Libertarian Paternalism, Information Sharing, and Finanial Deision-Making Brue Ian Carlin Simon Gervais Gustavo Manso February 16, 2011 Abstrat We develop a theoretial model to study the welfare effets of libertarian paternalism on information aquisition, soial learning, and finanial deision-making. Individuals in our model are permitted to appreiate and use the information ontent in the default options set by a soial planner. We show that in some irumstanes the presene of default options an derease welfare by slowing information propagation in the eonomy. An extension of the model shows that partial information dislosures by the soial planner an inrease individuals inentives for gathering and sharing information, but that this does not affet the set of irumstanes in whih the absene of default options is optimal. Our analysis also onsiders a setting in whih individuals an sell their information to others. We show that default options ause the quality (and prie) of advie to derease, whih may lower soial welfare. Finally, we study the effets of prorastination and exessive trust in the soial planner on our analysis. This paper is an updated version of a previous working paper, When Does Libertarian Paternalism Work?, by the same authors. We would like to thank Zvi Bodie, Mihael Brennan, James Choi, Mike Fishman, David Laibson, Brigitte Madrian, Randall Mork, Marus Opp, Adriano Rampini, Tano Santos, Hong Yan, and seminar partiipants at Duke University, the University of South Carolina, the University of Colorado at Boulder, Yale University, Northwestern University, Imperial College, University College London, Prineton University, University of Illinois at Urbana-Champaign, the University of British Columbia, DePaul University, Brigham Young University, the annual onferene of the Finanial Researh Assoiation, the annual meeting of the Amerian Eonomi Assoiation, the USC-UCLA Finane Day, and the UBC Summer Finane Conferene for their omments and suggestions. All remaining errors are the authors responsibility. Anderson Shool ofmanagement, University ofcalifornia, Los Angeles, 110Westwood PlazaSuite C413, Los Angeles, CA 90995, brue.arlin@anderson.ula.edu, (310) 825-7246. Fuqua Shool of Business, Duke University, One Towerview Drive, Durham, NC 27708-0120, sgervais@duke.edu, (919) 660-7683. MIT Sloan Shool of Management, 50 Memorial Drive E52-446, Cambridge, MA 02142-1347, manso@mit.edu, (617) 253-7218.
1 Introdution Finanial sophistiation has lagged behind the growing omplexity of retail markets (e.g., NASD Literay Survey, 2003). 1 This not only degrades personal welfare, but also affets the real eonomy. 2 What to do about this disparity between omplexity and sophistiation has reeived muh attention, but the optimal solution remains hotly debated. Whereas some are proponents of inreasing awareness through eduation (e.g., Lusardi and Mithell, 2007), others favor improving peoples hoies by offering well thought-out default options. Indeed, libertarian paternalism, as posed by Thaler and Sunstein (2003, 2008), makes sense in many venues and has been shown to improve some of the finanial deisions that people make (Thaler and Benartzi, 2004). Libertarian paternalism is provoative beause it links two ideas that are on the surfae ontraditory, but may indeed be an unompromising ompromise. In theory, it allows a soial planner to diret individuals through default options without imposing her will, so that everyone may enjoy the best of both worlds: guidane without the tax of obtrusion. This poliy needs to be implemented judiiously, however, rather than as a blanket poliy. Glaeser (2006) suggests that in some ontexts libertarian paternalism may be hard to publily monitor and may lead to hard paternalism. 3 Mithell (2005) questions the redistributive onsequenes of libertarian paternalism. Korobkin (2009) argues that, beause libertarian paternalism ignores the externalities that individuals reate for eah other, its poliies may not maximize olletive welfare even though they indue individuals to make optimal deisions for themselves. These observations raise an obvious question: Given its propensity for induing or exaerbating externalities, when do we expet libertarian paternalism to be welfare improving? To explore this, we analyze an important dimension of this problem: the effet of libertarian paternalism on the aquisition of finanial information and soial learning. Madrian and Shea (2001) show that 401(k) default options provide information to market partiipants, whih hanges both their pereptions and resultant investment deisions. 4 Individuals frequently interpret default op- 1 See Carlin (2009) and Carlin and Manso (2010) for further disussion. 2 For example, Ausubel (1991), Mithell et al. (1999), Baye and Morgan (2001), Brown and Goolsbee (2002), Christoffersen and Musto (2002), Hortaçsu and Syverson (2004), Green (2007), and Green et al. (2007) show that finanial sophistiation (or the lak of it) has important priing effets in various ontexts. Campbell (2006) and Calvet et al. (2006) disuss the signifiant welfare reperussions that this an have. Finally, other effets of finanial sophistiation are analyzed by Capon et al. (1996), Alexander et al. (1998), Sirri and Tufano (1998), Wilox (2003), Barber et al. (2005), Agnew and Szykman (2005), and Choi et al. (2009). 3 See also Rostbøll (2005), and Whitman and Rizzo (2007) for similar arguments. 4 Madrian and Shea (2001) reah this onlusion from the empirial observation that existing employees systematially hange their partiipation and investment deisions after a hange in the defaults provided to new employees. Mithell et al. (2009) reah a similar onlusion in their study of the mutual funds used by retirement plan partiipants. 1
