; .....
f
UNIVERSIDAD DE CHILE
FACULTAD DE CIENCIAS FÍSICAS Y MÁTEMATICAS
DEPARTAMENTO DE INGENERÍA INDUSTRIAL
A GENERAL APPROACH FOR SCREENING PROBLEMS WITHOUT THE
SINGLE-CROSSING PROPERTY
TESIS PARA OPTAR AL GRADO DE MAGÍSTER EN
ECONOMÍA APLICADA
TIBOR ALEJANDRO HEUMANN EPSTEIN
rRecmsaoo
Programe Formación Capital Humano Avanzado
26 DIC 2012
, !ngreso N~": ............... ).Q.Q._§.Q.............................
Destino: .......................................................................
PROFESOR GUIA:
NICOLÁS FIGUEROA GONZÁLEZ
MIEMBROS DE LA COMISION:
JUAN ESCOBAR CASTRO
ROBERTO COMINETTI COTTI-COMETTI
JUAN PABLO MONTERO A YALA
SANTIAGO DE CHILE
NOVIEMBRE 2011
Resumen
En este trabajo consideramos el modelo clásico de diseño de mecanismos, con un principal, que debe
tomar· una decisión o determinar la asignación de un bien, y agentes, que poseen información privada
que es relevante para el principal. Para utilizar de manera óptima la información de los agentes, el
principal diseña un menú de contratos, donde cada uno especifica la decisión que tomará el principal y
las transferencias que se le darán al agente. Dado este menú, cada agente elige el contrato que más le
favorece.
El objetivo del principal es diseñar un menú de contratos que maximice su bienestar, que puede
coincidir o no con el bienestar social. Existe una amplia literatura que considera este problema, sin
embargo la mayor parte de ésta toma como suposición fundamental que las preferencias de los agentes
satisfacen la propiedad de corte único (S. C.P. por sus siglas en ingles). Esta propiedad nos garantiza que
la valoración marginal de los agentes por el bien en cuestión cambia monótonamente con su información
privada. Para el principal esto simplifica significativamente el diseño del menú óptimo, ya que garantiza
que el problema de maximización que enfrentan los agentes, al elegir el contrato que más les favorece, es
un problema de maximización cóncavo. Como los agentes enfrentan un problema cóncavo, el principal,
al diseñar el menú de contratos, solo debe preocuparse localmente de la condición de primer y segundo
orden de los agentes.
En esta tesis consideramos el caso en que las preferencias de los agentes no satisfacen S. C.P.. Desde
un punto de vista técnico, al relajar este supuesto, se pierde la monotonicidad en las preferencias de los
agentes. Esto hace que para el principal no sea suficiente analizar las condiciones de primer y segundo
orden de los agentes, y deba analizar la decisión de cada agente globalmente. Por esto, el problema
de maximización para el principal es mucho mas complejo de analizar ya que no basta con maximizar
localmente los contratos para cada agente, si no que se debe considerar los efectos globales de cada
contrato.
En esta tesis, se introduce la condición de "doble cruce", que es un supuesto mas débil que S.C.P..
Asi, se encuentran condiciones necesarias para que un mecanismo sea implementable y también condición
necesarias para la optimalidad de éste. Estas condiciones son interpretadas desde un punto de vista
económico, lo que permite extender las intuiciones a una generalidad de problemas en que no se cumple
S.C.P. y entender las limitaciones que impone este supuesto.
Por otro lado, ocupando las condiciones necesarias, encontramos un nuevo método para solucionar y
encontrar contratos óptimos en el modelo que estudiamos. Este método permite transformar un problema
de dimensión infinita en un problema bidimensional, que se puede solucionar. Ejemplificamos el método
propuesto resolviendo dos ejemplos. La primera situación consiste en encontrar la forma optima de
arrendar una tecnología que queda obsoleta en el tiempo a una tasa desconocida para el principal. La
segunda es como regular la tecnología que usa un monopolio que produce externalidades negativas, pero
cuya eficiencia para implementar distintas tecnologías es información privada del monopolio.
1
Summary
In this thesis we consider the classic mechanism design model with a principal, that must make a
decision or determine the allocation of sorne good, and agents, which possess prívate information that is
relevant for the principal. To make the best use of the agents prívate information the principal designs
a menu of contracts, each of them specifying the decision the principal will make and transfers that will
be made to the agent. Given this menu of contracts the agents chooses the one that benefits him the
m os t.
The objective of the principal is to designa menu of contracts that maximizes his welfare, which might
or might not agree with the social welfare. The literature on the subject is extensive, nevertheless the
great majority is under the assumption that the agents preferences satisfy the Single Crossing Property
(S.C.P.). This property guarantee that the marginal utility of the agents for the good that is being
contracted upon changes monotonically with the prívate information. This simplifies the design of the
optimal menu of contracts for the principal since it guarantees that the agents will face a concave problem
when choosing the contract that benefits them the most, this in turn allows the principal to only consider
the first and second arder condition of the agents maximization problem when designing the menu.
In this thesis we consider the problem in which agents preferences do not satisfy the S.C.P. From a
technical point of view, relaxing this assumption implies that the monotonicity in the agents preferences
is lost. Thus, for the principal it is no longer sufficient to analyze only the first and second arder condition
that the agents will face, but the decision of each agent must be analyzed in a global way. This makes
the principal's maximization problem much harder since it is not enough to analyze the design of the
menu locally, but global effects must be taken in consideration for each contract.
In this thesis the Double Crossing Property is introduced, which is a weaker assumption than the
S.C.P. Necessary condition are found for the implementability and optimality of a menu of contracts.
This condition are interpreted from a economic standpoint, which allows to extend the intuitions to other
problem in which the S.C.P. is not fulfilled and understand the limitations that this assumption entail.
On the other hand, using the necessary conditions, a method is found that allows to find the optimal
menu in the model we study. This method allows to transform an infinite dimension problem in a two
dimensional problem, which can be solved. Finally the method is exemplified by solving two examples.
The first examples consists in finding the optimal way to lease a technology that becomes obsolete at
a time rate that is unknown to the principaL The second example is how to regulate a monopoly that
produces negative externalities, but the cost to implement different technologies, which produce different
amount of externalities, is unknown to the principaL
2
Agradecimientos
Le agradezco a mi familia por haberme apoyado, guiado y ayudado en todo lo que he hecho durante
mi vida, en especial a mis padres, que siempre me motivaron a buscar algo que me apasione, lo cual fue
un factor fundamental para que decidiera estudiar economía.
Le agradezco a Nicolás Figueroa por haber hecho de esta tesis una gran experiencia, por todo lo que
me ha enseñado y los innumerables consejos que me ha dado. Finalmente, sin su ayuda esta tesis no
habría sido posible.
Le agradezco a todos los profesores del CEA por ser parte fundamental de mi formación como
economista, en particular a Felipe Balmaceda por haberme ayudado y aconsejado en innumerables ocasiones durante el magíster.
Le agradezco a Olga Barrera y Julie Lagos por ayudarme, siempre con una gran disposición, con
todos los problemas administrativos que surgieron durante el programa.
Le agradezco a todos mis compañeros del MAGCEA, no solo por ayudarme en los distintos cursos
cuando lo necesite, si no que por convertir el programa de magíster en una gran oportunidad para hacer
excelentes amigos.
A las instituciones que me apoyaron en la realización de la tesis: CONICYT, por otorgarme la Beca
de Magíster; Vicerrectoría de Asuntos Académicos por otorgarme la Beca de Estadías Cortas.
3
Contents
1 Introduction
6
2 Model and Standard Results
7
3
Examples
3.1 Leasing . . . . . . . . . . . . .
3.1.1 Maximization Problem .
3.2 Externalities . . . . . . . . . .
9
9
11
14
4
Global Incentive Compatibility Constraints
16
5 Necessary Conditions for Optimality
19
6
21
Characterization of the Optimal Mechanism
6.1 Proof Strategy . . . .
6.2 Generalized Procedure ..
6.3 Increasing Policies . . . .
6.4 Optimal U-Shaped policy
6.5 Optimal Decreasing Policy .
21
23
27
27
32
7 Conclusions
44
8 References
44
9 Appendix
45
4
List of Figures
1
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7
9
9
Agents Preferences
U-Shaped I ....
Decreasing Policy I .
Agents Preferences Lease
Pointwise Maximization Lease .
Optimal Policy Lease . . . . . .
Pointwise Maximization Externalities.
Optimal Policy Externalities .
Iso-profit curve I ..
Iso-profit curve II ..
Decreasing policy II
Variational calculus.
Continuous policies .
Decreasing policy III
Function :E(·) I
Function :E (·) II .
Subset A
Subset B.
Subset C.
Subset D
Optimal U-Shape I
U-Shape policy II .
U-Shape example I
Optimal U-Shape example I
Optimal U-Shape example II
U-Shape example II . . . . .
Optimal U-Shape example III
Optimal Decreasing Policy I .
Bunclúng Policy I . . .
Discontinuous Policy I . . . .
Function x[~~:, 8](-) . . . . . .
Example Generalized Mínimum
Construction Decreasing Policies I
Construction Decreasing Policies II
Construction Decreasing Policies III
Construction Decreasing Policies IV
Construction Decreasing Policies V .
Construction Decreasing Policies VI
Construction Decreasing Policies VII
Decreasing Policy IV . . . . . . . . .
11
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5
1
Introd uction
Most of the mechanism design literature, and bayesian games in general, assumes that the agents'
preferences satisfy the single crossing property (S.C.P.). Under this assumption optimal contracts can
be easily characterized, and are in general always monotonic (the same thing happens with comparative
statics in bayesian games). Although this assumption is sometimes natural, this assumption involves
an important loss of generality, and the lack of tools to tackle more difficult problems is an important
bottleneck in the development of the field.
On top of the technical challenge, here are severa! reasons why understanding problems without the
S.C.P. is important. There are natural mechanism design problems that do not satisfy the S.C.P., in
section 3 we provide two. Quah and Strulovici [4] study and provide properties under which the sum of
functions that satisfy the S.C.P. inherit the S.C.P., and show that these conditions are quite restrictive.
Bernheim [2] and Bagwell and Bernheim [3] are two examples of signaling games in which, only by breaking the S. C.P., interesting phenomena arises. Finally, it is worth mentioning that severa! of the problems
and complexities that arise in a single dimensional model without S. C.P. also appear in multidimensional
screening, since both types of problems need to deal with non-local incentive compatibility constraints
. Thus, we believe that understanding mechanism design problems that do not satisfy the S.C.P. is a
fundamental step-stone to understand multidimensional screening.
If the S.C.P. is satisfied, we can arder types according to their marginal valuation of the allocation
(probability of winning in a auction, quantity produced in monopoly regulation, etc.), which in turn
allows to consider only local incentive compatibility conditions. Basically, the iso-profit curves, in the
space of the contracted good and transfers, can be ordered between types according on how "steep" they
are, therefore they always cross at most once. We study a model in which we can arder types according
to the concavity of the utility of the contracted good, therefore in this model the iso-profit curves always
cross at most twice. Although these problem might seem similar without the S.C.P. the local incentive
compatibility conditions are no longer sufficient to guarantee global incentive compatibility. This simple
extension leads to contracts that are qualitatively different to the ones that arise with S.C.P., allowing
to extract economic intuitions that could not be found otherwise. By giving a better understanding
of this model, in which the S.C.P. is not fulfilled, we can reach a better understanding of the real
implications that the assumption of the S.C.P. has on a model and how it restricts the richness of the
studied phenomena.
One of the main properties that is lost when the S. C.P. is not fulfilled is the monotonicity of the
optimal contract. As Arauja and Moreira [1] have shown, we can have optimal policies with a U-Shaped
form. Moreover, we show that without the S.C.P. there are two types of distortions that can be identified.
The first one, which is standard, is when sorne type is distorted because it is part of sorne active I.C.C.
(either is indifferent between his assignment and the assignment of sorne type iJ or vice versa), we cal!
this type of distortion a direct distortion. The second one, which is only found in problems in which
the S.C.P. is not satisfied, is given by types that are being distorted from the optimal assignment even
though there is no active I.C.C. involving that given type, which we callan indirect distortion. The latter
case is a very interesting situation in which a small variation in the assignment of sorne type could break
the I.C.C. between other two types, and highlights one of the main difficulties tackled in this paper.
Our closest reference on the literature is Arauja Moreira [1], in which they study almost the same
model, so naturally we use and extend severa! of their results (we also try to keep the notation as similar
as possible). They find necessary conditions for optimality when there is discrete pooling between types
(which is obviously a case of direct distortion) and find a specific setup in which their conditions are
sufficient for optimality. In this setup the only relevant global I.C.C. that need to be taken into account
to find the optimal policy arise from discrete pooling types. We are able to extended their results to
the general case of direct distortions, and also derive necessary conditions for optimality when indirect
distortions are optimal, which is one of our main theoretical results. Another important contribution
is the proposed method to find the optimal policy in a wide range of problems, almost completely
independent of the principal's objective function, which greatly generalize Arauja and Moreira's results.
Our proposed method breaks down the general problem into simpler problems, and the final result gives
us policies with discrete pooling, indirect distortion and direct distortion by types that are not being
pooled. Both examples in section 3 are solved using this method.
e
e
6
2
Model and Standard Results
The model is a basic principal-agent relationship. Agent's utility and the principal's utility depend on a
decision to be made by the principal, indexed by x E [;1:_, x] S: R, and sorne information privately know
by the agent, indexed by e E [fl, O] C R. Transfers can be made between the principal and the agent,
indexed by t E R, which are linear in the utility of the agent and principal. The principal has a prior
distribution on the agents prívate information, given by P(e) with P(fl) = 1- P(O) = O and density
p(e) >O V() E [fl, O]. The principal must designa menu of contracts that maximizes its expected utility.
Agents utility is quasi-linear in transfers, and is given by v(x, e)+ t, agent's have a reserve utility of
0 and the principal's utility has the form u(x, e)+ wv(x, e)- (1- w)t, with w E [0, 1].
Assumption l. We will assume that Vxxe(x, B) ;:::
o and Vxee(x, e) ;::: o for all x, e. 1
Remark 2. From the previous assumption, and using the implicit function theorem we can define a
unique decreasing function xa(e), such that Vxe(xa(e), e)
xa(e)
=o.
So we know that Vxe(x, e)
>o
{===}
X>
X
(}
Figure 1: Agents Preferences
\Ve define a mechanism as functions t(-) : 8-> R and x(·) : 8
{x(e), t(e)} is incentive compatible if and only if:
e
E
---+
[;1:_, x], and we say the mechanism
argmaxe•v(x(e'),e) +t(e')
By the revelation principle we know that the principal can restrict to I.C. mechanisms, and thus the
principal's problem is given by:
max
{x(B'),T(B')}
s.t.
(e u(x(e), e)+ wv(x(e), e)- t(e)dP(e)
}¡;;
v(x(e), e)+ t(e) ;::: o
{ e E argmaxB'E8 v(x(e'), e)+ t(e')
Using standard techniques, we can show any I.C. mechanism x(·) must fulfill the following conditions
Lemma 3. (Local J.C. C.)
Let {x( ·), t( ·)} be an J. C. mechanism then the following condition must be fulfilled:
1. (F.O.C.) VX(e) := v(x(e),e) +t(e) = VX@
+
J; v (x(z),z)dz
8
1
ve
E
8
This is a slightly stronger assumption than the one done my Arauja Moreira[l], they consider the case in
which Vx (x, O) is cuasi-convex in O instead of strictly convex. The reason why we need this slightly stronger
assumption is so we can get that the function o(O) = v(x1, O)- v(x2, 1!) is quasi-convex for al! x 1 , x 2 E [;l¿, x]
7
2. {S. O. C.) x( ·) is non-decreasing {non-increasing) in C S+ (C S_)
Using lemma 3 we can rewrite the principal's problem as follows :
max
{x(·)}
re [u(x(e), e)
}r,;
+ v(x(e), e)- (1- w )z(e)ve(x(e), e) ]p(e)de
f(x,e)=virtual surplus
s.t.
ve E e
ve E e
v(x(e, e)+ t(e) 2 o
e E argmaxB' v(x(e'), e)+ t(e')
{
x(-)is non-decreasing (non-increasing) in C S+ (C S_)
where z(e) is defined by:
2 _
z(e) -
P(e)
p(e)
_1-P(e)
{
p(e)
if ve ::; O
if ve 2 O
Proof. The previous results are standard in the mechanism design literature, and require no further
comment, the proofs can be found in Araujo and Moreira [1].
D
We will assume the virtual surplus f(x,e) is quasi-convex in x and it is maximized at x 1(e). Following
Araujo and Moreira [1] we can define the Global Incentive Function (GIF ) as follows:
iJ?"' (e, 0) := re
Je
1x~B) Vxe(x, 0)dxd0
(1)
x(e)
Araujo and Moreira [1] prove the following result:
Lernrna 4. A mechanism x(·) is J.C. if and only ifiJ?(e,e') 2 O for all e, e'
E
8
1.
