Insurance and Inequality With Persistent Private Information: Bloedel, Krishna & Leukhina (2025)

What this is. The paper’s core results, the recursive model, and the marginal cost martingale method with the defining equations: enough to know what was proved and how, without reading all 37 pages. To replicate or extend, read the original at https://doi.org/10.3982/ECTA20404.

TL;DR

The paper studies long-run welfare and inequality in optimal insurance contracts when the agent’s privately observed type follows an ergodic finite-state Markov chain, filling the gap between the i.i.d. benchmark of Thomas and Worrall (1990) and the permanent-shock benchmark of Williams (2011). A risk-neutral principal offers an infinite-horizon insurance contract to a risk-averse agent whose privately observed endowment evolves with arbitrary serial correlation bounded between these extremes.

Theorem 1 (the central result) shows that immiseration is universal under ergodic persistence: the agent’s promised utilities, flow utilities, and consumption all converge to their lower bounds in probability. Theorem 2 strengthens this under positive serial correlation (FOSD): the spread in continuation utility across types and the conditional variance of promised utility both diverge to infinity, reflecting “backloaded high-powered incentives.” The key insight is that ergodicity (mean-reversion) is the critical determinant: Williams (2011) shows bliss only at the knife-edge of zero mean-reversion (permanent shocks); any positive amount of mean-reversion restores immiseration.

The proofs construct a marginal cost martingale: a specific directional derivative of the principal’s value function that is a strictly positive martingale under the optimal contract. The Martingale Convergence Theorem combined with a “renewal property” of the Markov process shows this martingale converges to zero, implying immiseration. Numerical simulations with CARA utility and two endowment types show that greater persistence accelerates immiseration in the medium run, generates over-insurance (negative insurance wedge) after consecutive low shocks, and introduces order-dependence absent in the i.i.d. CARA case.

Core results

#	Result	Locator	Magnitude
R1	Theorem 1 (Immiseration): under any TVC-Regular environment with ergodic Markov types, the optimal contract generates immiseration	Theorem 1, p. 834	$v_i^{(t)} \to -\infty$ in probability for all $i \in S$; $u^{(t)} \to -\infty$ in probability; $c^{(t)} + \omega^{(t)} \to \underline{c}$ in probability; no stationary distribution exists
R2	Theorem 2 (Backloaded Incentives): under FOSD, the optimal contract exhibits growing cross-type utility spreads	Theorem 2, p. 836	$v_i^{(t)} - v_{i-1}^{(t)} \to +\infty$ in probability for all $i \geq 2$; conditional variance $\mathbf{V}(v_{s^{(t+1)}}^{(t)} \mid \mathbf{v}^{(t)}, s^{(t)}) \to +\infty$ in probability
R3	Theorem 3 (Recursive Domain): under MLRP or PPR type process, the implementable domain is characterized explicitly	Theorem 3, Appendix B, p. 853	$D = V_d = \{\mathbf{v} \in \mathcal{U}^d : v_d > v_{d-1} > \cdots > v_1\}$ under MLRP or PPR; under CARA + MLRP/PPR, $D = D^* = V_d$; an open, convex cone independent of $\alpha$ and $U$ (within DARA class)
R4	Numerical (§5.1): greater persistence yields faster immiseration in medium-run	Figure 3, p. 842	420,000 simulated paths of CARA/$d=2$ model: for $q = 0.8$ vs i.i.d. ($q = 0.5$), mean consumption $\mu_{C,t}$ declines faster and variance $\sigma^2_{C,t}$ grows faster in medium-run; patterns reverse in the first few periods
R5	Numerical (§5.2): persistence induces over-insurance and large intertemporal wedges after consecutive low shocks	Figure 5, p. 845	Insurance wedge turns negative (over-insurance) after strings of low shocks; intertemporal wedge grows to orders of magnitude larger than i.i.d. case; consumption depends on shock order (early bad luck penalized more than late bad luck)

Overall (paper’s conclusion). Immiseration is not an artifact of the i.i.d. assumption but is universal under ergodic persistence. The key determinant of long-run outcomes is mean-reversion in the type process. Any positive amount of mean-reversion is sufficient to generate immiseration, while the bliss result of Williams (2011) arises only at the knife-edge of zero mean-reversion. Persistence does affect the speed of immiseration and generates qualitatively new short-run distortions: over-insurance, large intertemporal wedges, and order-dependent consumption that are absent in the i.i.d. case.

