Mutual Savings and Ex-Ante Payments. Group Formation in Informal Insurance. Arrangements

Transcription

1 Mutual Savings and Ex-Ante Payments Group Formation in Informal Insurance Arrangements Tessa Bold University of Oxford Nuffield College Abstract In an environment of limited commitment, risk-sharing arrangements must be self-enforcing. This requires that insurance groups are robust not only to single-person deviations, but also to potential deviations by sub-groups and results in a tight bound on the size of insurance groups and the gains from risk-sharing. This paper shows how to enlarge the contract space in a manner that relaxes the constraints on the formation of insurance groups by introducing both mutual savings and ex-ante transfers and thus increases the benefits from mutual risk-sharing. Our main findings are that capital holding coupled with ex-ante transfers increase both the maximal stable size and the benefits from risk-sharing within a community. 1

2 1 Introduction In an environment of limited commitment, risk-sharing arrangements must be selfenforcing. This requires that insurance groups are robust not only to single-person deviations, but also to potential deviations by sub-groups. Genicot and Ray [6] show that stable groups are bounded in size and find that the stability of smaller groups destabilizes larger groups thus restricting the size of insurance arrangements we can expect to find in a community. Since the per-capita utility from risk-sharing increases with the size of an insurance arrangement, this significantly reduces the benefits from risk-sharing in the presence of limited commitment. Equally, there is ample empirical evidence that risk-sharing departs from first-best (Townsend [16], Udry [17], Dercon and Krishnan [3], Grimard [10], Ligon, Thomas and Worrall [14] and Gertler and Gruber [7]) and is often restricted to smaller networks in the community which may be based on ethnicity, kinship or geographic proximity (Murgai et al [15] and Fafchamps and Lund [4]). Given these findings, this paper explores whether an informal insurance arrangement can be structured in a manner that relaxes the limited commitment constraints and thus increases the benefits from risk-sharing by enlarging the contract space to include mutual savings and ex-ante transfers. Mutual savings are here defined as communally held assets, while ex-ante transfers are simply payments that are made before the state of the world is known. In bilateral risk-sharing arrangements the effect of the enlargement of the contracting space is well-documented. The constraint-efficient contract is in fact non-stationary and introducing such non-stationary transfers increases the benefits from risk-sharing compared with the stationary case (Ligon et al. [14], Attanasio and Rios-Rull [1] and Kocherlakota [12]). Gauthier, Poitevin and Gonzalez [5] show that this nonstationary contract can be improved upon via the introduction of ex-ante payments. Ligon, Thomas and Worrall [13] demonstrate that individual savings can assume the 2

3 role of ex-ante transfers, but that in general individual savings have an ambiguous impact on the extent of risk-sharing that can be sustained. This is so because on the one hand asset holding increases the resources available for future consumption, but on the other hand it also increases the benefit from deviation to autarky. Mutual savings are unequivocally welfare-increasing in the case of bilateral risk-sharing, since communally held savings do not affect the autarky payoff. The effects of enlarging the contract space are however less clear-cut when we extend our analysis to multi-lateral risk-sharing, because the degree of risk-sharing is in general non-monotonic in environmental parameters, and analytical results do not exist. 1 This paper however shows that introducing mutual savings and ex-ante transfers can significantly increase the maximal stable group size in a community and the benefits from insurance compared to risk-sharing without saving. To see why this is the case, note that without savings, ex-ante transfers can only play a limited role in alleviating limited commitment constraints. The reason for this is that while the agent who makes a positive ex-ante transfer relaxes his own expost enforcement constraint, the ex-ante transfer must not be so large as to violate the enforcement constraint of the agent who receives the ex-ante transfer. This is different in the presence of mutual savings, since ex-ante payments no longer have to be transferred between agents, but instead can be made into the communally held savings fund. Furthermore, even though in our model, a deviating sub-group does not forego its savings, but instead retains its per capita share of the assets after deviation 2, this nonetheless amounts to an additional penalty when deviating from an insurance arrangement in comparison to the case without capital. This is so because as the 1 Consider a simple example in which under stationary ex-post transfers an insurance arrangement must comprise a minimum of 3 members to achieve stability with respect to individual deviations. Introducing ex-ante transfers or capital increases the benefits from risk-sharing thereby reducing the minimum group size to a pair and this stable pair now destabilizes the group of size 3. 2 The case in which a deviation results in the loss of savings has been explored in Gobert and Poitevin[9] for the bilateral case. 3

4 size of an insurance group increases the same amount of consumption smoothing can be achieved with smaller per capita asset holdings, so that the deviating group is comparatively worse off. This makes the combination of ex-ante payments and mutual savings a potentially powerful tool for lessening the threat to the stability of an insurance arrangement posed by the presence of stable sub-groups. The paper proceeds by recursively setting up a model of group formation in informal insurance arrangements with mutual savings and stationary ex-ante and ex-post transfers in a dynamic programming framework and deriving Euler equations and conditions for the stability of such an arrangement. We focus on the stationary case here because the dynamics of group formation make it difficult to derive any generally applicable analytical results, so that we instead rely on a large number of simulations and careful sensitivity analysis to make some general predictions. For stationary transfers this can be done at small computational cost. We then present a variety of computed examples that study the interplay of mutual savings and ex-ante transfers. Since there is no formal financial sector in the model, we also impose an upper bound on capital that can be held in the communal savings fund and explore the implications of such a bound. 3 The simulations show that while the introduction of mutual savings increases the benefits of risk-sharing for a given group size as the upper bound on capital increases, stability continues to be threatened particularly for low values of the group s capital stock. This threat can only be alleviated by the introduction of ex-ante payments and it is this feature which increases the size of the maximum stable risk-sharing group in a community significantly. The paper proceeds as follows: The model is developed in Section 2. In Section 3, 3 A motivation for this could be that the risk of theft or embezzlement increases non-linearly with the size of capital holding. 4

