Risk sharing contracts with private information and one-sided commitment

Abstract

In a repeated unobserved endowment economy in which agents negotiate long-term contracts with a financial intermediary, we study the risk sharing implications of the interaction between incentive compatibility constraints (due to private information) and participation constraints (due to one-sided commitment). In particular, we assume that after a default episode, agents consume their endowment and remain in autarky forever. We find that once they are away from autarky today, if the probability of drawing the highest possible endowment shock is sufficiently small, the optimal contract prevents agents from reaching autarky tomorrow and, thus, from being “impoverished.” Moreover, an invariant cross-sectional distribution of lifetime utilities (or values) exists. A numerical example shows that the mass of agents living in autarky can be zero in the limit.

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Fig. 1
Fig. 2

Notes

  1. 1.

    Thomas and Worrall (1988) informally mention this result, although they perform their analysis assuming that the principal can also renege on the contract.

  2. 2.

    In related contexts, Hertel (2004) and Broer et al. (2017) consider both private information and limited commitment (or enforcement). Broer et al. (2017) study consumption risk sharing in a similar environment with persistent shocks, public insurance and ex ante participation constraints, whereas Hertel (2004) studies risk sharing contracts between two risk-averse agents, as in Kocherlakota (1996). Other papers that study the interaction between private information and limited commitment, but in different contexts, include Sleet and Yeltekin (2001) and Ales et al. (2014).

  3. 3.

    When private information is the sole friction in the model, Phelan (1998) argues that the crucial assumption to generate the immiseration result rests on preferences.

  4. 4.

    This result follows directly from the restrictions on continuation values and transfers that arise from the constraints in the recursive problem.

  5. 5.

    This result is reminiscent of the literature on dynamic risk sharing contracts with private information. In general, whenever a lower bound on continuation values is present, it is not an absorbing state. See, for example, Atkeson and Lucas (1995) and Wang (1995), among others.

  6. 6.

    A non-exhaustive list includes Thomas and Worrall (1988), Kocherlakota (1996), Attanasio and Ríos-Rull (2000), Kehoe and Levine (2001), Ligon et al. (2002), Krueger and Perri (2006), Krueger and Perri (2011), Tian and Zhang (2013) and Laczó (2014). See also Gobert and Poitevin (2006) who allow for savings in a model of dynamic risk sharing with limited commitment.

  7. 7.

    To be precise, neither contribution normalizes lifetime utility and profits by \((1-\delta )\). It is easy to verify that this normalization is innocuous, not altering the results in Ljungqvist and Sargent (2012) and Thomas and Worrall (1990).

  8. 8.

    In his model, this lower bound is endogenized by assuming that the outside option is the value of signing a long-term contract with another financial intermediary, which is determined in equilibrium. The main messages of the paper, i.e., some risk sharing occurs and a non-degenerate distribution of values exists in the limit, would follow if this lower bound were treated exogenously.

  9. 9.

    In contrast with this paper, Phelan (1995) simplifies his framework along two dimensions. First, for ease of exposition, he considers an economy with \(n=2\). Second, he assumes constant absolute risk aversion preferences.

  10. 10.

    One can show that the optimal contract at \(v=w_{\mathrm{aut}}\) maximizes the principal’s profits from transacting with \(j=n\), \(-(1-\delta ) b_n + \delta W(w_n)\), subject to (PC) binding at \(j=n\), \((1-\delta ) u(\theta _n + b_n) + \delta w_n = (1-\delta ) u(\theta _n) + \delta w_{\mathrm{aut}}\).

  11. 11.

    Indeed, fix \(w_n> w_{n-1}> \cdots > w_1 \ge w_{\mathrm{aut}}\), and let \(b_1\) be free to adjust. Then, pick \(b_2,\ldots , b_n\) (as “functions” of \(b_1\)) such that downward incentive compatibility constraints are binding, i.e., \((1-\delta )u(\theta _j + b_j) + \delta w_j = (1-\delta )u(\theta _j + b_{j-1}) + \delta w_{j-1}\). Hence, standard results imply that other incentive compatibility constraints are not violated. Finally, keeping in mind that \(b_2,\ldots , b_n\) are “continuous” and “monotone” in \(b_1\), adjust \(b_1\) such that (PK) is satisfied at \(v = w_{\mathrm{aut}}\).

  12. 12.

    This claim is not straightforward. Since \(\pi _n = 0\), the financial intermediary’s profits and (PK) do not depend on \(w_n\) and \(b_n\). Hence, one needs to show that there exist values for \(w_n\) and \(b_n\) that preserve (IC) and (PC). Given that \(b_{n-1} \le 0\) by Lemma 2, the concavity of u implies that this is accomplished by setting \(w_{n} = w_{n-1}\) and \(b_n = b_{n-1}\).

