Envelope Theorems for Arbitrary Parametrized Choice Sets

Transcription

1 Envelope Theorems for Arbitrary Parametrized Choice Sets Antoine LOEPER 1 and Paul Milgrom January 2009 (PRELIMINARY) 1 Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University, a-loeper@kellogg.northwestern.edu

2 Abstract Existing envelope theorems apply to fixed choice sets or to convex maximization problems (Milgrom and Segal 2002). We derive envelope theorems for parametrized choice sets without imposing any convex or topological structure on the choice sets. We show that the traditional envelope theorem formula hold at any point where the generalized Lagrange multipliers and the constraint are sufficiently smooth in the parameter. We provide conditions under which the value function is differentiable or absolutely continuous. We apply these theorems to mechanism design problems. JEL Classification Numbers: C60, D86 Keywords: Envelope theorems, constrained maximization, differentiable value function, Lagrange multiplier

3 1 Introduction In Milgrom and Segal 2002, it was shown that to obtain the usual envelope theorems, no structure had to be imposed on the choice set nor continuity of the maximand with respect to the choice variable. More precisely, they proved the differentiability of the parametrized program V (t) =supf (t, x), (1) x X without imposing conditions on the structure of X nor on the differentiability of x f (t, x). The basic intuition beeing that the sup of a function is independent of the labelling of the choice set. With parametrized choice sets, the problem becomes V (s) = sup f (x). (2) x X(s) where X (s) is usually defined as for some function g. X (s) ={x : g (s, x) 0} The main purpose of this paper is too weaken considerably the conditions imposed by the existing envelope theorems for the optimization programs of type (2). Contrary to the program with fixed choice set, one cannot completely dispense with conditions on the structure of the choice set or on the regularity of f (x). The reason is that variations in the choice set X (s) are formally equivalent to an onto mapping Ψ (s,.) : X (s o ) X (s) with which the program (2) can be re-written as V (s) = sup f (Ψ (s, x)). x X(s o) 1

4 This formulation equivalent to the program in (1) with the important difference that the maximand depends on s indirectly through the choice variable. Nevertheless, one of the main insight of this paper is that a sufficient condition, on top of the necessary regularity of g as a function of s is that the following program W (s, c) = sup f (x), g(s,x)+c 0 must be sufficienctly smooth in c. In particular, we do not necessarily need to impose a metric or topological structure on X. Of course, such a structure may help to establish the regularity of W is sufficiently smooth in c but it is not needed if the underlying economic problem guarantees the regularity of W, as in the case in some applications as we shall argue. Moreover, the differentiability of W with respect to c is essentially indispensible for our purpose because envelope theorems basically assert that whenever it exists, V s g is the product of and W. In other word, W plays the role of the s c c Lagrangean multiplier for convex programing. Section 2 lays out the notations and introduce the generalized lagrange multiplier. Section 3 states general results on the differentiability and absolute continuity of V and section 4 derives some applications. 2 Notations 2.1 The Maximization Problem Let S denotes the set of admissible parameters for the problem. We assume S is an open bounded interval of R. X is the set of admissible maximizer 2

5 of the maximand f. We are interested in the regularity of the parametrized program: V (s) = sup x X(s) f (x). (3) To be able to make statements about the differentiability of the correspondence X (s), we will consider maximization programs in which X (s) is defined as the lower contour sets of a function g: X (s) ={x X; g (s, x) 0}. (4) Whenever they exist, we denote the solutions of the above program as follows: and X = s S X (s). X (s) ={x X (s) :f (x) =V (s)}, 2.2 Differentiability: definitions For any map Φ : T Y where T R and Y is some arbitrary space, Φ t (t, y) will denote its partial derivative w.r.t. t at (t, y),andφ t (t+,y) (resp. Φ i (t,y)) denotes its right-hand (resp. left-hand) directional derivative. On top of the standard notions of differentiability and absolute continuity (see e.g. Royden 1988), we shall use the following conditions: Definition 1 A family of function (Φ (., y)) y Y ³ Φi (t,y) Φ i (t o,y) t t o y Y is equidifferentiable at to if converges uniformly as t t o.itisequidifferentiable on T if it is equidifferentiable for all t T. As argued in Milgrom and Segal 2002, equidifferentiability is implied by the equicontinuity of {Φ s (., y)} y Y. When the set Y indexing the family of 3