5 simulations for a wide range of parameters are presented, which show the effects of combining mutual savings with ex-ante transfers. In Section 4, we turn to a unique data set on funeral insurance in rural Ethiopia to examine whether these predictions are borne out in practise. We indeed find evidence that insurance arrangements exist, which hold assets and charge ex-ante transfers and that such insurance arrangements outperform those which restrict transfers to ex-post transfers both in the coverage offered and in the size of the insurance arrangement. Conclusions are drawn in Section 5. 5

6 2 The Model We begin by characterizing the symmetric and stationary stable contract by recursively setting up the model of group formation with mutual savings and ex-ante transfers and then show how to compute such a contract using standard dynamic programming. In our context mutual saving means that for each group size, capital holding is decided jointly, so that optimal capital only depends on group size and the aggregate shock. This implies that even though individuals can save in isolation, upon joining an insurance arrangement, savings are pooled at the group level. The ex-ante transfer is paid into the mutual fund before the state of nature is realized with the purpose of relaxing the agents ex-post enforcement constraints. 4 We consider a community of H households. Each period t = 1, 2,... household i receives an income y(s t ) > 0 where s is the state of nature drawn from a finite set S = {1,..., S} with S 2. The state of nature follows a Markov process with the probability of transition from state i to j given by π(s j s i ), and π(s j s i ) > 0 for all i and j. Individuals and insurance groups have access to a storage technology with interest rate r, where r > 0, with the maximum storage in any period K, where K is non-decreasing in n, the number of members in an insurance arrangement. This effectively puts a bound on per capita asset holdings in the insurance group. This seems a sensible assumption, since there is an opportunity cost of handing over money to the group s savings fund rather than using it for individual consumption. 5 The capital stock at the beginning of period t is denoted by k(t). All households have an increasing identical twice continuously differentiable utility function u i s(t), and c i s(t) > 0. Households are risk-averse, infinitely lived and discount the future with 4 Ligon, Thomas and Worrall [13] would not class such a payment as ex-ante, but rather as increased savings. Nonetheless, we refer to it as an ex-ante payment to indicate that it is chosen before the state of the world is known. 5 This is not the same as putting a bound on total assets held by a group, the implications of which will be discussed later on. 6

7 common discount factor β. In each period, a contract specifies the following structure for consumption for each household and mutual savings in an insurance arrangement: 1. The household s ex-ante consumption c i s(t) EA and the household s non-negative ex-ante transfer t EA, which is made before the state of nature is realized. 2. Its ex-post consumption c i s(t) EP, and (positive or negative) ex-post transfer t EP s, which is made after the state of nature is realized. 3. The optimal capital holding k s (t+1), which is a function of beginning of period capital k(t), the number of households in the insurance arrangement n, and the realized state of nature s. Consumption takes place at the end of the period, when the state of nature is known. We now proceed to define stability of an insurance arrangement recursively. An arrangement of size n is considered stable if there is no sub-group that can credibly deviate from the arrangement. A credible deviation requires that the sub-group is better off after deviation and that the subgroup itself is stable with respect to further deviations. We begin with an individual household. In each period, t, having observed the current state s, the household has zs(t) i = (1+r)k i (t)+ys(t) i total resources available. It then chooses ex-post consumption c i s(t) EP and the remainder k s (t + 1) is stored. For any beginning of period capital k(t), the household s expected utility from the beginning of period t onwards is U(z(t), 1) = E β τ t u(zs(τ) i k s (τ + 1)) (1) τ=t 7

8 where the expectation is taken over all possible evolutions of the capital stock and income realizations. Using standard dynamic programming arguments, optimal consumption and capital can be found as the arguments which maximize the following equation set up in terms of value functions: [ Ws(z i s(t), i 1) = max ui (zs(t) k i s (t+1))+β k(t+1) S π sr (Wr(k i r (t+1)+yr(t+1)), i 1) ] (2) where the second argument of W indicates the size of an insurance arrangement, which is 1 in autarky. Since there are no possible deviations, the set of stable payoffs for an individual contains the solutions W i s(z i s(t), 1) to the above problem for all possible states s and beginning of period resources z. Denote this set by V (1). Furthermore, since there are no enforcement constraints, there is no role for ex-ante payments in autarky. r=1 This definition of a stable payoff is now extended to an arrangement of size n. Recursively having defined sets of stable payoffs for arrangements of size m = 1,..., n 1, we consider a coalition of size n. The aggregate resources of such an arrangement are denoted by z s (t) = (1 + r)k(t) + n i=1 yi s(t). In each period, ex-ante consumption c i (t) EA, ex-post consumption c i s(t) EP and aggregate capital k s (t + 1) is chosen. The expected utility of being in an arrangement of size n for a given beginning of period capital stock and income realization is U(z(t), n) = E β τ t u(c i s(t) EP ) (3) τ=t Again this can be formulated in terms of value functions and consumption and capital are the maximizing arguments of [ Ws(z i s (t), n) = max ui (c i s(t) EP ) + β c,k(t+1) S π sr Wr(k i r (t + 1) + r=1 n yr(t i + 1), n) ] (4) i=1 8

