A nonsmooth approach to envelope theorems

Journal of Mathematical Economics 61 (2015) 157–165

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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

A nonsmooth approach to envelope theorems✩ Olivier Morand a , Kevin Reffett b , Suchismita Tarafdar c,∗ a

Department of Economics, University of Connecticut, United States

b

Department of Economics, WP Carey School of Business, Arizona State University, United States

c

Department of Economics, Shiv Nadar University, India

article

info

Article history: Received 2 April 2012 Received in revised form 31 August 2015 Accepted 2 September 2015 Available online 11 September 2015 Keywords: Constrained otimization with nonconvexities Envelope theorems Nonsmooth analysis Stochastic growth Lattice programming

abstract We develop a nonsmooth approach to envelope theorems applicable to a broad class of parameterized constrained nonlinear optimization problems that arise typically in economic applications with nonconvexities and/or nonsmooth objectives. Our methods emphasize the role of the Strict Mangasarian–Fromovitz Constraint Qualification (SMFCQ), and include envelope theorems for both the convex and nonconvex case, allow for noninterior solutions as well as equality and inequality constraints. We give new sufficient conditions for the value function to be directionally differentiable, as well as continuously differentiable. We apply our results to stochastic growth models with Markov shocks and constrained lattice programming problems. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Since the work of Viner (1931) and Samuelson (1947), the envelope theorem has become a standard tool in economic analysis. In its ‘‘classical’’ form an envelope theorem is simply a continuous derivative of the value function in a parameter. Sufficient conditions for its existence at first required a great deal of mathematical structure, including convexity, interiority, as well as the continuous differentiability (‘‘smoothness’’) of objectives and constraints (e.g., Samuelson, 1947, Rockafellar, 1970, Mirman and Zilcha, 1975, Benveniste and Scheinkman, 1979). In subsequent work, some of these assumptions were loosened. For instance, versions of Danskin’s Theorem in Clarke (1975) and Milgrom and Segal (2002) relax conditions on the structure of the choice set in unconstrained problems, with Clausen and Strub

✩ We thank Rabah Amir, Bob Becker, Madhav Chandrasekher, Andrew Clausen, Bernard Cornet, Manjira Datta, Amanda Friedenberg, Karl Hinderer, Felix Kubler, Rida Laraki, Cuong LeVan, Len Mirman, Andrzej Nowak, Ed Prescott, Manuel Santos, Carlo Strub, Yiannis Vailakis, and Lukasz Woźny, and especially Atsushi Kajii, and two referees for very helpful discussions and comments during the writing of this paper. Kevin Reffett thanks the Centre d’Economie de la Sorbonne (CES) and the Paris School of Economics for arranging his visits during the Spring terms of both 2011 and 2012. The usual cavaets apply. ∗ Corresponding author. E-mail address: [email protected] (S. Tarafdar).

http://dx.doi.org/10.1016/j.jmateco.2015.09.001 0304-4068/© 2015 Elsevier B.V. All rights reserved.

(2013) further extending these results to problems with interior solutions and integer decisions in very general dynamic settings. Recently, and more along the lines of this paper, RinconZapatero and Santos (2009) have extended the classical C 1 envelope theorem to infinite horizon stochastic dynamic programs with inequality constraints in the presence of noninterior solutions. These findings (as well as Milgrom and Segal’s results for the cases with inequality constraints), however, only concern convex programs and it is not clear how they can be extended to economic models with nonconvexities and/or non-differentiable objectives. At the same time, the optimization literature has made a lot of progress on stability bounds for nonconvex non-smooth programs (see, for instance, the monographs of Clarke, 1983 and Bonnans and Shapiro, 2000), although the focus has generally not been on simple and practical sufficient conditions for exact directional derivatives. In this paper we combine recent results from the optimization literature with sets of conditions easily verifiable in finite dimensional problems sufficient for the existence of generalized envelope theorems applicable to many economic models with nonconvexities or non-smooth objectives. When seeking envelope theorems for such programs, several important issues arise. First, since classical envelope theorems cannot be expected, one would like to propose an alternative notion of a ‘‘generalized’’ envelope that fits most applications and is a ‘‘substitute’’ for the classical envelope. Second, the proposed approach must work in settings with

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both equality and inequality constraints, and when optimal solutions are not necessarily interior. Third, when Slater’s condition is not appropriate, new constraint qualifications allowing for simple (and, if possible, exact) calculations of a generalized envelope or for the existence of differential bounds for the value function (the latter is often all that is needed in applications) need to be identified. Methodologically, we take a ‘‘nonsmooth’’ approach closely related to the work of Gauvin and Tolle (1977), Gauvin and Dubeau (1982) and Auslender (1979), and consider Lipschitz programs in finite dimensional spaces, in which objective functions are only assumed to be locally Lipschitz, thus not necessarily differentiable, and the constraints are continuously differentiable.1 We show how progressively stronger conditions on primitive data lead to progressively sharper characterizations of the differentiable properties of the value function. Specifically, we give conditions under which value functions admit differential bounds, are Clarke differentiable, directionality differentiable, and continuously differentiable. Our sharpest results focus on constraint systems satisfying the Strict Mangasarian–Fromovitz Constraint Qualification, a refinement of the Mangasarian–Fromovitz Constraint Qualification equivalent to the uniqueness of the Karush–Kuhn–Tucker multiplier in our setup. The remainder of the paper is laid out as follows. In Section 2, we describe the benchmark class of optimization programs we consider. In Section 3 we present our main results. In particular, we provide results on differential bounds for the value function, directional differentiability, and C 1 differentiability. Applications of some of these results make up the last section of the paper, and the Appendix briefly exposes some mathematical tools of nonsmooth analysis and some lattice programming notions. 2. Lipschitz programs Given a space A of actions (or controls), a parameter space S, and an objective function f : A × S → R, we consider the following parameterized Lipschitz program: V (s) = max f (a, s)

(2.0.1)

a∈D(s)

where the feasible correspondence is given by: D(s) = {a|g i (a, s) ≤ 0, i = 1, . . . , p, hj (a, s) = 0, j = 1, . . . , q} and the optimal solution correspondence is defined as: A∗ (s) = arg max f (a, s).

