Warm-Glow Giving and Freedom to be Selfish⇤ ¨ ur Evren † and Stefania Minardi‡ Ozg¨ September 25, 2015

Abstract Warm-glow refers to prosocial behaviour that causes the actor to experience positive feelings, apart from its social implications. We study an individual who enjoys taking a prosocial action that incurs a private cost because such actions improve her social image. A private cost obtains only in the presence of a more selfish option, implying a preference for freedom to behave selfishly. We provide behavioural foundations for this model by building upon the experimental findings on motivation crowding out. Our theory distinguishes between warm-glow and other motivations for giving, and subsumes Andreoni’s (1989, 1990) model of public good provision.

JEL Classification: D11, D64, D81 Keywords: Warm-Glow, Freedom of Choice, Motivation Crowding Out, Altruism, Public Goods, Voting

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We are grateful to Efe Ok and Debraj Ray for their continuous guidance and support. We owe special thanks to the editor Martin Cripps and two anonymous referees for their thoughtful suggestions which improved the paper significantly. We also thank Fuad Aleskerov, Rose Anne Dana, David Dillenberger, Itzhak Gilboa, Anna Gumen, Johannes H¨orner, Farhad H¨ usseinov, Edi Karni, Mark Machina, Massimo Marinacci, Sujoy Mukerji, Jawwad Noor, David Pearce, Leonardo Pejsachowicz, Philipp Sadowski, Kota Saito, Andrei Savochkin, Marciano Siniscalchi and Joel Sobel for very useful comments and discussions, as well as seminar participants at HEC Paris, HSE, NYU, PSE, 24th International Conference on Game Theory, NASM 2013, NES 20th Anniversary Conference, RUD 2013, and SAET 2013. When this project started, the authors were Ph.D. students at NYU, Department of Economics. All errors are our own. † New Economic School, 100 Novaya Street, Skolkovo, Moscow 143025, Russia. Email: [email protected] ‡ Economics and Decision Sciences Department, HEC Paris, 1, rue de la Lib´eration, 78351, Jouy-en-Josas, France. e-mail: [email protected]

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In the last decades, there has been a surge of interest in models of prosocial behaviour that depart from the classical approach based on the notion of altruism. In particular, since Olson (1965), Arrow (1972), and Becker (1974), it has been argued that social image concerns or a desire for acclaim may also drive prosocial behaviour. Motivated by similar observations, Andreoni (1989, 1990) proposes a model of warm-glow in which an individual’s personal donation to a charity or public good makes a positive impact on her utility, independently of how this action influences the social allocation. Throughout this paper, “warm-glow” refers to this e↵ect of prosocial behaviour that causes the actor to experience positive feelings, and our focus will be on the particular type of warm-glow that is associated with the pleasure of social acclaim.1 In contrast to warm-glow, pure altruism dictates that a person should care only about the social allocation. Distinct from these two categories, one can also envision a third type of individuals who perceive prosocial behaviour as an unpleasant obligation because of the negative feelings associated with acting selfishly, such as feeling ashamed (Dillenberger and Sadowski, 2012) or guilty (Noor and Ren, 2015). Identifying observable, behavioural distinctions between warm-glow and alternative motivations for prosocial behaviour is a task of major importance as it may facilitate welfare analysis in related problems. For example, a person who is known to be a generous individual in her community would typically dislike a tax increase for the provision of public goods if her generosity is driven solely by image concerns. A purely altruistic donor, however, may enjoy such a policy change if it truly increases the total amount of a public good. In turn, a person who is ashamed to behave selfishly might be better or worse o↵ depending on whether she prefers paying taxes as a legal obligation to making donations as a means to avoid shame. Traditionally, economists study agents’ consumption-giving choices in a static setup to understand the behavioural implications of the aforementioned motivations. Unfortunately, this approach has severe limitations. If the decision maker (DM) makes a donation to avoid negative feelings, such as shame or guilt, why can’t we think of this behaviour as a source of positive feelings, i.e. warm-glow?2 Distinguishing the relief of shame from the pleasure of 1

Some scholars (e.g., Glazer and Konrad, 1996) use the term warm-glow to refer to prosocial behaviour that is valuable per se, while Andreoni (1990) mentions several forms of image concerns to motivate his theory. In fact, Footnote 2 below indicates that Andreoni’s conception of warm-glow is even more general than our interpretation in this paper. Further remarks on the use of the term can be found in Andreoni (2006, 2008). 2 ‘[...] if we were to include warm-glow [in welfare analysis], we are not sure whether it should increase or decrease welfare. If giving to charity is relieving a guilty feeling, then although it certainly increases utility to give, it does not necessarily mean utility is higher than it would be if the government had forced the contribution through taxation’ (Andreoni, 2006, p. 1226). According to this perspective, warm-glow and relief of a guilty feeling are distinct for the purposes of welfare analysis, but they are behaviourally similar.

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acclaim—a particular form of warm-glow—is even more subtle because both originate from a desire for a good social image. On the other hand, as Andreoni (1990) shows, distinguishing warm-glow from altruism in a static setup is more feasible, but the static analysis has its limits even to this end. In particular, Andreoni’s findings demand certain conditions that rule out some reasonable forms of utility functions, including quasilinear forms. Shortly put, we still have a partial understanding of the driving forces behind prosocial behaviour and the welfare implications for the donor (DellaVigna et al., 2012). Focusing on a two-stage choice problem, this paper proposes a model of warm-glow that is clearly distinct from the modes of behaviour associated with the aforementioned, alternative motivations. In the first stage, the DM chooses a set of social allocations (i.e. a menu) from which she will select an element in the second stage. We assume that the recipient observes only the choice of an allocation in the second stage, but not the earlier choice of a menu. Our DM, on the other hand, experiences warm-glow to the extent she appears generous in the eyes of the recipient. Hence, observability of second-stage choices may lead to warm-glow in that stage, while the first-stage behaviour is influenced by the anticipated feeling of warm-glow. In contrast, by itself, the DM’s first-stage behaviour does not lead to warm-glow because it is not observed by the recipient.3,4 One example that fits the described setup is the case of a parent who would like to be remembered as a generous person by her children who may be able to observe (or fully appreciate) only the later decisions of the parent. Another example is the case of a person who is planning to make a large charitable donation from her savings in order to improve her social image. The DM’s earlier decision to save, by itself, is not likely to contribute to her image because, typically, individual saving decisions are not publicly observable. Roughly, our approach is a dual of Dillenberger and Sadowski’s (2012) model of shame. The key idea of Dillenberger and Sadowski is that if a person selects prosocial allocations to avoid shame inflicted by selfish behaviour, then she must be better o↵ upon removal of prosocial allocations because this would allow her to behave selfishly, without experiencing After all, inclusion of warm-glow in welfare analysis entails the possibility that the purpose of the behaviour attributed to warm-glow is actually to avoid a guilty feeling. Indeed, it is hard to think of a di↵erence between the two modes of behaviour in a static setup. 3 Glazer and Konrad (1996) and B´enabou and Tirole (2006) remark that charitable donations are rarely anonymous, pointing to the prevalence of image concerns. 4 We abstract from the possibility that the presence of an outside observer (the analyst or the experimenter) may also influence the DM’s behaviour. In an experiment, such e↵ects can be minimized by doubleblind techniques that allow the experimenter to elicit the subjects’ choices under anonymity. Moreover, the welfare of the outside observer is typically independent of the DM’s actions and, hence, the presence of the former would lead to weaker image concerns, if any. Furthermore, it seems reasonable to assume that the outside observer would influence both stages uniformly, thereby leading to stronger image concerns in the second stage due to the additional e↵ect of the recipient (see also Saito (2015, p. 341) for related remarks).

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negative feelings, as she actually wishes. Thus, their model is characterised by preference for smaller menus, just as in Gul and Pesendorfer’s (2001) theory of temptation and self control.5 By contrast, our warm-glow model depicts a decision maker who enjoys larger menus. In our model, the DM enjoys the presence of self-serving options even if she does not plan to select them. Indeed, selecting an other-serving option can meaningfully act as a signal of generosity only if the DM has the freedom to select a more selfish option. Put di↵erently, our DM exhibits a preference for “freedom to behave selfishly,” a particular form of preference for larger menus, in stark contrast with Dillenberger and Sadowski’s (2012) model of shame. In turn, a purely altruistic DM must care only about the final allocation, and hence, can be identified as a person who is neutral against the size of the menus. As a by-product, our menu-choice approach also makes more transparent well-known distinctions between warm-glow and altruism. In particular, in Section 3.1, we show that Andreoni’s (1990) predictions on non-neutrality of redistributive policies remain valid in a dynamic setup even with quasilinear temporal utility functions, despite the fact that such functions must be ruled out in a static setup. As discussed in Section 3, these predictions match the related empirical and experimental findings much better than those of purealtruism theories. Overview of the Results. Our main result (Theorem 2) provides axiomatic foundations for the warm-glow model outlined above. Instead of directly assuming the warm-glow hypothesis, our substantive axioms are simply designed to accommodate the experimental findings on motivation crowding out. Shortly put, these experiments show that utilising rewards and fines to incentivise prosocial behaviour can backfire on occasion and lead to more selfish behaviour. From our main result it follows that the findings in this strand of literature can also be interpreted as evidence for the adequacy of our warm-glow model. In other words, our theory undermines the concern that the warm-glow models are ad hoc.6 A key feature of our model is that the private cost incurred upon selecting a given allocation—instead of the most selfish option—has a positive e↵ect on the utility of the DM, apart from its material consequences. This cost increases with the maximum possible private consumption o↵ered by a menu, decreases with a reward attached to the allocation in question, and is zero when there is no other available option. On one hand, it follows 5 Conceptually, the main di↵erence between guilt and shame is that the former is a private emotion, while the latter is related to image concerns. Building upon Gul and Pesendorfer, Noor and Ren (2015) model guilt avoidance as a form of temptation to act selfishly which, itself, generates guilt. Similarly to Dillenberger and Sadowski, their model gives rise to a preference for smaller menus. 6 ‘Putting warm-glow into the model is, while intuitively appealing, an admittedly ad hoc fix’ (Andreoni, 2006, p. 1222).

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that the DM can select a prosocial allocation from a given set but switch to a more selfish allocation upon the introduction of a fine or reward to incentivise prosocial behaviour, in line with the findings on motivation crowding out. On the other hand, we can define a notion of warm-glow payo↵ as the di↵erence between the utility of selecting an allocation x under the presence of a more selfish option and the utility of selecting x when there is no other option. In Section 3, we show that this warm-glow model subsumes Andreoni (1989, 1990) in the context of public good provision. On the way to our main result, we also provide an auxiliary representation (Theorem 1) that accommodates only the adverse e↵ects of fines. In the remainder of this section, we discuss the related literature, including the findings on motivation crowding out. We introduce the formal setup in Section 1, and state our representation theorems in Section 2. In Section 3, we formally relate our theory to Andreoni (1989, 1990). Section 4 is devoted to a novel application on welfare analysis. In Section 5, we relate our model to Riker and Ordeshook’s (1968) theory of voter participation in large elections. We conclude in Section 6. All proofs are relegated to the appendix, and a further online appendix contains some other supplementary material. Related Literature. A plethora of evidence shows that restricting one’s freedom to behave selfishly may actually motivate that person to take more selfish actions. For example, Falk and Kosfeld (2006) examine a principal-agent problem in which the agent chooses a productive activity a that incurs costs for herself while increasing the principal’s payo↵. In turn, the principal determines the menu available to the agent. Falk and Kosfeld find that if the principal imposes a lower bound for a, then, compared to the case in which the principal does not impose such a restriction, a majority of the agents select a lower level of a. As Falk and Kosfeld note, a potential explanation of this finding is that ‘agents do not like to be restricted, and perceive control as a negative signal of distrust’ (pp. 1611-1612). Similarly, Fehr and Rockenbach (2003) and Houser et al. (2008) find that trustees may return less when investors use sanctions to enforce demanding outcomes. As for the dual evidence on the adverse e↵ects of rewards, a remarkable paper is due to Gneezy and Rustichini (2000), who show that o↵ering monetary rewards to children for volunteer work may decrease their performance. In another field study, Mellstr¨om and Johannesson (2008) find that the supply of female blood donors decreases almost by half when they are o↵ered monetary compensation. Moreover, in a laboratory experiment, Ariely et al. (2009) show that social image concerns are an important driver of prosocial behaviour and that the introduction of monetary incentives can have detrimental e↵ects. The findings on the adverse e↵ects of rewards and fines are usually interpreted as evidence 4

for “motivation crowding out,” which refers to the idea that extrinsic motivations may crowd out intrinsic or image related motivations.7 Our main representation establishes a precise relationship between these findings and some prominent warm-glow models. To the best of our knowledge, this is the first paper that formally explores the relationship between these two strands of literature. By now, there is a sizeable literature on preference for freedom of choice, which maintains that a large set of alternatives may be more valuable for a DM than the alternative that the DM plans to select from that set (see, e.g., Sen, 1985, 1988; Puppe, 1996).8 The focus of this strand of literature is the measurement of freedom of choice associated with menus. Apart from Dillenberger and Sadowski’s (2012) model of shame and Noor and Ren’s (2015) model of guilt that we discussed earlier, in a concurrent paper Saito (2015) studies the notion of “pride” in a menu-choice setup. In addition to pride, which is a form of warmglow, Saito’s model also accommodates temptation to act selfishly and shame associated with selfish behaviour. While Saito’s scope is broader, we propose a more comprehensive model of warm-glow. In particular, the second-stage behaviour corresponding to Saito’s model satisfies the weak axiom of revealed preferences (WARP).9 By contrast, our model allows for violations of WARP in the second stage, as the utility function that governs the choice of allocations is reference dependent. This is crucial for our purposes because neither the applied warm-glow models nor the related experimental findings are compatible with WARP. Yet, in dynamic consumption-saving problems, Saito’s model leads to interesting predictions (such as non-neutrality of redistributive policies). In that respect, our example in Section 3.1 with quasilinear utility functions resembles Saito’s approach. To provide foundations for warm-glow, Cherepanov et al. (2013) propose a static choice model. While they do not attempt to distinguish warm-glow from the relevant forms of non-altruistic behaviour, they point out that, unlike altruism, these modes of behaviour may violate WARP. Although the second-stage behaviour in our model also violates WARP, it is quite distinct from that in Cherepanov et al. Indeed, Cherepanov et al. think of the warm-glow payo↵ as a fixed number, whereas we model it as an increasing function of the private cost associated with a given allocation. This novelty makes our model compatible 7

This idea dates back to Titmuss (1970), who also considers the specific example of blood donation. Starting with Deci (1975), a group of cognitive psychologists advocate the same view. Frey and Jegen (2001) provide an extensive survey of the evidence collected in this literature. A related literature, which solely focuses on the freedom aspect, builds upon Brehm’s (1966) theory of reactance, which maintains that a threat to someone’s freedom to choose an option may lead that person to a psychological state which involves an enhanced level of attraction to that option. An early review of experiments on Brehm’s theory can be found in Berkowitz (1973). 8 This contention contrasts with the “preference for flexibility” approach that focuses on the instrumental value of larger menus driven by choice uncertainty (Kreps, 1979; Dekel et al., 2001). 9 The same is true for the most structured representation of Dillenberger and Sadowski (2012).

