An Experimental Test of Risk-Sharing Arrangements. Gary Charness. Garance Genicot

Transcription

1 An Experimental Test of Risk-Sharing Arrangements Gary Charness University of California, Santa Barbara Garance Genicot Georgetown University February 26, 2004 ABSTRACT We investigate risk-sharing without commitment by designing an experiment to match a simple model of voluntary insurance between two agents when aggregate income is constant. Participants are matched in pairs. Each period, they receive their income with or without a random component h that one person or the other receives; after observing own and counterpart income, each person in a pair can decide to make a transfer to the other person. It is common information that there is a given probability that all pairs will be dissolved at the end of each period, with participants re-matched. At the end of the experiment, one period is randomly drawn to count for cash payment. Participants all face the same variance in their income, but do not necessarily have the same mean income. This setting allows us to experimentally test different implications of risk-sharing without commitment. In particular, we find strong evidence of risk-sharing and reciprocal behavior, where transfers are higher with a higher continuation probability and with a higher degree of risk aversion. However, transfers are lower with inequality, in contrast with existing models of both risk-sharing and social preferences. JEL Classification Numbers: A49, C91, C92, D31, D81, O17. Key Words: experiments, gift exchange, informal insurance, risk-sharing, social preferences. We are grateful to the California Social Science Experimental Laboratory at UCLA, particularly to Patricia Wong for arranging the logistics and to Raj Advani for expert programming. We also thank seminar participants at the NEUDC at Yale University, the World Bank DECRG, Princeton University and Boston University for their helpful suggestions; all errors are of course our own. We thank the UCSB Academic Senate for partial funding for our experiments. Please address all correspondence to charness@econ.ucsb.edu and gg58@georgetown.edu.

2 1. INTRODUCTION Risk is a pervasive fact of life for most people, especially in developing countries. Individuals have often been shown to respond to the large fluctuations in their income by engaging in informal risk-sharing by providing each other with help in the form of loans, gifts and transfers in time of need. There is considerable empirical evidence that risk-sharing provides some limited insurance in village communities (Deaton, 1992, Townsend, 1994, Udry, 1994, Jalan and Ravallion, 1999, Ligon, Thomas and Worrall, 2002, Grimard, 1997, Gertler and Gruber, 1997, Foster and Rosenzweig, 2002). The most important limitation appears to arise from the lack of enforceability of these risk-sharing agreements. The fact that these agreements must be designed to elicit voluntary participation often seriously limits the extent of insurance they can provide. A growing theoretical literature provides a characterization of the optimal self-enforcing risk-sharing agreement and some of its consequences (Kimball, 1988, Coate and Ravallion, 1993, Kocherlakota, 1996, Kletzer and Wright, 2000, Ligon, Thomas and Worrall, 2002, Genicot and Ray, 2003, and Genicot, 2003). In this project, we experimentally test different implications of these models of risksharing without commitment. We chose a very simple model with two agents in which, each period, one of the two agents, selected at random, receives an amount of money h (in addition to his or her fixed income). The model is described in detail in the next section. We then designed an experiment that closely matches the model. Participants are matched in pairs. Each period, they receive their income with or without the random component h, and, after observing own and counterpart incomes, each person in a pair can decide to make a transfer to the other person. It is common information that there is a given probability that all pairs will be dissolved at the 1

3 end of each period, with participants re-matched. At the end of the experiment, one period is randomly drawn to count for cash payment. Participants all face the same variance in their income, but do not necessarily have the same mean income. Of course, positive experimental transfers do not by themselves prove that risk-sharing is taking place, since a transfer could instead reflect some form of altruism or social preference. If transfers do in fact represent reciprocal risk-sharing behavior, we would expect to see certain patterns in the data. For example, we would expect a longer time horizon to lead to a higher level of risk sharing, while recent models of (distributional) social preferences (e.g., Fehr and Schmidt, 1999, Bolton and Ockenfels, 2000, and Charness and Rabin, 2002) would predict no difference in behavior as we vary the expected length of the interaction. We would also expect to see a correlation between one s degree of risk aversion and one s attraction to a risk-sharing relationship; social-preference models would not predict this correlation, unless altruism is correlated with risk aversion. 1 Finally, do past transfers by the other individual in a match affect the transfers one makes? Again, this would be expected with reciprocal exchange, but not if transfers are driven by social preferences that reflect only distributive concerns. We find evidence in support of risk sharing in the laboratory. Average transfers double when there is a 90% likelihood that a match will continue into the next round compared to when there is an 80% likelihood of continuation. We measure risk aversion using a simple investment choice and find a significant positive relationship between one s degree of risk aversion and one s transfer choices. Moreover, reciprocal behavior is shown to be an important factor: the higher the first transfer made by an individual s partner within a match, the higher the individual s transfer made upon receiving a good shock. 1 We are unaware of any evidence on this point. 2

