Risk-averse asymptotics for reservation prices Laurence Carassus

∗

Mikl´os R´asonyi†

August 5, 2010

Abstract In a semi-martingale framework, we prove that the utility indifference price for an agent becoming infinitely risk adverse converges to the superreplication price, under suitable assumptions.

Keywords: Utility indifference price, Superreplication price, Convergence, Utility maximization, Risk aversion. MSC 2000 Subject Classification: Primary: 91B16, 91B28 ; Secondary: 93E20, 49L20 OR/MS Subject Classification: Primary: Utility / Value theory ; Secondary: Finance / Asset Pricing JEL classification: C61, C62, G11, G12

1

Introduction

In this article we investigate the effect of increasing risk aversion on utilitybased prices. We are dealing with the utility indifference price (or reservation price), defined in [13] for the first time. This is the minimal amount added to the initial capital of a seller of a contingent claim which allows her to attain the same utility that she would have attained from her initial capital without selling the option, see Definition 4.3 below. Intuitively, when risk aversion tends to infinity, the reservation price should tend to the superreplication price (i.e. the price of hedging the option without any risk). This question has been extensively treated when the agent has constant absolute risk aversion (i.e. for exponential utility functions). The convergence of reservation prices to the superreplication price was shown in [22] for Brownian models and in [8] in a general semimartingale setting. A nonexponential case was treated in [5], but with severe restrictions on the utility functions. The case of general utilities was considered in [6] and [7] for the first time, in discrete-time market models. Now we prove a corresponding continuous-time result in a semimartingale framework, under suitable hypotheses. ∗ Laboratoire de Probabilit´ es et Mod` eles Al´ eatoires, Universit´ e Paris 7 Denis Diderot, 16 rue Clisson, 75013 Paris, France. E-mail: [email protected] † University of Edinburgh, School of Mathematics, King’s Buildings, Mayfield Road, Edinburgh, EH 9 3JZ, UK. On leave from the Computer and Automation Institute of the Hungarian Academy of Sciences, Budapest. The second author dedicates this paper to Annam´ aria Brecz. E-mail: [email protected]

1

In section 2 we model agents’ preferences and introduce a growth condition related to the asymptotic elasticity of their utility functions. In section 3 the market model and a compactness assumption are discussed. In section 4 the concept of utility indifference price is formally defined and the main results are proved. We conclude with examples in section 5. Proofs of technical results are postponed to section 6.

2

Risk averse agents

We consider a sequence of agents trading in the market and assume that their preferences are described by von Neumann–Morgenstern-type utility functions (see section 2.2 of [11]). More precisely, Assumption 2.1 Un , n ∈ N are twice continuously differentiable, strictly concave and increasing functions on R such that for each x ∈ R, rn (x) :=

−Un′′ (x) −→ ∞. Un′ (x) n→+∞

(1)

The function rn is called the (absolute) risk aversion of an agent with utility function Un . This concept was introduced in [1] and [19]. In this paper we are interested in what happens when agents become infinitely risk-averse, i.e. when (1) holds. We introduce the Fenchel conjugates of Un : Vn (y) := sup{Un (x) − xy},

y ∈ (0, ∞).

(2)

x∈R

As easily checked, the Vn are finite, convex functions. We shall now stipulate a certain growth condition on the utility functions we consider, formulated in terms of the Vn . Similar assumptions are standard in the literature, as explained below. Assumption 2.2 For each [λ0 , λ1 ] ⊂ (0, ∞) there exist positive constants C1 , C2 , C3 such that for all n and for all y > 0, Vn (λy) ≤ C1 Vn (y) + C2 y + C3 .

(3)

holds for each λ ∈ [λ0 , λ1 ]. Remark 2.3 Let us drop n and consider a fixed utility function U with its conjugate V . It is proved in [12] that condition (3) (for this fixed V ) is equivalent to xU ′ (x) xU ′ (x) lim sup < 1, lim inf > 1. (4) x→−∞ U (x) U (x) x→∞ The first of the two conditions in (4) was introduced in [17], the second one in [23] and they have become standard assumptions in optimal investment problems since then. A utility function U satisfying (4) is said to have reasonable asymptotic elasticity (terminology of [23]). Thus Assumption 2.2 can be seen as a (dually formulated) reasonable asymptotic elasticity condition, uniform in n.

