1 Risk sharing with limited commitment and storage: Model
First, I review the risk-sharing model with storage under limited commitment, following the model in Ábrahám and Laczó (2018).
1.1 Set up
1.1.1 Model
As in the basic limited commitment model, a social planner maximized the discounted sum of utilities of households, subject to resource constraints and participation constraints. But now, the social planner is allowed to publicly save resources across times, and also, each household can privately save for future periods. Here,
- public storage is observed by households, and households cannot access once they leave the risk-sharing network (hence the public storage does not affect the autarky value), and
- private storage is unenforceable in that social planners cannot use the private storage for resource allocation, and also, households can access the private storage under autarky too.
Hence, the social planner solves the following maximization problem. \begin{aligned} &\max_{\{c_{it}(s^t)\}} \sum_i \lambda_i \sum_{t = 1}^{\infty} \sum_{s^t} \delta^t \pi(s^t) u(c_{it}(s^t)) \\ \text{subject to} &\sum_i c_{it} (s^t) \le \sum_i y_{it}(s_t) + (1+r) B \left(s^{t - 1} \right) - B\left(s^t \right)\quad \forall s^t, \forall t \\ &\sum_{r = t}^{\infty} \sum_{s^r} \delta^{r - t} \pi(s^r | s^t) u(c_{ir}(s^r)) \ge \tilde{U}_{i}^{aut}(s_t) \quad \forall s^t, \forall t, \forall i, \\ &B(s^t) \ge 0 \quad \forall s^t, \forall t \end{aligned} where the similar notations are used as in the case without storage, with a few exceptions that, B(s^t) is a public storage at the state history s^t, and \tilde{U}_i^{aut} is the autarky value with private storage. Also, for simplicity, assume that the income process is i.i.d..
Note that, here I do not consider the possibility of households relying on private storage while they are in the risk-sharing network. In principle, they can use private saving, and hence to the list of constraints, I would need to add intertemporal optimality conditions (that is, Euler equations of each household). However, Ábrahám and Laczó (2018) shows that private storage does not matter when public asset accumulation is optimal (Proposition 4). Therefore, I can safely ignore the possibility of private storage from the constraints, which simplifies the expression and also numerical solution greatly.
Using the recursive form in Marcet and Marimon (2019), the Lagrangian can be written as \begin{aligned} \mathcal{L} = \sum_{t = 1}^{\infty} \sum_{s^t} \beta^t Pr(s^t) \left\{ \sum_i \left[ M_i(s^t) u(c_i(s^t)) - \mu_i(s^t) U^{aut}_i(s^t) \right] + \gamma(s^t) \left( \sum_i (y_i(s_t) - c_i(s^t)) + (1 + r) B(s^{t - 1}) - B(s^t) \right) + \rho(s^t) B(s^t) \right\}, \end{aligned} where \mu_i(s^t), \gamma(s^t), and \rho(s^t) are Lagrangian multipliers of participation constraints, the resource constraint, and the non-negativity constraint of public storage, respectively. The variable, M_i(s^t), captures the evolution of the Pareto weights, and it evolves as M_i(s^t) = M_i(s^{t - 1}) + \mu_i(s^t).
The first-order conditions are M_i(s^t) u'(c_i(s^t)) = \gamma(s^t) \quad \forall i, which implies that, for any two households, i and j, the following holds: \frac{u'(c_i(s^t))}{u'(c_j(s^t))} = \frac{M_j(s^t)}{M_i(s^t)}.
From now on, I consider the case where there are two households in a village. Letting x(s^t) \equiv \frac{M_2(s^t)}{M_1(s^t)}, I obtain x(s^t) = \frac{u'(c_1(s^t))}{u'(c_2(s^t))}. Defining, \nu_i(s^t) = \frac{\mu_i(s^t)}{M_i(s^t)} for i = 1, 2, I obtain x(s^t) = x(s^{t - 1}) \frac{M_1(s^{t - 1})}{M_2(s^{t - 1})} \frac{M_2(s^t)}{M_1(s^t)} = x(s^{t - 1}) \frac{(M_1(s^t) - \mu_1(s^t)) / M_1(s^t)}{(M_2(s^t) - \mu_2(s^t)) / M_2(s^t)} = x(s^{t - 1}) \frac{1 - \nu_1(s^t)}{1 - \nu_2(s^t)}.