  13. 13.

    To do so, we define a relaxed version of the program by allowing the principal to choose a joint distribution probability over transfers and continuation values. In other words, we convexify the program so that the maximization problem is well defined and can be cast as a Lagrange functional. Then, we show that W is differentiable and, if \(\delta \) is high enough, strictly concave. Hence, the solution of the relaxed version must be deterministic and, thus, feasible within the original program.

  14. 14.

    A standard result states that the concavity of u implies that it is sufficient to account for local upward and downward (IC) constraints.

  15. 15.

    Indeed, suppose that (PC) binds at some \(k > 1\); then, Proposition 1 and (IC) imply that

    $$\begin{aligned} (1-\delta )u(\theta _k) + \delta w_{\mathrm{aut}} = (1-\delta )u(\theta _k + b_k) + \delta w_k > (1-\delta )u(\theta _{k} + b_{j}) + \delta w_{\mathrm{aut}}, \hbox { for all } j \ne k. \end{aligned}$$

    Hence, \(b_j < 0\) for \(j \ne k\). But Lemma 1 implies that \(b_j \ge 0\) for \(j=1,\ldots ,k-1\), a contradiction.

  16. 16.

    Note that \(w_{\max }\) is an absorbing state but is never reached along the optimal contract path.

  17. 17.

    Indeed, suppose that \(w_n \le v\). Equation (2) in “Appendix B,” used to prove Proposition 1, and the strict concavity of W imply that \(w_j = v\) and \(\varsigma _{j} = \xi _j =0\) for all j. In this case, a well-known result states that the principal would like to spread continuation values, \(w_1< v < w_n\), yielding a contradiction.

  18. 18.

    Note that since \(W'(w_{\mathrm{aut}}) > 0\) for \(\pi _n\) small enough, both the financial intermediary and the agents are better off by signing an optimal contract with \(v_0 > w_{\mathrm{aut}}\) rather than \(v_0 = w_{\mathrm{aut}}\). Hence, none of the agents would ever be in autarky.

  19. 19.

    “Appendix C.2” discusses to what extent Propositions 1 and 2 can be generalized if we assume that the agents and the principal do not have the same discount factors.

  20. 20.

    See, for instance, the discussion in Farhi and Werning (2007), page 383.

  21. 21.

    There are, of course, other differences. For example, regarding preferences, Phelan (1995) assumes CARA preferences, whereas we assume that \(\delta \) is high enough in this section.

  22. 22.

    To solve the problem numerically, we use the value function iteration method with interpolation. We consider a tolerance of \(10^{-6}\) for convergence. In all experiments, we allow for an equally spaced grid between \(w_{\mathrm{aut}}\) and \({\overline{w}}\) with 2001 gridpoints. Because \(w_{\mathrm{aut}}\) varies with some parameters of the model, the step size between adjacent gridpoints varies with the experiment. In each iteration, we use the sequential quadratic programming algorithm embedded in the fmincon command in MATLAB. In addition, we interpolate the value function using the shape-preserving piecewise cubic method embedded in the interp1 command in MATLAB.

  23. 23.

    Note that the optimal continuation values in the model with (PC) display kinks a bit above the lower bound \(w_{\mathrm{aut}}\), although barely visible for \(w_1\) in Fig. 1. These kinks are located at the threshold value of v, such that (PC) ceases to bind at \(j=1\) for higher promised values. Above this threshold value, the possibility that (PC) might bind tomorrow at \(j=1\) is encoded in the value function of the principal, generating other kinks in the optimal contract, such as the clearly visible one for \(w_1\). It is worth to mention that the clearly visible kinks for \(w_1\) in the models with and without (PC) are not located at the same v. In the latter case, the kink is associated with the threshold value of v, such that \(w_1 \ge w_{\mathrm{aut}}\) ceases to bind for higher promised values.

  24. 24.

    In our numerical example, the autarky state is reached after one period, but convergence could take longer if the initial promised value were set higher.

  25. 25.

    Note that, in our numerical example, this lower bound is only a bit above the autarky value. Therefore, it could be possible that this small difference is simply a numerical error. To address this concern, we compute lower bounds for specifications with 2001, 5001, 10001 and 20001 gripoints, as well as different interpolation methods, such as linear, piecewise cubic, cubic convolution and spline. We find that lower bounds vary little in between − 0.57358 and − 0.57338, still strictly above the autarky value of − 0.57422. Hence, it is unlikely that the convergence to a value strictly above the autarky value is stemming from a numerical error.