Proof. The proof can be found in Araujo and Moreira [1].
D
Rernark 5. Lemma 4 allow us to turn a two stage maximization problem into a iso-perimetric problem.
To explain the problems presented by the incentive compatibility constraints in this model we consider
continuous mechanisms. If the policy is non-decreasing in es+ and non-increasing in es_, there will be
a ea such that the policy is non-decreasing for types higher than ea and non-increasing for types lower
than ea. Thus we can see that we may have two separate types e1, e2 E
such that x(el) = x(e2) = ~,
and obviously there must be a unique transfer associated to allocation ~· The first challenge is to find a
policy that satisfies the local I.C.C. and keeps the same transfers for pooling types.
e
2
For simplicity we consider the case where ve(-) does not change sign, or w = 1
Note that the integrals of <Px(e,B) might be integrating in a negative direction, making the zones
integrate positive value. This will be the case when analyzing decreasing policies.
1
8
es_
X
xo(B)
()
Bo
Figure 2: U-Shaped I
The second challenge is presented by the I.C.C. for decreasing policies In the following figure we have
a decreasing policy x(-), that is locally I.C. and "close" to x 0 (B). For policy x(·) to be I.C., condition 1
must be satisfied. However, q,x(e 2 ,B 1 ) (represented by the shaded area) might be negative.
X
Xo(B)
x(B)
()
Figure 3: Decreasing Policy I
Let's consider the previous figure and take the case in which x(-) is I.C. and q,x(e2, 8¡) = O. Since
the value of q,x(e2, 81 ) depends on the assignments given to all types in between (8 1 , Bz) we might have
that there are types in (81 , 82 ) that are being distorted even though all I.C.C. involving these types are
not binding. So, even for types that don't have their I.C.C. binding with any other type, it might be
necessary to keep them " low" so that the shaded area is positive. This is the case we call indirect
distortion.
3
3.1
Examples
Leasing
Consider a principal who leases a technology from an agent. The expected lifespan of the technology is a
prívate information of the agent, as different technologies become obsolete at different rates. \Ve assume
that at time t a technology becomes obsolete for the agent with probability e-et' but becomes obsolete
e
9
for the principal ata higher rate with probability e-(B+r)t (r can also be a an extra discount rate). 1
A technology O, if not obsolete, gives an instantaneous income of Oto the agent and has a coste(O)
with e' (O) < O. Therefore, the utility for agent O of having access the technology O from time t on is:
v(t, O)=
1:
oe- 88 ds- e( O)= e-et- e( O)
The principal can extract a higher instantaneous income from the technology than the agent, (30.
Thus, the utility for the principal to keep the technology up to time t is:
0
u(t,O) = ¡t (30e-(B+r)sds = (3- s=O
O+ r
(1- e-(B+r)t)
~
=(3(8)
The principal offers a menu of contracts { t( 0), T( O)}eEe, specifying the time at which the lease ends
and the transfer made to the agent. The total income of agent O if he chooses contract O' is given by
V(O', O)= e-et(B') -e( O)+ T(O').
Since the principal maximizes expected utility, the principal's problem is given by:
U=
max
{t(B'),T(B')}
fe {(3(0)(1- e-(r+B)t(B'))- T(O')}dP(O)
}\-
s.t. I.C.C. & P.C.
Looking at the agent's preferences it is easy to see that the single crossing property is not fulfilled,
since Vte changes signs. This comes from the fact that it is unclear which agent is willing to receive less
for delaying the end of the lease in a amount of time flt. On the one hand, a technology of a higher
O receives a higher income stream and thus is willing to forfeit more transfers in arder to receive the
technology earlier. On the other hand, agents with with a higher O have a higher discount value (the
income stream is more front-loaded) and thus the present value of the income stream decreases with O,
making him less willing to forfeit transfers to receive the technology back earlier. In fact we can have
that the first effect dominates, in which case t andO are strategic substitutes (vte <O), or the second,
in which case t andO are strategic complements (vte > 0).
Another way to look at the problem is also interesting. The income that agent O gets from receiving
the technology back at time t is e-et, which is also the probability of technology O not being obsolete by
time t. Thus we can see that higher O's are technologies that become obsolete at a higher rate and thus,
conditional on not being obsolete at time t, have a higher probability of becoming obsolete in the extra
time flt, requiring a higher compensation to delay the end of the lease. However higher O's are already
obsolete by time t with a higher probability and, if that is the case, require no extra compensation to
delay the end of the lease. Then, they require a lower compensation to accept a delay in the expiration
date of the lease.
Noting that Vte = e-et(Ot- 1) we have the following figure:
1
The same model also applies to several other situations like exclusivity, copyright, venture capital etc.
10
t
(}
(}
Figure 4: Agents Preferences Lease
To simplify the analysis, we make the following assumptions:
l. If the principal did not exist, it would not be profitable for the agent to develop any technology.
3
That is: v(O,e) = 1- c(e)::; O.
2. Al! technologies would be implemented if there were fui! information. That is f3 e!r - e( e) ;:: O
3. Higher e's are more profitable for the agent. That is ve(t, e) = -te-et -e' (e) ;:: O
In particular, to get an exact solution, we assume that e( e)= 1- 2e, P(e) is uniform in the interval
and r = L With this, (1) to (3) are fulfilled,
[!, 1], f3 = 4.5
3.1.1
Maximization Problem
Lt's consider the case in which there is fui! information (FI), so the principal just maximizes the surplus,
since he can extract al! rents, We have:
e
tFI (e)= argmaXt /3--(1- e-(B+r)t)
e+r
-1
=?
+ e-et- c(e)
1
tF 1 (e) = -;¡:-Log(fJ)
The result is quite intuitive. From an efficiency point of view the optimal time at which the lease
must end is independent of and it is given by the time at which the extra stream that the principal
receives offsets the extra discount rate it confronts, Thus, tFI is the time in which the present value of
the stream that the principal and the agent get from the technology are the same,
e,
Going back to the problem with private information and using standard techniques we can rewrite
the principal's problem as:
U=«~)
k{
e-et(e)- f3(8)e-(r+O)t(e)
+ z(8)te-et(e)
}p(8)d(J + Ep[H(8)]
=f(t,e)
s.tJ.C.C.
1
where z(e) = ~feje), f(t, e) is the virtual surplus and HC) does not depend on t(} Pointwise
maximization of the virtual surplus leads to:
3
With this, the critica! type is not endogenous
11
This has a straightforward interpretation. The principal's income is given by the surplus minus the
informational rents. Maximizing the first term is equivalent to m=imizing total surplus (it is maximized
at tF 1 ) while the second part corresponds to minimizing the informational rents, which are minimized
at ta(e), where Vte =O.
We know proceed to show the optimal solution. As we can see in the next figure, we can divide
the type space e in two zones. Long lived technologies in [~,el] and short lived technologies in [el, B].
Pointwise maximization leads to a lease time t higher than tFI for long lived technologies, while the
opposite happens for short lived technologies.
t
1
1
(}
to((J) (}
(}
Figure 5: Pointwise Maximization Lease
However, t 1 (-) does not satisfy even the local I.C. constraints (it is decreasing in a region where
is positive). Moreover, a naive "fixing" of this does not solve the problem. Consider, for example,
the policy x(·) given by
VtB
t
1
1
(}
to((J) (}
(}
Figure 6:
Comparing the utility obtained by an agent of type Bthat tells the truth or declares sorne other type
[~, 1 ) one can verify the latter gives a greater utility. Using the local l. C. constraints the difference
between both utilities can be written as:
e2 E
e
12
<'I!x (e, e2)= v(x(e), e)+ T(e) - (v(x(e2), e)+ T(e2))
=
v(x(e), e)+ T(e) -v(x(e2), e)-
T(e2)
'-----v----'
....._,_.,
V(ii,e)
V(e2,Bz)-v(x(Bz),e2)
= v(x(e2), e2)- v(x(e2), e)+ V(e, e)- V(e2, e2)
- J/2 ve(x(02),s)ds
e
=-
x(e2)
11
e2
J/2 ve(s,s)ds
Vxe(z, s)dzds
x(z)
If éJ!X(e, e2) <o for sorne e2, then the an agent of type e would prefer to declare e2. In this case the
area that needs to be calculated is shown in the previous figure, and it is easy to note that for e2 close
enough to e1 the area in which Vxe > O will domínate the area in which Vxe < O, and thus the policy is
not incentive compatible.
Intuitively, what is happening is that policy x(·) assigns a higher t to types lower than e1 , which is
compensated with higher transfers. Locally, lower technologies need a smaller compensation for delaying
the end of the lease because the dominating effect is the bigger income stream. But a technology is
very likely obsolete by time tFI, and thus his owner is willing to get an extra transfer in exchange for
sorne extra leasing time.
e
The optimal policy, computed with methods developed in this paper, is shown in the next figure:
t
,...------~1 t*((})
1
:ti
~----~~------------~
:t¡((})
(}
Figure 7: Optimal Policy Lease
We can see that long lived technologies are distorted from tFI in the direction of the point-wise
maximization, although not as much as pointwise maximization would dictate. Short-lived technologies,
on the other hand, are distorted from tFI away from the pointwise solution. The intuition behind these
distortions líes in the global I.C. constraints. They force short lived technologies to be distorted in the
same direction as the long lived technologies, to avoid non-local deviations. For this reason, since the
distortion of the long lived technologies affects the choices that can be made about the short lived ones,
long lived technologies are distorted less. The resulting policy consists in two zones in which the policy
is bunching and a middle zone that has a U-shaped form, in which there is a discrete pooling between
long and short lived technologies.
Through out the paper we will show how to find the optimal U-Shaped form in a general context.
We show that the effect that the global LC.C. ha.ve in the policy, which leads to this kind of U-shaped
form in which there is discrete pooling, can be easily weighted.
13
3.2
Externalities
Let's consider a monopoly that has constant marginal costs equal to xe, where x is the technology
adopted by the firm and e is the firm's efficiency. The firm can decide its production leve!, but the
principal (e.g. a governmental agency) may force it to adopt a given technology. For simplicity we
assume that the firm faces a linear demand. Thus, the operational profits made by the firm are given
by:
v(x, e)= m;x (A- bq)q- exq
(A- ex) 2
4
= -'----b_:_
As a side effect, the firm produces a negative externality. More expensive technologies (higher x) are
cleaner.
We assume that externalities are proportional to the quantity produced, and are reduced proportionally to the technology adopted. That is,
=?
Externalities = q(E- ¡3x)
The principal offers a menu {x(e), T(e)}eEe and maximizes the expected value of
u(x, e) = Firm's Profits + Consumer's Surplus - Externalities
It is easy to see that: Firm's Profits
the principal's problem is given by:
2
= (A~:x) +T and Consumer's Surplus =
2
(A-;;:x)
-T, therefore
s.t. I.C.C. & P.C.
Unlike the previous example, since the principal doesn't care about transfers, the virtual surplus is
the same as the total surplus.
Again, it is unclear which firm is willing to pay more to decrease x, so the S. C.P. is not fulfilled. On
the one hand, more efficient firms are less affected by x (since x ande are complements), and thus more
efficient firms are less willing to pay for a marginal decrease in x. On the other hand, more efficient
firms want to produce more, and thus more efficient firms might be willing to pay more to reduce x.
Once again we can see that it is unclear what effect dominates. It could be that x and e are strategic
substitutes (vxe < 0), or strategic complements (vxe > 0).
Final!y it is worth mentioning that in case the marginal costs would have the form
x, one of the
effects would disappear and the S.C.P. would be fulfilled. If this would be the case, then more efficient
firms would always be more willing to pay for a decrease in x.
Going back to the maximization problem
e+
2
U=
max
{t(B'),T(e')}
{ { (A-ex)
} 8
4b
+ (A-ex)
2
-q(E-¡3x)}dP(e)
Sb
=f(x,e)
s.t. I.C.C. & P.C.
Pointwise maximization (which is the same as the first best in this case) leads to
Looking at the next figuré, we can see that
4
xFI (-)
All graphs are done for the case A= 5, b = 1, E=
satisfies all the local I. C. constraints.
Jt, {3 = 5, ft = 2, e= 4 and a uniform distribution for
p(B)
14
X
(}
(}
Figure 8: Pointwise Maximization Externalities
Nevertheless, xFI is not I.C. We compare the income that type e receives by declaring e or (]_ under
this policy. Using the local LC.C. the difference between both utilities can be written as
<PXFI
(e, fl)= v(xF I (e), e)
=
+ T( e) -
(v(xF I ((]_),e)
v(xFI (e), e)+ T(e) -v(xFI (r]_), e)-
+ T(fl))
T(fl)
"-v-'
V(.Ul)-v(x(f!_),f!_)
V(B,B)
= v(xF 1(f]_),f]_)- v(x(f]_),e)
+ V(e,e)- V(r]_,f]_)
'-----v---'
Jf ve(s,s)ds
It is unclear to the naked eye if this area is positive or negative in this case, but for these parameters
the corresponding value is -0.19393. This means that the difference between the transfers paid to (]_ and
e is bigger than the cost to e of implementing XF I (fl) instead of XF I (e)' which means XFI (-) is not LC ..
Note that the marginal rate of substitution between the transfer t and the technology x at sorne level
xF 1 ( íJ) is given by the marginal cost of the technology to the respective type íJ (to keep him indifferent
between telling the truth and making a local deviation). Therefore, the difference between the transfers
paid to (]_ and does not only depend on the difference between the technologies assigned to them, but
also on the rate of substitution between the technology and the transfers of all technologies in between
xFI (e) and xF 1 ((]_). Therefore, the difference in the transfers paid to (]_ and e depends on the whole path
of XFI (-) between (]_ and e.
Since the optimal policy is not I. C., it is clear that the optimal policy must lie below xF1( ·) at least in
sorne interval so that the computed area has bigger zone in which Vxe < O, which in turn would increase
the value of the computed area. Intuitively, this implies that more efficient firms are getting a given
technology x, which lowers the rate of substitution between technology and transfer, therefore lowering
the difference between the transfers paid to (]_ and
The following figure presents the optimal policy
e
e.
x*(·)
15
ro(O)
/'(0)
.r"(O)
Figure 9: Optimal Policy Externalities
We can see that the optimal policy has a bunching zone in [~, B] and a strictly decreasing zone [B, e].
Since the difference in transfers between and ~in x* (·) depends on the whole path of x* (·) in [B, e] this
zone is distorted from xFI (-) to lower this difference. On other hand, x* ( ·) can be seen as truncated at
B* because the higher the assignment for ~ the more difficult it is to keep the difference in transfers low
enough, and thus truncating the optimal policy allows to lower the distortion of interior types.
Intuitively in zones of the policy in which Vxe is bigger in absolute value we have that the difference in
preferences between types are more pronounced, and thus a bigger decrease in transfers can be achieved
with lower distortions. Geometrically, adding area where Vxe is bigger in absolute value is more efficient.
8
8
As a result in the optimal policy all the strictly decreasing part has the same ratio 1:~:~~ )(%?:0~ ), which
is the reason why the optimal policy in the precious figure has zones in which the distortion is bigger
than others.
So far we have only given the intuition on how the global I.C.C. between and (}_ is managed, which
is sufficient for all the global I.C.C. to be fulfilled in this particular example. This is not the general
case, and through out this paper we show how the global I.C.C. are managed in a general way, and we
characterize the optimal solution for these cases, but the previous intuitions are kept.
e
e
4
Global Incentive Compatibility Constraints
Before giving any results concerning the I.C.C. in the case of a continuum of types we will analyze what
kind of policy are implementable in this model but concidering just two types, eH > eL. To take a
case in which the S.C.P. is not fulfilled we will consider the case in which exists ~t E (;f, x) such that
Vx(~t, eH) = Vx(~t, eL)· Since in this model Vxx8 > o, we know that Vx(e, eH) > Vx(C eL) for e > ~t
and vice-versa. Looking at the I.C.C. between (JH andeL :
Thus, any assignments for eH and eL that are implementable through transfers must satisfy the
previous inequality. Just by looking at the previous inequality, we can see that for any given x(eH)
there is a bounded space where x(eL) can be implementable. Moreover, we know that x(BH) 2: (:S:)~t =?
x(eL) ::; (;:::)eH, so we can know if x(eL) is bigger or smaller than x(eH) based only in x(eH)· The
previous can easily be seen graphically, the following two plots show the curves of iso-profit of eH and
eL, and show the implementable space left for x(eL) and x(eH) when x(eH) and x(BL) are held fixed
respectively.