9 Such an arrangement will be stable if no stable subgroup of size m n 1 can credibly deviate from it. S Ws(z i s (t), n) u i (c i s(t) EA ) + β π sr Wr( i m m n k(t) + yr(t i + 1), m) (5) r=1 i=1 where Wr(, i m) is a stable payoff for a group of size m. The assumption here is that each individual consumes its own income less the specified ex-ante transfer c i (t) EA in the period of deviation and that the deviating sub-group of size m is excluded from the original group but keeps its share of assets m n k(t).6 Insurance is then continued within the sub-group of m individuals beginning in the period following deviation. The set of stable payoffs for a group of size n then contains all the solutions to (4), which satisfy the enforcement constraints in (5) for all possible states s and levels of beginning of period capital, denoted V (n). Thus stability can be defined in the following manner. Definition 2.1 An arrangement of size n is stable if and only if, for every possible beginning of period capital k(t) and state s, the following inequality constraints are satisfied (i) (ii) U i (z(t), n) U i S Ws(z i s (t), n) u i (c i s(t) EA ) + β π sr Wr( i m m n k(t) + yr(t i + 1), m) r=1 i=1 for every stable m n 1 This definition states that an arrangement is stable if at all times the individual prefers making the contractual transfers to reneging on the contract and continuing insurance within a stable arrangement of size m < n 1. Constraint (i) is an ex-ante 6 It is also possible to stipulate that a deviating sub-group loses all assets and starts with k = 0 after deviation. This would make the threat of deviation less powerful. 9

10 constraint, which must hold before the state of nature is observed. It can be understood as a promise-keeping constraint stating that each household in the arrangement must be given at least the level of expected discounted utility already promised. In general, there is an infinite number of promised utilities, however, sensibly, promised utility cannot exceed first-best utility that could be achieved in the absence of ex-post enforcement constraints and should be at least as large as the utility household i can achieve in any stable sub-group, otherwise the arrangement will never be stable. Constraint (ii) is an ex-post constraint and must hold after the state of nature is realized. This may possibly be achieved with the help of ex-ante transfers which reduce the one-period gain from reneging from the arrangement. 7 Given this definition of stability, we now show how to solve for the optimal constraint symmetric and stationary ex-ante and ex-post transfers and end of period capital by setting up a dynamic program problem for a social planner who seeks to maximize a weighted sum of expected utilities: W i s(z s (t), n) = max k s(t+1),c i s (t)ep,c i (t) EA n [ λ i u(c i s (t) EP )+β i=1 subject to the following constraints: µ i (s) : W i s(z s (t), n) u i (c i s(t) EA ) + β S n π sr Wr(k i r (t+1)+ yr(t+1), i n) ] r=1 i=1 S π sr Wr( i m m n k(t) + yr(t i + 1), m) r=1 i=1 (6) r, for every stable m n 1 (7) φ i (t) : U i (z(t), n) U i (8) ω i (t + 1) : n n (1 + r)k(t) + ys(t) i k s (t + 1) + c i s(t) EP (9) i=1 7 It is usually the case that the satisfaction of the ex-post constraints guarantees that ex-ante constraints hold. However, as we will show in our simulations, this is not the case when mutual savings are restricted to lie below an upper bound at all times. Such a specification seems to be a reasonable assumption for rural communities without access to formal financial institutions. 10 i=1

11 where equation (7) represents the ex-post enforcement constraint and equation (8) represents the promise-keeping constraint that has to hold in expectation. From constraint (7), it can be seen that when storage is increased this has a direct effect on both the value of remaining in a group of size n as well as the value of deviation. This mitigates the trade-off observed when saving takes place only in autarky as discussed by Ligon, Thomas and Worrall [13]. Finally, equation (9) represents the aggregate resource constraint. We have three choice variables, ex-ante consumption, ex-post consumption and capital in each period. The optimization is conducted in the following manner. Capital and ex-post consumption is chosen to maximize expected utility subject to the enforcement constraints for an arrangement of size n. Ex-ante transfers are used to relax the ex-post enforcement constraints. There are several ways in which this could be done. For example, ex-ante transfers could be used to achieve first-best risk-sharing in every state of the world. Alternatively, one could stipulate that ex-ante transfers cannot exceed an upper bound B, in which case ex-ante transfers would optimally be set to this bound and ex-post consumption is then chosen to maximize the degree of risk-sharing that can be sustained. We follow a third option, in which for each group size and beginning of period capital, we choose ex-ante transfers such that a group of size n can just achieve stability. This means in practise that if a group is stable without recourse to ex-ante transfer, ex-ante transfers are zero, and ex-ante consumption equals individual income. If any or all of the enforcement constraints are binding for all feasible choices of ex-post consumption thus rendering an arrangement unstable, ex-ante transfers are used to achieve stability. Since the ex-post constraints will vary depending on the realized state, but the ex-ante transfer is determined before the state of nature is realized, it will be chosen so as to satisfy the maximal ex-post constraint with equality. This 11