By taking a derivative with respect to B(s^{t + 1}), I get the planner’s Euler equation as \begin{aligned} &\quad \gamma(s^t) \ge \beta (1 + r) \sum_{s^{t + 1}} Pr(s^{t + 1}) \gamma(s^{t + 1}) \\ &\Leftrightarrow u'(c_i(s^t)) \ge \beta (1 + r) \sum_{s^{t + 1}} Pr(s^{t + 1}) \frac{u'(c_i(s^{t + 1}))}{1 - \nu_i(s^{t + 1})}, \end{aligned} where equality holds when B(s^{t + 1}) > 0. These conditions determine the policy functions, and the individual value functions can be written with three state variables, (y, B, x), as follows: V_i(y, B, x) = u_i(c_i(y, B, x)) + \beta \sum_{y'} Pr(y') V_i(y', B', x').
The interval update rule found in Ligon, Thomas, and Worrall (2002) holds in this context too, but interval bounds depend on public storage. Specifically, by defining the interval bounds, \underline{x}(y, B) and \overline{x}(y, B), V_2(y, B, \underline{x}(y, B)) = U_2(y) and V_1(y, B, \overline{x}(y, B)) = U_1(y), the relative Pareto weight x is updated such that the policy function x'(y, B, x) is characterized as follows: x'(y, B, x) = \begin{cases} \overline{x}(y, B) \quad &\text{if } x > \overline{x}(y, B) \\ x \quad &\text{if } x \in [\underline{x}(y, B), \overline{x}(y, B)] \\ \underline{x}(y, B) \quad &\text{if } x < \underline{x}(y, B). \end{cases}
1.1.2 Value of autarky with storage
The expected lifetime utility of an agent i under autarky with storage, at a state y with private savings b, is V_i^{aut}(y, b) = \max_{b' \ge 0} \left\{ u(y_i + (1 + r) b - b') + \beta \sum_{y'} Pr(y') V_i^{aut} (y', b') \right\}. The intertemporal optimality condition, that is, the (individual) Euler equation, is u'(c(y, b)) \ge (1 + r) \beta \sum_j Pr(y') c(y', b'(y, b)), where the equality holds if \beta'(y, b) \ge 0. Note that b'(y, b) = y_i + (1 + r) b - c(y, b). I use the policy function iteration method to calculate c(y, b).
What we are interested in is U_i^{aut} \equiv V_i^{aut}(y, 0), that is, the autarky value when there is no private saving, since this is the state households consider in participation constraints.
1.2 Transition of the states
First I review the solutions (policy functions and value functions) of the model and see how the states evolve in the model. In this document, I define the relative Pareto weight, x, as x \equiv \frac{u'(c_1)}{u'(c_2)}, which implies that higher x is in favor of Household 2. For policy functions, I consider x'(y, B, x), a current relative Pareto weight, B'(y, B, x), a next-period storage, and (\underline{x}(y, B), \overline{x}(y, B)), a interval specific to a state-storage pair. Note that x'(y, B, x) = \begin{cases} \overline{x}(y, B) \quad &\text{if } x \ge \overline{x}(y, B) \\ x \quad &\text{if } x \in (\underline{x}(y, B), \overline{x}(y, B)) \\ \underline{x}(y, B) \quad &\text{if } x \le \underline{x}(y, B). \end{cases} Individual value functions are denoted as V_1(y, B, x) and V_2(y, B, x).
By the intratemporal optimality conditions \begin{aligned} c_1(y, B, x) &= \frac{y + (1 + r)B - B'(y, B, x)}{1 + x'(y, B, x)} \\ c_2(y, B, x) &= \frac{y + (1 + r)B - B'(y, B, x)}{1 + 1 / x'(y, B, x)}, \end{aligned} and \nu_1(y, B, x) = \max \left\{ 1 - \frac{x'(y, B, x)}{x}, 0 \right\}. Note that \nu_1 is not 0 when participation constraint of Household 1 binds.
For each (y, B), consider the following cases:
1.2.1 Case 1: x < \underline{x}(y, B)
By the update rule of the relative Pareto weight, x' = \underline{x}(y, B). Since the participation constraint of Household 2 is binding, u(c_2(y, B, x)) + \beta \sum_{y'} Pr(y') V_2(y', B'(y, B, x), \underline{x}) = U_2^{aut}(y) \quad \left( = V_2(y, B, x) \right), and the planner’s Euler equation is u'(c_1(y, B, x)) \ge \beta (1 + r) \sum_{y'} Pr(y') \frac{u'(c_1(y', B'(y, B, x), \underline{x}(y, B)))}{1 - \nu_1(y', B'(y, B, x), \underline{x}(y, B))}. Note that, in this case, B'(y, B, x) does not depend on x: if you look at the binding participation constraint of Household 2, since x is updated to \underline{x}(y, B), the consumption c_2 does not depend on x, and since the right-hand side of the equation does not depend on x, the value function term should not depend on x either. Or, an alternative way to argue this is to refer to Lemma 1 of Ábrahám and Laczó (2018) stating that the current x is a sufficient statistic for B', given B (and actually y too because I consider a more general case that the aggregate income can differ across states).
