Yves Breitmoser
email: breitmoser@euv-frankfurt-o.de.

Steffen Schlächter

Extortion in the laboratory Friedel Bolle EUV Frankfurt (Oder)

Keywords: JEL classification: C72, C91, D74 laboratory experiment, mafia, extortion, punishment, finitely repeated game JEL classification: C72, C91, D74 laboratory experiment, mafia, extortion, punishment, finitely repeated game

In a laboratory experiment, we study a finitely repeated game (T = 15) under complete information. In each round, P demands tribute (cash transfer) from A, A complies or refuses, and after refusals P may punish A. In equilibrium (payoff maximization), P does not punish and A refuses any positive demand. In the experiment, P punishes increasingly often and increasingly severely as she gains experience; most As comply with P's demands. The observations are compatible with linear spite. In a finite mixture model, the types of P and A in the subject pool are characterized. An A that is resistant to extortion (declines all demands) is very rare, and hence the threat of punishment in general is effective, but all As either ignore actual punishment or react negatively to it. They accept to pay tribute but they are resistant to piecemeal expropriation.

A c c e p t e d M a n u s c r i p t

• laboratory experiment on finitely repeated extortion game

• P(rincipal) demands tribute from A(gent), and if A does not comply, P may punish A

• P punishes increasingly often and increasingly severely after rejections Many economic interactions are asymmetric in that one player, P(rincipal), is able to punish misbehavior of the other, A(gent), but effectively not vice versa. Such interactions can be found in social, military, and business hierarchies, between teachers and students, parents and children, or the mafia and shopkeepers. Formal analyses of such interactions usually build on the pair of assumptions that (i) punishment is sufficiently severe to destroy the benefits of misbehavior of opponents, and (ii) it is costly. The former implies that misbehavior is ruled out if the threat of punishment is credible, while the latter implies that it is non-credible in finitely repeated interactions. In turn, if the threat of punishment is non-credible (e.g. if the interaction has finite time horizon),

A has nothing to fear and misbehaves: students harass their teachers, children mess around, shopkeepers refuse payment, and so on. Experimental analyses of this prediction (under complete information) are rare, however. The only dedicated experiment seems to be the one of [START_REF] Jung | On the existence of predatory pricing: An experimental study of reputation and entry deterrence in the chain-store game[END_REF], who consider a game of P against a sequence of short-lived A's, but in the above examples, A is actually long-lived. 1 In our experiment, P faces a long-lived A. The subjects are matched to form pairs and play a finitely repeated extensive form game. The time horizon (T = 15) is common knowledge. In each round, P is endowed with 40 and A is endowed with 160.

They play a constituent game with three stages. First, P chooses a demand x ∈ [0, 160], second A decides whether to accept or decline, and third, if A declined, P may punish A through reducing his income by some amount y ∈ [0, 160]. Punishment (y > 0) bears costs of 10 for P. The game has been designed to distinguish demands that exploit P's strategic advantages from demands inspired by fairness concerns, and similarly to distinguish punishment in general from punishment to level incomes. For example, if P intends to level incomes, then she demands x = 60 ( = 160/2, these choices are separated to distinguish whether P wishes half of A's endowment rather than equality). Demands x > 60 suggest that P intends to exploit the strategic asymmetry and demands

x < 60 suggest that P feels weak. Analogously, P punishes in order to level incomes 1 There is more loosely related work, e.g. on entry deterrence without the possibility of punishment ex post [START_REF] Mason | An experimental analysis of subgame perfect play: the entry deterrence game[END_REF], chain store games under incomplete information [START_REF] Cooper | Signalling and adaptive learning in an entry limit pricing game[END_REF], repeated "reputation" games under incomplete information, e.g. [START_REF] Camerer | Experimental tests of a sequential equilibrium reputation model[END_REF], [START_REF] Neral | The sequential equilibrium theory of reputation building: A further test[END_REF], and [START_REF] Andreoni | Rational cooperation in the finitely repeated prisoner's dilemma: Experimental evidence[END_REF], and also the wealth of experiments on ultimatum games, but in none of these cases, P may punish A in every round and the threat of doing so is non-credible.

2

A c c e p t e d M a n u s c r i p t by choosing y = 130 in stage 3, she punishes "adequately" in relation to the rejected demand by choosing y = x (i.e. thus A does not profit through rejecting the demand), while the maximal and perhaps most effective punishment is y = 160.

The results can be summarized as follows. First, the principals (P) become increasingly inclined to punish (y > 0) as they gain experience, and eventually, they almost universally choose the maximal punishment y = 160 after declines of their demands. This inclination to punish is a divergence from subgame perfection2 that almost all subjects in the role of P adopt. There is no "end effect" in that behavior approaches subgame perfection in the last rounds of the repeated game. Second, the principals' demands are fairly moderate, often just high enough to equalize payoffs, and even such demands get rejected by some of the agents. This contrasts strikingly with the observed convergence toward maximal punishment. For, taking the latter as given essentially transforms the interaction into a repeated ultimatum game. In relation to ultimatum games, equalizing demands are moderate indeed and show that the Ps struggle to fully exploit their strategic advantage.

Third, we observe that both demands and accepted demands are significantly increasing in the course of time, while the econometric estimates of the demand strategies are independent of "exogenous inputs" such as time, acceptance decisions, and previous punishment. Econometrically, they simply constitute stationary ARMA processes.

To understand the discrepancy between overall trend and individual behavior, we reestimate the strategies controlling for subject heterogeneity (in a finite mixture model).

After segregating subject types, we find that that about 43% of the Ps actually do increase demands in time, and another 42% of them increase demands after acceptance (while acceptance rates happen to be increasing in time). In turn, only controlling for subject heterogeneity disentangles the diversity of behavior and explains the overall trend. Arguably, similar diversity may be hidden in many experimental analyses.

Fourth, our results confirm the suspicion that punishment is a two-edged sword (see e.g. [START_REF] Fehr | Detrimental effects of sanctions on human altruism[END_REF]. The anticipation of getting punished induces agents to accept moderate demands, but the actual execution of punishment does not improve acceptance rates. About 40% of the agents ignore the level of punishment, and the remaining 60% react negatively to getting punished (i.e. their acceptance rates A c c e p t e d M a n u s c r i p t drop). Overall, agents seem willing to comply with moderate requests, and in particular they give more than they would without the threat of punishment (i.e. in dictator games). But they ignore punishment as such, or at least seek to maintain a reputation of ignoring it, seemingly to prevent piecemeal expropriation. This resistance to punishment is sustained although P may punish at low costs, which suggests that it would be stable also under other cost structures.

Finally, let us comment on the persistence of punishment, despite its violation of subgame perfection. 3 This has been observed similarly in the context of "altruistic" punishment by Fehr andGächter (2000, 2002). Altruistic punishment differs from "egoistic" punishment in extortion games, however. Altruistic punishers intend to sustain efficiency in contributions to public goods, while egoistic ones intend to enforce redistribution of resources (or contracts, see e.g. [START_REF] Fehr | Reciprocity as a Contract Enforcement Device: Experimental Evidence[END_REF]. This affects the strategic consequences fundamentally, as for example [START_REF] Masclet | Threat and Punishment in Public Good Experiments[END_REF] report that threats of altruistic punishment effectively increase contributions, while in our case the previous period's punishment (= the threat for punishment in this period) decreases acceptance rates. The reason for the persistence of punishment in our experiment is therefore understood best in contrast to [START_REF] Jung | On the existence of predatory pricing: An experimental study of reputation and entry deterrence in the chain-store game[END_REF], where punishment also serves redistribution but disappears over time. The reason for the divergence seems to be the relative cost-effectiveness. In our case, a self-chosen damage can be inflicted at costs of 10, while Jung et al. give P the option to destroy 70 at costs of 90 for herself. The low costs of punishment in our experiment loosely reflect the examples cited above (e.g. mafia or parental punishment), but analyzing experiments with intermediately effective technologies may allow future research to locate the threshold between persistence and disappearance of punishment. As for altruistic punishment, [START_REF] Egas | The economics of altruistic punishment and the maintenance of cooperation[END_REF] and in particular [START_REF] Nikiforakis | A comparative statics analysis of punishment in public-good experiments[END_REF] have shown that its cost effectiveness is decisive.

The remainder of the paper is organized as follows. Section 2 describes the experimental game and the procedure. Section 3 describes the basic results. Section 4 estimates the subjects' strategies and the subject types. Section 5 discusses the results and concludes. The experimental game is a finitely repeated game (T = 15) of an extensive-form constituent game. The players are matched with the same partner for the duration of the experiment and the role assignments (principal and agent) are held constant. The number of rounds (T = 15) had been chosen such that subjects can be considered experienced in the final rounds, without risking that the subjects perceive it as an infinitely repeated game at any point. Thus, behavior is projected to stabilize in the sense of Nash equilibrium in the final rounds, which allows us to assess individual strategies and implicitly the possible interdependence of preferences.

The constituent game consists of three decision stages. We refer to this game as the extortion game, but alternative interpretations are possible.

The equilibrium predictions depend on the players' utilities. If players maximize pecuniary payoffs (i.e. if u i = π i for i = A, P) and act in accordance with subgame perfection, then the equilibrium payoffs are unique: π P = 40 and π A = 160. Along the equilibrium path, P demands any x ∈ [0, 160], A rejects if x > 0, and P chooses zero punishment y = 0. This equilibrium prediction holds invariantly in all repeated games with finite time horizon.

