teoria das organizações e contratos
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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]
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 2 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 3 / 25
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′))
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 4 / 25
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
∀k ∈ [k, k], k ∈ argmax{U(w(k′), e(k′)|k) : k′ ∈ [k, k]} (IC)
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 5 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 6 / 25
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+)
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 7 / 25
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)
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 8 / 25
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)}
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 9 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 10 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 11 / 25
Implementable effort functions
Definition
An effort function e(·) is implementable when there exists w(·) suchthat
∀k ∈ [k, k], k ∈ argmax{U(w(k′), e(k′)|k) : k′ ∈ [k, k]} (IC)
Proposition
An effort function e(·) is implementable if and only if e′(·) 6 0
We will only show that this condition is necessary
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 12 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 13 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 14 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 15 / 25
Participation constraints
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))
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 16 / 25
Participation constraints
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(·)
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 17 / 25
(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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 18 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 19 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 20 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 21 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 22 / 25
Monotonicity of the risk ratio
For which distributions does the hypothesis hold true?
Uniform, normal, exponential, logistic, Laplace, . . .
Every distribution characterized by f(·) decreasing
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 23 / 25
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
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 24 / 25
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)
EESP (FGV) Teoria das organizacoes e contratos 3o trimestre 2015 25 / 25
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