characterizations of simple nonsmooth envelope theorems for program (2.0.1) can be read from linearizations of the Lagrangian dual at its optimum. Letting the vector of dual variables be denoted by (λ, µ) ∈ p R+ × Rq , we conjugate program (2.0.1) with a classical Lagrangian duality scheme as follows2 : L(a, λ, µ; s)

 f (a, s) − λg (a, s) − µh(a, s) = −∞, ∞,

p

if a ∈ A, (λ, µ) ∈ R+ × Rq p if a ̸∈ A, (λ, µ) ∈ R+ × Rq otherwise

where: g (a, s) = [g 1 (a, s), . . . , g p (a, s)]; h(a, s) = [h1 (a, s), . . . , hq (a, s)]. A point a ∈ D(s) is a Karush–Kuhn–Tucker (KKT) point if there exists (λ, µ) ∈ Rp+ × Rq such that: 0 ∈ ∂a f (a, s) − (λ∇a g + µ∇a h) (a, s) and:

λg (a, s) = 0. We denote by K (a, s) the set of ‘‘KKT multipliers’’ (λ, µ) associated with the KKT point a. In constrained optimization, constraint qualifications are needed to guarantee that optimal solutions are KKT points, and that the feasible region around such points does not vanish under local perturbations of the parameters. The strongest of these constraints is the Linear Independence Constraint Qualification (LICQ), central to the work of Gauvin and Dubeau (1982) and the focus of RinconZapatero and Santos (2009), for example. Definition 1. A feasible point a ∈ D(s) satisfies the LICQ if the following vectors are linearly independent,

∇a g i (a, s), i ∈ I ,

∇a hj (a, s), j = 1, . . . , q

where I = {i : g i (a, s) = 0}. The LICQ is rather strong and plays no role in our argument. Rather, we focus a weaker constraint, the Mangasarian–Fromovitz Constraint Qualification (MFCQ), which is equivalent to Robinson’s constraint qualification in finite dimensional spaces (and with a finite number of constraints). Definition 2. A feasible point a ∈ D(s) satisfies the MFCQ if:

a∈D(s)

We will maintain some baseline assumptions throughout the paper: n

m

Assumption 1. (a) A and S are open convex subsets of R and R , respectively; (b) the objective function f : A × S → R is locally Lipschitz in ( a, s ) , (c) the constraints g i , i = 1, . . . , p and hj , j = 1, . . . , q are jointly C 1 , and n ≥ q.

(i) the following vectors are linearly independent,

∇a hj (a, s),

j = 1, . . . , q

(ii) there exists y ∈ Rn such that,

∇a g i (a, s)y < 0,

i ∈ I;

∇a hj (a, s)y = 0,

j = 1, . . . , q

where the set of active constraints are denoted by I = {i : g i (a, s) = 0}.

We use a standard Lagrangian duality scheme applied in a nonconvex context. Of course, the cost of relaxing convexity is immediate as we generally lose strong duality and the construction of a necessary and sufficient first order theory for optimal solutions generically becomes impossible. Nevertheless, this does not prevent us from providing mild conditions under which sharp

Because the MFCQ is not sufficient to guarantee the uniqueness of the KKT multipliers, it is very difficult to obtain directional derivatives and sharp characterizations of the generalized gradient. However, the MFCQ is sufficient for asserting the nonemptiness of the set of multipliers (as well as their compactness; e.g., Gauvin and Tolle, 1977, Corollary 2.8), as stated in the following proposition.

1 Problems with nonsmooth constraints as well as ‘‘mixed-integer programming’’ problems are studied in Morand et al. (2013).

2 If A is closed, then the abstract constraint a ∈ A induces an additional term in the Lagrangian (see, for instance, Clarke, 1983, Chapter 6).

O. Morand et al. / Journal of Mathematical Economics 61 (2015) 157–165

159

Proposition 3. Under Assumption 1, if the MFCQ holds at a∗ (s) ∈ p A∗ (s), then there exist λ ∈ R+ and µ ∈ Rq such that:

a∗ (s) ∈ A∗ (s), then V is locally Lipschitz near s and, in any direction x ∈ Rm :

(a) λi g i (a∗ (s), s) = 0

lim inf

(b) 0 ∈ ∂a f (a∗ (s), s) − (λ∇a g + µ∇a h) (a∗ (s), s).

≥

To obtain uniqueness of the KKT multiplier for each optimal solution we impose a slightly stronger condition than the MFCQ (in the sense that it implies the MFCQ). Although technically not exactly a constraint qualification (since it assumes the existence of a multiplier), this refinement of the MFCQ has been identified in the literature as the Strict Mangasarian–Fromovitz Constraint Qualification (SMFCQ) or simply the Strict Constraint Qualification (Bonnans and Shapiro, 2000, Definition 4.46). Definition 4. A feasible point a ∈ D(s) together with a multiplier (λ, µ) ∈ K (a, s) satisfies the SMFCQ if:

i ∈ Ib ;

∇a hj (a, s), j = 1, . . . , q

(ii) there exists y ∈ Rn such that,

∇a g i (a, s)y < 0, ∇a hj (a, s)y = 0,

i ∈ Is ;

∇a g i (a, s)y = 0,

t

sup

i ∈ Ib

j = 1, . . . , q

where the set of binding and saturated constraints is denoted by Ib = {i ∈ I : λi > 0}, Is = {i ∈ I : λi = 0} respectively. The equivalence between uniqueness of the KKT multiplier and the SMFCQ for smooth programs is demonstrated in Proposition 1.1 in Kyparisis (1985, Proposition 1.1), a result we state here without proof (see also Bonnans and Shapiro, 2000, Remark 4.49). Proposition 5. Assume that f is continuously differentiable in a, and that Assumption 1 holds. Then the SMFCQ holds for (a, (λ, µ)) iff K (a, s) is a singleton. 3. Differentiability of the value function This section derives successively stronger characterizations of the envelopes under increasingly stronger sufficient conditions, from Dini bounds for the value function to smooth classical envelopes. 3.1. Stability bounds Our first result pertains to the existence of bounds for the Dini derivatives of the value function in general Lipschitz programs. The most general result is established by Clarke (1983, Theorem 6.5.2 and Corollary 4) and also appears in a similar form in Bonnans and Shapiro (2000, Theorem 4.26), but their hypotheses involve properties of the value function (Clarke, 1983, Hypotheses 6.5.1, p. 241) rather than being stated directly in terms of restrictions on the primitives data. As an alternative, Proposition 6 presents a set of sufficient conditions for Clarke’s result to hold, namely the MFCQ and uniform compactness, that are easily checked in many economic models. Recall that a correspondence D is uniformly compact near s if there exists a neighborhood K of s such that the closure of ∪s′ ∈K D(s′ ) is compact, Gauvin and Dubeau (1982). Proposition 6. Under Assumption 1, if D(s) is nonempty and uniformly compact near s and the MFCQ holds at the optimal solution