2

3

Market model

Let us fix a time horizon T > 0. Our market is modelled by an adapted ddimensional semimartingale St , t ∈ [0, T ] on a given continuous-time stochastic basis (Ω, F, (Ft )t∈[0,T ] , P ). We think that S represents the evolution of the (discounted) prices of d risky assets. Assumption 3.1 We assume that S is locally bounded (to avoid technical complications) and that M= 6 ∅, (5) where M denotes the set of measures Q ∼ P such that S is a local martingale under Q. Intuitively, the condition (5) stipulates that there is absence of arbitrage in the market, this was made rigorous in [9]. We furthermore make the following Assumption 3.2 There exists Q0 ∈ M such that the sequence Vn (dQ0 /dP ) is uniformly integrable (with respect to P ). We denote by Mv the set of such Q0 s. Remark 3.3 The Assumption above is a certain kind of compactness requirement. Similar conditions have already appeared in investigations on the stability of optimal strategies with respect to perturbations of utility functions, see [18] and [16]. Both papers consider a sequence Un of utility functions converging to a limiting utility U and show convergence of the corresponding optimal strategies and utility-based prices. In [18] the sequence Un is assumed to be dominated by some U with conjugate function V such that V (Z) is integrable, where Z = dQ0 /dP for a particular Q0 ∈ M, the so-called “minimal martingale measure”.1 Translated into the present context, Assumption (UI) of [16] requires the uniform integrability of   dQ0 , n ∈ N, Vn y dP for each y > 0 for some Q0 ∈ M. It is immediate that, under Assumption 2.2, the latter condition is equivalent to Assumption 3.2. Note, however, that in both mentioned papers the Un are defined on the positive real axis only. Assumption 2.2 allows us to prove that the set Mv is, in fact, large. Lemma 3.4 Assume that there is x0 ∈ R such that Un (x0 ) is bounded from below. Under Assumptions 2.2 and 3.2, the set Mv is dense in M with respect to the total variation norm topology. Remark 3.5 This result was essentially reported in Proposition 6 of [3], but without a proof. See also Proposition 2.1.20 on page 89 of [14]. Proof. See Appendix 6.1.

2

1 In

fact, the hypothesis is slightly weaker there but it would lead us far from our subject to discuss these delicacies here.

3

4

Utility indifference prices

We define the set of admissible trading strategies stipulating that they have a finite credit line (to avoid doubling strategies). Definition 4.1 Let P denote the set of predictable processes (w.r.t. the given stochastic basis) and set A := {φ ∈ P : φ is S-integrable, for some w > 0, Vt0,φ ≥ −w for all t ∈ [0, T ]}, where we write Vtx,φ := x + from initial endowment x.

RT 0

φt dSt for the value process of strategy φ starting

Remark 4.2 When the existence of optimal strategies is investigated one needs to enlarge the class A, see e.g. [23]. For our present purposes this is unnecessary as the value of the supremum in (6) below would be the same over the extended class of strategies in [23]. Fix a bounded nonnegative FT -measurable random variable G, interpreted as a contingent claim to be delivered at the end of the period [0, T ]. We could substantially relax the boundedness assumption on G but do not do so for the sake of a simple presentation. Let us consider an agent with initial capital x ∈ R and utility function Un who tries to trade optimally on [0, T ] and who also delivers the claim G at T . The expected utility she may attain is un (x, G) := sup EUn (VTx,φ − G),

(6)

φ∈A

this is well-defined (but may be +∞) as admissible strategies have bounded from below value processes. Definition 4.3 The utility indifference price for Un and initial endowment x0 is pn (x0 , G) := inf{p ∈ R+ : un (x0 + p, G) ≥ un (x0 , 0)}, (7) i.e. it is the minimal extra capital that allows for delivering G while attaining the same utility as without claim delivery. Remark 4.4 Replacing Un (x) by affine transformations γn Un (x) + δn , γn ∈ (0, ∞), δn ∈ R does not change the corresponding utility indifference prices. Note also that Assumption 2.1 is also invariant under this type of affine transformations. The superreplication (or superhedging) price of G is defined as π(G) := inf{x ∈ R : there is φ ∈ A such that VTx,φ ≥ G a.s.}, this is a utility-free concept expressing the amount needed to hedge the claim without taking any risks. The following dual characterization is a fundamental result of mathematical finance, see e.g. [10] and the references therein.