4 Since we examine matches where the individuals have the same fixed income as well as matches featuring unequal fixed payoffs, we test for the effect of ex ante inequality on risk sharing. Genicot (2003) studies risk sharing between two agents facing the same income fluctuations and same preferences but differing in their fixed income, and shows that inequality helps risk sharing in a large range of cases. Since we keep the expected difference in post-shock payoffs the same in both types of match, social-preference models with linear specifications 2 (e.g., Fehr and Schmidt 1999; Charness and Rabin 2002) would predict no difference in transfers across conditions, if we consider the one-round payoffs. 3 We instead find that inequality actually decreases risk sharing. We discuss possible explanations based on both ease of coordination and one s identity (as in Akerlof and Kranton 2000) as an ex ante poor or rich person, consistent with results in social psychology such as Tajfel, Billig, Bundy, and Flament (1970), who find that subjects strongly favor members of their experimental ingroup, even in a situation devoid of the usual trappings of ingroup membership. We also find that males do significantly more transfers than females, especially when the female tendency towards greater risk aversion is taken into account. Previous experimental work on risk-sharing is rather limited. Bone et al. (2000) report an experiment designed to test whether pairs of individuals are able to exploit efficiency gains in the sharing of a risky financial prospect (taking advantage of their difference in risk aversion, with commitment). Their results indicate that fairness is not a significant consideration, but rather that having to choose between prospects diverts partners from allocating the chosen prospect efficiently. The pattern of agreements suggests that, where allocation is the sole issue, partners 2 The Bolton and Ockenfels (2000) and Cox and Friedman (2002) models can accommodate non-linearity in payoff ratios or differences; their predictions in this case depend on parameter values. 3 Moreover, if instead we would consider the basis for comparison in social preference models to be each person s expected payoff in the match, then these models would also predict more risk sharing with inequality. 3

5 largely favor ex ante efficiency over ex post equality. From the transcripts there is little indication that ex post fairness is a significant consideration. The most closely-related experiment to ours is perhaps the Selten and Ockenfels (1998) Solidarity Game since our experiment is effectively a solidarity game if the match continuation probability is zero. In their game, each of three players has an ex ante independent 2/3 chance of winning 10 DM and a 1/3 chance of receiving nothing. Before learning the outcome, each player decides on an amount that she commits to give to one loser or to each of two losers, if she actually won the 10 DM and there are one or two losers. The great majority of subjects were willing to make some conditional gifts. 4 They distinguish five behavioral types, with the most common type (36%) giving the same total amount to one loser and to two losers. This behavior is not readily explained by altruistic utility maximization. 5 In a field experiment in Zimbabwe, Barr (2003) conducted one-shot risk-sharing games among villagers who had been observed to share risk with each other. Prior to choosing a lottery in which they wish to participate, individuals are explicitly provided with the some risk-sharing option either with commitment to an equal split or, in other village, with the possibility of keeping one s return without this being directly disclosed to others (though they can infer some information from the payoff they receive, especially in small groups). She finds more participation in risk-sharing groups, larger groups and more risk taking in the first case. Looking at the possibility of publicly revealing withdrawals from risk sharing, she concludes that both intrinsic and extrinsic motivations are important. However, as the experiment was conducted 4 However the design of their experiment and the fact that transfers are elicited only from winners may be driving some of their results. 5 Subjects were also asked to estimate the average gifts of others. There is a significant positive correlation between the estimates and the subjects' own gifts. This is similar to the false-consensus effect known in the literature (e.g., Ross et al., 1977). Among male subjects, those studying economics show a more egoistical behavior than others. Among female subjects, no such education effect can be found. Females tend to give more than males. 4

6 without anonymity, among people who know each other, and without commitment, the interpretation of these results is unclear. Finally, the Goeree, Holt and Palfrey (2003) study might also be relevant. They examine experimental results for a variety of generalized matching-penny games, and find that a simple two-parameter model combining quantal-response equilibrium and risk aversion explains the observed choice patterns. They also conclude that alternative explanations based on inequality aversion are not supported by the data. Their risk aversion estimates, around 0.5, are significant and of approximately the same magnitude as the estimates in our experiment as well as the estimates recovered from several different auction experiments and auction field data. The remainder of our paper is organized as follows. The next Section lays out the basic model of risk sharing without commitment and describes some important implications. Section 3 then presents the experimental design. In Section 4, the main results of the experiment are presented. Section 5 discusses some implications of the paper, and Section 6 concludes. 2. A MODEL OF RISK-SHARING WITHOUT COMMITMENT A standard model of risk sharing without commitment goes as follows: Time is discrete and the number of period is infinite. In each period t, two agents, indexed by i Œ{1,2}, receive an income y i and one of them (chosen at random) also receives a fixed monetary gain h. They each have a probability 1/2 of receiving h but the aggregate income is constant at Y = y 1 + y 2 + h in each period. The following table summarizes the income distribution of the two agents: State 1 (p = 1/2) State 2 (p = 1/2) Agent 1 y 1 + h y 1 Agent 2 y 2 y 2 + h 5