o ∗ L− s (a (s), s, λ, µ; x)



inf

∗ a∗ (s)∈A∗ (s) (λ,µ)∈K (a (s),s)

t

t →0+

≤



V (s + tx) − V (s)

lim sup

sup

Los (a∗ (s), s, λ, µ; x)



sup

a∗ (s)∈A∗ (s) (λ,µ)∈K (a∗ (s),s)



where: o ∗ L− s (a (s), s, λ, µ; x)

=

min

ςs ∈∂s f (a∗ (s),s) ∗

   ςs − (λ∇s g + µ∇s h)(a∗ (s), s) · x

Los (a (s), s, λ, µ; x)

=

(i) the following vectors are linearly independent,

∇a g i (a, s),

V (s + tx) − V (s)

t →0+

max

ςs ∈∂s f (a∗ (s),s)

   ςs − (λ∇s g + µ∇s h)(a∗ (s), s) · x .

Proof. As suggested by Rockafellar (1982, p. 29),3 the original program can be re-written as: V (s) = max f (a, a′ ) g (a, a′ ) ≤ 0 h ( a, a′ ) = 0

−a′ + s = 0. To this ‘‘modified program’’ we associate the multipliers (λ, µ, θ ) and the Lagrangian: M ((a, a′ ), λ, µ, θ ; s) = f (a, a′ ) − λg (a, a′ )

− µh(a, a′ ) − θ (−a′ + s) and denote by KM (a∗ (s), s) the set of KKT multipliers associated with a solution (a∗ (s), s) ∈ A∗M (s). Note that, being identical, the ‘‘modified program’’ and the initial program have, for any given s, the same set of solutions (A∗ (s) = A∗M (s)) and of multipliers in the sense that (λ, µ, θ ) ∈ KM (a∗ (s), s) iff (λ, µ) ∈ K (a∗ (s), s). The assumption of uniform compactness of D near s implies that ∪s′ ∈S ′ D(s) is compact for some compact neighborhood S ′ of s. As a result:

  ∃δ > 0, s′ − s < δ H⇒ D(s′ ) is compact since D(s′ ) is closed and included in the compact cl [∪s′ ∈S ′ D(s)], which implies that A∗ (s′ ) is nonempty, and Hypothesis 6.5.1 in Clarke is therefore satisfied (by setting ε0 = δ and Ω = cl [∪s′ ∈S ′ D(s)]). The Lipschitz property of the value function V follows directly from Corollary 1 of Proposition 6.5.2 in Clarke (1983). The stability bound results follow from the application of Corollary 4 in Clarke (1983) to the modified program, as the MFCQ for the original program is equivalent to the MFCQ for the above program, which implies (in Clarke’s terminology) that the set of abnormal multipliers reduces to {0}. Corollary 4 thus implies that: sup

inf

∗ a∗ (s)∈A∗ (s) (λ,µ,θ )∈KM (a (s),s) M

θ ·x

≤ DV+ (s, x) ≤ DV + (s, x)

sup

sup

a∗ (s)∈A∗ (s) (λ,µ,θ )∈KM (a∗ (s),s) M

3 See also Clarke (1983) and Bonnisseau and LeVan (1996).

θ · x.

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O. Morand et al. / Journal of Mathematical Economics 61 (2015) 157–165

However, the Lagrange multiplier rule implies that 0 ∈ ∂a′ M ((a∗ (s), s), λ, µ, θ; s) that is: 0 ∈ ∂s f (a∗ (s), s) − (λ∇s g + µ∇s h)(a∗ (s), s) − θ or, equivalently:

θ ∈ ∂s f (a∗ (s), s) − (λ∇s g + µ∇s h)(a∗ (s), s) and, therefore: inf

(λ,µ,θ )∈KM (a∗ (s),s)

θ ·x =

inf

min

(λ,µ)∈K (a∗ (s),s) ςs ∈∂s f (a∗ (s),s)

[(ςs

  − (λ∇s g + µ∇s h)(a∗ (s), s) · x and: sup

(λ,µ,θ )∈KM (a∗ (s),s)

θ ·x =

sup

max

∗ (λ,µ)∈K (a∗ (s),s) ςs ∈∂s f (a (s),s)

[(ςs

  − (λ∇s g + µ∇s h)(a∗ (s), s) · x .  With these differential bounds in place, we are now ready to state sufficient conditions under which simple ‘‘min–max’’ directional envelopes exist for all class of Lipschitz programming problems.

solution a∗ (s) ∈ A∗ (s), then for any direction x ∈ Rm : V ′ ( s, x) =



max

a∗ (s)∈A∗ (s)

 (∇s f − λ∇s g − µ∇s h) (a∗ (s), s) · x .

This is a very important result. The key condition used in the argument is the SMFCQ, which in our context is equivalent to uniqueness of the KKT multipliers (in finite dimensional settings), which in turn allows us to write simple forms for exact directional derivatives (as opposed to just Dini’s bounds, as in the previous section). It also bears mentioning that Rincon-Zapatero and Santos (2009) emphasize the importance of having conditions under which the KKT points are globally unique in their proof of C 1 differentiability of the value function (e.g., see Rincon-Zapatero and Santos, 2009, Theorem 3.2). Certainly, the LICQ condition they use is one such condition (along with enough smoothness in the primitive data), but the SMFCQ should be considered an important alternative to the LICQ. We give an example highlighting this point in Section 4 of this paper. 3.3. Unconstrained programs

Additional assumptions are needed for the existence of directionality differentiable envelopes. The problem is that under only the MFCQ, neither ∂s f (a∗ (s), s) nor K (a∗ (s), s) are singletons for each optimal solution, which implies that exact directional derivatives are typically not available. In the next result, we assume that the set of multipliers is unique, and continuous differentiability (which is, in finite dimensional spaces, equivalent to strict differentiability) of f in s.