tions as the reommended ourse of ation (Brown and Krishna, 2004; MKenzie et al., 2006), whih may derease their willingness to aquire further information. At the same time, soial learning is an important mehanism by whih individuals aquire knowledge, espeially in settings in whih default options are onsidered (Duflo and Saez, 2002, 2003; Sorensen, 2006; and Beshears et al., 2009). When the use of default options redues information aquisition by individuals, this may negatively impat soial learning. If in turn knowledge deteriorates suffiiently when people are guided by a soial planner, whether they are fored to make hoies or not, total welfare may derease. This implies that in some irumstanes it may be optimal to either implement a limited form of libertarian paternalism or to leave market partiipants alone, even if some people s hoies end up regrettably suboptimal. We develop a theoretial model to analyze the effet of libertarian paternalism on information aquisition, soial learning, and finanial deision-making. We haraterize settings in whih providing default options may derease welfare beause information aquisition and aggregation slows. We do this both when information perolates aording to a soial learning tehnology (e.g., Ellison and Fudenberg, 1993 and 1995; Manski, 2004; Duffie and Manso, 2007) and in a setting in whih uninformed individuals an purhase information from informed ones (i.e., an advie market). In the base model that we analyze, eah individual must make a finanial deision whose payoff depends on his unknown type. The soial planner knows a harateristi that is ommon aross all the individuals in the market. She must deide between two poliies: (i) institute a default option that impliitly disloses her information; (ii) hold on to the information and let individuals make their own hoies without guidane from a default. Individuals an exert ostly effort to find out their own type, whih inludes the planner s information, so that they make an even better deision. Higher aggregate effort also inreases the probability that any one individual beomes informed. This form of soial learning provides an externality where one individual s effort affets other people s welfare and vie versa. We derive onditions under whih default options are optimal and desribe when they destroy soial surplus. The tradeoff revolves around the fat that the information ontained in the default option provided by the soial planner redues eah individual s inentive to gather and share any additional information. Thus, although the information in the default is useful to any one individual, it redues the positive externalities assoiated with soial learning. When the informationsharing tehnology is suffiiently effetive, the ost of information aquisition is low, and/or the individual-speifi information is more valuable, providing a default option is suboptimal. Under 2
these onditions, a soial planner maximizes welfare by letting market partiipants fend for themselves and allowing soial learning to take plae. Alternatively, if the information known by the planner is relatively more valuable and these other onditions do not hold, then default options add value. This sheds light on when libertarian paternalism is likely to add value. For example, default options are likely to be welfare-improving when individuals are suffiiently homogeneous. Consider the default option of a low-fee life yle fund that automatially realloates wealth to fixed inome assets as investors age. It is unlikely that there is muh variation in preferenes for suh agedependent realloations. Yet, people s ability to aess this information for themselves is limited. Therefore, in this ase, providing a default option is likely to add value. However, default options are unlikely to inrease soial welfare when people s needs are more heterogeneous or when the information aquired by individuals is relatively valuable ompared to the information ontained in the default option. An example of this might be a deision as to whether or not to purhase a life annuity. People s needs for these retirement vehiles are quite variable (e.g., simple life versus joint survivorship) and given the degree of adverse seletion assoiated with suh hoies, these deisions are diffiult to reverse ex post. Getting the hoie right on the first attempt is valuable: if providing defaults for this deision dereases some people s inentives to beome savvy, this may lead to a drop in welfare. We proeed to onsider the possibility that the soial planner aquires imperfet information about its onstituents. In this ase, systemati errors derease the auray for people who use the default options, but inrease the effort that individuals employ to aquire and aggregate information. We show that the latter effet dominates the former in that issuing no default is more likely to be of value when the planner s information is imperfet. Our analysis thus onfirms an objetion raised by Glaeser (2006) that soial planners are not immune from making errors or having biases. Given this, we then ask whether the soial planner would ever want to issue an imperfet default even though she has perfet information. We show this not to be the ase. That is, despite being given a broader ation spae inluding noisy defaults, the planner s optimal hoie is binary: either issue a fully informative default option or leave individuals to fend for themselves. The same omparative statis still hold as before, supporting the generality of our findings. We then haraterize an eonomy in whih information sales (i.e., advie) are allowed to take plae. A fration of the individuals are reognized as information gatherers, whereas the remainder rely on advie markets for guidane. The soial planner faes the same problem as before, and 3
information gatherers deide how muh ostly effort to employ in aumulating knowledge. The differene here is that the soial learning tehnology takes the form of information exhanges between information gatherers and the rest of the publi. 5 In this version of the model, the presene of a default option dereases the value of advie. That is, sine fewer information gatherers will beome knowledgeable, the quality of advie in the market suffers. As in the base model, not offering a default option sometimes dominates issuing a default option, espeially if the ost of effort is low and the value of individual-speifi (soial planner) information is high (low). Finally, we explore the effet of two behavioral onsiderations on our analysis. First, we onsider prorastination by embedding our model in a framework that is similar to that of Carroll et al. (2009). We determine when it is optimal for the soial planner to offer an aurate default (i.e., a entered default), an offset default that inreases information-gathering inentives, or fore its onstituents to make ative deisions. Besides showing that the information ontent of the default options remains a robust onsideration when adding a prorastination omponent to the model, we obtain several additional results. Speifially, we show that the optimal magnitude of an offset default option is dereasing in the degree to whih information is shared. That is, when soial learning is more potent, the soial planner does not need to indue people to aquire information as muh. As a result, even people who do not opt out of the default option are better off: even though they do not partiipate in the learning or information-sharing proess, the offset required in the default is lower, whih gives them a better outome when they remain in the default. In other words, soial learning affets the redistributive properties of libertarian paternalism. 