6 functions has a topological structure, the following notion can be viewed as a pointwise version of equidifferentiability. Definition 2 If Y is a topological space, a map Φ : T Y R has converging variations in t at (t o,y) (resp. (t o,y), resp. (t o +,y)) ifforany sequences t n t (resp. t n % t, resp. t n & t) andy n y, Φ(t n,y n ) Φ(t o,y n ) t n t o converges. Clearly, converging variations at (t o,y) implies Φ t (t o,y) exists, and with the notations of the definition, Φ(tn,yn) Φ(toyn) t n t o converges necessarily to Φ t (t o,y). Converging variations at (t o,y) holds whenever Φ t is continuous in both argument at (t o,y). Indeed, by the Mean Value Theorem, Φ(t n,y n ) Φ(t o y n ) t n t o = Φ t (s n,y n ) for some s n [t o,t n ]. As shown in Milgrom and Segal 2002, the following strengthening of absolute continuity is the key condition to get the absolute continuity of a parametrized maximization program. Definition 3 A family of function (Φ (., y)) y Y is uniformly absolutely continuous if for all y Y, Φ (., y) is absolutely continuous and there exists ameasurablefunctionb : T R such that for all y and almost all t, Φ t (t, y) b (t). 2.3 Generalized Langrange Multipliers The traditional envelope theorems factor in the variations of the choice set X (s) in the value function via the Lagrange multipliers. The later are defined as the saddle points of the Lagrangean of the program. However, when the 4

7 sets X (s) are not convex and the function f is not quasi-concave, the minmax theorems do not apply and the Lagrange multipliers may not be well defined. To get around this problem we define the following more general program: W (s, c) = sup g(s,x)+c 0 f (x). (5) By definition, V (s) =W (s, 0) and W is non increasing in c. As such, it is differentiable in c almost everywhere. When the maximization problem is convex and the Lagrangean has a unique saddle point, W c (s, 0) exists and is exactly equal to the Lagrangean multiplier at s. In the case of a convex problem whose Lagrangean admits (x, λ) as a saddle point, existing envelope theorems state that V 0 (s) = λg s (s, x) or equivalently, V 0 (s) =W c (s, 0) g s (s, x). Hence, the existence of the derivatives W c and g s are clearly necessary for the envelope theorem to hold. Our results will show that the differentiability of W and g with respect to c and s respectively (together with some continuity condition) are almost sufficient. In particular, no convexity or norm structure need to be imposed on the set X. We shall furthermore provide some conditions under which V is absolutely continuous, i.e. that the envelope theorem holds under its integral form. 5

8 3 General Results 3.1 Directional Derivatives Lemma 1 Let s, s 0 S, x X (s) and x 0 X (s 0 ) then V (s 0 ) V (s) W (s, g (s, x 0 )) W (s, g (s 0,x 0 )), (6) and V (s 0 ) V (s) W (s 0, g (s, x)) W (s 0, g (s 0,x)). (7) Proof. By definition of V and W, V (s 0 ) and since g (s 0,x 0 ) 0, sup f (y) =W (s, g (s, x 0 )), (8) g(s,y) g(s,x 0 ) V (s) sup g(s,y) g(s 0,x 0 ) f (y) =W (s, g (s 0,x 0 )). (9) Substracting (9) to (8), we get (6). By switching the role of (s, x) and (s 0,x 0 ) we get (7). The next two propositions express V s as a function of W c and g s,when these partial derivaives exist, as in the traditional envelope theorems. The first proposition imposes some further regularity on W. Proposition 1 Let x X (s o ), suppose that W (s o,c) has converging variations in c at (s o, 0). Whenever the following derivatives exist, V s (s o ) W c (s o, 0) g s (s o,x), V s (s o +) W c (s o, 0) g s (s o +,x). If V is differentiable and g s (s o,x)=g s (s o +,x), then the above inequalities hold with equality. 6

9 Proof. Suppose first g (s, x) =0.Lets n & s. From(7), V (s n ) V (s) s n s W (s n, g (s n,x)) W (s n,g(s, x)). (10) s n s Taking the limit, we get the first inequality. The second inequality obtains by considering s n % s. Suppose now g (s, x) < 0. This implies that W is constant for c [0, g (s, x)], sow c (s, 0) = 0. Since g (s, x) is continuous in s at (s, x), x X (s 0 ) for s 0 sufficiently close to s. Hence, whenever they exist, V s (s ) 0 and V s (s+) 0, so the inequalities of the proposition still hold. The second proposition imposes regularity conditions only on the primitive g, but requires X to have a topological structure. Proposition 2 If there exists s n % s o, x n X (s n ) such that x n x X (s o ) and g has converging variations in s at (s o,x), wheneverthefollowing derivatives exist, V s (s o ) W c (s o, 0) g s (s o,x). If there exists s n & s o, x n X (s n ) such that x n x X (s o ) and g has converging variations in s at (s o +,x), whenever the following derivatives exist V s (s o +) W c (s o, 0) g s (s o +,x). If V is differentiable and the two limits above are the same, the above inequalities hold with equality. Proof. Consider the case s n & s. Without loss of generality, suppose g (s, x n ) l R {± }. g (s n,x n ) l, sol 0. Since g has converging variations at (s, x), 7