The value function of Household 1 is V_1(y, B, x) = u(c_1(y, B, x)) + \beta \sum_{y'} Pr(y') V_1(y', B'(y, B, x), \underline{x}(y, B)). Notice that this also does not depend on x since c_1 and B' do not depend on x.
1.2.2 Case 2: x = \underline{x}(y, B)
By the update rule of the relative Pareto weight, x' = \underline{x}(y, B). Since the participation constraint of Household 2 is binding, u(c_2(y, B, \underline{x}(y, B))) + \beta \sum_{y'} Pr(y') V_2(y', B'(y, B, \underline{x}(y, B)), \underline{x}) = U_2^{aut}(y) \quad \left( = V_2(y, B, \underline{x}(y, B)) \right), and the planner’s Euler equation is u'(c_1(y, B, \underline{x}(y, B))) \ge \beta (1 + r) \sum_{y'} Pr(y') \frac{u'(c_1(y', B'(y, B, \underline{x}(y, B)), \underline{x}(y, B)))}{1 - \nu_1(y', B'(y, B, \underline{x}(y, B)), \underline{x}(y, B))}. The value function of Household 1 is V_1(y, B, \underline{x}(y, B)) = u(c_1(y, B, \underline{x}(y, B))) + \beta \sum_{y'} Pr(y') V_1(y', B'(y, B, \underline{x}(y, B)), \underline{x}(y, B)).
1.2.3 Case 3: x \in (\underline{x}(y, B), \overline{x}(y, B)
Since no participation constraint is binding in this case, x' = x. Therefore, the value functions of the households are \begin{aligned} V_1(y, B, x) &= u(c_1(y, B, x)) + \beta \sum_{y'} Pr(y') V_1(y', B'(y, B, x), x) \\ V_2(y, B, x) &= u(c_2(y, B, x)) + \beta \sum_{y'} Pr(y') V_2(y', B'(y, B, x), x). \end{aligned} And the planner’s Euler equation needs to be satisfied: u'(c_1(y, B, x)) \ge \beta (1 + r) \sum_{y'} Pr(y') \frac{u'(c_1(y', B'(y, B, x), x))}{1 - \nu_1(y', B'(y, B, x), x)}.
1.2.4 Case 4: x = \overline{x}(y, B)
By the update rule of the relative Pareto weight, x' = \overline{x}(y, B). Since the participation constraint of Household 1 is binding, u(c_1(y, B, \overline{x}(y, B))) + \beta \sum_{y'} Pr(y') V_1(y', B'(y, B, \overline{x}(y, B)), \overline{x}) = U_1^{aut}(y) \quad \left( = V_1(y, B, \overline{x}(y, B)) \right), and the planner’s Euler equation is u'(c_1(y, B, \overline{x}(y, B))) \ge \beta (1 + r) \sum_{y'} Pr(y') \frac{u'(c_1(y', B'(y, B, \overline{x}(y, B)), \overline{x}(y, B)))}{1 - \nu_1(y', B'(y, B, \overline{x}(y, B)), \overline{x}(y, B))}. The value function of Household 2 is V_2(y, B, \overline{x}(y, B)) = u(c_2(y, B, \overline{x}(y, B))) + \beta \sum_{y'} Pr(y') V_2(y', B'(y, B, \overline{x}(y, B)), \overline{x}(y, B)).
1.2.5 Case 5: x < \overline{x}(y, B)
By the update rule of the relative Pareto weight, x' = \overline{x}(y, B). Since the participation constraint of Household 1 is binding, u(c_1(y, B, x)) + \beta \sum_{y'} Pr(y') V_1(y', B'(y, B, x), \overline{x}) = U_1^{aut}(y) \quad \left( = V_1(y, B, x) \right), and the planner’s Euler equation is u'(c_1(y, B, x)) \ge \beta (1 + r) \sum_{y'} Pr(y') \frac{u'(c_1(y', B'(y, B, x), \overline{x}(y, B)))}{1 - \nu_1(y', B'(y, B, x), \overline{x}(y, B))}. The value function of Household 2 is V_2(y, B, x) = u(c_2(y, B, x)) + \beta \sum_{y'} Pr(y') V_2(y', B'(y, B, x), \overline{x}(y, B)). As in Case 1, B', V_1, and V_2 do not depend on x.