  26. 26.

    Recall that the value of autarky changes with \(\pi _j\), \(j=1,\ldots ,n\). In these cases, the autarky values are equal to \(-\,0.4645\) and \(-\,0.3548\), respectively.

  27. 27.

    Fernandes and Phelan (2000) show that serially correlated shocks increase the dimensionality of the state space. In particular, the number of state variables is \(n + 1\) with shocks that follow a first-order Markov process rather than one with shocks that are i.i.d. In this case, both the number of constraints and the number of programs that need to be computed increase substantially with the number of shocks. Hence, the assumption that \(n\ge 3\), which is crucial to deliver our main conclusions, poses some computational challenges under the empirically relevant case of persistent income shocks. See also Doepke and Townsend (2006).

  28. 28.

    Recall that strict concavity of W, \(w_n \in (w_{\mathrm{aut}}, w_{\max })\) and \(\lim _{v \rightarrow w_{\max }} W'(v) = - \infty \) imply that \(W'(w_n) > -\,\infty \). Since this result is valid for all distributions of \(\{\pi _j\}_{j=1}^n\), including those with \(\pi _n \rightarrow 0\), then \(\lim _{\pi _n \rightarrow 0} W'(w_n) > -\,\infty \).

  29. 29.

    Suppose this is not the case; then, the strict concavity of u, \(b_{j+1} \le b_j\) and \(\lambda _{j+1,j} > 0\) imply that

    $$\begin{aligned} (1-\delta )[u(\theta _{j} + b_{j}) - u(\theta _{j} + b_{j+1})] > (1-\delta )[u(\theta _{j+1} + b_{j}) - u(\theta _{j+1} + b_{j+1})] = \delta (w_{j+1} - w_{j}). \end{aligned}$$

    Hence,

    $$\begin{aligned} (1-\delta )u(\theta _{j} + b_{j}) + \delta w_{j} > (1-\delta )u(\theta _{j} + b_{j+1}) + \delta w_{j+1}, \end{aligned}$$

    and thus, \(\lambda _{j,j+1} = 0\), a contradiction.

  30. 30.

    Other papers in the literature also allow for different discount factors in related environments. Wang (2005), for instance, considers a model of dynamic risk sharing with private information and costly state verification. Opp and Zhu (2015) study the dynamics of long-term contracts when the agent is impatient in a general setting that nests Thomas and Worrall (1988), among others.

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Correspondence to Eduardo Zilberman.

Additional information

We are especially indebted to David Martimort, whose comments and discussions were invaluable. We are also grateful for helpful comments from Carlos E. da Costa, Lucas Maestri, Thierry Verdier, two anonymous referees, and participants at conferences and seminars. The views expressed in this paper are those of the authors and do not necessarily reflect the position of the Central Bank of Chile. All errors are ours.

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Appendices

Appendix A: Continuous types

In this appendix we show that, if \(\theta \) follows a continuous distribution F, with \(f = F' > 0\) and support \([{\underline{\theta }}, {\overline{\theta }}]\), i.i.d. over time and agents, rather than a discrete distribution with finite support, then autarky becomes an absorbing state, such that the principal cannot spread continuation values at all.

The remaining ingredients of the model are the same. Under this stochastic process, the principal’s problem can be rewritten as:

$$\begin{aligned}&W(v) = \max _{\{b(\theta ), w(\theta ) \in [w_{\mathrm{aut}}, w_{\max }]\}_{\theta }} \int _{{\underline{\theta }}}^{{\overline{\theta }}}\left[ -\,(1-\delta )b(\theta ) + \delta W(w(\theta ))\right] f(\theta ) \mathrm{d}\theta \\&\hbox {subject to} \\ \hbox {(PK)}&\int _{{\underline{\theta }}}^{{\overline{\theta }}} [(1-\delta )u(\theta + b(\theta )) + \delta w(\theta )] f(\theta ) \mathrm{d}\theta = v, \\ \hbox {(IC)}&(1-\delta )u(\theta + b(\theta )) + \delta w(\theta ) \ge (1-\delta )u(\theta + b({\hat{\theta }})) + \delta w({\hat{\theta }}), \\&\hbox { for all } \theta , {\hat{\theta }} \in [{\underline{\theta }}, {\overline{\theta }}]\times [{\underline{\theta }}, {\overline{\theta }}], \\ \hbox {(PC)}&(1-\delta )u(\theta + b(\theta )) + \delta w(\theta ) \ge (1-\delta )u(\theta ) + \delta w_{\mathrm{aut}}, \hbox { for all } \theta \in [{\underline{\theta }}, {\overline{\theta }}], \end{aligned}$$

where \(w_{\mathrm{aut}} = \int _{{\underline{\theta }}}^{{\overline{\theta }}} u(\theta )f(\theta )\mathrm{d}\theta \). This problem is basically the same as the problem described in Sect. 2, except that we substitute summations by integrals in order to deal with continuous types. The remainder of this appendix shows that \(w_{\mathrm{aut}}\) is an absorbing state.