7 In line with standard practice, let us assume that all agents have additively time-separable Von Neumann-Morgenstern utility functions defined over consumption, such that the expected lifetime utility at time t is given by  t j= 0 j E d u ( c ) "i Œ{1, 2}. i i t+ j where u i > 0, u i < 0, lim cæ0 u i (c) = -, and d Œ (0,1) is the discount rate. The operator E t is the expectation conditional on what is known at time t. Since individuals are risk-averse, optimality would require that the ratio of their marginal utilities remains constant over time and across states of nature. When the aggregate income is constant, this implies keeping each individual s consumption at a constant level. The exact levels depend on the welfare weights used, but must sum to the aggregate income and satisfy the voluntary participation constraint. If then it must be that * c is the optimal level of consumption for individual 1, Ï * 1 1 Ô u1( c ) u1( y1 + h) + u1( y1) 2 2 Ì Ô * 1 1 u2 ( Y - c ) u2 ( y2 + h) + u2 ( y Ó 2 2 As motivated and discussed in the Introduction, we focus on the theme that insurance arrangements must be self-enforcing, and that this requirement constrains the form of such arrangements. The enforcement constraint refers to the possibility that at some date, an individual who is called upon to make transfers to others in the community refuses to make those transfers. To be self-enforcing, a risk-sharing agreement must be such that the expected net benefits from participating in the agreement is at any point in time larger than the one time gain from defection. The literature on risk-sharing concentrates on the constrained optimal or second-best self-enforcing schemes. It follows that the constraint is modeled by supposing that 2. ) 6

8 a deviating individual is excluded from the insurance pool, and must then bear stochastic fluctuations alone (or given an equivalent continuation utility). That is, a risk-sharing agreement i resulting in a stream of consumption { } for individual i must be such that, at any period t, where u ( c i i t ) + E j t  j= 1 c t " t i i t+ j ) ui ( zt ) + j t Âd j= 1 d u ( c E u ( z ) "i Œ{1, 2} (1) i i zt is the total income of individual i at time t (the sum of yi and h, if received this period). If the power of such punishment is limited, then perfect insurance may not be possible. However, even when full risk sharing is not possible, individuals may be able to design a risksharing agreement by limiting transfers in states for which the enforcement constraint is binding (see Coate and Ravallion, 1993 and Kocherlakota, 1996, among others). It is well known that with this simple distribution, when a first-best solution is not incentive compatible, the constrained optimal agreement is characterized by two values, t* 1, the transfer made by 1 to 2 when 1 received h and, t* 2, the transfer made by 2 to 1 when 2 received h. 6 These transfers are such that the incentive constraints (1) hold with equality for both agents, that is t* = (t* 1, t* 2,) is defined by: i i t+ j d (1 - ) u1( y1 + h - t 2 d (1 - ) u2 ( y2 + h - t 2 * 1 * d * d d ) + u1( y1 + t 2 ) = (1 - ) u1( y1 + h) + u1( y1) d * d d 2 ) + u2 ( y2 + t 1) = (1 - ) u2 ( y2 + h) + u2 ( y2 ) (2) A first implication of this model is that, when full insurance is not achieved, a higher discount rate d increases the weight put on the long-term gain from insurance relative to the short-term gain from deviating. There are threshold values d and d, with 0 < d < d < 1, such 6 The constrained optimal scheme may specify a slightly smaller transfer than t*i to an agent as long as only she has received h in order to give to this agent a larger share of the surplus. However, as soon as the state in which the other agent receives h occurs then the constrained optimal agreement is stationary and consists of t*. 7

9 that for values of d smaller or equal to d no risk sharing is possible, for d Π( d, d ) there is some constrained insurance, and for values of d greater or equal to d first-best risk sharing can be achieved. When constrained, a higher d raises the transfers that individuals are able to make to each other and so the insurance that they can provide for each other. A second implication of this model is that an overall increase in the risk aversion exhibited by the agents increases t* and the level of risk sharing that individuals can achieve, by increasing the long term gain from insurance. For instance, if individuals have utility 1 u i (c) = c 1- r i 1-r i, where r i is the Arrow-Pratt coefficient of relative risk aversion, then it is easy to show that an increase r i relaxes i s incentive constraint (2), thereby improving insurance. Finally, what is the effect of inequality? Let s consider different values of y 1 and y 2, keeping the aggregate income Y constant. Clearly, if y 1 = y 2 both individuals are ex ante identical. Now, increasing y 1 and decreasing y 2 to keep Y unchanged would make 1 relatively wealthier than 2, while keeping the variance of their income constant. To be sure, the set of Pareto optimal allocations is unaffected since the aggregate income is the same. However, the division of wealth affects the autarchic utility and thereby does affect the set of self-enforcing allocations. Genicot (2003) shows that for a large range of utility functions such spreadpreserving inequality between the two agents increases the likelihood of first-best risk sharing and increases the transfers that agents make within the constrained optimal agreement. In what follows, we describe an experiment in which we replicated the setting of this model of risk sharing without commitment and tested specific features of risk sharing, such as the various effects of risk aversion, the time horizon, ex ante inequality, and beliefs. 8