To unify our approach with some existing results we prove the existence of directional envelopes for unconstrained programs without the assumption of a differentiable objective, a result related to Clarke’s version of Danskin’s Theorem (Clarke, 1975, Theorem 2.1). As in our previous results, some kind of compactness is needed (the unconstrained choice domain D(s) = A is not uniformly compact) so we assume the inf-compactness condition used in Bonnans and Shapiro (2000) and defined as follows. In the original program (2.0.1), the objective f satisfies the inf-compactness condition at s if there exist r ∈ R and a compact set Ω ⊂ A such that for every s′ in a neighborhood of s the set {a′ ∈ D(s′ ), f (a′ , s′ ) ≥ r } is a nonempty set contained in Ω .

Theorem 7. Under Assumption 1, if D(s) is nonempty and uniformly compact near s, f is C 1 in s, the MFCQ holds for every optimal solution a∗ (s) ∈ A∗ (s), and the set of multipliers, K (a∗ (s), s) is singleton for every optimal solution a∗ (s) ∈ A∗ (s), then for any direction x ∈ Rm :

Corollary 9. Under Assumption 1, if f is Clarke regular in s and if f satisfies the inf-compactness condition at s, then for every optimal solution a∗ (s) ∈ A∗ (s) and any direction x ∈ Rm , the directional envelope is given by:

V ′ (s, x) =

V ′ (s; x) =

3.2. Directional differentiability

max

a∗ (s)∈A∗ (s)

  (∇s f − λ∇s g − µ∇s h) (a∗ (s), s) · x .

Proof. The objective f is C 1 in s hence:

∇s L(a∗ (s), s, λ, µ; x) = (∇s f − λ∇s g − µ∇s h) (a∗ (s), s). By assumption, K (a∗ (s), s) is a singleton thus by Proposition 6: sup

a∗ (s)∈A∗ (s)

(∇s f − λ∇s g − µ∇s h) (a∗ (s), s) · x

= D+ V (s; x) = D+ V (s; x) = sup (∇s f − λ∇s g − µ∇s h) (a∗ (s), s) · x. a∗ (s)∈A∗ (s)

Noting that the supremum is attained (Gauvin and Dubeau, 1982, Corollary 4.3): D+ V (s; x) = D+ V (s; x) =

max

a∗ (s)∈A∗ (s)

{(∇s f − λ∇s g − µ∇s h)

(a∗ (s), s) · x .  

The next corollary provides a sufficient condition for the set of KKT multipliers to be unique for the existence of directionally differentiable envelopes. Corollary 8. Under Assumption 1, if D(s) is nonempty and uniformly compact near s, f is C 1 in (a, s), and the SMFCQ holds for every optimal

′

max fs a∗ (s)∈A∗ (s)

(a∗ (s), s; x).

Proof. By inf-compactness at s, there exists r such that if s′ belongs to the open neighborhood ε0 B(s) of s, then {a′ ∈ A, f (a′ , s′ ) ≥ r } is nonempty and included in Ω . The function f (., s′ ) must then necessarily attain its maximum in the nonempty set {a′ ∈ A, f (a′ , s′ ) ≥ r } ⊂ Ω , hence A∗ (s′ ) ⊂ Ω and A∗ (s′ ) is nonempty, thus Clarke’s Hypothesis is satisfied letting Λ = Ω . Proposition 6 therefore holds true and: V (s + tx) − V (s)

lim inf

t →0+

= lim inf

t →0+

≥ lim inf

t f (a∗ (s + tx), s + tx) − f (a∗ (s), s) t f (a∗ (s), s + tx) − f (a∗ (s), s) t

t →0+

= fs′ (a∗ (s), s; x) the last equality following from f being directionality differentiable. By Clarke regularity, the upper bound in Proposition 6 satisfies: lim sup

V (s + tx) − V (s) t

t →0+

≤

max

{ς · x} = f o (a∗ (s), s; x) = fs′ (a∗ (s), s; x).

ς ∈∂ fs (a∗ (s),s)

O. Morand et al. / Journal of Mathematical Economics 61 (2015) 157–165

Upper and lower bounds coincide for any direction x ∈ Rm , hence: V ′ (s; x) =

max

a∗ (s)∈A∗ (s)

 fs′ (a∗ (s), s; x) . 



A few remarks on this result. First, if we in addition assume that the family of functions {f (a, .)}a∈A is equidifferentiable at s, then we arrive at a multidimensional version of Theorem 3 of Milgrom and Segal (2002), itself a generalization of the classic smooth envelope theorem (e.g., Mirman and Zilcha, 1975, Lemma 1). Second, the result can be applied to infinite horizon dynamic programs as has been done for related nonsmooth envelopes based upon Clarke’s version of Danskin’s Theorem (e.g., see Askri and LeVan, 1998). Indeed, we give an application to monotone controls in stochastic growth models in the next section of the paper. 3.4. Clarke gradient of the value function The existence of bounds for the rate of growth of the value function clearly implies that the value function is locally Lipschitz, so we give below a characterization of its Clarke gradient with and without the assumption of continuous differentiability. Proposition 10. Under Assumption 1, if D(s) is nonempty and uniformly compact near s and MFCQ holds at every optimal solution a∗ (s) ∈ A∗ (s), the Clarke gradient of the value function satisfies:

 ∂s V (s) ⊂ co ∪a∗ (s)∈A∗ (s) ∪(λ,µ)∈K (a∗ (s),s) where co is the closure of the convex hull. If, in addition, f is C 1 in s and SMFCQ holds at every optimal solution, then:

  ∂s V (s) = co ∪a∗ (s)∈A∗ (s) (∇s f − λ∇s g − µ∇s h)(a∗ (s), s) . Proof. The first inclusion is demonstrated in Clarke (Clarke, 1983, Corollary 1, p 242). Given the differentiability and the uniqueness of multipliers implied by the SMFCQ it follows that:

  ∂s V (s) ⊂ co ∪a∗ (s)∈A∗ (s) (∇ fs − λ∇s g − µ∇s h)(a∗ (s), s) . (3.4.1) Reciprocally, consider any η, such that:

(3.4.2)

By construction, for all x:

η·x ≤

∇ fs (a∗ (s), s) − λ∇s g (a∗ (s), s)   − µ∇s h(a∗ (s), s) · x = V ′ (s; x) max

differentiability in Theorem 7 addresses the C 1 differentiability of the value function. Corollary 11. Under Assumption 1, if D(s) is nonempty and uniformly compact near s, if f is C 1 and strictly quasi-concave in a, g quasi-convex in a, h affine in a, and if SMFCQ holds for every a∗ (s) ∈ A∗ (s), then V is C 1 at s with:

∇ V (s) = (∇s f − λ∇s g − µ∇s h) (a∗ (s), s). Proof. By the strict quasi-concavity of f , the quasi-convexity of g, and the assumption that h is affine, A∗ (s) is a singleton, and SMFCQ guarantees uniqueness of the multiplier. The desired result then follows from Clarke (1983, Corollary 2, p. 242), noting that C 1 and strict differentiability coincide in Rn .  The difference between Corollary 11 and the envelope result of Rincon-Zapatero and Santos is meaningful. One can give simple economic examples (e.g., in consumer theory) in which the latter does not apply, as the LICQ is not satisfied, yet the value function is actually C 1 and the SMFCQ does hold (see next section). Alternatively, one can also construct examples in which the SMFCQ fails, the MFCQ holds, and the value function is not C 1 (see Tarafdar (2010) for such examples, or an earlier draft of this paper). So the importance of the SMFCQ seems very clear in obtaining sharp envelope theorems, at least in the finite dimensional parameter case. 4. Examples and applications

 (∂s f − λ∇s g − µ∇s h)(a∗ (s), s) .

 η ∈ ∪a∗ (s)∈A∗ (s) ∇s f (a∗ (s), s) − λ∇s g (a∗ (s), s)  − µ∇s h(a∗ (s), s) .

161



a∗ (s)∈A∗ (s)

the last equality following from Theorem 7. Since V ′ (s; x) V o (s; x), by definition of the Clarke gradient:

≤

η ∈ ∂s V (s) and ∂ Vs (s) contains any convex combination of such η, hence:

  ∂s V (s) ⊃ co ∪a∗ (s)∈A∗ (s) (∇ fs − λ∇s g − µ∇s h)(a∗ (s), s) . 

In this section we present an important example showing that the LICQ is not the most general sufficient condition for C 1 envelopes, followed by two applications of our results to lattice programming. 4.1. A consumer problem: SMFCQ vs. LICQ In the following example, the objective is C 1 and strictly quasiconcave, the LICQ fails, the SMFCQ holds, and the value function is C 1 (thus illustrating the importance of Corollary 11 in the preceding section). Consider a consumer endowed with income m ∈ [8, 12] and seeking to maximize utility u(a1 , a2 ) = a1 · a22 , in which ai is the quantity of good i. Up to (and including) 5 units of good i may be purchased at price of 1 per unit, but each additional unit (i.e., beyond 5) costs twice as much. The budget correspondence is thus given by: a1 + a2 − m ≤ 0

(4.1.1)

2a1 + a2 − 5 − m ≤ 0

(4.1.2)

a1 + 2a2 − 5 − m ≤ 0

(4.1.3)

2a1 + 2a2 − 10 − m ≤ 0.

(4.1.4)

Since it is convex and the objective is strictly quasi-concave, the optimal solution is unique. Denoting λ(m) the vector of KKT multipliers, the optimal solutions can be verified to be: a∗ (m); λ∗ (m)

 3.5. C 1 differentiability of the value function Finally, in setting with constraints and noninterior optimal solutions, perhaps the most general result in the literature concerning the existence of a classical C 1 envelope theorem appears in a recent paper by Rincon-Zapatero and Santos (2009, Theorems 3.1 and 3.2).4 A corollary of our main result on directional

4 Mirman and Zilcha (1975) and Benveniste and Scheinkman (1979) give conditions for the existence of C 1 envelopes for effectively unconstrained problems



 0)) , m < 10 ((m − 5, 5); (100 −   10m  , 0, 10m −275,    m + 5 ( m + 5 ) m + 5   , ; 0, 0 ,0 , m = 10 3 3 9 =    m    20m − m2 m2 − 10m   5, ; 0, 0, , , m > 10. 2

4

4

in dynamic programming framework. Milgrom and Segal (2002) provide similar condition for unconstrained static programming problems.

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For m < 10 constraints (4.1.1) and (4.1.3) are active, and the LICQ is satisfied. Symmetrically, for m > 10, constraints (4.1.3) and (4.1.4) are active, and the LICQ is satisfied. However, at m = 10, all constraints are active, yet only constraint (4.1.3) is binding (λ3 > 0), while all other constraints are saturated. The LICQ is clearly violated, but the vector y = (−2, 1) satisfies the conditions for the SMFCQ. Rincon-Zapatero and Santos main theorem (RinconZapatero and Santos, 2009, Theorem 3.1) does not apply; yet, we still have unique multipliers under Theorem 7. The value function for this problem is:

 (m − 5)25, m < 10      (m + 5)3 , m = 10 V (m) = 27   2  5m   , m > 10 4

Assumption 4.2.2. The production function f : R+ × Z → R+ is increasing, concave and C 1 in its first argument, supermodular and f (0, z ) = 0. In addition, there exists kmax > 0 such that ∀k > kmax , f (k, zmax ) < k. Let K = [0, kmax ]. Assumption 4.2.3. The utility function u : K → R+ is strictly increasing, continuous, concave, with u(0) = 0; 0 < β < 1. Assumption 4.2.4. For all M > 0, there exists x0 ∈ K with x0 > 0 such that ξ > M for all ξ ∈ ∂x u(x0 ). Assumption 4.2.5. Almost everywhere u′ (f (k, z ) − y)f1 (k, z ) is increasing in z. The last two assumptions are the non-smooth equivalent to u′ (0) = +∞ and the curvature assumption of Hopenhayn and Prescott, respectively. We prove the following result:

which is continuously differentiable with derivatives:

 25, m < 10      (m + 5)2 , m = 10 V ′ (m) = 9      5m , m > 10.