10 Suppose first l =0.From(6), V (s n ) V (s) s n s W (s, g (s n,x n )) W (s, g (s, x n )) g (s n,x n ) g (s, x n ) g (s n,x n ) g (s, x n ). s n s Since W c exists and g has converging variations, the right-hand side tends to W c (s, 0) g s (s, x). 1 Suppose l<0. IfV (s n ) is not constant for n N for some N N, necessarily there exists two subsequences ρ, σ N N such that g s ρ(n),x σ(n) > 0. Since g has converging variations, g s ρ(n),x σ(n) l 0, a contradiction. So V (s n ) is constant for n N and V s (s ) =0. Since g (s, x n ) l, this means that W (s, c) is constant on [0, l[, sow c (s, 0) = 0 and the equation of the proposition holds. Observe that the existence of a selection x n of X (s n ) which converge in X (s) is guaranteed whenever the condition of Berge maximum theorem are met, i.e. f upper semicontinuous and X (s) continuous and compact valued for some topology on X. Alternatively, we shall show in the next section that this condition is satisfied almost everywhere for all selection of X (t) whenever X is a second countable topological space. Thenextpropositionusesjointconditionsong and W to establish the directional differentiability of V. It does not impose any structure on X nor on the behavior of X (s) as s s o. Proposition 3 Suppose X (s) 6= for all s S, g (s o,x) and g s (s 0,x) are bounded on X. If W has converging variations in c at (s 0, 0) and 1 Note that if g (s n,x n )=g(s, x n ) for all n N for some N, thensinceg has converging variations at (s, x), g s (s, x) =0. Moreover g (s, x n ) 0 so x n X (s) for all n N, which implies V s (s+) 0 and the inequality of the proposition still holds. 8

11 (g (., x)) x X is equidifferentiable, then V is left-hand and right-hand differentiable at s o and for any such selection, V s (s 0 ) = lim W c (s 0, 0) g s (s 0,x(s)), s%s0 V s (s 0 +) = lim W c (s 0, 0) g s (s 0,x(s)). s&so Proof. Let s 0 n & s, s 00 n & s, x 0 n X (s 0 n) and x 00 n X (s 00 n). Since g (s 0,x 0 n), g (s 0,x 00 n), g s (s 0,x 0 n) and g s (s 0,x 00 n) arebounded,wecanrestrict attention to converging subsequences. We denote their limit g 0, g 00, g 0 s and g 00 s respectively. Without loss of generality, we can either assume that s 0 n <s 00 n for all n or the opposite. Let us consider the former case for concreteness. Suppose first that g (s 00 n,x 00 n) 6= g (s 0 n,x 00 n) for n sufficiently large. By equidifferentiability of g, g (s 0 n,x 00 n) and g (s 00 n,x 00 n) both converge to g 00.Letusfirst consider the case g 00 =0.SinceW has converging variations at (s o, 0), W (s 0 n, g (s 0 n,x 00 n)) W (s 0 n, g (s 00 n,x 00 n)) g (s 00 n,x 00 n) g (s 0 n,x 00 n) W c (s, 0). (11) Since g is equidifferentiable, g (s 00 n,x 00 n) g (s 0 n,x 00 n) s 00 n s 0 n = g s (s o,x 00 n)+o ( s 00 n s o + s 0 n s o ) g 00 s. (12) Combining (11) and (12) with (6), lim sup V (s00 n) V (s 0 n) s 00 n s 0 n W c (s, 0) g 00 s. (13) Suppose now that g 00 < 0. As argued in the proof of proposition 2, necessarily, lim V (s00 n) V (s 0 n) s 00 n s0 n =0and W c (s, 0) = 0, so (13) still holds. Finally, if g (s 00 n,x 00 n)=g (s 0 n,x 00 n) for all n sufficiently large, then x 00 n X (s 0 n) so V (s 0 n) V (s 00 n) and lim sup V (s00 n) V (s 0 n) 0. Moreover, by equidifferentiability of g, gs 00 =0, so (13) still s 00 n s 0 n holds. 9