First, note that incentive compatibility requires that \({\hat{\theta }} = \theta \) solves the following problem:

$$\begin{aligned} U(\theta ) = \max _{{\hat{\theta }} \in [{\underline{\theta }}, {\overline{\theta }}]} \left\{ (1-\delta )u(\theta + b({\hat{\theta }})) + \delta w({\hat{\theta }})\right\} , \hbox { for all } \theta \in [{\underline{\theta }}, {\overline{\theta }}]. \end{aligned}$$

A standard lemma implies that (IC) is verified if and only if:

$$\begin{aligned} \hbox {(LIC)}&{\dot{U}}(\theta ) = (1-\delta )u'(\theta + b(\theta )) \hbox { almost everywhere},\\ \hbox {(M)}&b \hbox { is non-increasing}, \end{aligned}$$

where \({\dot{U}}\) is the derivative of U with respect to \(\theta \). The first requirement is the local incentive compatibility constraint, which follows from applying the envelope theorem to the problem above [see Milgrom and Segal (2002)]. The second requirement is the monotonicity condition that guarantees that (IC) is globally satisfied.

Second, note that given incentive compatibility, (PC) can be rewritten as

$$\begin{aligned} U(\theta ) \ge (1-\delta )u(\theta ) + \delta w_{\mathrm{aut}}, \hbox { for all } \theta \in [{\underline{\theta }}, {\overline{\theta }}], \end{aligned}$$

and define \(\Omega \) to be the set of endowments for which the participation constraint is binding. Formally,

$$\begin{aligned} \Omega = \{\theta \hbox { } | \hbox { } U(\theta ) = (1-\delta )u(\theta ) + \delta w_{\mathrm{aut}}\}. \end{aligned}$$

The following proposition shows that the interaction among (IC), (PC) and (PK) eliminates the possibility of risk sharing when \(v = w_{\mathrm{aut}}\). Hence, \(w_{\mathrm{aut}}\) is an absorbing state, meaning that once in autarky, the agent remains in autarky almost surely. Algebraically, at \(v=w_{\mathrm{aut}}\), \(w(\theta ) = w_{\mathrm{aut}}\) and \(b({\theta }) = 0\) almost everywhere. Therefore, \(W(w_{\mathrm{aut}}) = 0\).

Proposition 3

If \(v = w_{\mathrm{aut}}\), then \(w(\theta ) = w_{\mathrm{aut}}\) and \(b({\theta }) = 0\) almost everywhere.

Proof

Note that at \(v = w_{\mathrm{aut}}\), (PC) is binding almost everywhere. Otherwise, (PK) would imply that

$$\begin{aligned} w_{\mathrm{aut}}= & {} \int _{{\underline{\theta }}}^{{\overline{\theta }}} [(1-\delta )u(\theta + b(\theta )) + \delta w(\theta )] f(\theta ) \mathrm{d}\theta \\> & {} \int _{{\underline{\theta }}}^{{\overline{\theta }}} [(1-\delta )u(\theta ) + \delta w_{\mathrm{aut}}] f(\theta ) \mathrm{d}\theta , \end{aligned}$$

which contradicts \(w_{\mathrm{aut}} = \int _{{\underline{\theta }}}^{{\overline{\theta }}} u(\theta )f(\theta )\mathrm{d}\theta \). Hence, almost every \(\theta \in \Omega \).

Let \(\mathring{\Omega }\) be the interior of \(\Omega \), which is clearly not empty when \(v = w_{\mathrm{aut}}\). Note that \( {\dot{U}}(\theta ) = (1-\delta )u'(\theta )\) for all \(\theta \in \mathring{\Omega }\). Hence, for each \(\theta \in \mathring{\Omega }\), (LIC), \(u'' < 0\) and (PC) imply that \(b(\theta ) = 0\) and \(w(\theta ) = w_{\mathrm{aut}}\). \(\square \)

In an environment with continuous types, almost no one draws the highest type. Hence, we conjecture that an extrapolation of Proposition 1 implies that, although an absorbing state, autarky will never be reached in equilibrium by agents who are outside the autarky state. Indeed, the same intuition applies. The impossibility of spreading continuation values in autarky and thus providing some risk sharing should make the autarky state even costlier in an environment with continuous types.