Theory / model

The environment is a discrete-time infinite-horizon insurance model (§2, p. 827). A risk-neutral principal with discount factor $\alpha \in (0,1)$ offers an insurance contract to a risk-averse agent (same discount factor). The agent’s Bernoulli utility is $U: (\underline{c}, \infty) \to \mathbb{R}$ satisfying Assumption DARA (p. 827): strictly increasing, strictly concave, satisfying Inada conditions $\lim_{c \to \underline{c}} U'(c) = +\infty$ and $\lim_{c \to \infty} U'(c) = 0$, bounded above and unbounded below ($\mathcal{U} = (-\infty, 0)$), and with decreasing absolute risk aversion. Standard CARA and HARA utilities satisfy DARA.

The agent’s type $\omega^{(t)} \in S := \{\omega_1, \dots, \omega_d\}$ ($\omega_d > \cdots > \omega_1$) evolves as a fully connected, time-homogeneous, first-order Markov chain (Assumption Markov, p. 828) with transition matrix $\mathbf{F} = (f_{ij})$, where $f_{ij} = \mathbf{P}(\omega^{(t+1)} = \omega_j \mid \omega^{(t)} = \omega_i) > 0$ for all $i, j$. This ensures the process is ergodic and bounded, allowing arbitrary serial correlation. Assumption NHB (p. 828) restricts the agent to under-reporting: he cannot report a type higher than his true type, capturing the idea that endowments are partially verifiable.

The principal’s goal is to minimize lifetime cost of delivering a vector of promised utilities $\mathbf{v}^{(0)} = (v_1, \dots, v_d) \in \mathcal{U}^d$ subject to promise keeping and incentive compatibility.

Recursive formulation (§3, p. 829). Following Green (1987) and Fernandes and Phelan (2000), the state variable for the recursive problem is $(\mathbf{v}, s) \in D \times S$, where $\mathbf{v} = (v_1, \dots, v_d) \in \mathcal{U}^d$ is the vector of interim promised utilities (the agent’s continuation utility $v_i$ conditional on his true current type being $i$, assuming truthful reporting in all future periods) and $s \in S$ is the agent’s report in the previous period.

At state $(\mathbf{v}, s)$, the contract offers a menu $(u_i, \mathbf{w}_i)_{i \in S}$ of flow utility $u_i$ and continuation utility vector $\mathbf{w}_i$. The recursive constraints (p. 830) are:

$$v_i = u_i + \alpha \mathbf{E}^{\mathbf{f}_i}[\mathbf{w}_i], \qquad \forall i \in S \tag{PK$_i$}$$ $$v_i \geq \psi(u_j, i, j) + \alpha \mathbf{E}^{\mathbf{f}_i}[\mathbf{w}_j], \qquad \forall i > j \in S \tag{IC$_{ij}$}$$

where $\mathbf{E}^{\mathbf{f}_i}[\mathbf{w}_j] := \sum_{k=1}^d f_{ik} w_{jk}$ is the expected continuation utility that a type-$i$ agent obtains by reporting $j$, and $\psi(u_j, i, j) := U(\omega_i + C(u_j, j))$ is the flow utility a true type-$i$ agent receives when the contract delivers consumption $C(u_j, j) := U^{-1}(u_j) - \omega_j$.

Substituting (PK$_j$) into (IC$_{ij}$) yields the combined incentive constraint (p. 836; specific d=2 case at p. 830):

$$v_i - v_j \geq \underbrace{\psi(u_j, i, j) - u_j}_{\text{i.i.d. info rent}} + \alpha\underbrace{\bigl[\mathbf{E}^{\mathbf{f}_i}[\mathbf{w}_j] - \mathbf{E}^{\mathbf{f}_j}[\mathbf{w}_j]\bigr]}_{\text{Markov info rent}} \tag{IC$_{ij}^*$}$$

The Markov information rent term is new relative to the i.i.d. case: it arises because the agent’s current type also determines his beliefs about future types, giving him intertemporal preferences over continuation contracts.

The principal’s recursive problem (p. 831) is to minimize expected discounted cost:

$$P(\mathbf{v}, s) := \inf_{\xi \in \Xi} \mathbf{E}\!\left[\sum_{t=0}^\infty \alpha^t C\!\left(u_\xi^{(t)}, s^{(t+1)}\right)\bigg|\,(\mathbf{v}^{(0)}, s^{(0)}) = (\mathbf{v}, s)\right] \tag{RP}$$

where $C(u, j) := U^{-1}(u) - \omega_j$ is the consumption cost to the principal and $\Xi$ is the set of feasible recursive contracts. The Bellman equation characterizing $P$ is (Proposition 3.2, p. 834):

$$P(\mathbf{v}, s) = \min_{(u_i, \mathbf{w}_i)_{i \in S} \in \Gamma(\mathbf{v})} \sum_{i \in S} f_{si}\bigl[C(u_i, i) + \alpha P(\mathbf{w}_i, i)\bigr] \tag{FE}$$

where $\Gamma(\mathbf{v})$ is the constraint correspondence of all menus satisfying (PK$_i$)-(IC$_{ij}$) with $\mathbf{w}_i \in D$ for all $i$. Under (TVC)-Regularity, $P(\cdot, s)$ is convex, and there exists a unique optimal contract $\xi^*$ that is continuous on $D \times S$ (Proposition 3.2(b)).