12 means that in effect the use of ex-ante transfers is minimized subject to achieving stability of an insurance arrangement. This seems to be a reasonable strategy, since otherwise first-best stability of any arrangement could always be guaranteed, which is not a very interesting problem, or alternatively, the optimal contract depends on an arbitrary upper bound for ex-ante transfers. Even so, minimizing ex-ante transfers in this manner may still require them to be large. The question is how large can they be? Answering this question requires some discussion of the timing of the income realizations. Certainly ex-ante transfers cannot exceed the highest possible income state, since otherwise consumption would definitely be negative. One possibility would be to require them to be less than mean income. The motivation for this would be an income distribution with the following timing: Each period, an agent obtains income ȳ. With probability p, he receives a positive shock of size ɛ 1 and with probability 1 p a negative shock of size ɛ 2 so that the maximum ex-ante transfer cannot exceed mean income. However, since we want to investigate what group sizes and extent of risk-sharing can be achieved with the minimal use of ex-ante transfers, we will require them to be no larger than the lowest income realization y 1. In that case, even if in each period, the agent had income y 1 and then received a positive shock with probability p, the ex-ante transfer could still be made. 12

13 This discussion leads to the following Lagrangian for the social planner in each period: L = + n i=1 λ i t=0 S [ β t π s u(c i s (t) EP ) + β r=1 β t π s µ i s(t) [ u(c i s(t) EP ) + β t=0 (u(c i s(t) EA ) + β S r=1 S π sr Wr(k i r (t + 1) + r=1 S π sr Wr(k i r (t + 1) + r=1 π sr max m n yr(t i + 1), n) ] i=1 n yr(t i + 1), n) i=1 W r( i m m n k(t) + yr(t i + 1), m)) ] + resource constraint + participation constraint + β t[ η(t)(y 1 τ(t) EA ) ρ(t)(τ(t) EA ) 2] (10) t=0 The last two terms indicate that ex-ante transfers are to be minimized and cannot exceed the lowest income realization y 1. Further, since all ex-post enforcement constraints have to be satisfied only the one representing the stable sub-group of size m with the largest gain from deviation is relevant for the maximization. i=1 Substituting for the aggregate resource constraint, we have the following n 1 first order conditions for aggregate capital: λ i Wk(z i s (t), n) + µ i s(t)(wk(z i s (t), n) Wk(z i s (t), m) = n β(1 + r)λ i Wk(k i r (t + 1) + yr(t i + 1), n)+ µ i s(t + 1)(W i k(k r (t + 1) + i=1 n yr(t i + 1), n) Wk(k i r (t + 1) + i=1 m yr(t i + 1), m) ] i=1 i = 1,..., n (11) If µ i s(t + 1) = µ i s(t) = 0, this leads to the familiar Euler equation for savings. The 13

14 first order condition for consumption states that u ic = λ n + µ n (s) u nc λ i + µ i (s) i = 1,..., n 1 (12) This first order condition says that an agent s ex-post consumption is determined by the size of the Lagrange multiplier on its enforcement constraint. If neither agent i nor agent n s enforcement constraint binds, their ratio of marginal utility remains the same. If the constraint binds for one agent, that agent s consumption share is increased and if it binds for both agents, the consumption share of the agent with the larger Lagrange multiplier is increased by more. If we define θ(0) as the initial ratio of Pareto weights and normalize the Lagrange multipliers to ν(s) = µ(s)/(λ + µ(s)), we can show that for any two agents, we have the following up-dating rule for the ratio of Pareto weights θ ji (t) = 1 v i(s) 1 v j (s) θ ji(0) (13) since 1 v i (s) 1 v j (s) θ ji(0) = λ i +µ i (s) µ i (s) λ i +µ i (s) λ j λ j +µ j (s) µ j = (s) λ i λ j +µ j (s) λ i λ i +µ i (s) λ j λ j = 1 v i(s) λ j (14) λ i 1 v j (s) λ i λ j +µ j (s) This is the updating rule for Pareto weights in the stationary case. In contrast to a dynamic contract where the ratio of Pareto weights is time-dependent, here the ratio of Pareto weights only depends on the current state of the system, which depends both on the income realization and beginning of period assets. Thus whenever there are no binding constraints, the initial ratio of Pareto weights remain unchanged, if the enforcement constraints are binding, they are changed so that all the enforcement constraints are satisfied with equality. Since we can solve for consumption from the first-order conditions in terms of Pareto weights, capital and current income, we are in fact seeking Pareto weights and capital allocations such that utility is maximized subject to the constraints. If there is no fea- 14

15 sible choice for capital and Pareto weights that satisfies the enforcement constraints, ex-ante constraints are used to satisfy the constraints with equality. If this ex-ante transfer exceeds y 1, no stable contract for a group of size n exists. We now turn to a variety of computed examples to illustrate the properties of this model. 15