Appendix B: Proof of Proposition 1

In this appendix, we prove Proposition 1. The first-order conditions (FOCs) of the Lagrangian in Sect. 4 with respect to \(w_j\) and \(b_j\) are:

$$\begin{aligned}&\pi _{j}[W'(w_j) + \mu ] = \lambda _{j+1,j} - \lambda _{j,j-1} + \lambda _{j-1,j} - \lambda _{j,j+1} - \varsigma _{j} - \xi _j, \hbox {and} \\&\pi _j[1- \mu u'(\theta _j + b_j)] = (\lambda _{j,j-1} + \lambda _{j,j+1} \\&\quad +\, \varsigma _{j})u'(\theta _j + b_{j}) - \lambda _{j+1,j}u'(\theta _{j+1} + b_j) - \lambda _{j-1,j}u'(\theta _{j-1} + b_j), \end{aligned}$$

respectively. Moreover, \(\mu = - \,W'(v)\) [envelope theorem; see Milgrom and Segal (2002)].

By summing the FOCs with respect to \(w_j\) in j and substituting \(\mu = -\, W'(v)\), one obtains:

$$\begin{aligned}&\sum _{j=1}^n \pi _{j}W'(w_j) + \sum _{j=1}^n \left( \varsigma _{j} + \xi _j\right) = W'(v). \end{aligned}$$
(2)

By using Lemmas 2 and 3, evaluate the equation above at \(v= w_{\mathrm{aut}}\). After rearranging the terms:

$$\begin{aligned}&W'(w_{\mathrm{aut}}) = W'(w_n) + \frac{1}{\pi _n}\sum _{j=1}^n \left( \varsigma _{j} + \xi _j\right) , \end{aligned}$$

where \(w_n\), with a slight abuse of notation, is the optimal continuation value for type-n at \(v = w_{\mathrm{aut}}\). Also, the multipliers are evaluated at \(v = w_{\mathrm{aut}}\).

We complete the proof in two steps. First, we show that at \(v = w_{\mathrm{aut}}\), the optimality conditions imply that \(\lim _{\pi _n \rightarrow 0} \sum _{j=1}^n \left( \varsigma _{j} + \xi _j\right) > 0\), such thatFootnote 28

$$\begin{aligned}&\lim _{\pi _n \rightarrow 0} W'(w_{\mathrm{aut}}) = \infty . \end{aligned}$$

Second, we show that the condition above implies that if \(v > w_{\mathrm{aut}}\), then \(w_{\mathrm{aut}}\) is not reachable within a neighborhood of \(\pi _n = 0\), i.e., there exists \({\underline{\pi }}(v) > 0\) such that \(w_j > w_{\mathrm{aut}}\) for all j and for all \(\pi _n < {\underline{\pi }}(v)\).

Step 1. At \(v = w_{\mathrm{aut}}\), \(\lim _{\pi _n \rightarrow 0} \sum _{j=1}^n \left( \varsigma _{j} + \xi _j\right) > 0\).

Proof

We prove that \(\sum _{j=1}^n \left( \varsigma _{j} + \xi _j\right) > 0\) for all distributions \(\{\pi _j\}_{j=1}^n\), including those with \(\pi _n \rightarrow 0\). Suppose by contradiction that at \(v=w_{\mathrm{aut}}\), \(\sum _{j=1}^n \left( \varsigma _{j} + \xi _j\right) = 0\). Hence, \(\varsigma _j = \xi _j = 0\) for all j. Note that the FOCs with respect to \(w_j\) and \(b_j\) (including \(j=n\)) become:

$$\begin{aligned}&\pi _{j}[W'(w_j) + \mu ] = \lambda _{j+1,j} - \lambda _{j,j-1} + \lambda _{j-1,j} - \lambda _{j,j+1}, \hbox {and} \\&\pi _j[1- \mu u'(\theta _j + b_j)] = (\lambda _{j,j-1} +\lambda _{j,j+1})u'(\theta _j + b_{j}) \\&\quad - \, \lambda _{j+1,j}u'(\theta _{j+1} + b_j) - \lambda _{j-1,j}u'(\theta _{j-1} + b_j), \end{aligned}$$

respectively. By summing the FOCs with respect to \(w_j\) in j and substituting \(\mu = - W'(w_{\mathrm{aut}})\), one obtains:

$$\begin{aligned} \sum _{j=1}^{n-1} \pi _{j}W'(w_j) + \pi _n W'(w_n) = W'(w_{\mathrm{aut}}). \end{aligned}$$
(3)

Whether \(\pi _n > 0\) or \(\pi _n \rightarrow 0\) does not matter for the arguments below. We break the analysis into two cases: \(W'(w_{\mathrm{aut}}) < 0\) and \(W'(w_{\mathrm{aut}}) \ge 0\).