10 3. EXPERIMENTAL DESIGN All the experiments reported here were conducted at the CASSEL Laboratory in UCLA. We had six sessions, with an even number of participants ranging from 12 to 18 in a session (depending on show-ups). Participants earned an average of about $17, including a $5 show-up fee, for about an hour of their time. The procedures that we followed are described below and the complete experimental instructions for the d =.9 treatment are shown in the Appendix. Note that participants were never told the nature of our research; in particular the terms risk sharing or insurance were never used during the experiment. Prior to the main experiment, we first asked people to complete an investment question. Each person was provisionally endowed with 100 units ($10) and could invest any portion of this amount in a risky asset that had a 50% chance of success and paid 2.5 times the amount invested if successful. The decision-maker retained the units not invested. We told the participants that we would later choose two people at random in each session for actual payoff implementation, and a coin was flipped after the session to determine success or failure for these investors. The objective of this investment question was two-fold. First, it provides us with a measure of risk aversion for each individual. 7 To be sure, the higher the investment the less risk averse the individual is. Second, we use the answer to this question in the main experiment to match individuals with similar degrees of risk aversion. 8 7 Naturally inducing risk aversion on the participants rather than relying on their preferences would allow us to have a more accurate measure of risk aversion. Unfortunately, there is little evidence that the method used to induce risk aversion (the binary lottery procedure) works, and there are a couple of studies indicating that it does not work (Camerer and Ho, 1994, Selten et al., 1999). The lack of reliable methods for inducing risk neutrality or controlling for risk attitudes and social preferences in experiments complicates direct empirical testing of theories based on richer behavioral assumptions. 8 To reduce heterogeneity, we paired people who had invested more (less) than 67 in the risky investment with other people who had invested more (less) than 67. 9

11 The body of the session then consisted of a number of matches. The features of this experiment were designed to closely fit the model presented in Section 2. For the duration of each match, every participant was paired with one other person. Each match was comprised of an uncertain number of periods or rounds that was determined as follows: After each round, the computer determined (for all current matches) whether another round would follow. In three of our sessions (Treatment 1), the continuation probability was 80%, and in the other three sessions (Treatment 2), the continuation probability was 90%. In the first case, the expected number of subsequent rounds in a match (at any point in time after the first round) was four, and in the latter case, the expected number of subsequent periods was nine. The participants in the corresponding sessions were informed of the relevant mathematical fact. The continuation probability is designed to play the same role in the decision process of the experimental subjects as the discount factor in Section 2. This also avoids the unraveling problem resulting when the number of rounds in any match is known in advance. Ex ante, we therefore expected each match to last five rounds in Treatment 1 and ten rounds in Treatment 2. When the matches ended, all participants were randomly re-matched for the next match. We had ten matches in each 80% session and seven matches in each 90% session. In each round, each person received income, which was comprised of a fixed portion and an amount that was added to the fixed income for that round for one of the people in each match. The person receiving this extra amount was randomly chosen in each pair for every round of the match. The fixed portions did not vary during the match, but did change from match to match. Over time, matches alternated between having both fixed portions be 70 units (equality) and having the fixed portions be 20 and 120 (inequality). In all cases, the amount randomly assigned 10

12 and added was 200 units. This income distribution corresponds to the two-state distribution with constant aggregate income described in Section 2. In the beginning of each round, each participant learned his or her fixed income, the fixed income of the other person in the match, and which one of them received the extra 200 in that round. Everyone then chose a non-negative amount, not to exceed the income received, to transfer to the other person; these designated amounts were then transferred. We allowed individuals to make transfers whether or not they had received the 200 units as we did not want to bias the experiment in favor of risk-sharing, and as we did not want the subjects to infer the main topic of the experiment. Participants saw a history of the income and transfers for each previous round in that match, and could also review their previous matches. Finally, we needed to avoid possible wealth effects in payoffs. If every round counted towards payoffs, individuals would care about the distribution of the sum of income net of transfers over all rounds instead of the income net of transfers earned in each round (and would also know accumulated income to date, possibly affecting behavior). To address this, and to make sure that the subjects face similar decisions as in the model described in Section 2, we chose only one round (of the many that were played in the session) for conversion of experimental payoff units to real dollars, at the rate of 17 experimental units to $1, as is clearly indicated in the instructions. We also asked participants for some information about their decisions and expectations in the beginning of the first and fourth matches. Specifically, the individuals who received h were asked: what motivates your choice of transfer? and the others (who did not receive h) were asked: what transfer do you expect the other person to give you? At the end of the session, participants answered questions concerning their gender and major before receiving their payoff. 11

13 4. MAIN RESULTS We first present some summary statistics about our data and then discuss in turn several important questions in relation to our results: Question 1: Does a higher continuation probability increase the amount of risk sharing? Question 2: Does a higher degree of risk aversion increase insurance? Question 3: What is the effect of ex ante inequality on risk sharing? Question 4: How do time and past transfers affect risk sharing? Question 5: Do demographics such as gender and major affect the transfer chosen? Question 6: Are transfers sensitive to whether expectations are met? Table 1 shows the average transfer made in each session, and the overall average for each treatment: Table 1: Average Transfer, by Session and Treatment Avg. Transfer # Observations Std. Dev. Session Session Session Treatment 1 (d =.8) Session Session Session Treatment 2 (d =.9)

14 A first observation is that a substantial amount of transfer takes place. With a continuation probability of 80% (Treatment 1), we see an average transfer of about 15 and, with a continuation probability of 90% (Treatment 2), the average transfer is around 30. The overall transfer is more than twice as high when the continuation probability is 90% instead of 80%. The Wilcoxon-Mann-Whitney ranksum test (see Siegel and Castellan, 1988) on session-level data, a most conservative approach that considers each session as only one observation, finds that transfers are significantly higher in Treatment 2 (p = 0.050, one-tailed test). 9 This aggregation ignores the substantial heterogeneity present in the population, indicated by the large standard deviation. Figure 1 shows the frequency with which each range of transfer is made in the two treatments: Figure 1 - Distribution of Avg. Individual Transfers 50% Proportion in Range 40% 30% 20% 10% Treatment 1 Treatment 2 0% Avg. Individual Transfer 9 Throughout the paper, we round p-values to three decimal places. 13