Next, consider Fehr-Schmidt inequity aversion [START_REF] Fehr | A theory of fairness, competition, and cooperation[END_REF]. If i = j ∈ {P, A} denotes the players and (π i , π j ) the payoffs, then the Fehr-Schmidt utility of i is (using the guilt weight a i ≥ 0 and the envy weight b i ≥ a i )

u i = π i -a i • max{π i -π j , 0} -b i • max{π j -π i , 0}. (1)
A c c e p t e d M a n u s c r i p t and rejecting is the maximal demand that he would accept.

x max = (130 + b A • 120)/(1 + 2b A ) (2)
Hence, x max also is the choice of P in all SPEs if b A is common knowledge (assuming

a P ≤ 1/2; she demands x = 60 if a P > 1/2). It equates with 130 if A is egoistic (b A = 0)
and it approaches 60 as b A tends to infinity. 4Finally, assume that the players' preferences exhibit "linear altruism" u i = π i + α i π j for i ∈ {A, P}. In this case, P punishes a non-compliant agent if α P < -1/16, and if she does, she chooses maximal punishment y = 160. Anticipating such punishment, in the unique SPE, an A with α A ≥ 1 would accept any demand, and an A with α A < 1 would accept demands up to

x ′ max = (160 + 10α A )/(1 -α A ). (3)
That is, an altruistic agent (α A ≥ 0) would accept any demand, and spiteful agents with for example α A = -10/7 would accept demands up to 60. In contrast to inequity aversion, a spiteful principal does not generally prefer the equitable payoff allocation (100, 100) over the disagreement payoff, which is (30, 0) if P is spiteful, nor does she prefer the maximal sustainable allocation (160x ′ max , 40 + x ′ max ) over disagreement in general. If she does not, namely if x ′ max < (-10-160α P )/(1-α P ), then P deliberately makes an offer that A will reject.

There are at least two other reasons why P may punish non-compliance of A: P may try to build a reputation in the sense of [START_REF] Kreps | Reputation and imperfect information[END_REF] and [START_REF] Milgrom | Predation, reputation and entry deterrence[END_REF], or P may be boundedly rational. The former may be relevant even in a game that is repeated only 15 times, e.g. if the share of boundedly rational players to be mimicked is sufficiently high, which seems reasonable in laboratory experiments.

A c c e p t e d M a n u s c r i p t

Experimental procedure

The experiment took place in the computer laboratory of the European University Viadrina in Frankfurt (Oder), Germany. Each computer terminal in the laboratory was partitioned, so that subjects were not able to look at other computer screens, or to communicate via audio or visual signals. Subjects were recruited from an email list consisting of students from the faculties of Cultural Science, Business and Economics, and Law. We conducted eight sessions, each with either 10 or 12 participants. Altogether, 92 subjects were randomly and anonymously matched to form 46 pairs who played 15 rounds of the above game. Our data set therefore comprises observations from 46 stochastically independent finitely repeated games.

Each session proceeded as follows. First, subjects were randomly allocated to their seats, second they were provided with the experimental instructions and a short control questionnaire (translations of both are provided in the appendix). Their answers to the control questionnaire allowed us to verify their understanding. Subjects in doubt were verbally advised by assistants before the experimental game began. At the end of the experiment, subjects were informed of their payments, and asked to privately choose a code name and password. This was used to anonymously collect their payments from a third party about a week after the experiment.

Throughout the experiment, "mildly" loaded language was used. For example, the game was not referred to as "mafia game" or "extortion game" and the term "punishment" was avoided, but we describe the stage-3 action of P as "If the A-participant rejects the demand, then the B-participant has the option to destroy an arbitrary number of points (maximally 160) of the A-participant." The experimental instructions and the full data set are provided as supplementary material. The experiment was conducted using z-tree [START_REF] Fischbacher | z-tree: Zurich toolbox for ready-made economic experiments[END_REF], the programs are available from the authors upon request. Every session lasted about 30-40 minutes. The average income was e 12.07 and varied between e 6.45 and e 18.45.

Basic results

The average results are displayed in Figure 1. Almost all demands are positive (99%), and the most frequent demand is 60. When accepted, a demand of 60 equates the final A c c e p t e d M a n u s c r i p t incomes at 100 each. Principals usually punish rejections of their demands (in 90% of the cases), and in 70% of these cases the maximal punishment y = 160 had been chosen. The occurrences of maximal punishment are compatible with linear altruism if α P < -1/16. Punishment below the principal's demand (0 < y < x) was observed in only 3.6% of the cases, and similarly y = x (relative frequency 1%) or the SPE punishments of egoistic players (y = 0, relative frequency: 10.4%) and Fehr-Schmidt players (y = 130, 3.6%) had not been chosen systematically. Apparently anticipating severe punishment, agents typically accept moderate demands. Overall, 72% of the demands were accepted, while the average accepted demand was 64.64, and the average rejected demand was 85.38. These values differ significantly (p < .001),5 and in this sense agents are more likely to reject high demands. As we will see below, demands and acceptance decisions are compatible with linear altruism, too.

The paths of play of the 46 pairs are highly diverse. In some cases, the agent seems to try to enforce demand reductions by rejecting previously accepted demands, in other cases, the principal seems to be attacking (by raising demands after acceptances), and in some pairs, the demands had actually been held constant for all 15 rounds and got accepted throughout. Such variations in the course of action, and its structure, suggest that the subject pool is not homogenous. This will be discussed in detail once the basic structure of the data over time has been described.

Figure 2 presents box-and-whisker plots of demands and accepted demands over time. It shows that the quartiles of the demands remained fairly constant for the duration of the experiment, and in particular the lower quartile is at x = 60 from round 3 on. Overall, demands are increasing in time, however. Tables 1-3 describe how demands, acceptance rates, and punishment changed over time. In these three tables, we have segregated the 15 rounds into three phases of five rounds each. Table 1 shows that the relative frequency of demands x < 60 decreases over time, from 23% initially to 6% eventually, and the relative frequency of demands x > 60 increases over time, from 37% initially to 55% eventually. These changes are significant (p < .001 and p < .001, respectively, in Wilcoxon matched pairs tests, using the 46 pairs as independent observations). In parallel to this development, the "accepted demands" changed over time. Figure 2 shows that the eventual lower quartile is above the initial upper quartile. Similarly, Table 2 shows that the acceptance rates with respect to most levels Overall, the principals' incomes increase from 79.6 initially to 90.1 eventually (p < .01 in a Wilcoxon paired sample test, using the n = 46 individual means as independent observations) and the agents' incomes decrease from 84.1 to 71.2 (p = 0.046), see Table 1. This shows that the extortion games converge to a state favoring P, but the sources of this development (i.e. the time dependence of demands and acceptance rates) are not obvious. This is analyzed in the next section.

The aggregate behavior does not converge toward the exact predictions for either payoff maximizing players or Fehr-Schmidt players. The former is incompatible with the observation that both demands and acceptance rates are clearly positive and even increasing in time, and the latter is not exactly compatible with the joint distribution of demands and punishments. In the last third of the experiment, most subjects choose the maximal punishment y = 160 (82%, see Table 3)-when punishment does not serve reputation building anymore-while the majority (55%) of demands is x > 60.

Demands x > 60 suggest that most players are at most partially inequity averse, but if so, they should punish by y ≤ 130. A possible explanation is that principals are inequity averse in general and spiteful after rejections (i.e. angered), but our experiment was not designed to distinguish this from the simpler notion that Ps are spiteful throughout. Specifically, maximal punishment results for α P < -1/16 in linear altruism, demands

x ≥ 60 result for α P ≥ -7/10, and the intersection of these conditions is non-empty.

Strategy estimation and latent type classification

To understand why extortion is profitable and how individuals act in detail, we now estimate the subjects' strategies. For a wide range of utility functions, the SPE predictions are unique for finitely repeated extortion games. This follows from the uniqueness of the constituent game equilibria, and implies in our case that the SPE strategies are even stationary (i.e. time invariant). Assuming stationarity seems a little restrictive when estimating behavioral strategies, however, and we therefore allow for a variety of alternative influences.

In modeling the demand strategy (X i,t ) 15 t=1 of principal i, we allow for correlation with last round's demand X i,t-1 (and for this reason, we skip the observation for t = 1), • P i,t-1 -0.5584

(0.0532) * * * • ε i,t-1 + ε i,t
where ε i,t has standard deviation σε = 18.87. This suggests that demands are best described as ARMA processes-the "exogenous inputs" such as acceptance decisions, time, and punishment are insignificant with respect to the "representative" subject.

The obvious issue with this conclusion is that the assumed existence of a "representative" subject is invalid. For this reason, we now relax the assumption of subject homogeneity. The econometric technique to do so is finite mixture modeling [START_REF] Peel | Finite Mixture Models[END_REF]. In the existing economic literature, finite mixture modeling is still used irregularly, but the applications that exist have shown that latent heteregoneity is a concern in experimental analyses. For example, [START_REF] Bardsley | The experimetrics of public goods: Inferring motivations from contributions[END_REF] and [START_REF] Fischbacher | Social preferences, beliefs, and the dynamics of free riding in public goods experiments[END_REF] show heterogeneity of strategies in public goods contributions, [START_REF] Conte | Mixture models of choice under risk[END_REF] and [START_REF] Harrison | Expected utility theory and prospect theory: One wedding and a decent funeral[END_REF] show heterogeneity of preferences in choice under risk, and heterogeneity of "levels of reasoning" in pbeauty contest is observed by [START_REF] Stahl | Boundedly rational rule learning in a guessing game[END_REF] and [START_REF] Ho | Iterated dominance and iterated best response in experimental "p-beauty contests[END_REF] and in other games by [START_REF] Kübler | Limited depth of reasoning and failure of cascade formation in the laboratory[END_REF] and [START_REF] Crawford | Level-k auctions: Can a nonequilibrium model of strategic thinking explain the winner's curse and overbidding in private-value auctions?[END_REF], to name just a few.

These studies suggest that heterogeneity of subject pools is the rule rather than the exception, and thus also that finite mixture modeling should be considered routinely.

Technically, we consider a finite mixture of tobit models to represent individual demand strategies (similar to e.g. [START_REF] Bardsley | The experimetrics of public goods: Inferring motivations from contributions[END_REF], for public goods contributions) and finite mixtures of logit models to represent acceptance strategies. The details are summarized in Appendix A. The results are in Tables 4 and5.

Before we discuss the estimated demand strategies, let us note that the Bayes information criterion (BIC, [START_REF] Schwarz | Estimating the dimension of a model[END_REF]

13

A c c e p t e d M a n u s c r i p t type that we add (until level 3). Considering that improvements of about 20 points are highly significant in adding six-parametric types (according to likelihood ratio tests), improvements of 200 points are striking and confirm the observations on heterogeneity made in the aforementioned studies. In particular, note the significance of heterogeneity in contrast to the insignificance of exogenous inputs such as A j,t-1 , R t , and P i,t-1 .

We identified three types of principals in our subject pool (according to the BIC).