4

Theorem 4.5 If M = 6 ∅ then we have π(G) = sup EQ G. Q∈M

We are now ready to state our main result i.e. the convergence of utility indifference prices to the superreplication price when risk aversion tends to infinity. We will present this result under two types of assumptions. The first one (see Assumption 4.6) refers to the existence of a initial wealth x0 for which all the investors have (asymptotically) a common preference and also a common non-zero growth rate. The second type of assumptions imposes the elasticity Assumption 2.2 and the compactness Assumption 3.2 on a normalized family of utility functions (see Assumptions 4.9, 4.10 and Theorem 4.11 below). Assumption 4.6 There exists some x0 , β ∈ R and α ∈ (0, ∞) such that Un′ (x0 ) Un (x0 )

−→

α,

−→

β.

n→+∞ n→+∞

Theorem 4.7 If Assumptions 2.1, 2.2, 3.1, 3.2 and 4.6 hold then the quantities un (x0 , G), n ∈ N are finite and the corresponding utility indifference prices pn (x0 , G) tend to π(G) as n → ∞. Proof. It is a standard fact that pn (x0 , G) ≤ π(G), see page 152 of [6] for a proof. For the reverse inequality, we argue by contradiction. Suppose that for some ε > 0 and a subsequence nk one has pk := pnk (x0 , G) ≤ π(G) − ε for each k. We may and will suppose that nk = k and pk → π(G) − ε, k → ∞. Take arbitrary φ ∈ A and Q ∈ Mv . Vt0,φ , t ∈ [0, T ] is a stochastic integral with respect to a Q-local martingale that is bounded from below, so it is a Q-supermartingale, see Theorem 2.9 in [9] or Corollaire 3.5 in [2]. Hence EQ (VTx0 +pk ,φ − G) ≤ x0 + pk − EQ G. (8) Define   vkφ (Q) := E Vk (dQ/dP ) + (dQ/dP )(VTx0 +pk ,φ − G) . Note that the bound in (8) is independent of φ. Proposition 4.8 below, Q ∈ Mv and pk → π(G) − ε imply lim sup sup vkφ (Q) ≤ β − x0 E k→∞ φ∈A

dQ + (x0 + π(G) − ε − EQ G). dP

By the definition of conjugate functions we have, lim sup sup EUk (VTx0 +pk ,φ − G)

≤

k→∞ φ∈A

inf lim sup sup vkφ (Q)

Q∈Mv

k→∞

φ

≤ β + π(G) − ε − sup EQ G. Q∈Mv

5

(9)

It follows that uk (x0 , G) are finite. Lemma 3.4 shows that supQ∈Mv EQ G = supQ∈M EQ G, thus (9) and Theorem 4.5 imply lim sup uk (x0 + pk , G) ≤ β − ε. k→∞

But uk (x0 + pk , G) ≥ uk (x0 , 0) ≥ Uk (x0 ), thus lim inf k→∞ uk (x0 + pk , G) ≥ β, a contradiction. 2 The following proposition was needed in the proof above. Proposition 4.8 Under Assumptions 2.1 and 4.6, for each y > 0, Vn (y) −→ β − x0 y. n→+∞

Proof. See Appendix 6.2. 2 We now turn to our second Theorem, which is merely a reformulation of Theorem 4.7 under conditions that are sometimes easier to check. To state its hypotheses, we first need to introduce some normalization of the functions Un . Fix some initial wealth x0 ∈ R and set : ˜n (x) := Un (x) − Un (x0 ) , U Un′ (x0 )

n ∈ N, x ∈ R.

(10)

Define the Fenchel conjugates ˜n (x) − xy}, V˜n (y) := sup{U

y ∈ (0, ∞).

(11)

x∈R

˜n ), n ∈ N. We now restate Assumptions 2.2 and 3.2 for the family (U Assumption 4.9 For each [λ0 , λ1 ] ⊂ (0, ∞) there exist constants C1 , C2 , C3 such that for all n and for all y > 0, V˜n (λy) ≤ C1 V˜n (y) + C2 y + C3 .