16
X
X
~
~
~--------------------T
Figure 10: Iso-profit curve I
T
Figure 11: Iso-profit curve II
From the previous analysis it is easy to see that the point at which the isocurve between two types
are tangent is important,
Definition 6. Let e E [fl, B], for a given e E [;,¿, x] if there exists e' i= e such that Vx(e, e)= Vx(e, e'), we
will define é(e, e) = e', Just by the assumptions made of the agent's preferences we know that if é(e, e)
exists, then it is unique. Moreover ifé(e,e) <e then we know that Vxe(e,e) >o> Vxe(e,é(e,e)), and
viceversa,
just using the previous intuitions we can make the following observation on the I.C.C. for a continuum
of types:
Proposition 7. Let x(·) be an J. C. mechanism, and let e' be such that é(x(e'), e') exists and é(x(e'), e')
e'. Then the following must hold true:
::; x(e')
x(e)
{
2: x(e)
= x(e')
<
e E [é(x(e'), e'), e']
e E [fl, é(x(e'), e')]
e= é(x(e'),e')
Proof. The previous result is direct from the I.C.C. between two types, described previously. It can be
deducted from the fact that for a given e and x(e) in which exists é(e, e) <e, then the tangency point
is below x(e) for e'> é(e,e) and above x(e) for e'< é(e,e), and the rest is trivial from the the I.C.C.
between two type (just rename e= eH, and any eL< e, and see where the tangency point is with respect
tox(e)),
D
The previous proposition shows that the assignment of any given type induces a shape on al! lower
types, which consists in separating the lower types in a zone in which the assignments are lower and a
second zone in which the assignments are higher. The following theorem will be very useful in characterizing the optimal policy since it helps disconnect the optimization problem by allowing us to find the
optimal policy for the types that have lower and higher assignment independently.
Theorem 8. Let i:(·) be an J. C. policy, then for any given interval [e1, e2]such that
we have that the policy i:' defined by:
Xa
= x(ez) = x(e1),
is also J.C.
The opposite is also true. That is to say, let i:(·) be J.C. and such that the policy is bunching in an
interval [el, ez]. For any policy x(-) defined in [e1, e2], such that x(-) is J. c. and such that X a = x( e2) =
x(e 1) = x(el) , then policy i:'(-) defined by:
17
x'(e) = {x(e)
x(e)
e E ¡.e_, e1] u ¡e2, e¡
e E ¡e1, e2]
is also J. C.
o
Proof. The proof can be found in the Appendix.
As we explained in the the description of the model the I.C.C. in this model presents two challenges,
one concerning with discrete pooling and the second one concerning decreasing policies. The previous
result will allow us to disentangle the complete optimization in subproblems, which will allow us to
separate the challenge concerning discrete pooling with the one concerning decreasing policies. The next
theorem will be useful to characterize decreasing policies, it will ensures that binding global I.C.C. are
always between pairs of types, moreover it will allowing us to find a order in the active I.C.C. This will
allow us to describe the global I.C.C. as "nested", which significantly limits the possible combinations
of active constraints.
Theorem 9. Let x(e) be an J. C. mechanism, such that x(e) is decreasing in some interval [e,, e13]. For
any (el < e2::; es < e4) E [e,, e¡3] it can never hold true that <I>(e4, e2) =O and <I>(es, el)= O.
Proof. The proof of theorem 9 can be explained easily geometrically. Consider the following figure with
a decreasing policy:
X
e
Figure 12: Decreasing policy II
The GIF function between e1 and es is given by area A+C weighted by Vxe and the GIF function
between e2 and e4 is given by area C+D weighted by VxB· So jf the I.C.C. between el and es is active,
and the I.C.C. between e2 and e4 is active it means that the areas A+C weighted by Vxe are O and the
areas C+D weighted by Vxe are O. We also know that for the policy to be I.C.C. the area C weighted
by Vxe must be positive, so we have that areas A +C+ D weighted by Vxe is positive. Finally, it can be
shown that area D weighted by Vxe, must be negative, and thus A+B+C+D weighted by Vxe is negative,
which means that 4 would want to jump to e1 , and thus the mechanism is not I.C.
The details can be found in the Appendix.
O
e
So far we have used the characteristics of the functional form of the agents preferences to find global
properties of implementable policies. The next lemma is result found by Araujo and Moreira [1], which
we will present here since we will use it. This lemma ensure that the local necessary conditions of types
that are binding by global I.C.C. are compatible.
Lemma 10. Let x(·) be an implementable decision ande,(} E e be such that <I>(e, B) =o
{i} Jf x(·) is strictly monotonic and continuous at iJ then
. {ii} If x(·) is continuous ate , then
ve(x(B), e)= v 8 (x(e), e)
18
To ihterpret lemma 10 let's consider x(·) to be a policy that fulfills al! hypothesis of 10. Equation
<I>(e, e)= o means that e is indifferent between the contract offered to him and the contract offered toe.
\Ve have that x(·) is strictly monotonic and continuous ate, which means that the policy is screening
locally at
Since e is indifferent between the contract offered to him and the contract offered to
if
the contracts offered Iocally around were to be offered to e we must have that e chooses
contract
(else x(·) would not be LC.). Hence preferences of e ande must be indistinguishable around x(e), which
is condition (i). Mathematically, the marginal change in transfers for a marginal change of allocation at
x(e) is exactly vx(x(e), B), and <I>(e, B) =O implies that e is indifferent between his contract and the one
given to so condition (i) guarantees that e will not want to change to a marginally lower allocation or
a marginally higher allocation than x(e). In other words condition(i) guarantees that e doesn't want to
change to a neighborhood near
Since x(·) is I.C.C., locally around e al! types receive the same utility as if they were choosing x(e)
(at least in a first order approximation), and thus choosing x(e) cannot yield a higher utility than x(e)
foral! types in a neighborhood around e. Hence, condition (i) states that al! types in some neighborhood
around e al! types must receive (almost) the same utility from choosing x(e) and x(e). Mathematically
to interpret the second part I will rewrite the equations as follows:
e.
e
e's
e,
e,
e
ve(x(e),e) = ve(x(e),e) ~
Notice that the term
e
I:c~l Vx(z, e)dz
a ¡x(B)
-a
'P
vx(z,rp)dzl 10 =e =O
x(B)
gives exactly how much e values the difference between the
e,
allocation given to and to him. Since e is indifferent between his contract and the one given to
condition (ii) guarantees that no type in a neighborhood near e values this difference more than e,
because in this case he would rather contract than his. In other words condition(ii) guarantees that
no type in a neighborhood of wants to change to
e
e
5
e
N ecessary Conditions for Optimality
The next theorem is one of our main results. It gives an optimality condition for al! types whose I.C.C.
have some slack in the optimal policy. This is the case as we mention in the introduction of indirect
distortion since non of the global I.C.C. involving the given type is active, nevertheless they need to be
distorted to ensure the fulfillment of the I.C.C. among other types.
Theorem 11. Let x*(·) be an optimal policy, which is continuous in (e 1 , e2).
1. Suppose that\:fe E (el,e2)VB' E e q,x*(B,B') >O, then fx(x(~~~B)p(B) is non-increasing in (8 1 ,82)
2. Suppose that \fe E (81, B2)VB' E
e
q,x* (B', e) >O, then fx(x(~~:)p(B) is non-decreasing in (B1, B2)
o
Proof. See Appendix.
Remar k 12. It is worth mentioning that theorem ?? holds for any functional form of the agents pref-
erences v(x, B) or the prinicpal's objective function f(x, B).
Corollary 13. Suppose that x*(·) is continuous in some interval (8 1 , e2), and <I>(B, e')> O and <I>(B', B) >
O for all BE (el, B2) and B' E e. Then,
fx(x*(B), B)p(e) .
.
Vxe(x*(B), B) ~s constant m (B1, B2)
Note that corollary 13 implies that the optimal policy must keep the value of fv(~~~~(/ constant unless
there is an incentive compatibility constraint that is binding, that is:
Corollary 14. Let x* be the optimal policy. Then for all B' E [Q, B] such that x* is continuous at B' one
of the following two condítions must hold:
19
J
_,¿_ fx(x*(B),B)p(B)
0
· dB Vxe(x*(B),B) B=B J
,
_
2. 3()"E8suchthat<I>x'(e',()")=O
V
<I>x*(e",()')=O
Corollary 13 has an interesting interpretation. First note that Vxe(x*(()), ()) can be interpreted as
the "price" of distorting type e. Consider a policy such that there are no binding I.C.C. for al! types
in (ell () 2), and a principal who wants to increase x(() 2) in dx' without affecting the I.C.C. between
types that are outside (() 1 ,()2 ) (we are disregarding the local I.C.C.). This change would would increase
<I>x* (() 2, () 1 ) by an amount of Vxe(x* (() 2), () 2)dx' d() (which in turns increases <I>x* (()A, ()B) by an amount of
Vxe(x*(()2),()2)dx'd() for all types ()A< ()1 and ()B > ()2 ). In arder to keep <I>x*(eA,()B) constant, it is
necessary to decrease x* ( ·) at sorne other type ()1 by an amount dx such that:
Vxe(x* (el), ()l)dxd()
+ Vxe(x*(()2), ()2)dx' d() =O
X
()
Figure 13: Variational calculus
We make the analogy to an agent maximizing utility under a fixed budget. An increase dx' in the
consumption of one good (x*(() 2)) times the price ofthe good must be the same as a decrease of dx in the
consumption of sorne other good (x*(()l)) times its price. In this case Vxe (x* ({}z), ()2) and Vxe(x* (el), ()1)
can be interpreted as these relevant prices. Moreover, fx(x*(()),())p(()) is the marginal utility obtained
by the principal by a marginal in crease of the "consumption of the good x* (e)". Therefore, keeping the
ratio 1:~:~~J(;j:O\e) constant is analogous to keeping the ratio between marginal utility and price constant
in classical consumer theory.
It is important to highlight that in the case in which the S.C.P. holds the feasibility of a policy is
independent of the term Vxe (it is only necessary to keep the monotonicity of the policy). Thus, the
optimal policy is affected by the term Vxe only through the informational rents.
There is an implication of corollary 13. Under those conditions and 88X fx(x,~)~\B)
<O it turns out that
Va;(} X
1
the curve that keeps fx(x,~)pe)B)
constant is a mínimum, and thus it is always better to have discontinuous
Vxe x,
jumps. But note that:
_!!__ fx(x, ())p(()) <O ~ _ VxxB > _ fxx
8x Vxe(x,())
Vxe
fx
Noting that the informational rents of type ()' are given by the rents given to sorne other type ()'- !:!,.()
plus ve (x* (()'), ()') · !:!,.(), we can interpret - v"" 8 as the agents risk aversion, while - f xxX is the principal'S
1
risk aversion. Therefore, the principal only decides to "smooth" it's risk if he is more risk averse than
the agents.
V~
The next theorem gives us an optimality condition for types with an active I.C.C .. This theorem is
an extension of a result from Arauja Moreira [1], the difference is that we use both conditions of lemma
10 , and we ensure no other global I.C.C. is broken.
20
Theorem 15. Let x*(·) be the optimal implementable convex-valued correspondence, ande", e' E
such that q.x* (e", e') = O. Jf x* (-) is strictly monotonic and continuous at e' ande" , then:
fx(x*(e"), e")p(e")
Vxe(x*(e"),e")
6
e
be
fx(x*(e'), e')p(e')
Vxe(x*(e'), e')
Characterization of the Opthnal Mechanism
6.1
Proof Strategy
In this section we will use the implementability conditions from section 4 and the optimality conditions
from section 5 to characterize the optimal policy. It is important to note that the optima.lity conditions
from section 5 are derived using variational calculus, and thus it is natural to reduce the space of policies
to continuous policies. However, the space of continuous feasible policies endowed with the sup-norm is
not closed. Noting that discontinuous feasible policies that are the limit of feasible continuous policies
can be extended to a convex-valued correspondence by adding the values between the left and right limits
without breaking any I.C.C., as seen in figure 14.
X
Figure 14: Continuous policies
Following Araujo and Moreira [1] we define the extended version ofx(·) as the correspondence that
contains all the values between the right and left limit of x(· ).
Definition 16. We define X as the space of extended policies, or in other words the space of convexvalued correspondence.
We willlook for the optimal policy in x 5 •
First, note that for any feasible policy x(·) E X there exists ea such that ve > ea x(·) líes above xaO
and \le < ea x(·) lies below xa(-), therefore x(·) E X is quasi-convex. Thus we can identify 4 types of
geometries:
• non-increasing
• non-decreasing
• U-Shaped in which x(B) ;:: x(fl)
• U-Shaped in which x(B)
< x(fl)
In subsection 6.2 we show that the problem of finding the optimal policy can be reduced to finding
the optimal policy in the following three types of policies :
• Increasing policies, which are given by the policies x(-) E X such that xe(e) ;:: O for all e E e
• U-Shaped policies, which are given by policies x(·) E x in which for all ~ in the codomain of x(·)
(except for x(ea)) there exists e, e', such that x(e) = x(e') = ~ and vxe(~,e) <O< vxe(~,e').
• Decreasing policies, which are given by the policies x(-) E X such that xe(e) ::; O for all e E
5
Araujo and 1\.foreira [1] show this restriction may actually reduce the principal's utility
21
e
The I.C.C. of the three types of policies previously explained present different challenges, and thus
we develop different techniques for each of them. V\Te proceed to give an overview of the challenges each
type of policy presents and the respective characterization we give of the optimum.
Increasing Policies: For the increasing policies the local I.C.C. are sufficient to satisfy the global
I.C.C., and thus they can be solved as if the S.C.P. was fulfilled. For this reason we do not speak
at length of this kind of policies.
Decreasing Policies: The decreasing policies present the problem explained section 1, which we will
now revise. It can be seen geometrically as trying to find the curve as close as possible to xo, but
keeping the shaded area positive for all e, e'.
X
rTi"t"t-1-1--
xo (B)
x({})
()
Figure 15: Decreasing policy III
It is easy to see that in this case it may be possible to modify a policy locally around sorne e, and
breaking the I.C.C. between sorne e' <<e ande">> e, therefore in this case the global I.C.C.
are the hardest to manage since changes affect in a "global" way.
The characterization of the optimal solution in this case is done with the intuition behind corollary
13, all types e that are kept "low" to keep the I. c. between other pair of types e', e" should keep a
fxP. Therefore the method consists in finding a family of policies that keep the ratio
constant ratio Vx8
fxP constant (we call them isocurves), and it is obvious that among these policies the one with
Vx8
higher ratio are better (as long as the pt best is not achieved). For each of these policies that is
not I.C. we identify critical parts of the policy that need to be "fixed" to keep the I.C.C. Through
this method, for each isocurve we know that the optimal policy must lay below the isocurve in
the critica! parts and above it in the parts that leaves slack for improvement. This allow us to
parametrize the optimal policy as a fnnction of these isocurves.
To interpret the solution it is necessary to recall the intuition behind corollary 13, in which each
type is a "good" being consumed, assignment x( e) is the level of consumption, fxP is the marginal
utility of consumption and Vxe is the price paid. Therefore, the previous method is simply stating
that the optimal consumption is achieved whenever all the budget is spent and the ratio between
the marginal utility of a good and it's price is constant across goods. Whenever this is not possible
we want to keep this ratio as high as possible.
U-Shaped Policies: In this case there exists a e0 such that x(·) is increasing for e> eo and decreasing
for e < e0 , and therefore there are pooling types between types greater and smaller than e0 . Thus,
the main challenge is to keep the local I.C.C. for types bigger and smaller than e0 , but having
both parts of the policy agree on the transfers assigned to pooling types.
The key feature for solving this type of policy is noticing that to keep the global I.C.C. the policy for
types bigger than e0 pin down completely andina unique way all types lower than e0 . Therefore,
we show that this is equivalent to solving only for types bigger than 0 , in which case only local
I.C.C. need to be taken into consideration, and modifying the objective function to weight for all
e
22
types smaller than 8 0 that are being pinned down. Summing up, we show that solving this type
of policy is equivalent to solving a problem in which the S.C.P. is fulfilled, but with a modified
objective function.
6.2
Generalized Procedure
Now we explain the general procedure, and how the optimization problem is broke down into simpler
subproblems. Using lemma 7 for any feasible policy x(·) Ex and () > ()' the following must hold:
That is to say, for any given type B such that the point (x(B), B) is in es+ and the point (x(B), B') is
its reflection, type ()' must be pooled with B. For this reason, it is important to define the sub-region of
es+ in which the points have a reflection. For this reason we introduce the function I:(-), which gives
the biggest assignment such that types have a reflection.
Definition 17. Let's define the function L:(o) implicitly by Vx (L:(B),f!_) =
It is easy to notice the following properties:
$
Vx
(L:(B), B)
Ij xo(f!.) exists, then and L:(fl.) = xo(f!.)
• VB E 6\{f!_} L:(B) > xo(B)
• 2::( ·) is strictly decreasing
X
X
L(O)
L(O)
Xo(O)
x 0 (0)
+-------------o
Figure 16: F\mction L::(·) I
Note that for any type () and for all assignments
a refiection.
Figure 17: Function L::(·) II
X
E
(xo(B), 2::(8)] the point (x, B) is in
es+
and has
We can identify 4 different types of policies:
(A) Let x(·) Ex be a feasible policy such that x(O) E [;¡;_,x 0 (0)]. In this case it is clear that x(·) must
be non-increasing.