After eliminating insignificant parameters (again according to BIC) from the type definitions, we obtained model C reported in Table 4. Type 1 has relative frequency 28.7%

and a demand strategy with residual standard error σε 1 = 32.49. That is, about two in seven subjects are of type 1. The long-term mean of X 1 i,t is µ 1 = 56.8399/(1 -.3658) = 90.9, which is high in relation to the other two types. Hence, we refer to them as aggressive subjects. About one in nine subjects (10.8%) is of type 2-with the extremely low residual standard error σ ε 2 = 0.172. Due to the low residual variance, and the longterm mean at µ 2 = 60, these subjects will be referred to as fair subjects. The remaining approximately three in five subjects (60.6%) are of type 3. They have the residual standard error σ ε 3 = 10.61, and most interestingly they strategically react to acceptance decisions (raising the demand by about 10 after acceptance). The long-term mean in response to rejecting agents, i.e. if A j,t-1 = 0 and P i,t-1 = 160, is µ 3 = 67.2. The long-term mean in response to accepting agents, i.e. if A j,t-1 = 1 and P i,t-1 = 0, is µ 3 = 76.7. We refer to them as adaptive subjects. 7 The agent strategies are estimated similarly. Agent j responds with either "accept" or "reject" to the principals demand, which we represent as a finite mixture of logit models with the following independent variables: the current demand X i,t , the demand change ∆ i,t = X i,t -X i,t-1 in relation to the previous round, the number of rounds left R t = 15t, and last round's punishment P i,t-1 . Again three types are identified. 8 The estimates are reported in Table 5, and the average marginal effects required to interpret the estimated strategies are reported in Table 6. The average ac-7 Since only 3/5 of the subjects react to rejections and punishments, this explains why these terms had been insignificant in the global regression model.

8 A referee raised the concern that the behavior (and type) of a given A depends on behavior (type) of the P he is matched with, or vice versa. After computing the posterior classifications for all subjects, see Appendix A, we verified the stochastic independence between A's classification and P's classification in linear probability regression models. We found that there was no significant interaction effect between the types (at the 10% level), i.e. the classifications of A and P do not violate stochastic independence.

14

A c c e p t e d M a n u s c r i p t Note: The marginal effect is the first derivative of the acceptance probability with respect to the respective independent variable. We computed these derivatives for all data points and report their averages.

ceptance probabilities of types 1 through 3 are 0.873, 0.388, and 0.711 respectively (these probabilities are the predictions averaged over all data points). Overall, Type 1 (share 28.8%) is largely acceptive with little response to the demand itself and intermediate response to the number of rounds left. Type 2 (share 9.75%) is on average rejective with intermediate (yet insignificant) response to the demand, and stronger response to demand changes and the number of rounds left. Finally, the majority of agents is of type 3 (share 61.5%). They are particularly responsive, with intermediate acceptance rates and strong negative responses to the level of the demand, to demand increases, and to punishment. The behavior of type 3 can be interpreted as reciprocal and the negative response to punishment resembles counter-punishment in the sense of [START_REF] Nikiforakis | Punishment and counter-punishment in public good games: Can we really govern ourselves[END_REF]. Note that the counter-punishment is substantial. After maximal punishment y = 160, acceptance rates drop by roughly 14% on average.

These strategy estimates allow us to shed light on the observations made in Section 3. Table 2 showed that for most levels of demands, the acceptance rates are increasing in time. This increase of acceptance probabilities in time has been found for the agents classified as "acceptive" or "rejective". The majority of agents is classified as "responsive," however, and they respond to the demand, to demand changes, and to punishment, but not to time. The univariate tests also had shown that principals' demands are increasing in time, while the estimated representative demand strategy was stationary. After disaggregating subject types, we can now see that time is relevant only for aggressive principals. In addition, "adaptive" principals increase demands (by about 10 points) when last round's demand had been accepted, and since acceptance rates are increasing in time, this implies that "adaptive" demands correlate with time, too-without causality. Finally, principals classified as "adaptive" seem to believe that A c c e p t e d M a n u s c r i p t punishment improves acceptance rates (see that P i,t-1 is significant for them). Agents do not respond positively to punishment, though. The first two types of agents do not respond to punishment at all, and the majority of agents (the "responsive" ones) react negatively.

Conclusion

In our experiment, a powerful player called P(rincipal) interacts with a second player called A(gent) for 15 rounds. In each round, P may demand a transfer from A, e.g.

in terms of money, housework, homework, quality time, and so on. A may accept or decline the transfer, and after declines, P may punish A at comparably small costs for herself. The induced valuations have it such that subjects maximizing pecuniary payoffs do not make any transfer from A to P under subgame perfection, and that P would never punish A after rejections (due to the costs). We found that P's demands are largely moderate, As tend to accept moderate demands, but As that decline a demand are punished harshly. Furthermore, one type of Ps (accounting for 3/5 of the population) raise demands after harsh punishment, seemingly expecting A to give in, and yet all types of A respond non-positively to punishment. The acceptance rates remain either constant or drop after punishment.

Overall, however, the subjects' interactions do not display excessive disagreement. The maximal combined earnings of P and A are e 30 in the experiment, and the average combined earnings were e 24.14-about 80% of the maximum, disagreement occurred on average only every fourth round. Hence, the fairness standards of the various P and A in our experiment are largely compatible with one another, which one would not expect unless these standards (or beliefs) are "realistic" and apply similarly outside the laboratory. In contrast to this overall compatibility of fairness standards, the convergence toward maximal punishment after rejections seems particularly surprising. Further research may investigate how frequency and extent of punishment depend on the efficiency of the punishment technology and on the possibility to punish agents also after acceptance.

Finally, let us emphasize the relevance of controlling for subject heterogeneity.

We did so non-parametrically, by finite mixture modeling, which strikingly improved the goodness-of-fit and resolved the (counter-intuitive) insignificance of state variables

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A c c e p t e d M a n u s c r i p t such as time and previous acceptance decision with respect to the decisions of the "representative" subject. There simply is no representative subject, and in general, analyses assuming otherwise risk getting misled. In our case, accounting for heterogeneity recovered the state variable effects, and amongst others it allowed us to pin down why demands are increasing in time-although exogenous inputs are insignificant for the representative subject. From a more general point of view, this shows that modeling heterogeneity is important not only for technical econometric reasons, but also to understand how individual behavior relates to the overall patterns of the data.

The approach of finite mixture modeling, which has yet to gain wide application (but see [START_REF] Bardsley | The experimetrics of public goods: Inferring motivations from contributions[END_REF][START_REF] Conte | Mixture models of choice under risk[END_REF][START_REF] Harrison | Expected utility theory and prospect theory: One wedding and a decent funeral[END_REF], seems particularly helpful in this respect.

Appendix A Finite mixture modeling

A detailed review of finite mixture modeling can be found in [START_REF] Peel | Finite Mixture Models[END_REF]. The following intends to provide a brief introduction with all the information needed to follow the above analysis. The basic idea is to model a (heterogenous) population with a multiplicity of discrete player types. The main alternative approach would be to assume that subject heterogeneity can be represented using parametric distributions of random effects on the intercept (as in mixed effects models) or on other model parameters (as in say mixed logit models). The finite mixture model is more flexible in describing interdependence of parameters across player types, but it works well only if the actual distribution of types is indeed clustered in the parameter space, which then allows us to approximate it using discrete types. In our analyses, the distinction of discrete types did improve the log-likelihood strikingly, and hence we concluded that it works well.

To define the finite mixture model formally, consider a population with K ∈ N types. Each type k ≤ K is defined by P parameters (e.g. coefficients of a tobit or logit model), denoted as α k := {α k,p } p≤P . The aggregate parameter profile is α := {α k } k≤K . The prior probability that a subject is of type k is denoted as ρ k , for all k ≤ K; the posterior follows from Bayes' Rule as shown below. Now, let o s,t denote the action of subject s ≤ S in round t ≤ T of the experiment, and let Pr(o s,t |α k ) denote the probability/density that a player of type k (i.e. with parameters α k ) chooses o s,t .

A c c e p t e d M a n u s c r i p t

The log-likelihood of the model, represented by its parameters (α, ρ), given the data

set o = (o s,t ) is LL(α, ρ|o) = ∑ s≤S ln ∑ k≤K ρ k • ∏ t≤T Pr(o s,t |α k ). (4)
In turn, the posterior probability that subjects s is of type k, conditional on the actions

o s , is Pr(Type = k | o s ) = ρ k • ∏ t≤T Pr(o s,t |α k ) ∑ k ′ ≤K ρ k ′ • ∏ t≤T Pr(o s,t |α k ′ ) . ( 5
)
For the maximum likelihood estimates of the parameter profile (α, ρ), the aggregate posterior share of type k equates with its prior share. For every individual, the posterior classification is probabilistic (although it may be rather decisive, assigning probabilities close to either 0 or 1 for all types to every subject). The posteriors for our subjects are listed in the supplementary material.

We obtained the parameter estimates by maximizing the full likelihood function jointly over all parameters (to obtain efficient estimates, see [START_REF] Arcidiacono | Finite mixture distributions, sequential likelihood and the em algorithm[END_REF]. We used the Nelder-Mead algorithm as implemented in GNU R. The standard errors were derived from the information matrix, which in turn was computed numerically. Model selection was based on the Bayes information criterion (BIC, [START_REF] Schwarz | Estimating the dimension of a model[END_REF]. In our case, the BIC of a given model can be defined as

BIC = -LL + (K * P + K -1)/2 * ln(S * T ), (6)
where K * P + K -1 is the number of parameters and S * T is the number of observations.

A c c e p t e d M a n u s c r i p t

Supplementary material

Extortion in the laboratory

Overview

The supplementary material provides translations of both the instructions and the control questionnaire used in the experiment (the originals, in German, are available from the authors) as well as various figures and tables with the content described next.

• Figures 1-4 plot all demands and acceptance decisions for all P-A pairs. For each plot, the header displays the unique identifiers of P and A in the respective match (the same identifiers are used in the tables following below), and the plot contains a connected line representing the various demands of P and for each demand it displays a symbol representing either acceptance or rejection of A (a triangle pointing upwards indicates acceptance, and a little cross indicates rejection).

• Table 1 provides a tabular representation of all observations. Each "cell" in this table consists of two lines and lists the actions of a particular P-A pair. The order of P-A pairs is the same as in Figures 1234. The upper line in each cell shows the identifier of the repsective P and all demands she made. The lower line shows the identifier of A and his corresponding acceptance decisions ("1" is acceptance, "0" is rejection).

