Ambiguity, optimism, and pessimism in adverse selection models

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ScienceDirect Journal of Economic Theory 171 (2017) 64–100 www.elsevier.com/locate/jet

Ambiguity, optimism, and pessimism in adverse selection models ✩ Raphaël Giraud a , Lionel Thomas b,∗ a LED, Université Paris 8-Vincennes-Saint Denis, France b CRESE, EA 3190, Univ. Bourgogne Franche-Comté, F-25000 Besançon, France

Received 29 March 2016; final version received 23 April 2017; accepted 16 June 2017 Available online 27 June 2017

Abstract We investigate the effect of ambiguity and ambiguity attitude on the shape and properties of the optimal contract in an adverse selection model with a continuum of types, using the NEO-additive model. We show that it necessarily features efficiency and a jump at the top and pooling at the bottom of the distribution. Conditional on the degree of ambiguity, the pooling section may be supplemented by a separating section. As a result, ambiguity adversely affects the principal’s ability to solve the adverse selection problem and therefore the least efficient types benefit from ambiguity with respect to risk. Conversely, ambiguity is detrimental to the most efficient types. © 2017 Elsevier Inc. All rights reserved. JEL classification: D81; D82 Keywords: Adverse selection; Ambiguity; Ambiguity aversion; NEO-additive model; Non-expected utility models; Behavioral economics

✩

We thank two anonymous reviewers and the Editor Marciano Siniscalchi for their careful reading and very useful comments. We thank Christian At, Alain Chateauneuf, François Cochard, Ulrich Horst, David Kelsey, Peter Klibanoff, Michael Mandler, Sujoy Mukerji, Florence Naegelen, Luca Rigotti, Tomasz Strzalecki, and audiences of the Exeter Prize in Decision Theory and Experiments 2014, the Knightian Uncertainty in Strategic Interactions and Markets Workshop at Bielefeld University, and the CRESE seminar at the University of Franche-Comté. * Corresponding author. E-mail addresses: [email protected] (R. Giraud), [email protected] (L. Thomas). http://dx.doi.org/10.1016/j.jet.2017.06.004 0022-0531/© 2017 Elsevier Inc. All rights reserved.

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1. Introduction Motivation. Regulation models with asymmetric information usually assume that the regulator knows the distribution of a natural monopoly’s marginal cost. However, in some cases (Sappington and Weisman, 1996, p. 115), the information gathering process might be conflicting, even when the information providers are not biased. At a broader level, most principal-agent models with adverse selection share the assumption that the principal knows the probability distribution of the agent’s type. In other words, in standard contract theory, the outcome of the information gathering process by the principal is a precise probability distribution. However, it is likely that this process will provide contradicting evidence. In that case, the principal faces a trade-off: • either summarize all the evidence by a single probability distribution, at the cost of omitting all conflicting evidence, however valuable it may be, • or make do with a less precise representation of uncertainty so as to keep as much of the information as possible. It might therefore be wiser to work with an imprecise probability distribution in order to keep more of the available information, even though it is potentially conflicting. In this paper, our aim is to study the impact of an imprecisely known probability distribution on the second-best optimal contract in an adverse selection model with a continuum of types. Poor knowledge of the probability distribution of efficiencies is an instance of the general concept of ambiguity (Ellsberg, 1961). Among the many models addressing it,1 the NEO-additive model (Chateauneuf et al., 2007) has particularly appealing properties. It ties the concepts of ambiguity and ambiguity attitude to two additional parameters, without abandoning the probabilistic approach altogether. Comparison with the expected utility model is therefore made easier in this model. Main findings. We consider a Baron–Myerson’s model where the agent privately knows his type (i.e. his marginal cost), distributed over a real interval. The principal’s beliefs about this parameter are ambiguous and modeled according to the NEO-additive model. With no ambiguity, equivalently in the expected utility model, the contract exhibits the wellknown rent extraction-efficiency trade-off: downward distortions from efficiency are imposed on production (except for the most efficient type) in order to reduce the adverse selection agency cost (i.e. the information rent). Moreover, under usual monotonicity conditions, the contract is smooth and separating. By contrast, the optimal contract under ambiguity exhibits the following properties: there is still no distortion at the top, but a jump at the top and pooling at the bottom. These departures from the expected utility model arise because the presence of ambiguity, combined with ambiguity seeking, or optimism, (resp. ambiguity aversion, or pessimism), leads to the introduction of a mass point for the most (resp. the least) efficient agent. This has two consequences. First, this places a positive weight on the principal’s need to secure efficient production for those types. She is thus tempted to eliminate output distortions at the top and bottom. Second, this increases the agency cost of adverse selection with respect to the expected utility case. The principal is then induced to increase distortions for all types in order to limit the rent of the most efficient type. 1 See Etner et al. (2012) for a survey.

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The two consequences work in opposing directions. The simplest way for the principal to satisfy these opposing effects would be to give up the continuity of the production schedule. More precisely, she would like to implement a downward jump at the top, and an upward jump at the bottom. But, to get a truthful report from the agent, the contract must ensure a production scheme which is non-increasing with efficiency (i.e. the implementability condition). It follows that the former jump does not violate this condition, whereas the latter does. So the jump at the top is optimal, but the jump at the bottom is not. The principal keeps the first jump in the contract and replaces the second by pooling for a whole range of inefficient types. We can conclude that ambiguity and optimism sharpen the rent extraction-efficiency trade-off, whereas ambiguity and pessimism blunt it. Related literature. Combining ambiguity aversion and mechanism design involving adverse selection has already been the topic of a number of papers (De Castro and Yannelis, 2010; Bergemann and Schlag, 2011; Vierø, 2012, 2014; Bodoh-Creed, 2012; Mondello, 2012; Lopomo et al., 2014; Bose and Renou, 2014; Di Tillio et al., 2014; Wolitzky, 2016).2 Except Mondello (2012), none of these papers considers NEO-additive beliefs in the canonical adverse selection model, in which the principal entertains some belief about the agent’s type, all the uncertainty is on the side of the principal, and production is available in multiple units (actually, is divisible). So they are unable to fully address the question of the rent extraction-efficiency trade-off changes. However, two papers are relatively close. In Bergemann and Schlag (2011), a firm sells a single unit of production to a customer with privately known valuation. The trade-off boils down to the identification of a cut-off price, and the implementability condition is not an issue, whereas it is crucial in our paper. With ambiguity this price is lower than without, thus the agent consumes more, whereas in our case the agent may produce less or more than without ambiguity depending on its type. Vierø (2012, 2014)’s setup is similar to ours. But she assumes that both the agent and the principal perceive ambiguity and moreover that they have the same set of priors, which makes it impossible to compare her results to ours. Finally, unlike Mondello (2012), our main contribution is to study the continuous case instead of the two-types case. It turns out that considering intermediary types actually provides new insights: jump at the top and pooling at the bottom cannot be established in Mondello (2012)’s setting. Moreover, Mondello (2012) claims that the optimal contract under ambiguity leads to upward distortions above the efficient production level for the inefficient type. However, this is based on the misassessment that the adverse selection agency cost is negative instead of positive.3 The consequences of the existence of mass points in the distribution of types have already been addressed in the standard adverse selection literature with an expected utility principal by Hellwig (2010) and Lewis and Sappington (1993). In the former, the mass point is given as exogenous. In the latter, it emerges endogenously due to the incentive constraints because the agent may be ignorant about his type. Three differences emerge in our model. First, despite their unsurprising locations, the mass points are the result of the principal’s optimization. They are therefore endogenous. Second, by contrast with Lewis and Sappington (1993), we introduce two mass points that are also endogenously located at the bounds of the support of the distribution, 2 The impact of ambiguity on contract theory has also focused on moral hazard. Pioneering papers are Ghirardato (1994) and Mukerji (1998). Other references on moral hazard or mixed agency problems include Rigotti (1998), Karni (2009), Weinschenk (2010), Gottardi et al. (2012), Lopomo et al. (2011). 3 See his point i in Proposition 1, p. 13 and the proof p. 23–24. Details are available upon request.

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while the mass point is at an interior point in their study. Third, we find that under certain conditions, these mass points can interact with each other, to the point that the contract reduces to the first best at the top and pooling anywhere else. This clearly cannot arise as an optimal choice when there is only a single mass point. Organization of the paper. The model is laid out in section 2. In section 3, the characteristics of the optimal contract under ambiguity are presented. Comparative statics on these characteristics are discussed in the final section. Proofs and additional material are presented in the appendices. 2. The model 2.1. Background A risk-neutral principal derives utility S(q) from the production of a quantity q of a good by a risk-neutral agent. We assume that S is twice continuously differentiable with S  > 0, S  < 0. The principal pays a transfer t to the agent, and thus the principal’s net payoff is v(q, t) = S(q) − t.

(1)

The agent’s privately known marginal cost (or efficiency type) is c ∈ C = [c, c]. Agent c’s net payoff is thus u(q, t) = t − cq.

(2)

Adding (1) and (2), the social surplus is S(q) − cq. It attains its maximum at the first-best level of production given, for all c ∈ C, by q F B (c) = S  −1 (c). It is well known that efficiency is achievable if the principal is able to observe c. Indeed, in that case, she keeps all the surplus because no information rent has to be left to the agent. Let us now study the second-best setting where the principal cannot observe the agent’s efficiency. 2.2. Second-best problem Principal’s beliefs. Let (C) be the set of cumulative distribution functions on C. We assume that the principal believes there exists a distribution F on C, with density f , f > 0 on C, and α ∈ [0, 1),4 such that the true distribution of types lies in the set5 F,α = {G ∈ (C) | G(B)  (1 − α)F (B), for every Borel set B} = {αH + (1 − α)F | H ∈ (C)} = α(C) + (1 − α)F. Clearly, the larger α, the larger that set is. In particular, it can be shown6 that, if G ∈ F,α , 1 then DKL (F || G)  ln 1−α , where DKL (F || G) is the relative entropy or Kullback–Leibler 7 divergence of G from F . 4 We exclude the case α = 1. Indeed, it requires a separate proof, while providing no additional intuition, as the optimal

contract is the limit as α goes to 1 of the optimal contract in the case α < 1.  5 For G ∈ (C), abusing notation we denote G(B) = c 1 dG for every Borel set B ⊆ C. That is, we denote both c B a distribution function and the measure it induces with the same letter. The equality of the two expressions of the set requires a (simple) proof, available upon request. 6 See Appendix section A.1. dF 7 If G ∈  F,α , then F is absolutely continuous with respect to G. Denote dG its density with respect to G, then  c dF DKL (F || G) = c ln dG dF .

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Since the relative entropy is a measure of the divergence of the distribution G from the dis1 . Hence, the larger tribution F , this set is included in a “ball”8 centered on F with ray ln 1−α α is, the larger the ray is. Therefore, α can be interpreted as measuring the degree of ambiguity, and equivalently 1 − α as the degree of confidence the principal has when she estimates the distribution of types using F . We assume that the structure of the principal’s beliefs is common knowledge. This means that the agent knows that the principal believes that the true probability distribution lies in F,α , that the principal knows that the agent knows, etc. Objective function. The principal’s objective function is a convex combination of the optimistic and pessimistic criteria (the best and worst possible expected utilities in F,α ). Assuming that she can offer the direct revelation mechanism t (·), q(·),9 her objective function is therefore c W = β max

G∈F,α

c V (c)dG(c) + (1 − β) min

G∈F,α

c

V (c)dG(c),

(3)

c

where V (c) := v(q(c), t (c)) = S(q(c)) − t (c).

(4)

Thus, β denotes the degree of the principal’s ambiguity loving (or optimism) and 1 − β her degree of ambiguity aversion (or pessimism). As it turns out, given the shape of the set of priors, when V is bounded and reaches its bounds, the objective function can be rewritten (Chateauneuf et al., 2007, Remark 3.2, p. 543) c W = α[β max V (c) + (1 − β) min V (c)] + (1 − α) c∈C

c∈C

V (c)f (c)dc c

= α[β max (S(q(c)) − t (c)) + (1 − β) min (S(q(c)) − t (c))] c∈C

c∈C

(5)

c (S(q(c)) − t (c)) f (c)dc.

+ (1 − α) c

Contract. A contract stems from a communication game in which the principal asks the agent for some message about his type and maps it into a contingent payoff structure. In order to focus without loss of generality on direct revelation mechanisms, we need a version of the Revelation Principle applying to our setting with ambiguity. In the standard Bayesian analysis (Myerson, 1979), it is a simple application of Bayesian Nash equilibrium to the communication game alluded to. In particular, to prove it, one usually assumes that all players: • share the same prior belief as to the probability distribution from which nature will select a type for each player at the beginning of the game, • form their posterior belief regarding the other players’ type given their own type by updating their prior belief using Bayes’ rule. 8 This is not strictly speaking a ball as the relative entropy is not a distance in the mathematical sense. 9 We will discuss this assumption in the next paragraph.