6 Seond, we reonsider our analysis when individuals put too muh trust in the soial planner. Speifially, as suggested by the empirial work of Madrian and Shea (2001), we investigate a situation in whih individuals overweight the importane of the soial planner s information. Our analysis shows that over-dependeny on the soial planner makes individuals follow the path of least resistane and herd into the defaults, as doumented by Choi et al. (2002) and Johnson and Goldstein (2003). 7 This further leads to suboptimal information aquisition and soial learning, ausing the benefits of default options to erode. As a result, default options are less likely to inrease welfare than in our base model. 5 Beause individuals learn to make better deisions by interating with their skilled peers, our approah is similar in spirit to work by Glaeser (1999) and Glaeser and Maré (2001) in whih agents beome more produtive when working with others who are skilled. 6 For a disussion of the redistributive effets of libertarian paternalism, see Mithell (2005) and Zanitelli (2009). 7 Similarly, Korobkin (1998) argues that ontrat defaults rules often lead to herding and suboptimal ontratual terms as a result of a status quo bias. 4
Soial interations have been shown to affet a variety of deisions that have a signifiant impat on the finanial well-being of individuals and households: their deisions to partiipate in markets (Hong et al., 2004; Brown et al., 2008; Kaustia and Knüpfer, 2009), to enroll in retirement plans (Madrian and Shea, 2001; Beshears et al., 2009), to buy stoks (Shiller and Pound, 1989), to selet health plans (Sorensen, 2006), to purhase ars (Grinblatt et al., 2008), and to use welfare programs (Bertrand et al., 2000). 8 Based on our analysis, given that individuals learn from their peers when they make important finanial deisions, a systemati implementation of default options to all of these (and other) domains may not be optimal. In partiular, as implied by Arrow s (1994) arguments about soial knowledge, it is important to weigh the soial multiplier effets of learning (e.g., Glaeser et al., 2003) when onsidering the design of default options or more generally the adoption of poliies based on libertarian paternalism. 9 The remainder of the paper is organized as follows. Setion 2 outlines our basi model and determines when it is optimal to use default options. Speifially, Setion 2.1 analyzes the ase when the soial planner an only issue fully informative default options, whereas Setion 2.2 allows for imperfet default options. In Setion 3, we turn to information sales to endogenize the mehanism underlying information propagation. Setion 4 explores the effets of behavioral biases on our analysis. Finally, Setion 5 provides some onluding remarks. All proofs are in the appendix. 2 Soial Learning 2.1 Basi Model The market is omposed of a soial planner and a ontinuum (a non-atomi finite measure spae (I, I, γ)) of heterogeneous, rational individuals who all fae a signifiant finanial deision. Examples of suh a deision might be an investment-onsumption hoie, a apital alloation deision, or a hoie of insurane. For simpliity, but without loss of generality, we set the total measure γ(i) of individuals to 1 (i.e., a unit mass). The ex post utility from the deision for eah individual i I is given by Ũ i (x i ) = ( τ i x i ) 2, (1) where x i R is a hoie variable and τ i is the individual s true (but unknown) type. The type τ i is the sum of a omponent g that is ommon to all individuals and an idiosynrati omponent t i that 8 For a survey of the literature on soial interations, see Manski (2000). 9 Similarly, Ahdieh (2009) stresses the importane for any publi intervention aimed at individuals to internalize the soial dynamis that it may affet. Also, Camerer et al. (2003) propose a form of asymmetri paternalism aimed at minimizing the externality distortions of regulation. 5
isspeifitoindividuali. Weassumethat g and t i aretwoindependentnormallydistributedrandom variables, eah with zero mean and respetive varianes Σ g and Σ t, and that Cov( t i, t j ) = ρσ t, with ρ [0,1), for any {i,j} I 2 with i j. 10 Thus, for eah individual i, τ i is normally distributed with a mean of zero and a variane of Σ τ Σ g +Σ t. As (1) is a quadrati loss funtion, the goal of eah individual is to hoose x i to be as lose to τ i as possible in order to minimize his expeted loss. Before hoosing x i, eah individual i an exert some effort in order to improve the probability that he finds out about his own type. An individual s effort of e i [0,1] omes with a personal utility ost of C(e i ) = 2 e2 i, (2) where is a positive onstant. An individual who selets an effort level e i observes his true type τ i (i.e., reeives an informative signal) with probability e i +αē, (3) where ē I e idγ and α [0,1), and observes nothing otherwise. Individuals know when they did not reeive an informative signal. Given that ē represents the average effort exerted by individuals in the population, the signal speifiation in (3) implies that an individual is more likely to learn his own type when many individuals seek to learn theirs. This positive externality of effort aptures the idea that as more people exert effort and more of the population beomes informed, their interations lead to more spillovers in the learning proess; this ultimately makes it easier for anyone to learn about the finanial deision that they have to make. While not speifially modeled, the miro-foundation for this setup might be a model of searh in whih individuals are more likely to learn from eah other as more of the population is informed. Setion 3 provides an alternative miro-foundation in whih the information externality omes from the exhange of information between skilled and unskilled individuals. In this base model, the parameter α measures the degree of this information externality. The soial planner ostlessly observes the ommon omponent g of the individuals types. For example, this ould orrespond to the planner having an informed opinion about the optimal average savings rate for a group of individuals. The planner then hooses whether to set a default option that takes g into aount or to leave individuals to their own devies. 