(12)

holds for each λ ∈ [λ0 , λ1 ]. Assumption 4.10 There exists Q0 ∈ M such that the sequence V˜n (dQ0 /dP ) is uniformly integrable (with respect to P ). Theorem 4.11 Assume that Assumptions 2.1, 3.1, 4.9 and 4.10 hold. Then the utility indifference prices pn (x0 , G) tend to π(G) as n → ∞. ˜n (x0 ) = 0 and U ˜n′ (x0 ) = 1, thus Assumption 4.6 is satisfied Proof. Obviously, U ˜ for the sequence Un at x0 . Let u ˜n be defined as ˜n (V x,φ − G). u ˜n (x, G) := sup E U T φ∈A

˜n are affine transforms of the Un , we may alternatively write Since the U pn (x0 , G) = inf{p ∈ R+ : u ˜n (x0 + p, G) ≥ u˜n (x0 , 0)}, ˜n , too (recall Remark 4.4). and Assumption 2.1 holds true for U ˜n at x0 allows us to conclude. Applying Theorem 4.7 for U 6

2

Remark 4.12 Let us now assume that Un , n ∈ N satisfies Assumption 2.1 and consider Assumption 4.13 There exists Q0 ∈ M such that dQ0 /dP and dP/dQ0 are both bounded. Note that a convex function attains its maximum on an interval at one of the endpoints. If Assumption 4.13 holds then 1/K ≤ dQ0 /dP ≤ K for some K > 0 and hence ˜n (x0 ) − |x0 |K ≤ V˜n (dQ0 /dP ) ≤ |V˜n (K)| + |V˜n (1/K)|. −|x0 |K = U By Proposition 4.8, it follows that the sequence |V˜n (dQ0 /dP )| is uniformly bounded. This argument shows that Assumption 4.13 is stronger than Assumption 4.10 and it is actually independent of the choice of x0 . We claim that, replacing Assumption 4.10 by Assumption 4.13, one may drop Assumption 4.9 from the hypotheses of Theorem 4.11 and the conclusions remain true, for all x0 ∈ R. Indeed, in this case Corollary 1.2 of [15] directly implies that the measures Q0 ∈ M satisfying Assumption 4.13 are dense in M hence one does not need to appeal to Assumption 4.9. The rest of the proof is identical (Assumption 4.9 is only used to prove Lemma 3.4). It is very easy to construct models which satisfy Assumption 4.13: consider a locally bounded semimartingale St , t ∈ [0, T ] on the stochastic basis (Ω, F, (Ft )t∈[0,T ] , P ) such that M = {Q0 } is a singleton (as e.g. in the Black-Scholes model) and choose an arbitrary P0 ∼ Q0 with dP0 /dQ0 , dQ0 /dP0 bounded (e.g. take P0 = Q0 ). Looking at the price process S on the stochastic basis (Ω, F, (Ft )t∈[0,T ] , P0 ) Assumption 4.13 will clearly be satisfied. It has to be pointed out, however, that the previous construction is somewhat artificial and in most of the naturally arising continuous-time models Assumption 4.13 does not hold. In discrete-time models Assumption 4.13 is true under reasonable conditions, see Remark 7.2 and Theorem 6.2 of [20] for examples and [21] for a more extensive discussion.

5

Examples

Example 5.1 The case of the exponential utility was already covered in previous work, we show that it is subsumed in our main theorem, too. Let 1 − exp{−αn x} Un (x) = , αn with some αn > 0 tending to ∞ as n → ∞. Proposition 5.2 If the market model satisfies Assumption 3.1 and there exists Q0 ∈ M such that dQ0 dQ0 ln <∞ (13) E dP dP then pn (0, G) → π(G), n → ∞. Proof. Choosing x0 = 0 it is straightforward that Un (0) = 0, Un′ (0) = 1, rn (x) = ˜n satisfy Assumption 2.1. Moreover, direct calculation gives αn and thus Un = U 7