6
By definition e(x(e), B) =e'
23
X
Figure 18: Subset A
x as follows:
We will define the subset A e;;;
x(·)
E
A
{=}
x(·) is feasible
1\
x(B) :S: x 0 (B)
(B) Let x(·) Ex be a feasible policy such that x(B) E (x 0 (B), ~(B)]. In this case we have that the point
(x(B), B) has a refl.ection, given by (x(B), B(x(B),
and therefore x(B) = x(B), moreover using the
local I.C.C. we know that:
en
• x(-) lies below x(B) for all (}
E
[B, B]
• x (·) is non-increasing for all (} E [~, B]
X
-+----'-------...__0
8
Figure 19: Subset B
We will define the subset B e;;;
x(·)
E
B
x as follows:
{=}
x(·) is feasible
1\
xo(B) < x(B) :S: ~(B)
So far we have identified all policies x(·) such that x(B) :S: ~(B), so now we proceed to describe the
policies x(·) such that x(B) > ~(B). We will consider the sub-cases in which x(·) and ~(-) intersects and
the case they do not (note that x(·) can intersect ~(·) only if x(B) > ~(8)) 8 .
Ex be a feasible policy such that x(·) intersects with ~(·), that is to say 3(}' E 8\{~,B}
such that x(e') = ~((}')
(C) Let x(·)
By definition of ~(· ), the refiection of (x((}'), (}') is (x((}'), ~), therefore x@ = x((}'), moreover using
the local I.C.C. we know that:
7
8
e
To avoid excess notation we will reefer to ÉÍ(x(iJ), B) simply as
We consider that x(·) and :S(-) intersects if and only if there is an interior type
24
e such that :S(B) =
x(B)
• x(-) lies below x(B') for all BE [.Q, B']
• x (·) is non-decreasing for all B E [B', 8]
X
f-----------~-------0
()'
Figure 20: Subset C
We will define the subset
x(·) E
e
e ~ X as follows:
~ x(·) is feasible
1\
:JB' E 8\{.Q,O} such that x(B') = l:(B')
(D) Let x(·) Ex be a feasible policy such that x(O) > 2:(0) and x(·) does not intersect with 2:(-). Then
it is easy to note that in this case x(_Q) 2:: x 0 (_Q) and x(·) must be non-decreasing.
X
~x((J}
'
:E(())
xo(6)
~---------------------0
~
Figure 21: Subset D
We will define the subset D
~
x as follows:
x(·) E D
~
x(-) is feasible
1\
x(ft) 2:: xo(.Q)
A direct proposition from the previous description is the following:
Lemma 18. Let x(·) E
hold:
x
be a feasible policy, then one and only one of the following conditions must
• x(·) E A, or equivalently x(O) ::; x 0 (0).
In this case x(·) must be non-increasing.
< x(O)::; 2:(0).
In this case x(·) must be U-Shaped in [B, 8] and non-increasing in [.Q, B]
• x(·) E B, or equivalently x 0 (0)
• x(·) E
e,
or equivalently :lB' E 8\{.Q,O} x(B') = 2:(8').
In this case x(·) must be U-Shaped in [.Q,B']and non-decreasing in [B',O]
• x(·) E D, or equivalently x(ft) > x 0 (_Q).
In this case x(-) must be non-decreasing.
25
o
Proof. Direct from the previous description.
Now we will proceed to explain how to find the optimal policy in X· Consider the following procedure:
• Sweep over all values of ~ E [±, x] and proceed as follows:
- If ~ is in [;r.,x 0 (B)] denote by A~(·) the optimal policy x(·) E A such that x(B) = ~- If ~ is in (x 0 (B),:E(B)] denote by A~(-) the optimal policy x(·) E E such that x(B) = ~-
Using theorem 8 we can see that the problem is separable and we can find, independently,
the optimal non-increasing part Xa(·) for e E [fl,B] and the optimal U-Shaped part Xf3(-) for
e E [B,BJ.
To find the non-increasing part, we consider that xa(e) = ~ for all e E [B, B] and proceed
to find the optimal policy in [fl, B]. To find the optimal U-Shaped part, we find the optimal
policy in [B, e] disregarding all I.C.C. with e E [fl, B].
Then we define M (·) as follows:
e E [8,8]
e E [fl, B]
Being able to separate both problems allow us to disregard sorne of the global l. C. C. when
finding the optimal policy. This is very useful since the challenges that present the I.C.C.
in the non-increasing and the U-Shaped part are quite different, therefore separating the
problem allow us to use different techniques for the different parts of the policies
- If ~ is in (:E(B), :E(fl)) denote by A~(-) the optimal policy x(·) E C such that x(·) intersects
with :E(·) ate'= ¿:- 1 (~), that is to say x(:E- 1 (~)) = ~-
Just like befare, using theorem 8 we can see that the problem is separable and we can find,
independently, the optimalnon-decreasing part X o: (.) for e E [e'' B] and the optimal U-Shaped
part Xf3(·) for e E [fl, e'].
To find the non-decreasing part, we consider that xa(e) = ~ for all e E [fl, e'] and proceed
to find the optimal policy in [e', B]. To find the optimal U-Shaped part, we find the optimal
policy in [fl, e'] disregarding all l. C. C. with e E [e', B].
Then we define A~(-) as follows:
- If ~ is in [x 0 (fl),x] denote by A~(-) the optimal policy x(-) E D such that x(fl) = ~-
• Find the optimal policy x*(·) by maximizing over all ~in [;r., x], that is to say:
C=argmax~
r f(A~(e),e)p(B)de
le
x*(-) =A e(·)
It is easy to see that the previous procedure reduces the problem of finding the optimal policy x* (·)
into finding three types of policies:
26
Decreasing Policies: For all ~in [;,¿, l:(B)] finding the optimal non-increasing policy x(·) Ex such that
x(B) = ~Increasing Policies: For all ~in [I:(B),x] find the optimal non-decreasing policy x(·) E
x such
that
x(~) = ~-
U-Shaped Policies: For all ~in [x 0 (B),x 0 (~)] find the optimal U-Shaped policy x(·) Ex such that
w E [~.el] u [ez, B] x(e) = ~- \Vhere {el, ez} are given by:
{e .e}= {{B,B}
2
l.
{.fl_,E- 1 (~)}
~E
[xo(B),I:(B)]
~E [I:(e),xo(~)]
Vve now proceed to explain how to find each type of policy.
6.3
Increasing Policies
Lemma 19. Let x( ·) E x be a non-decreasing policy, then the local J. C. C. are necessary and sufficient
to guarantee the global J. C. C.
Proof Using lemma 19 it is clear that finding the increasing policies can be done as if the S.C.P. was
fulfilled, and thus require no extra explanaition.
D
6.4
Optimal U-Shaped policy
In this section we will find the optimal U-Shaped policy for the zone B of the ironing (the U-shaped form
ofzone C can be solved the same way). Therefore, we will find the optimal policy M(-) in [B(B,O,B],
such that A~(B) =~-In what follows we will denote A~(-) by A(-) and B(B,~) simply by
We will use a
different approach than the one found in Araujo and Moreira [1] which will allow us to cover a broader
range of problems, although the basic intuitions are similar.
Intuitively, in any incentive compatible policy in X the part that lies in es_ is the reflection of the
part that lies in es+, thus any incentive compatible policy in X is uniquely determined by the section
that lies in es+. Therefore we just need to find the intersection e0 between the policy and x 0 (-), and
the optimal shape of the policy on es+, x(·). With this, and considering a modified objective function
that takes into account the effect of x(·) 011 the policy on es_ we can solve the problem.
e.
Lemma 20. The optimal U-shaped part of A(·), such that A(B) = ~. can be defined by a type eo anda
non-decreasing policy x(·) such that x(e 0 ) = x 0 (e 0 ) and x(B) = ~. where A(·) is given by:
A e _ {x(e)
( ) - x(B(A(e),B))
W E [eo,B]
WE[B,eo)
e 0 and x(·) are the solution to the following problem9 :
~
¡x-'(z) fx(z, y)p(y)- fx(z, e(z,
" y))p(e(z,
" y)) Vxe(z,y)
,
dzdy
¡
Bo,x(-) xo(Bo) x
Vxe(z, e(z, y))
\[1 := m~n
0
1 (z)
s.t.
{
~(·) is non-decreasing
x(eo) = xo(Bo)
x(B) = ~
Problem \[1 consists in minimizing the area enclosed by the curves xo(·), x(·) and ~ (shaded area in
figure 22), weighted by fx(x, e)p(e)- fx(x, IJ(x, B))p(IJ(x, e)) vx~(;,~·::e)) ·
9
If x(·) has a bunching zone, the inverse
we consider the smallest element in :r;- 1 (-)
:r- 1 (·)
is not uniquely defined. In case x- 1 (-) is not uniquely defined
27
X
xo(O)
x<e>
e
Figure 22: Optimal U-Shape I
Proof. The proof of lemma 20 consists of two parts. We first prove that any I.C. policy ¡{) such that
is determined by a type e0 and a non-increasing policy x(-) defined in [e 0 , iJ]. Then we prove
that the characterization given by \]i yields the optimal policy.
¡(iJ)
=e
Stepl
• Since Vxe(e, iJ) >O and Vxe(e, é) <O, then the policy ¡(·) must pass through x 0 (·), then there
exists ea such that ¡(ea)= xo(eo).
• xo(-) is decreasing and ¡(·) must be non-increasing in es+, thus ea is uniquely defined and
¡(·) lies in es+ for all e> e0 .
• We will denote by x(-) the section of ¡(·) that lies in es+ (e> ea).
• It is easy to note that for all e in [ea, iJ] the policy x(·) must have a refiective type. To
see this note that é(e, e) is continuous in both arguments, B(x(eo), ea) =ea, é(x(iJ), iJ) =e.
Moreover, for any type e' E [é,e 0 ) the policy ¡(e') can be found as a refiection of a type
e E (eo,iJ], that is:
• Therefore, the policy ¡( ·) is uniquely defined by a type e0 and a non-increasing policy x(·)
in es+. Moreover, since 801~·~) < O and 801~·~) < O, x(·) non-decreasing implies that it's
refiection is non-increasing, and thus ¡(-) satisfies the local I.C.C.
• By construction pooling types have the same transfers. Using lemma 19 it is easy to see
that global I.C.C. are satisfied for types greater than 0 , and thus we only need to show that
global I.C.C. are satisfied for types smaller than e0 • Let's take e' < e" < e0 , in this case it
is easy to see that the local I.C.C. are a sufficient condition for iP 7 (e',e") >O, on the other
hand we have that:
e
e"
iP'(e",e')=
r
}8'
7 (e")
7 (y)
1
Vxe(y,z)dzdy=1
'Y(e')
'Y(e')
e"
1
Vxe(z,y)dydz
¡- 1 (z)
Noting that:
B(z,,- 1 (z))
1
Vxe(z, y)dy =O (By definition of B(·, ·))
,-'{z)
we get,
28
l
B(z,-y- 1 (z))
e"
Vxe(z, y)dy=
-y-l(z)
1
1
B(z,-y- 1 (z))
Vxe(z, y)dy-
-y-l(z)
1
=O
0(-y(e"),e")
=B(
Note that fe, 'Y
(e") e")
'
Vxe(z, y)dy
e"
1
iB(z,-y- (z))
Vxe(z,y)dy- _
Vxe(z,y)dy
e(-y(e"),e")
e"
B( ·(e") e")
Vxeb(fJ''), y)dy =O, and since Vxxe >O we know that JB', 1
'
Vxe(z, y)dy >
-
1
O (remember z > !'(8")), moreover it is easy to see that J;(~C;'~).~~)} Vxe(z, y)dy >O, therefore,
'iz
1
e"
-y-'(z)
-y(e') (
e"
)
Vxe(z,y)dy<0=>1
-1
Vxe(z,y)dy
-y(e")
-y-l(z)
dz>O=>iJ!~'(8",8')>0
Step 2
Now we will prove that the characterization given for 80 and x(·) yields the optimal policy.
• Note that by subtracting a constant to the maximization problem we can rewrite the problem
as follows:
M(·)
E
argmax-y(-)
lJ
f('y(8),8)p(8)d8
{==?
A~(-) E argmax-y(-)
s.t. J.C.C.
lJ
-l
5
f('y(8),8)p(8)d8
f((,8)p(8)d8
s.t. J.C.C.
{==?
A~(-)
E
rij Jr<e)
~
f'x ('y( z), B)p( 8)dzd8
argmax,1c ) Je
s.t. J.C.C.
• Rewriting the term J/ Jte) fxb(z), 8)dzd8 conveniently.
Let's denote 80 implicitly by 1'(80) = x 0 (80), note that ·y(80) is the lowest assignments of
policy !'(·). Note that for all (E (1'(80),(] the policy 1'0 assigns ( to at least two types.
Let's define 1'.:¡:: 1 (-) and 1'= 1 0 as follows:
X
Xo(O)
(}
Figure 23: U-Shape policy II
29
Rewriting the integrals:
l
e¡~
8
fx(z, y)p(y)dzdy=
¡~
¡
¡x
7
J'(8b) 7:::
/'(Y)
~
¡J'(8b)
Note that for all z E [!(Bb), ~]
for the last integral:
0
1
_¡
1
(z)
fx(z, y)p(y)dydz
(z)
1
1'::: 1 (z)
fx(z, y)p(y)dydz +
B(z, 1¡ 1 (z))
Vx(z, y) = Vx(z, y')
=?
=
¡~
¡1'_¡
1
(z)
fx(z, y)p(y)dydz
?(8b) x0 1 (z)
1:::: 1 (z). Using the following change of variable
Vxe(z, y)dy = Vxe(z, y')dy'
but by definition y= B(z, y'), thus:
=?
~~
Jx;)' fx(z,y)p(y)dydz =
J'(Bb) 1'::: 1 (z)
~~
Jx;)' fx(z,B(z,y'))p(B(z,y')) Vxe(::,y') dy'dz
?(8b) 1'_¡ 1 (z)
Vxe(z, B(z, y'))
thus,
re¡~ fx(z,y)p(y)dzdy= ¡~ r~'(z) fx(z,y)p(y)- fx(z,B(z,y))p(B(z,y)) Vxe(::,y) dydz
Je J'(Y)
?(8b) lx 0 (z)
Vxe(z, 8(z, y))
We have that 1¡ 1 (z) = :r- 1 (-), andas we previously showed x(-) non-decreasing is sufficient
for implementability. Thus we get the following:
x(·) E argminx(·)
¡?(8b)
~
¡x- (z) fx(z,y)p(y)- fx(z,B(z,y))p(B(z,y))
1
x0
1
(z)
(::·Y )
Vxe
dydz
Vxe(z, 8(z, y))
s.t. x(·) non-decreasing
~ A~(·) E argmax?(·) lee j(!(8),8)d8
s.t. I.C.C.
D
Lemma 20 turns what may seem a very difficult problem into a very tractable one. There are several
cases that can easily be solved using this approach, we will give a couple of examples.
Example 21. We will now show we can recover the case shown in Arauja Moreim {1}. This is the case
in which there is a unique non-decreasing curve xu(·), such that xu(·) divides the space CS+ in two, such
that fx(x, 8)p(8)- fx(x, B(x, B))p(B(x, 8)) Vx8v("~,e)e))
is negative "above" curve xu(-) and positive "below"
X)
X)
xu(-)10
X
~--------------~--.__0
(}'
6
Figure 24: U-Shape example I
----------------------------
10In the following figures we use the definition :=:(x, e) = fx(x, B)p(e)- fx(x, B(x, B))p(B(x, B)) Vx:c:.~(~;e)).
30
In this case, depending if ~ is in zone B or C of the procedure, it is easy to see we recover both
U-Shaped parts described by Araujo Moreira [1}.
If ~
< Xu(fl)
In this case it is easy to notice that e0 solves the equation xa(eo) = xu(eo) and x(·) is given by:
and therefore A~ is given by:
A~
e _ {x(e)e > 80
( ) - x(e(A~(e),e))
e< ea
X
,í:.(ll)
'
§---~<<e:
:~:'
'
'
:xo(ll)
-+--~--------------~e
¡¡
0(§,71)
Figure 25: Optimal U-Shape example I
In this case it is easy to notice that e0 also solves the equation xo (ea)
by:
and therefore A~ is given by:
A~(e)- {x(e)
-
x(e(A~(e),B))
e > ea
e< ea
X
,:E(II)
~xo(ll)
-+--'----------.....__
e
Figure 26: Optimal U-Shape example II
31
= X u (ea)
and x(-) is given
Example 22. Let's consider the the case in which there is a unique non-increasing curve xu(·), such
that xu(-) divides the space CS+ in two, such that fx(x,e)p(e)- fx(x,B(x,e))p(B(x,e))vx:c:.~·::e)) is
negative "below" curve X u (·) and negative "above" X u(-)
X
~------------------~8
o
Figure 27: U-Shape example II
In this case we have that x(·) must have the following form:
x(e) =
{~
e E [ea, 8]
e= ea
[xo(eo),~]
and therefore Af. is given by:
Af.(e) =
{x(~)
x(e(x(e),e))
e> ea
e< ea
Moreover, if fx(x,e)p(e)- fx(x,é(x,e))p(B(x,e)) vc-~· 81
Vxe x, x,
satisfies the following equation:
811
l
e
A
A
fx(x,e)p(e)-fx(x,e(x,e))p(e(x,e))
x0 (8 0 )
is non-increasing in e we have that e 0
Vxe(x e)
A'
dx=O
Vxe(x, e(x, e))
X
,:E(O)
!A"co:
§
'
''
'
:xu(ll)
:xo(ll)
o
A
llo
-
- 11(§,11)
o e
Figure 28: Optimal U-Shape example III
6.5
Optimal Decreasing Policy
In this section we will find the optimal decreasing policy for the zone B of the ironing. Therefore, we
will find the optimal policy Af.(·) in [~, é(e, 0], such that Af. (O) = ~- We will disregard the case in which
the maximum between x 1 (·) and ~ is implementable since this case the problem is trivial. That is, if the
policy y(·) given by:
32
y(e)
=
{~max{xl(e),O
e>iJ(B,~)
e~ iJ(il,e)}
is implementable, then the trivially we have that ve< B(B,~) Ae(e) = y(e).