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To extend this approach to our setting, it is thus necessary to come up with the right generalization of the concept of (common) prior, on the one hand, and of Bayesian updating, on the other hand. The crucial point is that these generalizations must affect neither the players’ attitude towards ambiguity, nor the NEO-additive form of their payoff function. This is what we shall now informally discuss.10 In our framework with ambiguity, a “prior” is in fact a set of priors: players believe that types will be selected according to a probability distribution that belongs to a certain set of distributions of the form F,α for some F and α. The corresponding notion of common prior means here that this set is the same for all players. Regarding the updating procedure, the most natural way of extending Bayes’ rule to a set of priors is to update each prior in it using Bayes’ rule, the so-called full Bayesian updating rule.11 As it turns out, if the conditioning event E is such that (1 − α)F (E) > 0, then,12 for a NEO-additive decision-maker, this set is equivalent, from the decision theoretic point of view, to a set of the form FE ,αE , where FE is the Bayesian update of F conditional on E, for some unique αE ∈ [0, 1). In other words, if the decision maker, with the same ambiguity attitude parameter β, replaces the set F,α by its full Bayesian update or by FE ,αE , in (3), the value obtained is the same. Using for E the type of each player thus yields a foundation for interim beliefs. Equipped with these two concepts, it is pretty simple to generalize Bayesian Nash equilibrium in the NEO-additive framework and to mimic the standard proofs of the Revelation Principle. Principal’s problem. Since the agent cannot be forced to accept the contract at a lower utility than its reservation level, normalized to 0, the principal faces, for each c ∈ C, the agent’s participation constraint U (c) := u(q(c), t (c)) = t (c) − cq(c)  0.

(PC)

Moreover, due to the non-observability of efficiency, the principal must offer a contract such that the agent gets a higher net payoff when he reports truthfully. It follows that the incentive constraints are, for all c, cˆ ∈ C: U (c)  t (c) ˆ − cq(c). ˆ

(IC)

The principal’s problem (SBP) is to maximize her objective function (5) with respect to t (.) and q(.), subject to (PC) and (IC). This needs to be reformulated before it can be solved. First, following standard computations in incentives theory, the incentive and participation constraints are respectively for all c ∈ C U  (c) = −q(c),

(IC1)

q(.) non-increasing,

(IC2)

U (c) = 0.

(PC1)

Second, let us, for the sake of exposition, focus on the case where c ∈ arg maxc∈C V (c) and c ∈ arg minc∈C V (c). Since this will actually hold at the optimal contract (see Appendix, Claim 6), it is not restrictive to focus on this case. c Combining (IC1) and (PC1), the agent gets an information rent given by U (c) = c q(ε)dε. Then, using (4) and (PC), we get 10 Technical details are beyond the scope of the present paper but are available upon request. 11 See Fagin and Halpern (1991), Jaffray (1992), and Walley (1991). 12 See Eichberger et al. (2012, Equation (3), p. 250) and Eichberger et al. (2010, Proposition 1, point 3).

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c V (c) = S(q(c)) − cq(c) −

(6)

q(ε)dε. c

Plugging (6) into (5), and using (3), the objective function becomes, after integration by parts ⎛ ⎞ c ⎜ ⎟ W = αβ ⎝S(q(c)) − cq(c) − q(c)dc⎠ + α(1 − β) (S(q(c)) − cq(c)) c

c

+ (1 − α)

 F (c) q(c) f (c)dc. S(q(c)) − cq(c) − f (c)

c

Collecting terms, we obtain, since α < 1

W = αβ S(q(c)) − cq(c) + α(1 − β) (S(q(c)) − cq(c))   c αβ F (c) + 1−α + (1 − α) S(q(c)) − cq(c) − q(c) f (c)dc. f (c)

(7)

c

The second-best problem (SBP’) consists, when c ∈ arg maxc∈C V (c) and c ∈ arg minc∈C V (c), in maximizing (7) with respect to q(.) subject to (IC2). F (c)+

αβ

The term f (c)1−α q(c) appearing in (7) is the agency cost of adverse selection, i.e. the cost of information rent. It can be decomposed into two parts F (c) f (c) q(c); αβ ambiguity: (1−α)f (c) q(c).

• the standard no ambiguity cost • an additional cost due to

The former is the cost incurred by the principal when using the distribution F , even if this distribution is the right one; the latter can be interpreted as the additional cost incurred when it is the wrong distribution. We assume that the cost of information rent satisfies the following property. Assumption 1 (Monotone adjusted hazard rate). For all α ∈ [0, 1) and all β ∈ [0, 1], the hazard rate T (c, α, β) :=

αβ F (c)+ 1−α f (c)

is increasing in c over C.

This is a version of the Monotone Hazard Rate (MHR) used in contract theory. It avoids pooling contracts due to non-monotonicity.13 Preliminary observations. From (IC2), any solution to (SBP’) is differentiable almost everywhere. But, we restrict our investigation to solutions that are piecewise continuously differ-

13 With differentiable f this assumption is equivalent to



 αβ f  (c) F (c)   f (c) > 1−α (f (c))2 . If f  0, this assumption is stronger

than the standard MHR assumption. If f   0, it is implied by it. In general, however, they are independent. The uniform distribution satisfies both.

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entiable,14 while paying special attention to bounds because NEO-additive beliefs in adverse selection problems introduce mass points at c or c whenever αβ > 0 or α(1 − β) > 0. However, when αβ = 0 or α(1 − β) = 0, it will always be possible to find smoother solutions. For instance, when α = 0, replacing a solution q ∗ by q  such that q  (c) = q ∗ (c) for c ∈ / {c, c} and q  (c) > q ∗ (c)  q ∗ (c) > q  (c) would neither change the value of the objective function (since we are modifying q ∗ on a set of F -measure 0) nor the value of the potential rent given (IC1).15 But, if q ∗ is continuous, its replacement by q  seems economically awkward. So we would be tempted to regard q ∗ as optimal, rather than q  . In this spirit, we will focus on production schedules satisfying the following property. Definition 1. A production schedule q is admissible if the associated function V is such that arg maxc∈C V (c) = ∅ and arg minc∈C V (c) = ∅. 3. Properties of the optimal contract 3.1. No ambiguity: α = 0 When there is no ambiguity, the objective function (7) is the usual one in expected utility models. Let us present the optimal production as a benchmark. It is well known (see e.g. Laffont and Martimort, 2002, p. 87) that the optimal production q EU in this case is: 

F (c) q EU (c) = S  −1 c + . (EU) f (c) It reflects a conflict between incentives and efficiency. To ensure a truthful report, the principal must leave an information rent to the agent. Consequently, the production is distorted downward from its efficient level in order to reduce this agency cost. The quantity q EU (c) reflects the standard rent extraction-efficiency trade-off in expected utility models. Three other properties must be noted. First, the most efficient agent, c, has an efficient production. Second, the production schedule is continuous and differentiable everywhere. Third, under Assumption 1, the contract is separating. The robustness of these properties to the introduction of ambiguity must be investigated. 3.2. Ambiguity: 0 < α < 1 3.2.1. General properties of the optimal contract Proposition 1. Let q ∗ be an admissible solution to (SBP). Then, for all β ∈ [0, 1]: (i) No distorsion at the top: q ∗ (c) = q F B (c). (ii) Jump at the top if and only if β > 0: if β > 0, lim q ∗ (c) < q ∗ (c); if β = 0, lim q ∗ (c) = q ∗ (c).

c↓c

c↓c

14 A function g is piecewise continuously differentiable on an interval [a, b] if there exists a finite subdivision a = x < 1 . . . < xi < · · · < xn = b such that g is continuously differentiable (and thus continuous) on each interval (xi , xi+1 ), and both g and g  have left and right limits at each xi (e.g., Muir, Jr., 2015, p. 71). 15 See Hellwig (2010) for a full discussion of this problem in the expected utility case.

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(iii) Pooling at the bottom if and only if β < 1: There exists c∗ ∈ C such that q ∗ (c) = q ∗ (c) for all c ∈ (c∗ , c]. Moreover, c∗ < c if and only if β < 1. Two observations can be made compared to the no ambiguity case: a discontinuity at c and pooling over (c∗ , c] emerge. How can this be explained? Jump at the top. While the absence of distorsion at the top is standard, the jump is not, and can be understood in the following way. The presence of ambiguity (α > 0) and of a certain degree of ambiguity seeking (or optimism) (β > 0) sharpens the standard conflict between efficiency and rent extraction for efficient types. Indeed, it introduces (in the NEO-additive functional) a mass point at c.16 Both the social surplus S(q(c)) − cq(c) and the rent, U (c) are allocated a strictly positive weight. For the principal, the first effect reinforces (with respect to EU) the will to get efficient production at c. The second effect, on the other hand, increases (with respect to EU) the agency cost that is the sum of the standard no ambiguity cost and the additional cost due to ambiguity. This implies the need for a further distortion (with respect to EU) for types c > c in order to limit the information rent of the type c. This exacerbated conflict between efficiency and rent extraction results in a jump. Note that when ambiguity is present but the principal is fully ambiguity averse (pessimistic, β = 0), the jump disappears because the mass point does. Thus it requires some amount of ambiguity seeking or optimism. Pooling at the bottom. The presence of ambiguity and a certain degree of ambiguity aversion or pessimism blunt the conflict between efficiency and rent extraction for inefficient types by introducing (in the NEO-additive functional) a mass point at c. This raises the importance for the principal of the social surplus at c. By contrast, it does not affect the information rent that is null at that point because of the participation constraint (PC1). This effect implies that the principal wishes to eliminate production distorsions at this point. But, because of the agency cost, downward distortions exist for types slightly more efficient than c. A separating contract would violate incentive compatibility (IC2): slightly more efficient types would have an incentive to claim to be the most inefficient type. The principal must therefore give up the separation of inefficient types and offer a pooling contract. The principal gives up important distorsions with respect to EU, so that the conflict between efficiency and rent extraction is blunt for inefficient types. Again, note that if ambiguity is present but there is no ambiguity aversion at all (β = 1), pooling disappears, because the principal no longer cares about efficient production of the least efficient type c. 3.2.2. Full shape of the optimal contract under ambiguity To study the full shape of the optimal contract, we need a lemma and a notation. Lemma 1. For all β ∈ [0, 1), the equation in α (1 − αβ)αβ = (1 − α)f (c)(c − c)

(8)

has a unique solution in (0, 1], that we denote α ∗ (β). Moreover, α ∗ (β) = 1 if and only if β = 0. 16 Note that this is possible here only because the beliefs of the principal are actually nonadditive (or even better put this mass point creates the nonadditivity). Thus the presence of a mass point here is a genuine consequence of ambiguity.

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This lemma allows us to distinguish between low ambiguity whenever 0 < α < α ∗ (β) and high ambiguity whenever α ∗ (β)  α < 1. Low ambiguity: 0 < α < α ∗ (β). Proposition 2. Under low ambiguity, the admissible optimal production, q ∗(c), is given by ⎧  −1 S (c) ⎪ ⎪

 ⎪ αβ ⎪ F (c)+ 1−α ⎨  −1 c + f (c) S q ∗ (c) =

 ⎪ ⎪ αβ ∗ ⎪ ⎪ ⎩S  −1 c∗ + F (c )+∗ 1−α f (c )

if c = c, if c ∈ (c, c∗ ], if c ∈ (c∗ , c],

with c∗ ∈ (c, c] implicitly defined by   ∗ ) + αβ F (c 1−α α(1 − β) c∗ + −c f (c∗ ) 

c

(1 − α) c +

=

F (c) +

αβ 1−α

f (c)

c∗

− c∗ −

F (c∗ ) +

αβ 1−α

f (c∗ )

 f (c)dc

(9)

whenever β < 1 and c∗ = c whenever β = 1.17 Under low ambiguity, the effect of ambiguity on the conflict between efficiency and rent extraction, and particularly its blunting effect for inefficient types, is not strong enough to completely preclude the possibility of screening some of the efficient types. When α < α ∗ (β), the ambiguity is sufficiently low for the conflicts created by the mass points at c and c not to interact which each other. In the interval of types between the jump and the pooling zone, ambiguity implies a usual rent extraction-efficiency trade-off, except that the agency cost is higher than without ambiguity. The contract is separating. The rent extraction-efficiency trade-off implies that production is such (c) that marginal social surplus S  (q(c)) − c equals the sum of marginal information cost Ff (c) and αβ marginal ambiguity cost (1−α)f (c) . The limit between the separating part of the contract and the pooling part is determined by (9). To interpret this equation, let

  −1

q (c) := (S ) s

c+

F (c) +

αβ 1−α



f (c)

which is a decreasing function of c because of Assumption 1 and the fact that (S  )−1 is decreasing. 17 According to the Preliminary observations paragraph in subsection 2.2, when β = 1, any q ∗ (c)  S  −1 (c + T (c, α, β)) can be chosen as this does not affect the ex ante expected payoff of the principal. However, it is thus w.l.o.g. to choose q ∗ (c) = S  −1 (c + T (c, α, β)).

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Thus, (9) can be rewritten 



∗



α(1 − β) S (q (c )) − S (q s

FB

 c

(c) = (1 − α) S  (q s (c)) − S  (q s (c∗ )) f (c)dc.