Case 1\(W'(w_{\mathrm{aut}}) < 0\). Since W is strictly concave, \(w_{\mathrm{aut}} \le w_1\) and \(w_j \le w_{j+1}\), Eq. (3) implies that \(w_j = w_{j+1} = w_{\mathrm{aut}}\) for \(j=1,\ldots ,n-2\). The FOCs with respect to \(w_j\) (excluding \(j=n\)) become:

$$\begin{aligned} \lambda _{j+1,j} - \lambda _{j,j-1} + \lambda _{j-1,j} - \lambda _{j,j+1} = 0. \end{aligned}$$

Given that \(\lambda _{1,0} = \lambda _{0,1} = 0\), a simple iterative argument implies that \(\lambda _{j+1,j} = \lambda _{j,j+1}\) for all j. Moreover, \(\lambda _{j+1,j} = \lambda _{j,j+1} = 0\).Footnote 29 Hence, the FOCs with respect to \(b_j\) (excluding \(j=n\)) become:

$$\begin{aligned} u'(\theta _j + b_j) = -\frac{1}{W'(w_{\mathrm{aut}})} > 0. \end{aligned}$$

Hence, since \(u'' < 0\), \(b_{j} > b_{j+1}\), for \(j=1,\ldots ,n-2\). But from (IC), \(w_{j} = w_{j+1}\) implies \(b_{j} = b_{j+1}\), a contradiction.

Case 2\(W'(w_{\mathrm{aut}}) \ge 0\). Since W is strictly concave, \(w_{\mathrm{aut}} \le w_1\) and \(w_j \le w_{j+1}\), Eq. (3) implies that \(w_1 = w_{\mathrm{aut}}\). Given that \(\lambda _{1,0} = \lambda _{0,1} = 0\), the FOC with respect to \(w_1\) and the arguments in footnote 29 imply that \(\lambda _{2,1} = \lambda _{1,2} = 0\). Hence, the FOC with respect to \(b_1\) becomes:

$$\begin{aligned} u'(\theta _1 + b_1) = -\frac{1}{W'(w_{\mathrm{aut}})} \le 0, \end{aligned}$$

in contradiction to \(u' > 0\). \(\square \)

Step 2.For each\(v > w_{\mathrm{aut}}\), \(\lim _{\pi _n \rightarrow 0} W'(w_{\mathrm{aut}}) = \infty \)implies that there is\({\underline{\pi }}(v)\)such that\(w_j > w_{\mathrm{aut}}\) for \(j=1,\ldots ,n\)and for all \(\pi _n < {\underline{\pi }}(v)\).

Proof

Suppose by contradiction that at \(v > w_{\mathrm{aut}}\), \(w_j = w_{\mathrm{aut}}\) is an optimal choice for some j. Since \(w_{\mathrm{aut}} \le w_1\) and \(w_j \le w_{j+1}\), it must be the case that \(w_1 = w_{\mathrm{aut}}\). Consider the FOC with respect to \(w_1\) after plugging \(\mu = - W'(v)\) and evaluating at \(w_1 = w_{\mathrm{aut}}\):

$$\begin{aligned} \xi _1 + \varsigma _1 = \pi _{1}[W'(v) - W'(w_{\mathrm{aut}})] + \lambda _{2,1} - \lambda _{1,2}. \end{aligned}$$

Given that the maximization problem is well defined, \(\lambda _{2,1} < \infty \) for all distributions \(\{\pi _j\}_{j=1}^n\), including those with \(\pi _n \rightarrow 0\). Hence, \(\lim _{\pi _n \rightarrow 0} W'(w_{\mathrm{aut}}) = \infty \) and \(\lim _{\pi _n \rightarrow 0} W'(v) < \infty \) (recall that \(v > w_{\mathrm{aut}}\) and W is strictly concave) imply that \(\xi _1 + \varsigma _1 < 0\) in a neighborhood of \(\pi _n = 0\), a contradiction. \(\square \)

Appendix C: Additional results

Within the context of the simple numerical example described in Sect. 5, this appendix reports the optimal contracts and value functions in the models with and without (PC). Then, with some algebra and the simple numerical example, we discuss to what extent our conclusions can be generalized if we assume that the agents and the principal do not have the same discount factors.

C.1     Numerical example: Optimal contracts and value functions

In this section, within the context of the simple numerical example from Sect. 5, we show the optimal contracts and the principal’s value functions that arise in the models with and without (PC).