In what follows we will denote A~(·) by A(·) and B(B,~) simply by B. As we explain in subsection
6.2 this problem can be simplified by considerA( e) = ~ for all types in [B, B]. The decreasing policy in
zone A of the ironing can be found in a similar way, although it is possible to make sorne simplifications
which we explain in remark 39.
Befare discussing the optimal decreasing policy we will go back to sorne intuitions concerning the
I.C.C. of decreasing policies
Lernma 23. Let x'(·) and x"(·) be two policies such that for all e in interval
and for all e E e x' (e) 2: x" (e) . Then the following condition holds:
[B, B] x'(B) = x"(B) = ~
if x '() is an l. C. policy then x" () is also an J. C. policy
Intuitively, a policy that makes higher assignments in the decreasing part is closer to curve xa(e) and
thus it is more likely that some global J. C. C. is broken (see fig 15)
Proof. By looking at fig 15 we can see that the global I.C.C. between e 1 and 82 depends on the shaded
are, and thus the closer the policy is to x 0 (e) the smaller this area is.
O
Definition 24. Since the ratio M
will play a crucial role on the optimal policy we will make the
VxB
following definition:
r(x,e) = !x(x,e)p(e)
Vxe(x,e)
We will refer to r(·,·) as the critica[ ration
Now we will make a couple of technical assumptions.
Assumption 25. The following two assumptions are made in what follows of this subsection :
ar(x,B)
----¡¡;;---
>O For all (x, e) in
es_
ar(x.e)
------¡¡8
.
>O For all (x, e) m
es_
The domain of the assumption is slightly stronger than we need. Since in the decreasing policies
it is never optimal for the policy to be above x 1 (·) we could restrict the domain to all (x, e) such that
x ~ x 1 (e). In this case we would need to recurrently make special mentions in the proofs so to avoid
unnecessary difficulties we make a stronger assumption, the method for the weaker assumption is the
same and should become apparent to the reader. Finally, note these two assumptions are natural when
Xl (.) is completely contained in e S- but it is not implementable.
From the assumptions 25 a first straight forward conclusion can be obtained about the shape of the
optimal decreasing policy
Lemma 26. The optimal decreasing policy consists of bunching parl in an interval [é, B], a continuous
and strictly decreasing part in [ii, é] and a bunching parl in [~,e] , where é E (~, B] ande E [~, 6Í) 11 . In
what follows we will refer to é as the point where the policy starts being strictly decreasing, and éx as the
point é corresponding to policy x(-), likewise we will refer toe as the point where the policy stops being
strictly decreasing, and ijx as the point Bcorresponding to policy x(·).
0nly if A(fl_) 2: 2:(0) we can have that i/' > 1}_, otherwise it is easy to see we can apply the same reasoning as
in the proof of this lemma to discard the bunching zone in [fl_, §A]
11
33
X
xo(B)
x(B)
e"
Figure 29: Optimal Decreasing Policy I
Proof. The reasoning is the sarne as in lemma 28, but using the assumptions 25. It is easy to convince
oneself in any bunching zone, which is encompassed by two decreasing zones, can have the value of the
critica! ratio srnoothen by increasing the value of the policy in the beginning of the bunching zone and
decreasing the value of policy at the end of the bunching zone (see fig 30):
X
Figure 30: Bunching Policy I
Likewise, srnoothening a discontinuous jurnp by increasing the policy at the right of the discontinuous
jump and decreasing it at left of the jurnp effectively srnoothens the value of the value of the critica!
ratio in the policy (see fig 31):
X
Figure 31: Discontinuous Policy I
The techincal details can be found in the appendix.
34
o
The next lemma (which is a direct consequence of theorem 11) states that the optimal policy tends
to "evens up" the value of the critical ratio across the policy, which only changes when the I.C.C. are
binding.
Lemma 27.
Let e1 be such that r(A(e), e) is strictly increasing in e 1 , then e1 must be indifferent between its
assignment and the assignment given to some type e2 < e 1 . On the other hand, if el is such that
r(A(e), e) is strictly decreasing in e1, then some type e2 > e1 is indifferent between its assignment and
the assignment given to el. That is to say:
• lf e1 E (fl, é] is such that ar(Ad~~),e 1 ) >O, then exists a e2 < e1 such that <I>A(e 1, e2) =O
• If e1 E (fl, é] is such that ar(Ai:~),el) <O, then exists a e2 > e1 such that <I>A(e2, e¡)= O
o
Proof. It is a reformulation of theorem 11
The previous lemma states that the variations of the value of the critical ratio must be as small as
possible. The next lemma states two properties on types that are binding. The first, which is a direct
consequence of lemma 15, says that binding types can always be jointly varied, thus it is never optimal
for them to have different value of the critical ratio The second property says that types in-between a
pair of types that are binding must have lower value of the critical ratio than the binding types on the
edges, otherwise this would also leaves slack to have lower variations in the value of the critica] ratio by
averaging the changes in the middle with the edges.
Lemma 28.
Consider e2, el E 8 such that el < e2 and e2 is indifferent between its assignment and the assignment
given to el, that is <l>A(e2, el)= 0.
1. If e1, e2 E (¡jA, iJA), then the value of the critical ratio must be the same for both types, moreover
all types in-between must have a lower value of the critical ratio. That is to say:
2. If e2 E (¡jA, iJA) and e1 :::; ¡jA all types smaller than
ratio than e2 • That is to say:
e2
must have a smaller value of the critical
3. lf e1 E (¡jA, iJA) and e2 2 iJA all types in between e1 and iJA must have a smaller value of the critical
ratio than e1 . That is to say:
Proof. \Ve will start by proving item l. The first part of 1 is direct from theorem 15, the second
part can be proved by contradiction.
Suppose there exists e' E (e 1, e2 ) such that r(A(e'), e') > r(A(e 1), el)
e" be the biggest type in [e', e2] such that r(A(e"), e") = r(A(e'), e'). That is to say, e" =
max{e E [e',e 2]¡r(A(e),e) =r(A(e'),e')}
• Let
35
• From lemma 26 we have that discontinuous jumps are never optimal, and thus we know that
e" < e2 , moreover by continuity df'(~~),e) le=W' < O.
• If di'(~~),e) le=e" < 0., then theorem 11 there must exists a
but by theorem 9 we know that e"' E [e 1 ,e 2 ].
e"'> e" such that ifJA(e"',e") =O,
• From theorem 15 we know that f(A(e'"), e"')= f(A(e"), e")= f(A(e'), e').
• Thus we arrive to a contradiction
The proof of ítem number 2 is similar as befare.
• For all e E
befare).
(B, e 2 ) we must have that r(A(e2 ), e 2 )
:::::
f(A(e), e) (the proof can be done exactly as
(B, e 2 ) we know that r(A(e 2 ), e2 )
::;:.:
f(A(B), B)
• By continuity in
The proof of ítem number 3 is similar as befare.
• For all e E (el, é) we must have that f(A(el), el) 2': f(A(e), e) (the proof can be done exactly as
before).
• By continuity in (e 1 , é) we know that r(A(e 1 ), el) 2': f(A(é), é)
D
So far we have that lemma 26 gives us a general overview of how the optimal policy looks. On the
other hand, lemmas 28 and 27 gives us an intuition on how the optimal policy looks like in the strictly
monotonic zone, which minimizes the variations of the value of the critical ratio. Using both of these
insights we define a family of policies which have the general form given by lemma 26 and keeps the
value of the critical ratio constant on the strictly decreasing part.
Definition 29. Let ¡[x, e'](e) be the isocurve that passes throught (x, e') and keeps f(x, e) constant.
That is :
r(¡[x,e'](e),e) =f(x,B')
ve
which is uniquely. defined and decreasing in e beca use of assumption 25.
As we previously explained, because of lemma 27 it is natural to define a function that is equal to ~
in an interval ¡é,eJ, keeps r(-,·) constantfor all e E [B,é] and is bunching in some interval [ft,B]. Let
x(¡¡;,e'](·) be afunction that is bunching in (min{B,e'},O], keeps r constant and is truncated by¡¡;. That
is to say:
~
x[¡¡;,e'](e) =
{ Min(¡¡;,¡((e'](e)]
e> min{B,e'}
e:::; min{B,e'}
X
X[K¡,O¡){-)t----.
X[Kz,Oz](-)
e
Figure 32: Function x[¡¡;, B] (-)
36
e
The method will consist in taking a fixed assignment for (}_ and characterizing the optimal policy for
the given assignment, then we will maximize over al! possible assignments of (}_.
Definition 30. Let A[ K](·) be the optimal policy x(·)
Ex
such that x(f}_) =
n;
12
The optimal policy will be characterized using the family of policies x[K, B](·), and finding al! the
intersections between the functions A[li](·) and x[K,B](·), foral! B. That is to say, for each function
x[n;, B](·) we will find a collection of types that have the same value in functions A[n;](·) and x[n;, B](-). To
identify these critica! collection of types we make the following definition:
Definition 31. For a policy z(-) consider the following problem 13 :
We call the solutions of Qz genemlized mínimum.
Remark 32.
• Note that if Qz is attained ata unique collection of types then <P"(Bfj, B2j) <O
Vj E {1, .. , nz}
• If z(·) is J. C. then n =O is a solution of Qz and Qz =O.
• The solution of Qx[~<,e) is uniquely defined for almost for every B (almost en el sentido discreto de
verdad).
x[~<
• Let{B11
'
e')
x[~<
, ... ,B2 n,'
. l y. IfB'
spectwe
<
e')
x[~<
} and{B 11 '
e")
B" t h en {Bx[K,e') ,
11
x[~<
, ... ,B2 n,;
... ,
gY[e')} an d {Bx[~<,e") ,
wz·zz say {By[e')
11 , ... , Zn'
11
gx[x;,e")
gx[K,e")} )
{ 11
' ··· ~ 2n"
e")
[ e')
} besolutionsofproblemsQx",
gx[~<,e')}\{B
Zn'
_, B-} e
un" (Bx[K,e")
l=l
11
, gx[K,e"))
.
21
[ e")
andQx"·
re(}n t h'zs case we
gx[".e")} are nested , an d we wz·zz d enote {Bx[,.,e') ,
11
... , Zn"
gx[,.,e')}
... , Zn'
Problem Qz gives a basic measure on how much eJ\.'tra rent is a policy leaving for agents to attain by
lying, or in other words how much incentives are there to deviate for the agents. If the solution of Qz
is O then the policy z(·) is I.C., and thus no agent can get more than the informational rents, therefore
there are no incentives for deviation. On the other hand if the solution of Qz is strictly positive then
there exists an agent that has incentives to deviate, moreover the incentives to deviate for any agent are
bounded by Qz.
Definition 33. We will define J'x[,;,e') as the difference of the interior between the right and left limit
of Qx[x;,e) at 8' 14 . That is to say:
x[~<,e')w h ere {e lj
,
Qx(I\O,e'). That is:
... ,
gx[~~:,e')- }
Zn'-
,
gx[~~:,e')+ }
. ht z·zmz't s respec t'zve ly of
an d {Bx[~<,e')+
lj
, ... , Zn'+
,
are th e lejt an d ng
{ gx[~~:,e')lJ
gx[~<,e'JJ ••• ,
2n 1 -
}
'
= r
e_:~:-
{Bx[~~:,e)
lJ
'···~
gx[~~:,e)
2n
}
'
Remember there is a superscript ~ in A(-) we are already omitting
The complex domain in the maximization problem is to avoid degeneracies in the solution arising from pooling
types
14
Using the properties of Qx[~~:,e] described in 32 it is easy to see that we can find the right and left limit of the
solution at any ()
12
13
37
-<
Note that if Qx[~<,B') has a unique solution, then J'x[~<,B') is the given solution. Note that if Qx[~<,B') does
not have a unique solution, then J'x[~<,B') is the smallest subset of 8 that contains all of Qx[~<,B') solutions.
For this reason we call Jx[no,e) the extended generalized mínimum of x[n:, e]
Lemrna 34. For all e' E (fl., lJ) there exists a unique e" such that e E J'x[~<,B"]
Proof.
Let 's take sorne e' E (fl., lJ), we will show there exists a unique e" such that e' E ?1 K,B"J.
• Note we can find a e big enough such that {fl.,lJ} is a unique solution of Qx[~<,BJ.
• Note we can find a e srnall enough such that nx[K,B) =O is a unique solution of Qx[~<,B)_
• Thus, we can find a e" anda Esrnall enough such that for all e E (e", e" +E) e' E u~¡ex[~<,B)- , ex[no,B]-l
J=l lJ
2]
ande E (e"- E, e") e' E
u;:l [e~J"· l+, e;J"· 8l+J.
8
• Using the upper herni-continuity of Qx[~<,B) we know that e' E
n"- (e'x[K,B")- ex[K,B")-)
u
J=l
lJ
'
u;=l ¡e~t e"] +, e;t[
u+
[
8"]
+] and e'
1-
2]
• Thus by definition
e' E J'x[no,e"J.
o
• Since the solutions of Qx[K,B) are nested it is easy to see that e" is unique.
The next lernrna will be the rnain result that will allow us to characterize the optirnal policy.
Lernrna 35. If A[n:](e)
> x[n:,e'](e) for some e E (fl.,iJ), then
(ve E J'x[~<,B'l)\{fl.,lJ}
1.
A[n:](e) = x[n:,e'](e)
2.
ve
E
u;: (elj, e0) \
{fl., lJ}
3.
ve
tj_
u;:
A[n:](e) ~ x[n:, e'](e)
1
1
[eij, etJ
A[n:](e) < x[n:, e'](e)
Proaf.
We will prove this lernrna in two steps, first for the case in which Qx[K,B') is uniquely defined and
then the case in which Qx[~<,B') is not uniquely defined.
Taking the case in which nx1"· 8'l > O and Qx[K,B') is uniquely defined:
• Since A[n:](e) > x[n:, e'](e) for sorne e E (fl., iJ), then it is easy to notice that exists sorne
(ex[~<,B'l, gxl"· 8'l) such that A[n:](e) > x[n:, e'](e)
eE
• But, nx[K,B') >O implies that x[n:, e'](·) is not I.C. Since A[n:](·) is l. C., the I.C.C. constraint irnplies
that there rnust exists sorne e E (ex[,.,B'l, jjx[K,B'J) such that A[n:](e) < x[n:, e']( e)
• We can define the collection of types
TI= {1r11, 1r21 , ... , 7r1m, 7r2m} such that
m
m
j=l
j=l
Note '<hE
ll\{7rn,7r2m}
7f
E (ex[K,B'J,{}x[~<,B'J)
• Using lernrna 27 we know that q,A[~<) (7rlj, 'if2j) = O Vj E {1, 2 .... , m} (in case
have that q,A[~<) ( 'iflm, lJ) =O and if 7rn ::; jjx[><,B') we have that q,A[><) (1r21.fl.) = O)
38
7r2m
=
gx[~<,B') we
• Remember that the I.C.C. are given by computing sorne areas weighted by Vx&· Therefore, the
sum of these areas between types J in policy A[K] can be at most the areas between types J in
policy x[K, e'], plus the areas added in between if. But, from the previous item we know that the
areas added in between if are given by I;j: 1 q,x[to,B'l(711j,71zj)· therefore, we have the following
inequality:
But, ifll =/= J*(e') we would have the following inequality (remember Qx[K,B'J is uniquely defined):
~ q,A["] (ex[to,B']
ex[to,B']) -< ,¿:_.
;f-. q,x[to,B'] (ex[to,B']
ex[",e']) + ,¿:_.
~ q,x[~<,B'] (711],. 712]·) <O
l]
, 2]
1]
' 2]
,¿:_.
j=l
j=l
j=l
So there must exist a pair of e that are not I.C.