• Table 2 provides the posterior classifications for all P based on their actions.

• Table 3 provides the posterior classifications for all A based on their actions. The entries are ordered such that the first P listed in Table 2 ("S16") is the one who was matched with the first A listed in Table 3 ("S11"), the second P ("S17") was matched with the second A ("S12"), and so on. The posterior classifications are probabilistic, as described in Appendix A of the paper.

1

Table(s)

A c c e p t e d M a n u s c r i p t

Experimental instructions Instructions

You are participating in an experiment on decision making. The computer will match you randomly in groups of two. You remain in the same group for the duration of the whole experiment. One member of each group will be referred to as A-participant and the other member is referred to as B-participant. These assignments are random, too, and held constant for the duration of the experiment.

The experiment consists of 15 stages. The procedure in each stage is identical to that in all other stages. In each stage, you earn points that will be converted into Euro and paid to you after the end of the experiment. Here, the sum of the points earned in all stages is relevant. For each point, you obtain 1 cent. The payment can be collected a week after the experiment from the secretary of the chair.

The procedure of single stages The endowment of the A-participant is 160 points and that of the B-participant is 40 points. First, the B-participant may choose a number of points that he demands from the A-participant. Second, the A-participant has to decide whether to comply with the demand or not.

• If the A-participant accepts the demand, then the respective number of points is transferred to the B-participant.

• If the A-participant rejects the demand, then the B-participant has the option to destroy an arbitrary number of points (maximally 160) of the A-participant.

Exercising this option bears costs of 10 points for the B-participant.

Then the stage ends and the resulting scores are saved for both participants. If the 15th stage is not reached yet, then a new stage begins (with constant assignments of roles).

Example 1 The B-participant demands x 1 points and the A-participants accepts the demand. The resulting scores for this stage are as follows.

A-participant: 160 - The resulting scores for this stage are as follows.

A-participant: 160 B-participant: 40

Example 3 The B-participant demands x 3 points, the A-participant rejects the demand, and the B-participant destroys y 3 points of the A-participant. The resulting scores for this stage are as follows.

A-participant: 160 -y 3 B-participant: 40 -10

Questionnaire

Please tick the correct answers.

1. The person that is matched with me in a group changes after each stage.

is always the same.

may change, but may also be the same.

2. The number of stages overall is equal to 1.

finite, but random and not known in the beginning.

equal to 15.

3. The assignments of A-and B-participants in a group is held constant in all stages.

changes after each stage.

may change, but may also be kept constant.

May payment in this experiment

•

  eventually, most As comply with P's demands• finite mixture modeling of type heterogeneity is applied to understand behavior 1

0.

  The principal P is endowed with 40 Euro-cent, the agent A is endowed with 160 Euro-cent 1. P chooses a demand x ∈ [0, 160] 2. A decides whether to accept or reject the demand; if A accepts, then the round ends and the payoffs are π P = 40 + x and π A = 160x 3. If A rejects, then P chooses a punishment y ∈ [0, 160]; if P chooses y = 0, then π P = 40 and π A = 160 results, and otherwise π P = 30 and π A = 160y results

  A c c e p t e d M a n u s c r i p t

Figure 1 :

 1 Figure 1: Game time line and average decisions

  A c c e p t e d M a n u s c r i p t of demands increased over time.

  A c c e p t e d M a n u s c r i p t

Figure 2 :

 2 Figure 2: Box-and-whisker plots of demands and accepted demands

  The B-participant demands x 2 points, the A-participant rejects the demand, and the B-participant chooses not to exercise the option of destroying points.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :

 1234 Figure 1: Individual demands and acceptance (triangle)/rejection (+) decisions

  An inequity averse P with Fehr-Schmidt preferences punishes non-compliance of A in equilibrium if and only if b P ≥ 1/12. If b P > 1/12, then P chooses y = 130 in

	every SPE, and (30, 30) results when stage 3 is reached. In Stage 2, A will therefore
	accept certain positive demands. For example, an inequity averse A knowing b P > 1/12
	generally accepts 60, since he prefers (100, 100) to (30, 30), but he would never accept
	a demand greater than 130. The demand x that makes A indifferent between accepting

Table 1 :

 1 Demand categories and incomes over time

	Rounds Rel. frequency of demand x	Incomes
		x < 60 x = 60	x > 60	Agent Principal	Σ
	1-5	22.6% 40.0%	37.4%	84.1	79.5	163.7
	6-10	12.6% 38.3%	49.1%	75.0	82.8	157.8
	11-15	6.1% 38.7%	55.2%	71.2	90.2	161.3

Table 2 :

 2 Frequencies of demands and acceptance rates (in brackets) over time

	Rounds	[0, 45)	[45, 55)	[55, 65)	[65, 75)	[75, 85) [85, 105) [105, 160]
	1-5	26(89%) 23(74%) 95(86%) 24(58%) 25(56%) 22(50%) 12(17%)
	6-10	12(92%) 13(62%) 93(97%) 36(50%) 31(55%) 31(39%) 11(36%)
	11-15 8(100%)	1(0%)	96(97%) 34(71%) 41(66%) 26(42%) 18(44%)

Table 3 :

 3 Frequencies of punishment (conditional on demand rejection) with last round's acceptance decision A j,t-1 ∈ {0, 1} of her agent j, with the number of rounds left R t = 15 -t, with last round's punishment P i,t-1 , and for serial correlation of individual errors ε i,t-1 . Such econometric models are known as censored "ARMAX"

	Rounds	y = 0	y ∈ (0, 160)	y = 160	Σ
	1-5	10 (14.9%) 26 (38.9%) 31 (46.3%) 67
	6-10	7 (10.1%) 18 (26.1%) 44 (63.8%) 69
	11-15	3 (5.4%)	7 (12.5%) 46 (82.1%) 56

Table 4 :

 4 Maximum likelihood estimates of the proposer models

	Model Type	Interc	Dem	Accept	RoundsLeft	Punish	Theta	Sigma	Share	BIC LL
	A	1	3.017 (3.9259)	0.9289 (0.0314) * * *	4.1128 (3.039)	-0.0578 (0.1021)	0.0084 (0.021)	-0.5584 (0.0532) * * *	18.8722 (0.5258) * * *	2828.33 (-2805.69)
	B	1	9.9768 (7.5578)	0.9033 (0.0574) * * *	2.3947 (5.2418)	-0.2064 (0.2181)	-0.0305 (0.0357)	-0.5754 (0.092) * * *	26.0469 (1.1605)	* * *	(0.0766) 0.4667 * * *
		2	-3.4758 (3.1001)	0.9389 (0.0291) * * *	8.6467 (2.3766) * * *	-0.0573 (0.0705)	0.0549 (0.0164) * * *	-0.3431 (0.0576) * * *	6.9881 (0.3628) * * *	2638.34 (-2589.83)
	C	1	55.7812 (17.7765) * *	0.3865 (0.2156) .	0 (-)	-0.8942 (0.5668)	0 (-)		-0.0433 (0.2282)	32.4928 (1.7722)	* * *	(0.069) 0.2865 * * *
		2	13.6486 (0.0032) * * *	0.7726 (2e-04) * * *	0 (-)	0 (-)	0 (-)		-0.7119 (0.2173) * *	0.1724 (0.0151) * * *	(0.0456) 0.108 *
		3	9.0971 (4.1901) *	0.7426 (0.0547)	* * *	10.6469 (2.8261) * * *	0 (-)	0.0513 (0.0193)	* *	-0.0354 (0.0864)	10.6122 (0.4943)	* * *	2448.12 (-2373.74)

Table 5 :

 5 Maximum likelihood estimates of the responder models

	Model Type	Int		Dem	∆Dem			RoundsLeft	Punish	Share	BIC LL
	A		4.2034 (0.4319)	* * *	-0.03187 (0.00483) * * *	-0.02716 (0.00723)	* * *	-0.07437 (0.02311)	* *	-0.00857 (0.00149)	* * *	348.72 (-332.38)
	B	1	7.8422 (1.4654)	* * *	-0.0505 (0.0104) * * *	0.0115 (0.0084)			-0.2435 (0.0865) * *	0.0046 (0.0059)	(0.0894) 0.2958 * * *
		2	5.3390 (0.6966)	* * *	-0.0520 (0.0087) * * *	-0.0621 (0.0132)	* * *	-0.0677 (0.0299) *	-0.0097 (0.0019) * * *	342.66 (-306.71)
	C	1	7.6883 (1.4034)	* * *	-0.0452 (0.009) * * *	0 (-)				-0.2736 (0.0828) * * *	0 (-)	(0.0825) 0.2874 * * *
		2	4.0678 (1.8914) *	-0.0425 (0.0256)	-0.1132 (0.0491)	*	-0.2697 (0.1088)	*	0 (-)	0.0974 (0.0557)
		3	6.9618 (0.9611)	* * *	-0.078 (0.0131)	* * *	-0.0501 (0.0143)	* * *	0 (-)	-0.0073 (0.0024) * *	337.05 (-294.56)

Table 6 :

 6 Average marginal effects in the logit models for A's acceptance Type Demand X i,t Change ∆ i,t Rounds left R t Punishment P i,t-1

	1	-0.00378	0	-0.02291	0
	2	-0.00637	-0.01698	-0.04043	0
	3	-0.00919	-0.00590	0	-0.00086

Table 3 :

 3 Posterior classification of agents

			Types				Types
	Subject Aggressive	Fair	Adaptive	Subject Aggressive	Fair	Adaptive
	S11	0	1	0	S52	1	0	0
	S12	0.9982	0	0.0018	S53	0.1282	0	0.8718
	S13	0.6353	0	0.3647	S54	0.0129	1e-04	0.987
	S14	0.5884	0	0.4116	S55	0	0	0.9999
	S15	0.7651	0	0.2349	S56	0.0174	3e-04	0.9823
	S21	0.0386	8e-04	0.9606	S61	0.0271	0	0.9729
	S22	0.1729	0.001	0.8261	S62	0.1106	0	0.8894
	S23	0.1383	0	0.8617	S63	0.0469	1e-04	0.953
	S24	0	0.2127 0.7873	S64	0	0.9999	1e-04
	S25	0.0051	0.3444 0.6505	S65	0	0.9968 0.0032
	S26	0	0	1	S66	0	0.0772 0.9227
	S31	1	0	0	S71	0.0014	0	0.9986
	S32	0.0239	5e-04	0.9756	S72	1	0	0
	S33	0.0143	0	0.9857	S73	1e-04	0.8956 0.1043
	S34	0.4323	0.001	0.5667	S74	0.2504	0	0.7496
	S35	0.8879	0	0.1121	S75	0	2e-04	0.9998
	S41	0.0966	0	0.9034	S76	0.9703	2e-04	0.0295
	S42	0	0	1	S81	0.5998	0	0.4002
	S43	1	0	0	S82	0.8233	0	0.1766
	S44	1e-04	1e-04	0.9997	S83	0	0.0363 0.9637
	S45	0.0528	0	0.9472	S84	0.2317	0.0236 0.7447
	S46	0.0013	0.0038 0.9949	S85	0.3773	0	0.6227
	S51	0.8698	0	0.1302	S86	0	2e-04	0.9998

Unless stated otherwise, by "subgame perfection" we refer to the subgame perfect equilibrium (SPE) assuming that it is common knowledge that all players maximize payoffs.