15 We see a great diversity of average individual transfers, particularly in Treatment 2. The Wilcoxon ranksum test on individual average transfers in the two treatments confirms that these are higher with the higher continuation probability (Z = 3.54, p = 0.000). 10 While overall individual transfers are an important metric, it is not clear what to make of positive transfers when a participant has received less income in the round than the other matched person. These may have some value through signaling one s cooperative nature, or may simply represent confusion. A better metric may be average transfers made when the chooser has the higher income. Table 2 shows these by session and treatment, and Figure 2 gives the distribution of individual average transfers made when ahead: Table 2: Average Transfer when receiving h, by Session and Treatment Avg. Transfer # Observations Std. Dev. Session Session Session Treatment 1 (d =.8) Session Session Session Treatment 2 (d =.9) Since there is interaction between participants, these average individual transfers are not completely independent. 14

16 Figure 2 - Dist. of Average Transfers, good shock 50% Proportion in Range 40% 30% 20% 10% Treatment 1: delta=0.8 Treatment 2: delta=0.9 0% Avg. Individual Transfer The average transfer made when ahead is higher in every session in Treatment 2, when the continuation probability is 0.9, than it is in Treatment 1, with a continuation probability of 0.8, and this average transfer when ahead is always higher than the corresponding overall average transfer in each session. Figure 2 shows a pattern similar to that seen in Figure 1, but with higher levels of average individual transfers. In fact, the average transfer when ahead was higher than the average transfer when behind for 85 individuals, and this was reversed for eight people. 11 A simple binomial test (See Siegel and Castellan, 1988) finds this to be very significant (Z = 7.98, p = 0.000). It is clear that people are not just randomly and arbitrarily transferring money, but are instead quite sensitive to which matched person receives higher income in the round. It is also useful to consider the average transfer made in each match. The distribution of these average transfers is shown in Figure 3: 11 The remaining person always chose a transfer of 0. 15

17 Figure 3 - Distribution of Avg. Transfers, by Match 50% Proportion in Range 40% 30% 20% 10% Treatment 1: delta=0.8 Treatment 2: delta=0.9 0% Avg. Individual Transfer The difference between treatments in Figure 3 is perhaps even stronger than in Figures 1 and 2. A Wilcoxon test confirms the difference is highly significant (Z = 5.96, p = 0.000). Another way to see that individuals in a match are insuring each other is to consider the standard deviation of realized consumption for a pair of individuals in a match. In the absence of transfers, their income distribution has a standard deviation of Table 3 illustrates clearly that, by making transfers to each other, individuals are sharing risk: the standard deviation of their consumption within a match is less than the standard deviation of their realized income. This is particularly true under Treatment 2 (high continuation probability) and when the fixed income is equal (y 1 = y 2 =70) as opposed to unequal (y i =20, y -i =120 for i =1,2). 12 Note that in this simple model the standard deviation of consumption of individuals matched with each other will be the same. 16

18 Table 3: Average Standard Deviation of Consumption Avg. Std. Dev. of Income Avg. Std. Dev. of Consumption # of Matches Treatment 1 Equal (y i = y i =70) (d=0.8) Unequal (y i =120, y j =20) Treatment 2 Equal (y i = y i =70) (d=0.9) Unequal (y i =120, y j =20) Thus far we have established that we see significant transfers and that these transfers are highly dependent on whether the chooser has received h or not. Hence, individuals are sharing risk. Moreover, the transfers and risk-sharing are definitely higher when the continuation probability increases, and when individuals have the same fixed-income. In the aggregate, the average transfer from the h person was with inequality and was with equality. Table 4 provides a breakdown, by continuation probability and fixed income, of average transfers after receiving h: Table 4: Average Transfer when receiving h, by Fixed Income and Treatment Fixed Income Treatment 1 (d = 0.8) Treatment 2 (d = 0.9) We see that average transfers are lowest for people with a fixed income of 20, and are actually a bit higher with a fixed income of 70 than with a fixed income of 120. To address the questions listed at the beginning of this section and study the determinants of the transfers that individuals make, we supplement our non-parametric statistical analysis with 17

19 some regression analysis. We use a Tobit model to account for the censored nature of the transfers, and report both standard Tobit regressions and random-effects Tobit regressions to account for unobserved individual characteristics. Table 5 presents the results of four different specifications of independent variables on all transfers made, while Table 6 presents the corresponding regressions for transfers made by people who received h. Note that the coefficient represents the effect on the latent variable. [Table 5 and 6 here] Before we examine the effects listed in the beginning of the section, note that Table 5 shows that receiving h clearly increases the transfers. The coefficient on Variable income (equal to 200 if the individual received h and 0 otherwise) is strongly statistically significant and implies that receiving h increases the average transfer by about 22 units. Question 1: Continuation probability. In both Table 5 and Table 6, we see that a higher continuation probability substantially increases the transfers. According to specification (3) in Table 5, increasing the continuation probability from 0.8 to 0.9 increases transfers by 18.6 units; specification (3) in Table 6 indicates that transfers made by people who receive h increase by 29.7 units. A closer examination shows that there are significantly different patterns over time with the different continuation probabilities. The coefficient for Delta=.9 is considerably smaller in Tables 5 and 6 for specifications (2) and (4), which include an interaction term Round*delta=.9. In Table 5, when d = 0.8 and individuals are equal, transfers decrease by 3.9 units per round within a match, while when d = 0.9, transfers actually increase by 0.3 units per round within a match; we see a similar pattern when unequal and in Table 6. Thus, a major part of the treatment effect appears to stem from differences over the course of matches. 18