• Therefore we must have that if = J• (e'), moreover by continuity we can see that
A(K](e) =x[K,e'](e)
ve E (e~J''· 8 'l,e;J"• 0 '])
Vj E {1,2 .... ,n*}
• The last point is trivial by the construction of the proof.
Now analyzing the case nx[~<,e'] = O and Qx[K,B'] is uniquely defined
< ¡;ix[~<,e'J such that A[K](e1) = x[K, e'](el), then by lemma 27 there must
exista type e2 such that A[K](e 2) = x[K,e'](e 2 ) and q,A["l(el.e 2) =O or q,A["l(e2,el) =O
• If there exists a type e1
• By lemma 11 we know that q,A[~<l(e2 ,el) > q,x["· 8'l(e 2 ,e1)
• But, if nx[~<,e'] =O and uniquely defined, then Ve1, e2 E
e
q,x[~<,e'] (e2, el) >O
• Thus, we arrive to a contradiction.
Now taking the case in which Qx[~<,B'] is not uniquely defined:
• From lemma 32 we can find e" < e' such that ¡jx[~<,B"J is uniquely defined and Jx[~~:,B"] is arbitrarily
clase to { e~I"·e']-, ... , e;l"·e']- }. Using the previous result for when ¡jx[t<,B'lis uniquely defined,
Ve rj:. u;:-1 (e~J"·e'J-,e;J"· 8 ']-] A(K](e) 2: x[K,e'](e).
• From lemma 32 we can find e" > e' such that ¡jx[~~:,e"] is uniquely defined and Jx[~~:,e"] is arbitrarily
clase to { e~I"·e']+, ... , e;l"·e']+}. Using the previous result for when ¡jx[~~:,e'] is uniquely defined,
ve E u;:~ [e~y"·e']+, e;]"· e']+]
• Thus, V8 E
A[K](e) ~ x[K, e'](e)
u;:~(e~J"·e'J+,e;J"• 8 ']+J\ u;:~(e~J",B'l-,e;J"· 0 ']-]
Definition 36. We define junction Y(·)
:e---+ e
A(Kj(e) = x[K,e'](e)
D
implicitly as follows:
e E ¡jx[~~:,T(8)]
The next theorem is the final characterization of the optimal policy. Given lemma 28 and 27 we
can see that for all types A[K](-) is bounded by x[K, Y( e)](·), and A[K](e) = x[K, Y(e)](e) for all types
is I.C.C. Thus, it is clear that the optimal policy A[K](·) will be x[K, Y(e)](e) whenever the x 1(·) is not
achievable, therefore we have the following theorem:
39
Lemma 37. A[K)(·) is given by:
A[K)(8) = Min{x[K, Y(e)](8)},xl(e)} 'v'B E [fl,B]
Proof.
t
• Let's denote by the maximum of r(·,.) in policy A[K) in [fl, B], and
that r(A(K)((}), e) =f. That is:
ethe respective type such
• If f =o (r(A[K)(8) =o for a given e is equivalent to A[K)(e) = xl(e)), then using lemma 35 it is
easy to see that the optimal policy would be given by the characterization of lemma 37. It is also
easy to see that f S O, so we now take the case f <O .
e,
• If f < o and {fl, e} E Jx[~<,B], then for all e E [fl, B] we have that Y(8) ::::
and thus we can use
directly lemma 35 to see that the optimal policy would be given by the characterization of lemma
37.
• If f
< O and {fl, e}
f:_ Jx[~<,B] then:
- we can find an interval [el, ez] that doesn't intersect U7~+1 [e~J"·e'J+, 8~J"·e'J+]. That is to say:
- In this case, using lemma 35 it is easy to see that 'v'B E [el, 8z]
A[K](e) = x[K, e], (8)
- By construction oflemma 35 it is easy that for any (8' <e") E 8 such that [8', e"] n¡el, ez]-/=
</;, q,A[~<] ( 8", e') > O.
- Thus we can find a function h( ·) > O defined in [e 1 , e 2 ) and "small enough" such that function
A(K)(-) + h(·) is I.C.
- Since f <O for ah(·) "small enough" it is easy to see that it is an improvement on A[K](-).
- Thus, we arrive to a contradiction, therefore it is never possible to have f < O and {fl, B} f:~k~.
o
As we previously explained, the optimal policy can be.found by maximizing A[K)(·) over all possible
K.
Theorem 38. The optimal policy A(·) is given by:
A(·)= maxA(K)(·)
"
Remark 39. It is easy to see that in case we are solving for the zone A ofthe ironing the same procedure
could be done just considering B= B and repeating it the same way. Nevertheless, there is a simplification
that can be done for this case. It is easy to see that lemma 26 can be extended, and the optimal policy in
zone A of the procedure, if x(B) > ;r_ consists in a bunching zone [Q, (jA[~<IJ anda strictly decreasing zone
[tJA[~<], B]. Thus, the optimal policy in zone A of the ironing can be done just by repeating the previous
algorithm for the case x( B) = ;r_ {if in the optimal policy x( B) were to be greater than ;r_ then there will be
a discontinuous jump ate which is irrelevant from the optimization point of view)
40
Summing up, the previous method is based in finding the generalized minimums for the family of
functions x[~V, 11](·). All types that are part of the generalized mínimum for sorne function x[~V, 11](·) have
the same value in the optimal policy than in the respective function x[~V,11](·). Al! types that do not
belong to the generalized mínimum of any function x[~V, 11](·) are found from the closest type that is
part of the generalized mínimum by staying on the respective function x[~V, 11](· ). \Ve proceed to show a
pictographic example on how the optimal policy is found.
Consider the following figure in which we consider the functions x[~V, 11 1 ](·), x[~V, 11 2 ](·), x[~V, 11 3 ](· ), x[~V, 114 ](·),
x[~V, 115 ](·) and x[~V, 116](·) which will help us characterize the optimal policy. We identify the solutions of
Qx[~<,Bi] with the circles, and we use black circles if the solution consist of one pair of types, and white
circles if the solution consists of two pair of types 15 .
X
Figure 33: Example Generalized Mínimum
There are sorne characteristics we would like to highlight:
• Note that the solution of Qx[~<,B3 ] is not unique
• Note that f!. is part of the solution of Qx[~<,B6 ] and Qx[~<,Bs].
• To exemplify we will assume 11 5 is the smallest type such that f!. belongs to it's generalized mínimum
and Qx[~<,Bs] has a unique solution.
• To exemplify we will also assume that nx["• 82 l = O is also a solution of Qx[~<, 82 l
• To exemplify we will assume x 1 (-) is above
x[~V,11 6 ](·)
foral! 11
Now we will show how the optimal policy would look like based on the solutions of Qx[~<,ei]. \Ve
will explain step by step how the optimal policy would be constructed, we will start by showing the
assignments of types that do not belong to the generalized mínimum of any function x[n,, 11](·) and then
explain how the rest is found by finding the generalized mínimum of all functions x[~V, 11](·) . As we
proceed we will show the optimal policy with a thick line.
• First note that x[~V, 113 ](·) has multiple solutions, therefore the right and left limit ofthe generalized
mínimum in this case corresponds to both of it's respective solutions , and therefore al! points in
between both solutions are in J'x[~<,e 3 ]. So, necessarily the optimal solution passes through the
points that are in the difference between the interior of both solutions.
15
Note that the figure is a pictographic representation that isn't scaled to a situation in which we could have
the marked generalized minimums and the necessary hypothesis fulfilled
41
X
~----------------------0
Figure 34: Construction Decreasing Policies I
• The reasoning for x[K,e2)(-) is the same as for x[K,e3](·) only the left limit of the generalized
mínimum is empty and thus the points that are in the difference between both solutions in this
case are the same than the points that are in between the only non empty solution.
X
.J..-------------0
Figure 35: Construction Decreasing Policies II
• Since es is the smallest type such that fl. belongs to it's generalized mínimum, we know that fl.
will be part of the right limit of the generalized mínimum but will not be part of the left limit of
the solutions, thus ex[~<,B 5 ] will be the smallest type in the left limit of the generalized mínimum.
Therefore, all e E [fl., ex[~<,B 5 ]J will be in Jx[~<,B 5 ], and thus the optimal solution passes through
x[K, es](·) for all these e. Therefore for all e E [fl., ¿;x[~<,Bó]J the optimal solution has a value of K;
42
X
~----------------------8
Figure 36: Construction Decreasing Policies III
• All types that have not yet being assigned belong to the generalized minimum of sorne function
x[K, B] that are in between the functions shown in the figure. Thus all types that need to be
assigned would be found by finding the generalized minimum of all functions x[K, B], which would
connect all the marked dots. Connecting the dots from x[K, B6 ] to x[K, B2 ] we would have the
following sequence
X
X
~X[K,66]
~~~:Z:l
K,64J
X
~~~f~:z=¡
X[K,63
K,(h}
X
X K,(},
x¡K,62
K,6¡
X[K,6;
X
Figure 37: Construction Decreasing PoliciefN
Figure 38: Construction Decreasing
Policie~ V
X
X
~
~~f~:Z:I
~X[K,64]
~X[K,66]
X[K,65]
X[K,64]
X[K,6J]
X[K,62]
x[K,6¡]
X[K,6J]
X[K,62]
X[K,6¡]
8
Figure 39: Construction Decreasing Policies VI
Figure 40: Construction Decreasing Policies
43
8
vrr
Finally we would like to give an overview of the solution of the second example explained in section
3. This problem corresponds to a particular case in which there is a unique type iJ such that {f!_, e} and
nx[~<,B] =O are solutions of Qx[~<,Bl, and thus in this case the optimal policy is obviously equal to x[K, iJ](}
This is the easiest possible case to analyze, nevertheless the parameterizations needed to have multiple
solutions to Qx[~<,e] are not simple so, even though it is the easiest case, it is a usefull case to understand.
7
Conclusions
The difficulty in approaching screening problems in which the S.C.P. is not fulfilled has lead to lack
of research of this kind of problems. This does not only limit our understanding of particular models
that should be studied, but also undermine our ability to understand other complex problems, like
multidimensional screening. In this paper we introduce new techniques to tackle screening problems that
do not satisfY the S.C.P. To approach this problem, we exploit the structure of the agent's preferences
to derive necessary conditions for optimality, and variationalmethods to derive necessary conditions for
optimality.
One of our main contributions is the new necessary conditions for implementability and optimality of
a policy. We were able to interpret these conditions economically, which allows to extend the intuitions
to other economic problems in which there is incomplete information. These necessary conditions also
help understand the limitations that the S.C.P. poses on screening problems, thus helping understand
in a better way other problems in which the standard techniques cannot be used, like multidimensional
screening.
We also propase a method to find optimal policies in this model which is robust and can be applied
almost independently of the principal's objective function. The general optimization problem is reduced
to finding optimal U-Shaped forms, decreasing policies and increasing policies. The U-Shaped forms are
characterized in the same way as problems in which the S. C.P. is fulfilled, but with a modified objective
function. On the other hand, the characterization of decreasing policies is based on the concept of
generalized minimums, which is a natural concept in this kind of problems, and thus we believe it could
be useful to salve multidimensional screening. We exemplifY the method with two examples, both of
them being very natural problems, nevertheless yielding complex but interesting solutions.
The first direction in which our work could be extended is by relaxing assumption 25, this assumption
guarantee the variational calculus approach is sufficient, and thus relaxing them would require new tools
to approach the problem. Another interesting problem is to use this approach to tackle multidimensional
screening. Since the main difficulty of multidimensional screening is the lack of the S. C.P., we believe
this approach could useful in the characterization of the solution.
8
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238-263.
8. G. Maggi, A. Rodrguez-Clare, On countervailing incentives, J. Econ. Theory 66 (1995) 238-263.
9. J.-C. Rochet, P. Chon, Ironing, sweeping and multidimensional screening, Econometrica 66 (1998)
783826.
44
10. W.P. Rogerson, The rst-order approach to principalagent problems, Econometrica 53 (1985)
13571367.
11. A. Arauja, D. Gottlieb, H. Moreira, A model of mixed signals with applications to countersignalling, The RAND Journal of Economics Vol. 38 (2007) 10201043
9
Appendix
Theorem 8: For any given interval [éil, Bz]such that
policy x' defined by:
Xa
= x( Bz) = x( éil)'
we have that the
is also I.C.
It is easy to see that the opposite proposition is also true. That is to say, let x(B) be
I.C. and such that the policy is bunching in an interval [8 1 ,82 ]. For any policy x(B) defined
in [éil, Bz], such that X is l. C. and such that Xa = x(Bz) = x(BI) = x(BI) ' then policy x' defined
by:
is also I.C.
Proof. First, note that W,éi' E [~,8 1 ] U (éi 2 ,B) we have that if>±(B,B') = q,x'(B,B'), thus the I.C.C. are
fulfilled. We only need to prove that (ve E [~,8 1 ] U (Bz,B)) (ve' E ((h,Bz)) v(xa,B') +T"' ~ v(x(B),B') +
T(B). Since x(-) is incentive compatible we have the following:
1
x(e)
{=}
1
x(e)
Vx(z, el)dz::; Te>- T(B)
1\
Xa
Vx(z, ez)dz::; Te>- T(B)
Xu
If x( B) ~ XC> and using Vxee ~ o we know that
E (81, Bz) J:~e) vx(z, B')dz ::; max{f:~e) Vx(z, Bz)dz, J:?) vx(z, B1)dz}
and thus,
ve'
¡a
x(8)
=?ve'
If x(B) ::;
XC>,
using that Vxee
ve'
E (81, Bz)
therefore,
E (81, Bz)
vx(z, B')dz::; T"'- T(B)
=? v(x"', B') + T"' ~ v(x(B), B') + T(B)
then we know that max{fx(B) Vx(z, Bz)dz, t(B) Vx(z, el)dz} ::; Jx(B) Vx(z, B)dz. Again
~O we know that ve' E (8 1 ,~:) the function J:~e) vx(z, B') is mon~~onic in él', and thus
J:~e) vx(z, B')dz::; max{f:~e) Vx(z, Bz)dz, J:~e) vx(z, B1)dz}
la
x(e)
=?ve' E (éil, Bz)
Vx(z, ()')dz::; Te>- T(B)
45
=? v(xa, B') + Ta ~ v(x(e), B') + T(B)
To prove the second part, note that (ve' E [ti, el] u (e2' e)) (W E (el' e2)) the following is true:
v(x'(e),e)
+ T'(e)
>
This is because
x(O)
>
v(ec, e)+ Ta
'-V-'
is !.C.
This is
..._;:;::...because x(O)
v(x(e'), e)+ T(e')
is !.C.
therefore (W E [ti, el] U (e2, e)) (W' E (el, e2)) we have that q,x' (e, e') 2 O.
Thus, we need to prove that (W' E [ti, el] u (e2' e)) (W E (el' e2)) we have that v( x' (e)' e') + T' (e) 2
v(x(e'), e)+ T(e'). Note that since x(·) is I.C. we have that
ve E (el, ez) I:(o) Vx(z, e)dz ::; min{f;,(o) Vx(z, e2)dz, I:(o) Vx(z, el)dz }, but since VxOO > o we
must have that X1 (e) S: X a which implies
'Ve' E [ti,el] U (e2,e) J;(o)vx(z,e')dz 2 min{f;,(0 )vx(z,e2)dz,J;(0 )vx(z,e1)dz} 2 T'(e)- Ta,
where the last inequality is given by the implementability of x(· ). Thus we have the following
(ve' E [ti, el] U (e2,e)) ('Ve E (el,e2)):
v(x'(e'), e')+ T'(e')
>
>
..._;:;::...-
v(x'(e), e')+ T' (e)
..._;:;::...-
Using that x(·) is !.C.
o
Theorem 9: Let x(e) be an I.C. mechanism, such that x(e) is decreasing in sorne interval
[ea, e,B]· For any (el < e2 S: e3 < e4) E [ea, e,B] it can never hold true that <P(e4, e2) = O and
<P(e3, e1) =o.
Proof. We will prove that for given e1 < e2 ::; e3
< e4 it cannot hold true that:
. I will prove it by assuming that there exists e1 < e2 ::; e3
and showing that the mechanism is not LC. between land 4.