In most cases, convergence toward Nash equilibrium is observed in the final rounds of finitely repeated games, e.g. in Prisoner's dilemmas[START_REF] Selten | End behavior in sequences of finite Prisoner's Dilemma supergames A learning theory approach[END_REF], public goods games[START_REF] Muller | Strategic behavior and learning in repeated voluntary contribution experiments[END_REF][START_REF] Neugebauer | Selfish-biased conditional cooperation: On the decline of contributions in repeated public goods experiments[END_REF], market games[START_REF] Loomes | Do Anomalies Disappear in Repeated Markets?*[END_REF], and investment games[START_REF] Cochard | Trusting behavior in a repeated investment game[END_REF].

If b A is private information of A, then P maximizes her expected utility according to her risk attitude and the distribution of b A in the population, and ends up choosing some x ∈ [60, 130] as well.

in a Mann-Whitney-U test using the mean demands within the 46 pairs as independent observations

In this section, we use " * " to denote significance at α = .05 (in two-sided tests), " * * " to denote significance at α = .01, and " * * * " to denote significance at α = .001.

A c c e p t e d M a n u s c r i p t

100 100 100 120 100 100 100 100 120 120 120 120 100


Johannes Binswanger
email: j.binswanger@uvt.nl.

Lans Bovenberg

Simon Gächter

David Laibson

Wieland Müller

Manuel Oechslin

Ward Romp

Eytan Sheshinski

Dynamic decision making with feasibility goals: A procedural-rationality approach *

Keywords: Feasibility goals, planning, bounded/procedural rationality, life cycle saving, portfolio choice. JEL classification: D03, D81, D91

This paper proposes a new framework of intertemporal choice: decision making by means of so-called feasibility goals. These refer to the feasibility of certain target levels of consumption associated with specific future benchmark scenarios. The feasibility goals framework takes into account bounded/procedural rationality without stepping outside the realm of an optimization framework. The paper has two aims. First, it provides a general characterization of the feasibility goals framework. Second, it provides a simple life cycle model with feasibility goals to illustrate the applicability of the concept. The life cycle model is promising for better understanding individual portfolio choice.

A c c e p t e d M a n u s c r i p t 1 Introduction

Most economic models rely on the assumption of unbounded rationality. This may be natural in the case of simple choices. The plausibility of this assumption has been questioned, however, in the case of complex dynamic decisions under uncertainty such as portfolio choice or life cycle saving [START_REF] Pemberton | Attainable non-optimality or unattainable optimality: A new approach to stochastic life cycle problems[END_REF][START_REF] Thaler | Psychology and savings policies[END_REF][START_REF] Laibson | Self-control and saving for retirement[END_REF]. Advocates of the assumption of unbounded rationality point out that this assumption should not be taken literally. Rather, people are meant to behave as if they had unbounded cognitive capacities. They argue that people have been led to make rational optimizing decisions thanks to learning. This would happen in the same way as a billiard player learns to hit balls precisely without mastering the differential equations describing the balls' movements [START_REF] Friedman | The methodology of positive economics[END_REF]. However, two recent papers [START_REF] Lettau | Rules of thumb versus dynamic programming[END_REF][START_REF] Allen | Individual learning about consumption[END_REF] have put into question whether, from a theoretical perspective, boundedly rational agents can plausibly learn the type of dynamic behavior predicted by models of unbounded rationality. [START_REF] Barberis | A survey of behavioral finance[END_REF], [START_REF] Campbell | Household finance[END_REF], and, in particular, [START_REF] Choi | Reinforcement learning and savings behavior[END_REF] provide empirical evidence that many people fail to learn unboundedly rational behavior in the domain of portfolio choice.

In the presence of uncertainty, a key feature of decision making under unbounded rationality is the requirement of a complete action plan. This plan specifies an optimal action at each future node of an event tree that may be reached with positive probability.

It is noteworthy that, in the absence of such a plan, an expected utility function could not be evaluated and would thus be vacuous. Pre-specifying actions for all possible future circumstances is therefore an essential element of the standard model. Under realistic circumstances, the requirement of a complete action plan makes decision making according to the standard model very complex [START_REF] Pemberton | Attainable non-optimality or unattainable optimality: A new approach to stochastic life cycle problems[END_REF]. Realistic circumstances entail the existence of transitory and permanent shocks to income, of liquidity constraints and the opportunity to invest in multiple financial assets.

If the complete plan of optimal actions -according to the standard model -had a simple structure, then it would be straightforward for agents to learn to behave "as if" 1 A c c e p t e d M a n u s c r i p t following the standard model. However, [START_REF] Allen | Individual learning about consumption[END_REF] have come up with an astonishing finding: under plausible assumptions, it would take an agent about one million years to learn a reasonable approximation to behavior as predicted by the standard model. This suggests that its predicted behavior is indeed complex under realistic circumstances.

This conclusion is further underlined by the observation that the standard model predicts a rather counter-intuitive (to those not familiar with the theory) pattern of portfolio choice over the life cycle. Typically, the young (at ages below 40) would invest 100 percent of their savings in stocks. Thereafter, the share of savings invested in stocks would gradually decrease to about 50 percent around retirement [START_REF] Cocco | Consumption and portfolio choice over the life cycle[END_REF][START_REF] Gomes | Optimal life-cycle asset allocation: Understanding the empirical evidence[END_REF]. In contrast, empirically, equity shares are very low for the young and then increase to a moderate level (see Section 5.2). The fact that the prediction of the standard model differs markedly from the empirical pattern gives rise to a puzzle.

In light of the above, a natural first step towards a model that describes decision making of boundedly rational agents is to eliminate the requirement of complete contingent planning. In order to deviate from the classical model in only one direction, it is valuable to continue relying on the other pillars of the standard model: preferences, constraints and optimization. In this paper, I put forward the framework of so-called feasibility goals that does exactly this.

In the feasibility goals framework, an individual has certain target consumption levels that she wants to remain feasible in certain future benchmark scenarios. For instance, an individual may choose her current consumption such that, in a future worst-case scenario, her minimum feasible future consumption is a certain proportion of her current consumption.

The essential property of feasibility goals is that meeting them does not pre-determine actual future decisions, unlike in the case of the standard model. To understand this, suppose that an agent has the feasibility goal of consuming at least 60 percent of current consumption in a worst-case scenario next period. What she will actually consume if the worst-case scenario materializes in that period is left open. This decision will be made 2 A c c e p t e d M a n u s c r i p t only once this scenario has materialized. She may choose an actual consumption level that turns out to be higher or lower than 60 percent of the level of consumption in the previous period. However, having pursued the feasibility goal in the previous period, she ensures that she ends up with the possibility to consume at least 60 percent of her previous consumption in a worst-case scenario. In other words, the agent is prepared for a worst-case scenario.

The fact that ex-post choices may not be equal to ex ante feasibility goals for a particular scenario does not reflect dynamic inconsistency. The reason is that achieving the feasibility goal does not mean pre-specifying any future choice. Rather, it means preparing for bad future circumstances. The fact that decisions are made prudently, but sequentially "on the spot," reflects the principle of procedural rationality that characterizes actual planning. This paper consists of two parts. In the first part, I introduce the concept of a feasibility goal. This is first done in an intuitive way by means of a few simple examples.

I then provide a formal characterization of the feasibility goals framework. The second part of the paper presents a specific tractable life cycle model with feasibility goals that is calibrated and compared to the data. The aim of this second part is to show that the feasibility goals approach has bite for applied work. The analysis of the life cycle model shows that forward looking behavior when guided by simple feasibility goals is sufficient for very smooth consumption profiles. Furthermore, the life cycle model explains empirical asset allocation patterns better than a large variety of existing models.

The paper is organized as follows. Section 2 outlines the related literature. Section 3 introduces the concept of feasibility goals in an intuitive way. Section 4 characterizes the concept formally. Section 5 presents a life cycle model with feasibility goals. Section 6 concludes.

A c c e p t e d M a n u s c r i p t 2 Related Literature

This paper contributes to the small but growing literature of intertemporal models in behavioral economics. Existing behavioral life cycle models include the mental accounting model of [START_REF] Shefrin | The behavioral life-cycle hypothesis[END_REF], models of hyperbolic discounting [START_REF] Laibson | Golden eggs and hyperbolic discounting[END_REF], and the loss aversion model of [START_REF] Bowman | Loss aversion in a consumption-savings model[END_REF]. The aim of this literature is to study the effect of psychological biases on savings behavior. In this paper, the issue is bounded rationality rather than psychological biases. Gabaix andLaibson (2000, 2005) provide models of bounded rationality where agents use short-cuts when working through a decision tree. In particular, agents remove branches with low probability. An important difference between the feasibility goals framework and the models of Gabaix and Laibson is that they study decisions where a choice is made only once at the beginning of a probabilistic event three. In contrast, the feasibility goals framework applies to dynamic situations with a multitude of follow-up decisions. [START_REF] Bolton | Thinking ahead: The decision problem[END_REF] provide a model of procedural rationality in dynamic decision making to analyze endogenous information processing. In contrast, the feasibility goals framework describes prudent and forward-looking decision making taking limited information processing capabilities as exogenously given. [START_REF] Pemberton | Attainable non-optimality or unattainable optimality: A new approach to stochastic life cycle problems[END_REF] develops a life cycle model that shares the spirit of feasibility goals models. However, neither does he provide a general characterization of the feasibility goals framework nor does he analyze asset allocation.