(10)

c∗

Let c∗∗ be such that q s (c∗∗ ) = q F B (c) whenever this equation has a solution on C,18 and let = c otherwise. Equation (9) tells us that the contract is separating on [c∗∗ , c∗ ] and pooling on [c∗ , c]. How is c∗ defined though? Let us examine the LHS of (10), then the RHS. Since c∗ ∈ [c∗∗ , c] and q s is decreasing, we have q s (c∗ )  q s (c∗∗ )  q F B (c).19 Thus, since  S is decreasing, the LHS is nonnegative. More precisely, the LHS is null if c∗ = c∗∗ , positive if c∗ > c∗∗ , and increases as c∗ moves away from c∗∗ . So the more separating the contract is, the higher the LHS is. The LHS is thus the cost of separating on [c∗∗ , c∗ ], i.e. the cost of not having efficient production at c. The RHS is nonnegative because q s and S  are decreasing. More precisely, the RHS is null if ∗ c = c, strictly positive if c∗ < c, and increases as c∗ moves away from c. So the more pooling the contract is, the higher the RHS is. The RHS is thus the cost of pooling on [c∗, c], i.e. the cost of giving up producing q s (c) over this interval. Specifically, at c, the decrease in marginal utility suffered by the principal if she pools is S  (q s (c)) − S  (q s (c∗ )). Since the cost of separating on [c∗∗ , c∗ ] is increasing in c∗ and the cost of pooling on [c∗ , c] is decreasing in c∗ , there is a tradeoff between these two costs. Equation (9) tells us that c∗ is the result of resolving this conflict by equating these costs. c∗∗

High ambiguity: α ∗ (β)  α < 1. Proposition 3. For all β > 0 and α such that α ∗ (β)  α < 1,20 the optimal production, q ∗ (c) is given by  ∗

q (c) =

S  −1 (c)  S  −1

c+



αβ 1−αβ (c − c)

if c = c, if c ∈ (c, c].

(11)

Compared with low ambiguity, optimal production is still characterized by a jump and pooling, but there is no longer separation. In this case, the ambiguity is high enough for the conflicts generated by the mass points at c and c to interact with each other. In other words, ambiguity creates an overall conflict between optimism, pessimism, and incentives. Because of the presence of optimism (resp. pessimism), there is a jump (resp. pooling). But, a high level of ambiguity would imply a downward jump at c at least as large as the efficient production gap between the most and the least efficient agents. Note that the principal behaves under high ambiguity as she would if she considered only two types and maximized expected utility: the production corresponds to the standard expected utility model with two types, c and c, with probabilities αβ and 1 − αβ. αβ 18 That is, whenever q s (c)  q F B (c), i.e. c − c  (1−α)f (c) . 19 This is obvious if q s (c∗∗ ) = q F B (c); if c∗∗ = c, then q s (c∗∗ ) = q s (c) < q F B (c) by definition of c∗∗ . 20 Note that under high ambiguity, β cannot be null since α ∗ (0) = 1.

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3.3. Shutdown Given the properties of the optimal contract and Assumption 1, it is easy to check that the integrand in (7) is decreasing. Therefore, it can be optimal for the principal to allow for some shutdown of inefficient types. Let c∗ be the threshold type such that all types c > c∗ are excluded from the contract by the principal. The objective function becomes

W = αβ S(q(c)) − cq(c) + α(1 − β) (S(q(c∗ )) − c∗ q(c∗ ))   c∗ αβ F (c) + 1−α q(c) f (c)dc. (12) + (1 − α) S(q(c)) − cq(c) − f (c) c

We state the following proposition. Proposition 4. The threshold c∗ is implicitly defined by   αβ F (c∗ ) + 1−α ∗ ∗ ∗ (1 − α) S(q (c∗ )) − c∗ q (c∗ ) − q (c∗ ) f (c∗ ) = α(1 − β)q ∗ (c∗ ). f (c∗ )

(13)

The marginal benefit from c∗ corresponds to the payoff it provides to the principal. This is the LHS of (13). But the presence of ambiguity implies that the principal would like this type to produce an efficient quantity. This entails that she must incur the marginal cost with respect to efficiency of this production. This corresponds to the RHS of (13). It is interesting to note that (13) can be rewritten as 

F (c∗ ) ∗ α ∗ ∗ q (c∗ ) f (c∗ ) = q ∗ (c∗ ) > 0. S(q (c∗ )) − c∗ q (c∗ ) − f (c∗ ) 1−α Thus, the principal excludes more types than with expected utility (where the RHS is 0). 3.4. An example To illustrate the results and summarize them in a graph, consider the following specification. √ We let c = 1, c = 2, S(q) = q and F (c) = c − 1, the uniform distribution on C = [1, 2]. Then, the first best contract is given by 1 , 4c2 while the optimal second-best contract under expected utility is given by q F B (c) =

q EU (c) =

1 . 4(2c − 1)2

The optimal second-best production under ambiguity is given by, in the case of low ambiguity ⎧ 1 ⎪ if c = c ⎪ ⎨4 (1−α)2 ∗ if c ∈ (c, c∗ ] q (c) = 4[(2c−1)(1−α)+αβ]2 ⎪ 2 ⎪ (1−α)√ ⎩ if c ∈ (c∗ , c], 4[3−α(1+β)−2 α(1−β)]2

where

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Fig. 1. The optimal contract for β = .1 and β = .9 and various values of α (subscripted: for instance in the case β = .1, ∗ and c∗ denote the optimal contract for α = .5). q.5 .5

√ 2 − α(1 + β) − α(1 − β) c = . 1−α ∗

Moreover, low ambiguity arises when  1 + β − 1 + 2β − 3β 2 ∗ α < α (β) = . 2β 2 In the case of high ambiguity, optimal production is such that 1 if c = c, ∗ q (c) = 4(1−αβ)2 if c ∈ (c, c]. 4[2−αβ]2 Fig. 1 shows the shape of the optimal contract under different configurations of ambiguity and ambiguity attitude. 4. Comparative statics Ambiguity and ambiguity attitude have two consequences for the objective function (7). They both modify the agency cost and the weights assigned to its three components. They imply: 1) qualitative effects, through the trade-off between separating and pooling, 2) quantitative effects, through the production distortions, 3) allocative effects, through the payoffs secured by the principal and the agent. These are studied in turn. Qualitative effects. Notice that the trade-off between separating and pooling is relevant only for low ambiguity, since in this case, c∗ ∈ (c, c). Proposition 5. Under Assumption 1, when 0 < α < α ∗ (β): • c∗ is a decreasing function of α, • there exists a unique βˆ ∈ [0, 1] such that c∗ is a decreasing function of β if β < βˆ and an ˆ increasing function of β if β > β. Recall that c∗ is such that the cost of separating below c∗ is equal to the cost of pooling above c∗ (cf. equation (10)). The effects of α and β on these two costs are the following.

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On the cost of separating side, a rise in α or β has two different effects. On the one hand, it leads to a decrease in the level of production q s (c) for all c, in particular at c∗ , because the agency cost is increased. This raises the cost of separating because production is further away from the first best at c. On the other hand, α raises the weight attached to getting efficient production from the least efficient agent, whereas β decreases it. So α reinforces the former effect on the cost of separating; the opposite is true for β. Taken together, these two effects lead to an increase in the cost of separating in the case of α, and a U-shaped effect in the case of β. On the cost of pooling side, a rise in α and β reduces q s (c∗ ), but also q s (c) for all c  c∗ because the agency cost is raised. Thus, the effect on the difference between these two levels, which contributes to the cost of pooling, is not clear-cut. Moreover, when α increases, the cost of pooling falls because the principal’s confidence in the distribution F is reduced. By contrast, β is neutral. Ultimately, α tends to reduce the cost of pooling while β has an unclear effect. Therefore, the overall effect on c∗ is the following. An increase in ambiguity lowers c∗ because it increases the cost of separating and reduces the cost of pooling. The pooling zone is thus increased.21 By contrast, a decrease in ambiguity aversion (an increase in optimism) does not have an unequivocal overall effect on c∗ . As a result, the c∗ function is a U-shaped function of β. Quantitative effects. Two aspects are examined: (1) the size of the jump (2) the level of the constant production on the pooling zone. This is done in the next three propositions. The first two are devoted to the case of low ambiguity, the last one to the case of high ambiguity. When ambiguity is low, recall that the size of the jump is given by q F B (c) − q s (c), and the constant production q¯ is such that, using (11),   αβ F (c∗ ) + 1−α ∗ ∗  −1 ∗ q¯ = q (c ) = (S ) . c + f (c∗ ) Proposition 6. When 0 < α < α ∗ (β), the size of the jump at the top is increasing in α and β. When α and β rise, q s (c) decreases for all c because the agency cost increases, in particular at c, whereas q F B (c) remains unchanged. So the jump increases. As mentioned above, ambiguity and ambiguity seeking (optimism) sharpen the rent extraction efficiency trade-off at the top. Thus, when the corresponding parameters increase, the conflict is heightened and the jump increases. Proposition 7. Let q¯ be the production of inefficient types when 0 < α < α ∗ (β). Then • •

∂ q¯ ∂α ∂ q¯ ∂α ∂ q¯ ∂β

> 0 if and only if F (c∗ ) > β. Equivalently, for all β there exists αˆ ∈ (0, α ∗ (β)] such that ∂ q¯  0 on (0, α) ˆ and ∂α  0 on (α, ˆ α ∗ (β)]. < 0.

As α or β increase, the agency cost rises, ceteris paribus. Therefore, constant production tends to be reduced. However, as previously noted in Proposition 5, a rise in α or β affects the trade-off between separating and pooling at the bottom of the distribution, and therefore the size of the pooling zone through the value of c∗ . In the case of a rise in α, c∗ falls, which tends to 21 This is in line with other similar results in the literature, for instance the fact that ambiguity aversion leads to absence of trade under certain circumstances (Dow and da Costa Werlang, 1992).

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increase constant production. The overall effect of ambiguity on constant production thus results in an approximately inverted U-shaped curve: the agency cost effect trumps the effect on c∗ for small values of α and is trumped by it for large values. In the case of a rise in β, c∗ can rise or fall. However, some effects offset each other, resulting in an overall decrease in constant production. Let us analyze the case of high ambiguity. Using Proposition 3, the jump is equal to  αβ F B  −1 q (c) − q, ¯ with q¯ = S c + 1−αβ (c − c) for all c ∈ (c, c]. In this case, the size of the jump and the constant production covary in opposite directions. We have the following proposition. Proposition 8. Let q¯ be the production of all types except c when α ∗ (β)  α < 1. Then and ∂∂βq¯ < 0.

∂ q¯ ∂α

<0

As in the preceding proposition, a rise in α and β increases the agency cost. So this reduces the constant production. This corresponds also to the overall effect now because, unlike in the ¯ As mentioned, the jump varies in previous proposition, c∗ no longer contributes to defining q. the opposite direction and thus increases when ambiguity and ambiguity seeking increase. Allocative effects. The way the size of the jump and the quantity produced by inefficient types in the optimal contract vary with ambiguity and ambiguity attitude suggests, given the connection between quantity produced and rent, that it is worth studying the effect of ambiguity and ambiguity aversion on the rent of the agent and the surplus of the principal. c The agent’s rent. Let U ∗ (c) = c q ∗ (ε)dε be the rent of an agent of type c ∈ C. We then have the following proposition. Proposition 9. • For all α ∈ (0, 1),

∂U ∗ (c) ∂β

< 0 for all c ∈ (c, c].

< α ∗ (β),

• If 0 < α then there exists K > 0 such that ∗)  1 + K. if F (c β • If α ∗ (β)  α < 1,

∂U ∗ (c) ∂α

∂U ∗ (c) ∂α

> 0 for all c ∈ (c, c] if and only

< 0 for all c ∈ (c, c].

First, all agents benefit from the fact that the principal is ambiguity averse, independently of the level of ambiguity she perceives. Indeed, ambiguity aversion leads to the principal being more attentive to the objective of getting the least efficient type to produce the first best quantity and raises the agency cost. Both effects work in the direction of raising the production of the least efficient types, which in terms of rent also benefits the most efficient types. From a more psychological point of view, a more ambiguity averse principal will prefer less ambiguous, i.e. “flatter” contracts, to more variable ones, which also benefits all agents. Second, this proposition shows that, when the principal perceives a low ambiguity, more ambiguity favors all the agents as long as the principal is sufficiently ambiguity averse. Indeed, more ambiguity will lead the principal to prefer even more unambiguous contracts if she is sufficiently ambiguity averse. Conversely, what this proposition says is that if she is not sufficiently ambiguity averse, then some agents, at the top of the distribution, i.e. efficient ones, might suffer from an increase in ambiguity, while the agents at the bottom of the distribution, inefficient ones, might still benefit from it. Finally, when the principal perceives a lot of ambiguity, we know from Proposition 3 that she behaves as if there were only two types and the probability of the efficient type c were αβ, and

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the probability of the inefficient type(s) 1 − αβ. Thus rises in α (more ambiguity perceived) or β (less ambiguity aversion) lead to a decrease in the latter probability, and as a consequence, the principal will allocate a lower rent to the inefficient type, since the objective of productive efficiency of the less efficient type is less pressing. Social surplus and principal’s payoff. Using (6), recall that V ∗ (c) = S(q ∗ (c)) − cq ∗ (c)) − c ∗ c q (ε)dε is the principal’s payoff when she actually faces an agent of type c ∈ C. The effects of ambiguity and of her ambiguity attitude on this payoff are not as clear cut as the effects they have on the agent. However, since they affect the total surplus V ∗ (c) + U ∗ (c), we can draw the consequences from this and Proposition 9. Proposition 10. • For all α ∈ (0, 1), • If 0 < α < α ∗ (β), • If α ∗ (β)  α < 1,

∂V ∗ (c)+U ∗ (c) ∂β ∂V ∗ (c)+U ∗ (c) ∂α ∂V ∗ (c)+U ∗ (c) ∂α

< 0 for all c ∈ (c, c]. < 0 for all c ∈ (c, c] if and only if F (c∗ ) < β. < 0 for all c ∈ (c, c].