20 Question 2: Risk aversion. Our initial investment question provides us with an estimate of individual attitude towards risk. 13 As illustrated in Figure 4, we observed a large range of answers to this investment question. In both Table 5 and 6, the coefficients on Investment (the amount invested in the risky investment) are significantly negative in all specifications. Since a larger investment in the risky asset indicates less risk aversion, a negative impact on the transfers is exactly what we would expect. We conclude that a higher degree of risk aversion increases the transfer that one chooses when high and therefore increases risk-sharing. Specification (3) in Table 6 indicates that, among people receiving h, a person who chose to invest 30 would transfer 43 units more than other person who chose to invest 80. Specification (4) in Table 6 indicates that the other person s investment in the risky asset also has a negative and substantial impact on the transfers. The patterns are similar in Table 5, except that the coefficient on the other person s investment is not significant in specification (4). Question 3: Inequality. By looking at the effect of the fixed income on transfers, we can assess the overall effect of equality. Recall that changes to an individual s fixed income are concurrent with changes in his partner s income. Table 5 indicates that total transfers are 40% higher with equality than with inequality (twice the coefficient of D70 compared with the coefficient of D120); Table 6 shows that a h-receiving person with a fixed income of 70 transfers 13 Assuming constant relative risk aversion, the risk aversion would be given by Ï Ô ln(1.5) if inv <100 r Ì. An investment choice of 25 thus corresponds to r = 0.67, an ln(inv * inv) - ln(100 - inv) 0 if inv =100 Ó Ô investment choice of 50 corresponds to r = 0.32, and an investment choice of 75 corresponds to r =

21 21 more than a person with a fixed income of 20, but only four less than a person with a fixed income of We can supplement our regression analysis here by considering whether each individual made larger transfers on average with equality or inequality, since each person participated in both environments. It turns out that average transfers were higher for 60 people with equality and for 32 people with inequality, with no difference for the other two people. The binomial test finds this difference highly significant (Z = 2.92, p = 0.004, two-tailed test). Question 4: Past transfers and time. Risk sharing requires some reciprocity. We see that the first transfer made by the individual s partner within a match has a strong positive effect on his transfers. In Table 5, we see that an additional 10 units of transfer in the first period by one s counterpart (Other s 1 st tr) increases an individual s average transfer in each period by 3.5 units; in Table 6, the corresponding figure is 4.5 units. Interestingly, the coefficient on Other s 1 st tr h is significantly negative, indicating that transfers made by an individual when he has not received h are particularly effective, perhaps being interpreted as a form of signal. We also see in Table 5 that there is a very different evolution of transfers over time within matches, depending on whether the individuals fixed incomes are the same or not. The coefficient of Round*equal is significantly positive, and in specification (3) more than offsets the negative coefficient of Round. Thus, when the individuals have the same fixed income, transfers 14 There was not much difference in the transfers made by the person not receiving h with equality (10.40) and inequality (9.20). 20

22 within a match are fairly flat over time. In contrast, transfers decrease over time when the fixed incomes differ. 15 Question 5: Demographics. The average transfer by males was 28.9 while the average transfer by females was This is surprising given that females are more risk averse than males in our experiment, 16 so that we might have expected females to make higher transfers. In any case, we observe in specifications (1) and (3) of both Tables 5 and 6 that females make significantly smaller transfers, even when we control for risk aversion. The coefficient on Female is much lower when we include the interaction term Round*female, in specifications (2) and (4), as it appears that transfers decrease over time relative to male transfers. The lower transfers by females results in a significantly lower net consumption for males (Z = 2.68, p = 0.007, using a Wilcoxon test on individual average consumption. 17 We also tested (not shown) for the role of students major on the transfers made, and found virtually no effect. Question 6: Beliefs. In the first round of a couple of matches per session, we elicited the beliefs of subject who did not receive the 200 regarding the transfer they expect to receive from the other. We find that, out of 94 observations, 38% of them received more than they did expect, 9% received exactly what they expected and 53% received less than expected. Table 8 shows the effect of meeting individual s expectations or not on the transfers, as we include the difference between an individual s expectations and the transfer actually received. Bdiff takes values +1, 0, -1 if the transfer received earlier in the current match was above, at or 15 We also find (not shown) that transfers significantly increase with a higher number of rounds in the previous match. This suggests that participants views on the likely length of a match are sensitive to their own experience, despite the statements of mathematical expectations given in the instructions. 16 This gender investment result is extremely robust, as is discussed in Charness and Gneezy (2004). 17 Note that this is not due to females having better draws; females comprised 56.4% of the population and females had the larger endowment 55.6% of the time. 21