11 The planner s goal 10 Note that the positive orrelation aross the idiosynrati omponent t i of individuals types does not play a role until we allow for information sales, in Setion 3. 11 Note that, beause individuals are rational and do not fae any ost for hoosing x i, the soial planner ould 6
in this hoie is to maximize total welfare. Sine individuals are rational, they are able to glean information about g from a default option if it is offered. 12 This in turn will affet their hoie of effort in gathering further information. Let S i denote the information set of an individual i at the time he must make his deision x i. This set is equal to { τ i } if the individual observes his true type, whether or not the soial planner sets a default option. 13 When there is a default option and the individual does not observe his type, S i = { g}. Finally, when there is no default option and the individual does not observe his type, S i =. The following lemma defines the optimal hoie of x i, given the information set S i. Lemma 1. The optimal hoie of x i for individual i is E [ τ i S i ]. With a default option, eah individual i who observes an informative signal opts out of the default and hooses x i = τ i, whereas any individual who remains uninformed does not opt out, i.e., hooses x i = g, as presribed by the soial planner. If no default option is offered by the planner, any individual i who beomes informed still hooses x i = τ i, and hooses x i = 0 if he does not get to observe an informative signal. Consistent with Madrian and Shea s (2001) empirial findings, there is information ontent in the default options that the soial planner provides, as uninformed individuals optimally (and rationally) hoose to use it. Before hoosing x i but after the soial planner s deision to announe a default option, eah individual i hooses the effort level e i that maximizes his expeted utility. This hoie takes into aount the fat that he will subsequently hoose x i aording to Lemma 1. It also depends on individual i s information set Si 0, whih is then g if the planner makes a default option available and is empty otherwise. The following lemma summarizes and simplifies this maximization problem. Lemma 2. Individual i hooses his effort level e i to maximize E[Ũi (x i ) C(e i ) S 0 i ] ] = (1 e i αē) [(1 δ)σ g +Σ t 2 e2 i, (4) where δ = 1 when a default option g is offered by the soial planner and δ = 0 when people are left to their own devies. equivalently announe an uninformative default or fore individuals into a deision. The distintion between these poliies will beome more important when we add prorastination to our model, in Setion 4.1. 12 As long as the planner s hoie for the default option is one-to-one with g, every individual an infer g perfetly. Thus, it is without loss of generality that we assume in what follows that the planner announes g as the default option when she makes suh an option available. 13 Tehnially speaking, the information set is { g, τ i} when the soial planner announes a default option and individual i observes his own type, but the additional information provided by g (i.e., knowing g and τ i separately) is not useful for any of the deisions that this individual must make. 7
This result highlights the tradeoff faed by eah individual. Effort is ostly (seond term in (4)) but it redues the variane that the individual is subjet to (first term in (4)). At the same time, the onerted effort of every individual reates a publi good, ē, that redues the variane for everyone. Going forward, we make the following assumption, whih guarantees an interior solution to the effort problem but does not affet the eonomis of the analysis. Assumption 1. The ost parameter is suh that > 2(Σ g +Σ t ). The following proposition haraterizes the effort hoie of individuals, with and without a default option. Proposition 1. If the soial planner adopts a default option, eah individual hooses effort e i = Σ t ed, (5) whereas if the soial planner does not adopt a default option, eah individual hooses effort e i = Σ g +Σ t e N. (6) Inspetion of (5) and (6) shows that individuals exert more effort with higher Σ t and lower. That is, the more variane about an individual s type that is resolved when an informative signal is obtained and the lower the ost of aquisition, the more effort eah individual is willing to employ. Importantly, it is also the ase that e N = e D + Σ g. This implies that people exert more effort without a default option, and that the differene between e N and e D inreases as Σ g gets larger and as gets smaller. Sine the positive externality ē omes from the average effort of individuals in the eonomy, it follows that there are greater opportunities for people to learn from eah other when default options are not provided by the soial planner. In this sense, whether a default option is welfare improving depends on the strength of the learning externality relative to the value of the information that the soial planner has in her possession. Given Proposition 1 and Lemma 2, we an ompute the total welfare with a default option as W D = Σ t + (1+2α)Σ2 t 2 and the total welfare without a default option as, (7) W N = (Σ g +Σ t )+ (1+2α)(Σ g +Σ t ) 2. (8) 2 The next proposition ompares welfare with and without a default option. 8