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with Andreoni (1989, 1990). 1.

Setup

An allocation refers to a generic element of X := R2+ . The DM faces a two-stage choice problem: In the first stage, she selects a set of allocations and, in the second stage, she chooses an allocation from the set that she has selected earlier. As noted in the introduction, only the second stage is publicly visible. We do not explicitly model the DM’s second-stage behaviour—it is left as a part of the interpretation of first-stage behaviour. (A formal model of second-stage behaviour can be found in the online appendix.) The allocations are denoted by x, x0 , y, z, etc. Given an allocation x := (x1 , x2 ), the first component x1 stands for our DM’s private consumption. In turn, x2 represents a second consumption variable, which can be the private consumption of another individual, or the amount of a public good. In either case, the recipient refers to a second individual who benefits from x2 . For any x, y 2 X, x y means xi yi for i = 1, 2 while x > y means x y and xi > yi for some i. Let K be the collection of all nonempty, compact subsets of X. We equip K with the Hausdor↵ metric dH (induced by the Euclidean norm k·k).10 A generic element of K is referred to as a menu. We denote by KP the collection of all menus A that consist of Pareto efficient allocations—x, y 2 A and x y imply x = y. Our primitive is a preference relation % over a collection A of menus such that A ◆ KP . The relation % describes the DM’s first-stage (private) choice behaviour. We shall set A := K in our first representation theorem, while A := KP in our main representation. 2.

Representation Theorems

Our first axiom is a standard rationality property which, inter alia, rules out a potential reference dependence in the first stage. A1: Weak Order (WO). % is a complete and transitive binary relation on A. Our second axiom asserts that increasing the size of a menu cannot harm the DM. A2: Setwise Monotonicity (SM). For any A, B 2 A, if A ◆ B, then A % B. As a key feature, our representations depict a DM who enjoys her freedom to behave selfishly. The next axiom entails that only the most selfish option (i.e. the option that o↵ers 10 dH (A, B) := max {maxx2A miny2B kx pact sets A, B ✓ R2 .

yk , maxy2B minx2A kx

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yk} for any pair of nonempty, com-

the maximum possible private consumption) influences the DM’s perception of freedom.11 A3: Weak Instrumentalism (WI). Let A, B, C 2 A and suppose C = A [ B. If there exists a y 2 A \ B such that y1 x1 for every x 2 C, then C ⇠ A or C ⇠ B. The name of this axiom is inspired by a stronger property that characterises the standard model of menu-choice when combined with (WO) and (SM): A [ B ⇠ A or A [ B ⇠ B.

(I)

The logic behind property (I) consists of two parts. First, if the DM can anticipate which alternative she will select from a menu, she should evaluate this menu solely with that particular alternative. In this sense, property (I) describes a purely instrumentalist person who views a menu solely as a means to her final choice. Second, the DM’s behaviour in the second stage must be compatible with WARP: If she plans to select a given alternative from A [ B, she must also select it from any subset (A or B) that contains the alternative.12 (WI) dispenses with both of these assumptions that underlie property (I). In particular, (WI) is compatible with the findings on the adverse e↵ects of restricting one’s freedom to behave selfishly, which point to a non-instrumentalist mode of behaviour that also violates WARP. For example, consider a set of three allocations C := {x, y, z} (that belongs to A), where each allocation represents a split of a dollar between the DM and the recipient. Suppose that y is the most selfish option for the DM, and x is the most generous one, so that y1 > z1 > x1 . Then, (WI) implies either C ⇠ {x, y} or C ⇠ {y, z}. If only the former equivalence holds, we may as well have {y, z} C {x, z}. In view of the property (SM), this is the pattern that one would expect if the DM is planning to select z from {x, z} while selecting x from C. Indeed, the findings on the adverse e↵ects of restricting one’s freedom to behave selfishly point precisely to such a switch towards more selfish behaviour in response to the restriction. In line with the discussion above, we are particularly interested in cases of the form {x} {x, y} {y} for some x, y 2 X with y1 x1 and {x, y} 2 A. We will write xf y to denote this pattern of preference. Our interpretation of this pattern is that the DM plans to select x from {x, y}, and this act is more valuable than the mere consumption of x because 11 In that respect, our approach is akin to that of several other papers, including Gul and Pesendorfer (2001, 2005a) and Dillenberger and Sadowski (2012), although these papers are concerned with modelling preference for smaller menus. 12 Gul and Pesendorfer (2005b) shows that in the absence of (SM), the property (I) characterises a DM who expects her preference relation to change in a certain way by the second stage. In this case, the logic behind property (I) remains the same except that it becomes necessary to distinguish between the DM’s future and current selves. Specifically, the future self of the DM still satisfies WARP while the current self is solely concerned with the allocation that the future self will select from a given menu.

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the menu {x, y} o↵ers the freedom to select a more selfish option y. Although (WI) gives a special role to the most selfish option in the evaluation of a menu, the axiom is silent about the nature of this relationship. The next property fills this gap. A4: Freedom to Be Selfish (FS). If xf y, then {x, y 0 } % {x, y} for any y 0 2 X such that y10 y1 and {x, y 0 } 2 A. According to this axiom, if the DM plans to select an allocation x over a more selfish option y, then she should be better o↵ whenever y is replaced with a y 0 such that y10 y1 . That is, the freedom to choose a higher level of private consumption leads to higher welfare. In terms of image concerns, our interpretation of this pattern is that the DM can convey a stronger signal of generosity as the maximum possible private consumption increases. Next, we rule out negatively interdependent preferences. A5: Singleton Pareto (SiP). {x} % {y} for any x, y 2 X with x

y.

It is worth noting that (SiP) also allows for a purely selfish attitude over singletons represented by the function {x} ! x1 . We proceed with a standard continuity property. A6: Continuity (C). For each A 2 A, the sets {B 2 A : B % A} and {B 2 A : A % B} are closed in A. The properties introduced so far are necessary conditions in both of our representations. The main di↵erence between our warm-glow model and our basic representation of preference for freedom to be selfish lies in how the utility of {x, y} changes with the alternative that the DM plans to select from this set. We now turn to this issue and state our representation theorems. 2.1.

A Representation of Preference for Freedom to Be Selfish

Our first representation requires the following axiom, which we state for the case of a preference relation over K. A7: Coordinatewise Monotonicity (CM). If xf y, then {x0 , y} % {x, y} for any x0 2 X such that x0 x. This axiom asserts that if x0 x, and if the DM plans to select x from a menu that contains a more selfish option, then replacing x with x0 should make her better o↵. The intuition is that such a change should not have negative implications for the DM’s perception of freedom to behave selfishly while improving the set of available allocations from a material point of view. 8

We are now ready to state our first representation theorem, which is based on (CM). Theorem 1. A binary relation % on A := K satisfies the axioms (A1)-(A7) if and only if there exists a weakly increasing and continuous function Uf : X ⇥ R+ ! R such that, for every A, B 2 K, A%B

i↵

max Uf (x, max y1 ) x2A

y2A

max Uf (x, max y1 ). x2B

y2B

(1)

In what follows, we refer to such a function Uf as an f-index for %. This representation suggests that, given a menu A, the DM plans to select the allocation that maximises Uf (x, max y1 ) over A. Accordingly, Uf (x, max y1 ) is interpreted as the utility y2A

y2A

of selecting x from A. Since Uf is weakly increasing in its last argument, a key implication is that the utility of selecting a given allocation x increases with the maximum possible private consumption, max y1 . This makes (1) a representation of preference for freedom to be selfish y2A

because max y1 can be viewed as a measure of DM’s freedom to behave selfishly. Moreover, y2A

the e↵ect of max y1 can be related to image concerns: As the maximum possible private y2A

consumption increases, the act of selecting a prosocial allocation conveys a stronger signal of generosity. It is easy to verify that the representation above is consistent with the evidence on the adverse e↵ects of using punishments to foster prosocial behaviour. For example, given a menu {x, y} with y1 > x1 and y2 < x2 , a fine t > 0 on the act of selecting y reduces not only the utility of the most selfish option, but also that of the prosocial allocation x. Indeed, the utility of selecting x declines from Uf (x, y1 ) to Uf (x, y1 t), assuming that the fine is small enough so that y1 t x1 . Although (CM) is certainly appealing from a normative point of view, the findings on the adverse e↵ects of rewards cast doubt about the descriptive power of this axiom. Using the words of Gneezy and Rustichini, ‘if the reward directly a↵ects the utility of an individual in a negative way (because it reduces the intrinsic motivation), then performance may decline with the increase in monetary incentive’ (2000, p. 793). Put di↵erently, an ill-advised reward r > 0 for selecting a prosocial allocation x over a selfish allocation y may decrease the DM’s utility, leading to the pattern {(x1 + r, x2 ), y} {x, y}. This pattern would imply xf y by (SM) and (SiP), and hence, is not compatible with (CM). Thus, we utilise a modified version of (CM) in our model of warm-glow.

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2.2.

Warm-Glow Representation

Throughout this section, we focus on Pareto efficient allocations and assume A : = KP .13 The next assumption modifies (CM) in a suitable way. A7*: Compensated Coordinatewise Monotonicity (CCM). If xf y, then {x0 , (a, y2 )} % {x, y} for any (x0 , a) 2 X ⇥ R such that x0 x and a y1 + (x01 x1 ). Observe that this axiom allows for the pattern {(x1 + r, x2 ), y} {x, y}. Thus, rewarding a prosocial action can make the DM worse o↵. Moreover, the axiom prescribes that the DM would certainly enjoy a “present” that is not contingent on her behaviour. That is, {(x1 + r, x2 ), (y1 + r, y2 )} % {x, y}. Gneezy and Rustichini (2000) point out that their experimental findings confirm this hypothesis.14 On a technical note, the restriction that we have imposed on the domain of % reduces the power of property (C). We solve this issue with an additional continuity axiom, called Extension Continuity, which is discussed in Appendix A. In what follows, (A8) denotes this additional continuity axiom. We are now ready to state our warm-glow representation, which is our main result. Theorem 2. A binary relation % on A := KP satisfies the axioms (A1)-(A6), (A7*) and (A8) if and only if there exists a weakly increasing and continuous function Uw : X ⇥R+ ! R such that, for every A, B 2 KP , A%B

i↵

max Uw (x, max y1 x2A

y2A

x1 )

max Uw (x, max y1 x2B

y2B

x1 ).

(2)

This representation suggests that the utility of selecting a given allocation x from a menu A is an increasing function of max y1 x1 , the private cost that the DM incurs if she decides y2A

to select x from A. Put di↵erently, max y1 y2A

x1 tells us how much more private consumption

the DM could enjoy if she were not to select x. Thus, the representation establishes an alternative relation between the DM’s welfare and her freedom to behave selfishly, which is also consistent with the evidence on the adverse e↵ects of sanctions against selfish behaviour. Moreover, as in representation (1), the role of the private cost in representation (2) can be attributed to social image concerns: An other-serving action that incurs a higher cost makes a stronger impact on the image of the DM. The di↵erence between representations (1) and (2) lies in their implications about the 13

The online appendix discusses two alternative methods to extend the domain, each featuring some pros and cons. 14 ‘The negative e↵ect is significant only if the reward is contingent on the performance; subjects who are paid a fixed positive amount, independent of their performance, do not display reduction in intrinsic motivation’ (Gneezy and Rustichini, 2000, p. 793).