12 Withthesamereasoningasabove,(7)impliesthatifg (s 0 n,x 0 n) 6= g (s 00 n,x 0 n) for n sufficiently large, V (s 00 n) V (s 0 n) s 00 n s 0 n If g 0 =0, taking the limit we get W (s00 n, g (s 0 n,x 0 n)) W (s 00 n, g (s 00 n,x 0 n)) g (s 00 n,x 0 n) g (s 0 n,x 0 n) g (s 00 n,x 0 n) g (s 0 n,x 0 n). s 00 n s 0 n lim inf V (s00 n) V (s 0 n) W s 00 n s 0 c (s, 0) gs. 0 (14) n If g 0 < 0, forthesamereasonasabove,lim V (s00 n) V (s 0 n) =0and W s 00 n s 0 c (s, 0) = 0, n so (14) still holds. Finally, if g (s 0 n,x 0 n)=g (s 00 n,x 0 n) for all n sufficiently large, then x 0 n X (s 00 n) so V (s 00 n) V (s 0 n). Moreover, by equidifferentiability of g, gs 0 =0so (14) still holds. To conclude the proof, observe that by inverting the role of s 0 n and s 00 n,we getfrom(13)and(14)that W c (s, 0) g 00 s = W c (s, 0) g 0 s. 3.2 Absolute Continuity The following propositions give conditions under which V (s) is absolutely continuous. Since absolute continuity implies differentiability almost everywhere, combining any of these propositions with the results of the previous sections give conditions under which for all s, s 0 S, and for any selection x (s) of X (s) satisfying the corresponding conditions, we have Z s 0 V (s 0 ) V (s) = W c (s, 0) g s (s, x (s)). s 10

13 Lemma 2 If there exists (h (., x)) x X λ 0 such that for all s S, uniformly absolutely continuous and then V is absolutely continuous. V (s) =supf (x) λh (s, x), x X Proof. Theorem 2 in Milgrom and Segal Notice that the h function used in lemma 2 need not be the g function of the original formulation of the maximization problem as it is the case for the Lagrangean of a convex program. The next proposition gives a condition under which h (s, x) =max(0,g(s, x)) works. h i Proposition 4 If sup W (s,c) W (s,0) s is bounded as c 0 and (g (., x)) c x X is uniformly absolutely continuous, then V is absolutely continuous. Proof. Let h (s, x) =max(0,g(s, x)). Since0 and g (s, x) are absolutely continuous for all x, soish (s, x). The integral bound condition clearly holds, so h is uniformly absolutely continuous. For all λ 0, let us denote Ω λ (s, c) = sup f (x) λh (s, x). x X:g(s,x)+c 0 Suppose firstthatthereexistsλ 0,c < 0 such that for all s, Ω λ (s, c) = V (c). By uniform absolute continuity, for all s S, thereexistsε>0 such that s s 0 ε implies sup g (s, x) g (s 0,x) c, which shows that σ [s ε,s+ε] X (σ) σ [s ε,s+ε] {x X : g (σ, x)+c 0}. 11

14 So lemma 2 for X = σ [s ε,s+ε] X (σ) implies that V is absolutely continuous on ]s ε, s + ε[. By compactness of S, one can extract a finite subcover which establishes absolute continuity. Suppose now that for all λ 0,c < 0, thereexistss S such that Ω λ (s, c) > V (c). Then there exists two sequence s n and x n such that g (s, x n ) & 0 and f (x n ) ng (s n,x n ) >V(s n ).Therefore,foralln W (s n, g (s n,x n )) W (s n, 0) > ng (s n,x n ), which is impossible under our assumptions. Observe that the first condition on W is satisfied in particular if W is Lipshtiz at any (s, 0). 4 Applications 4.1 Second Countable Spaces In this section, we show how standard topological assumptions on X can strengthen our results. For future references, a topological space is second countable if it has a countable base. In particular, any separable metric space, and thus any Euclidean space, is second countable. It is Lindelof if every open cover has a countable cover. It is implied by second countability. Lemma 3 Suppose T is Lindelof and T 0 is secound countable, then a set of isolated points of T T 0 (for the product topology) is at most countable. Proof. Let I = {(x i,x 0 i):i I} be a set of isolated points in T T 0.Let (B n ) n N be a countable base of T 0. Under our assumptions, for all i I there 12