Proposition 2. The total welfare W N without a default option is higher than the total welfare W D with a default option if the ost parameter is in the following region: This region is non-empty if and only if 2(Σ g +Σ t ) < < (Σ g +2Σ t ) 1+2α. (9) 2 Γ Σ g Σ g +Σ t < 2(2α 1) 1+2α. (10) Aording to Proposition 2, welfare without a default option may be higher than welfare with a default option. This arises beause the presene of a default option redues people s inentives to learn about the eonomi problem they fae, whih in turn slows the pae of information propagation throughout the eonomy. In other words the very presene of a default option reates an inentive for the population to herd into it, a damaging effet when people an learn a lot from eah other (i.e., when α is greater than 1 2 and large), and when the ost of information aquisition is low (i.e., when is small). As shown in (10), the availability of a default option is more likely to be detrimental if the portion Γ of the volatility that the soial planner an eliminate with her information about g is small relative to the extent of information externalities. To gain further insight into this result, let us use (7) and (8) and define the differene W W N W D = Σ g + (1+2α)Σ g(σ g +2Σ t ). (11) 2 Notie that, sine Σ g = ΓΣ τ and Σ t = (1 Γ)Σ τ, we an rewrite this expression as W = ΓΣ τ + (1+2α)Γ(2 Γ)Σ2 τ 2 It is easy to verify that, holding the total variane Σ τ fixed, we have. (12) ( W) Γ = Σ τ + (1+2α)(1 Γ)Σ2 τ, (13) and this quantity is positive if and only if Γ < 1 (1+2α)Σ τ. That is, an inrease in the ability of the soial planner to urb variane by revealing its knowledge of g through a default option makes this option relatively less appealing when Γ is small or the total variane Σ τ is large. In other words, when important information about individuals is unobservable to the soial planner (small Γ) or when there is a lot of unertainty about the individuals finanial 9
deision (largeσ τ ), inreasingthepreisionofthisinformationmakes defaultoptionslessappealing, as suh options then have a partiularly detrimental effet on information gathering inentives, and in turn on information sharing. Similarly, after fixing the proportion Γ of the total variane that the soial planner an ontrol, we have whih is positive if and only if ( W) Σ τ = Γ+ (1+2α)Γ(2 Γ)Σ τ Σ τ > (1+2α)(2 Γ)., (14) Thus an inrease in overall unertainty renders the presene of default options detrimental to welfare when this unertainty is large to begin with (large Σ τ ) and when the soial planner s ability to redue unertainty is limited (small Γ). The former effet has two potential interpretations. First, Σ τ might proxy for the amount of heterogeneity in the population: when people s needs or attributes differ a lot, default options are more likely to be suboptimal. Seond, Σ τ might also proxy for the eonomi value at risk in eah individual s deision: when deisions are more important, the soial planner should refrain from issuing a default in order to promote learning and information sharing by individuals. The latter effet is diretly related to the information gathering inentives of individuals: an inrease in Σ τ makes the default option damaging when Γ is small beause the importane of the information that individuals forego by exerting less effort to gather it, (1 Γ)Σ τ, is large relative to the preision of the information they learn from the default option, ΓΣ τ. Together, these omparative statis suggest venues in whih default options are likely to add value. For instane, default options are more likely to add value when there is little ross-setional variation in the population than when this variation is higher. By inspetion of (11), the relationship between W and Σ g is non-monotoni. Similarly, our analysis of (13) shows that W is non-monotonially affeted by hanges in Γ. Based on this, it is feasible that the soial planner an optimize welfare by limiting her information olletion to an imperfet signal and by offering to the population a default option that is not perfetly orrelated with g. We explore this next. 2.2 Imperfet Soial Planner One of Glaeser s (2006) objetions to the optimality of libertarian paternalism is that the soial planner may, like individuals, make errors in judgement and deision-making. For example, the planner may have limited preision when gathering information about its onstituents. In this ase, 10
default options reveal an imperfet, yet unbiased, signal about g. 14 Alternatively, the planner may gather perfet information about g, but wish to dislose an imperfet signal of this information through a default option. Charaterizing these issues is the purpose of this setion. Suppose that the soial planner only observes a noisy signal s = g + ɛ, where ɛ is normally distributed with mean zero and variane Σ ɛ, and is independent from g and t i for all i I. As before, eah individual i an exert effort e i for a ost given by (2) and learns his type τ i with probability e i + αē. If the planner issues a default option that onveys her noisy signal, rational individuals will take this into aount when hoosing how muh information to aquire and share. We haraterize this effet in the following proposition. Proposition 3. If the soial planner implements an imperfet default option with noise Σ ɛ, eah individual i hooses effort e i = (1 δ)σ g +Σ t, (15) where δ Σg Σ g+σ ɛ. An individual i who observes a fully informative signal opts out of the default option and hooses x i = τ i = g + t i. An individual i who does not beome informed hooses x i = δ s = δ ( g+ ɛ ), the default option offered by the soial planner. As in Proposition 1, the optimal hoie of effort is dereasing in and inreasing in Σ t and Σ g. Additionally, as the amount of noise in the default inreases (higher Σ ɛ, and thus lower δ), the higher is the effort that eah individual is willing to exert to learn about τ i. Therefore, the preision of information ontained in the default option drives the inentives of individuals to aquire information, whih in turn affets how muh is learned via information sharing. Given Proposition 3, we an ompute the total welfare with a noisy default option as W D (Σ ɛ ) = [ [ ] ] 2 (1 δ)σg +Σ t (1 δ)σ g +Σ t + (1+2α) 2 ( ) ( ) ΣɛΣ 2 g Σɛ Σ g Σ g+σ ɛ +Σ t = +Σ t + (1+2α). (16) Σ g +Σ ɛ 2 The next proposition ompares welfare with and without a default option when the soial planner s information is imperfet. 14 Suh mistakes might also result from a systemati bias in the soial planner s information gathering proess. Of ourse, sine individuals in our model are fully rational, they would orretly interpret the information ontained in default options and remove the effets of these systemati biases. With an unbiased but noisy signal about g, improving preision is not possible but does indue individuals to employ more effort in aquiring their own information. Therefore, the model as posed ould inlude a systemati bias, but this would not hange the eonomis of our results. Only if individuals ould not understand and adjust for the planner s biases would suh mistakes hange the analysis and lead to lower welfare. 11