Figure 3 plots the optimal contract (continuation values and transfers) for each type, along with \(45{^{\circ }}\) lines (dashed lines) and horizontal lines (dotted lines) highlighting the autarky state. Because participation constraints tend to bind for lower values of v, the optimal contracts with and without (PC) are similar for higher values of v. Figure 1 in the main text plots in larger-scale optimal continuation values as functions of v near its lower bound, \(w_{\mathrm{aut}}\).

Fig. 3
figure3

Optimal contract. The top and bottom plots consider the optimal choices of \(w_j\) and \(b_j\) for all j as functions of v, respectively. The left and right plots consider the model with and without (PC), respectively. Full lines, dashed lines and dotted lines represent the optimal contract for each endowment shock j, the \(45{^{\circ }}\) line and the autarky state, respectively

The left graph of Fig. 4 plots the value functions of the principal in the models with and without (PC), represented by the full and dashed lines, respectively. Because differences between them are not visible, the right graph reproduces at a larger scale their shape near the lower bound \(w_{\mathrm{aut}}\). Note that slightly above \(w_{\mathrm{aut}}\), the slope of the value function is positive in the model with (PC) but negative in the model without (PC). Given that spreading continuation values becomes markedly costly near \(w_{\mathrm{aut}}\) once (PC) is accounted for, the principal can increase his profits by promising more consumption to the agent in the future.

Fig. 4
figure4

Value functions of the principal. Full and dashed lines represent the models with and without (PC), respectively. The left graph plots the value functions, whereas the right graph plots them at a larger scale near \(v=w_{\mathrm{aut}}\)

C.2     Different discount factors

Assume that the discount factor of the principal, say \(\beta \), differs from that of the agents, \(\delta \).Footnote 30 Because Lemmas 1 and 2 do not rely on the preferences of the principal, they are still valid even if \(\beta \ne \delta \). Hence, the impossibility of spreading continuation values at \(v = w_{\mathrm{aut}}\) for lower realizations of the endowment shocks, which is the main driving force behind our results, is still present if discount factors are allowed to differ freely. In addition, if we assume that the agents are more patient than the principal, i.e., \(\beta < \delta \), an inspection of the proof of Lemma 3 reveals that this lemma remains valid.

In this section, we discuss to what extent Propositions 1 and  2 can be generalized if \(\beta < \delta \). It turns out that they are fairly robust. Intuitively, if the agents are more patient, the force at play to postpone consumption near autarky is reinforced. This does not mean that our conclusions would not be valid if the opposite holds. Even if \(\beta > \delta \), due to Lemmas 1 and 2, the main driving force behind our analytical results still applies. On the one hand, the impossibility of properly spreading continuation values at the autarky state introduces a force to backload consumption. On the other hand, more-impatient agents introduce a motive to frontload consumption. We provide a numerical example showing that the force to backload dominates near the autarky state. Hence, our conclusions can still be valid even if the principal is more patient.

A close inspection of the online appendix reveals that the intermediate steps used to derive a Lagrangian functional for the principal’s problem still apply. Hence, the Lagrangian would be analogous to that in the main text, except that \(\beta \) substitutes \(\delta \) in the principal’s objective function. Following the steps outlined in “Appendix B”, one can sum the first-order conditions with respect to \(w_j\), substitute the envelope condition \(\mu = - W'(v)\), and use Lemma 2 to evaluate the resulting equation at \(v = w_{\mathrm{aut}}\) to obtain

$$\begin{aligned}&W'(w_{\mathrm{aut}}) = \frac{\pi _n \frac{\beta }{\delta }}{1-(1-\pi _n)\frac{\beta }{\delta }} W'(w_n) + \frac{1}{1-(1-\pi _n)\frac{\beta }{\delta }} \sum _{j=1}^n (\varsigma _{j} + \xi _j), \end{aligned}$$
(4)

where \(w_n\), with a slight abuse of notation, is the optimal continuation value for type-n at \(v = w_{\mathrm{aut}}\). Multipliers are also evaluated at \(v = w_{\mathrm{aut}}\).

If \(\delta = \beta \), the equation above collapses to the one in the main text. Assume \(\beta < \delta \) instead, such that Lemma 3 is still valid. In this case, one can follow similar steps as those in “Appendix B” to show that

$$\begin{aligned}&\lim _{\pi _n \rightarrow 0} W'(w_{\mathrm{aut}}) = \frac{1}{1-\frac{\beta }{\delta }} \sum _{j=1}^n (\varsigma _{j} + \xi _j) > 0. \end{aligned}$$

Since this limit is not infinity, this result does not generalize Proposition 1. However, as we argue in the main text, it is the possibility to make positive profits by increasing continuation values that prevents the principal from promising the autarky value in the next period. Hence, this force at play behind Proposition 1 is still present in this case.