An Intuitive Introduction to Feasibility Goals

I begin the analysis by developing the concept of feasibility goals in an intuitive way. This prepares the formal characterization of the concept in Section 4. A feasibility goal intrinsically relates to the notion of a (future) scenario. Imagine that a scenario refers simply to one specific path through an event tree. It starts at current time t. Denote the (deterministic) terminal time by T and denote a particular scenario starting at t by s t .

A c c e p t e d M a n u s c r i p t A scenario s t "passes" Tt time periods and nodes of the event tree and ends at time T . Naturally, there is uncertainty about the future if and only if there is more than one scenario from a node at t onwards.

A feasibility goal assigns to each point in time along the scenario s t a level of consumption that is to be affordable. A feasibility goal does not specify any actual future action. Thus, it does not constitute a contingent plan. It only specifies feasibility of a certain level of consumption along s t .

How the feasibility goal concept works is best understood by means of a set of simple examples. The below examples are intentionally kept stylized in order to illustrate the concept. In Section 5 below, I present a more sophisticated model of life cycle saving and asset allocation that illustrates the usefulness of the concept for applied work and of its empirical bite.

Example 1: An Absolute Feasibility Goal for a Worst-Case Scenario

Consider an agent living for three periods. Assume that the agent has an endowment of y 1 in the first period. Furthermore, she can transfer resources to the future by saving.

The gross interest rate is equal to r.

This first example of a feasibility goal is a particularly simple one. It entails affordability of specific absolute consumption levels in a future worst-case scenario. These are c2 for the second and and c3 for the third period, respectively. The worst-case scenario is defined as the path through the event tree where future income is lowest. Consider the extreme case that the worst-case scenario entails no income in the future at all (except from current savings). Assume that y 1 > c2 r + c3 r 2 , i.e. the present value of minimum lifetime income exceeds the present value of the future consumption levels associated with the feasibility goal. Assume finally that the agent has a preference ordering according to which she likes most the choice with the highest level of current consumption, out of all A c c e p t e d M a n u s c r i p t choices that allow for meeting the feasibility goal. 1

In the first period, the agent's decision problem looks as follows. Denote first-period consumption by c 1 . The agent meets her feasibility goal in the first period if and only

if y 1 -c 1 ≥ c2 r + c3 r 2 , i.e. c 1 ≤ y 1 -c2 r -c3 r 2 .
This means that savings in the first period are sufficient for financing the feasibility goal. Whenever c 1 > y 1 -c2 r -c3 r 2 , then it is not feasible to consume at least c2 in the second and c3 in the third period and so the feasibility goal is not achieved. Since the decision maker maximizes current consumption subject to meeting the feasibility goal, she chooses c 1 = y 1 -c2 r -c3 r 2 and saves the rest of her income.

Meeting the feasibility goal does not pre-specify any decisions in the second (or third) period. Instead, the agent follows a procedural rationality approach. She simply applies the same decision mechanism again upon entering period two or three. Since she made a prudent forward-looking decision in the first period, it is assured that her later consumption will indeed not fall below a certain lower bound.

Specifically, the agent enters the second period with a level of resources of X 2 . The latter depends on the particular scenario that has emerged. In the worst-case scenario, it consists only of savings. Otherwise, it may also comprise other sources of income.

Whatever X 2 amounts to, the agent's feasibility goal calculus, as applied in the second period, requires to consume at least c3 in the third period. Since, in the worst-case scenario, the agent earns no income apart from returns on her savings in the third period, she needs to save at least c3 r in order to meet the feasibility goal. Since she maximizes current consumption subject to meeting the feasibility goal, she will choose a second-

period consumption of c 2 = X 2 -c3
r . In the third period the agent just consumes whatever resources are left.

As has been illustrated by this example, intertemporal decision making by means of feasibility goals means applying the same decision algorithm every period anew. Apart from this, there is no functional link between decision making at different points in time, 1 Of course, other preference orderings would be possible.

6

A c c e p t e d M a n u s c r i p t unlike in the case of a contingent action plan in the standard model. However, by their very nature, feasibility goals allow for making prudent and forward-looking decisions in spite of this seemingly missing link. To see this, note that the agent will indeed have the possibility of consuming at least c2 in the second period and c3 in the third.

Decision making by means of feasibility goals does not lead to dynamic inconsistency.

The reason is that achieving feasibility goals does only mean to ensure feasibility of certain future actions, not pre-specifying any actual future consumption levels. Thus, there is, by the very nature of the feasibility goals concept, no room for dynamic inconsistency.

Rather, it reflects a procedural-rationality approach.

Example 1 is particularly simple and stylized, but it provides a clean illustration of how the feasibility goals concept works.

Example 2: A Relative Feasibility Goal for a Worst-Case Scenario

In this example everything is identical to the first one except that the feasibility goal is now relative in nature. This relative feasibility goal entails consuming at least a fraction α of current consumption in every future period if a worst-case scenario emerges. It is natural to assume that 0 < α ≤ 1, although this restriction is not necessary from a technical point of view.

In the first period, the feasibility goal is again met if and only if savings are sufficiently high, i.e.

y 1 -c 1 ≥ αc 1 r + αc 1 r 2 , or c 1 ≤ 1 + α r + α r 2
-1 y 1 . If the decision maker's aim is to maximize current consumption subject to meeting the feasibility goal then she will choose

c 1 = 1 + α r + α r 2
-1 y 1 and save the rest of her income.

In the second period, the agent ends up with a level of resources X 2 . She meets her feasibility goal in the second period if X 2 -c 2 ≥ αc 2 r . If she maximizes current consumption subject to achieving the feasibility goal, she chooses c 2 = r α+r X 2 . In the third period, the agent will consume whatever resources are left.

7

A c c e p t e d M a n u s c r i p t

Example 3: Feasibility Goals for a Worst-Case and a Normal Scenario

This example illustrates that it is straightforward to have multiple feasibility goals. The first feasibility goal coincides with the one of the previous example and may be dubbed safety goal. The second feasibility goal relates to a "normal" or "average" scenario.

Specifically, the agent's goal is the feasibility of a sequence of consumption levels that increases by at least a factor β ≥ 1 from period to period. In other words, this means feasibility of a higher standard of living in the future in a normal scenario. There are again three periods. Non-financial income in the normal scenario is y n in both future periods. In the worst-case scenario non-financial income is zero as in the previous two examples.

In the first period, meeting the normal-scenario goal means assuring feasibility of a consumption profile of the form (c 1 , βc 1 , β 2 c 1 ). This requires that c 1

+ βc 1 r + β 2 c 1 r 2 ≤ y 1 + y n r + y n r 2 ≡ Y n or c 1 ≤ 1 + β r + β r 2 -1
Y n . In contrast, the feasibility goal associated with the worst-case scenario requires that

c 1 ≤ 1 + α r + α r 2 -1 y 1 (see Example 2 above).
If the agent chooses the highest level of first-period consumption that is compatible with both goals, she chooses

c 1 = min 1 + β r + β r 2 -1 Y n , 1 + α r + α r 2 -1 y 1 .
In the second period, achieving the normal-scenario goal requires that c 2 + βc 2 r ≤ X 2 + y n r . Achieving the feasibility goal for the worst-case scenario requires c 2 + αc 2 r ≤ X 2 . The highest level of second-period consumption that allows the agent to meet both goals is given by c 2 = min r β+r X 2 + y n r , r α+r X 2 Let us start with a formal description of an event tree. Time t runs from one to T . At t = 1, an event e 1 ∈ E 1 occurs, where E 1 is the set of all possible events at t = 1. Given the realization of a particular e 1 ∈ E The "hats" on the actions ât+τ constituting the feasibility goal indicate that these may be different from the actions a t+τ that are actually chosen at times t + τ . It is important to note that, according to Definition 1, "meeting a feasibility goal" is a qualification of a current action a t .

Let us illustrate this definition by means of Example 2 of the previous subsection.

Consider the decision problem in period one. The action a t corresponds to the choice of consumption c 1 . The feasibility goal for the worst-case scenario is (â t+1 , ât+2 ) = (αc 1 , αc 1 ).

For concreteness, suppose that y 1 = 100, r = 1, α = .5. Let us ask whether c 1 = 50 fulfills the feasibility goal. In case that the worst-case scenario emerges, this results in a level of resources of (100 -50) × 1 = 50 for the second period. (Recall that there is no nonfinancial income in the worst-case scenario.) The target consumption level for the second period for this scenario is .5 × 50 = 25. A consumption level of 25 is thus feasible. It leaves another 25 of savings, leading to a level of resources of 25 in the third period in case of the worst-case scenario. This is just the target level specified by the feasibility goal. Thus, the choice c 1 = 50 does fulfill the feasibility goal in the first period. If we had c 1 = 60, then the level of resources in the second period would be 40 and the target level of consumption for the second period would be 30. Thus, the target could still be met in the second period. However, there would remain savings of only 10 in the second period.

A c c e p t e d M a n u s c r i p t

This would lead to a level of resources of only 10 in the third period, not allowing for a target consumption of 30. Thus, the choice c 1 = 60 would not fulfill the feasibility goal.

The case of Example 2 highlights the fact that, for the feasibility goal ft (s t ) = (â t+1 , ât+2 , . . . , âT ), the target actions ât+τ , 1 ≤ τ ≤ Tt may be a function of the current actions a t . In fact, whenever feasibility goals are relative goals with respect to current consumption, then the target actions ât+τ are functions of current action a t . On the other hand, feasibility goals need not be functions of current actions. This is illustrated by Example 1.

Preferences over Feasibility Goals

Feasibility goals by themselves do not yet imply a specific choice. Rather, they partition feasible choices in period t in a subset of choices that meet the feasibility goals and a subset that does not. A preference relation is required to rank the choices that do fulfill the feasibility goals.