Proposition 4. The total welfare W N without a default option is higher than the total welfare W D (Σ ɛ ) with a noisy default option if the ost parameter is in the following region: ( ) 2Σɛ +Σ g 1+2α 2(Σ g +Σ t ) < < Σ g +2Σ t. (17) Σ ɛ +Σ g 2 This region is non-empty if and only if where Γ = Σg Σ g+σ t and Φ = Σg Σ g+σ ɛ. ΓΦ < 2(2α 1) 1+2α, (18) Comparing the result in Proposition 4 with that in Proposition 2, the region in whih no default dominates default is larger when the soial planner s information is impreise. In fat, Proposition 4 shows that this region gets larger as Σ ɛ inreases (and Φ dereases). This result is not obvious: an impreise default hurts individuals who deide take the default option, but also provides inentives for individuals to searh more intensively, whih improves information sharing. Comparison of Propositions 2 and 4 shows that the first effet dominates the seond, onfirming Glaeser s (2006) onjeture that the ase for libertarian paternalism is weaker if the soial planner makes errors in judgement or has impreise information. Clearly, this motivates an analysis of whether the soial planner would optimally hoose to issue a noisy default, even when she has free aess to perfet information about g. Thus, let us onsider a broader ation spae for the soial planner in whih she an issue default options that do not onvey a preise signal regarding g. As suh, the planner ould still hoose to issue a default option that onveys g perfetly, but we now allow the planner to instead issue a default option that onveys g + ɛ, in whih she hooses the variane Σ ɛ > 0 of ɛ. If a finite Σ ɛ is hosen, individuals an learn some (i.e., inomplete) information about their deision from the default. Of ourse, as before, the planner an still make the default option perfetly informative about g by hoosing Σ ɛ = 0, and effetively refrain from making a default option available by hoosing Σ ɛ =. Given our previous disussion, the soial planner s hoie of Σ ɛ affets welfare through two hannels. A higher preision improves the hoies that individuals make when they do not observe an informative signal, but it dereases the inentives of individuals to ollet and share information in the first plae. Taking these two fores into aount, the next proposition haraterizes the soial planner s optimal default poliy. Proposition 5. The optimal hoie of noisy default poliy is given by { Σ 0, if > (Σ g +2Σ t ) 1+2α ɛ = 2, otherwise. (19) 12
Proposition 5 implies that our analysis in Setion 2.1 holds even when we onsider a broader ation spae for the soial planner. That is, the planner s deision is effetively binary: she either hooses a fully informative default option or offers no default whatsoever. Again, if the ost of information aquisition is suffiiently high (high ), the size of the variation or value at risk is suffiiently low (low Σ g +Σ t ), or the information sharing tehnology is suffiiently weak (low α), the soial planner issues a fully informative default option (i.e., Σ ɛ = 0). Otherwise, the planner lets individuals fend for themselves (i.e., Σ ɛ = ). 3 Information Sales So far, our model shows that the adoption of default options is ostly and potentially suboptimal when individuals in the eonomy an help eah other learn about their own type. In this setion, we show that the externality need not be of the form speified in Setion 2. In partiular, we show that allowing individuals to sell their information to uninformed individuals an generate similar results. That is, the presene of default options redues the inentive for individuals to gather and resell their information, potentially leading to an overall redution of information in the eonomy and to lower welfare. In essene, formal exhanges of information aross individuals in the eonomy, as modeled in this setion, provide a miro-foundation for our base model. To establish our results, we adapt the basi model of Setion 2 to a ontext in whih some individuals an (and will) purhase information from other individuals in the eonomy. More speifially, we assume that a subset I µ I, with γ(i µ ) = µ, of individuals are skilled in the sense that they an gather information about their type with the same tehnology as before, exept that we set α = 0 in (3) to emphasize the fat that externalities derive purely from information sales. That is, for a ost of C(e i ) = 2 e2 i, individual i I µ reeives a signal that reveals his type g + t i with probability e i. The other individuals j I \ I µ are unskilled in that gathering information about their own type is prohibitively ostly. Instead, these unskilled individuals an purhase information from skilled individuals. Although everyone s skill is publily observable, the private information of any one skilled individual is not. That is, no one an tell if individual i learned g + t i or not. Thus, for a prie p (to be determined shortly), an unskilled individual j an purhase a signal from a skilled individual i, but does not know if he learns g + t i (whih is orrelated with his own type g + t j ) or noise (whih is not) in the proess. 15 Throughout this setion, we go bak to the assumption that the soial planner s 15 We assume that skilled individuals who do not learn their own type sell an uninformative signal that is randomly 13