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e↵ects of rewarding prosocial actions. In the former representation, rewarding an action can only make that action more attractive. By contrast, according to representation (2), rewarding an action may reduce the overall utility of that action by decreasing the cost that the DM incurs. In particular, rewards can also backfire in line with the related evidence. As we noted earlier, this di↵erence between the two representations is driven by the axioms (CM) and (CCM). Now, to see why representation (2) embodies a notion of warm-glow, consider a pair of allocations x, y such that x1 < y1 and x2 > y2 . Then, the act of selecting x over y is a prosocial action taken by free will, which makes the recipient believe that the DM is a generous person. For the DM, this action yields the utility Uw (x, y1 x1 ), including the pleasure of being known as a generous person. On the other hand, if x is the only available option, the act of selecting x becomes a necessity with no value for the sake of the DM’s image, leading to utility Uw (x, 0). Therefore, Uw (x, 0) gives the instrumental payo↵ resulting from the consumption of x, while Uw (x, y1 x1 ) Uw (x, 0) can be considered as the warmglow payo↵ associated with the act of selecting x over y. Indeed, as we will see in Section 3, the second-stage behaviour implied by representation (2) coincides with that in Andreoni’s (1989, 1990) theory. In what follows, we refer to representation (2) as a warm-glow representation, while the associated function Uw is referred to as a w-index. Observe that if Uw (x, ) is constant in for each x, then the DM never experiences warm-glow. In this case, the representation reduces to classical utility maximisation: A % B i↵ maxx2A Uw (x, 0) maxx2B Uw (x, 0). We refer to the corresponding DM as purely altruistic. Another case of special interest is when Uw depends only on x1 and . This corresponds to a purely egoistic DM who is motivated solely by warm-glow and her private consumption. Finally, we can think of a strict warm-glow representation in which Uw (x, ) is strictly increasing in (x, ). The next remark describes the characterisations of these special cases. Remark 1. (a) For pure altruism, the property (I) must hold for all A, B 2 KP . (b) For pure egoism, given any {x, y}, {x0 , y 0 } 2 KP with x1 = x01 and y1 = y10 , we must have {x, y} ⇠ {x0 , y 0 }. (c) For a strict warm-glow representation, all the following conditions must hold: (i) x > y implies {x} {y} for any x, y 2 X; (ii) {x, (x1 + ", 0)} {x} for any " > 0 and x 2 X with x2 > 0; (iii) xf y implies {x, y 0 } {x, y} for any y 0 2 X with y10 > y1 and y20 < x2 ; (iv) xf y implies {x0 , (a, y2 )} {x, y} for any x0 > x and a y1 + x01 x1 . (For brevity, we omit the proofs of these assertions.) To conclude this section, note that the domain KP in Theorem 2 is rich enough to cover many applications. For example, in a typical model of charity or public good provision, the 11

DM has an initial endowment, and the menu A that she faces consists of all allocations that she can obtain by donating some of her endowment, given other factors such as government transfers. Naturally, these choice sets contain only efficient allocations. In the remainder of the paper, we turn our attention to such applications. 3.

Relations to Andreoni’s Model

Andreoni (1989, 1990) studies a game on public good provision between a set of individuals {1, ..., I}. He assumes that there is one public good and one private good and that one unit of the private good can be converted into one unit of the public good with a linear technology. Each individual i is endowed with an amount wi of the private good that she can allocate between her private consumption, xi , and her gift to the public good, gi . Moreover, the P government levies lump sum taxes ⌧i . So, G := Ii=1 gi is the total private contributions to P the public good, and T := Ii=1 ⌧i is the total tax receipts that are fully used for the provision of the public good. A generic agent, say agent 1, takes as given the private consumption and gifts of others, (x2 , g 2 ), ..., (xI , g I ), and chooses a consumption-gift pair (x1 , g1 ) so as to solve an optimisation problem of the following form: max U (x1 , G + T, g1 ) s.t. x1 + g1 + ⌧1 = w1 and 0  x1  w1

⌧1 .

(3)

Here, U is a weakly increasing function on R3+ , which captures altruistic concerns15 and warm-glow experience by its second and third arguments, respectively. In our terminology, then, agent 1 faces the menu A := {(x1 , G + T ) : 0  x1  w1

⌧1 , x 1 + g 1 + ⌧1 = w 1 , P G + T = ⌧1 + g1 + Ii=2 ⌧i + g i }.

Clearly, the menu A belongs to KP and max y1 = w1 y2A

the budget constraint, we see that g1 = max y1 y2A

⌧1 . Thus, upon solving for g1 in

x1 . That is, g1 is simply the last argument

of a w-index in our terminology. The function Uw (x, ) := U (x, ), defined on X ⇥ R+ , qualifies as a w-index, and agent 1’s allocation choice maximises Uw (x, max y1 x1 ) over the y2A

menu A. To summarise, in the present setup, the second-stage behaviour implied by our warm-glow representation reduces to Andreoni’s model. It should be remarked, however, that the two models treat subsidies di↵erently: Remark 2. Suppose that the government subsidises agent 1’s gift at the rate s 2 (0, 1), 15

In the related literature, it is customary to view one’s concern for the public good as a form of altruism. We also follow this convention.

12

so that the agent faces the budget constraint x1 + (1 s)g1 = w1 . Then, according to our theory, the warm-glow component in agent 1’s utility function is given by w1 x1 = (1 s)g1 . This amount equals precisely the agent’s net contribution to the public good because the subsidy sg1 is paid from the government’s budget. By contrast, according to Andreoni’s model the warm-glow component should be g1 , i.e. the agent’s gross contribution. We leave it for further research to investigate which approach to subsidies is more suitable. The main contribution of Andreoni’s model is that, under suitable assumptions, it makes the equilibrium amount of the public good sensitive to fiscal policies and income distribution. This di↵ers from the corresponding models of pure altruism, in which a government expenditure on the public good (financed by lump-sum taxes) crowds out voluntary contributions dollar-for-dollar. Moreover, according to models of pure altruism, the total amount of the public good is also independent of the income distribution.16 Andreoni’s predictions on incomplete crowding out and non-neutrality of income distribution are strongly supported by empirical studies (see, e.g., Hochman and Rodgers, 1973; Abrams and Schmitz, 1978, 1984; Clotfelter, 1985) as well as laboratory experiments on public good provision and dictator games (see, e.g., Andreoni, 1993; Chan et al., 1996; Eckel et al., 2005). Andreoni formalises his notion of warm-glow by assuming that the function U is strictly increasing in its last argument. However, his analysis of the distinctions between altruism and warm-glow demand some further assumptions. Most notably, Andreoni assumes that the private consumption and the gift of an agent are both strictly increasing functions of her wealth, which rules out quasilinear utility indices, among others. Indeed, it can easily be seen that in problem (3), the allocation choice implied by the purely altruistic utility index U = u(x1 ) + G + T would simply coincide with that induced by the purely egoistic utility index U = u(x1 ) + g1 . On a related note, Bergstrom et al. (1986, Section 2) emphasise that with quasi-homothetic utility indices, income transfers would be neutral even in a model of impure altruism like that of Andreoni. (For a related finding, see, also, Proposition 2 of Andreoni (1990)). Finally, we should recall that taking into account the boundary solutions further complicates the task of distinguishing between pure and impure altruism on the basis of allocation choice (see Footnote 16). In view of these remarks, our menu-choice approach not only provides foundations for Andreoni’s model, but also arms us with clear-cut distinctions between purely altruistic agents and those motivated by warm-glow. In particular, we have uncovered that a DM 16

For theoretical findings on crowding out under pure altruism, see Warr (1982), Roberts (1984), Bernheim (1986), and Andreoni (1988), among others. Neutrality of income distribution under pure altruism has been demonstrated by Warr (1983) and Bergstrom et al. (1986). However, these findings are subject to some exceptions: If only a subset of the agents make donations, government spending and income distribution may influence the equilibrium amount of the public good (Bergstrom et al., 1986).

13

motivated by warm-glow must value larger choice sets because she enjoys taking a prosocial action with her free will, while a purely altruistic DM is neutral towards the size of the choice set that she faces, as in property (I). Next, we show that when applied to consumptionsaving problems, our theory can reproduce Andreoni’s (1990) findings on the distinctions between warm-glow and altruism even with a utility index that is quasilinear in the warmglow component (which must be ruled out in a static setup). 3.1.

Bequeathing with Quasilinear Utility Indices

Consider two generations within a family: parents and an heir. In period 1, the parents allocate their wealth, w0 > 0, between their private consumption, x0 , and saving, w1 = w0 x0 . At the beginning of period 2, they receive an income support ⇢(w1 ) 2 [0, w0 ], which is financed by a tax on the heir. The parents allocate their adjusted income between their period 2 consumption, x1 , and a bequest, g1 = w1 + ⇢(w1 ) x1 . The heir’s initial wealth also equals w0 . She moves last and consumes all of her adjusted income, x2 = w0 + g1 ⇢(w1 ) = w0 + w1 x1 . We now examine the parents’ behaviour in a subgame perfect equilibrium. First of all, the menu that the parents face in period 2 takes the form A(x0 ) := {(x1 , x2 ) : 0  x1  w1 + ⇢(w1 ), x2 = w0 + w1

x1 }.

This menu belongs to KP , and the associated maximum possible private consumption, m1 , equals w1 + ⇢(w1 ). Thus, we also see that g1 = m1 x1 . The parents’ problem in period 1 is to make a choice among the pairs of the form (x0 , A(x0 )) for x0 2 [0, w0 ]. Let V be a utility function over {(x0 , A) : x0 2 R+ , A 2 KP } that represents the parents’ preferences. Then, the parents’ problem can be formalised as follows: max V (x0 , A(x0 )) s.t. 0  x0  w0 . (4) Moreover, following the proof of Theorem 2 (cf. Remark 3), if the parents’ preferences restricted to {(x0 , A) : A 2 KP } satisfy the properties (A1)-(A6), (A7*) and (A8) for each x0 , then we can find a w-index Ux0 : R3+ ! R such that V (x0 , A(x0 )) = max{Ux0 (x1 , x2 , g1 ) : (x1 , x2 ) 2 A(x0 )}.

(5)

Let us now consider a purely egoistic utility index Uxe0 = u(x0 ) + u(x1 ) + g1 , and a purely altruistic one Uxa0 = u(x0 ) + u(x1 ) + x2 , where u : R+ ! R+ is a function that satisfies the Inada conditions. Just as in the corresponding model of Andreoni, after substituting for g1 and x2 , we immediately see that for any fixed (x0 , w0 ), the maximisers of Uxa0 and Uxe0 over 14

the set A(x0 ) coincide. That is, in this setup, we cannot distinguish between the two types of parents based on period 2 behaviour. However, the saving behaviour of the two types are typically di↵erent because the income support influences the marginal value of saving for egoistic parents by altering their perception of freedom in period 2. For a concrete example, let us pick a pair of numbers (b, k) 2 (0, w0 ) ⇥ (0, 1) such that b kw0 , and set ⇢(w1 ) := b kw1 for w1 2 [0, w0 ]. A key feature of this specification is that ⇢(w1 ) is a decreasing function, which means that the income support is progressive. Moreover, w1 + ⇢(w1 ) is increasing, so that higher savings imply higher income in period 2. In what follows, for i = a, e, we denote by xi0 the solution of the problem (4) for the case Ux0 = Uxi 0 . Similarly, (xi1 , xi2 ) denotes the value of (x1 , x2 ) that solves the problem on the right side of (5) for x0 = xi0 and Ux0 = Uxi 0 . Finally, w1i and g i stand for the corresponding values of w1 and g, respectively. The next proposition shows that, under the given specification, altruistic parents tend to save more than egoistic parents. Moreover, among interior solutions, the slope of ⇢ has no e↵ects in the case of altruistic parents, while the saving of egoistic parents, their gift to their heir, and the heir’s consumption decrease with k. Proposition 1. (i) w1a > w1e unless w1a = w1e = 0. Moreover, u0 (w0 )  1 implies w1a > 0. (ii) For large values of w0 (which imply interior solutions), w1e , g e and xe2 are decreasing functions of k, but the behaviour of the altruistic parents and the consumption of their heir do not depend on k. Qualitatively, part (ii) of this proposition shows that egoistic parents adopt a more selfish behaviour as the income support becomes more progressive (corresponding to larger values of k). In particular, egoistic parents are sensitive to redistributive policies, unlike altruistic parents. Moreover, with the given quasilinear specification, this dichotomy is driven solely by the dynamic nature of the exercise. We relegate the formal arguments to Appendix D. 4.

Welfare Analysis: Designing Budget Sets

Policy tools studied by public economists, such as taxes or subsidies, directly influence agents’ budget sets. In this section, we will demonstrate with a simple example how our theory can be utilised to conduct welfare analysis in related problems. In particular, we will show that the warm-glow motive may make inefficient a traditional recipe that simply relies on lump-sum taxes (or quotas) for redistribution of income. Consider a planner and two agents. Agent 1 owns 1 dollar that can be allocated between the two agents. The planner wants to make sure that agent 2 receives at least ↵2 2 (0, 1) 15

dollars. More specifically, the planner’s problem is to design a budget set for agent 1 that maximises agent 1’s welfare subject to the constraint that consuming not more than ↵ := 1 ↵2 must be optimal for agent 1. Let K1 denote the collection of nonempty, closed subsets of [0, 1]. A budget set B refers to an element of K1 , while a point x1 2 B represents agent 1’s consumption. The planner is non-wasteful in the sense that every penny that is not consumed by agent 1 is transferred to agent 2. Thus, a budget set B induces the menu A(B) := {(x1 , 1 x1 ) : x1 2 B}. Note that this menu belongs to KP for any B 2 K1 . Agent 1 is endowed with a preference relation %1 over KP that satisfies the hypotheses of Theorem 2. V denotes a continuous function over KP that represents %1 , and U (x1 , x2 , ) is the associated w-index. Thus, agent 1 evaluates the budget sets with the function W (B) := V (A(B)). We define m(B) := max B = max {x1 : (x1 , x2 ) 2 A(B)}, and c (B) := arg max{U (x1 , 1 x1 , m(B) x1 ) : x1 2 B}, so that W (B) = U (x1 , 1 x1 , m(B) x1 ) for any x1 2 c (B). The planner knows agent 1’s preference relation and believes that given a budget set B, agent 1 will select an x1 2 c (B). Moreover, if c (B) contains multiple points, agent 1 will minimise her consumption over c (B) (which is equivalent to maximising agent 2’s consumption). Thus, the planner’s problem can be formalised as follows: max W (B) s.t. B 2 K1 (↵),

(P)

where K1 (↵) := {B 2 K1 : c (B) \ [0, ↵] 6= ?}.17 The problem (P) has a solution because K1 (↵) is a compact set, and W is a continuous function over K1 . It is important to observe that if agent 1 were purely altruistic, then a set B 2 K1 (↵) would solve the problem (P) i↵ B \ c ([0, ↵]) 6= ?. In particular, the set [0, ↵] would be a solution, which means that the planner can simply impose a consumption quota of ↵, or a lump-sum tax of 1 ↵. We shall next show that if agent 1 is not purely altruistic, then there may not exist any convex set that solves the problem (P), at least if the desired transfer 1 ↵ is relatively small. To this end, we assume the following: (i) U is continuously di↵erentiable on R3+ (with possibly infinite partial derivatives when the corresponding coordinate is 0). (1,0,0) (ii) U is strictly increasing in , and @U (1,0,0) > @U@x . @ 1 17

In the online appendix, we show that the second-stage choice correspondence associated with a preference relation that admits a warm-glow representation is essentially unique. Thus, given the preference relation %1 , the problem (P) does not depend on the functions V or U , subject to some minor regularity conditions.