This proposition shows that ambiguity aversion increases the size of the total surplus. We know from Proposition 9 that ambiguity aversion raises the agent’s rent, and this proposition shows that, it may or may not lower the principal’s payoff, but in any case the size of the rent increase is always sufficient to compensate for it. As far as ambiguity is concerned, let us consider first the case of high ambiguity. In this case, more ambiguity entails a decrease in total surplus. Therefore, since we know from Proposition 9 that the rent of the inefficient agents in this case decreases with ambiguity, we see that, again, even though the principal might gain from ambiguity, her gain can never exceed the absolute value of the loss incurred by the agent. Consider now the case of low ambiguity. In this case, the total surplus increases with ambiguity, if and only if β < F (c∗ ), i.e. if and only if the principal is sufficiently ambiguity averse. ∗ In that case, let K be the constant identified in Proposition 9 such that ∂U∂α(c) > 0 if and only if ∗ F (c ) β > 1 + K. Then, we have the following corollary. Corollary 1. If 0 < α < α ∗ (β) and 1 <

F (c∗ ) β

< 1 + K, then

∂V ∗ (c) ∂α

> 0 for all c ∈ (c, c].

This corollary identifies a sufficient condition for which the principal’s payoff might increase with ambiguity. She should be ambiguity averse but not too much so. 5. Conclusion The optimal contract in an adverse selection model with a continuum of types and a parametric model of ambiguity and ambiguity aversion, namely the NEO-additive model, necessarily involves efficiency and a jump at the top of the distribution and pooling at the bottom of the distribution. As a result, ambiguity adversely affects the principal’s ability to solve the adverse selection problem and therefore, if the principal is not very ambiguity averse, the least efficient types benefit from ambiguity with respect to risk, while ambiguity is detrimental to the most efficient types.

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Appendix A. Appendices A.1. Result on the Kullback–Leibler divergence c G(B)  (1 − α)F (B) ∀B ⇔

c 1B dG  (1 − α)

1B dF ∀B

c

c

c

c

⇔

1B dG  (1 − α) c

dF dG ∀B dG

c



c ⇔

1B

1B

 dF dG  0 ∀B. 1 − (1 − α) dG

c

This implies



 dF 1 dF 1 dF 1 − (1 − α)  0 G-a.s. ⇒  G-a.s. ⇒ ln  ln G-a.s. dG 1 − α dG 1−α dG



 1 dF ⇒ ln  ln F -a.s., 1−α dG      c 1 dF and thus: ln 1−α  c ln dG dF = DKL (F || G). A.2. Proof of Propositions 1–3 A.2.1. Proof strategy Since the objective function (5) is in principle neither concave nor convex, we cannot solve (SBP) using standard methods. However, for each location of the argmax and argmin of V , we obtain, rewriting the objective function along the line of (7), an instance of a family of concave control problems, denoted [P1 ]. Our proof strategy is therefore to find the admissible solution to this family (section A.2.2) and to show that for this solution, V is non-increasing. This implies that (SBP) is indeed equivalent to (SBP’), and we can thus leverage the analysis of section A.2.2 to solve it. A.2.2. Preliminary optimal control results In this section, we shall state and solve [P1 ]. Statement. Let • c1 , c2 ∈ C, with c  c1 < c2  c, • λ 1 , λ2 ∈ R + , • T1 , T2 , T3 be three functions from C to R+ , nondecreasing and of class C 1 , with T1 (c) = 0, Ti (c) > 0 for all i ∈ {1, 2, 3} and c > c, and T1 (c1 )  T2 (c1 ), T2 (c2 )  T3 (c2 ).

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Consider the following optimal control problem c1 max W (q(.), c1 , c2 ) = (S(q(c)) − cq(c) − T1 (c)q(c))f (c) dc q(.)

c

+ λ1 (S(q(c1 )) − c1 q(c1 )) c2 + (S(q(c)) − cq(c) − T2 (c)q(c))f (c) dc + λ2 (S(q(c2 )) − c2 q(c2 )) c1

c (S(q(c)) − cq(c) − T3 (c)q(c))f (c) dc

+

[P1 ]

c2

s.t.

(I C2),

q(c) free ,

q(c) free.

Results. In this paragraph, we state and prove two propositions associated with [P1 ]. Propositions 11 and 12 solve [P1 ] respectively when (c < c1 and c2 < c) and (c = c1 and c2 = c). However, Proposition 11 extends to the case c2 = c because its point (v) g) holds then vacuously and the rest of its proof remains unchanged. Assuming c2 < c just makes for a more fluid and homogeneous proof. As for the case (c1 = c and c2 < c), it is covered by replacing point (iv) c) in Proposition 12 by point (v) g) of Proposition 11. Again, this would not affect the rest of the proof. To begin, let c2 ψ(c) := λ2 (c + T2 (c) − c2 ) + (F (c2 ) − F (c))(c + T2 (c)) − (ε + T2 (ε))f (ε) dε.

(14)

c

The following proposition presents the properties of the optimal production in problem [P1 ] when c < c1 and c2 < c. Proposition 11. When c < c1 and c2 < c, there exists c1∗ ∈ (c, c1 ], c2∗ ∈ [c1 , c2 ] and q¯2 ∈ R+ such that: (i) c1∗ < c1 if and only if either λ1 > 0 or λ1 = 0 and ψ(c1 ) > 0; (ii) c2∗ < c2 if and only if λ2 > 0; (iii) c2∗ > c1 if and only if ψ(c1 ) < 0; (iv) c1 (λ2 + F (c2 ) − F (c1 ))S (q¯2 ) = (c + T1 (c))f (c) dc + λ1 c1 

c1∗

c2 + (c + T2 (c))f (c) dc + λ2 c2 c1

− (λ1 + F (c1∗ ) − F (c1 ))(c1∗ + T1 (c1∗ )); (v) any admissible solution q ∗ to problem [P1 ] has the following properties:

(15)

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a) q ∗ (c) = q F B (c); b)  (S  )−1 (c + T1 (c)) ∗ q (c) = (S  )−1 (c1∗ + T1 (c1∗ ))

if c < c < c1∗ , if c1∗  c < c1 ;

c) q ∗ (c1 ) = (S  )−1 (c1∗ + T1 (c1∗ )) if c1∗ < c1 , otherwise, q ∗ (c1 )  (S  )−1 (c1 + T1 (c1 )); d) if c2∗ > c1  (S  )−1 (c + T2 (c)) if c1 < c < c2∗ , ∗ q (c) = (S  )−1 (c2∗ + T2 (c2∗ )) if c2∗  c < c2 ; e) q ∗ (c) = q¯2 if c2∗ = c1 < c  c2 ; f) q ∗ (c2 ) = (S  )−1 (c2∗ + T2 (c2∗ )) if c1 < c2∗ < c2 , and q ∗ (c2 )  (S  )−1 (c2 + T2 (c2 )), if c1 < c2∗ = c2 ; g) q ∗ (c) = (S  )−1 (c + T3 (c)) if c2 < c < c and q ∗ (c)  (S  )−1 (c + T3 (c)); h) if ci∗ = ci for i = 1 or i = 2, any value q ∗ (ci ) compatible with (IC2) can be chosen. Proof. We proceed in seven steps. Step 1: Necessary and sufficient conditions of optimality. Problem [P1 ] is a multi-stage control problem. For notational convenience, let c0 := c and c3 := c. Let, for i ∈ {1, 2, 3}, Ci := (ci−1 , ci ) and Ci = [ci−1 , ci ]. Let N ⊂ C be the (possibly empty) finite set of points where (q ∗ ) is not continuous. Let d Ci := Ci  N and C d = C1d ∪ C2d ∪ C3d . We are looking for a piecewise C 1 solution q ∗ . Let y ∗ : C → R− be the control such that ∗ (q ) = y ∗ whenever this applies. For each i ∈ {1, 2, 3}, let μi : C i → R be the absolutely continuous costate variable associated with state q on C i . On C i , the Hamiltonian Hi is defined by Hi (c) = (S(q(c)) − cq(c) − Ti (c)q(c)) f (c) + μi (c)y(c).

(16)

From now on, to make equations lighter, we abuse notation and drop the ∗ from optimal variables. All variables, unless otherwise specified, are to be considered as optimal. So, from the maximum principle (see Amit, 1986; Makris, 2001), we know that, for all i ∈ {1, 2, 3}, the optimal q, μi , and y satisfy, for all c ∈ Ci ∂Hi (c) = μi (c)  0, ∂y

y(c)

∂Hi (c) = μi (c)y(c) = 0, ∂y

(17)

and, for all c ∈ Cid

∂Hi (c) = − S  (q(c)) − c − Ti (c) f (c) ∂q

= f (c) c + Ti (c) − S  (q(c)) .

μi (c) = −

(18)

Transversality conditions are μ1 (c) = 0, μ3 (c) = 0.

(19)

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Finally, transition conditions are

μ1 (c1 ) − λ1 S  (q(c1 )) − c1 = μ2 (c1 ),

μ2 (c2 ) − λ2 S  (q(c2 )) − c2 = μ3 (c2 ).

83

(20) (21)

Let us formulate three important remarks for the following analysis. First, (20) and (21) can be rewritten, for all i ∈ {1, 2}, as

(22) μi (ci ) − μi+1 (ci ) = λi S  (q(ci )) − ci , so that ⎧ FB ⎪ ⎨μi (ci ) − μi+1 (ci ) > 0 ⇔ λi > 0 and q(ci ) < q (ci ), μi (ci ) − μi+1 (ci ) = 0 ⇔ λi = 0 or q(ci ) = q F B (ci ), ⎪ ⎩ μi (ci ) − μi+1 (ci ) < 0 ⇔ λi > 0 and q(ci ) > q F B (ci ).

(23)

Second, for all c ∈ C, let

and

qis (c) := S  −1 (c + Ti (c)) ,

(24)

⎧ s ⎪ ⎨q1 (c) s q (c) := q2s (c) ⎪ ⎩ s q3 (c)

(25)

on [c, c1 ] on (c1 , c2 ] on (c2 , c]

Then, combining (18), (24), and (25), we observe that for all c ∈ C d μi (c)  0 ⇔ q(c)  qis (c).

(26)

Third, for every i, the Hamiltonian Hi being linear in y, it is concave in q since S  < 0. Moreover, scrap values are also concave in the state variable because S  < 0. So conditions (17)–(21) are also sufficient (see Seierstad and Sydsaeter, 1987, Theorem 6, p. 186). We will examine first the path followed by the solution(s) on each set Ci . We will then examine their values at the boundaries. Before proceeding however, let us introduce a few notations and simple preliminary results, the straightforward proof of which is omitted. Step 2: Simple preliminary results. Lemma 2. (i) q F B is differentiable on C and (q F B ) (c) < 0 for all c ∈ C. (ii) q s is differentiable on Ci and (q s ) (c) < 0 for all c ∈ Ci . Moreover, q s is subject to downward jumps, at ci , if Ti (ci ) < Ti+1 (ci ). In other words, q s is decreasing. Lemma 3. Under Assumption 1, ψ is increasing on [c, c2 ]. To begin with, let us study the path followed on each set Ci . We will first examine the path of μi , and then deduce from it the path of q. Step 3: Properties of μi .