default option is perfet (i.e., equal to g) when it is made available; that is, we refrain from showing as in Setion 2.2 that this hoie is optimal even if the planner an hoose the preision of her information. The following lemma haraterizes the value derived from the information by an unskilled individual who onsults a randomly seleted skilled individual. Lemma 3. If the soial planner does not adopt a default option, the maximum amount that an unskilled individual is willing to pay for the information sold by a randomly seleted skilled individual is ν 0 = (Σ g +ρσ t ) 2 ē µ = [ Γ+ρ(1 Γ) ] 2 Στ ē µ, (20) Σ g +Σ t where ē µ 1 µ I µ e i dγ, Σ τ = Σ g +Σ t, and Γ = Σg Σ g+σ t. If the soial planner adopts a default option, the maximum amount that an unskilled individual is willing to pay for the information sold by a randomly seleted skilled individual is ν 1 = ρ 2 Σ t ē µ = ρ 2 (1 Γ)Σ τ ē µ. (21) Unskilled individuals are willing to pay more to learn a skilled individual s type when they know that skilled individuals exert a lot of effort to learn their own type, i.e., ν 0 and ν 1 are both inreasing in ē µ. This makes sense as a fration ē µ of the µ skilled individuals will be informed in equilibrium, while the other (1 ē µ )µ skilled individuals sell useless noise. From (20) and (21), we an also see that unskilled individuals are willing to pay a higher prie for a skilled individual s information when their type is highly variable (large Σ τ ) and when it is more highly orrelated with that of other individuals (large ρ). This last result is onsistent with the fat that, keeping Σ τ fixed, ν 0 is inreasing in Γ, as types are more orrelated when the ommon omponent g aounts for a larger portion of eah individual s type. This is also onsistent with ν 1 being dereasing in Γ as, when the soial planner announes g, the unknown portion of an individual s type orrelates with someone else s type only to the extent that the default option leaves unertainty regarding t i. In fat, using (20) and (21), it is straightforward to verify that ν 0 > ν 1 for a given total variane Σ τ and aggregate level of effort ē µ. Indeed, beause types are more orrelated aross individuals when g is unknown, it is the ase that unskilled individuals are willing to pay more to learn a skilled individual s type when there is no default option offered. As we shall see below, this differene between ν 0 and ν 1 is exaerbated by the fat that the equilibrium effort level of skilled individuals is greater in the absene of a default option. drawn from a normal distribution with a mean of zero and a variane of Σ g + Σ t, whih makes it impossible for information buyers to tell noise from real information. The skilled individuals have nothing to gain from doing anything else. 14
The prie that a skilled individual will end up harging for his information will in general depend on how muh ompetition he faes from other information sellers or, alternatively, on how easy it is for unskilled individuals to onsult another skilled individual. To apture these possibilities in a tratable manner, we assume that eah unskilled individual meets with one randomly seleted skilled individual, and that the eonomi surplus from their transation is split as a Nash bargaining outome. More speifially, we assume that a skilled individual harges p = θν δ for the information he sells to an unskilled individual, where θ [0,1] and δ = 1 if a default option is made available (δ = 0 otherwise). When θ = 1 (θ = 0), the skilled (unskilled) individual extrats all the surplus from the transation. 16 Setting θ (0,1) allows us to apture any intermediate market power senario. As the following analysis shows, our results are unaffeted by the size of θ, as money exhanges between individuals anel out in the total welfare funtion that the soial planner seeks to maximize. We start with the following result, whih desribes the equilibrium in the absene of a default option. Proposition 6. If the soial planner does not adopt a default option, then eah skilled individual i I µ hooses an effort level e i = Σg+Σt = Στ, and hooses x i = τ i or x i = 0, depending on whether or not he observes τ i. Eah unskilled individual j I \I µ purhases a signal s j (whih is τĩ or noise) from a randomly seleted skilled individual ĩ I µ for a prie p = θν 0, with ν 0 given by (20), and hooses x j = Σ g +ρσ t Σ g +Σ t ē µ s j = [ Γ+ρ(1 Γ) ] ē µ s j. (22) The skilled individuals behavior is the same as in Setion 2.1. In partiular, their behavior is not affeted by the possibility of reselling their information to unskilled individuals. This is due to the fat that unskilled individuals annot distinguish between skilled individuals who learn their type and skilled individuals who do not. That is, they pay θν 0 to the one skilled individual they enounter, informed or not. As we see from (22), the extent to whih unskilled individuals rely on the information they purhase depends on its orrelation with their type, as inreases in ρ, Γ and ē µ all ultimately lead to a higher orrelation between s j and τ j. The following result is the analogue of Proposition 6 when the soial planner makes a default option g available. Proposition 7. If the soial planner adopts a default option, then eah skilled individual i I µ hooses an effort level e i = Σt = (1 Γ)Στ, and hooses x i = τ i or x i = g, depending on whether or 16 Note that when θ = 0, the transation an be interpreted as a free information exhange between two individuals with different skills. For example, this aptures the situation in whih a new employee asks an existing employee of the same firm about his hoies in the ompany s 401(k) plan. 15
not he observes τ i. Eah unskilled individual j I \I µ purhases a signal s j (whih is τĩ or noise) from a randomly seleted skilled individual ĩ I µ for a prie p = θν 1, with ν 1 given by (21), and hooses x j = g+ρē µ ( s j g). (23) The omparative statis on the individuals hoies with respet to Σ τ, ρ and ē µ are similar to those in Proposition 6: more risk (large Σ τ ) leads to more effort, and more orrelation (large ρ and ē µ ) leads to heavier reliane on purhased information. When Γ is large, skilled individuals do not gain muh from learning their type perfetly, as the default option already reveals a large portion of their type. As suh, they work less. Although Γ affets the prie of information (as shown in Lemma 3), it does not affet the weight that unskilled individuals put on the information they aquire from skilled individuals. Instead, they use the default option to remove the ommon omponent inluded in the signal and plae weight only on the idiosynrati omponent. Finally, note that as in Proposition 1, the skilled individuals exert a higher level of effort in the absene of a default option sine the inentive to gather information