16 3 Simulated Model In this section we present simulations for a wide range of parameters. A variety of summary measures are computed to illustrate the properties of a model of informal insurance with mutual savings and ex-ante transfers and to distinguish between the effects of capital holding alone and capital holding combined with ex-ante transfers. The effects of imposing an upper bound on capital are also explored. The model to be estimated is of the following form. We solve for ex-ante and ex-post transfers for a risk-sharing arrangement of n individuals in a community of N = 10 with individual utility function u(c) = 1 1 ρ c1 ρ (15) where ρ is the coefficient of relative risk aversion. In each period, individual income is y(1) with probability (1 p) and y(2) with probability p, with y(2) > y(1). Aggregate shocks therefore follow a binomial distribution with parameters n and p. The discount rate is β and the aggregate resource constraint is given by: n c i (t) + k(t) = i=1 n y i (t) + (1 + r)k(t 1) (16) i=1 where k [0, K(n)] and r is the interest rate. K(n) is non-decreasing in n. In all simulations, we attempt to minimize the use of ex-ante transfers. We now present simulations for two cases: In the first case, the maximum capital K(n) a group can hold is fixed and in the second, maximum capital is increasing linearly in n. The motivation for examining those two cases is that communities in developing countries often do not have access to a formal financial sector and we want to explore what impact this has on an informal insurance arrangement with assets. Presumably, in the absence of a formal financial sector, there will always be a bound 16

17 on the total assets an insurance arrangement can hold, so the first case proxies for the situation, in which this restriction binds for the relevant parameters and group sizes, whereas in the second case it does not. We report the following properties for the stable insurance arrangements in a community: The relative gain from being in a stable arrangement of size n compared to the maximal per capita payoff that can be achieved, i.e. the first-best in a community of all N = 10 when capital holding is unconstrained, starting with no initial capital: U (0, n) U (0, 1) 100 (17) Ũ(0, N) U(0, 1) where Ũ(0, N) is the expected value of being in a group of size 10 with first best transfers and unconstrained capital. This gives a measure of the return to insurance in percentage terms. Furthermore, we report how far an arrangement diverges from first best risk-sharing by calculating the Euclidian distance of the feasible transfers from the first-best transfer for a range of beginning of period capital holdings. This provides a useful way to measure to what extent capital holding increases the benefits from risk-sharing merely through providing an opportunity for consumption smoothing across time versus increasing the opportunity to insure within a time period by reducing the gain from deviation. Finally, we calculate the maximum ex-ante transfer for each stable arrangement as a proportion of the lowest income realization y 1. Thus there are two scenarios for which transfers and stable sizes are calculated varying the discount factor, income distribution and coefficient of risk aversion. The effect of 17

18 varying ρ and y is summarized by the need for insurance ψ which is defined as ψ = u (y(1)) u (y(2)) u (y(2)) (18) The need for insurance increases for a mean preserving spread between y(1) and y(2). Moreover, for the same income distribution, a utility function that exhibits higher risk aversion will lead to a higher need for insurance. It is hence a summary measure for environmental uncertainty and attitudes towards risk. The model is calculated for the following parameterizations: β = {0.8, 0.81, 0.82, 0.83, 0.84, 0.85, 0.86, 0.87, 0.88, 0.89, 0.9}, individual income y and the coefficient of risk aversion ρ are varied resulting in the following ten values for the need for insurance ψ = { 0.561, 0.628, 0.646,0.713, 0.732, 0.808, 0.811, 0.875, 0.889, 0.938}. r = 11% and p = 0.5. To keep the computations tractable, for each parametrization we choose capital optimally and then vary ex-ante and ex-post consumption to achieve stability of an arrangement as discussed in Section 2. In each case, results are compared to a model of informal insurance without assets and ex-ante transfers. Figure 1-4 show the largest stable group both with and without the aid of ex-ante transfers, the stable gain for these two group sizes, the distance from first best risksharing for these two group sizes for a range of beginning of period capital, and the maximum ex-ante transfer for every stable group. The two upper panels in Figure 1 and 2 present the results without ex-ante transfers, while the bottom panels show the results with ex-ante transfers. Results for all group sizes are presented in the appendix. It is notable from Figure 1, that capital holding alone in the majority of cases does not increase the maximal size of stable insurance arrangements in a community in the absence of ex-ante transfers. In fact, for some parameters it even decreases it. 18

19 No ex ante transfers No ex ante transfers Groupsize Groupsize No capital constrained capital unconstrained capital discount factor need for insurance Ex ante transfers Ex ante transfers Groupsize 6 4 Groupsize discount factor need for insurance Figure 1: Largest stable group size 19

20 For example when β = 0.82, the maximal stable size is 10 when there is no capital holding and 3 when saving is possible and capital is unconstrained. This is so because the higher benefit that can be achieved from risk-sharing within a small group when mutual savings are possible destabilizes larger groups. Furthermore, without ex-ante transfers, the maximal stable size is significantly smaller than the community in the majority of cases. Once we introduce ex-ante transfers, the largest stable size - at least when capital is unconstrained - increases significantly. Thus, even though the use of ex-ante transfer is minimized and we have imposed a tight bound on the largest ex-ante transfer, large groups can become stable with the aid of relatively small exante transfers. Nonetheless, even with ex-ante transfers the largest stable group size is not always equal to the community for all parameters. This indicates that capital holding and ex-ante transfers can mitigate the limited commitment problem, but cannot completely avoid it. A comparison of the maximal stable group size when capital is constrained and when it is unconstrained also gives interesting results. Fixing the maximal capital stock significantly reduces the largest stable size in a given community even in the presence of ex-ante transfer. This is not due to strategic effects, but is a result of the fact that the expected payoff from being in an insurance arrangement is now decreasing as n grows larger. To put it differently, a group of size 3 with a capital stock of K = 1 is better off in expectation than a group of size 10 with the same capital, because for a group of size 3, K provides much more scope for consumption smoothing than for a group of size 10. Hence, when capital holding is restricted, it is the ex-ante promise-keeping constraint, not the ex-post enforcement constraint that fails. While capital holding alone does not increase group sizes, it does however increase the gain from risk-sharing over and above what can be achieved in groups without asset holding. This can be seen in Figure 2. For the majority of groups, which do not 20