Theorem 13. Under Assumptions 4.2.1–4.2.5, the value function V is supermodular, Y ∗ (k, z ) is strong set order ascending, and the extremal optimal investment policies ∨Y ∗ and ∧Y ∗ are increasing in (k, z ). Proof. Recall that V is the unique continuous bounded function satisfying the Bellman equation:

2

V (k, z ) = T (V )(k, z ) = 4.2. Monotone controls in stochastic growth models with Markov shocks In this example we use the Dini bounds of Proposition 6 to prove the existence of monotone controls in stochastic growth models with Markov shocks without the ‘‘strict cardinal complementarity’’ condition used by Hopenhayn and Prescott (1992, Proposition 2). That condition is very restrictive because it requires the graph of the feasible correspondence to be a sublattice which, in practice, excludes all but Leontief production functions. The dynamic program (i.e., Bellman’s equation) associated with the social planner’s program for these economies involves continuous and concave (hence locally Lipschitz) functions that are well suited for the application of our results on Dini bounds (Proposition 6). For such functions, we note that Clarke gradients and superdifferential sets coincide, and that the following property holds5 : Proposition 12. Let F : K ⊂ R → R+ an increasing, concave and continuous function, and x′ , x ∈ K with x′ > x. Then ∀ξ ∈ ∂x F (x) and ∀ξ ′ ∈ ∂x F (x′ ), ξ ′ ≤ ξ . Proof. Follows from the supergradient being a maximal antitone operator (see Section 3 in Rockafellar, 1969).  We impose the same assumptions on utility and production as in Hopenhayn and Prescott (1992, Section 6, Part B), except that we do not restrict our attention to production functions with sublatticed graph correspondence, nor do we require the utility function to be C 1 or even differentiable.6 Assumption 4.2.1. Uncertainty is a random production shock z ∈ Z = [zmin , zmax ] ⊂ R++ following a Markov process with transition function Q assumed to be stochastically increasing.

5 For differentiable concave continuous function, this property is equivalent to the statement that the derivative is decreasing. 6 We could do away with the differentiability assumption on the production function as well.

+β



 max

0≤y≤f (k,z )

u(f (k, z ) − y)

V (y, z ′ )Q (z , dz ′ )

 (4.2.1)

and is obtained as the pointwise limit of the sequence {Vn }∞ n =1 defined by: Vn+1 (k, z ) =

 max

0≤y≤f (k,z )

+β



u(f (k, z ) − y)



Vn (y, z ′ )Q (z , dz ′ )

(4.2.2)

and given V0 = 0. We prove in the next lemma by a recursive argument that each Vn is supermodular; obviously, as the pointwise limit of a sequence of supermodular functions, V inherits that property. To conclude the proof, we note that the supermodularity of V implies that the objective in (4.2.1) has increasing differences in (k, z ) while the choice correspondence is strong set increasing in (k, z ), so by Theorem 2.8.3 in Topkis (1998) Y ∗ is strong set order increasing and both ∨Y ∗ and ∧Y ∗ are increasing in (k, z ).  Lemma 14. Each Vn is supermodular in its arguments. Proof. Assuming that the continuous concave function Vn is supermodular we show that Vn+1 inherits that property. The desired result follows by induction. Denote by Yn∗+1 (k, z ) the set of optimal solutions for program represented by expression (4.2.2). Because of the concavity of u and f , for each n and each z the functions Vn (., z ) are continuous and concave (thus locally Lipschitz), and so are all  the functions β Vn (., z ′ )Q (z , dz ′ ). The objective in (4.2.2) is therefore continuous, concave and locally Lipschitz, while the constraints are C 1 . By Berge’s Theorem of the Maximum, each optimal correspondence Yn∗+1 (k, z ) is non-empty, compact-valued and upper hemicontinuous in k.  If Vn is supermodular, then by Assumption 4.2.4 β Vn (y, z ′ ) ′ Q (z , dz ) has increasing differences in (y, z ). By concavity of u and f function u(f (k, z ) − y) has increasing differences in (y, z ), thus the objective in (4.2.2) has increasing differences in (k, z ) while the choice correspondence is strong set order increasing in (k, z ). By Topkis (1998, Theorem 2.8.3), Yn∗+1 is then strong set order

O. Morand et al. / Journal of Mathematical Economics 61 (2015) 157–165

increasing and both ∨Yn∗+1 and ∧Yn∗+1 are increasing in (k, z ) (and so are f − ∨Yn∗+1 and f − ∧Yn∗+1 by a similar argument). We note that the hypothesis of Proposition 6 is trivially satisfied while Assumptions 4.2.2–4.2.4 insure interiority of solutions. Thus, for any x > 0 and for each z 7 : t inf

t →0+

≥

Vn+1 (k′ , z ′ ) − Vn+1 (k, z ′ ) =

sup

≥

y∈Yn∗+1 (k,z ) ξ ∈∂k u(f (k,z )−y)

ξ ·x≥0

sup

sup

y∈Yn∗+1 (k,z ) ξ ∈∂k u(f (k,z )−y)

=

sup ξ ∈∂k u(f (k,z )−∧Yn∗+1 (k,z ))

ξ ·x

ξ ·x

in which the equality follows from the property of Clarke gradients established in Proposition 12 and applied to the increasing concave and continuous function k −→ u(f (k, z ) − y). Next, recalling that f (k, z )−∧Yn∗+1 (k, z ) is increasing in k, again by Proposition 12 for any k′ ≥ k: sup ξ ∈∂k u(f (k,z )−∧Yn∗+1 (k,z ))

ξ≥

sup ξ ′ ∈∂k u(f (k′ ,z )−∧Yn∗+1 (k′ ,z ))

ξ ′.

As a result, for any θ > 0 and any k ∈ [θ , kmax ], for any x > 0: Vn+1 (k + tx, z ) − Vn (k, z )

0 ≤ lim inf

t →0+

≤ lim sup

t Vn+1 (k + tx, z ) − Vn (k, z ) t

t →0+

≤

sup ξ ∈∂k u(f (θ,z )−∧Yn∗+1 (θ ,z ))

ξ · x.

The existence of a uniform upper bound on all Dini derivatives D+ Vn+1 (k, z ; x) and D+ Vn+1 (k, z ; x) for all k ∈ [θ , kmax ] then implies that for each z the function Vn+1 (., z ) is (globally) Lipschitz on any [θ , kmax ]; since it is also increasing and continuous on [0, kmax ] it is therefore absolutely continuous on [0, kmax ] (see Problem 37 in Royden and Fitzpatrick, 2010). By the fundamental theorem of integral calculus (Theorem 10 in Chapter 6 of Royden and Fitzpatrick, 2010), Vn+1 (., z ) is differentiable almost everywhere on [0, kmax ] and, more importantly, its derivative is integrable over [0, k] for any 0 < k ≤ kmax with: Vn+1 (k, z ) =

k



Vn′ +1 (x, z ′ )dx

k

Vn′ +1 (x, z )dx = Vn+1 (k, z ) − Vn+1 (k, z ).