16

Intuitively, the role of property (ii) is to ensure that the warm-glow motive is important. In particular, the second part of (ii) implies that when given the opportunity, agent 1 would prefer making a small sacrifice to consuming the whole pie. This is especially plausible if U is a concave function. Since U is continuously di↵erentiable, the second part of property (ii) can also be extended to smaller consumption levels. That is, (i) and (ii) imply that there exists a 2 [0, 1) such that @U (x01 , 1 x01 , 0) @U (x01 , 1 x01 , 0) > for every x01 2 ( , 1]. (6) @ @x1 Note that we can select a relatively small in many cases of interest. For example, if @U (x01 ,1 x01 ,0) 0 = 1 for x1 2 (0, 1], we can simply let = 0. @ We are now ready to state our first result in this section. Proposition 2. Suppose that properties (i) and (ii) hold, and let 2 [0, 1) be as in (6). If ↵ > , and if [0, 1] does not solve the problem (P), then there does not exist a convex set B 2 K1 that solves the problem (P). The message of this result is that when the desired transfer 1 ↵ is relatively small, then we cannot simply rely on quotas or lump-sum taxes to reach an efficient outcome. Rather, as we will momentarily see, we should consider non-convex budget sets that o↵er a target consumption level coupled with a suitably selected maximum level of consumption. The first step in the proof of Proposition 2 is to show that if a convex set solves the problem (P), then there exists a solution of the form [0, m] such that m ↵. In turn, if ↵ > , then property (6) implies that m cannot belong to c ([0, m]). Hence, for any x1 2 c ([0, m]) \ [0, ↵], we must have U (x1 , 1 x1 , m x1 ) > U (m, 1 m, 0). Since U is continuous, it then follows that for all sufficiently small " > 0, agent 1 would select x1 also from the set B 0 := {x1 , m + "}, and would prefer this set to [0, m]. Thus, a set of the form [0, m] cannot solve the problem (P) unless m = 1. We present the details of this proof in Appendix D. We shall now provide a tractable characterisation of the sets that solve the problem (P). Suppose that the planner wants to implement a consumption level x1 2 [0, ↵]. Since U is increasing in , the optimal way of doing so is to couple x1 with the largest consumption level m that satisfies the incentive compatibility constraint U (x1 , 1 x1 , m x1 ) U (m, 1 m, 0). Put di↵erently, if we let m⇤ (x1 ) := max{m 2 [x1 , 1] : U (x1 , 1 x1 , m x1 ) U (m, 1 m, 0)}, we have that W ({x1 , m⇤ (x1 )})

W (B) for every B 2 K1 s.t. x1 2 c(B).

17

The next step is to select a target consumption level x1 in [0, ↵]. To this end, the planner can solve the following problem of a single variable: max U (x1 , 1

x1 , m⇤ (x1 )

x1 ) s.t. x1 2 [0, ↵].

(P1)

In Appendix D, we will show that a doubleton {x⇤1 , m} with m x⇤1 solves the problem (P) i↵ x⇤1 solves the problem (P1) and m = m⇤ (x⇤1 ). More generally, we have the following characterisation. Proposition 3. A set B 2 K1 solves the problem (P) i↵ c (B) contains a solution x⇤1 of the problem (P1) such that m(B) = m⇤ (x⇤1 ). It follows that a generic solution of (P) contains a solution of (P1), say x⇤1 , coupled with the point m⇤ (x⇤1 ), and perhaps, some additional points x1  m⇤ (x⇤1 ) such that U (x1 , 1 x1 , m⇤ (x⇤1 ) x1 )  U (x⇤1 , 1 x⇤1 , m⇤ (x⇤1 ) x⇤1 ). It is also worth noting that if agent 2 takes the role of the planner, then the problem that we have just studied can also be thought of as a principal-agent problem (where agent 2 is the principal). In particular, if we denote by 1 x1 a level of activity, and let agent 1 and 2’s material payo↵s be x1 and 1 x1 , respectively, we simply obtain a version of Falk and Kosfeld’s (2006) setup.18 In this case, a lower bound for the level of the activity corresponds to an upper bound for agent 1’s material payo↵. To see why such upper bounds may backfire, suppose that agent 1 is endowed with a purely egoistic w-index U (x1 , ). Then, given a set of the form [0, m], agent 1’s problem in the second stage becomes to maximise U (x1 , ) subject to the constraint x1 + = m. Analytically, this is equivalent to a utility maximisation problem in standard consumer theory with two goods. In particular, the finding of Falk and Kosfeld, i.e. that restrictions may backfire, corresponds to those cases in which the “Marshallian demand” for the material payo↵ x1 decreases with m.19 For the purposes of welfare analysis, an interesting implication of such cases is that relaxing the upper bound m may actually lead to Pareto improvements, by securing a larger menu for agent 1 and a larger material payo↵ for agent 2. 18

The only di↵erence is that in our example, the marginal cost of the activity for agent 1 coincides with the principal’s marginal gain (and both are equal to 1). Our analysis can easily be modified for alternative scenarios. 19 For example, let ↵, ↵0 , , 0 and ✓ be positive numbers such that ↵ > > 0 > ↵0 , and define U (x1 , ) := min{↵x1 + , ↵0 x1 + 0 + ✓}. Then, U is a concave and strictly increasing function on R2+ , while the associated Marshallian demand for x1 is a decreasing function of m for ↵ ✓↵0  m  ✓ 0 . Further examples of this sort can be found in the survey of Heijman and Mouche (2012), and references therein.

18

5.

An Application to Participation in Large Elections

Explaining voter turnout in large elections has been a major challenge for political economists. The difficulty stems from the fact that when many people vote, the probability of a single voter being decisive (pivotal) is close to zero, whereas voting incurs significant costs. In an earlier attempt to resolve this paradox, Riker and Ordeshook (1968) suggested that the act of voting may be valuable per se, as the citizens may perceive it as a civic duty.20 As a notable shortcoming, the turnout rates predicted by Riker and Ordeshook’s theory are not compatible with some empirical findings on the role of voting costs and image concerns. For example, Funk (2010) shows that the introduction of mail voting in Switzerland failed to raise the aggregate turnout rate, while actually reducing the voter participation in some communities. Although mail voting must decrease the voting costs, it also makes unobservable who adheres to her “civic duty.” Thus, as B´enabou and Tirole (2006) also point out, Funk’s findings suggest that motivation crowding out and image concerns are also relevant for voters’ behaviour. Based on our warm-glow representation, in this section, we will provide an extension of Riker and Ordeshook’s theory along these lines. Let us first take a look at the calculus of voting proposed by Riker and Ordeshook. Suppose that there are two candidates, L and R. Our DM is a voter who prefers the candidate L. Assume that the victory of L will bring a material payo↵ u > 0 to the DM, whereas the victory of R is worth 0. Given other voters’ behaviour, we denote by pL the probability of being decisive for the DM if she votes (for candidate L), while P stands for the probability of winning for candidate L if the DM abstains. Finally, c denotes the cost of voting, and d the intrinsic payo↵ associated with the act of voting, as posited by Riker and Ordeshook. The implied expected payo↵ scheme reads as follows: (P + pL ) u Pu

c+d

if the DM votes, if the DM abstains.

Thus, the DM’s decision between voting and abstaining is determined by the following rule: vote if and only if pL u + d

c.

In particular, no matter how small pL might be, the DM would vote if d c. To see how our warm-glow representation can generate a similar mode of behaviour, 20 Evren (2012) provides a detailed discussion of more recent models on voter turnout that build upon warm-glow theory or classical altruism.

19

assume that in a paternalistic fashion, the DM perceives the victory of L as a public good. This means that the DM is facing a trade-o↵ between her expected (material) payo↵, ⇡1 , and the well-being of the society which depends on the winning probability of candidate L. Let us denote the latter probability by Q. Suppose that the DM resolves this trade-o↵ with a w-index U that is defined over vectors of the form (⇡1 , Q, ), where stands for DM’s sacrifice in terms of her material payo↵.21 As we explained earlier, ⇡1 associated with the act of voting equals (P + pL ) u c, while abstaining leads to ⇡1 = P u. It follows that when pL is small (as in a large election), abstaining yields a higher payo↵. In such cases, abstaining acts as the most selfish option, and = c pL u gives the cost of voting in terms of the foregone payo↵. At the same time, voting and abstaining, respectively, lead to Q = P + pL and Q = P . We thus conclude that the DM would vote if and only if U ((P + pL ) u

c,P + pL , c

pL u)

U (P u,P, 0) .

(7)

Suppose now that U (⇡1 , Q, ) := ⇡1 + ( ) for an increasing function : R+ ! R, so that we have a purely egoistic DM at hand. Then, (7) reduces to pL u + (c pL u) (0) c. Also note that (c pL u) will be approximately equal to (c) for small values of pL . Thus, the parameter d in the Riker-Ordeshook model corresponds to the warm-glow payo↵ (c pL u) (0) ⇡ (c) (0). Compared with Riker and Ordeshook, the distinctive feature of the model outlined above is that the warm-glow payo↵, (c) (0), depends on the cost of voting. Indeed, if citizens were forced to vote—say, by a prohibitively high fine on abstention—it would seem reasonable to assume that they would not attribute an intrinsic value to the act of voting. This is precisely what our model predicts: Given a fine on abstention, the cost of voting in terms of the foregone material payo↵ reduces to = c pL u , leading to a smaller warm-glow payo↵ (c pL u ) (0). Thus, a fine on abstention or a policy that reduces the voting costs, as in the case of mail voting, may actually decrease turnout rates. 6.

Concluding Remarks

This paper investigates the behavioural content of warm-glow. Our axioms generalise the standard model of menu-choice so as to accommodate the evidence on the adverse e↵ects of using reward-punishment mechanisms to incentivise prosocial behaviour. Our main representation (Theorem 2) shows that this mode of behaviour is closely related to earlier 21

In the online appendix, we discuss how our representations can be extended to such alternative domains.

20

warm-glow models, such as Riker and Ordeshook (1968) and Andreoni (1989, 1990). Moreover, our menu-choice approach opens a door to further applications on welfare analysis and consumption-saving problems. In the online appendix, we provide an extension of our theory that explicitly models second-stage behaviour. Our analysis shows that the second-stage behaviour corresponding to our theory satisfies a weak form of WARP, consistently with the experimental evidence on motivation crowding out, and in stark contrast with Saito (2015). More surprisingly, our characterisation of second-stage behaviour is remarkably similar to that in Noor and Takeoka’s (2015) model of temptation and menu-dependent self-control. Thus, it appears to be difficult to distinguish these two models based only on second-stage behaviour, while the first-stage preferences exhibit very distinct features. These findings reinforce the merits of the menu-choice framework as a tool to model warm-glow giving. A key feature of Theorem 2 is that the corresponding notion of warm-glow does not directly depend on how others are influenced by the DM’s behaviour. Although this idea is at the very core of the warm-glow literature, one can envision alternative models in which others’ payo↵ also enter the calculus of warm-glow. An extension of Theorem 2 along these lines can be found in Evren and Minardi (2013). Note, also, that we have focused on two-dimensional allocations for ease of exposition—multi-dimensional versions of our representation theorems can be proved without altering our axioms. If one views “giving” as an act of free will, as opposed to a compulsory transfer of resources, our warm-glow model can be seen as a theory of “preference for giving.” On the other hand, negative feelings, such as shame, may also motivate prosocial actions, even if the DM dislikes such actions per se. As discussed in the introduction, earlier axiomatic models have focused on such negative feelings, which are characterised by preference for smaller menus. Accordingly, to incorporate shame into our model, we would need to weaken the assumption of preference for larger menus, so that the DM can also exhibit a preference for smaller menus. The reason is that, on one side, a larger menu may enhance the DM’s freedom to behave selfishly and the potential impact of warm-glow. On the other side, a larger menu may also make the DM worse o↵ if she is attracted by a selfish option but feels ashamed to select it. These two motives would interact together giving rise to possibly di↵erent patterns of behaviour, depending on the prevailing force. Moreover, this sort of an extension may also require a weakening of our Weak Instrumentalism axiom because of the need for additional reference points. For example, in Dillenberger and Sadowski (2012) the best alternative with respect to a normative ranking of the allocations acts as a reference point. Saito (2015) pursues some of these ideas and combines both types of motivations in a particular setting. His model may help design experiments to assess the relative importance 21

