1. Introduction - Yale Economics

Journal
of Mathematical
Economics
16 (1987) 191-200.
A NECESSARY
AND
RATIONALIZABILITY
North-Holland
SUFFICIENT
CONDITION
FOR
IN A QUASI-LINEAR
CONTEXT
Jean-Charles
ROCHET*
Ecole Polytechnique, 75230 Paris, France
Universitt! Paris IX-Dauphine, Paris, France
Submitted
March
1986, accepted
May 1987
The aim of this note is to give a simple characterization
of the rationalizability
of decision rules
(or action profiles). The necessary
and sufficient condition
we obtain
suggests interesting
analogies between the Implementation
Problem and Revealed Preference Theory. Two particular
cases are examined:
(a) The one-dimensional
context, which shows that our condition
is a generalization
of the
monotonicity
condition of Spence-Mirrlees,
(b) The linear set-up, which shows that rationalizability
in multiple dimension
requires more
than monotonicity:
it implies also symmetry
conditions
which are translated
by Partial
Differential
Equations
(analogue
in this context of Slutsky equations
for Revealed Preference
Theory).
1. Introduction
The formalism of Social Choice Theory is now well understood
and widely
used in Economic Theory, especially when incomplete
information
problems
are at stake. In particular,
the concept of implementability
by a mechanism
has been fruitfully applied to many different questions.
For instance Baron
and Myerson (1982) and Guesnerie and Laffont (1984), in a context of public
control
of firms, characterize
the class of regulatory
policies
that are
available to an uninformed
Public Authority.
Another example is non-linear
prices: Goldman,
Leland and Sibley (1980), and Maskin and Riley (1984)
examine the selling strategies that are attainable
by a uninformed
monopolist. Similar problems occur in Optimal Tax Theory [Mirrlees (1976), Seade
(1977)], Product Differentiation
[Mussa and Rosen (1978)] and more generally in all models that can be set-up in a Principal-Agent
framework.
In a quasi-linear
context it is tempting
to split the mechanism
into two
parts: the ‘physical’ part, that corresponds
to the action or the choice of the
agent and the monetary
part, or compensatory transfer. Following
Guesnerie
*I am very grateful to Tyrrell Rockafellar who suggested to me the characterization
proved in this paper. I also benefited from the comments
of Andreu
Mas-Cole11
anonymous
referees.
03044068/87/%3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland)
theorem
and two
192
J.-C. Rochet, A condition for rationalizability in a quasi-linear context
and Laffont (1984) we will call ‘action profile’ or ‘decision rule’ any mapping X(a)
from the set of agents’ characteristics to the set of possible actions. We will
say that such a mapping is rationalizable if it is implementable via compensatory transfers, that is if there exists a transfer function t( .) which makes
the mechanism (X( .), t( +)) truthfully implementable in dominant strategies.
This paper is concerned with the characterization of such action profiles.
2. Notations and characterization
Let agents
function:
be characterized
theorem
by
a parameterized
quasi-linear
utility
(1)
V( 0, x, t) = u( 0, x) - t,
where the parameter 0 belongs to a given set a, x denotes the ‘physical
allocation’ vector, belonging to a given subset % of IWMand corresponding to
the action or choice of the agent. t is the compensatory transfer. We say that
an action profile X(.), that is a mapping X from S2 to SF, is rationalizable if
and only if
3t:8+R
V(0,8’)EQ2u(o,x(o))-t(o)~u(o,X(8’))-t(B’).
(2)
In other words there exists a compensatory transfer t(.) such that the
mechanism (X, t) is truthfully implementable in dominant strategies.
The ‘Taxation Principle’ [cf. Guesnerie (1981), Hammond (1979) and also
Rochet (1985)] states that (2) is equivalent to the following very similar
condition:
3&SY+lR VOEQ max,,,(u(O,x)-4(x)}
That is to say : X(s) is rationalizable
a non-linear price scheme.
Theorem 1.
is attained for x=X(O).
if and only if it can be decentralized by
A necessary and sufficient condition for X(.) to be rationalizable:
For allfinite cycles
OO,O,,...,ON+l=O,
in Sz 1
I>
j. I44 +
1,
Proo$
X(0,)) -40/c>X(0,)))s 0.
(3)
J
[It is adapted from Rockafellar (1972, p. 238).] Let X(.) be rationalizable via t( .) and O,, O,, . . . , ON=Oo be a finite cycle. Then
J.-C. Rochet, A condition for rationalizability
Thus by adding
in a quasi-linear
context
193
up these inequalities:
or
which implies (3).
Conversely,
in 52
0
let (3) be fulfilled,
Condition
construction
(3) implies
of V( .):
vu
which implies
that
take an arbitrary
V(Q,) equals
8, in Q and set for any 0
zero. Now
for any 8 we have by
2 v(e)+ u(e,,x(e))-44 x(e))
that V(e) is finite. We also have for any 8 and 8
v(e) 2 V(Q) + u(e, x(ey)-u(8’,
x(e,)).