Increasing differences and supermodularity are equivalent properties on R2 hence Vn+1 is a supermodular function.  4.3. Minimax lattice programming

t

t →0+

k′

k

Vn+1 (k + tx, z ) − Vn (k, z )

lim sup



k′



(the second inequality results from the monotonicity of u(f (k, z )− y) in k), and

≤

these two inequalities following from the property that ∧Yn∗+1 (k, z ) is increasing in z and from Assumption 4.2.5, respectively. As a result, for k′ > k and z ′ > z:

Vn+1 (k + tx, z ) − Vn (k, z )

lim inf

163

Vn′ +1 (x, z )dx.

We develop a minimax lattice programming method combining envelope theorems and duality to generate strong monotone comparative statics in a class of simple consumer optimization problems. Such sharp characterization, which implies that least and greatest selections are increasing, cannot be obtained by Quah (2007). His flexible set order methods only guarantee the existence of a monotone selection, unless all controls are flexible set ordered (see Quah, 2007). Consider a standard n good consumer problem for which the commodity space is the sublattice Rn+ endowed with the componentwise partial order, and s ∈ R+ is income. Good 1 is the numeraire, and 0 < pi i = 2, . . . , n are the relative prices. Denote by c−1 the vector (c2 , . . . , cn ) and by P the vector (p2 , . . . , pn ). Assume the following complementarity conditions as in Quah (2007), on top of some regularity conditions: Assumption 4.3.1. The function u(c1 , c−1 ) has strict increasing differences in (c1 , c−1 ), is concave in c−1 for each c1 , strictly increasing in all its arguments, and continuously differentiable in its first argument. In addition, there exists some j ≥ 2 such that u is continuously differentiable in cj and limcj →0 ucj (c1 , c−1 ) = +∞. Given P ≫ 0 and s > 0 consider the consumer’s problem: V (s) = max u(c ) c ∈D(s)

where: D(s) = {c ∈ Rn+ |c1 + p2 c2 + · · · + pn cn ≤ s} and C ∗ (s) is the set of optimal solutions so that: C ∗ (s) = arg max u(c ). c ∈D(s)

Denote by C1∗ (s) = {c1 , ∃c−1 (c1 , c−1 ) ∈ C ∗ (s)} the consumer demand correspondence for good 1. We prove the following result:

Clearly, where Vn+1 (., z ) is differentiable, both Dini derivatives coincide, and:

Theorem 15. Under Assumption 4.3.1 C1∗ : R+ → L(R+ ) is ascending in Veinott strong set order, which implies that ∨C1∗ and ∧C1∗ are increasing selections.

Vn′ +1 (k, z ) = u′ (f (k, z ) − ∧Yn∗+1 (k, z ))f1 (k, z )

Proof. For each P ≫ 0 and s > 0:

since u is locally Lipschitz hence differentiable almost everywhere. Note that Vn′ (k, z ) is an increasing function of z since, for z ′ ≥ z:

V (s) = max V A (c1 , s)

u′ (f (k, z ′ ) − ∧Yn∗+1 (k, z ′ ))f1 (k, z ′ )

in which:

0

c1 ∈[0,s]

≥ u′ (f (k, z ′ ) − ∧Yn∗+1 (k, z ))f1 (k, z ′ )

V A (c1 , s) =

≥ u′ (f (k, z ) − ∧Yn∗+1 (k, z ))f1 (k, z )

with:

max

c−1 ∈D(c1 ,s)

u(c1 , c−1 )

(4.3.1)

(4.3.2)

D(c1 , s) = {c−1 ≥ 0|P .c−1 ≤ s − c1 }. 7 A symmetric argument can easily be made for any x < 0 but is omitted for simplicity.

Assuming that V A is supermodular in (c1 , s), since the feasible correspondence of program (4.3.1) is strong set order ascending in

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s, the desired result follows directly from Theorem 2.8.2 in Topkis (1998). The rest of this section is devoted to the proof of the supermodularity of V A , and it is where our results become useful. We first note that this supermodularity is immediate in the case of 2 goods, since, by substitution: V A (c1 , s) = u(c1 , (s − c1 ))/p2

(4.3.3)

and the assumed concavity of u together with its strict increasing differences lead to the desired result. With three or more goods, however, the argument needs to be generalized, and this is we do next.  Program (4.3.2) is Lipschitz and satisfies the assumptions of uniform compactness and the SMFCQ at any optimal solution. In addition, the assumption of strict concavity implies uniqueness of ∗ 8 the optimal solution, c− 1 (c1 , s), which necessarily satisfies :

(4.3.4)

and:

λ∗ (c−∗ 1 (c1 , s)) c1 + P .c−∗ 1 (c1 , s) − s = 0. 



If pj > 0 and u is continuously differentiable in ci at least for some i > 2, then equality (4.3.4) guarantees the uniqueness of the multiplier. As a result, by Theorem 7: ∗ VsA (c1 , s; x) = λ∗ (c− 1 (c1 , s)) · x.

For a given s if the budget constraint is not binding, then

λ∗ (c−∗ 1 (c1 , s)) = 0 in which case λ∗ is trivially weakly increasing

f (x0 + td) − f (x0 )

D+ f (x0 ; d) = lim inf

t

t ↓0

.

The radial left Dini derivatives are defined similarly simply changing t ↓ 0 to t ↑ 0. The directional derivative at x0 ∈ X in the direction d ∈ Rn is defined as: f ′ (x0 ; d) = lim

f (x0 + td) − f (x0 ) t

t ↓0

,

and the Clarke upper and lower generalized directional derivatives at x0 in the direction d ∈ Rn are respectively: f (y + td) − f (y)

f o (x0 ; d) = lim sup

t

y→x0 t ↓0

f −o (x0 ; d) = lim inf

∂cj u(c1 , c−∗ 1 (c1 , s)) − λ∗ (c−∗ 1 (c1 , s)).pj = 0 for some j ≥ 2

and the lower radial right Dini derivative as:

f (y + td) − f (y) t

y→x0 t ↓0

.