of shame and preference for giving in subjects’ behaviour. Our analysis in the online appendix also shows that the second-stage behaviour implied by our theory is essentially unique. However, the utility indices in our representations do not possess strong uniqueness properties. One way to solve this problem may be to study preferences over lotteries of menus. Another promising approach is to focus on menus of lotteries, as in Saito (2015). However, by itself, enlarging the domain (and invoking usual expected utility hypotheses) may not be sufficient to obtain uniquely defined representations. Rather, it may be necessary to impose further structure on utility indices, which might rule out some important applications. For example, Andreoni (1989, 1990) simply works with quasiconcave utility indices that induce a “normal” demand function, which are outside the scope of Saito’s additively separable form. We leave the investigation of this uniqueness problem to further research. Appendix A: Extension Continuity The axiom (A8) in the statement of Theorem 2 reads as follows: A8: Extension Continuity (EC). Let x0 , y 0 2 X be such that {x0 , y 0 } 2 K\KP . Then, for any A, B 2 KP with A B, there exists a neighbourhood N of {x0 , y 0 } such that one of the following holds: (i) A {x, y} for every {x, y} 2 N \ KP . (ii) {x, y} B for every {x, y} 2 N \ KP . The warm-glow representation in Theorem 2 requires the existence of a continuous weak order %0 on K that coincides with % on KP . This can be assured only if % is well-behaved on KP . It turns out that property (EC) is all we need to this end. The axiom tells us that if A B, then all doubletons in KP that are sufficiently close to a set {x0 , y 0 } 2 K\KP must either be strictly worse than A (as in the case B %0 {x0 , y 0 }) or strictly better than B (as in the case {x0 , y 0 } 0 B). Appendix B: Proof of Theorem 1 As it is a routine exercise, we omit the “if” part of the proof of Theorem 1. To prove the “only if” part, let % be a binary relation on A := K that satisfies the axioms (WO), (SM), (WI), (FS), (SiP), (C), and (CM). For each A 2 K, pick a point y ⇤ (A) 2 K such that y1⇤ (A) = maxy2A y1 . Define A := {{x, y ⇤ (A)} : x 2 A}, and note that A (equipped with the Hausdor↵ metric) is homeomorphic to A and compact. Thus, % admits a maximal set in A by (C). That is, 22

there exists an allocation x(A) in A such that {x(A), y ⇤ (A)} % {x, y ⇤ (A)} for every x 2 A. The following claim proves a related observation. Claim 1. For any A 2 K, we have A ⇠ {x(A), y ⇤ (A)}. Proof. Fix a set A 2 K, and let {x1 , ..., xn , ...} be a countable, dense subset of A. For every n 2 N, put An := {x1 , ..., xn } [ {x(A), y ⇤ (A)}. Then, (SM) implies A1 % {x(A), y ⇤ (A)}. Moreover, by (WI), we have A1 ⇠ {x1 , y ⇤ (A)} or A1 ⇠ {x(A), y ⇤ (A)}. As {x1 , y ⇤ (A)} {x(A), y ⇤ (A)}, either equivalence implies A1 - {x(A), y ⇤ (A)}, that is, A1 ⇠ {x(A), y ⇤ (A)}. Similarly, either A2 ⇠ {x2 , y ⇤ (A)} or A2 ⇠ A1 , and in both cases, we have A2 ⇠ {x(A), y ⇤ (A)}. Inductively, it follows that An ⇠ {x(A), y ⇤ (A)} for every n. Moreover, since the sequence A1 , A2 , ... is uniformly bounded in Euclidean norm and increases with respect to set inS n clusion, it is well known that An ! cl( 1 n=1 A ) in Hausdor↵ metric (see, e.g., Dekel et S n al., 2001, Lemma 5). In turn, cl( 1 n=1 A ) equals A by construction. Hence, (C) implies A ⇠ {x(A), y ⇤ (A)}, as we sought. ⇤ For future use, note that in the above proof we have not utilised (FS) or (CM). Next, we prove another general claim that does not require these two axioms. Claim 2. Let x, x0 , y, y 0 2 X be such that x0 {x} {x, y} {y}.

x and y 0

y. Then, {x0 , y 0 } % {x, y} unless

Proof. Suppose that {x} {x, y} {y} does not hold. By (SM), this means that either {x, y} ⇠ {x} or {x, y} ⇠ {y}. Moreover, (SM) and (SiP) imply {x0 , y 0 } % {x0 } % {x} and {x0 , y 0 } % {y 0 } % {y}. As % is transitive, we conclude that {x0 , y 0 } % {x, y}. ⇤ Next claim makes use of (FS) and (CM) to strengthen the conclusion of Claim 2. Claim 3. Let x, x0 , y, y 0 2 X be such that x0

x and y 0

y. Then, {x0 , y 0 } % {x, y}.

Proof. In view of Claim 2, without loss of generality we can assume {x} {x, y} {y}. 0 By relabelling if necessary, assume also y1 x1 . Then, (FS) implies {x, y } % {x, y} since y10 y1 . It remains to show that {x0 , y 0 } % {x, y 0 }. By applying Claim 2 again, we can assume {x} {x, y 0 } {y 0 }. Since y10 x1 and x0 x, the desired conclusion follows from (CM): {x0 , y 0 } % {x, y 0 }. ⇤ Recall that when endowed with the Hausdor↵ metric, the space of all nonempty, compact subsets of R2 is separable. Then, as a subspace, K is also separable. Hence, Debreu’s classical theorem implies that there exists a continuous function V : K ! R such that A % B i↵ V (A) V (B), for every A, B 2 K. The following claim characterises the main feature of f-indices compatible with V . Claim 4. Let U : X ⇥ R+ ! R be a weakly increasing function. Then, properties (ii)-(iv) below hold simultaneously if and only if property (i) holds. 23

(i) max U (x, y1⇤ (A)) = V (A) for all A 2 K. x2A

(ii) U (x, x1 ) = V ({x}) for all x 2 X.

(iii) U (x, )  V ({x, ( , 0)}) for all (x, ) 2 X ⇥ R+ with

(iv) U (x, ) = V ({x, ( , 0)}) for all (x, ) 2 X ⇥ R+ with {( , 0)}.

x1 . x1 and {x}

{x, ( , 0)}

Proof. First, suppose that (i) holds. By letting A := {x}, we directly see that (ii) must also hold. Now, take any (x, ) 2 X ⇥R+ with x1 , so that = y1⇤ (A0 ) where A0 := {x, ( , 0)}. Then, by (i) and (ii), V (A0 ) = max{U (x, ), U (( , 0), )}, implying that V (A0 ) U (x, ). 0 0 This verifies (iii). Moreover, A {( , 0)} means V (A ) = max{U (x, ), U (( , 0), )} > U (( , 0), ). In turn, the latter statement is equivalent to V (A0 ) = U (x, ) > U (( , 0), ). In particular, {x} A0 {( , 0)} implies V (A0 ) = U (x, ), which verifies (iv). Conversely, suppose that (ii)-(iv) hold, and take any A 2 K. Let us write x instead of x(A), and y ⇤ instead of y ⇤ (A). From Claim 1 and the definition of x, it follows that V (A) = V ({x, y ⇤ })

V ({x, y ⇤ }) for x 2 A.

(B.1)

Moreover, Claim 3 implies V ({x, y ⇤ })

V ({x, (y1⇤ , 0)}) for x 2 A.

(B.2)

By property (iii), we also have V ({x, (y1⇤ , 0)})

U (x, y1⇤ ) for x 2 A.

Thus, by combining (B.1)-(B.3), we see that V (A)

(B.3)

sup U (x, y1⇤ ). x2A

To prove the converse inequality, it suffices to show that V ({x, y ⇤ })  max⇤ U (x, y1⇤ ) .

(B.4)

x2{x,y }

Note that max {V ({x}), V ({y ⇤ })} = max {U (x, x1 ) , U (y ⇤ , y1⇤ )}  max⇤ U (x, y1⇤ ) by propx2{x,y }

erty (ii) and weak monotonicity of U . Hence, (B.4) directly holds if max {V ({x}), V ({y ⇤ })} = V ({x, y ⇤ }). Assume max {V ({x}), V ({y ⇤ })} < V ({x, y ⇤ }). Then, (FS) implies V ({x, y ⇤ })  V ({x, (y1⇤ , 0)}). Since V ({(y1⇤ , 0)})  V ({y ⇤ }), it follows that max {V ({x}), V ({(y1⇤ , 0)})} < V ({x, (y1⇤ , 0)}). By (iv), we conclude that U (x, y1⇤ ) = V ({x, (y1⇤ , 0)}), implying V ({x, y ⇤ })  U (x, y1⇤ ). This proves (B.4). Hence, V (A) = sup U (x, y1⇤ ) = max⇤ U (x, y1⇤ ). Finally, the latter equality implies that x2A

x2{x,y }

24

the function x ! U (x, y1⇤ ) attains its maximum over A (either at x or y ⇤ ). This proves the property (i). ⇤ We complete the proof of Theorem 1 with the following claim. Claim 5. Define a function U : X ⇥ R+ ! R as U (x, ) := V ({x, ( , 0)}) for all (x, ) 2 X ⇥ R+ . The function U is an f-index for % . Proof. In view of Claim 4, it suffices to show that U is a weakly increasing, continuous function that satisfies the properties (ii)-(iv) in Claim 4. Claim 3 and the definition of V directly imply that U is weakly increasing. Moreover, since V and the function (x, ) ! {x, ( , 0)} are both continuous, so is U . To verify property (ii) in Claim 4, fix an x 2 X. Note that {x, (x1 , 0)} % {x} by (SM), and {x} = {x, x} % {x, (x1 , 0)} by Claim 3. Thus, {x, (x1 , 0)} ⇠ {x}, implying that U (x, x1 ) := V ({x, (x1 , 0)}) = V ({x}) , as we seek. Finally, note that U trivially satisfies the properties (iii) and (iv) in Claim 4. ⇤ Appendix C: Proof of Theorem 2 We omit the “if” part of the proof which is a routine exercise. Let % be a binary relation on A := KP that satisfies the axioms (WO), (SM), (WI), (FS), (SiP), (C), (CCM), and (EC). For each A 2 KP , let y ⇤ (A) denote the unique element of A such that y1⇤ (A) = maxy2A y1 . Notice that if a set A belongs to KP , any compact nonempty subset of A also belongs to KP . Hence, we can repeat the arguments in the proof of Claim 1 to show that for each A 2 KP , there exists a point x(A) 2 A such that A ⇠ {x(A), y ⇤ (A)} % {x, y ⇤ (A)}

for every x 2 A.

(C.1)

Let us set x _ y := (max{x1 , y1 }, max{x2 , y2 }) for every x, y 2 X, and x ⇧ := (x1 + , 0) for every (x, ) 2 X ⇥ R+ . Note that for any (x, ) 2 X ⇥ R+ , the point x ⇧ belongs to X, while the set {x, x ⇧ } belongs to KP i↵ (i) = 0 and x2 = 0 or (ii) > 0 and x2 > 0. Observe that, as in the proof of Theorem 1, there exists a continuous function V : KP ! R such that A % B i↵ V (A) V (B), for every A, B 2 KP . The following claim is an analogue of Claim 4 for the present setup. Claim 6. Let U : X ⇥ R+ ! R be a weakly increasing function. Then, properties (ii)-(iv) below hold simultaneously if and only if property (i) holds. (i) max U (x, y1⇤ (A) x2A

x1 ) = V (A) for all A 2 KP .

(ii) U (x, 0) = V ({x}) for all x 2 X.

(iii) U (x, )  V ({x, x ⇧ }) for all (x, ) 2 X ⇥ R+ such that {x, x ⇧ } 2 KP . 25

(iv) U (x, ) = V ({x, x ⇧ }) for all (x, ) 2 X ⇥ R+ such that {x, x ⇧ } 2 KP and {x} {x, x ⇧ } {x ⇧ }. Proof. As in the proof of Claim 4, (ii), (iii), and (iv) easily follow from (i) upon letting A := {x} and A0 := {x, x ⇧ }. To prove the converse implication, suppose that (ii)-(iv) hold. Take any A 2 KP , and set x := x(A), y ⇤ := y ⇤ (A). The expression (C.1) means that V (A) = V ({x, y ⇤ }) = max V ({x, y ⇤ }).

(C.2)

x2A

For each x 2 A, set x := y1⇤ x1 . Note that (x, x ) 2 X ⇥ R+ . Moreover, if x is distinct from y ⇤ , then x and x2 are positive numbers because A consists of Pareto efficient allocations. Hence, {x, x ⇧ x } = {x, (y1⇤ , 0)} 2 KP for every x 2 A\{y ⇤ }. Thus, property (iii) implies V ({x, (y1⇤ , 0)}) U (x, x ) for x 2 A\{y ⇤ }. (C.3) Furthermore, we have that V ({x, y ⇤ })

V ({x, (y1⇤ , 0)}

for x 2 A\{y ⇤ }.

(C.4)

Indeed, if V ({x, (y1⇤ , 0)}) > max{V ({x}), V ({(y1⇤ , 0)})}, then (C.4) is a direct consequence of (FS). Otherwise, it is implied by the analogue of Claim 2. From (C.2)-(C.4), it follows that V (A) sup U (x, x ). Furthermore, since y⇤ = 0, (SM) and property (ii) imply V (A) sup U (x, x ), as we seek.

x2A\{y ⇤ } ⇤

V ({y }) = U (y ⇤ ,

y ⇤ ).

We conclude that V (A)

x2A

As in Claim 4, we will complete the proof by showing that V ({x, y ⇤ })  max⇤ U (x, x2{x,y }

x ).

Since max {V ({x}), V ({y ⇤ })} = max {U (x, 0) , U (y ⇤ , 0)}  max⇤ U (x, x2{x,y }

(C.5) x ),

without loss of

generality, we can assume max {V ({x}), V ({y ⇤ })} < V ({x, y ⇤ }). In this case, x and y ⇤ must be distinct, and hence, {x, (y1⇤ , 0)} belongs to KP . Moreover, (FS) implies V ({x, y ⇤ })  V ({x, (y1⇤ , 0)}). Since V ({(y1⇤ , 0)})  V ({y ⇤ }), it follows that max {V ({x}), V ({(y1⇤ , 0)})} < V ({x, (y1⇤ , 0)}). Then, by property (iv), we have U (x, y1⇤ x1 ) = V ({x, (y1⇤ , 0)}). We conclude that V ({x, y ⇤ })  U (x, y1⇤ x1 ). This proves (C.5). ⇤ In what follows, we will define a continuous function U on X ⇥ R+ . The construction of U will build upon Claim 6, but first we need to extend the function V to include the 26

doubletons in the closure of KP . Set KP2 := cl{{x, y} : x, y 2 X, {x, y} 2 KP }.