Vn (y) = (1/αn )[y ln y + 1 − y] showing that Assumption 4.10 holds provided that there is Q0 ∈ M satisfying (13), i.e. whenever a finite-entropy martingale measure exists (which is a very modest assumption). Looking at the form of Vn we see that it is enough to show (3) for one function V (y) = y ln y + 1 − y only since 1/αn is a bounded sequence. Take 0 < λ0 ≤ λ ≤ λ1 < ∞ and denote C˜ := sup0 1, V (λy) ≤ λ1 V (y) + (1 + λ1 )C. V (λy) = λy ln(λy) + 1 − λy ≤ λy ln y + λ| ln λ|y + 1 − λy ≤ λ1 y ln y + λ1 [| ln λ0 | + | ln λ1 |]y + 1 = λ1 (V (y) − 1 + y) + λ1 [| ln λ0 | + | ln λ1 |]y + 1 ≤ λ1 V (y) + λ1 [| ln λ0 | + | ln λ1 | + 1]y + 1 ˜ hence we may choose C1 := λ1 , C2 := λ1 [| ln λ0 |+| ln λ1 |+1], C3 := 1+(1+λ1 )C. Theorem 4.11 applies and the respective reservation prices pn (0, G) converge to π(G). 2 In fact, one can check, doing some more calculations, that Theorem 4.11 applies for each x0 ∈ R and pn (x0 , G) converge to π(G), for all x0 . Example 5.3 Now we present a case that has never been tackled before in continuous-time models. 1 1 [(1 − x)βn − 1]1{x≤0} Un (x) := − [(x + 1)−αn − 1]1{x>0} − αn βn with αn > 0, βn > 1, both tending to ∞ as n → ∞. These functions are continuously differentiable with strictly monotone derivatives, hence they are strictly concave. They are also twice continuously differentiable on R \ {0}. Take now βn := αn + 2 then Un′′ exists in 0, too, and Un′′ is continuous. Proposition 5.4 If the market model satisfies Assumption 3.1 and there exists Q0 ∈ M such that dQ0 1+ε E <∞ (14) dP for some ε > 0 then pn (0, G) → π(G), n → ∞. Proof. Calculate Un′ (x)

=

(x + 1)−(αn +1) 1{x>0} + (1 − x)αn +1 1{x≤0}

Un′′ (x)

=

−(αn + 1)(x + 1)−(αn +2) 1{x>0} − (αn + 1)(1 − x)αn 1{x≤0} .

1 1{x≤0} ˜ Thus Un (0) = 0, Un′ (0) = 1, rn (x) = (αn + 1)( {x>0} x+1 + 1−x ) and Un = Un clearly satisfy Assumption 2.1. It remains to check Assumptions 4.10 and 4.9. We have, for y ∈ (0, ∞),         αn +2 αn 1 1 1 1 y αn +1 1{y>1} + y αn +1 1{y≤1} . Vn (y) = +y− 1+ −y+ 1− αn + 2 αn + 2 αn αn

Thus if there exists Q0 ∈ M such that dQ0 /dP ∈ L1+ε (P ) for some ε > 0 (i.e. (14) holds) then Assumption 4.10 will be satisfied for n large enough since (αn + 2)/(αn + 1) converges to 1. Finally, Assumption 4.9 is satisfied by the scaling properties of the power functions. We may conclude by Theorem 4.11 that the reservation prices pn (0, G) tend to π(G) as n → ∞. 2 8

Example 5.5 We sketch a method to generate further examples by “stitching together” the previous two. Take Un , Vn as in Example 5.3. Define an (y) := Vn′′ (y), y ≤ 2,

an (y) =

2Vn′′ (2) , y > 2, y

this is a continuous function that is nonnegative (by convexity of Vn ). (The choice of 2 is arbitrary here, it may be changed to any other constant > 1.) Define Z yZ z Vˆn (y) := an (u)dudz, y ∈ (0, ∞), 1

1

hence Vˆn is a convex function with Vˆn (1) = Vˆn′ (1) = 0. Note that an (y) for y > 2 coincides with the second derivative of the Fenchel conjugate of 1 − exp{−κn x} κn 1

for the choice κn := (αn + 1)/2 αn +1 → ∞, n → ∞, hence the present example is indeed a mix of Examples 5.1 and 5.3. ˆn (x) := inf y>0 {Vˆn (y) + xy}. By the bidual theorem (see e.g. TheDefine U orem 4.2.1 on page 76 of [4]) we have that Vˆn is just the Fenchel conjugate of ˆn . One may check U ˆn (0) = 0, U ˆ ′ (0) = 1 and, by inverting the argument of U n ˆn satisfies Assumption 2.1. Proposition 4.8 one may show that U Choose x0 = 0. Assuming the existence of Q0 ∈ M satisfying (14) (for example, Q0 ∈ L2 is sufficient), Assumptions 4.10 and 4.9 can be verified for Vˆn = V˜n just like in Examples 5.1 and 5.3, involving more tedious calculations. The procedure just presented can be used to get a plethora of new examples falling into the scope of Theorem 4.11.