It is important to note that Clarke generalized directional derivatives of Lipschitz functions always exist, while directional derivatives of such functions need not. A locally Lipschitz function f is Clarke regular if its directional derivative exists in all directions and if it coincides with its Clarke generalized directional derivative (i.e., f o (x; d) = f ′ (x; d)). A function f is differentiable at x0 ∈ X if its directional derivatives exist in all direction and if f ′ (x0 ; d) = ∇x f (x0 ) · d. Function f has a strict derivative at x0 , denoted by Dx f (x0 ), when for all d ∈ Rn : f (x + td) − f (x)

in c1 since positive. Alternatively, if the budget constraint is ∗ binding, c− 1 must be decreasing in c1 given the assumption is strict increasing difference for the objective of program (4.3.2). This property together with the concavity of u(c1 , c−1 ) in c−1 and ∗ its strict increasing differences imply that ∂ci u(c1 , c− 1 (c1 , s)) is increasing in c1 . ∗ Condition (4.3.4) therefore implies that λ∗ (c− 1 (c1 , s)) is increasing in c1 , which is precisely the statement that V A is supermodular. We note that the optimal solutions in this example are not necessarily interior, so the results of Clarke (1975) and Milgrom and Segal (2002) do not apply. Also, the value functions are not concave in c1 , so the results of Gol’stein (1972) and Milgrom and Segal (2002, Theorem 5) do not apply when computing the single crossing property.

⟨Dx f (x0 ), d⟩ = xlim →x

Appendix. Mathematical appendix

A.2. Lattices and supermodularity

A.1. Lipschitz functions

(X , ≥) is a partially ordered set if the binary relation ≥ is reflexive, transitive and antisymmetric. An upper (resp. lower) bound of B ⊂ X is an element xu (resp. xl ) in B such that ∀x ∈ B, x ≤ xu (resp. xl ≤ x). A lattice is a partially ordered set (X , ≥) such that any two elements x and x′ in X have a least upper bound in X , denoted x ∧ x′ , and a greatest lower bound in X , denoted x ∨ x′ . B ⊂ X is a sublattice of X if it contains the sup and the inf (with respect to X ) of any pair of points in B. A mapping f : (X , ≥X ) → (Y , ≥Y ) is isotone (or increasing) if f (x′ ) ≥Y f (x) when x′ ≥X x, for x, x′ ∈ X . A correspondence F : (X , ≥X ) → 2Y is ascending in the set relation on 2Y denoted by ≥S if F (x′ ) ≥S F (x), when x′ ≥X x. A particular set relation of interest is Veinott’s strong set order (see Veinott, 1992, Chapter 4) defined as: A1 ≥S A2 if ∀(a, b) ∈ A1 × A1 , a ∧ b ∈ A2 and a ∨ b ∈ A1 . Let (X , ≥) be a lattice. A function f : X → R is supermodular (resp., strictly supermodular) in x if ∀(x, y) ∈ X 2 , f (x ∨ y) + f (x ∧ y) ≥ (resp., >) f (x) + f (y). An important property of the class of supermodular functions is they are closed under pointwise limits

Given the metric spaces (X , ρX ) and (Y , ρY ), a function f : X → Y is (locally) Lipschitz near x with modulus k(x) ≥ 0, if for all x′ , x′′ in a neighborhood of x:

ρY (f (x′′ ), f (x′ )) ≤ k(x)ρX (x′′ , x′ ). There are several notions of differentiability for a Lipschitz function f : I ⊂ Rn → Rm . First, given x0 ∈ I and a direction d ∈ Rn , define the upper radial right Dini derivative as: D+ f (x0 ; d) = lim sup t ↓0

f (x0 + td) − f (x0 ) t

,

8 The assumed Inada condition with respect to c simplifies the argument by j insuring that the multiplier associated with the inequality constraints cj ≥ 0 is zero.

0 t ↓0

t

and it is continuously differentiable if Dx f (x) : Rn → Rn×m is continuous at x0 . In a finite dimensional domain, a strictly differentiable function is continuously differentiable. Finally, recall that the subgradient of a convex function f : Rn → Rm is the set of p ∈ Mm×n satisfying: p · d ≤ f (x0 + d) − f (x0 ) for all directions d ∈ Rn . The subgradient of a Lipschitz function may not necessarily exist, but its Clarke generalized gradient always does and is defined as the set:

∂x f (x0 ) = {ξ ∈ Rn , f o (x0 ; d) ≥ ξ · d}.

O. Morand et al. / Journal of Mathematical Economics 61 (2015) 157–165

(Topkis, 1998, Lemma 2.6.1). Let (P , ≥P ) be a partially ordered set, and B ⊂ X × P. The function f : B −→ R has (strictly) increasing differences in (x1 , p) if for all p1 , p2 ∈ P, p1 ≤P p2 H⇒ f (x, p2 ) − f (x, p1 ) is (strictly) increasing in x ∈ Bp1 , where Bp is the p section of B. References Askri, K., LeVan, C., 1998. Differentiability of the value function of nonclassical optimal growth models. J. Optim. Theory Appl. 97, 591–604. Auslender, A., 1979. Differentiable stability in non convex and non differentiable programming. Math. Program. Stud. 10, 29–41. Benveniste, L., Scheinkman, J., 1979. On the differentiability of the value function in dynamic models of economics. Econometrica 47, 727–732. Bonnans, J.F., Shapiro, A., 2000. Perturbation Analysis of Optimization Problems. Springer-Verlag New-York, Inc.. Bonnisseau, J.M., LeVan, C., 1996. On the subdifferential of the value function in economic optimization problems. J. Math. Econom. 25, 55–73. Clarke, F., 1975. Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 247–262. Clarke, F., 1983. Optimization and Nonsmooth Analysis. SIAM. Clausen, A., Strub, C., Envelope Theorems for Nonsmooth and Nonconcave Optimization, MS, 2013. Gauvin, J., Dubeau, F., 1982. Differential properties of the marginal function in mathematical programming. Math. Program. Stud. 19, 101–119. Gauvin, J., Tolle, J.W., 1977. Differential stability in nonlinear programming. SIAM J. Control Optim. 15, 294–311.

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