21 100 No Ex ante transfers 100 No Ex ante transfers Stable Gain discount factor Stable Gain No capital restricted capital unrestricted capital need for insurance 100 Ex ante transfers 100 Ex ante transfers Stable Gain Stable Gain discount factor need for insurance Figure 2: Relative payoff in the largest stable group 21

22 require the use of ex-ante transfers to achieve stability, the maximal stable gain under capital holding is larger than without. This is the case for two reasons. Firstly, any group can do at least as well with mutual savings as without since mutual savings enlarge the contract space. Furthermore, the state of the world now consists both of the capital holding of a group and the number of individuals with high income k. This means that in general, stability only requires departure from first-best transfer when assets are low, since as capital increases, so does the penalty for deviation. This further increases the benefits from risk-sharing compared to insurance without asset holding. Finally, combining ex-ante transfers with capital holding dramatically increases the benefits from risk-sharing that can be achieved. The effect of capital holding on feasible transfers is explored in Figure 3. It graphs the difference between groups that hold capital and those that do not, of the distance from first best transfers for the largest stable group in each case. A positive difference indicates that transfers for groups with mutual savings are closer to first-best than for those without capital holding. For most parameter values, the difference is indeed positive and increasing in the beginning of capital period stock, even though we do find some negative values for low values of the capital stock when the capital stock is unrestricted. However, this is largely accounted for by the fact that in this case, the largest stable group with mutual savings is much larger than the largest stable size for those without capital. So while we observe that a group of size 10 with capital is further away from first-best transfers than a group of size 2 without capital, this greater distance from first-best transfers is amply compensated for by the increased insurance offered by a much larger group. Thus capital holding and ex-ante transfers do not only have a positive effect on the benefits from risk-sharing by increasing the opportunity for consumption smoothing across periods and increasing the stable largest stable size in a community but also by rendering feasible transfers, which are closer to first best. 22

23 No Ex ante transfers, restricted capital No Ex ante transfers, unrestricted capital Distance capital 2 Distance discount factor capital need for insurance Ex ante transfers, restricted capital Ex ante transfers, unrestricted capital Distance Distance capital discount factor capital need for insurance Figure 3: Distance from first-best transfers in the largest stable group 23

24 Finally, Figure 4 shows the maximum ex-ante transfers required for each group size to achieve stability. Unsurprisingly, the maximum ex-ante transfer is increasing in group size, indicating that the stability of larger groups is more easily threatened than that of smaller groups. As is discussed further in the appendix, this is also due to the fact that ex-ante transfers make intermediate group sizes stable so that the largest stable group has to operate close to first-best risk-sharing in order to be robust to deviations by sub-groups. This then requires larger ex-ante transfers. Still, ex-ante transfers do not hit their upper bound of y 1 for most group sizes, showing that even relatively small ex-ante transfers can have a profound impact on the benefits of risk-sharing. 24

25 maximum ex ante transfer groupsize discount factor maximum ex ante transfer groupsize need for insurance 0.9 Figure 4: Maximum ex-ante transfer by beginning of period capital 25

26 4 Empirical Example: Funeral Insurance in Ethiopia We now turn to an empirical example to examine whether any of the numerical results presented in the previous section are borne out in practise. To assess this, we use a unique data set on funeral insurance in Ethiopia. These data originate from a number of communities in rural Ethiopia, studied as part of the Ethiopian Rural Household Survey (ERHS). This survey has been collecting panel data on households and communities since 1989, focusing on 15 communities from across the country. In this study - referred to as the Funeral Insurance Survey (FIS) in the following - the data are from a sub-sample of funeral societies in these villages for which key figures could be interviewed in the village. In total, detailed data has been collected on 78 funeral societies in seven villages - about half the number of funeral societies present in these villages. In two villages, the data were matched to the households in the household survey, allowing some more detailed analysis. The villages in question consist of the communities in Sirbana Godeti and Tirufe Kechema, two relatively prosperous villages in central Ethiopia. In Ethiopia, the funeral associations are known as iddir - associations that ensure payout in cash and in kind at the time of a funeral for a deceased member of the family of a member of the group. Membership is clearly defined, with written lists, and by no means do these groups consist of loose, rapidly changing associations of people. Payments are made when members incur costs related to funerals related to the death of a well-defined set of relatives. The actual payout is conditional on the relationship of the member to the deceased: for example, the payment for a spouse is typically different from the payout for a child or for uncles and aunts. The insurance groups are not only remarkable for their functioning; their membership is also widespread. In the sample of 15 Ethiopian villages (the ERHS), it was found 26