< e4 such that q,x(eb e3) = q,±(e2, e4) =O,
1°
4
q,x(e4, el)=
=
=
=
1x(y) Vxo(z, y)dzdy
o, x(O,)
o x(y)
o. x(y)
0a
x(y)
31
Vxo(z,y)dzdy+
Vxo(z,y)dzdyVxo(z,y)dzdy
o, x(O,)
Oz 1x(O,)
Oz 1 x(O,)
o x(y)
o. x(y)
0a
x(y)
31
Vxo(z,y)dzdy+
Vxo(z,y)dzdyVxo(z,y)dzdy
o,
x(O,)
Oz 1±(Oz)
Oz 1 x(Oz)
o. x(Oz)
0a
x(Oz)
+
Vxo(z,y)dzdyVxo(z,y)dzdy
~ 1 ±~)
~ 1 ±~)
o x(y)
o. x(y)
e3 x(y)
31
Vxo(z,y)dzdy+
Vxo(z,y)dzdyVxe(z,y)dzdy
e, x(e,)
Bz 1x(Bz)
Bz x(Bz)
1
1
1
1
1
1
1
1
1
1
~----~~----~
11
=1>"(83,8z)20(By !.C. between 2 and 3)
So, we have the following:
46
Since we are considering a decreasing policy we have that x(el) > x(e2), and using that Vxee >O we
have that:
But, we know that I:,3 I:(~!) Vxe(z, y)dzdy =o, so we must have that I:(~l; Vxe(z, e3)dydz 2':
equality if the policy is continuous at e3. Now, using that Vxxe > o we have that
l
x(el)
o (with
x(e 1 )
Vxe(z, e3)dydz 2': 0
=?
x(Bs)
1
Vxe(z, e3)dydz 2': 0
x(82)
and thus we have that
So, the mechanism is not I.C. between
e1 and e4 •
o
Theorem 11: Let x(e) be the optimal policy.
l. Let the interval (ebe2) be such that
ve
E
(e 1 ,e2 )ve'
E e <J?X*(e,e') >O, then fx(x(~:·:)p(e)
ve
E
(e 1 ,e2 )ve'
E
must be non-increasing
2. Let the interval (el>e2) be such that
must be non-decreasing
e
<J?x*(e',e) >O, then fx(x(e),e)p(e)
VxB
Proof. \Ve will first prove corollary 13 and then make sorne considerations to prove Theorem 11.
Let's take a interval (e1 ,e2 ), such that for all e in this interval the I.C.C. of x*(-) are not active 16 .
That is:
Let's considera perturbation h(·), such that
• ve
E [~,
e1] u ¡e2, e¡ h(e) =o
• h(-) is continuous in
• j:, J0
2
h(y)
Vxe(x*(y)
e.
+ z, y)dzdy =O
Let's define x(e) = x*(e) +ah(e). We will prove that for ah(·) small enough x(-) is I.C., and thus
h(·) is an admissible perturbation
• First note that for a small enough perturbation all I.C.C. concerning types in (er, e2) are also
fulfilled (including the local I.C.C. in (el: e2 ) ). That is:
3a such that
ve E (el,e2)ve E e <J?x(B',e)
• The I.C.C. between types greater than
fulfilled. That is:
16
e2
or smaller than
>O
1\
<J?x(e,e') >O
e1 remain unchanged,
and thus are also
Note that a necessary condition for this is that x* (-) must be strictly monotonic in (8 1 , 82 )
47
• Let's take
e> e2 ande'< e1 , we have that:
iJ>x(e, e')= re ¡x(y) Vxe(z, y)dzdy
Je, lx(B')
e 1x*(y)
1ez1x*(y)+h(y)
=
Vxe(z,y)dzdy+
Vxe(z,y)dzdy2::0
e' x* (B')
e1 x• (y)
l
.....
~------~r-
=0
----~
e' x(y)
if>X(8,e')= r r
Vxe(z,y)dzdy
le lx(e)
e' 1x*(y)
1ez1x*(y)+h(y)
=
Vxe(z,y)dzdyVxe(z,y)dzdy2::0
e, x*(y)
e x*(B)
l
~------~r-------~
=0
Therefore h(·) is an admissible perturbation. So, if x*(-) is the optimal policy, then we must have
that J:,2 V(x(e) + h(e), e)p(8)d8 has it's maximum value at h(·) = O. In this case the utility to the
principal of policy x(·) is given by:
ux=
=
lj
f(x(y), y)p(y)dy
{[j j(x*(y),y)p(y)dy+ ¡ez j(x(y),y)p(y)dy- ¡ez j(x*(y),y)p(y)dy
Jfl.
=
ux*
le,
+
l
ezlh(e)
e,
o
le,
fx(x* (y)+ z, y )p(y)dzdy
=UX· +fez t(B) fx(x*(y}+z,y)p(y)Vxe(x*(y)+z,y)dzdy
Je, Jo
Vxe(x (y)+ z, y)
Optimizing with respect to h(·) we get the following optimization problem:
max
+
ux*
h(·)
s.t.
l
l
ezlh(e)
e,
ezlh(y)
e,
o
fx(x*(y)
Vxe(x*(y)
o
+ z, y)p(y)dzdy
+ z, y)dzdy =O
this leads to the following pointwise F.O.C.
fx(x*(y)
+ h(y),y)p(y)- AVxe(x*(y) + h(y),y) =O
which has it's optimum at h(y) =O, and thus we get the following equation:
fx(x*(y),y)p(y) =
Vxe(x*(y), y)
.>..
The demonstration of Theorem 11 is basically the same than corollary 13 only that it is necessary
to be a more careful with the variation h(·).
Let's consideran interval (el, e2) such that VB E (el, e2)VB' E 8 <px* (8, e') >O (condition q;x· (8', e) 2::
O is guarantee by the I.C.C. of x*(·)).
Let's considera perturbation h(-), such that
48
• h(·) is continuous in e.
• f:.
2
J0h(y) Vxe(x*(y)
• ve E [Bl, (}2]
+ z, y)dzdy =O
I:, foh(y) Vxe(x*(y) + z, y)dzdy:::; o
Let's define x(B) = x*(B) + ah(B). We need to prove that for ah(·) small enough x(-) is I.C., and
thus h(·) is an admissible perturbation.
• It is easy to see that the cases in which B', e E [eb e2] ande', e E e\[ell e2] can be proved the same
way as corollary 13.
• If (e E (el, e2))
(e'
E e) then for ah(·) small enough q,x* (e, e')
>
0, which can also be proven
the same way as as corollary 13.
• Thus we need to prove that
(ve'
E (el, e2))
(ve
E e) exists a
h(-) small enough such that
q,x* (e, e').
If e' E (B1,e2)ve E [~,e1]:
<J?x(e,e')= re ¡x(y) Vxe(z,y)dzdy
Jo-
lx(e')
e ¡x*(y)
=
¡ e'
Vxe(z, y)dzdy-
x*(B')
¡e' ¡x*(y)+h(y)
e,
x*(y)
Vxe(z, y)dzdy ;=::O
~-------v--------~
::;o
<J?x(e,e')= re ¡x(y) Vxe(z,y)dzdy
Jo-
=
l
lx(B')
e ¡x*(y)
8'
Vxe(z,y)dzdy+
x•(e')
¡e' ¡x*(y)+h(y)
e2
x*(y)
Vxe(z,y)dzdy ;=::O
~-------v--------~
2:0
Thus, h(·) is a admissible perturbation. Optimizing with respect to h(·) we get the following optimization problem:
ux*
max
h(·)
s.t.
ve
E
+
e2
h(e)
11
e,
o
fx(x*(y) +z,y)p(y)dzdy
J:,2 J~(y) Vxe(x* (y)+ z, y)dzdy =O
e h(y)
(el, 82] fe, fo
Vxe(x*(y) + Z, y)dzdy:::; 0
this leads to the following lagrangian:
L
=
fx(x*(y)
+ h(y),y)p(y) + AVxe(x*(y) + h(y),y) + {Y v(w)dwvxe(x*(y) + h(y),y)
le,
with v(w) ;::: 0\iw E [e1, e2]and the pointwise F.O.C. leads to:
fx(x*(y)
+ h(y), y)p(y) + (A+ 1~ v(w)dw) Vxe(x*(y) + h(y), y)= O
49
which has it's optimum at h(y) =O, and thus we get the following equation:
r
fx(x*(y), y)p(y) =-.Av(w)dw
Vxe(x*(y),y)
Je 1
since
Ji, v( w )dw is not decreasing we get the result.
D
The second case can be done the same way.
Theorem 15: Let x* O be the optimal implementable convex-valued correspondence,
and B", B' E e be such that q,x* (B", B') =O. If x*O is strictly monotonic and continuous at B'
and B" , then:
fx(x*(B"), B")p(B")
Vxe(x*(B"),B")
fx(x*(B'), B')p(B')
Vxe(x*(B'),B')
Proof. Since the case x*(B") = x*(B') can be found in Araujo and Moreira [1] we will only analyze the
case in which x*(B") i= x*(B'). It is easy to see from the analysis made in section 6 that if x*(·) is an
implementable convex-valued correspondence the only way we can have x*(B") i= x(B') and q,x* (B", B') =
O is if B" > B' and x*(B') > x(B"). Moreover, using the analysis made in section 6 and using theorem 9
to know
(VB E 8\[B', B"])(VB E [B', B"]) q,x* (B, B) >O
1\
q,x* (B, B) >O
In what follows we will assume we can find intervals (B¡, B2) and (Bs, B4) such that:
• B" E (Bs, B4) and B' E (B1, B2)
• x*(-) is strictly monotonic and continuous in (B1, B2 ) and (B 3 , B4).
• Foral! B" E (B 3 , B4 ) there exists a B' such that q,x* (B", B') =O, and for all B' E (B 1 , B2) there exists
a B" such that q,x* (B", B') = O
• q,x* (B1, B4) = q,x* (B2, Bs) =O
We can always find intervals (B 1 , B2 ) and (B 3 , B4 ) that satisfY the first two conditions. If we can't
satisfY the third condition, then we could prove the theorem with a variation similar to the one made in
theorem 11. If the third condition can be satisfied then the fourth condition can also be satisfied, and
then we can there is a unique bijection between B' E (B 1 , B2) and B" E (B 3 , B4 ) where each pair is defined
by <I>(B", B') =O. For each pair the following conditions must be true:
Vx(x(B'), B')
= Vx(x(B'), B")
(2)
ve(x(B'), B")
=
ve(x(B'), B")
(3)
\\lhere conditions 2 and 3 are given by lemma 10. Now we will make a variation around x*(·),
so x(B) = x*(B) + h(B). For x(B) to be I.C. we will define h(B) continuous and arbitrary in (B 3 , B4 )
such that h(Bs) = h(B4 ) = O. x(B) in (B1,B 2) will be obtained from condition 2 and 3, and VB tJ_
(B 1 , B2 ) U (B 3 , B4 ) h(B) =O. First we need to show x(·) is I.C.:
• VB',B" E 8\[B1,B4] q,x*(B",B')
• VB', B" E [B2, B3 ] q,x* (B", B')
=
=
<I>x(B",B')
<I>x(B", B')
• Using using the analysis made in section 6 and using theorem 9:
thus, for ah(·) small enough we have that the conditions are also satisfied for x(·)
50
So, with condition 2 and 3 we can define the function <p(x, 8) and 7/J(x, 8) by:
ve(x, 8) =ve( 7/J(x, 8), 8)
Vx(7/J(x, 8), 8) = Vx(7/J(x, 8), <p(x, 8))
So we have the following:
e•
G(h)=
l
l
¡
fx(x(8), 8)p(8)d8
e3
+
e.
fx(x(8), 8)p(8)d8-
e3
1
fx(7/J(x(8), 8), <p(x(8), 8))p( <p(x(8), 8))( 'Px(x(8), 8)xe(8)
+ <pe(x(8), 8) )d8
e3
e.
=
fx(x(8), 8)p(8)d8
<p(x(e2)h)
e.
=
l<p(x(e,),er)
w(x( 8), 8) - w( 7jJ(x(8), 8), <p(x( 8), 8)) ('Px (x( 8), 8)xe (8) +<pe (x( 8), 8) )d8
e3
where fx(x(8), 8)p(8) = w(x(8), 8). Now, taking the variation in the direction of h:
6hG( h)=
fe:• [Wx (x* (8), 8)h- w ('Pxx (x* (8), 8)x{¡ (8)h + 'Pex (x* (8), 8)h + 'Px(x* (8), 8)he)
-h (Wx7/Jx(x* (8), 8)
+ We<px (x* (8), 8)) ('Px(x* (8), 8)x{¡ (8) + <pe(x* (8), 8))] d(J
Where w means that it is evaluated in (7/J(x*(8), 8), <p(x*(8), 8)). Integratating the term with he by
parts we get:
6hG(h )=
fe:• [Wx(x* (8), 8)h - hw ('Pxx (x* (8), 8)x{¡ (8) + 'Pex(x* (8), 8)) + hw ('Pxx(x* (8), 8)x8(8) + 'Pex (x* (8), 8))
+h<,ox(x* (8), 8) (Wx( 7/Jx (x* (8), 8)x{¡ (8)
-h (Wx7/Jx (x* (8), 8)
+ 7/Je(x* (8), 8)) + we ( ('Px (x* (8), 8)x{¡ (8) + <pe (x* (8), 8))))
+ We'Px (x* (8), 8)) ('Px (x* (8), 8)x{¡ (8) + <pe (x* (8), 8))] d8
Simplifying terms ...
6hG(h)=
fe:• [Wx(x* (8), 8)h
+hwx ('Px (x* (8), 8)7/Je (x* (8), 8)) - <pe(x* (8), 8)7/Jx (x* (8), 8)))] d8
Since x*(-) is optimal we must have that 6hG(h) =O, and since h(8) is arbitrary we have that
wx(x*(8),8) +wx ('Px(x*(8),8)7/Je(x*(8),8)- <pe(x*(8),8)7/Jx(x*(8),8)) =O
Finally we know that <p(x, 8) and 7/J(x, 8) are defined by:
ve(x, 8) = ve(7/J(x, 8), 8)
Vx(7/J(x,8),<p(x,8))
=
Vx(7/J(x,8),8)
So, we have that:
7/Je(x, 8) = vee(x, 8) - vee( 7/J(x, 8), 8)
Vxe( 7/J(x, 8), 8)
51
'1/Jx (x,e )
=
Vxe(x,e)
Vxe('I/J(x,e),e)
n) Vxe('I/J(x,e),e) +'1/Je(x,e) (vxx('I/J(x,e),e) -Vxx('I/J(x,e),<p(x,e)))
<pe (x u 'Vxe('I/J(x,e),<p(x,e))
n) _ '1/Jx(x,e) (vxx('I/J(x,e),e) -Vxx('I/J(x,e),<p(x,e)))
Vxe ('1/J(x, e), <p(x, e))
'Px (X, u -
So, finally we have that:
e)·l· (*(e) e))_ (*(e) e)·1· (*(e) e))= '1/Jx(x,e)(vxx('I/J(x,e),e) -vxx('I/J(x,e),<p(x,e)))"1' ( e)
( *(e) '<p8X
l.f!xX
'
<peX
''f/xX
'
Vxe('I/J(x,e),<p(x,e))
<p8X,
_ Vxe ('1/J(x, e), e)
+ '1/Je (x, e) (Vxx ('1/J(x, e), e) Vxe( '1/J(x, e), <p(x, e))
=?
l.f!x(x*(e),e)'ljJe(x*(e),e))- <pe(x*(e),e)'I/Jx(x*(e),e))
Vxx ('1/J(x, e), <p(x, e))) Wx (x, e)
·
Vxe('I/J(x,e),e) '1/Jx(x,e) = _
Vxe(x,e)
Vxe( '1/J(x, e), <p(x, e))
Vxe ('1/J(x, e), <p(x, e))
therefore,
(W" E (e 3,e4))(ve' E (e 1,e2)) such that cpx*(e",e') =O we have that <p(x*(e"),e") = e1 and
'1jJ(x*(e'1 ), e")= x*(e'). Thus we get the result.
D
Lemma 26: The optimal decreasing policy consists of bunching part in an interval [e' e]
and a continuous and strictly decreasing part in [f!_,
where E (f!_,
In what follows we
will refer to as the point where the policy starts being strictly decreasing, and
as the
point corresponding to policy x(·).
e],
e
e
e
e].
ex
<e.
Prooj. Let's first consider the case of a I.C. mechanism x(·) with discontinuous jump in e1
We will
use e'+ and el- to denote that a policy is being evaluated at the right and left limit of e' respectively.
Let's consider a variation of the form h(-) such that:
• h(·) is continuous in [e'- E, e') U (e', e'+ E]
• h1 (·) <O
This assumption is just to ensure the mechanism x(e)
is I.C. if there is a bunching zone near
<0
• h(e)
=
{
>O
=o
+ h(e)
e'
veE(e'-E,e')
ve E (e', e' +E)
ve rf: (e'- E, e'+ E)
rB'+e ¡x<!)+h(ií)
( ií)d dií _O
• Je'-e x(B)
Vxe z, u z u -
We will denote p(-) = x(·) + h(-). Note that the I.C.C. between pairs of e not in (e' -E, e' +E) remain
unchanged by variation h(e), and thus it is only necessary to check that h(e) improves the principals
utility and the LC.C. with e E (e'- E, e'+ E) are not broken. The change in the principals utility is given
by:
52
¡B'+e
¡B'+e:
~U= lw-e f(p(B),B)p(B)dB- lw-e f(x(B),B)p(B)dB
e'+e¡p(e)
1
-1·
=
-
=
=
_ fx(z, e), B)p(B)dzde
8'-e x(e)
e' +e p( e)
- - - ¡
( e-)fx(.-,,e),e)p(e)d-d()Vxe z'j
""
e'-e x(e)
Vxe(z,e)
e'
p(e)
- - -
r r
Je'-e lx(e)
Vxe(z,e/x(z,e),()~p(e)dzdB+
Vxe(z, (})
e' ¡p(e)
1
8' -e
_ vxe(z,B)r(z,B)dzdB+
x(B)
e' +e: p(e)
- - r
r
Vxe(z,e/x(z,e),e~p(e)dzdB
Je,
lx(e)
Vxe(z, e)
le'+" ¡p(e)
_ vxe(z,B)r(z,B)dzdB
8'
x(B)
Using that the variation h(e) satisfies:
e' ¡p(e)
le' +e ¡p(e)
_ Vxe(z, B)dzde +
_ Vxe(z, e)dzde =O
l 8'-e x(B)
8'
x(B)
Note that the first term is positive and the second one is negative. Using that ar~~,e) > O we know
that for a h(()) small enough we have that
and thus,
~U=
e' ¡p(e)
_ Vxe(z, e)r(z, e)dzde +
e'-e x(B)
1
e'+" ¡p(e)
1
8'
_ Vxe(z, e)r(z, e)dzde >O
x(B)
thus, it is a profitable variation.