23 below expectations. Clearly, its effect in the random-effects specifications is quite significant and positive on the transfers made by an individual. It is interesting to see that the effect of the round within the match is not significant when we control for the difference between expectations and received transfers, so that we don t observe a pure time-decay effect (within a match) even in the d = 0.8 treatment. 5. DISCUSSION Our experiment provides some strong support for the model of risk-sharing without commitment. First, there is evidence that individuals are providing limited insurance to each other. Net positive transfers are going from individuals receiving a high shock to the other and these transfers substantially reduces the standard deviation of consumption. Second, transfers are limited in a way that is consistent with the modest levels of risk aversion exhibited by the participants. 18 A longer expected time horizon has a strong positive effect on transfers, as does a higher degree of risk aversion. None of these effects would be expected to arise if transfers are motivated purely by altruism or distributive preferences. As we also find that one s chosen transfer depends positively on the other party s first transfer and on receiving transfers that meet or exceed expectations, there are clear evidence of reciprocal relationships. While all of these factors support the risk-sharing model presented in Section 2, the effect of inequality on transfers is negative rather than positive. For utility functions of the HARA class (hyperbolic absolute risk aversion), inequality should improve risk sharing and not decrease it; with decreasing risk aversion (as traditionally assumed) we should observe that individuals with lower fixed income making higher transfers when receiving h than individuals with higher 18 These levels of risk aversion are similar to the ones found in other experiments. 22

24 fixed income make when receiving h, as the poorest agent trades mean consumption in exchange for insurance. Yet, in our experiment we observe that high-fixed-income individuals actually transfer more than low-fixed-income individuals. Of course, individual preferences may be very different from HARA utility functions. The fact that the person with more fixed income makes higher transfers is consistent with social preferences. However, the fact that overall transfers are lower with inequality also goes against the consensus of the social-preference models. In some sense, equal fixed payoffs may make coordinating on reciprocal transfers easier. Table 4 clearly suggests that the inequality results stem from people with low fixed income making substantially lower transfers. This may reflect the fact that even when the person with the low fixed income receives h, he or she is nevertheless poorer in expected terms during the remainder of the match. Since we pay for only one period, this story might not apply, but it could still influence behavior. In any event, by this logic we should also see substantially higher transfers with the high fixed income than with equal fixed incomes, and we don t. Perhaps one views one s local wealth in a self-serving manner, and this divergence in perspectives makes risk sharing more difficult. In another sense equal fixed payoffs make it easier for one to identify with the other person and interpret the transfers made. Previous studies have shown that participants in experiments are prone to identification or solidarity with an arbitrary group. Tajfel, Billig, Bundy, and Flament (1970) find that subjects treat people who have been designated to be part of their group quite differently than people not in their group. Charness, Rigotti, and Rustichini (2003) achieve strong group identification and coordination in Battle-of-the-Sexes games with partisan audiences. If, as Akerlof and Kranton (2000, p. 748) assert, a person s self is 23

25 associated with different social categories and how people in these categories should behave, perhaps it is reasonable to expect that people with similar average income levels would more readily form reciprocal relationships than people with very different average incomes. Ex ante equality may also set up an egalitarian frame, leading to more sharing. More tests are clearly needed to better understand these results. 6. CONCLUSION In this paper, we test experimentally for risk sharing without commitment and some of its implications. The experiment was designed to fit as closely as possible the models of risksharing without commitment used in the literature, and, following the literature, we focused on the constrained optimal equilibrium. We find strong evidence of risk sharing in the laboratory, including significant support for some important features of the models of risk sharing without commitment. It is striking that both a higher continuation probability and a higher degree of risk aversion greatly increase the level of risk sharing that individuals choose. We also find that beliefs matter, in that how actual transfers compare to expected transfers plays a role in subsequent transfers. Moreover, a form of reciprocity is shown to be important for risk sharing: The higher the first transfer made by an individual s partner within a match, the higher the individual s transfer, especially upon receiving a good shock. Outside the model, we find that, far from increasing risk sharing, inequality between individuals in a match actually decreases it, suggesting the influence of some form of social cohesion or identification process. We also observe that the person with less total income in a round frequently makes a small positive transfer to the person with more total income. Apparently such a transfer is seen 24

26 as a signal of intent, and it is particularly interesting that such transfers in the first round of a match are (dollar-for-dollar) more effective in increasing the transfers subsequently made by one s counterpart than transfers made in the first round when receiving the larger total income. More research is needed to explore the systematic use of signaling in risk sharing and contracting environments. REFERENCES Akerlof, G. and R. Kranton (2000), Economics and Identity, Quarterly Journal of Economics, 115, Barr, A. (2003), Risk Pooling, Commitment, and Information: An experimental test of two fundamental assumptions, mimeo. Binswanger, H.P. (1980) Attitude Toward Risk: Experimental Measurement in Rural India, American Journal of Agricultural Economics 62, Bolton, G., and A. Ockenfels (2000), ERC A Theory of Equity, Reciprocity, and Competition, American Economic Review 90, Bone, J., J. Hey and J. Suckling (2003), A Simple Risk-Sharing Experiment, forthcoming Journal of Risk and Uncertainty. Camerer, C. and T. Ho (1994), Isolation Effects and Violation of Compound Lottery Reductions, mimeo. Charness, G. and U. Gneezy (2004), Gender and Risk Attitudes in Investment Decisions, mimeo. Charness, G. and M. Rabin (2002), Understanding Social Preferences with Simple Tests, Quarterly Journal of Economics, 117, Charness, G., L. Rigotti, and A. Rustichini (2003), They are Watching You: Audience Effects in Economic Institutions, mimeo. Coate, S. and M. Ravallion (1993), Reciprocity without Commitment: Characterization and Performance of Informal Insurance Arrangements, Journal of Development Economics, 40, Cox, J. and D. Friedman (2002), A Tractable Model of Reciprocity, mimeo. 25

