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

© Copyright 2022 Paperzz