In the feasibility goals framework, the domain of preferences is what I dub the feasibilitygoals induced choice (FIC) set. To state the formal definition of the FIC set, assume that there are K feasibility goals as defined by Definition 1 related to K scenarios. We have then:

Definition 2 (FIC set) Define F t ⊂ B t to be the set of all current choices a t that fulfill the K feasibility goals. Using a shortcut notation, write the budget sets as

B t+τ s k t , a t , 1 ≤ τ ≤ T -t, 1 ≤ k ≤ K. The feasibility-goals induced (FIC) choice C t set is defined as the set F t × B t+1 s 1 t , a t × B t+2 s 1 t , a t × . . . × B T s 1 t , a t × B t+1 s 2 t , a t × B t+2 s 2 t , a t × . . . × B T s 2 t , a t × . . . × B t+1 s K t , a t × B t+2 s K t , a t × . . . × B T s K t , a t (1)
for which a t ∈ F t .

Thus, the FIC set is the Cartesian product of the set of current actions that fulfill the A simple instance of this is given by Examples 2 and 3 above. The elements of the FIC set C t have the typical form

a t , f 1 a t ; s 1 t , f 2 a t ; s 2 t , . . . , f K a t ; s K t .
(2)

The next definition characterizes preferences in the feasibility goals framework.

Definition 3 In the feasibility goals framework, preferences F G t are a complete and transitive relation over the FIC set C t .

In contrast to the standard microeconomic framework, preferences are only relevant for ranking choices within the narrow FIC set, not the overall choice set. In other words, the agents has a preference order only over those choices that fulfill the feasibility goals.

This reflects bounded rationality.

It is noteworthy that preferences may not simply be defined over F t but over the larger set C t .3 This simply means that preferences F G t may not simply be preferences over current actions that fulfill the feasibility goals. Rather, preferences may also directly be responsive to the induced level of future target actions ât+τ . The life-cycle model in Section 5 will provide an instance of this.

As in the case of standard microeconomic theory, it is required that the preference relation F G t is complete and transitive over the FIC set C t . However, this rationality requirement for preferences over C t is weaker than in the case of the standard model To illustrate the domain C t of preferences, consider again Example 3 from the previous subsection. It has been assumed that the agent prefers the choice with the highest level of current consumption among all choices fulfilling the feasibility goals. There, we have, for the first period,

F 1 =    c 1 : c 1 ≤ min   1 + β r + β r 2 -1 Y n , 1 + α r + α r 2 -1 y 1      .
Furthermore, the elements of the set C 1 have the form

c 1 , (αc 1 , αc 1 ) , βc 1 , β 2 c 1 where c 1 ∈ F 1 . Denote the elements of C 1 by c 1 . Consider c ′ 1 , c ′′ 1 ∈ C 1 with first elements c ′ 1 and c ′′ 1
, respectively. In the case of Example 3, we have

c ′ 1 F G 1 c ′′ 1 iff c ′ 1 ≥ c ′′ 1 .
(3)

In this example, F G 1 is, in fact, a preference relation only over F 1 .

It is noteworthy that the fact that the feasibility goals framework continues to rely on the preference concept, although in a very restricted sense, distinguishes it from a rule-of-thumb approach. The latter would lead to a complete departure from classical decision theory, whereas the feasibility goals framework deviates insofar as there is no contingent action plan. The contingent action plan is substituted by a procedural rationality approach. This notwithstanding, the feasibility goals framework still relies on the classical pillars: preferences, constraints and optimization.

13

A c c e p t e d M a n u s c r i p t normal scenario (cf. Example 3 above). The goal referring to the worst-case is dubbed safety goal. For any period t, the safety goal is given by

(α 1 c t , α 2 c t , . . . , α T -t c t ) . (6)
Thus, it entails feasibility of (potentially differing) target fractions α τ of current consumption in the future in a worst-case scenario. This is a simple extension of the feasibility goal in Examples 2 and 3 above.

The second goal relates to a normal scenario and is reminiscent of the normal-scenario goal in Example 3. It is modeled in a more flexible way, however. This goal is dubbed the future standard of living (FSL) goal. It entails a standard of living of c + from a future period T * on for the case that the normal scenario materializes. Specifically, the FSL goal c + is defined as a level of consumption that is feasible from T * onward if (i) consumption is equal to the current level c t between t and T * , (ii) the normal scenario materializes, and (iii) savings are invested in a portfolio of bonds and stocks between t and T * in such a way that the safety goal is always met given a level of consumption of c t .

Formally, the FSL goal is stated as c t , c t , . . . , c + , c + , . . . ,

where c t appears T * -t -1 times and c + appears T -T * + 1 times. The FSL goal c + is determined as the constant consumption stream that can be financed by X n T * , where the superscript indicates the normal scenario. X n T * is determined by the accumulation process

X n t+τ +1 = b t+τ r + s t+τ ρ n + y n t+τ +1 , (8)
where ρ n denotes stock returns in the normal scenario and 0

≤ τ ≤ T * -t-1. Furthermore, b t+τ + s t+τ = X n t+τ -c t (9)
A c c e p t e d M a n u s c r i p t and the choice (c t , b t+τ , s t+τ ) is such that it fulfills the safety goal for all τ , 0 ≤ τ ≤ T * -t -1, along the projected normal scenario.

The relevant preference order F G t is defined as follows.

Definition 4 (Life Cycle Preferences)

Among two choices z = (c t , b t , s t ), z ′ = (c ′ t , b ′ t , s ′ t )
that fulfill all relevant safety goals, z is preferred to z ′ if z leads to a higher value of the quasi-utility function u F G = c t (c + ) γ , γ > 0.

In contrast to the normal-scenario goal in Example 3, there is thus an explicit trade-off between current consumption and the FSL goal c + .

There are three ways in which bounded rationality shapes preferences F G t . First, the agent only considers two future scenarios. Second, the agent immediately eliminates all choices that do not allow for meeting the safety goal. Third and most crucially, all arguments of the preference relation that refer to the future are feasibility goals (i.e. the safety goals α τ c t and c + ). There is no direct functional relationship between these goals and actual future decisions. This reflects that there is no contingent planning. However, meeting the feasibility goals means that a certain minimum level of consumption in the worst-case scenario and a certain (higher) standard of living in the normal scenario are indeed feasible in the future. Thus, the agent is prepared for the future thanks to her feasibility goals. Non-financial income is calibrated using estimations of the labor income profile for high-school graduates reported in [START_REF] Cocco | Consumption and portfolio choice over the life cycle[END_REF]. Labor income is subject to permanent and transitory shocks. These are assumed to be log-normally distributed. There is also a medical expenditure shock M that reduces non-financial income y t during retirement to M y t . It is realized upon retirement and is fully permanent. The expected value of M is one and its lower bound M is set to .5. Details about the calibration of non-financial income can be found in the Appendix. Real bond returns are assumed to be risk-free and to equal 2 percent per year. Stock returns are assumed to be iid over time and lognormally distributed with an expected annual return of 6 percent and a standard deviation of .157.

Calibrating the Model

The normal scenario is defined as the one corresponding to a state of the world where labor income, stock returns and the medical shock take on their expected values at all future dates, given current information. The worst-case scenario is defined as the one corresponding to future states of the world where labor income shocks and stock returns take on the value of their first percentile. Furthermore, the medical shock M takes on the value M upon retirement. The calibrations are run for the case where, ex post, the normal scenario materializes. Equity shares are defined as the ratio of total stock holdings to total financial assets, i.e.

A c c e p t e d M a n u s c r i p t s t / (b t + s t ).

Figure 1 shows that consumption follows income very closely until the age of about 35. Furthermore, the level of savings is very low during these years and there is only a small "buffer stock." Thereafter, savings start to increase and consumption levels off.

The dominant savings motive is preparation for retirement. 7

The first important observation is that the consumption profile is virtually as smooth as it can be in the presence of binding liquidity constraints. In fact, the predicted profile resembles very closely the shape of consumption profiles for expected utility models with buffer stock saving [START_REF] Carroll | Buffer-stock saving and the life cycle/permanent income hypothesis[END_REF][START_REF] Cocco | Consumption and portfolio choice over the life cycle[END_REF]. Thus, the analysis reveals that no complex contingent action plan is necessary in order to achieve consumption smoothing.

A key insight is thus that the simple procedural-rationality approach of the feasibility goals model is sufficient for consumption smoothing.

The second key observation is that the feasibility goals model matches the shape of empirical equity shares better than existing models. Table 1 shows how empirical equity shares vary over age, based on the 2004-wave of the Survey of Consumer Finances (SCF).

These equity shares contain direct as well as indirect equity holdings through mutual funds and retirement accounts. Empirical equity shares are very low under the age of 30.

Thereafter they increase until the age of 55 and decrease slightly thereafter.8

Existing models have major difficulties of explaining this empirical pattern. Equity shares are typically predicted to be close to 100 percent until the age of about 40. Thereafter, they decrease until retirement, where they reach a level of about 50 percent, on average [START_REF] Cocco | Consumption and portfolio choice over the life cycle[END_REF]. The prediction that equity shares are far higher than empirical estimates over a large part of the life cycle is very robust across existing models.

Very high equity shares arise even when taking into account the possibility of very low re-7 After the age of 70, consumption increase further whereas equity shares continue to decrease. The model is less suited, however, to predict savings behavior during later phases of retirement since important determinants of choices during that phase are missing. These include savings for nursery homes and bequest motives.

A c c e p t e d M a n u s c r i p t alizations of labor income or endogenous borrowing constraints [START_REF] Cocco | Consumption and portfolio choice over the life cycle[END_REF], fixed costs of stock market participation and Epstein-Zin preferences [START_REF] Gomes | Optimal life-cycle asset allocation: Understanding the empirical evidence[END_REF] or habit formation (Gomes andMichaelides, 2003, Polkovnichenko, 2007).

The feasibility goals model captures well the general shape of the empirical equity shares profile. In particular, equity shares are predicted to be very low until the thirties.

Thereafter, they increase to a level that is somewhat higher than observed in the data. At the age of 55, equity shares start to decrease slightly. In sum, the feasibility goals model explains the data better than a large variety of existing models.

The intuition behind the predicted equity share profile is as follows. The young face a substantial amount of career uncertainty. The downside risk associated with future income declines substantially over time. This happens because the effects of a sequence of low realizations of the permanent income shock accumulate over time (see equation ( 11) in the Appendix). Hence, the uncertainty about future income and its downside risk is the lower, the less periods there are between a current period t and retirement. Put simply, more things can still go wrong for a 25-year old than for a 50-year old.