15 exists an open set O i of T and n (i) N such that O i B n(i) I = {(x i,x 0 i)}. Let I (n) ={i I : n (i) =n}, by construction, the points {x i : i I (n)} are isolated in T. Since T is Lindelof, I (n) must be countable. Since I (n) is countable for all n, soisi. Proposition 5 Suppose X is second countable, W c (s, 0) exists for almost all s, andx (s) is a selection of X (s) such that g has converging variations in s at almost all (s, x (s)), then for almost all points of differentiability of V, V s (s) =W c (s, 0) g s (s, x). Proof. S endowed with the lower limit topology is Lindelof. From the previous lemma, the set of isolated point of the graph G = {(s, x (s)) : s S} of a selection x (.) of X (.) has at most a countable number of isolated points. This means that for almost all s, thereexistss n % s and x (s n ) x (s). A similar argument for the upper limit topology shows that for almost all s, thereexistss n & s and x (s n ) x (s). SinceV is differentiable almost everywhere, proposition 2 follows. 4.2 Continuous Functions on Compact Choice Sets In this section, X is a compact set. Proposition 6 Suppose f is upper semicontinuous, g s is continuous on S X, andw c is continuous on [, + ] S, thenv is absolutely continuous, and V 0 (s+) = W c (s, 0) max x X s (s, x), (s) V 0 (s ) = W c (s, 0) min x X s (s, x), (s) 13

16 and V is differentiable if and only if {g s (s, x) :x X (s)} is a singleton. Proof. To be finished Mechanism Design The first application of our theorems concern the integral representation of the agents utility in classical principal agent problems where the type of the agent enters in his maximization program through a constraint. This step is key in the usual optimal mechanism deisgn approach pioneered by Mirless (1971). A less obvious application concerns the characterization of pairwise strategyproof mechanism on connected type spaces. Let N denote the set of participant to the mechanism, X the set of possible alternatives, {U i (θ i,.): i N,θ i Θ i } the set of admissible utility functions of the agents. A mechanism Φ maps profile of type θ into an outcome Φ (θ) for all admissible profiles of type. The range of the mechanism is denoted R Φ. Definition 4 AmechanismΦ is strategy-proof if for all i N, θ i,θ 0 i Θ i, and θ i Θ i, U i (θ i, Φ (θ i,θ i )) U i (θ i, Φ (θ 0 i,θ i )). It is pairwise strategy-proof if for all i, j N, θ i,θ 0 i Θ i, θ j,θ 0 j Θ j and θ i,j Θ i,j, either U i (θ i, Φ (θ i,θ j,θ i,j )) U i θi, Φ θ 0 i,θ 0 j,θ i,j, or U j (θ j, Φ (θ i,θ j,θ i,j )) U j θi, Φ θ 0 i,θ 0 j,θ i,j. 14

17 Proposition 7 Suppose for all i N,θ i Θ i, U i θ i is continuous in (θ i,x) and the Pareto frontier U i (U i ) (on R Φ )islocallylipshitzineachu j for all j 6= i whenever U j < sup U j (R Φ ), then for all i, j N, U i (θ i, Φ (θ)) is absolutely continuous in θ j and at any point θ of differentiability, either U i θ j (θ) =0or U j (θ) is maximized on R Φ. Proof. Consider the pair i =1,j =2. Since Φ (θ) is pairwise strategyproof for {1, 2}, it solves the following program: P (θ) = max x R Φ : U 2,θ2 (x) U 2,θ2 (Φ(θ)) 0 U 1,θ 1 (x), otherwise there could be a jointly profitable deviation. We fix θ 2.Byconstruction, there exists a solution for all θ 2. The constraint of the program has a derivative with respect to θ 2 which is continuous in (θ 2,x). Sincewehave assumed that U 1 (U 1 ) (on R Φ ) is Lipshitz in U 2 whenever U 2 < sup U 2 (R Φ ), from theorem 4, P (θ) is absolutely continuous. is given by the traditional envelope formula. From 5, whenever P(θ) θ 2 The term g s (s, x) is given by θ 2 [U 2,θ2 (x) U 2,θ2 (Φ (θ))], which is zero at the truthful equilibrium by virtue of the unconstrained envelope theorem. In words, this proposition shows that for a mechanism to be pairwise strategy-proof, it must be the case that at almost all points, either i value function is locally independent of the type of j or j is a local dictator. References [1] Milgrom, P. and I. Segal, 2002, Envelope Theorems for Arbitrary Choice Sets, Econometrica, Vol. 20, No. 2, pp

18 [2] Mirrlees, James A, An Exploration in the Theory of Optimum Income Taxation, Review of Economic Studies 38, [3] Royden, H. L. (1988): Real Analysis, Third Edition. Englewood Cliffs: Prentice-Hall 16