(C.6)

We proceed with some technical observations. Claim 7. KP is a connected set. Proof. Let us show that KP is path-connected. Fix a pair of sets A, B 2 KP and a point x 2 X. Note that for any C 2 KP and ↵ 2 [0, 1], the set ↵C + (1 ↵){x} := {↵z + (1 ↵)x : z 2 C} belongs to KP . For each t 2 [0, 1], put f (t) :=

(

(1 2t)A + 2t{x} if 0  t  1/2, (2 2t){x} + (2t 1)B if 1/2 < t  1.

As we just noted, the function f maps [0, 1] into KP . Moreover, f is a continuous mapping, for limt!0.5+ f (t) = limt!0.5 f (t) = {x}. Finally, note that f (0) = A and f (1) = B. This completes the proof. ⇤ Claim 8. Let {x, y} 2 KP2 , and take a sequence {xn , y n } in KP that converges to {x, y}. Then: (i) V ({xn , y n }) converges to a finite number.

(ii) lim V ({xn , y n }) = lim V ({x0n , y 0n }) for any other sequence {x0n , y 0n } in KP that converges to {x, y}. Proof. By relabelling if necessary, assume y1n xn1 and xn2 y2n for every n. Note that by definition of the Hausdor↵ metric, lim xn and lim y n exist, and we have {x, y} = {lim xn , lim y n }. We shall now show that the sequence V ({xn , y n }) is bounded. Since they are convergent, the sequences xn and y n are bounded. Let x0 2 R2++ be such that x0 xn _ y n for every n. Clearly, {x0 , (2x01 , 0)} 2 KP . Moreover, {x0 , (2x01 , 0)} % {xn , y n } for every n.

(C.7)

Indeed, for any n, if {xn , y n } ⇠ {xn } or {xn , y n } ⇠ {y n }, then (C.7) follows from (SiP), (SM) and (WO) as in Claim 2. On the other hand, if {xn } {xn , y n } {y n }, then xn and y n are distinct, and {xn , (y1n , 0)} belongs to KP . Thus, in this case, {xn , (y1n , 0)} % {xn , y n } by (FS). In turn, this implies that {xn } {xn , (y1n , 0)} {(y1n , 0)} because % is transitive and {y n } % {(y1n , 0)} by (SiP). Now, since x0 xn and 2x01 x01 y1n xn1 , we can apply (CCM) to obtain {x0 , (2x01 , 0)} % {xn , (y1n , 0)}. Thus, (C.7) holds. Also note that (C.7) simply means V ({x0 , (2x01 , 0)}) V ({xn , y n }) for every n. 27

Since V ({xn , y n }) is bounded, the proof of (i) will be complete if we can show that for any two convergent subsequences V ({xnk , y nk }) and V ({xnl , y nl }), we have lim V ({xnk , y nk }) = lim V ({xnl , y nl }). k

l

(C.8)

Assume by contradiction that limk V ({xnk , y nk }) > liml V ({xnl , y nl }) for a pair of convergent subsequences V ({xnk , y nk }), V ({xnl , y nl }). Pick two numbers "1 and "2 such that limk V ({xnk , y nk }) > "1 > "2 > liml V ({xnl , y nl }). Then, V ({xnk , y nk }) > "1 > "2 > V ({xnl , y nl }) for all sufficiently large k and l.

(C.9)

Since V is a continuous function on the connected set KP , the set V (KP ) is an interval. Thus, (C.9) implies that there exists a pair of sets A, B 2 KP such that V (A) = "1 and V (B) = "2 . Moreover, we obviously have limk {xnk , y nk } = {x, y} = liml {xnl , y nl }. In view of (C.9), it follows that any neighbourhood N of {x, y} contains a pair of sets {xnk , y nk }, {xnl , y nl } such that {xnk , y nk } A B {xnl , y nl }. Since this contradicts (EC), we conclude that part (i) of the claim holds. To prove part (ii), take any other sequence {x0n , y 0n } in KP that converges to {x, y}. By part (i) of the claim, lim V ({x0n , y 0n }) exists. Suppose that this limit is distinct from lim V ({xn , y n }), say lim V ({xn , y n }) > lim V ({x0n , y 0n }). As in the proof of (C.8), we can find A, B 2 KP such that {xn , y n } A B {x0n , y 0n } for all sufficiently large n. In turn, this contradicts (EC) since lim{xn , y n } = {x, y} = lim{x0n , y 0n }. ⇤ For each {x, y} 2 KP2 , define V ({x, y}) := lim V ({xn , y n }) for every sequence {xn , y n } in KP that converges to {x, y}. In view of Claim 8, V (·) is a welldefined function on KP2 . Moreover, for any {x, y} 2 KP , we have V ({x, y}) = V ({x, y}) since a constant sequence that equals {x, y} trivially converges to {x, y}. In particular, V ({x}) = V ({x}) for every x 2 X. It should also be noted that {x, x ⇧ } 2 KP2 for any (x, ) 2 X ⇥ R+ .

(C.10)

To see this point, let (x, ) 2 X ⇥ R+ . For each n 2 N, set xn := (x1 , x2 + n1 ) and y n := (x1 + + n1 , 0). Then, {xn , y n } is a sequence in KP converging to {x, x ⇧ }, as we seek. In view of (C.10), for any (x, ) 2 X ⇥ R+ , the set {x, x ⇧ } belongs to the domain of 28

the function V . The next claim establishes some useful facts about the function V . Claim 9. (i) V is continuous on KP2 . (ii) V ({x, y})

V ({x}) for any {x, y} 2 KP2 .

(iii) V ({x0 , x0 ⇧ 0 }) V ({x, x ⇧ }) for all (x, ), (x0 , 0 ) 2 X ⇥ R+ such that (x0 , 0 ) (x, ). Proof. We start with the proof of (i). Let {xn , y n } be a sequence in KP2 that converges to {x, y}. By definition of V , for each n, there exists a set {x0n , y 0n } 2 KP such that dH ({x0n , y 0n }, {xn , y n }) < 1/n and

V ({x0n , y 0n })

V ({xn , y n }) < 1/n.

(C.11)

Since lim{xn , y n } = {x, y}, the former expression in (C.11) implies lim{x0n , y 0n } = {x, y}. Thus, V ({x, y}) := lim V ({x0n , y 0n }). Hence, from the latter expression in (C.11), it follows that lim V ({xn , y n }) exists, and we have lim V ({xn , y n }) = V ({x, y}). This proves (i). Now, take any {x, y} 2 KP2 , and let {xn , y n } be a sequence in KP that converges to {x, y} so that {x, y} = {lim xn , lim y n }. Without loss of generality, assume x = lim xn . Then, V ({x, y}) := lim V ({xn , y n }) lim V ({xn }) := V ({x}), where the weak inequality follows from (SM). This proves (ii). It remains to prove part (iii). Let (x, ), (x0 , 0 ) 2 X ⇥ R+ be such that (x0 , 0 ) (x, ). Note that if V ({x, x ⇧ }) = max{V ({x}), V ({x ⇧ })}, (SiP) and part (ii) of the claim imply V ({x, x ⇧ })  V ({x0 , x0 ⇧ 0 }). Therefore, without loss of generality, assume V ({x, x ⇧ }) > max{V ({x}), V ({x ⇧ })}.

(C.12)

Let {xn , y n } be a sequence in KP such that {x, x ⇧ } = lim{xn , y n }.

(C.13)

By relabelling if necessary, assume y1n xn1 and xn2 y2n for every n. Since x1 = min{z1 : z 2 {x, x ⇧ }} and xn1 = min{z1 : z 2 {xn , y n }} for each n, from (C.13) it follows that x1 = lim min{z1 : z 2 {xn , y n }} = lim xn1 . Similarly, (x ⇧ )1 = lim max{z1 : z 2 {xn , y n }} = lim y1n , while x2 = lim max{z2 : z 2 {xn , y n }} = lim xn2 and (x ⇧ )2 = lim min{z2 : z 2 {xn , y n }} = lim y2n . Hence, x = lim xn

and x ⇧

= lim y n .

(C.14)

By definition of V , (C.12)-(C.14) imply {xn } {xn , y n } {y n } for all sufficiently large n. Without loss of generality, assume {xn } {xn , y n } {y n } for every n. Then, {xn , (y1n , 0)} belongs to KP for every n because xn and y n are distinct, meaning that y1n > xn1 29

and xn2 > y2n 0. For each n, set n := y1n xn1 . Observe that lim n = (x ⇧ )1 x1 = by (C.14). Moreover, xn ⇧ n = (y1n , 0). Hence, (FS) and (SiP) imply {xn } {xn , xn ⇧ n } {xn ⇧ n } for every n. n Finally, for each n, set x0n := x0 _ xn and 0n := max{ 0 , n }. Then, 0n > 0 and x0n xn2 > 0. Hence, {x0n , x0n ⇧ 0n } 2 KP for every n. Furthermore, by construction, 2 x0n xn and (x0n ⇧ 0n )1 x0n (xn ⇧ n )1 xn1 . Therefore, (CCM) implies {x0n , x0n ⇧ 0n } % 1 {xn , xn ⇧ n } for every n. To complete the proof, observe that lim x0n = x0 _ x = x0 , lim 0n = max{ 0 , } = 0 , lim{x0n , x0n ⇧ 0n } = {x0 , x0 ⇧ 0 } and lim{xn , xn ⇧ n } = {x, x ⇧ } by continuity of the operators _, max and ⇧. Hence, we conclude that V ({x0 , x0 ⇧ 0 }) := lim V ({x0n , x0n ⇧ 0n }) lim V ({xn , xn ⇧ n }) =: V ({x, x ⇧ }). ⇤ The following claim completes the proof of Theorem 2. Claim 10. Define U : X ⇥ R+ ! R as U (x, ) := V ({x, x ⇧ }) for every (x, ) 2 X ⇥ R+ . Then, U is a weakly increasing and continuous function on X ⇥ R+ that satisfies properties (ii)-(iv) of Claim 6. Proof. That U satisfies the properties (iii) and (iv) of Claim 6 simply follows from the definition of U . To verify property (ii) of Claim 6, first note that Claim 9(ii) implies U (x, 0) V ({x}) for any x 2 X. For the converse inequality, suppose, by contradiction, that there exists an x 2 X such that U (x, 0) > V ({x}), i.e. V ({x, (x1 , 0)}) > V ({x}). By (SiP), the latter inequality implies V ({x, (x1 , 0)}) > max{V ({x}), V ({(x1 , 0)})}. Hence, as in the proof of Claim 9(iii), there exists a sequence {xn , y n } in KP such that {xn } {xn , y n } {y n } for every n and lim{xn , y n } = {x, (x1 , 0)}. Observe that we must have lim xn1 = lim y1n = x1 . Moreover, by relabelling if necessary, we can assume y1n > xn1 and xn2 > y2n 0 for every n. Then, n n n n n n n n {x , (y1 , n+1 x2 )} 2 KP for every n, and (FS) implies {x , (y1 , n+1 x2 )} % {xn , y n } for every n n n. Furthermore, x2 = lim xn2 = lim n+1 xn2 , and hence, {x} = {x, x} = lim{xn , (y1n , n+1 xn2 )}. n It follows that V ({x}) := lim V ({xn , (y1n , n+1 xn2 )}) lim V ({xn , y n }) =: V ({x, (x1 , 0)}), a contradiction. Thus, property (ii) of Claim 6 is proven. To prove that U is continuous on X ⇥ R+ , note that if (x, ) := lim(xn , n ) for a sequence (xn , n ) in X ⇥ R+ , then {x, x ⇧ } = lim{xn , xn ⇧ n }, and hence, V ({x, x ⇧ }) = lim V ({xn , xn ⇧ n }) by continuity of V . Finally, Claim 9(iii) directly implies that U is a weakly increasing function on X ⇥ R+ . ⇤ Remark 3. The proof of the following observation is implicit in the arguments above: Let % be a binary relation on A := KP that satisfies the axioms (A1)-(A6), (A7*) and (A8). 30

Then, for any continuous function V : KP ! R that represents %, there exists a w-index Uw such that V (A) = max Uw (x, max y1 x1 ) for every A 2 KP . x2A

y2A

Appendix D: Remaining Proofs Proof of Proposition 1. Let x1 be the number that satisfies u0 (x1 ) = 1. Since both utility functions are additive, we can separate parents’ problems into two periods. For both types of parents, as a function of their savings, optimal period 2 consumption is given by x1 (w1 ) = w1 + ⇢(w1 ) if w1 + ⇢(w1 )  x1 , and x1 (w1 ) = x1 if w1 + ⇢(w1 ) > x1 . For altruistic parents, the corresponding period 2 utility is as follows: Va (w1 ) :=

(

u(w1 + ⇢(w1 )) + w0 ⇢(w1 ) u(x1 ) + w0 + w1 x1

if w1 + ⇢(w1 )  x1 , if w1 + ⇢(w1 ) > x1 .

In turn, egoistic parents’ period 2 utility is given by: Ve (w1 ) :=

(

u(w1 + ⇢(w1 )) u(x1 ) + w1 + ⇢(w1 )

x1

if w1 + ⇢(w1 )  x1 , if w1 + ⇢(w1 ) > x1 .