Thus if we set t(e) = u(0, X(0)) - V(d) we have
u(e,
x(e))-t(e)
2
qe, x(q) - t(e,).
Cl
Condition
(3) is the best that can be hoped for without
additional
structure
on 52 and regularity
assumptions
on u. In fact our Theorem
appears as a generalization
of two classical results:
_ On the one hand the characterization
of the subgradients
of convex
functions by Rockafellar (1972); this aspect is developed in section 4.
_ On the other hand the equivalence
between monotonicity
and rationalizability in the one dimensional
context under the Mirrlees-Spence
condition
[Mirrlees
(1976), Spence (1974)]. Section
3 is dedicated
to this one
dimensional
context.
J.-C. Rochet, A condition for rationalizability in a quasi-linear context
194
3. The one dimensional
Proposition 1.
that
case
Let R = [&,, 8,] and M = 1. Assume moreover that u is g2 and
V(8,X)E51X
R-di:x(O,
X) > 0
(Spence-Mirrlees
condition)
(4)
Then condition (3) is fulfilled if and only $ X(.) is non-decreasing.
Consequently, our theorem appears as a generalization of the well-known
result of Spence (1974) and Mirrlees (1976): under the assumptions of
Proposition 1, an action profile is rationalizable if and only if it is nondecreasing.
Proof
We are first going to prove that X(.) is non-decreasing iff condition (3) is
fulfilled for all cycles of order 2. Let
We have to prove that d(f$,,O,) is non-positive
forward computation we obtain
Ate,,
0,)
= _ y
for all t?e, Or. By a straight-
‘7’)-L?_?!_
xco,)
aeax
(‘yt,dtds.
By condition (4) this implies
In order to complete the proof, we have to show that if condition (3) is
fulfilled for cycles of order 2, then it is also fulfilled for cycles of arbitrary
order N. We prove it by induction on N. For N =2 there is nothing to
prove. Let us assume the property true for N and prove it for N + 1. Let
=8,, be an arbitrary cycle of order N + 1. We have to prove
eO,e,,...,e,+,
that
, we,)) -
k=O
to,,we,))) 5 0.
This sum is of course independent of the starting point of the summation
so that we can ask with no loss of generality that
eN=
SUP
06kSN
ek.
(5)
J.-C. Rochet, A condition for rationalizability
in a quasi-linear
context
195
Let us define a cycle of order (N- 1) by setting
ksN-1,
8;=&
u;,=e,.
By assumption, we have
N-l
A’= 1 {UC%+,,
We;)) - u(e;,X(W) 5 0
k=O
so that
+
{“(eO~
x(eN))
-“(eN~x(eN))}
or after simplification
but by (5): 13~5 ONand 8, _ 15 ONand by monotonicity
so that ASO.
0
of X( .): X(0, _ r ) 5 X(0,)
There are interesting similarities between our rationalizability problem and
revealed preference theory. To clarify this point, let us give two formal
definitions:
Definition 1. A set of data D= {(Bi,Xi) i= 1,. . . , K) is called rationalizable if
and only if there exists: X:52+% which is implementable via compensatory
transfers and such that
X(Oi)=xi,
i=l ,..., K.
Definition 2. Given a set of data D = {(I$, xi), i= 1,. . . , K} we define a binary
relation on {x1,. . . , xK} as follows:
Xj RMV
Xi 0
U(8i,
Xj)
>
U(8i,
Xi),
where RMV means ‘is revealed more valuable than’.
(6)
196
J.-C. Rochet, A condition for rationalizability
in a quasi-linear context
By the ‘Taxation Principle’ (cf. $2) a set of data D is rationalizable
only if there exists a price schedule 4 such that
Vi=1 , . . . , K,
max {u(Oi,x) - 4(x)> is attained for X= xi.
x
if and
(7)
Suppose now that xi RMV Xj and use (7):
which implies
4txj)
>
4(xi)+
(8)
Thus a necessary condition for D to be rationalizable is that RMV admits
no cycles [condition (3)]. Condition (3) is thus the analogue of the Strong
Axiom of Revealed Preferences, and our theorem is the analogue of Afriat’s
result [Afriat (1965)], which shows how to compute, for any set of data
satisfying SARP, a utility function which rationalizes the data. In the one
dimensional context, one can restrict oneself to cycles of order 2: condition
(3) for 2-cycles is the analogue of the Weak Axiom of Revealed Preferences
(WARP).
4. The linear case
Proposition 2. Let 52 be a convex subset of Rk and u be linear w.r.t. 9 and %T1
w.r.t. x. Then an action profile X( .) is rationalizable if and only if there exists
a convex function V:s2-+R such that
where V(e) is the [email protected] of Vat B [cf. Rockafellar (1972, p. 215)].