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Lemma 4. For all i ∈ {1, 2, 3}, if μi is constant on some interval I ⊂ Ci , then μi (c) = 0 for all c ∈ I . Moreover, in this case, q = qis on I . Proof. Assume μi is constant on I . Then, for all c ∈ I ∩ Cid , μi (c) = 0, and thus by (26) q(c) = qis (c). Therefore, by Lemma 2, q  (c) < 0 for all c ∈ I ∩ Cid , and thus by (17) μi (c) = 0 for all c ∈ I ∩ Cid . Since μi is continuous on I and N is finite, this implies μi (c) = 0 for all c ∈ I . 2 Condition (18) holds at every point of C d . Hence, since q is continuous on every interval of C d , this implies that μi exists and is continuous on every interval of Cid . ˆ > 0 for some cˆ ∈ Cid , i ∈ {1, 2, 3}, then μi (c) > 0 for all c ∈ [c, ˆ ci ) ∩ Cid and Lemma 5. If μi (c) μi (c) > 0 for all c ∈ [c, ˆ ci ]. ˆ > 0 for some cˆ ∈ Cid , i ∈ {1, 2, 3}. Since Cid is open as a finite union of Proof. Assume μi (c) open intervals and μi is continuous at c, ˆ μi > 0 on an open neighborhood of cˆ in Cid . Since μi  0, we may therefore w.l.o.g. assume that μi (c) ˆ > 0. Now, let c˘1 ∈ (c, ˆ ci ]. Since μi is continuous over the compact interval [c, ˆ c˘1 ], it has a global maximum there, reached at c˘2 . Since μi (c) ˆ > 0, μi (c˘2 )  μi (c) ˆ > 0. Assume that c˘2 ∈ (c, ˆ c˘1 ). Then, there exists an open interval (c˘2 − ε, c˘2 + ε) such that μi > 0 over this interval. By (17), this implies that q is constant over this interval. On the other hand, this also implies, by Lemma 4, that μi is not always 0 over this interval. Therefore, since μi is piecewise C 1 , there exist c˘3 ∈ (c˘2 − ε, c˘2 ) ∩ Cid and c˘4 ∈ (c˘2 , c˘2 + ε) ∩ Cid such that μi (c˘3 ) > 0 and μi (c˘4 ) < 0. Thus, q(c˘3 ) > q s (c˘3 ) > q s (c˘4 ) > q(c˘4 ) and q is not constant on (c˘2 − ε, c˘2 + ε), which is a contradiction. Therefore, since μi (c) ˆ > 0, c˘2 = c˘1 . Since c˘1 was chosen arbitrarily in (c, ˆ ci ], this implies that μi is positive and strictly increasing on [c, ˆ ci ], proving the lemma. 2 Now define Mi := {c ∈ Ci | μi (c) > 0}. Step 4: Properties of Mi . By convention, we set λ3 equal to 0. For i ∈ {1, 2, 3}, Mi is bounded below by ci−1 . Therefore, if it is nonempty, it has a lower bound, that we denote ci∗ ; if Mi = ∅ by convention we set ci∗ = ci . Thus ci∗ < ci if and only if Mi = ∅. Lemma 6. At the optimal contract, we have: (i) (ii) (iii) (iv)

if μi (ci−1 ) = 0, then Mi = (ci∗ , ci ); for every i ∈ {1, 2}, if λi > 0 then ci∗ < ci ; for every i ∈ {1, 2}, if λi = 0, then Mi = ∅ if and only if μi+1 (ci ) = 0; M3 = ∅ if and only if μ3 (c2 ) = 0.

Proof. (i): Let us start by showing that μi (ci∗ ) = 0 when ci∗ > ci−1 . If μi (ci∗ ) > 0, then since μi is continuous on Ci , if ci−1 < ci∗ , one can find ε > 0 such that ci∗ − ε ∈ Ci and μi (ci∗ − ε) > 0, so that ci∗ is not the lower bound. Thus μi (ci∗ ) = 0 if ci∗ > ci−1 . If μi (ci−1 ) = 0, therefore, μi (ci∗ ) = 0 whether ci∗ > ci−1 or not. Then, let c˘ ∈ Mi be such that ∗ (ci , c) ˘ ⊆ Cid . It exists because Mi is open, ci∗ is its lower bound and μi is piecewise C 1 . Then, there exists cˆ ∈ (ci∗ , c) ˘ such that (μi ) (c) ˆ =

∗ μi (c)−μ ˘ i (ci ) c−c ˘ i∗

=

μi (c) ˘ c−c ˘ i∗

> 0. By Lemma 5, this implies

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that μi (c) > 0 for all c ∈ (c, ˆ ci ]. Since this is true for every c˘ ∈ Mi such that (ci∗ , c) ˘ ⊆ Cid , μi (c) > 0 for all c ∈ (ci∗ , ci ), i.e. Mi = (ci∗ , ci ). (ii): Assume λi > 0 for some i ∈ {1, 2}. Let us prove that Mi = ∅. Assume μi (c) = 0 for all c ∈ Ci . Then, on the one hand, by continuity μi (ci ) = 0 and, on the other hand, μi (c) = 0 for all c ∈ Ci , since Ci is a nonempty open interval. The latter implies q(c) = qis (c) on Cid by Lemma 4. Since the set of points of possible discontinuities of q is finite and q is nonincreasing, at any such point22 k, qis (k) = q− (k)  q(k)  q+ (k) = qis (k), therefore, q = qis on the whole Ci . In particular, by (IC2): q(ci )  q− (ci ) = qis (ci ) < q F B (ci ), since ci > c.

Now, (22) implies that −λi S  (q(ci )) − ci = μi (ci ) − λi S  (q(ci )) − ci = μi+1 (ci )  0, thus, since λi > 0, q(ci )  q F B (ci ), a contradiction. This proves that Mi = ∅. (iii): Assume λi = 0 for some i ∈ {1, 2}. If Mi = ∅ then μi (c) = 0 for all c ∈ Ci , therefore μi (ci ) = 0 by continuity. But, since λi = 0, (23) implies that μi+1 (ci ) = 0. Conversely, assume μi+1 (ci ) = 0. Appealing to Lemma 5, there cannot exist cˆ ∈ Ci∗ such that μi (c) ˆ > 0, otherwise μi (c) > 0 for all c ∈ [c, ˆ ci ] and, since λi = 0, by (23) this implies that μi+1 (ci ) > 0, a contradiction. Therefore, μi (c)  0 for all c ∈ Cid . But since λi = 0, μi+1 (ci ) = 0 and (23) imply μi (ci ) = 0, and since μi  0 and is continuous on Ci , it follows that μi (c) = 0 for all c ∈ Ci , i.e. Mi = ∅. (iv): If M3 = ∅, then by continuity μ3 (c2 ) = 0. Conversely, if μ3 (c2 ) = 0 and M3 = ∅ then repeating the argument in the proof of point (i), M3 = (c3∗ , c3 ), and, by Lemma 5, μ3 is increasing on M3 , hence by Lemma 5 and continuity μ3 (c) > 0, a contradiction with transversality condition (19). 2 Lemma 7. For all i ∈ {1, 2, 3}, if Mi = (ci∗ , ci ), the optimal production q on Ci is constant on (ci∗ , ci ], and such that q(c)  q s (ci∗ ) for all c ∈ (ci∗ , ci ]. Proof. Let i ∈ {1, 2, 3} be such that Mi = (ci∗ , ci ). Mi = ∅ if and only if ci∗ = ci if and only if (ci∗ , ci ] = ∅. Assume therefore that Mi = ∅. From the proof of Lemmas 5 and 6, we know that μi is increasing on Mi , thus μi > 0 on (ci∗ , ci ] by continuity. Thus by (17), q is constant on (ci∗ , ci ]. Denote q¯i its value on Mi . Since, therefore, q  exists and is continuous on (ci∗ , ci ], (18) applies. Since μi is increasing on this interval, μi > 0 on (ci∗ , ci ], so that, by (18), for all c ∈ Mi , q s (c) < q¯i . As c → ci∗ , therefore, this implies q s (ci∗ )  q¯i . 2 ∗ ,c ∗ Lemma 8. For all i ∈ {2, 3}, if Mi−1 = (ci−1 i−1 ) with ci−1 < ci−1 and μi (ci−1 ) > 0, then Mi = (ci−1 , ci ).

Proof. If μi (ci−1 ) > 0, then ci∗ = ci−1 . Let ci = inf{c ∈ Ci | μi (c) = 0}. By construction and continuity of μi , ci−1 < ci . Assume ci < ci . Then, μi > 0 on [ci−1 , ci ), therefore q is constant on this interval by (17). Moreover, there exists ci ∈ (ci−1 , ci ) such that μi (ci ) < 0 and therefore by (18) (that applies on (ci−1 , ci ) since q is constant there) q(ci ) < qis (ci ). On the other hand, ∗ ,c ∗ ∗ if Mi−1 = (ci−1 i−1 ) and ci−1 < ci−1 , by (17) again, q is also constant on (ci−1 , ci−1 ] and ∗  ∗  since (ci−1 , ci−1 ] ∩ [ci−1 , ci ) = ∅, q is constant on (ci−1 , ci ). By Lemma 7, therefore, q(c)  s (c∗ ) on (c∗ , c ). But, using Lemma 2, q s (c ) < q s (c s s ∗ qi−1 i i i i−1 )  qi−1 (ci−1 ) < qi−1 (ci−1 )  i−1 i−1 i   ∗ q(ci ), a contradiction. Therefore, ci = ci , hence Mi = (ci−1 , ci ) = (ci , ci ). 2 22 For any function h defined on some open subset I of C, and c˜ in the closure of I , we denote h (c) ˜ the − ˜ and h+ (c) left and right limits of h at c, ˜ whenever they exist.

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We are now able to fully elucidate the shape of Mi , in relation to the values of λi . Claim 1. (i) M1 = (c1∗ , c1 ) with c1∗ < c1 if and only if either λ1 > 0 or λ1 = 0 and ψ(c1 ) > 0; (ii) M2 = (c2∗ , c2 ) with c2∗ < c2 if and only if λ2 > 0; (iii) M3 = ∅. Proof. Let us investigate the four possible cases for the values of each λi in turn, and analyze the corresponding consequences for M1 , M2 and M3 in the most convenient order. (a) Case λ1 > 0 and λ2 > 0. (M1 ): Since μ1 (c0 ) = μ1 (c) = 0, it follows from (i) and (ii) of Lemma 6 that c1∗ < c1 and M1 = (c1∗ , c1 ). (M2 ): For i = 2, if μ2 (c1 ) = 0, by the same argument the conclusion follows. If μ2 (c1 ) > 0, the conclusion follows from the previous result and Lemma 8. Thus M2 = (c2∗ , c2 ). (M3 ): Assume M3 = ∅. By (iv) of Lemma 6, this implies that μ3 (c2 ) > 0 and c3∗ = c2 . By Lemma 8 and the previous result, M3 = (c2 , c3 ). But then, by continuity and monotonicity of μ3 on M3 this is going to imply that μ3 (c3 ) > 0. However, μ3 (c3 ) = μ3 (c) = 0. Hence M3 = ∅. (b) Case λ1 > 0 and λ2 = 0. (M1 ): For i = 1 nothing changes from the preceding case and M1 = (c1∗ , c1 ). (M2 ): If M2 = ∅, then, as in (a), M2 = (c2∗ , c2 ), hence, since λ2 = 0, by (iii) in Lemma 6, μ3 (c2 ) > 0. Then by Lemma 8 and the previous result, M3 = (c2 , c3 ). But then, by continuity and monotonicity of μ3 on M3 , this is going to imply that μ3 (c3 ) > 0. However, μ3 (c3 ) = μ3 (c) = 0. Therefore, M2 = ∅. (M3 ): Since λ2 = 0, this implies by (22) that μ3 (c2 ) = 0, which by (iv) in Lemma 6 is equivalent to M3 = ∅. (c) Case λ1 = 0 and λ2 > 0. (M2 ): Since λ2 > 0, by (ii) in Lemma 6, c2∗ < c2 . We know that μ2 (c1 )  0. If μ2 (c1 ) = 0, M2 = (c2∗ , c2 ) follows by (i) in Lemma 6. If μ2 (c1 ) > 0, then, since λ1 = 0, by (iii) in Lemma 6, M1 = ∅ and, since μ1 (c0 ) = 0, this implies, by (i) in Lemma 6 that M1 = (c1∗ , c1 ). Lemma 8 thus applies and M2 = (c1 , c2 ) = (c2∗ , c2 ). (M3 ): This, in turn, implies, by the argument given in (a), that M3 = ∅. (M1 ): By (iii) in Lemma 6, M1 = ∅ if and only if μ2 (c1 ) = 0. Now, by (18), (21) and Claim 2, μ2 (c) = 0 for c ∈ [c1 , c2 ] is equivalent, using (14) and given that S  (q2s (c)) = c + T2 (c), to ψ(c) = 0. By Lemma 3, since ψ(c2 ) = μ2 (c2 ) > 0, there exists c ∈ [c1 , c2 ] such that μ2 (c) = 0 if and only if ψ(c1 )  0. In other words, M1 = ∅ if and only if ψ(c1 ) > 0. (d) Case λ1 = 0 and λ2 = 0. (M2 ): If M2 = ∅, then by (iii) in Lemma 6 μ3 (c2 ) > 0 so that we are in the configuration of case (M3 ) in (a), and M3 = (c2 , c3 ). But this implies μ3 (c3 ) = μ3 (c) > 0, violating (19). Therefore, M2 = ∅. (M3 ): By Lemma 6 point (iii) and (iv) this implies M3 = ∅. (M1 ): If M1 = ∅ then by (iii) in Lemma 6 μ2 (c1 ) > 0, i.e. M2 = ∅. This contradicts the previous conclusion. So M1 = ∅. Finally, gathering the point (M1 ), (M2 ), and (M3 ) from (a), (b), (c), and (d), we obtain respectively (i), (ii), and (iii) in the claim. 2 Step 5: Properties of q.