It is easily verified that Va and Ve are concave, di↵erentiable functions of w1 2 [0, w0 ]. With these definitions, the problem (4) for type i parents is equivalent to maximisation of u(x0 )+Vi (w0 x0 ) subject to x0 2 [0, w0 ]. The first order conditions for the latter problem are u0 (x0 ) Vi0 (w0 x0 ) with equality if x0 < w0 . (D.1) To prove the first statement in (i), suppose xa0 xe0 . Then, u0 (xe0 ) u0 (xa0 ) and Ve0 (w0 xa0 ) Ve0 (w0 xe0 ) by concavity of u and Ve . Moreover, it is easily checked that Va0 (w1 ) Ve0 (w1 ) = ⇢0 (w1 ) = k for every w1 2 [0, w0 ]. In particular, Va0 (w0 xa0 ) > Ve0 (w0 xa0 ). Finally, (D.1) implies u0 (xa0 ) Va0 (w0 xa0 ). By combining these inequalities, we see that u0 (xe0 ) > Ve0 (w0 xe0 ). From (D.1) it follows that xa0 xe0 implies xe0 = w0 = xa0 , as we sought. Next, note that b x1 implies Va0 (0) = 1 u0 (b) > u0 (w0 ), where the last inequality follows from strict concavity of u. Hence, in this case, x0 = w0 does not satisfy (D.1) for i = a. Conversely, if b < x1 , it readily follows that Va0 (0) = u0 (b)(1 k) + k > 1. Hence, (D.1) fails again at x0 = w0 provided that u0 (w0 )  1. Thus, we conclude that u0 (w0 )  1 implies xa0 < w0 . This completes the proof of (i). To prove (ii), observe that Ve0 (w1 ) 1 k for every w1 2 [0, w0 ]. Let wk be the number such that u0 (wk x1 ) = 1 k. Then, w0 > wk and xe0 w0 x1 would imply u0 (xe0 )  u0 (w0 x1 ) < 1 k, which contradicts (D.1). Hence, w0 > wk implies xe0 < w0 x1 , i.e. 31

x1 < w1e . Since Va0 > Ve0 , by the same logic we also see that x1 < w1a for every w0 > wk . Thus, for any such w0 , (D.1) implies u0 (xa0 ) = 1 and u0 (xe0 ) = 1 k. It follows that xe0 is an increasing function of k, while xa0 = x1 (over the region w0 > wk ). Consequently, w1e and w1e + ⇢(w1e ) are decreasing with k, while w1a and w1a + ⇢(w1a ) are constant. Since xe1 = xa1 = x1 , it also follows that g e and xe2 are decreasing with k, but g a and xa2 remain constant. ⇤ Proof of Proposition 2. Take a convex set B 2 K1 (↵) that solves the problem (P), and put m := m (B). We shall first show that [0, m] 2 K1 (↵). Pick an x1 2 c (B) \ [0, ↵] and an x01 2 c ([0, m]). Observe that since B is a convex set, either x01 2 B or x01 < min B. Moreover, the former case implies x1 2 c ([0, m]), while the latter case implies x01  ↵. Thus, c ([0, m]) \ [0, ↵] is nonempty in either case. That is, [0, m] 2 K1 (↵), as we sought. Next, note that W (B)  W ([0, m]) since B ✓ [0, m]. It follows that [0, m] solves the problem (P) as well. Moreover, m < ↵ implies W ([0, m])  W ([0, ↵]) . Hence, in this case [0, ↵] also solves the problem (P). Thus, without loss of generality we can assume that B = [0, m] and that m ↵. Now, suppose that [0, 1] is not a solution to (P), so that m < 1. Also assume ↵ > . Observe that at " = 0, the derivative of the function U (m ", 1 m, ") with respect to " m,0) @U (m,1 m,0) is equal to @U (m,1 . Thus, by (6), this derivative is positive, which implies @ @x1 that U (m ", 1 m, ") > U (m, 1 m, 0) for all sufficiently small " 2 (0, m). Pick such an " and note that U (m ", 1 (m "), ") U (m ", 1 m, ") because U is weakly increasing. Since m " 2 B, it follows that m 2 / c (B). That is, U (x1 , 1 x1 , m x1 ) > U (m, 1 m, 0). Then, by continuity of U , there exists a 2 (0, 1 m) such that U (x1 , 1 x1 , m + x1 ) > U (m + , 1 (m + ), 0). The latter inequality means that c ({x1 , m + }) = {x1 }. Since m +  1 and x1  ↵, we conclude that {x1 , m + } 2 K1 (↵). Moreover, U (x1 , 1 x1 , m + x1 ) > U (x1 , 1 x1 , m x1 ), i.e. W ({x1 , m + }) > W (B), because U is increasing in . As we sought, this contradicts the hypothesis that B solves the problem (P). ⇤ Proof of Proposition 3. For any x1 2 [0, 1], set (x1 ) := {m 2 [x1 , 1] : U (x1 , 1 x1 , m x1 ) U (m, 1 m, 0)}, and observe that is a compact valued, upper hemicontinuous correspondence. Thus, m⇤ (x1 ) := max (x1 ) is a well-defined, upper semicontinuous function. Since U is weakly increasing and continuous, the function x1 ! U (x1 , 1 x1 , m⇤ (x1 ) x1 ) is also upper semicontinuous. Hence, the problem (P1) has a solution x1 2 [0, ↵]. Let W ⇤ be the value of the problem (P). It is clear that {x1 , m⇤ (x1 )} 2 K1 (↵), and hence, W ⇤ W ({x1 , m⇤ (x1 )}). To establish the converse inequality, let B 2 K1 (↵) be a solution of the problem (P), and pick a point x1 2 c (B) \ [0, ↵], so that W ⇤ = U (x1 , 1 x1 , m(B) x1 ). Observe that x1 2 c (B) implies m(B) 2 (x1 ). Thus, U (x1 , 1

x1 , m(B)

x1 )  U (x1 , 1

x1 , m⇤ (x1 ) 32

x1 )  U (x1 , 1

x1 , m⇤ (x1 )

x1 ).

Here, the former inequality holds because U is increasing in , and the latter inequality follows from the definition of x1 . Also note that W ({x1 , m⇤ (x1 )}) = U (x1 , 1 x1 , m⇤ (x1 ) x1 ). Hence, W ⇤ = W ({x1 , m⇤ (x1 )}), as we sought. It follows that for any B 0 2 K1 , if c (B 0 ) contains a solution x⇤1 of the problem (P1) such that m(B 0 ) = m⇤ (x⇤1 ), then W (B 0 ) = U (x⇤1 , 1 x⇤1 , m⇤ (x⇤1 ) x⇤1 ) = U (x1 , 1 x1 , m⇤ (x1 ) x1 ) = W ⇤ . Thus, any such B 0 solves the problem (P). Conversely, for any B that solves the problem (P) and any x1 2 c (B) \ [0, ↵], we have U (x1 , 1 x1 , m(B) x1 ) = W ⇤ = U (x1 , 1 x1 , m⇤ (x1 ) x1 ) U (x1 , 1 x1 , m⇤ (x1 ) x1 ). Since U is increasing in , it clearly follows that m(B) = m⇤ (x1 ). Combining this observation with the equality U (x1 , 1 x1 , m(B) x1 ) = U (x1 , 1 x1 , m⇤ (x1 ) x1 ), we also see that x1 solves the problem (P1). This completes the proof. ⇤

33

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36

Online Appendix (Not for publication) This document contains the following supplementary material: (i) a study of second-stage choice behaviour; (ii) two representation theorems which extend our warm-glow model (Theorem 2); and (iii) a discussion of how our theory can be extended to alternative sets of social outcomes. Appendix E: On Second-Stage Behaviour In this section, we take as an additional primitive a choice correspondence C on KP . For any A 2 KP , C(A) is a nonempty subset of A that represents the set of allocations that the DM may choose from A in the second stage. Our purpose is to explore the conditions under which we can find a w-index that induces the same choice correspondence as C in the second stage.22 That is, we seek a w-index U such that C = CU , where CU is defined as, for every A 2 KP , CU (A) :=

⇢

x b2A:U

✓

x b, max y1

x b1

y2A

◆

= max U x2A

✓

x, max y1 y2A

x1

◆

.

Observe that for any A 2 KP , there exists a unique point y ⇤ (A) 2 K such that y1⇤ (A) = maxy2A y1 . We start by imposing two properties, Weak WARP and Sophistication. A similar formulation of these assumptions appears in Noor and Takeoka (2015). Weak WARP (H1). Let A, B 2 KP such that B ✓ A. Then, y ⇤ (A) 2 B and C(A)\B 6= ; imply C(A) \ B = C(B). In our warm-glow representation, the most selfish option acts as a reference point that influences the agent’s second-stage behaviour. Thus, (H1) asserts that the conclusion of classic WARP holds necessarily only if the most selfish option in B coincides with the most selfish one in the larger set A. In particular, the DM may switch from a prosocial allocation to a more selfish one upon a change in the most selfish option. As noted in the paper, this is in stark contrast with Saito (2015), which satisfies the classical WARP in the second stage. Next, consider two menus A and B such that B ✓ A. If C(A) \ B = ;, then it must be that A is strictly preferred to B. The following axiom formalises this observation. Sophistication (H2). Let B 2 KP and x 2 X\B such that B [ {x} 2 KP . Then, 22

A similar analysis can be carried out for our first representation.

1

y ⇤ (B [ {x}) 2 B implies B [ {x}

B

if and only if C(B [ {x}) = {x}.

By definition of a w-index U , for such B and x, the conditions B [ {x} B and CU (B [ {x}) = {x} are equivalent to each other. So, (H2) entails that CU (B [ {x}) = C(B [ {x}) in such cases. The following result extends this observation to all instances in which the DM does not select the most selfish option. Proposition 4. Let U be a w-index for % and C a choice correspondence on KP that satisfies (H1). Then, the pair (%, C) satisfies (H2) if and only if the following two properties hold for any A 2 KP : (i) y ⇤ (A) 2 / CU (A) implies CU (A) = C(A).

(ii) y ⇤ (A) 2 CU (A) if and only if y ⇤ (A) 2 C(A). Proof. We omit the “if” part of the proof, which is a routine exercise. For the “only if” part, pick A 2 KP . Let us write y ⇤ instead of y ⇤ (A), and suppose y ⇤ 2 / CU (A). Pick any x 2 CU (A). Then, U (x, y1⇤ x1 ) > U (y ⇤ , 0), so that {x, y ⇤ } {y ⇤ }. Sophistication implies C({x, y ⇤ }) = {x}. So, by Weak WARP, we must have y ⇤ 2 / C(A). Now, pick any x0 2 C(A), and suppose, by contradiction, that x 2 / C(A). From Weak WARP, it follows 0 ⇤ 0 that C({x , x, y }) = {x }, while Sophistication implies {x0 , x, y ⇤ } {x, y ⇤ }. In turn, the latter condition implies U (x0 , y1⇤ x01 ) > U (x, y1⇤ x1 ), which contradicts the hypothesis that x 2 CU (A). We conclude that CU (A) ✓ C(A). To prove the converse inclusion, suppose x0 2 / CU (A). Then, U (x, y1⇤ x1 ) > max{U (y ⇤ , 0), U (x0 , y1⇤ x01 )}. This amounts to saying {x, x0 , y ⇤ } {x0 , y ⇤ }, while Weak WARP implies x0 2 C({x, x0 , y ⇤ }), a contradiction to Sophistication. Thus, C(A) ✓ CU (A), as we sought. This completes the proof of (i). It remains to show that y ⇤ 2 / C(A) implies y ⇤ 2 / CU (A). Assume y ⇤ 2 / C(A) and pick 0 0 ⇤ 0 an arbitrary x 2 C(A). Then, C({x , y }) = {x } by Weak WARP and {x0 , y ⇤ } {y ⇤ } by Sophistication. Hence, U (x0 , y1⇤ x01 ) > U (y ⇤ , 0) and y ⇤ 2 / CU (A). ⇤ Proposition 4 shows that if % admits a w-index U and the pair (%, C) satisfies properties (H1) and (H2), then C is almost identical with CU . Specifically, for any A 2 KP , if the most selfish option y ⇤ (A) does not belong to CU (A), then we have CU (A) = C(A). In turn, if CU (A) contains y ⇤ (A), then this is also the case for C(A), and vice versa. What remains undetermined is if (and which) other allocations can be selected along with the most selfish option. This partial identification is due to the potential multiplicity of w-indices compatible with %. To gain insight, suppose {x} {x, y} ⇠ {y} and y1 > x1 . Depending on the choice 2

of U , we may have either U (x, y1 x1 ) = U (y, 0) or U (x, y1 x1 ) < U (y, 0). In both cases, the DM may select y from {x, y}, but whether she would also select x depends on the choice of the w-index U . However, when {x, y} {y} and y1 > x1 , we must certainly have U (x, y1 x1 ) > U (y, 0), so that x is identified as the unique choice from {x, y}. A full characterisation of (%, C) is obtained by imposing the following additional property. Choice Regularity (H3). Take any {x, y} 2 KP with y1 > x1 , and suppose that there exists a neighbourhood N of {x, y} in KP such that {x0 , y 0 } ⇠ {y 0 } for every {x0 , y 0 } 2 N with y10 > x01 . Then, C({x, y}) = {y}. We say that a w-index U is regular if the pair (%, CU ) satisfies (H3). The notion of regularity is a variant of the local non-satiation property familiar from standard consumer theory. Indeed, in each of the following cases, the w-index in question is regular. Example 1. (Classical Altruism) Let U be a w-index that is constant in , and suppose that U (b x, 0) > U (x, 0) whenever x bi > xi for i = 1, 2. Example 2. (Pure Egoism) Let U be a w-index and u : R2+ ! R a strictly quasiconcave function such that U (x, ) = u(x1 , ) for every (x, ) 2 X ⇥ R+ . Example 3. (Impure Altruism) Let U be a w-index such that U (b x, ) > U (x, ) whenever x b1 x1 and x b2 > x2 .