Proposition 2 states that a non-degenerate invariant distribution exists. The part of the proof regarding the existence of an invariant distribution easily generalizes to different discount factors. The non-degeneracy follows from Lemma 3, which guarantees that none of the states reached is absorbing.

In principle, as Eq. (4) highlights, different values of \(\beta \), \(\delta \) and \(\pi _n\) could make the slope of profits, \(W'\), positive near \(v = w_{\mathrm{aut}}\), even if the principal is more patient than the agents. Indeed, as we show below, this is true in our simple numerical example for a wide range of discount factors. In this context, the main conclusions of this paper still apply, even if \(\beta > \delta \). As we argued above, the motive to postpone consumption near the autarky state can dominate the motive to anticipate it due to impatience.

Figure 5 (top graphs) reproduces the behavior of optimal continuation values, as functions of v, near \(w_{\mathrm{aut}}\) for \(\beta =0.7\) (left graphs) and \(\beta =0.9\) (right graphs). Recall that we set \(\delta = 0.8\). The figure also plots \(45{^{\circ }}\) lines (dashed lines) and horizontal lines (dotted lines) highlighting the autarky state. Note that continuation values are still strictly increasing functions of promised values, although not clearly visible in the right graph. Figure 5 (bottom graphs) also shows the trajectory of promised values over time after the realization of a sequence of the lowest endowment shock \(\theta _1\). The initial promised value \(v_0\) was such that it maximizes the principal’s profits in the benchmark parametrization with \(\beta = 0.8\). Again, promised values converge to a lower bound strictly above the autarky value in finite time. These numerical results generalize for a wide range of values for \(\beta \).

Fig. 5
figure5

Optimal continuation values near \(w_{\mathrm{aut}}\). The top plots consider the optimal choices of \(w_j\) for all j as functions of v near \(v= w_{\mathrm{aut}}\). Full lines, dashed lines and dotted lines represent \(w_j\) for each endowment shock j, the \(45{^{\circ }}\) line and the autarky state, respectively. The bottom plots consider the path of promised values over time after the realization of a finite sequence of the lowest endowment shock \(\theta _1\). The left and right plots consider the model with \(\beta =0.7\) and \(\beta =0.9\), respectively

Fig. 6
figure6

Comparative statistics: \(\beta \). The top graph plots the numerical right-derivative of W at \(v=w_{\mathrm{aut}}\) as a function of \(\beta \). The bottom graph plots the limit aggregate transfers as a function of \(\beta \)

Hence, even if the principal is more patient than the agents, the conclusions from Lemma 3 and Proposition 1 still apply in the context of this simple numerical example. Namely, some intertemporal trade occurs between the financial intermediary and the agents who draw \(\theta _n\), and the optimal contract prevents the agents from reaching the autarky state.

Indeed, Fig. 6 (top graph) plots the numerical right-derivative of W at \(v = w_{\mathrm{aut}}\) as a function of \(\beta \), ranging from 0.7 to 0.9, with step size 0.01. Recall that we set \(\pi _n = 0.2\). In all cases, the slope of \(W(w_{\mathrm{aut}})\) is positive, implying that the mass of agents living in autarky is zero according to our numerical simulations. This slope increases as the financial intermediary becomes more impatient. Hence, as the discussion above emphasizes, the force at play behind our main results is stronger for lower values of \(\beta \).

Figure 6 (bottom graph) also plots aggregate transfers, computed using the limit invariant joint distribution of types and values. Transfers decrease with the principal’s degree of patience. Moreover, there exists \(\beta \) above \(\delta \), such that aggregate transfers are zero. In particular, \(\beta \approx 0.82\). Hence, if we allow the interest rate embedded in the principal’s discount factor to adjust to equalize aggregate consumption and aggregate endowment, a market-clearing interest rate along with a stationary equilibrium arises in the context of this simple numerical example without changing our main conclusions. Again, as the main driving force behind our results is still present if discount factors are allowed to differ freely, this numerical result is likely to be valid more generally.

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Zilberman, E., Carrasco, V. & Hemsley, P. Risk sharing contracts with private information and one-sided commitment. Econ Theory 68, 53–81 (2019). https://doi.org/10.1007/s00199-018-1112-1

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Keywords

  • Risk sharing contracts
  • private information
  • One-sided commitment

JEL Classification

  • D31
  • D82
  • D86