If u is linear w.r.t. 0 condition (9) is equivalent to Rockafellar’s cyclical
monotonicity
condition
[Rockafellar
(p. 238)] for the mapping f3+
(&/%)(O, X(0)). Because of Proposition 2, Theorem 1 appears as a generalization of Rockafellar’s Theorem 24.8 (p. 238) which states that a mapping
is (the selection of) the subgradient of a convex function if and only if it is
cyclically monotone.
J.-C. Rochet, A condition for rationalizability in a quasi-linear context
Proof.
Let X( .) be rationalizable
function. We have
and V( .) the corresponding
v4 = ;;; {u(@.
X(W)- W))
197
indirect utility
(10)
which implies the convexity of V as a supremum of linear functions. (10) can
also be written
or by linearity of u w.r.t. 8:
v e,8’
v(e) 2 V(P) +
$ (ef, x(e,))(e- e,)
which is exactly what is asserted by (9). Conversely if (9) is fulfilled for all 0,
we can set
t(e) = u(e, x(e)) - v(e)
and we have by definition of aV(@‘)
ve,e’
v(e)?
v(v)+
$e:x(e.))(e-e.)
which implies (10) by linearity of u w.r.t. 8.
We also have another characterization
0
when X( .) is V’:
Proposition 3. Let 52 be a convex subset of Rk, u be linear w.r.t. tI and 9S2
w.r.t. x. Then a @ mechanism X(.) is rationalizable if
(11)
4b we,)) + ~(4, x(e,)) 2 48,, we,)) + 401, x(ed
(12)
V(&,WEQ2
Proof:
that
By convexity of s2, (11) implies the existence of a V ~%7~(52,
R) such
198
J.-C. Rochet, A condition for rationalizability in a quasi-linear context
Vf3Ef-2
~(e,x(e))=pI/(e).
Now by linearity of u, (12) can be restated as
Thus VV(e) is monotonic
and V is convex.
Conversely if X(.) is rationalizable,
convex function V such that
t/e4
0
Proposition
2 implies the existence of a
$e,x(ej)tav(s),
X(e) being continuous by assumption we have in fact
vue) =
$8,x(e)),
X being 59 and u being %Z2 this implies (11). Now (12) is simply a
restatement of the monotonicity of W(O). 0
Condition (ll), which is thus necessary for rationalizability in the linear
case, can be formulated as a (system of) partial differential equation(s). They
are the analogue of Slutsky equations for Revealed Preference Theory. The
following example is taken from Laffont, Maskin and Rochet (1985):
a={e=(a,b;E[o,i-p)
and
M=l,
u(a, b, x) = ax - +(!J+ 1)x2,
rot
1
z(e,x(e)) =$[x(a,b)l-$C-tx2(a,b)l
[a,
which gives the partial differential equation
J.-C. Rochet, A condition for rationalizability in a quasi-linear context
199
In fact a slight weakening of condition (11) can be proved to be necessary
for implementability under very general assumptions:
Proposition 3.
that
Let ~2 he a convex subset of R, u be g2 w.r.t. (0,X)
and such
(13)
(in the sense of positive semi-definite matrices).
rationalizable it is necessary that
Then for
X:Q+X
to be
is a.e. differentiable, and
1rot(&(l3,X(O)))=O
a.e. on !2
N.B. When u is not linear w.r.t. 8, it is very difficult to get sufficient
conditions for rationalizability.
Proof: The matrix A which appears in condition (13) can be taken
symmetric without loss of generality. Let I/ be the indirect utility function
corresponding to the implementable mechanism (X, t):
V(O) = 46 X(O)) - t(O),
By assumption we have for all 0
V(O)= s;,p {a(& X(6) - t(e)}
or
w(e)=v(e)-~(Ae,e)=s~~{u(e,x(~))-)(Ae,e)-t(e’)}.
(14)
By assumption (13) the mapping in the right-hand side is convex w.r.t. 19.
Thus the left-hand side is also convex as a supremum of convex functions.
Relation (14) can also be written
- +(A& 0) + +([email protected]‘, 6’).
(15)
J.-C. Rochet, A condition for rationalizability in a quasi-linear context
200
For all X the mapping ,9+u(O,X)-$(&I,
we have
2
(
[email protected],X(O’))-A&B-O’
13) is convex, thus for all (O,O’)
>
.
Together with relation (15) this implies
Consequently
A convex function on a finite dimensional convex set being a.e. twice
differentiable the mapping &+(&/~O)(O,X(~)) is a.e. differentiable and
moreover, for a.e. 8 in 52:
rot
(
$e,x(e))-Ae
)
=rot
*(e
be
x(e)
’
=0.
)
0