The conditional variance of continuation utility, used in Theorem 2, is defined at (4.1, p. 835):

$$\mathbf{V}\!\left(v_{s^{(t+1)}}^{(t)} \;\Big|\; \mathbf{v}^{(t)}, s^{(t)}\right) := \sum_{i=1}^d f_{s^{(t)},i} \left(v_i^{(t)} - \sum_{k=1}^d f_{s^{(t)},k}\, v_k^{(t)}\right)^2$$

Method

The core methodology is the marginal cost martingale (§4.3, pp. 837-840). It builds on the mechanism-design framework and value-function-iteration (Bellman equation) ideas, extending the martingale approach of Thomas and Worrall (1990) for i.i.d. types to the general Markovian setting via the Fernandes and Phelan (2000) recursive formulation. The paper’s stated primary methodological contributions are (i) the recursive formulation using interim promised utilities (extending Fernandes and Phelan (2000)) and (ii) the marginal cost martingale for analyzing long-run outcomes.

Let $DP(\mathbf{v}, s) = (P_1(\mathbf{v}, s), \dots, P_d(\mathbf{v}, s))$ denote the gradient of $P$ with respect to $\mathbf{v}$. The directional derivative in direction $\mathbf{1} = (1, \dots, 1) \in \mathbb{R}^d$ is:

$$D_{\mathbf{1}} P(\mathbf{v}, s) := \sum_{i \in S} P_i(\mathbf{v}, s)$$

This direction is unique in that increasing $\mathbf{v}$ along $\mathbf{1}$ raises every type’s continuation utility by the same amount $\varepsilon$, leaving all downward incentive constraints (IC$_{ij}^*$) unchanged (because the left-hand side $v_i - v_j$ is unaffected). Consequently, $D_{\mathbf{1}} P$ captures the marginal cost of increasing the agent’s ex ante promised utility without distorting his information rents.

Proposition 4.4 (p. 837): Under (TVC)-Regularity, the process $(D_{\mathbf{1}} P(\mathbf{v}^{(t)}, s^{(t)}))_{t=0}^\infty$ induced by the optimal contract is a strictly positive martingale.

The martingale property follows from an envelope argument on (FE): at the optimum,

$$D_{\mathbf{1}} P(\mathbf{v}, s) = \sum_{i=1}^d f_{si}\, D_{\mathbf{1}} P(\mathbf{w}_i, i)$$

which is precisely the martingale condition $\mathbf{E}[D_{\mathbf{1}} P(\mathbf{v}^{(t+1)}, s^{(t+1)}) \mid \mathbf{v}^{(t)}, s^{(t)}] = D_{\mathbf{1}} P(\mathbf{v}^{(t)}, s^{(t)})$. Strict positivity holds because the cost function $C(\cdot, j)$ is convex and the cost-smoothing motive always pushes $$D_{\mathbf{1}} P > 0$.

Proof sketch for Theorem 1 (§4.3):

Step 1 (Marginal cost martingale, p. 837). By Proposition 4.4, $D_{\mathbf{1}} P(\mathbf{v}^{(t)}, s^{(t)})$ is a strictly positive martingale. By the Martingale Convergence Theorem, it converges a.s. to some non-negative limit $Z \geq 0$.

Step 2 (Convergence to zero, p. 839). The key step is showing $Z = 0$ in probability. Assumption Markov implies the agent’s highest-type realization $\omega_d$ occurs infinitely often along any sample path. At such “renewal” histories the optimal contract is efficient (renegotiation-proof): the principal does not need to screen through continuation contracts, so the marginal cost martingale splits like in the i.i.d. case. At these histories, if the martingale were to converge to a strictly positive number, then $\mathbf{v}^{(t)}$ would converge to some interior point of $D$, implying the optimal contract perfectly stabilizes consumption, which contradicts incentive compatibility (Lemma C.18). Thus the martingale must converge to zero at renewal histories, and the Markov ergodicity extends this to all histories.

Step 3 (Convergence of allocations, p. 840). $D_{\mathbf{1}} P(\mathbf{v}^{(t)}, s^{(t)}) \to 0$ implies that the Lagrange multipliers on the incentive constraints converge to zero. This in turn implies that the agent’s consumption converges to the level that the first-best contract would deliver if cost were zero, which is $\underline{c}$.

