k′)|k) : k′ ∈ [k, k]} (IC)

whereU(w(k′), e(k′)|k) = u(w(k′))− kv(e(k′))


The principal’s maximization problem

max{e(·),w(·)}

∫ k

k[Π(e(k))− w(k)]f(k)dk

subject to

the participation constraint

∀k ∈ [k, k], u(w(k))− kv(e(k)) > U (PC)

the incentive compatibility constraint



The Spence-Mirlees condition (CS)

Consider general utility functions U(w, e|k)

Definition

The agent’s preferences U(·, ·|k) satisfy the Spence-Mirlees condition(CS+) if

∂

∂k

(∂U/∂e

∂U/∂w

)> 0

while they verify the (CS−) condition if

∂

∂k

(∂U/∂e

∂U/∂w

)< 0


The Spence-Mirlees condition (CS)

The previous definition is slightly informal

The rigorous statement is as follows

Definition

The agent’s preferences U(·, ·|k) satisfy the Spence-Mirlees condition(CS+) if, for any contract (w, e), the mapping

k 7−→ ∂U/∂e

∂U/∂w(w, e|k)

is strictly increasing

Replacing “strictly increasing” by “strictly decreasing”, we havethe definition of (CS+)


The Spence-Mirelees condition

The condition (CS−) indicates that the lower is an agent’sparameter k (the more efficient is the agent), the higher is themarginal rates of substitution between effort and wage

The monotonicity of the MRS with respect to k is also called thesingle crossing condition, the sorting condition, or the constantsign condition

If U(w, e|k) = u(w)− kv(e) then

∂U/∂e

∂U/∂w= −k v′(e)

u′(w)


Single crossing property

Proposition

Under condition either (CS−) or (CS+), two indifference curves of twodifferent types only cross once

Fix an arbitrary contract (w, e) and let Iso(k) be the indifferencecurve of type-k passing through (w, e)

Iso(k) := {(w, e) : U(w, e|k) = U(w, e|k)}

We have to show that

Iso(k) ∩ Iso(k′) = {(w, e)}


Single crossing: proof

There exists a function e 7→ h(e|k) such that

(w, e) ∈ Iso(k)⇐⇒ w = h(e|k)

Assume (CS−) and choose k′ > k (the other cases can be analyzedin a similar way)

Let ∆(e) be defined by

∆(e) := h(e|k)− h(e|k′)

We have ∆(e) = 0

Assume, by way of contradiction, that ∆(e) = 0 for some e 6= e

Choose e the first value after e for which we have ∆(e) = 0


Single crossing: proof

Observe that∂h

∂e(e|k) = −

∂U∂e∂U∂w

(h(e|k), e|k)

It follows from (CS−) that

∆′(e) > 0 and ∆′(e) > 0

There exists ε > 0 small enough such thatI ∆(·) is positive on (e, e + ε)I ∆(·) is negative on (e− ε, e)

It follows that we must have ∆(e) = 0 for some e ∈ (e + ε, e− ε)

We get a contradiction with the property that e is the first valueafter e for which we have ∆(e) = 0


Implementable effort functions

Definition

An effort function e(·) is implementable when there exists w(·) suchthat


Proposition

An effort function e(·) is implementable if and only if e′(·) 6 0

We will only show that this condition is necessary


Implementable effort functions: necessary condition

Assume that (w(·), e(·)) satisfies (IC)

LetV (`, k) := U(w(`), e(`)|k)

We must have

∀k ∈ [k, k], k ∈ argmax{V (`, k) : ` ∈ [k, k]}

The first and second order conditions are

∂V

∂`(k, k) = 0 and

∂2V

∂`2(k, k) 6 0

The above conditions are valid for all k

Deduce that∂2V

∂`∂k(k, k) > 0


Implementable effort functions: necessary condition

Express the conditions

∂V

∂`(k, k) = 0 and

∂2V

∂`∂k(k, k) > 0

Using the definition

V (`, k) = U(w(`), e(`)|k)