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We can now focus on the optimal production. We will discuss the shape of the optimal contract on each Ci . Claim 2. For all i ∈ {1, 2, 3}, the optimal production q on Ci (i) is such that q(c) = qis (c) for c ∈ (ci−1 , ci∗ ); (ii) is constant on (ci∗ , ci ] if Mi = (ci∗ , ci ), and such that q(c)  q s (ci∗ ) for all c ∈ (ci∗ , ci ], with equality if ci−1 < ci∗ ; (iii) q(ci∗ ) = qis (ci∗ ) whenever ci−1 < ci∗ . Proof. (i): For i ∈ {1, 2, 3}, if ci−1 = ci∗ , then (ci−1 , ci∗ ) = ∅ and the result holds vacuously. Assume therefore ci−1 < ci∗ . Then, since ci∗ is the lower bound of Mi , and ci−1 < ci∗ , μi = 0 on (ci−1 , ci∗ ]. Thus on this nonempty interval μi = 0, therefore, on any interval included in (ci−1 , ci∗ ) ∩ C d , we can apply Lemma 4 to obtain that q = qis on this interval. But since q may be discontinuous only at a finite set of points, and must be nonincreasing by (IC2), it must be that q = qis on the whole (ci−1 , ci∗ ). (ii): The first part is Lemma 7. Moreover, if ci∗ > ci−1 , since q is non-increasing, we have q(c) = q s (c)  q¯i by point (i) for all c ∈ (ci−1 , ci∗ ). Thus, as c goes to ci∗ , q s (ci∗ )  q¯i . Finally, q¯i = q s (ci∗ ). (iii): For all c ∈ (ci−1 , ci∗ ) and c ∈ (ci∗ , ci ), by (IC2) we have q(c)  q(ci∗ )  q(c ). Hence, by (i) and (ii): qis (c)  q(ci∗ )  qis (ci∗ ), thus as c goes to ci∗ we get the result. 2 At this stage, the path of the optimal contract is fully determined on each Ci . It remains to determine its values at boundary points ci , i ∈ {0, 1, 2, 3}. Claim 3. (i) For i ∈ {1, 2}, q(ci ) = qis (ci∗ ) if ci−1 < ci∗ < ci , q(ci )  qis (ci∗ ) if ci∗ = ci−1 , q(ci )  qis (ci∗ ) if ci∗ = ci . (ii) q(c)  q3s (c). Proof. This follows from (IC2) and Claim 2.

2

The following claim compares the first and second-best solutions. Claim 4. The optimal production satisfies q(c) < q F B (c),

∀c = c

and

q(c)  q F B (c).

Moreover, c1∗ > c. Proof. We will prove this starting from c ∈ (c2 , c] and working our way back to c. (a) c ∈ (c2 , c]. Claims 2 and 3 tell us that q = q3s on (c2 , c) and that q(c)  q3s (c). Thus, since s q (c) < q F B (c) except at c, the property holds on (c2 , c]. Moreover, q+ (c2 ) = q s (c2 ). (b) c ∈ (c1 , c2 ]. If M2 = ∅, by the same argument as above, the property holds. Thus, if there exists c ∈ (c1 , c2 ) such that q(c)  q F B (c), it must be that c ∈ M2 and therefore q(c) = q¯2 which is the constant value of q on M2 by Claim 2. In particular, since μ3 (c2 ) = 0, if λ2 = 0, by Claim 1, then M2 = ∅, contradicting the existence of c. Thus λ2 > 0. Now, q(c2 ) = q(c) =

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q¯2  q F B (c) > q F B (c2 ) by Lemma 2. Therefore, S  (q(c2 )) − c2 < 0 and, since λ2 > 0, we have, by (23), μ2 (c2 ) < μ3 (c2 ) = 0, which is impossible since μ2 (c2 )  0. Thus the property holds on (c1 , c2 ) and also at c2 since q(c2 ) = q¯2 . (c) c ∈ [c, c1 ]. Mutatis mutandis, we know that if there exists c ∈ (c, c1 ) such that q(c)  q F B (c), it must be that c ∈ M1 . In this case, since q is constant on M1 by Claim 2, q(c1 ) = q(c)  q F B (c) > q F B (c1 ), thus S  (q(c1 )) − c1 < 0. Now, assume that λ1 = 0. Then, we must have μ1 (c1 ) = μ2 (c1 ) > 0 since M1 = ∅. But this implies that M2 = (c1 , c2 ) by Lemma 8. Thus, q is constant on (c1∗ , c2 ], and this constant is equal to q¯2 . Thus by the previous result, q¯2 < q F B (c) for all c ∈ M2 , and by monotonicity of q F B this extends to (c1∗ , c1 ] too, contradicting the assumption. Therefore λ1 > 0. Then, μ1 (c1 ) < μ2 (c1 ). But μ1 (c1 ) > 0 since M1∗ = ∅, and therefore μ2 (c1 ) > 0, and we can repeat the preceding argument. This proves the result on (c, c]. To address the case c = c, let us show first that c1∗ > c. If c1∗ = c, by Claim 2 q(c1 ) = q¯1  s q1 (c1∗ ) = q1s (c) = q F B (c) > q F B (c1 ). This contradicts the previous results. Therefore, c1∗ > c, thus on (c, c1∗ ), q(c) = q s (c), and thus q(c)  q s (c) = q F B (c) since q is nonincreasing. 2 Step 6: Proof of points (i)–(iv) and (v), b)–h) in Proposition 11. To do this, we summarize the results so far. (i)–(ii): They follow from Claim 1 points (i) and (ii). (v), b)–e): By Claim 4, we have c1∗ > c0 = c. Therefore, using (i) and (ii) in Claim 2 and letting i = 1 we get b). The first part of c) makes use of (ii) in Claim 2 when i = 1 since c1∗ > c. Claim 2-(iii) leads to the second part of c). The optimal production q ∗ in d) and e) comes from (i) and (ii) in Claim 2 when i = 2. (iv): If c2∗ = c1 , then, by (19) 0 = μ3 (c) − μ1 (c), and thus also 0 = μ3 (c) − μ3 (c2 ) + μ3 (c2 ) − μ2 (c2 ) + μ2 (c2 ) − μ2 (c1 ) + μ2 (c1 ) − μ1 (c1 ) + μ1 (c1 ) − μ1 (c). Moreover, using (18), (20), (21), Claim 2 and the points above, we get c2



0 = −λ2 (S (q¯2 ) − c2 ) +



f (c) c + T2 (c) − S  (q¯2 ) dc

c1

− λ1 (S



(q1s (c1∗ )) − c1 ) +

c1



f (c) c + T1 (c) − S  (q(c)) dc

c

or 

c2

0 = −λ2 (S (q¯2 ) − c2 ) +



f (c) c + T2 (c) − S  (q¯2 ) dc

c1

− λ1 (c1∗

+ T1 (c1∗ ) − c1 ) +

c1



f (c) c + T1 (c) − c1∗ − T1 (c1∗ ) dc.

c1∗

Standard computations then lead to (15) and prove (iv). (iii): If c2∗ = c1 , since q2s (c1 )  q¯2 by Claim 3 and since S  is decreasing, c2 ψ(c1 ) = λ2 (c1 + T2 (c1 ) − c2 ) + (F (c2 ) − F (c1 ))(c1 + T2 (c1 )) −

f (c) (c + T2 (c)) dc c1

R. Giraud, L. Thomas / Journal of Economic Theory 171 (2017) 64–100





89

c2

 λ2 (S (q¯2 ) − c2 ) + (F (c2 ) − F (c1 ))S (q¯2 ) −

f (c) (c + T2 (c)) dc c1

= −λ1 (c1∗

+ T1 (c1∗ ) − c1 ) +

c1



f (c) c + T1 (c) − c1∗ − T1 (c1∗ ) dc

c1∗

= μ2 (c1 ) − μ1 (c1 ) + μ1 (c1 ) − μ1 (c1∗ ) = μ2 (c1 )  0. Conversely, if c2∗ > c1 , by point (d), (18), and (21), ψ(c2∗ ) = μ2 (c2∗ ) = 0. By Lemma 3, therefore, ψ(c1 ) < ψ(c2∗ ) = 0. (v), f )–h): The first part of f ) follows from (ii) in Claim 2, while the second part follows from (i) in Claim 3. The first part of g) follows from (i) in Claim 2, while the second part comes from (ii) in Claim 3. For h), since ci∗ = ci for i = 1 or i = 2 can only happen if λi = 0, any value compatible with (IC2) can be chosen. It remains to prove (v)-(a), i.e. q(c) = q F B (c). Step 7: Proof of point (v)-(a). At this point we only have q(c)  q F B (c) by Claim 4. We will show that equality is a consequence of restricting ourselves to admissible solutions. For that, we will prove a lemma. c From (6), for a solution q of [P1 ], V (c) = S(q(c)) − cq(c) − c q(ε)dε is the corresponding principal’s optimal payoff. Lemma 9. If q is a solution to problem [P1 ], V is non-increasing on (c, c], Proof. q(c) is differentiable on C  {c, c1∗ , c1 , c2∗ , c2 , c}, and on the relevant intervals, V (c) is too, and (V ) (c) = (S  (q(c)) − c)q  (c). Moreover, using Lemma 2, Claim 4 and results (ii) to (vi) above, we know that: 1) q(c)  q F B (c) except maybe at c, 2) q  (c) < 0 if c ∈ (c, c1∗ ) ∪ (c1 , c2∗ ) ∪ (c2 , c), and q  (c) = 0 if c ∈ (c1∗ , c1 ) ∪ (c2∗ , c2 ). So V (c) is decreasing on each interval of C {c, c1∗ , c1 , c2∗ , c2 , c}. To prove that V is globally non-increasing, we must therefore examine its behavior at the points cj∗ , j = 1, 2 and ci , i = 0, 1, 2, 3. Note that they are distinct only if the cj∗ , j = 1, 2 are interior points. If cj∗ is interior, since the production is continuous there, V is also continuous, therefore V is non-increasing on each interval (c, c1 ), (c1 , c2 ) and (c2 , c). Moreover, at ci , q(ci ) may be discontinuous. Let us show that when i ∈ {1, 2, 3} i = c V− (ci ) − V+ (ci ) > 0, where V− (ci ) = S(q− (ci )) − ci q− (ci ) − ci q(ε)dε and V+ (ci ) = c S(q+ (ci )) − ci q+ (ci ) − ci q(ε)dε. By the Mean Value Theorem, therefore, we get i = S(q− (ci )) − ci q− (ci ) − (S(q+ (ci )) − ci q+ (ci )) = (S  (q) ˜ − ci )(q− (ci ) − q+ (ci )) for some q˜ such that q− (ci ) > q˜ > q+ (ci ). Now, by points (ii) to (v) above, q− (ci )  qis (ci∗ ) if ci−1 < ci∗ , and q− (ci ) = q(ci ) if ci−1 = ci∗ . In both cases, using Claim 4, we know that qis (ci∗ ) < q F B (ci ) and that q(ci ) < q F B (ci ). Therefore so is q. ˜ It follows that i is the product of two positive factors, so it is positive. 2 What about at c? Since q F B (c0 ) maximizes S(q) − c0 q with respect to q 0 = V (c0 ) − V+∗ (c0 ) = S(q(c0 )) − c0 q(c0 ) − (S(q s (c0 )) − c0 q s (c0 )) = S(q(c0 )) − c0 q(c0 ) − (S(q F B (c0 )) − c0 q F B (c0 ))  0.

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Therefore, V (c)  V+ (c). If the inequality is strict, then we have two cases. First, if V (c) < V (c), then infc∈C V (c) = V (c) by Lemma 9, thus arg min V (c) = {c}. Moreover, supc∈C V (c) = V+ (c) by Lemma 9 and since, by Claim 4, c < c1∗ , V is decreasing on (c, c1∗ ]. Thus, there is no c0 ∈ C such that supc∈C V (c) = V (c0 ), hence arg maxc∈C V (c) = ∅, contradicting admissibility. Second, if V (c)  V (c), then arg minc∈C V (c) = {c} or {c, c}, and supc∈C V (c) = V+ (c) and again arg maxc∈C V (c) = ∅, contradicting admissibility. Therefore V (c) = V+ (c), i.e. q(c) = q F B (c). This completes the proof of Proposition 11. 2 The following proposition states the optimal production when (c1, c2 ) = (c, c). Proposition 12. If c1 = c and c2 = c, then, there exists c∗ ∈ [c, c] and q¯ such that (i) c∗ < c if and only if λ2 > 0; (ii) c∗ > c if and only if ψ(c) < 0, and in this case: ∗

∗

c

(1 + λ2 )(c + T2 (c )) =

(c + T2 (c))f (c) dc + λ2 c;

(27)

c∗

(iii) 

c

(1 + λ2 )S (q) ¯ =

(c + T2 (c))f (c) dc + λ2 c;

(28)

c

(iv) for any admissible solution q ∗ to problem [P1 ], a) q ∗ (c) = q F B (c); b) if c  c∗ > c  (S  )−1 (c + T2 (c)) if c < c  c∗ , ∗ q (c) =  −1 ∗ ∗ (S ) (c + T2 (c )) if c∗  c < c; otherwise q ∗ (c) = q; ¯ c) q ∗ (c) = limc→c q ∗ (c) if λ2 > 0 and q ∗ (c)  limc→c q ∗ (c) if λ2 = 0. Proof. If c1 = c and c2 = c, the maximum principle (see Amit, 1986; Makris, 2001) still applies without changes. The only things to do are to check whether the lemmas and claims in the proof of Proposition 11 go through in this case, and to adapt the proofs and maybe the results if they do not. Now, inspection of their proofs shows that: first, Lemmas 2, 4, and 5 go through; second, points (i), (ii), and (iii) of Lemma 6 hold for i = 2; third, Claim 2 holds for i = 2 and is vacuous for i ∈ {1, 3}; fourth, Lemma 8 becomes vacuous and has no bite anymore, which is why a separate proof is needed; fifth, Claim 4 holds for c ∈ (c, c]; and sixth, Lemma 9 holds. Claim 1 is recast as follows.