6

Appendices

6.1

Proof of Lemma 3.4

Fix Q0 ∈ Mv . Corollary 1.2 of [15] states that the set Mb := {Q ∈ M : dQ/dQ0 is bounded } is dense in M. It follows that also Mbb := {αQ0 + (1 − α) Q : 0 < α < 1, Q ∈ Mb } is a dense subset of M. Indeed, every Q ∈ Mb can be approximated (in the total variation norm) by the sequence Qn := (1/n)Q0 + (1 − 1/n)Q, n → ∞ and Qn ∈ Mbb by convexity of M. We shall show that Mbb ⊂ Mv . Fix an arbitrary Q ∈ Mb and 0 < α < 1. Let K > 0 be such that dQ/dQ0 ≤ K. We have, by the definition of Vn ,     dQ0 dQ0 dQ dQ Vn α ≥ Un (x0 ) − x0 α . (15) + (1 − α) + (1 − α) dP dP dP dP

9

In Assumption 2.2 choose C1 , C2 , C3 corresponding to λ0 := α and λ1 := α + (1 − α)K. As the random variable α + (1 − α)dQ/dQ0 falls into [λ0 , λ1 ] almost surely, it follows from the chain rule for Radon-Nikodym derivatives that      dQ0 dQ dQ0 dQ Vn α α + (1 − α) = Vn + (1 − α) dP dP dQ0 dP     dQ0 dQ0 + C2 + C3 . (16) ≤ C1 Vn dP dP Assumption 3.2 implies that the sequence C1 Vn (dQ0 /dP ), n ∈ N is uniformly integrable. Adding the fixed integrable random variable C2 (dQ0 /dP ) + C3 doesn’t change this, so the sequence Vn (αdQ0 /dP + (1 − α)dQ/dP ), n ∈ N is bounded from above by a uniformly integrable sequence, by (16). Recalling (15), the assumption on Un (x0 ) and |x0 (αdQ0 /dP + (1 − α)dQ/dP )| ≤ |x0 |[dQ0 /dP + dQ/dP ] we get that that Vn (αdQ0 /dP +(1−α)dQ/dP  ), n ∈ N is bounded from below by dQ 0 a fixed integrable random variable. Thus Vn α dQ is uniformly dP + (1 − α) dP integrable, Mbb ⊂ Mv follows and the Lemma holds.

6.2

Proof of Proposition 4.8

We claim that Assumption 2.1 implies Un′ (x) → ∞, x < x0 ,

Un′ (x) → 0, x > x0 .

(17)

The proof of this fact borrows the argument of Lemma 4 in [6]. First take x < x0 . lim inf Un′ (x) n→∞

=

lim inf Un′ (x0 )e

≥

αe

n→∞ Rx 0

x

Rx

0

x

rn (t)dt

lim inf n→∞ rn (t)dt

,

so Un′ (x) → ∞ as n → ∞, by Assumptions 2.1, 4.6 and the Fatou-lemma and we can conclude the first item of (17). Now take x > x0 . If second item of (17) were not the case, along a subsequence nk , for all k we would have Un′ k (x) ≥ ε > 0. By monotonicity Un′ k (y) ≥ ε, for all y ≤ x, so rn (y) → ∞ implies that −Un′′k (y) ≥ εrnk (y) → ∞, k → ∞. Then necessarily lim inf (−Un′ k (x)) k→∞

= ≥

lim

k→∞

(−Un′ k (x0 ))

−α +

Z

+ lim inf k→∞

Z

x

(−Un′′k (y))dy

x0

x

lim inf (−Un′′k (y))dy.

x0 k→∞

Thus 0 ≤ Un′ k (x) → −∞, k → ∞, a contradiction proving the second assertion of (17). 10

Let In be the inverse of Un′ (which exists by strict concavity of Un ). We claim In (y) → x0 , for y > 0. Indeed, let y > 0. If we had Ink (y) ≥ x0 + ε for some ε > 0 along a subsequence nk then Un′ k (Ink (y)) = y ≤ Un′ k (x0 + ε), but this latter tends to 0 by (17), a contradiction. Hence lim supn→∞ In (y) ≤ x0 . The proof of the reverse inequality for liminf is similar and we get In (y) → x0 . By definition, Vn (y) ≥ Un (x0 ) − x0 y. Since Vn can be calculated as Vn (y) = Un (In (y)) − In (y)y; concavity of Un implies that Vn (y) ≤ Un (x0 ) + (In (y) − x0 )Un′ (x0 ) − In (y)y. Thus 0 ≤ Vn (y) − (Un (x0 ) − x0 y) ≤ (In (y) − x0 )(Un′ (x0 ) − y) → 0, when n → ∞, showing the claim.

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