Deduce that∂U

∂w× e′ × ∂

∂k

(∂U/∂e

∂U/∂w

)> 0


Participation constraints

Fix k 6 k′

Observe that (ICk) and (PCk′) imply (PCk)

We can restrict our attention to the participation constraint of theleast efficient agent

u(w(k))− kv(e(k)) > U

Proposition

The participation constraint of the least efficient agent is binding, i.e.,

u(w(k))− kv(e(k)) = U



Denote by I(k) the type-k agent’s utility obtained from thecontract (w(k), e(k)):

I(k) := U(w(k), e(k)|k)

Lemma

I ′(k) = −v(e(k))



Proposition

I(k) = u(w(k))− kv(e(k)) = U +

∫ k

kv(e(x))dx

The informational rent of type-k agent is∫ k

kv(e(x))dx

Given an effort function e(·), the above equation determines theprice function w(·)


(PC) and (IC)

We have proved that if (PC) and (IC) are satisfied thenI the function e(·) is decreasingI the function w(·) is given by

u(w(k)) = kv(e(k)) + U +

∫ k

k

v(e(x))dx

The converse is true

We can now restate the principal’s problem in a simpler way


The principal’s maximization problem

max{e(·),w(·)}

∫ k

k[Π(e(k))− w(k)]f(k)dk

subject to

u(w(k)) = kv(e(k)) + U +

∫ k

kv(e(x))dx

ande′(k) 6 0

The more efficient is the agent, the greater is the effort that theprincipal demands of him

The wage will be greater the more efficient is the agent

Greater efficiency corresponds to greater informational rent


Risk-neutral agents

To solve the principal’s problem, we assume that u(w) = w

The problem of the principal reduces to

max{e(·)}

∫ k

k

[Π(e(k))− kv(e(k))−

∫ k

kv(e(x))dx

]f(k)dk

subject toe′(k) 6 0

Denote by (P) the above problem

Once we have found a solution e(·) of (P), we can derive thecorresponding optimal wage function

w(k) = kv(e(k)) + U +

∫ k

kv(e(x))dx


Risk-neutral agents

We propose to solve an alternative problem without imposing therestriction e′(·) 6 0

We denote by (Prelax) the following problem

max{e(·)}

∫ k

k

[Π(e(k))− kv(e(k))−

∫ k

kv(e(x))dx

]f(k)dk


Risk-neutral agents

Show that∫ k

k

[∫ k

kv(e(x))dx

]f(k)dk =

∫ k

kv(e(k))

F (k)

f(k)f(k)dk

Deduce that the solution e(·) of (Prelax) satisfies

Π′(e(k))−(k +

F (k)

f(k)

)v′(e(k)) = 0

Proposition

The solution e(·) of (Prelax) is also a solution of (P) when thehypothesis of monotonicity of the risk ratio holds:

d

dk

(F (k)

f(k)

)> 0


Monotonicity of the risk ratio

For which distributions does the hypothesis hold true?

Uniform, normal, exponential, logistic, Laplace, . . .

Every distribution characterized by f(·) decreasing


When e is non-increasing

The solution e(·) of the planner’s problem is characterized by theequation

v′(e(k)) =Π′(e(k))

k + F (k)f(k)

Observe that the solution e?(·) under symmetric information ischaracterized by

v′(e?(k)) =Π?(e?(k))

k

There is no distortion for the most efficient agent: e?(k) = e?(k)

For all other agents, the principal demands less effort underadverse selection


Optimal distortion

∀k > k, e(k) < e?(k)

By distorting the effort of type-k agent, the principal reduces theinformational rent of all more efficient agents k′ < k∫ k

kv(e(x))dx instead of

∫ k

kv(e?(x))dx

There is a loss of efficiency due to the deviation from e?(k)

The principal must balance both effects: the optimal condition isgiven by

v′(e(k)) =Π′(e(k))

k + F (k)f(k)


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