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Claim 5. (i) M1 = M3 = ∅; (ii) M2 = (c2∗ , c2 ), with c2∗ < c2 if and only if λ2 > 0. Proof. (i): This point is immediate since C1 = C3 = ∅. (ii): Let us investigate different cases in turn. (a) λ1 = 0. In this, case it follows directly from (19) and (20) that μ2 (c1 ) = μ2 (c) = 0. Thus, (ii) follows by points (i), (ii), and (iii) of Lemma 6 for i = 2. (b) λ1 > 0. If λ2 = 0, then since μ3 (c2 ) = μ3 (c) = 0, by point (iii) of Lemma 6, M2 = ∅, so the result holds. Assume therefore that λ2 > 0. Then, by point (ii) of Lemma 6, c2∗ < c. If c = c1 < c2∗ , then, by definition of M2 and c2∗ , μ2 (c1 ) = μ2 (c) = 0. Thus, (ii) follows by points (i), (ii), and (iii) of Lemma 6 for i = 2. The same happens if c = c1 = c2∗ and μ2 (c1 ) = μ2 (c) = 0. If μ2 (c1 ) = μ2 (c) > 0 then c2∗ = c1 = c. Let c2 = inf{c ∈ C2 | μ2 (c) = 0}. By construction and continuity of μ2 , c1 < c2 . Assume c2 < c2 = c. Then, μ2 > 0 on [c1 , c2 ), therefore q is constant on this interval by (17). Moreover, there exists c2 ∈ (c1 , c2 ) such that μ2 (c2 ) < 0 and therefore by (18) (that applies on (c1 , c2 ) since q is constant there) q(c2 ) < q2s (c2 ) < q F B (c2 ) < q F B (c1 ) by Lemma 2. On the other hand, by (19), (20), and (23), since λ1 > 0, q(c1 ) > q F B (c1 ). Since q(c2 ) = q(c1 ), q(c2 ) > q F B (c1 ), a contradiction. Therefore, c2 = c2 , hence M2 = (c1 , c2 ) = (c2∗ , c2 ), and (ii) holds. 2 Let us now prove the proposition itself. Let c∗ := c2∗ . Then take each point in turn. (i): This point follows from point (ii) of Claim 5. (iv) b)–c): These follow from point (ii) of Claim 5 and case i = 2 of Claim 2. (iv) a): If λ1 > 0, by (19) and (20), q(c)  q F B (c), so we must show the reverse inequality. If c ∈ C d , then it follows by continuity from Claim 4. So assume that c ∈ / C d and q(c) > q F B (c). ∗ Then, by (19) and (20), since λ1 > 0, μ2 (c) > 0. Thus, c = c, therefore q is constant on C, thus q  is too, and c ∈ C d , contradicting the initial assumption. If λ1 = 0, the proof follows from admissibility and Lemma 9 exactly as in the proof of Proposition 11. (iii): If c∗ = c, then, by Claim 2 for i = 2, q is constant on C2  {c}. Let q¯ be the value of q on this interval. Then, by (18), (19), (20), and (21) c μ2 (c) − μ2 (c) =



f (c) c + T2 (c) − S  (q) ¯ dc

c

¯ − c) + λ1 (S  (q(c)) − c) = λ2 (S  (q) ¯ − c), = λ2 (S  (q) since either λ1 = 0 or q(c) = q F B (c) as proved above, and standard computations yield point (iii) of Proposition 12. (ii): If c∗ = c then, since as shown in the proof of point (ii) in Claim 2, q¯  q s (c) and S  is decreasing, S  (q) ¯  S  (q s (c)), and, using (14), it follows that 



c

ψ(c) = λ2 (S (q (c) − c) + S (q (c)) − s

f (c)(c + T2 (c)) dc

s

c

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c



 λ2 (S (q) ¯ − c) + S (q) ¯ −

f (c)(c + T2 (c)) dc = 0, c

as required by (ii). Conversely, if c∗ > c, then by (18), (19), (20), (21), and (iv) c μ2 (c) − μ2 (c) =



f (c) c + T2 (c) − S  (q(c)) dc

c

c =



f (c) c + T2 (c) − S  (q s (c∗ )) dc

c∗

= λ2 (S  (q(c)) − c) + λ1 (S  (q(c)) − c) = λ2 (S  (q s (c∗ )) − c) = λ2 (c∗ + T2 (c∗ ) − c). Standard computations yield (27), which is equivalent to ψ(c∗ ) = 0. Thus, by Lemma 3, ψ(c) < ψ(c∗ ) = 0. This completes the proof of Proposition 12. 2 A.2.3. Argmax and argmin of V We now go back to our original problem (SBP) and we prove the following. Claim 6. For any admissible solution of (SBP) , c ∈ arg maxc∈C V (c) and c ∈ arg minc∈C V (c). Proof. Let cM ∈ arg max V (c) and cm ∈ arg min V (c). They exist by admissibility. First assume cM < cm . After similar computations as in (7), we obtain cM

W (q) = (1 − α)

S(q(c)) − cq(c) −

 F (c) q(c) f (c)dc + αβ(S(q(cM ) − cM q(cM )) f (c)

c

 cm αβ F (c) + 1−α q(c) f (c)dc S(q(c)) − cq(c) − + (1 − α) f (c) cM

+ α(1 − β)(S(q(cm )) − cm q(cm ))  c  α F (c) + (1−α) + (1 − α) q(c) f (c)dc. S(q(c)) − cq(c) − f (c) cm

Let c1 = cM , c2 = cm , λ1 = ⎧ F (c) ⎪ ⎪ ⎨ f (c)

αβ 1−α ,

αβ F (c)+ 1−α Ti (c, α, β) = f (c) ⎪ α ⎪ ⎩ F (c)+ (1−α) f (c)

λ2 =

α(1−β) 1−α

and

if i = 1 if i = 2 if i = 3.

(29)

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Then, for all α ∈ [0, 1) and all β ∈ [0, 1], we have, T1 (c) = 0, and, for all i ∈ {1, 2, 3}, Ti (c) > 0 for all c > c and Ti (c)  0 for all c by Assumption 1. This implies that T1 (c1 ) = T1 (cM ) = F (cM ) f (cM )



αβ F (cM )+ 1−α f (cM )

= T2 (cM ) = T2 (c1 ), and T2 (c2 ) = T2 (cm ) =

T3 (cm ) = T3 (c2 ). Therefore, the problem of maximizing lem [P1 ]. Similarly, assume cM > cm . We get cm

W (q) = (1 − α)

S(q(c)) − cq(c) −

αβ F (cm )+ 1−α f (cm )

1 1−α W (c)



α F (cm )+ 1−α f (cm )

=

is an instance of prob-

 F (c) q(c) f (c)dc f (c)

c

+ α(1 − β)(S(q(cm )) − cm q(cm ))  cM F (c) + α(1−β) 1−α q(c) f (c)dc S(q(c)) − cq(c) − + (1 − α) f (c) cm

+ αβ(S(q(cM ) − cM q(cM )) c

+ (1 − α)

S(q(c)) − cq(c) −

F (c) +

α 1−α

f (c)

 q(c) f (c)dc.

cM

Let c1 = cm , c2 = cM , λ1 =

α(1−β) 1−α , λ2

⎧ F (c) ⎪ ⎪ ⎨ f (c)

F (c)+ α(1−β) 1−α Ti (c, α, β) = f (c) ⎪ α ⎪ F (c)+ ⎩ (1−α) f (c)

=

αβ 1−α

and

if i = 1 if i = 2 if i = 3.

Then, for all α ∈ [0, 1) and all β ∈ [0, 1], we have, T1 (c) = 0, and, for all i ∈ {1, 2, 3}, Ti (c) > 0 for all c > c and Ti (c)  0 for all c by Assumption 1. This implies that T1 (c1 ) = T1 (cm ) = α F (cM )+ 1−α f (cM )

F (cm ) f (cm )



F (cm )+ α(1−β) 1−α f (cm )

= T2 (cm ) = T2 (c1 ), and T2 (c2 ) = T2 (cM ) =

F (cM )+ α(1−β) 1−α f (cM )



1 = T3 (cM ) = T3 (c2 ). Therefore, the problem of maximizing 1−α W (c) is again an instance of problem [P1 ]. Thus, if cm = cM , by Lemma 9 and the fact that q(c) = q F B (c) if q is admissible, V is non-increasing on C, so that c ∈ arg maxc∈C V (c) and c ∈ arg minc∈C V (c). If cm = cM , on the other hand, then V is constant and therefore automatically c ∈ arg maxc∈C V (c) and c ∈ arg minc∈C V (c). 2

We may now proceed to prove the propositions in the text. A.2.4. Proof of Proposition 1 By Claim 6, the optimal production problem is any admissible solution of problem [P1 ] when αβ , λ2 = α(1−β) c1 = c, c2 = c, λ1 = 1−α 1−α , and (29) holds. Therefore, T2 = T , and we can use Proposition 12 to prove Proposition 1. Indeed: (i): This point follows from Proposition 12 (iv)-a).

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(ii): Consider β > 0. This follows from Proposition 12 (iv)-b). If c∗ > c, limc↓c q ∗ (c) = < q F B (c) when β > 0. If c∗ = c, limc↓c q ∗ (c) = q¯ which satisfies (28). But then, since  T2 > 0 and T2 (c) > 0, (1 + λ2 )S  (q) ¯  c + T2 (c) + λ2 c  c + T2 (c) + λ2 c > c + λ2 c, when β > 0. Therefore, S  (q) ¯ > c, i.e. limc↓c q ∗ (c) = q¯ < q F B (c). Consider β = 0. Let us show first that c∗ > c, i.e., by Proposition 12 - (ii), ψ(c) < 0. Indeed, (c) if β = 0, then T2 (c) = Ff (c) , so that q s (c)

c ψ(c) = λ2 (c − c) + c −

(cf (c) + F (c))dc = (1 + λ2 )(c − c) < 0. c

But limc↓c q ∗ (c) = q s (c) = q F B (c) when β = 0. (iii): This follows from Proposition 12 - (i), (iv)-b) and (iv)-c). A.2.5. Proof of Lemma 1 For all x ∈ R, let ϕβ (x) = β 2 x 2 − (β + f (c)(c − c))x + f (c)(c − c). Consider β > 0. α ∈ (0, 1) is a solution to (8) if and only if it is a root of the polynomial ϕβ lying in (0, 1). Since ϕβ is a convex second degree polynomial it has at most two roots. Since ϕβ (0) > 0 and ϕβ (1) < 0, one of them is located in (0, 1), and is on the decreasing branch, so the second one must be on the increasing branch, so the root in (0, 1) exists, is unique, and is the smaller of the two roots.23 The discriminant of this equation of degree two in x is  = β 2 + (f (c)(c − c))2 + 2βf (c)(c − c)(1 − 2β). The smaller root is thus  β + f (c)(c − c) − β 2 + (f (c)(c − c))2 + 2βf (c)(c − c)(1 − 2β) ∗ α (β) := . 2β 2 Consider β = 0. We have ϕ0 (x) = −f (c)(c − c)x + f (c)(c − c), thus the only root is x = α ∗ (0) = 1. A.2.6. Proof of Propositions 2 and 3 Propositions 2 and 3 correspond to the cases c∗ > c and c∗ = c of Proposition 12, respectively. By Proposition 12 - (ii), these cases correspond to ψ(c) < 0 and ψ(c)  0, respectively. Recall αβ F (c)+λ1 , λ2 = α(1−β) that λ1 = 1−α 1−α and T (c) = f (c) . Note that the integrand in the definition of ψ (see (14)), (c + T2 (c))f (c) = cf (c) + F (c) + λ1 , is the derivative of c(F (c) + λ1 ). Then, standard computations yield ψ(c) < 0 ⇔

(1 − αβ)αβ (1 − αβ)αβ +c−c<0⇔ < f (c)(c − c). (1 − α)f (c) (1 − α)

Using Lemma 1, this is equivalent to α < α ∗ (β). Propositions 2 and 3 are just a way of rewriting Proposition 12 using the definitions of T2 , λ1 and λ2 . The value of q¯ can be computed using (28) 

c

(1 + λ2 )S (q) ¯ =

(c + T2 (c))f (c) dc + λ2 c c

23 Standard computations show that if α and α are the roots, then α + α = 1 + α α . Thus, α + (1 − α )α > 1, 1 2 1 2 1 2 1 1 2 β and if α1 < 1 then α2 > 1.

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c =

(cf (c) + F (c) + λ1 ) dc + λ2 c = c(1 + λ1 ) − cλ1 + λ2 c, c

thus

S  (q) ¯ =c+

αβ λ1 1+λ2 (c − c) = c + 1−αβ (c − c).