Since we have shown that h( ·) is a profitable variation we now need to show that it is I. C., for this
it is necessary to check the following:
First note that if x(·) is I.C. then J:(~:/ Vxe(z, e')dz < O, thus the local I.C.C. guarantee that the
I.C.C. are not broken for all
8 E (e'- E,()'+ E).
We also know that ve E [~,e'- f] (f>P(e, e) ~ q,x(e, B) ~o, and for e E (e'+ f, e) we know that the
local I.C.C. guarantee that <I>P(e, e) ~ O . So, if x(-) is I.C. then the condition iJ>P(e, B) ~ O is satisfied
for all E 6
e
Thus we need to check <I>P(B, e) ~O, which only needs to be checked for e<() (likewise for e> e the
local I.C.C. guarantees the global I.C.C.). The proof proceeds as follows:
• Check that iPP(e'+}) ~O
1\
(f>P((}'-,e) ~O=? p(·) is I.C. for all points between e'+ ande'-
• Check that if q,x(e, e'-) = O, then for a E small enough for all e in (e'iJ>P(e'-,e) >O then the implication is trivial)
-53
E,
e']
iJ>P(e, e) = O (if
• Check that if if>X(e, e'+) =o, then for a E srnall enough for all e in [e'-, e'+ e)
if>P(e'+,e) >O then the irnplication is trivial)
if>P(e, e)
=o
(if
To check that if>P(e, e'+) :::: 0 1\ if>P(e, e'-) :::: 0 =? The rnechanisrn is I.C. for all points between
e'+ ande'- note that the I.C.C. of any point between e'+ and e'- can be written as follows:
with J.l E [0, 1], in particular the cases J.l
we will show that
if>P(e, e'+)+
B¡p(e'+)
1
8'
= 1 and J.l =o are if>P(e, e'-) and if>P(e, e'+)
vxe(z, y)dzdy:::: o
1\
respectively. So
if>P(e, e'+) ::::o=>
p(O'-)
Bjp(O'+)
if>P(e, e'+)+
1
8'
Vxe(z, y)dzdy:::: 0
'if¡.¿ E
(0, 1)
(1-¡¡.)p(O'+)+¡¡.p(O'-)
Too proove this let"s suppose otherwise, that is to say for sorne
¡..tE
(0, 1):
but, if if>P(e, e'+):::: 0 then,
8jp(e'+)
1
8'
Vxe(z, y)dzdy <O
(1-¡¡.)p(O'+)+¡¡.p(O'-)
but, using that Vxxe(z, y) >O we can see that
Thus, we arrive to a contradiction
\Ve will now Check that if ii>x(e, e'-) = O the variation is I.C. in (e' - E, e'). First note that
ii>x(e,e'-) =O irnplies that x(·) is bunching in sorne interval (e1 ,e'). To check this notice let's assurne
othenvise.
54
Thus, the mechanism is not l. c. So, if cf>X (e'-' e) = o implies that x( o) is bunching in sorne interval
(rh,e'). Now we will check that if cf>x(fJ'-,e) = O for a é small enough the variation is I.C. for a
e E (e'- é,e').
e¡x(B-)
cf>x+h(e, e)= cf>x(e, e'-)+
Vxe(z, y)dzdy-
l B' x(B-)+h(B)
B'+E¡x(e'+)
Vxe(z, y)dzdy
l8
x(B'+ )+h(y)
lB' ¡x(B-)+h(B)
B
Vxe(z, y)dzdy
x(B-)+h(y)
1
=
l B'
x(B-)+h(e)
e'+E¡x(e'+)
=
e¡x(B-)
cf>x(e, e'-)+
Vxe(z, y)dzdy-
Vxe(z,y)dzdy±
l B'
lB' ¡x(r)+h(B)
B
x(B-)+h(y)
lB' ¡x(B-)
Vxe(z, y)dzdy
Vxe(z,y)dzdy±
10 ¡x(B-)
Vxe(z,y)dzdy
x(B'+)+h(y)
B x(B-)+h(B)
B-E x(e-)+h(y)
e x(B-)
B' x(B-)
cf>x(e, e'-)+
Vxe(z, y)dzdy +
Vxe(z, y)dzdy
lw lx(B-)+h(B)
le lx(B-)+h(B)
rr
0' ¡x(B-)
-
1 B-e x(B-)+h(y)
rr
Vxe(z,y)dzdy-
lB'+< ¡x(B'+)
le ¡x(e-)
Vxe(z,y)dzdy+ ·
Vxe(z,y)dzdy
x(B'+)+h(y)
e-E x(B-)+h(y)
0'
=0
=
cf>x(e,e'-)
+
e ¡x(e-)
l B x(B-)+h(B)
vxe(z,y)dzdy+
But, since Vx(x(B'-), e') < Vx(x(B'-), e) for a
( \ly E (B'-
é,
B')) ( \lz E [0, h(y))])
e¡x(e-)
le
x(B-)+h(B)
Vxe(z,y)dzdy+
é
le ¡x(e-)
0-< x(B-)+h(y)
vxe(z,y)dzdy
and h(-) small enough we have that:
vx(x(e'-)
+ z, y) < vx(x(B'--) + z, e)=?
B ¡x(B-)
r
Vxe(z,y)dzdy > 0
le-E x(e-)+h(y)
=?
The case cf>x(e,e'+) =O can be proved in a similar way.
The bunching case can be proved in the same way as the discontinuity, it is easy to note that a
bunching zone is a discontinuity in case the axis are exchanged. The optimizing problem an be thought
as finding the optimal type for each assignment, changing the problem into finding the optimal function
B(O. If there is a strictly decreasing part at the edges of a bunching zones, then the same variations
done for the discontinuities can be done in the bunching zones, and the problem is completely analog.
Nevertheless, if the bunching zone ends at (j_ the the problems are no longer analog, and as a matter of
fact these type of bunching zones may be optimal.
O
Lemma 32: The following observation can be made of the solution of Qx[~<,B').
• The solution of Qx[~<.B) is uniquely defined for almost for every B.
• Let {B~l"·e'j, ... , B~},~·B')} and {B~l"· 0 "), ... , (J~~~;B"]} be solutions ofproblems Qx[~<,B') and Qx[~<,B")
. l y. If (}'
respective
case we
no t e
·¡¡ say
WI
1
< e" t h en {Bx[~<,B')
_, e-} e un"
I=I (Bx[~<,B")
, (}x[~<.e"))
•
, ... , ex[~<.e'j}\{e
11
11
2 n'
21
{ex[~<,B') ... , (}x[~<,B')}
,
11
2 n'
an d
{Bx[~<,B") ... , (}x[~<,e"j}
,
11
2 n"
--" {ex[~<,e"j
(}x[~<,e"j}
{e 11x[~<,B ) ' ... , ex[~<,B')}
2n'
'
11
' ... , 2n"
55
)
are nes t e d , an d we
(r
n t h"IS
·¡¡ d e-
WI
Proof. To avoid excess notation we will denote x[K, B] by y[B]. Let's denote the set of all possible solutions
of QY[e'] by S(B'). The proof of lemma 32 will consist of 3 steps.
l. We will show that S(·) is an upper hemicontinuous correspondence.
2. For any given B' such that S(B') is not a singleton we can find §., s E S(B') such that for all
sE S(B')\{2., s} §.-< s-< s
3. S(-) is a singleton in a neighborhood around B', and using a comparative static analysis we will
show they are nested.
Note that for all B' < Bo we can find an E such that for all B E [~, B], y[B' + 17](B) < x 0 (B). Thus
for all B' < B0 we can find a bound fi such that for all B' nY[B'] < fi in sorne neighborhood around B'. We
will define problem C(B') as follows:
ñ
C(B')=
min
Lq,y[e'](B2j,B1j)
tt~ef~ :;e~~::; ... ::;ef~ ::;eg~::;;e j=l
It is easy to see that a solution to C(B') will consist of all the elements of a solution of problem
QY[e'] plus elements of the form B~~ = Bg~ for sorne k E {1, .. , n}. Using Berge's Maximum Theorem
we can see that the solutions to problem C(B') are upper hemicontinuous in B'. But, we can take a
subset of solutions of C(B') in which for all elements such that Bf~ = Bg~ we have that B~~ = Bg~ =
and this subset must also be upper hemicontinuous. Thus, the solutions to problem QY[B'] are upper
hemicontinuous in B'. The formal definition of C(B') that allows to easily see that satisfies the hypothesis
of Berge's Maximum Theorem goes as follows:
Let's define the set He R 2n, such that Vh = (hn, h12, .. , h1n, h2n) EH ~:::; h1 :::; h2 :::; ... :::; hm:::;
and let's define function \fl : H x [~, B0 ) ->Ras follows:
e,
e,
n
\fl(h, B) =
L
q,y[B] (h2i, h1i)
i=l
we can see by the maximum theorem that C*(B) = argmax{\fl(h,B)Jh EH} is a upper hemicontinuous correspondence in B.
Now we will show that for any B' such that S(B') is nota singleton, there exists §.,s. First we will
show that for any pair of types Blk, B2k, r.p 1j, r.p 2j that belong toa solution of S(B'), then it can never hold
true that Blk < r.p 1j ::=; B2k < r.p 2 j. We will prove this by contradiction
• Let's considera solution {Bn,Bz1, ... ,B1n,Bzn} E S(B'), and 'Pl.'P2 Es E S(B'), such that Blk <
'P1 :::; Bzk < r.pz
• There are two subcases that need to be consider, for sorne l > k
B2 cz- 1) ::=; r.p 2 < Bu or Bu ::=; r.p 2 <
B2z
In the former case it can be shown that <I>(r.p2,B1k)
contradiction.
< ¿~:,~<I>(B2j,Bl}), thus arrive toa
In the latter case it can be shown that <I>(Bzz, B1k) < ¿~=k <I>(B2 j, Bl)), thus arrive to a
contradiction.
ViTe will show how to prove a particular example of the second subcase previously mentioned, the
general case can be proved in the same way. Let's take the case in which S(B') is not a singleton, and
let tln,821,B1z,B22,813,823 Es E S(8') and r.p1,r.p2 Es' E S(8') such that 8n < '1'1 < B21 < B12 < 822 <
B13 < '{Jz < 823 ( see fig 41)
• Using the same argumentas in theorem 9 we can show that areas A,B,C must be strictly negative.
-56
X
x(O)
e
Figure 41: Decreasing Policy IV
• Since 'Pl, '(!z belong toa solution in S(B') it must hold true <I>('f!z, '(! 1) ::; <I>(Bz¡, '(! 1) + <I>(B22, B12 ) +
<I> ('Pl, B13 ). Thus, he shaded area must be less or equal to O
• Therefore, <I>(Bu, 823) < <I>(Bz1, Bu)+ <I>(Bz2, B12)
+ <I>(B23, Bl3) +A+ E+ C + shaded area
• Thus we arrive to a contradiction.
8
Now we will show that for any B such that S(e) is nota singleton Vr E S(B)\{.,!,s}
3Qy[B] 1
-----a¡¡s=r
>
~~ ls=..: >
181
8Qy[B] 1
'
.
{
} ( ) .
-----a¡¡s=s· Let s take sorne solutwn Bu, ez1, ... , B1n, Bzn -pE S e .
8QY[8]
i[>x[8](B 2 . B ·)
n~ (1e2j ( 18 l(B)
- - 8 [e](O) - 182j
- -8 [8J(e ·))
8
--1
_ = 8'\'n~
L...J=l
ae s-p
ae J' lJ = '\'
L..,
Vxe y
' 8)-y--deae
Vxe (y 1 l(B lJ·) ' B)dB y ae lJ
j=l
•
ayi•Jo
38
8,j
81j
>0
-
• The local I.C.C. guarantees that Vx8(yle] (8), e)
is least for p = s and greatest for p = §.
< o. Thus, the term z=;~l I:,~ Vx8(yle] (0), e) By~~(ii) dO
n'
re .
[]
-
-ayiBI(8 ·)
• On the other hand, by the first arder condition we have that - I;j~l Je:; Vxe (y 8 (B1j ), B)dB~ =
r821
[8]
- - ayl•l ce )
.
re
¡e¡
-
-a, 1•1 ce )
- Je 11 Vxe(y (Bu),B)dB~. OnceagambytheF.O.C.theterm- Je,:'vxe(Y (Bu),B)dB~
is different than O only if Bu = O, in which case it is least for p = s and greatest for p = §.
Thus we have the following
8QY[B]
'ifr
E
S(B)\{.,!,s}
8QY[B]
8QY[B]
--agls=.§: > --agls=r > --agls=ii
Using that S(B) is upper hemicontiuous we can see that for any B' such that S(B') is nota singleton
there exists a neighborhoods such that for all e E (B'- E, e') the correspondence S(B) is a singleton and
is arbitrarily clase to §. and for all BE (B', B' +E) the correspondence S(B) is a singleton and is arbitrarily
clase tos.
57
Let e', e" E (OY[K],BY[K]) be such that <f>Y[KJ(e',e") is a local mínimum (for local minimums in
whích e" ore' are a corner solution the comparative statics is basically the same). We have the following
,
( rememb er"'
rt-y[KJ(e' , e") = Jeu
re' ly[K](y)
equat Ions
y[KJ(e") VxB' (z, Y )dz dY )·.
8<f>Y[K] (e' e")
8e' '
=
jy[KJ(e')
y[K](e")
Vxe{r, e')dx
= v~(y[K](e'), e')- v~(y[K](e"), e')= o
8<f>Y[KJ(e',e") =-le'
'( [K](e") if')dB'8y[K](e") = ( ( [K](e") e")- ( [K](e") e')) 8y[K](e") =O
e" Vxe Y
,
Vx Y
,
Vx y
,
8 e"
8 e"
8 e"
'---y-'
#O
Now making the comparative statics with respect to K, we derive the previous equations with respect
to K and we get the following equations:
( ae')
(a 88'f!K
a 88'88' azq,vlKI(e',e"))
oe'f!e" ,e")
8ft, + o <!>y[Kf(e'
,e")
.pY[K[
,e")
( oe"oB' oe"oe"
8K
oe"oK
2
2
q,•lKI(e',e")
i] 2.pY[KJ(e'
~
(0 1
i] 2
q,•lKI(e',e"))
0
=
2
inverting the matrix we get the following:
ae')
a~
(~
ok
_1
=
8 2 <I>•lKJ (8' ,e")
ae' ae'
8
2
<I>•lKI(e',e")
ae"BB"
8 2<I>•lKJ (e'.e")
(
azq,vlKJ (O' ,e") azq,v[KJ (8' ,e") 8 2 <I>vlKJ (O' ,e")
ae" ae"
ae' ae"
aw ae"
aw ae"
o2 <I>•lKI(e'.e"))
ae'ae"
a 2q,vlKJ (e' .e")
·
ae' ae'
(8
2
<I>•lKI(e',e"))
ae'BK
2
8 <I>•lKJ (e' ,e")
<O(S.O.C.)
Calculating the terms:
82q;y[KJ(e',e")
8 e, 8 e"
= _
, ( [
ve x y K
](e") e') 8y[K](e")
,
>O
8 e,
'---y-'
>0
<O
82q;y[Kl(e',e") = (v ( [K](e") e")( [K](e") e')) 8y[K](e") 8y[K](e") >O
8e"8K
xx Y
'
Vxx Y
'
8K
8e'
' - - - y - '' - - - y - '
<O
>O
<0
82q;y[KJ(e',e") = v ,(y[K](e') e') 8y[K](e') -v ,(y[K](e") e') 8y[K](e")
8e'8K
xe
'
8K
xe
'
8K
'-v--'
'---y-'
>0
>O
<o
replacing the signs we have the following:
(~i')
Thus e" is decreasing in K and
they are unique.
=- (
~~) . ( ~) = ( ~)
e' is increasing in K
58
which means the solutions are also nested when
D
ae" aK