is stronger when they do not have a default option to fall bak on. This in turn auses the quality of their advie to derease, and further amplifies the previously disussed differene between ν 0 and ν 1. That is, unskilled individuals do not benefit as muh from a skilled individual s information, and are thus inlined to pay less for it. As in Setion 2, to assess the pros and ons of the planner s default option, we ompare total welfare with and without this option. In this ase, welfare must be aggregated over skilled and unskilled individuals. This is done in the following lemma. Lemma 4. The total welfare without a default option is The total welfare with a default option is W N = (Σ g +Σ t )+ µ 2 (Σ g +Σ t ) 2 + 1 µ (Σ g +ρσ t ) 2. (24) W D = Σ t + µ 2 Σ2 t + 1 µ ρ 2 Σ 2 t. (25) In Setion 2, an inrease in α enhanes overall welfare through the larger information gathering externalities that individuals have on eah other. We an now see from (24) and (25) that inreases in ρ have a similar effet in the presene of information sales. More preisely, straightforward differentiation of these two expressions with respet to ρ lead to W N ρ = 2(1 µ) (Σ g +ρσ t )Σ t > 0 (26) 16
and W D ρ = 2(1 µ) ρσ 2 t > 0. (27) That is, a larger orrelation aross individuals types leads to more welfare when a formal advie hannel, like information sales, is inorporated. We an also see that the inrease in welfare aommodated by this advie hannel is more important when a sizeable fration of the population is unskilled (i.e., 1 µ is large). Finally, it is lear that (26) is greater than (27): the advie hannel is more ruial and the role of ρ greater when the soial planner refrains from making a default option available, as unskilled individuals an then rely only on the skilled individuals information for their deisions. The next proposition is the analogue of Proposition 2 when we allow for information sales. Proposition 8. The total welfare W N without a default option is higher than the total welfare W D with a default option if the ost parameter is in the following region: ( Σ g +Σ t < < 1 µ ) Σ g + [ µ+2(1 µ)ρ ] Σ t. (28) 2 This region is non-empty if and only if ρ > 1 2 and Σ g < 2(1 µ) (2ρ 1). (29) Σ t µ As mentioned above, ρ plays an espeially important welfare role in information sales when the soial planner does not make a default option available. Proposition 8 formalizes this by showing that ρ 1 2 always makes the availability of a default option optimal. That is, unskilled individuals are better off learning the ommon omponent of their type perfetly from the soial planner when the information that an be aquired from other individuals is not all that useful. This implies that default options are espeially valuable when the needs of an individual are unlikely to be similar to those of his peers, inluding the ones who an advise him. We an also see from (29) that default options are less valuable when Σ t is large and Σ g is small, whih is similar to our findings in Setion 2. The extent to whih the soial planner an resolve the unertainty faed by the population is still an important determinant of the usefulness of default option. Interestingly, however, default options are more valuable when a larger fration of the population is skilled (large µ), even when ρ is large. This arises beause the information externalities that skilled individuals bring to the eonomy through information sales is limited: the small number of unskilled individuals leads to a small number of information sales, and so the effort hoies of skilled individuals with and without a default option (as derived in Proposition 7) do not lead to signifiantly different externalities. 17
4 Behavioral Considerations 4.1 Prorastination The default option models developed by Choi et al. (2003) and Carroll et al. (2009) revolve around the idea that individuals tend to prorastinate when they have to make deisions. In these models, prorastination derives from hyperboli disounting preferenes by whih individuals do not value the future benefits of optimal finanial deisions as muh as the urrent utility ost of making these deisions. That is, as suggested by Akerlof (1991), making deisions today appears more painful and ostly than the future losses that result from postponing these deisions. As we show in this setion, the same prorastination fores an be aommodated in our model without affeting our onlusions regarding the information ontent of default options. In this sense, our model omplements these existing models by adding informational onsiderations to the set of tradeoffs involved in the optimal design of default options. To illustrate our point, we return to the model of Setion 2 and assume that in order to make an ative deision eah individual i faes an additional ost { 0, prob. φ κ i = k, prob. 1 φ, where φ (0,1] and k > 0. This ost is observable by individual i at the outset, but is potentially misinterpreted by him. In partiular, every individual i thinks that his ost is κ i +b, where b 0. That is, although this individual will experiene a ost of κ i when he hooses to make an ative deision, hethinksthat doingso will ost himmore. As aresult, heis naturally inlined to postpone ative deision-making and instead rely on the default set by the soial planner. Therefore, as with hyperboli disounting, individuals are mistaken by the ost of making a deision today relative to the future gains that ome with this deision. As before, the soial planner hooses her default option poliy in order to maximize total welfare. As in Carroll et al. (2009), if the planner does not provide a default option, eah individual i must atively make a deision and therefore inur the ost κ i. Alternatively, the planner an announe a default option that individuals an adhere to without inurring any osts, or hange at their will. Beause the hange entails a utility ost for eah person, the planner s hoie of default now serves two purposes. First, as before, it serves to ommuniate the information that the planner has about g. Seond, as in the work of Choi et al. (2003) and Carroll et al. (2009), it serves to affet individuals inentive to exert effort. Speifially, the planner an now hoose a enter default of g or an offset default of g γ where, without loss of generality, γ > 0. In both ases, the default 18 (30)