In view of these examples, we have that (1) in the classical model, monotonicity of a utility index implies its regularity; (2) under pure egoism, strict quasiconcavity implies regularity; and (3) even an impure form of altruism suffices for regularity.23 Thus, it appears that one would rarely encounter a non-regular utility index in applications. Finally, we assume that second-stage choices are continuous in a standard sense: Closed Graph (H4). {(x, A) : x 2 C(A), A 2 KP } is a closed subset of X ⇥ KP . The promised characterisation of the pair (%, C) reads as follows. Proposition 5. Let C be a choice correspondence on KP and suppose that % admits a regular w-index U . Then, the pair (%, C) satisfies (H1)-(H4) if and only if C = CU . Proof. Let U be a continuous regular w-index for %. We omit the “if” part. For the “only if” part, assume that (C, %) satisfies (H1)-(H4). Consider A 2 KP and let us write y ⇤ instead of y ⇤ (A). By Proposition 4, we can assume y ⇤ 2 CU (A) \ C(A). Take any x 2 C(A) that is distinct from y ⇤ , and suppose, by contradiction, that x 2 / CU (A). Then, ⇤ ⇤ U (x, y1 x1 ) < U (y , 0). By continuity of U and definition of a w-index, there exists a neighborhood N of {x, y ⇤ } in KP such that {x0 , y 0 } ⇠ {y 0 } for every {x0 , y 0 } 2 N with 23

A particular implication of point (3) is that every strict warm-glow representation is regular.

3

y10 > x1 . Then, Choice Regularity implies C({x, y ⇤ }) = {y ⇤ }, whereas Weak WARP implies C({x, y ⇤ }) = {x, y ⇤ }, a contradiction. Conversely, take any x b 2 CU (A) that is distinct from y ⇤ so that U (b x, y1⇤ x b1 ) = U (y ⇤ , 0). From regularity of U , there exists a sequence {xn , y n } in KP that converges to {b x, y ⇤ } such that U (xn , y1n xn1 ) > U (y n , 0) and y1n > xn1 for every n. Clearly, we must also have limn xn = x b. Moreover, Proposition 4(i) implies C({xn , y n }) = {xn } for every n. So, by Closed Graph, we have x b 2 C({b x, y ⇤ }), and the desired conclusion follows from Weak WARP: x b 2 C(A). ⇤

It follows that given a preference relation % that admits a regular w-index, the pair (%, C) satisfies (H1)-(H4) if and only if C = CU for an arbitrarily selected regular w-index U . In particular, CU = CU˜ for any pair of regular w-indices U and U˜ . That is, the second-stage behaviour induced by regular w-indices for a given preference relation is uniquely defined. As mentioned in Section 6, Noor and Takeoka (2015, Section 3.3) present a closely related analysis of second-stage behaviour associated with a class of temptation driven preference relations over menus. In their model, the most tempting option acts as a reference point: Holding fixed the alternative x that will be selected, the DM’s welfare decreases with the di↵erence between the maximum possible temptation utility and that of x, leading to a preference for smaller sets. Thus, the two models are quite distinct, and yet, our axioms that link second-stage choices to preferences over menus are very similar to those of Noor and Takeoka. In particular, they postulate conceptually equivalent versions of Weak WARP and Sophistication, under the same names. In turn, one of their axioms (Ex Post Decreasing SelfControl) ensures that a functional in their representation is strictly increasing and, therefore, takes the role of our regularity property. Hence, it appears that, rather than the second-stage choices, it is mainly the properties of the first-stage preference % that distinguishes the two models. Appendix F: Extensions with Pareto Dominated Alternatives This section discusses the restriction imposed on the preference domain in Theorem 2 and presents two possible extensions. To gain insight, we focus on the DM’s behaviour when faced with a menu of the form {x, (x1 + r, x2 ), y}. We know that Theorem 2 allows for the pattern {(x1 + r, x2 ), y} {x, y}, which corresponds to an unwanted reward r. The question is how the DM would behave when she is also given the option to reject this unwanted award. That is, how she would evaluate {x, (x1 + r, x2 ), y} relative to {x, y} and {(x1 + r, x2 ), y}. According to one possible scenario, the DM would prefer rejecting the reward to accepting it if she dislikes the reward in the first place. In this case, we would 4

expect the pattern {x, (x1 + r, x2 ), y} ⇠ {x, y} {(x1 + r, x2 ), y}. Another scenario is that the DM would prefer accepting the reward to rejecting it because the former option Pareto dominates the latter in terms of material consumption. This corresponds to the pattern {x, (x1 + r, x2 ), y} ⇠ {(x1 + r, x2 ), y}. Hence, if {(x1 + r, x2 ), y} {x, y}, it would also follow that {x, (x1 + r, x2 ), y} {x, y}. These two alternative scenarios lead to di↵erent extensions of our warm-glow representation. Let us now focus on the first approach which implies that adding the more costly option x to the menu {(x1 + r, x2 ), y} makes the DM better o↵. At the same time, we wish to rule out the cases in which the DM enjoys the presence of a costly option that also reduces x2 . To this end, let us say that an allocation x 2 A is weakly efficient if there does not exist an x0 2 A such that x0i > xi for i = 1, 2. For any A 2 K, we denote by WP(A) the set of all weakly efficient allocations in A; note that WP(A) is a nonempty, compact set. In turn, KWP stands for the collection of all A 2 K such that A = WP(A). That is, a menu belongs to KWP i↵ it consists of weakly efficient allocations. Our first extension of Theorem 2 demands the following property. A9: Weak-Pareto Indifference. A ⇠ WP(A) for any A 2 K. Theorem 2a. The following statements are equivalent: (i) % is a complete and transitive binary relation on K that satisfies (A9), while % restricted to A := KWP satisfies (A2)-(A6) and (A7*). (ii) There exists a weakly increasing and continuous function Uw : X ⇥ R+ ! R such that, for every A, B 2 K, A%B

i↵

max Uw (x, max y1

x2WP(A)

x1 )

y2A

max Uw (x, max y1

x2WP(B)

y2B

x1 ).

This representation suggests that the DM follows a two-step choice procedure when evaluating a menu A. First, she eliminates those alternatives in A that are not weakly efficient. Then, among the remaining options, she selects an allocation that maximises a w-index Uw . To see the behavioural implications of Theorem 2a, consider a pair of allocations x, y and a number r > 0 such that y1 x1 + r and y2  x2 . Then, {x, y} {(x1 + r, x2 ), y} implies Uw (x, y1 x1 ) > Uw ((x1 + r, x2 ), y1 x1 r). Moreover, the latter inequality entails that {x, (x1 + r, x2 ), y} ⇠ {x, y}. Thus, the theorem formalises the first approach outlined above: The DM does not mind the presence of an unwanted reward if she is given the opportunity to reject the reward. Observe that, given any allocation x, if a menu A contains an x0 such that x0 x, then the pattern A [ {x} A implies x01 > x1 and x02 = x2 . Thus, the DM may enjoy the presence of a Pareto dominated alternative only if the alternative allows her to incur a higher private 5

cost without reducing the second consumption variable. Letting A = {x0 }, we also see that the representation allows for the pattern {x0 , x} {x0 }. This corresponds to rejection of a “reward” that is entirely unrelated to the DM’s contribution to the second variable. Hence, according to Theorem 2a, the DM may experience an extreme version of warmglow, which is absolutely independent of whether another person benefits from the DM’s sacrifice. If the DM is concerned with her social image, this scenario does not seem to be very plausible since others may not appreciate the DM for making a self-sacrifice that does not help them. Our next representation suits better for such a DM. For any A 2 K, let P(A) stand for the Pareto frontier of A; that is, P(A) := {x 2 A : @y 2 A s.t. y > x}. For ease of exposition, we focus on a binary relation % defined on Kc , where Kc stands for the collection of all A 2 K such that P(A) is a closed set. (In particular, Kc contains all finite menus as well as those menus in KP .) A10: Pareto Indifference. A ⇠ P(A) for any A 2 Kc . Our next representation, based on (A10), formalises the second approach outlined earlier. Theorem 2b. The following statements are equivalent: (i) % is a complete and transitive binary relation on Kc that satisfies (A10), while % restricted to A := KP satisfies (A2)-(A6), (A7*) and (A8). (ii) There exists a weakly increasing and continuous function Uw : X ⇥ R+ ! R such that, for every A, B 2 Kc , A%B

i↵

max Uw (x, max y1

x2P(A)

x1 )

y2A

max Uw (x, max y1

x2P(B)

y2B

x1 ).

Suppose that the DM is concerned with her image in the eyes of a recipient (who benefits from x2 and) who can observe the DM’s behaviour in the second stage. The distinctive feature of the representation above is that for r > 0, a menu of the form {x, (x1 + r, x2 ), y} is always indi↵erent to {(x1 + r, x2 ), y}. This property holds because the DM eliminates the Pareto dominated option x, knowing that the recipient would not appreciate the DM for selecting x over (x1 + r, x2 ). On the other hand, we may have {(x1 + r, x2 ), y} {x, y} if x1 + r  y1 and x2 > y2 . This pattern is driven by the fact that the recipient may appreciate the DM very much for selecting x over y, which is more costly for the DM than selecting (x1 +r, x2 ) over y. In such cases, it also follows that {x, (x1 +r, x2 ), y} {x, y}, which means that the presence of a reward may hurt the DM even if she is given the opportunity to reject the reward. In turn, from the perspective of revealed preferences, this corresponds to a form of preference for commitment, which is quite intuitive if the recipient cannot observe the DM’s behaviour in the first stage. 6

More generally, Theorem 2b rules out the cases of the form A [ {x} A if the menu A contains an allocation that Pareto dominates x. Because of this feature, we believe that Theorem 2b is more suitable than Theorem 2a as a model of warm-glow driven by image concerns. Yet, Theorem 2b also calls for further experimental analysis because of its complicated structure that combines preference for commitment with preference for larger menus. Moreover, the associated second-stage choice correspondence (i) may exhibit cycles (even for a fixed reference point), as in the sequential choice model of Manzini and Mariotti (2007); (ii) does not have a closed graph because the correspondence A ! P(A) is not upper hemicontinuous. As for the proofs, Theorem 2b is an immediate consequence of Theorem 2.1 Theorem 2a can also be proved easily, by a suitable modification of the proof of Theorem 2. One notable di↵erence is that Theorem 2a does not require the property (A8) because KWP is a closed set that contains KP . For brevity, we omit the details of the proof of Theorem 2a. Appendix G: Alternative Sets of Social Outcomes Our theory can be extended to alternative sets of social outcomes. Suppose that the set X of allocations is of the form X = X1 ⇥ X2 , where Xi is a connected, separable metric space for i = 1, 2. Then, under suitable assumptions on the behaviour of % over the collection of singletons {{x} : x 2 X}, we can find an aggregator ' : R2 ! R and functions ⇡i : Xi ! R, for i = 1, 2, such that {x} % {y} if and only if '(⇡1 (x1 ), ⇡2 (x2 )) '(⇡1 (y1 ), ⇡2 (y2 )).2 If x2 represents the private consumption of a second individual, just as in Harsanyi’s (1953, 1955) theory of utilitarianism, it may be appropriate to interpret ⇡i as a measure of well-being of individual i = 1, 2 from the perspective of the decision maker in question, who acts as a social planner. In fact, that ⇡i depends solely on xi would suggest that one views this function as the material payo↵ of individual i. In turn, if x2 represents a public good (which can also be a random variable), then we can view ⇡2 as the DM’s assessment of the social value of the public good while ⇡1 can be interpreted as before. Given these interpretations, we can restate our axioms in terms of the payo↵ vectors (⇡1 (x1 ), ⇡2 (x2 )) and utility possibility sets of the form {(⇡1 (x1 ), ⇡2 (x2 )) : x 2 A} ✓ R2 . In particular, by pursuing this approach, it is a straightforward exercise to obtain an extension 1

In the statement of Theorem 2b, upon writing “sup” in place of “max,” one can extend the domain of % to the collection of all bounded subsets of X with a nonempty Pareto frontier. To do so, it suffices to extend the domain in Theorem 2 accordingly (to all nonempty, bounded and efficient sets). The latter problem is straightforward because bounded sets can be approximated by compact sets. 2 A large body of literature is devoted to the study of axiomatic foundations of such representations that also demand the aggregator to be additive (see Wakker (1989, Chapter 3) and references therein). In turn, a nonadditive form of the representation can be derived by imposing a weak separability property along the lines of Mak (1984).

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of Theorem 2 that delivers a utility representation of the form V⇡ (A) := max U x2A

✓

⇡1 (x1 ), ⇡2 (x2 ), max ⇡1 (y1 ) y2A

⇡1 (x1 )

◆

for a function U : R2 ⇥ R+ ! R and compact sets A ✓ X which do not contain Pareto dominated allocations (as determined by the functions ⇡1 and ⇡2 ). (We omit the details of this derivation.) References Harsanyi, J. C. (1953). ‘Cardinal Utility in Welfare Economics and in the Theory of Risk-Taking’, Journal of Political Economy, vol. 61(5), pp. 434-435. ——— (1955). ‘Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparison of Utility’, Journal of Political Economy, vol. 63(4), pp. 309-321. Mak, K.-T. (1984). ‘Notes on Separable Preferences’, Journal of Economic Theory, vol. 33(2), pp. 309-321. Manzini, P. and Mariotti, M. (2007). ‘Sequentially Rationalizable Choice’, American Economic Review, vol. 97(5), pp. 1824-1839. Noor, J. and Takeoka, N. (2015). ‘Menu-Dependent Self-Control’, Journal of Mathematical Economics, vol. 61, pp. 1-20. Saito, K. (2015). ‘Impure Altruism and Impure Selfishness’, Journal of Economic Theory, 158(Part A), 336-370. Wakker, P. P. (1989). Additive Representations of Preferences, Berlin: Springer.

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