27 that about 80 % of households were members of at least one iddir. Payouts occur in cash and in kind as well as in the form of labour services. The insurance groups have written statutes, bylaws and records of contributions and payouts. The rules define membership procedures, payout schedules, contributions and also a set of fines and other measures for non-payment of contributions. The insurance that these groups provide is substantial with payouts at approximately $20 on average per group, which provides important protection for the insured households. While it is hard to estimate the full cost of a funeral, it is certain that they form a significant proportion of a month s income. The average cash payout per iddir is about 40% of monthly household consumption, so iddirs are crucial to help households to cover these expenses. There are three clearly distinguishable types of insurance groups in the survey, ones that make contributions only when a funeral occurs, those that in addition collect a small regular payment, and those that only collect regular payments from their members, who then make a claim to the insurance fund when they incur burial expenses. 8 A consequence of this is that the latter two types of insurance groups retain substantial savings. In the sample, asset holdings were on average about 1900 birr ($217). Average contributions for those charging a regular contribution were 1.64 birr per month. There is also a substantial entrance fee for anyone wanting to join: about 42 birr, or 25 times the average monthly contribution. The average group size of funeral societies is about 85. All the groups are substantially smaller than the community. These findings are summarized in Table 1. There are a number of predictions arising from the simulations in the previous section and we now proceed to examine whether we can find any evidence for these in our 8 Only one type is usually present in a village but there is some variation within villages. 27

28 Mean Median Standard deviation FIS Size of Iddir Monthly contribution Payout Assets ETB 9 = $ 1 (Source: FIS) Table 1: Descriptive Statistics of the Iddir sample: 1. Insurance groups that hold assets and charge ex-ante transfers offer larger coverage than those that do not. 2. With unrestricted capital holding, insurance groups that hold assets and charge ex-ante transfers should on average be larger than those that do not. 3. With restricted capital holding, insurance groups that hold assets and charge ex-ante transfers may be smaller than those that do not follow this practise, but will still offer higher coverage on average. The three types of insurance arrangements found in the survey vary in the degree to which they make use of ex-ante transfers. In the first type, only ex-post transfers, when a shock occurs, are used. In the second type a mixture of ex-post and exante transfers is used. Finally, the third type uses largely ex-ante transfers. Our model predicts that the timing of transfers, i.e. whether they are made ex-ante or ex-post should be correlated with the coverage offered by an insurance arrangement. Furthermore, if capital holding is not restricted, we would also expect to find that insurance groups are larger, the more they can make use of ex-ante transfers. The relationship between the timing of payment and both group size and coverage is shown in Figure 5. In accordance with our simulations, we indeed find a positive albeit small 28

29 relationship for both variables, with the effect of timing of contributions on payouts substantially larger than the effect on group size. 9 Furthermore, our simulations show a relationship between the upper bound on capital holding and both group size and the extent of coverage offered, i.e. if a group is not limited in the amount of capital it can hold, we would expect it to be larger and offer more coverage. Obviously, there is no way of knowing what the upper bound is on capital holding, since we only know what the current assets of a group are, which can vary substantially through time. Nonetheless, assuming that groups that hold a larger amount of current assets are less limited in the maximum amount of capital they can hold than those with fewer assets, we plot the relationship between per capita assets and group size and the extent of payout in Figure 6. The relationship between group size and current per capita assets is very small and slightly negative, but the plot for current per capita assets and the size of payouts shows a strong positive relationship broadly in line with the predictions from the simulation section. Obviously, this is not to say that the timing of transfers and the assets a group holds are the only predictors of group size and the extent of coverage, and that such factors as the composition and wealth of members do not matter, but it does show that irrespective of such factors the general pattern found in the data is similar to that predicted by our model. Finally, our model predicts that the performance of insurance groups can be improved upon through the design of transfer schemes alone rather than by resorting to other forms of incentives, which have been described as social pressure. In the literature on assortative matching (for example Ghatak [8] and Hoff [11]), it has been asserted that such social pressure may be particularly large among people of similar ethnic origin or kinship. While this is by no means to say that social pressure is not 9 Of course, the size of payouts is only an imperfect measure of the actual coverage of the insurance arrangements, but in the absence of precise figures on the cost of a funeral, we use it as a proxy for coverage. 29

30 group size timing of contributions size of payout timing of contributions Figure 5: Group size and coverage by timing of contributions 30

31 group size per capita assets size of payout per capita assets Figure 6: Group size and coverage by asset-holding 31

32 important in the formation and stability of funeral insurance groups, it is interesting to note that in contrast to other studies, analysis of group composition in the matched data of households and groups in Tirufe and Sirbana found only limited evidence of matching as shown in Dercon et al.[2]. In both these villages, the third type of insurance arrangement is prevalent. In other words, this may suggest that these more formal organizations can afford to allow people from a more diverse background to become members, presumably because the sophisticated transfer schemes can compensate for some of the informational and enforcement advantages of social and geographic proximity. 5 Conclusion This paper has shown that in the presence of limited commitment the combination of mutual savings and ex-ante transfers allows higher quality of coverage and increases the maximal stable size of an insurance arrangement, particularly when the amount of savings that can be communally held is not restricted. An empirical example using data on funeral insurance groups in rural Ethiopia shows that the structure and performance of these groups is similar to that predicted by our model. This suggests that savings and insurance can be structured to complement each other in order to increase the benefits from informal insurance. References [1] O. Attanasio and J.-V. Rios-Rull, Consumption smoothing in island economies: Can public insurance reduce welfare?, European Economic Review 44 (2000),