27 Deaton, A. (1992), Understanding Consumption, Oxford: Clarendon Press. Fehr, E. and K. Schmidt (1999), A Theory of Fairness, Competition, and Cooperation, Quarterly Journal of Economics, 114, Foster, A. and M. Rosenzweig, Imperfect Commitment, Altruism, and the Family: Evidence from Transfer Behavior in Low-Income Rural Areas, Review of Economics and Statistics, 83, Genicot, G. (2003), Inequality and Informal Insurance, mimeo. Genicot, G. and D. Ray (2003) Endogenous Group Formation in Risk-Sharing Arrangements, Review of Economic Studies 70 (1), Gertler, P. and J. Gruber (2002), Insuring Consumption Against Illness, American Economic Review, 92, Goeree, J., C. Holt and T. Palfrey (2003), Risk Averse Behavior in Generalized Matching Pennies Games, Games and Economic Behavior, 45, Grimard, F. (1997), Household Consumption Smoothing through Ethnic Ties: Evidence from Côte D'Ivoire, Journal of Development Economics, 53, Jalan, J. and M. Ravallion (1999), Are the Poor Less Well Insured? Evidence on Vulnerability to Income Risk in Rural China, Journal of Development Economics, 58, Kimball, M. (1988), Farmer Cooperatives as Behavior Toward Risk, American Economic Review, 78, Kletzer, K. and B. Wright (2000), Sovereign Debt as Intertemporal Barter, American Economic Review, 90, Kocherlakota, N. (1996), Implications of Efficient Risk Sharing without Commitment, Review of Economic Studies, 63(4), Ligon, E., J. Thomas and T. Worrall (2002), Mutual Insurance and Limited Commitment: Theory and Evidence in Village Economies, Review of Economic Studies, 69, Selten, R., and A. Ockenfels (1998), An Experimental Solidarity Game, Journal of Economic Behavior and Organization 34, Selten, R., A. Sadrieh, and K. Abbink (1999), Money Does Not Induce Risk Neutral Behavior, But Binary Lotteries Do Even Worse, Theory and Decision, 46,

28 Tajfel, H., M. Billig, R. Bundy and C. Flament (1970), Social Categorization and Intergroup Behavior, European Journal of Social Psychology, 1, Townsend, R. (1994), Risk and Insurance in Village India, Econometrica, 62(3), Townsend, R. (1995), Consumption Insurance: An Evaluation of Risk-Bearing Systems in Low- Income Economies, Journal of Economic Perspectives, 9(3), Udry, C. (1994), Risk and Insurance in a Rural Credit Market: An Empirical Investigation in Northern Nigeria, Review of Economic Studies, 61(3), Udry, C. (1995), Risk and Saving in Northern Nigeria, American Economic Review, 85(5),

29 APPENDIX Instructions Welcome to our experiment. For showing up on time, we will pay you a $5 show-up fee. In addition, you may receive additional earnings as the result of the outcomes in the experimental session. Today s session will take about an hour. To begin, we ask you to complete a brief questionnaire. The body of the session will be comprised of a number of segments. In each of these segments, each participant will be matched with one other person. Each segment is comprised of an uncertain number of periods. The number of periods in a segment is determined as follows: After each period, the computer will roll a die (for the entire room) to see whether another period will follow, with a 90% chance that another period will follow, and a 10% chance that the segment ends immediately. The computer will roll the die after every period. With this continuation probability, the expected number of subsequent periods in a segment, at any point in time, is 9. When the segment ends (10% chance after each period), all participants will be randomly rematched with other participants for the next segment. We anticipate that there will be approximately 7 segments in the session, but this will vary according to how many periods there are in the segments we aim to complete the session in about an hour. In each segment, you and the person with whom you are matched will receive income. This income is composed of a fixed portion and an amount (200) that is added to the fixed income for that period for one of the people in each match; the person receiving this extra amount is randomly chosen in each pair for every period of the segment. The fixed portions will not vary during the segment, but will change from segment to segment; these fixed portions may or not be the same for the two people matched. In all cases, this fixed portion will be considerably smaller than the 200 units that are randomly assigned. In the beginning of the period, you will learn your fixed income, the fixed income of the person with whom you are paired, and who received the extra 200 in the period. At this point, you choose to transfer money to the other person. This amount must be non-negative and no more than the total income you received in that period. The other person in your match simultaneously chooses to transfer money to you, subject to the same restrictions on the amount to be transferred. The designated amounts are then transferred, and the computer then determines whether another period follows in this segment. You will see a history of the income and transfers for each previous period in that segment. Thus, you will be involved in many periods. We wish to make it clear that only one of these periods will be chosen at random for conversion to real dollars, at the rate of 17 experimental units to one cash dollar. 28