After the age of 45, the medical expenditure shock M comes into play. To understand this, recall that agents do not want to let their future feasible standards of living fall below (α * ) i c t , where i represents the distance between the current and a particular future period. In early ages, retirement and hence the date of the realization of the medical expenditure shock lie far ahead in the future, such that α R-t is very low, where R denotes retirement age. When agents get closer to retirement, α R-t increases. This prevents equity shares from further increasing and eventually leads to a decrease in stock holdings and an increase in bond holdings. This leads to a decrease of the equity share.

Conclusion

This paper puts forward a new approach of modeling intertemporal choice: decision making by means of feasibility goals. This approach represents a natural alternative to the standard modeling approach for two reasons. Second, the framework stays close to the standard approach. It is still based on an optimization framework relying on the concepts of preferences and constraints, unlike, for instance, a rule-of-thumb approach. This paper has characterized the conceptual properties of feasibility goals models.

Furthermore, it provides a comparatively simple life cycle model that is illustrates the applicability of the concept. The key insights from the life cycle analysis are that: (i) the resulting consumption profiles are virtually as smooth as for the standard model; (ii) empirical patterns of asset allocation are far better explained than by the standard model.

y t = F t P t V t , (10)
for 1 ≤ t < R, where R denotes the age of retirement, and

P t = P t-1 U t . (11)
F t represents a non-stochastic component of non-financial income that determines the hump-shaped profile of "labor" (or non-financial) income over the life cycle in the absence of shocks. V t and U t are mutually independent iid random variables that are lognormally distributed with a mean of one. U t represents a shock component that has a permanent effect on labor income. V t represents a transitory shock component. [START_REF] Cocco | Consumption and portfolio choice over the life cycle[END_REF] report estimations for ( 10), ( 11), as well as for the standard deviations of the logs of V t and U t .9 The calibrations here are based on their estimations for high school graduates.

During retirement, it is assumed that non-financial income is given by

y t = δy R-1 M t , (12)
The parameter δ represents a "replacement rate" and is set to .6. M t ∈ M , M represents a medical expenditure shock. The idea is that the medical expenditure shock reduces the amount of income that is available for normal spending. For simplicity, it is assumed that this shock is realized at age R and is fully permanent. This means that M R+i = M R for

1 ≤ i ≤ T -R. Furthermore, it is assumed that E t M R = 1 for 1 ≤ t ≤ R -1.
A c c e p t e d M a n u s c r i p t

  and the set of induced target actions. Recall that, in general, current actions may affect the target actions ât + τ ∈ B t+τ , constituting a feasibility goal.

  with full contingent planning. In the standard model, the domain of preferences is the Cartesian product of all current and contingent future actions in all possible states of the world. In contrast, in the feasibility goals framework, the domain of the preference relation F G t is only the restricted set C t . It is restricted by the fact that an agent only A c c e p t e d M a n u s c r i p t considers choices that meet the feasibility goals in the first place. Second, concerning the future, preferences are only about feasibility goals, not actually planned contingent actions. Both restrictions reflect procedural rationality.

I

  now proceed by calibrating the model. This allows for getting an insight about the smoothness of the resulting consumption profile and whether the model has any power to explain the empirical asset allocation profile. I consider a setup where an agent starts working at age 21, retires at the beginning of age 66, and dies with certainty after age 85. 5 For the safety goal, I use the one-parameter 5 Considering uncertain lifespans is beyond scope of this paper.Page 18 of 26 A c c e p t e d M a n u s c r i p t specification α τ = (α * ) τ where α * is set to .97. 6 This specification has the advantage of being very parsimonious. Furthermore, it captures the idea that decision makers may be less concerned, or may think less carefully, about negative events in the far-distant future than in the near future. I set γ to .85. Finally, I set T * = (t + T ) /2, i.e. it refers to the middle of the remaining life time.

Figures 1

 1 Figures 1 and 2 present the simulation results. The solid line in Figure 1 refers to consumption. The dashed line represents non-financial income. The dotted line represents total income including financial income, which is endogenous since it depends on the agent's savings and asset allocation choices. Figure 2 shows calibrated equity shares.

  First, it takes into account bounded or Page 21 of 26 A c c e p t e d M a n u s c r i p t procedural rationality by eliminating the requirement of a full contingent action plan.

  20

  of Income ProcessFollowing[START_REF] Cocco | Consumption and portfolio choice over the life cycle[END_REF], I assume that

Figure

  Figure 1: Consumption and income

Figure

  Figure 2: Equity shares

  1 , the set of possible follow-up events for t = 2 is generally a function of the realization of e 1 . It is denoted E 2 (e 1 ) and has a typical element e 2 . Iterating further, a particular node of the event tree at time t is characterized by the event e t ∈ E t (e t-1 ), where e t-1 ∈ E t-1 (e t-2 ), e t-2 ∈ E t-2 (e t-3 ), . . ., e 2 ∈ E 2 (e 1 ), e 1 ∈ E 1 .For a more compact notation, it is convenient to define a history h t := (e 1 , e 2 , . . . , e t ), e 1 ∈ E 1 , . . ., e 2 ∈ E 2 (e 1 ), . . ., e t ∈ E t (e t-1 ) and to denote the set of possible events at t by E The budget set in period t + 1 generally depends on the action chosen at t, as well as on the event that has materialized in t + 1. Generically, one would describe this budget set as B t+1 (h t , e t+1 , a t ), where e t+1 ∈ E t+1 (h t ), a t ∈ B t (h t ). Similarly, the budget set in t + 2 would be described as B t+2 (h t , e t+2 , a t+1 ), where e t+2 ∈ E t+2 (e t+1 ), t+1 ∈ E t+1 (h t ), a t+1 ∈ B t+1 (h t , e t+1 , a t ), a t ∈ B t (h t ). In order to simplify notation, I will use a shortcut and describe the budget set τ periods ahead as B t+τ (s t , a t+τ -1 ), where a t+τ -1 ∈ B t+τ -1 (s t , a t+τ -2 ), a t+τ -2 ∈ B t+τ -2 (s t , a t+τ -3 ) etc. Similarly, A t+τ (s t , a t+τ -1 ) is a shortcut for the set of potentially feasible actions at time t + τ , given history h t and scenario s t .A feasibility goal is then defined as follows.Definition 1 (Feasibility Goal) A feasibility goal ft (s t ) for scenario s t is a mapping S t (h t ) → A t+1 (s t ) × A t+2 (s t ) ×, . . . , ×A T (s t ) assigning to s t a sequence of future target actions (â t+1 , ât+2 , . . . , âT ). A feasibility goal is met at time t if a t is such that ât+1 ∈ B t+1 (s t , a t ), ât+2 ∈ B t+2 (s t , ât+1 ), . . ., âT ∈ B T (s t , âT-1 ).

	M a n u s c r i p t
	A c c e p t e d

t (h t-1 ).

Given a history h t , a scenario at time t is defined as a sequence of follow-up events of the history h t . Generically, a scenario is thus described as a sequence (e t+1 , e t+2 , . . . , e T ) with e t+1 ∈ E t+1 (h t ), e t+2 ∈ E t+2 (e t+1 ), . . ., e T ∈ E T (e T -1 ). The set of all possible scenarios following history h t is denoted by S t (h t ). The typical element of this set is denoted by s t .

Before defining feasibility goals, we need to formally describe sets of feasible actions, i.e. dynamic "budget constraints." 2 Denote A t (h t ) ⊂ R n the set of potentially feasible actions at time t, given history h t . The typical element of A t (h t ) is a t and may be a multi-dimensional vector. Denote by B (h t ) ⊂ A t (h t ) the set of all feasible actions, i.e. a "budget" set. e

These may be proper budget constraints or represent any other restrictions on an agent's actions that may be relevant.

Of course, the framework allows for the possibility that the domain of preferences is, in fact, only F t ; this simply provides a special case. See below.

The model described below is discussed at greater length in a companion working paper[START_REF] Binswanger | Towards understanding life cycle saving of boundedly rational agents: A model with feasibility goals[END_REF] and for some specific details the reader is referred to the latter. The below discussion benefits from the concepts developed in Section 4 and the discussion here is more streamlined than in[START_REF] Binswanger | Towards understanding life cycle saving of boundedly rational agents: A model with feasibility goals[END_REF].

Calibrations for parameter values others than the ones discussed here can be found in[START_REF] Binswanger | Towards understanding life cycle saving of boundedly rational agents: A model with feasibility goals[END_REF]. The results discussed below do not hinge upon the specific parameter values chosen.

It is noteworthy that Table1is based on observations from only one particular year. Therefore, the shape of the equity share profile cannot be biased by what is known as time effects.

See their Tables 1, 2, 3.

A c c e p t e d M a n u s c r i p t 5 A Life Cycle Model with Feasibility Goals

The Model

In this section I present a model of life cycle saving and asset allocation. The main purpose of this section is to illustrate the applicability of the feasibility goals framework and its empirical bite. More specifically, the life cycle model demonstrates that, in spite of only procedural rationality, the resulting consumption path is very smooth. Second, the model shows that the feasibility approach is promising for better understanding individual portfolio choice. As will be discussed below, existing models have major difficulties explaining how portfolio structures vary over the life cycle. 4

Consider an agent who, at each time t, makes a decision about how much to consume, how much to invest in a safe asset, called bond, and and in a risky asset, called stock. Time t runs from 1 to T . Consumption is denoted by c t , bond holdings by b t and stock holdings by s t . In each period, budget constraints are given by the two equations

X t denotes total available resources in t (where X 1 is given). The variable y t denotes labor income, r denotes the risk-free interest rate and ρ t stock returns in period t. Both y t and ρ t are random variables. The distribution of y t may depend on previous realizations as well as on t itself. Thus, in expectations, y t may follow a typical hump-shaped profile over the life cycle (see Appendix). For simplicity, I assume that ρ t is iid.

Following much of the literature (see e.g. [START_REF] Cocco | Consumption and portfolio choice over the life cycle[END_REF], I assume that bond and stock holdings must be nonnegative, i.e. b t ≥ 0, s t ≥ 0.

There are two feasibility goals relating to two different scenarios: a worst-case and a

A c c e p t e d M a n u s c r i p t