33 [2] S. Dercon, T. Bold, J. de Weerdt, and A. Pankhurst, Group-based funeral insurance in Ethiopia and Tanzania, Centre for the Study of African Economies, Working Paper Series 27 (2004). [3] S. Dercon and P. Krishnan, Risk-sharing and public transfers, Economic Journal (2003). [4] M. Fafchamps and S. Lund, Risk-sharing networks in rural Phillipines, Journal of Development Economics 71 (2003), [5] C. Gauthier, M. Poitevin, and P.Gonzalez, Ex ante payments in self-enforcing risk-sharing contracts, Journal of Economic Theory 76 (1980), [6] G. Genicot and D. Ray, Group formation in informal risk-sharing arrangements, Review of Economic Studies 70 (2003), [7] P. Gertler and J. Gruber, Insuring consumption against illness, American Economic Review 92 (2002), [8] M. Ghatak, Group lending, local information and peer selection, Journal of Development Economics 60 (1999). [9] K. Gobert and M. Poitevin, Non-commitment and savings in dynamic risksharing contracts, [10] F. Grimard, Household consumption smoothing through ethnic ties: Evidence from Cote d Ivoire, Journal of Development Economics 53 (1997), [11] K. Hoff, Informal insurance and the poverty trap, [12] N.R. Kocherlakota, Implications of efficient risk-sharing without commitment, Review of Economic Studies 63 (1996),

34 [13] E. Ligon, J.P.Thomas, and T. Worrall, Mutual insurance, individual savings and limited commitment, Review of Economic Dynamics 3 (2000), [14] E. Ligon, J.P. Thomas, and T. Worrall, Informal insurance arrangements with limited commitment: Theory and evidence from village economies, Review of Economic Studies 69(1) (2002), [15] R.Murgai, P. Winters, E. Sadoulet, and A. de Janvry, Localized and incomplete mutual insurance, Journal of Development Economics 67 (2002), [16] R.M. Townsend, Risk and insurance in village India, Econometrica 62 (1994), [17] C. Udry, Risk and insurance in a rural credit market: An empirical investigation in northern Nigera, Review of Economic Studies 61 (1994),

35 6 Appendix In this appendix, we present results from our simulations for all group sizes in a community of size 10. In the tables below we present the stable gain and the maximum ex-ante transfer for all stable groups without capital holding and restricted and unrestricted capital holding. The results conform with the discussion in Section 3 and are robust in the sense, that for the vast majority of parameterizations groups that hold capital offer a greater gain from insurance than those that do not irrespective of the presence of ex-ante transfers. Another interesting property deserving some discussion is that with ex-ante transfers and capital holdings, the set of stable sizes tends to be connected, while this is not generally the case for groups that do not hold capital. The fact that so many groups of intermediate size are stable in the presence of ex-ante transfers has the added benefit that the largest stable group has to be very close to first-best risk-sharing in order to be robust to deviations, which further increases the benefits from risk-sharing. Capital No Capital constrained unconstrained n\β = 0.81 stable gain τ ea stable gain τ ea stable gain 2 70 % 0 % 69 % 0% 55 % % 25 % % 40 % % 44 % % 52% % 56 % % 60 % % 64 % % 68 % - 35

36 Capital No Capital constrained unconstrained n\β = 0.82 stable gain τ ea stable gain τ ea stable gain 2 70 % 0 % 69 % 0% 55 % % 25 % % 40 % % 44 % % 52% % 56 % % 60 % % 64 % % 68 % 89 % Capital No Capital constrained unconstrained n\β = 0.83 stable gain τ ea stable gain τ ea stable gain 2 72 % 0 % 71 % 0% 54 % % 25 % % 40 % % 48 % % 52% % 56 % % 64 % % 68 % % 72 % - Capital No Capital constrained unconstrained n\β = 0.84 stable gain τ ea stable gain τ ea stable gain 2 75 % 0 % 73 % 0% 54 % % 25 % 69 % % 44 % % 52 % % 52% % 60 % % 64 % % 72 % % 76 % - 36

37 Capital No Capital constrained unconstrained n\β = 0.85 stable gain τ ea stable gain τ ea stable gain 2 76 % 0 % 76 % 0% 54 % % 0 % 72 % % 44 % % 52 % % 56% % 60 % % 64 % % 72 % % 76 % - Capital No Capital constrained unconstrained n\β = 0.86 stable gain τ ea stable gain τ ea stable gain 2 78 % 0 % 78 % 0% 54 % % 0 % 72 % % 48 % % 56 % % 56 % % 64 % % 72 % % 76 % % 80 % - Capital No Capital constrained unconstrained n\β = 0.87 stable gain τ ea stable gain τ ea stable gain 2 81 % 0 % 81 % 0% 54 % % 0 % 72 % % 48 % % 56 % % 60 % % 64 % % 72 % % 76 % % 80 % - 37