A.3. Proof of Proposition 4 Let R(q, c) := α(1 − β) (S(q) − cq). The objective function (12) assigns the scrap value R(q(c∗ ), c∗ ) to the type c∗ . So according to the maximum principle (Seierstad and Sydsaeter, 1987, p. 184), the condition defining c∗ is H (c∗ ) + ∂R ∂c (q(c∗ ), c∗ ) = 0. From (16), we get (1 − α) (S(q(c∗ )) − c∗ q(c∗ ) − T (c∗ , α, β)q(c∗ )) f (c∗ ) + μ(c)y(c) − α(1 − β)q(c∗ ) = 0, but, μ(c)y(c) = 0 by (17). A.4. Comparative statics A.4.1. Proof of Proposition 5 Let (c, α, β) = c + (1 − αβ − (1 − α)F (c))T (c, α, β) − c. By Proposition 2 and (9), after standard computations, c∗ is given by (c∗ , α, β) = 0. Now, ∂ ∂T (c, α, β) = 1 − (1 − α)f (c)T (c, α, β) + (1 − αβ − (1 − α)F (c)) (c, α, β) ∂c ∂c ∂T (c, α, β) > 0 = 1 − (1 − α)F (c) − αβ + (1 − αβ − (1 − α)F (c)) ∂c by Assumption 1; on the other hand, ∂ ∂T (c, α, β) = (1 − αβ − (1 − α)F (c)) (c, α, β) + (F (c) − β)T (c, α, β) ∂α ∂α (1 − α)F (c) + αβ β + (F (c) − β) = (1 − αβ − (1 − α)F (c)) (1 − α)f (c) (1 − α)2 f (c) [1 − αβ − (1 − α)F (c)]β + (1 − α)(F (c) − β)[(1 − α)F (c) + αβ)] = (1 − α)2 f (c) β − β 2 + (1 − α)2 (β − F (c))2 > 0, = (1 − α)2 f (c) ∂c∗ hence, by the implicit function theorem, (α, β) < 0. ∂α In turn, ∂ ∂T (c, α, β) = (1 − αβ − (1 − α)F (c)) (c, α, β) − αT (c, α, β) ∂β ∂β α αβ + (1 − α)F (c) = (1 − αβ − (1 − α)F (c)) −α (1 − α)f (c) (1 − α)f (c) α (1 − 2(αβ + (1 − α)F (c))). = (1 − α)f (c) Thus ∂ (c, α, β) > 0 ⇔ 1 − 2(αβ + (1 − α)F (c)) > 0, ∂β

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and, by the implicit function theorem again, ∂c∗ 1 1 1−α (α, β) ≶ 0 ⇔ αβ + (1 − α)F (c∗ ) ≶ ⇔ β ≶ − F (c∗ ). ∂β 2 2α α From this we can derive the existence and uniqueness of a βˆ ∈ [0, 1] such that c∗ is a deˆ an increasing function of β if β > βˆ and ∂c∗ (β) ˆ = 0. Indeed, let creasing function of β if β < β, ∂β

g(β) := αβ + (1 − α)F (c∗ ) − 12 . Clearly, g is continuous and differentiable and

∂c∗ (α, β)f (c∗ ) ∂β α(1 − 2(αβ + (1 − α)F (c∗ ))) =α− (1 − α)f (c∗ )

∗ , α, β) (1 − (αβ + (1 − α)F (c∗ ))) (1 − α)f (c∗ ) 1 + ∂T (c ∂c   (1 − 2(αβ + (1 − α)F (c∗ ))) =α 1−

∗ ∗ 1 + ∂T ∂c (c , α, β) (1 − (αβ + (1 − α)F (c )))

∗ ∗ ∗ α 1 + ∂T ∂c (c , α, β) (1 − (αβ + (1 − α)F (c ))) − 1 + 2(αβ + (1 − α)F (c ))) =

∗ ∗ 1 + ∂T ∂c (c , α, β) (1 − (αβ + (1 − α)F (c ))) ∂T ∗

α ∂c (c , α, β)(1 − (αβ + (1 − α)F (c∗ ))) + αβ + (1 − α)F (c∗ )) = > 0.

∗ ∗ 1 + ∂T ∂c (c , α, β) (1 − (αβ + (1 − α)F (c )))

g  (β) = α + (1 − α)

Three cases may appear. ˆ = 0. (a) g(0) < 0 and g(1) > 0. In this case, there is a unique βˆ ∈ [0, 1] such that g(β) ˆ Moreover, since g is increasing this implies that g(β) < 0 for all β ∈ [0, β) and g(β) > 0 for all ˆ 1], thus, ∂c∗ (α, β) < 0 for all β ∈ [0, β) ˆ and ∂c∗ (α, β) > 0 for all β ∈ (β, ˆ 1]. β ∈ (β, ∂β ∂β (b) g(0)  0. Then, we can set βˆ = 0. (c) g(1)  0. Then we can set βˆ = 1. A.4.2. Proof of Proposition 6  Let eff := q EU (c) − q s (c) = S  −1 (c) − S  −1 c + top. Then ∂eff β 1   =− 2 ∂α (1 − α) f (c) S  S  −1 c +

αβ (1−α)f (c)

αβ (1−α)f (c)

 be the size of the jump at the

 > 0

and ∂eff α 1   =− ∂β (1 − α)f (c) S  S  −1 c +

αβ (1−α)f (c)

 > 0.

A.4.3. Proof of Proposition 7 If α < α ∗ (β), then q¯ = q s (c∗ ) = S  −1 (c∗ + T (c∗ )). Thus, 

∗ ∂ q¯ ∂T ∗ ∂c ∂c∗ ∂T ∗ = + (c , α) + (c , α) (S  −1 ) (c∗ + T (c∗ , α)) ∂α ∂α ∂α ∂c ∂α  ∗ 

∂c 1 ∂T ∗ ∂T ∗ . (c , α) + (c , α) = 1+ ∂c ∂α ∂α S  (S  −1 (c∗ + T (c∗ , α)))

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But, as was established in the proof of Proposition 5,  ∗

∂c β − β 2 + (1 − α)2 (β − F (c∗ ))2 ∂T ∗ (c , α) =− 1+ ∂c ∂α (1 − α)2 f (c∗ )(1 − (αβ + (1 − α)F (c∗ ))) and

β ∂T ∗ ∂α (c , α) = (1−α)2 f (c∗ ) ,

therefore

∂ q¯ ∂α β(1 − (αβ + (1 − α)F (c∗ ))) − (β − β 2 + (1 − α)2 (β − F (c∗ ))2 ) = (1 − α)2 f (c∗ )(1 − (αβ + (1 − α)F (c∗ )))

S  (S  −1 (c∗ + T (c∗ , α))).

(1 − α)β 2 − (1 − α)βF (c∗ ) − (1 − α)2 (β − F (c∗ ))2 (1 − α)2 f (c∗ )(1 − (αβ + (1 − α)F (c∗ ))) (1 − α)(β − F (c∗ ))(αβ + (1 − α)F (c∗ )) . = (1 − α)2 f (c∗ )(1 − (αβ + (1 − α)F (c∗ ))) =

∂ q¯ Thus ∂α < 0 ⇔ β > F (c∗ ). Now, let h(α) := F (c∗ ) − β. Then, h(0) = 1 − β > 0 and h(α ∗ (β)) = −β < 0. Moreover, ∗  ∗ h (α) = ∂c ˆ ∈ (0, α ∗ (β)) such that h(α(β)) ˆ = 0, and ∂α f (c ) < 0. Therefore, there exists α(β) ∂ q¯ ∗ α ≷ α(β) ˆ ⇔ F (c ) ≶ β ⇔ ∂α ≶ 0. As for β, 

∗ ∂ q¯ ∂T ∗ ∂c∗ ∂T ∗ ∂c = + (c , β) + (c , β) (S  −1 ) (c∗ + T (c∗ , β)) ∂β ∂β ∂β ∂c ∂β

 ∗  ∂T ∗ ∂c 1 ∂T ∗ . = 1+ (c , β) + (c , β)   −1 ∗ ∂c ∂β ∂β S (S (c + T (c∗ , β)))

But, as was established in the proof of Proposition 5,  ∗

∂c α(1 − 2(αβ + (1 − α)F (c))) ∂T ∗ (c , β) =− 1+ ∂c ∂β (1 − α)f (c∗ )(1 − (αβ + (1 − α)F (c∗ ))) and

∂T ∂β

(c∗ , β) =

α (1−α)f (c∗ ) ,

therefore

 ∂ q¯ α 1 − 2(αβ + (1 − α)F (c)) 1 = 1 − ∂β (1 − α)f (c∗ ) 1 − (αβ + (1 − α)F (c∗ )) S  (S  −1 (c∗ + T (c∗ , α)))

 αβ + (1 − α)F (c) 1 α < 0. = (1 − α)f (c∗ ) 1 − (αβ + (1 − α)F (c∗ )) S  (S  −1 (c∗ + T (c∗ , α)))

A.4.4. Proof of Proposition 8   c−c  −1 By (11), q¯ = S c + 1−αβ . Thus ∂ q¯ 1 (c − c)β  = 2 ∂α (1 − αβ) S  (S  −1 c +

c−c 1−αβ

1 ∂ q¯ (c − c)α  = ∂β (1 − αβ)2 S  (S  −1 c +

c−c 1−αβ

 < 0 and  < 0.

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A.4.5. Proof of Proposition 9 c  c∗ Combining U ∗ (c) = c q ∗ (ε)dε and Propositions 2–3, we get U ∗ (c) = c (S  )−1 (ε + T (ε, α, β))dε +(c − c∗ )q. ¯ Thus, ∂U ∗ (c) = ∂β

c∗ c

α ∂ q¯ 1 dε + (c − c∗ )   −1 (1 − α)f (ε) S (S ∂β (ε + T (ε)))

∂ q¯ and, by Propositions 7 and 8, we have ∂β < 0 for all α ∈ (0, 1). ∗ On the other hand, assume 0 < α < α (β). Then,

∂U ∗ (c) = ∂α

c∗ c

1 β ∂ q¯ dε + (c − c∗ ) 2   −1 ∂α (1 − α) f (ε) S (S (ε + T (ε)))

∂ q¯ Note that, when 0 < α < α ∗ (β), by Proposition 7 we have ∂α < 0 if and only if F (c∗ ) < β. ∗ Therefore, if F (c )  β, any K works. Assume on the contrary that F (c∗ ) > β. Then, note that 2 U ∗ (c) ∂U ∗ (c∗ ) 1 > 0 and ∂ ∂α∂c = − (1−α)β2 f (c) S  (S  −1 (c+T > 0. Then there exists c ∈ (c, c∗ ) such ∂α (c))) ∂U ∗ (c) ∂U ∗ (c) ∂U ∗ (c) ∂α = 0 if and only if ∂α  0. In other words, ∂α > 0 for all c ∈ ∗ ∂U (c) ∗ ∂α > 0. Now, by the Mean Value Theorem, there exists cˇ ∈ (c, c ) such that

that

c∗ c

C if and only if

(c∗ − c)β β 1 , dε = (1 − α)2 f (ε) S  (S  −1 (ε + T (ε))) (1 − α)2 f (c)S ˇ  (q s (c)) ˇ

and (1 − α)(β − F (c∗ ))(αβ + (1 − α)F (c∗ )) 1 (1 − α)2 f (c∗ )(1 − (αβ + (1 − α)F (c∗ ))) S  (S  −1 (c∗ + T (c∗ , α))) (β − F (c∗ ))T (c∗ ) 1 = , (1 − (αβ + (1 − α)F (c∗ ))) S  (q) ¯ therefore (β − F (c∗ ))T (c∗ ) ∂U ∗ (c) (c∗ − c)β 1 + >0⇔ >0 ∂α (1 − (αβ + (1 − α)F (c∗ ))) S  (q) ¯ (1 − α)2 f (c)S ˇ  (q s (c)) ˇ 1 (c∗ − c)β (β − F (c∗ ))T (c∗ ) > − ⇔ (1 − (αβ + (1 − α)F (c∗ ))) S  (q) ¯ (1 − α)2 f (c)S ˇ  (q s (c)) ˇ ∗ ∗  ¯ (c − c)β(1 − (αβ + (1 − α)F (c )))S (q) ⇔ β − F (c∗ ) < − T (c∗ )(1 − α)2 f (c)S ˇ  (q s (c)) ˇ ¯ (c∗ − c)(1 − (αβ + (1 − α)F (c∗ )))S  (q) F (c∗ ) <− ⇔1− β T (c∗ )(1 − α)2 f (c)S ˇ  (q s (c)) ˇ F (c∗ ) ¯ (c∗ − c)(1 − (αβ + (1 − α)F (c∗ )))S  (q) < . ⇔1+ β T (c∗ )(1 − α)2 f (c)S ˇ  (q s (c)) ˇ (c∗ −c)(1−(αβ+(1−α)F (c∗ )))S  (q) ¯ > 0 we are done. T (c∗ )(1−α)2 f (c)S ˇ  (q s (c)) ˇ ∂ q¯ ∂U ∗ (c) ∗ assume α (β)  α < 1. Then, ∂α = (c − c∗ ) ∂α

Letting K = Finally,

< 0 by Proposition 8.

R. Giraud, L. Thomas / Journal of Economic Theory 171 (2017) 64–100

99

A.4.6. Proof of Proposition 10 and its Corollary ∗ ∗ (c) ∂ q¯ Let γ = α or β. Clearly, ∂V (c)+U = ∂γ (S  (q ∗ (c)) − c). Now, (S  (q ∗ (c)) − c) > 0, hence ∂γ the results follow from Propositions 7 and 8. ∗) ∗ ∗ ∂V ∗ (c)+U ∗ (c) For the corollary, if 1 < F (c > 0, i.e ∂V∂α(c) > − ∂U∂α(c) . On the other hand, β , then ∂α if

F (c∗ ) β

< 1 + K,

∂U ∗ (c) ∂α

< 0. Combining the two we get the result.

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