For Theorem 2 (backloaded incentives), the argument uses Theorem 1(b) (flow utility $u^{(t)} \to -\infty$) combined with the incentive constraint (IC$_{ij}^*$): for FOSD type processes, the Markov information rent (the second bracketed term) is non-negative (Theorem 3 in Appendix B guarantees $\mathbf{E}^{\mathbf{f}_i}[\mathbf{w}_i] \geq \mathbf{E}^{\mathbf{f}_j}[\mathbf{w}_i]$ whenever $i > j$). Since the i.i.d. information rent grows without bound (from Theorem 1(b)) and the Markov rent is non-negative, the spread $v_i^{(t)} - v_{i-1}^{(t)}$ must also grow without bound.

Empirical specifications

Section 5 presents numerical simulations for the CARA / binary-type ($d = 2$) / symmetric-transitions special case, using (p. 841):

$$U(c + \omega) = -e^{-(c + \omega)}, \quad \omega_1 = -\log 5, \quad \omega_2 = \log 10, \quad \alpha = 0.5$$

with symmetric transition matrix $f_{11} = f_{22} = q$. Three persistence levels plus a very high case are studied: $q \in \{0.5 \text{ (i.i.d.)}, 0.65 \text{ (low)}, 0.8 \text{ (high)}, 0.95 \text{ (very high)}\}$. Under CARA utility and FOSD ($q \geq 0.5$), Theorem 3 gives $D = V_2 = \{(v_1, v_2) : v_2 > v_1\}$, so optimal contracts are homogeneous of degree 1 in $\mathbf{v}$ (property HD1, p. 843).

Speed of immiseration (§5.1, Figure 3, p. 842). For each $q$, 420,000 sample paths of the optimal contract are simulated for 40 time periods, with 21 initial $\mathbf{v}^{(0)}$ points drawn from a grid on $D = V_2$. Mean consumption $\mu_{C,t} := \mathbf{E}[c^{(t)} + \omega^{(t)}]$ and variance $\sigma^2_{C,t} := \mathbf{V}[c^{(t)} + \omega^{(t)}]$ serve as proxies for open-economy aggregate consumption and cross-sectional inequality. Key patterns:

1. $\mu_{C,t} \to -\infty$ and $\sigma^2_{C,t} \to +\infty$ at all persistence levels (Theorem 1).
2. Medium-run: greater $q$ yields faster decline in $\mu_{C,t}$ and faster growth in $\sigma^2_{C,t}$.
3. Short-run (first few periods): greater $q$ initially slows the decline of $\mu_{C,t}$ and the growth of $\sigma^2_{C,t}$.

Short-run distortions (§5.2, Figure 5, p. 845). The HD1 property implies the contract’s dynamics trace a countable set of rays in $V_2$: ray $E_2$ (continuation state after a high shock) and rays $\{B_k\}_{k \geq 1}$ (after $k$ consecutive low shocks, $B_k$ strictly below $B_{k-1}$). Along sequences of consecutive high and low shocks starting from $E_2$ (Figure 5):

• Insurance wedge $\equiv U'(c^{(t)} + \omega_1)/U'(c^{(t)} + \omega_2) - 1$: always positive in the i.i.d. case (under-insurance, consistent with Thomas and Worrall (1990)). Under persistence, it remains positive after high shocks but turns negative (over-insurance, $u_1 > u_2$) after consecutive low shocks.
• Intertemporal wedge $\equiv \mathbf{E}[U'(c^{(t+1)} + \omega^{(t+1)}) \mid \omega^{(t)}]/U'(c^{(t)} + \omega^{(t)}) - 1$: always positive in i.i.d. case (consumption drift downward). Under persistence, becomes orders of magnitude larger after consecutive low shocks.
• Order-dependence: unlike the i.i.d. CARA case (where Atkeson and Lucas (1992) show order-independence), consumption under persistence depends on the sequence of shocks, not just their frequency; early bad luck is penalized more than late bad luck.

Datasets used

No empirical datasets are used. All results are analytical (Theorems 1-3) or based on numerical simulations of the theoretical model. The replication code is publicly available.

Dataset	Role in paper	Wiki page
Synthetic model simulation (CARA utility, d=2 types, symmetric Markov, q in {0.5, 0.65, 0.8, 0.95})	Numerical illustrations of speed of immiseration and short-run distortions (Figures 3-5, §5)	No page (theoretical model; no external data source)

Sample: 420,000 paths per persistence level, 40 time periods, 21 initial state points (§5.1). Replication code: https://doi.org/10.5281/zenodo.14720557.

When to read the full paper

Read the original at https://doi.org/10.3982/ECTA20404 if you:

• need the formal proofs of Theorems 1, 2, or 3 (in Supplemental Appendices C, E, and I-J of the companion working paper Bloedel, Krishna, and Leukhina (2025b));
• are extending the recursive formulation to other environments (Appendix A covers equivalence between sequential and recursive contracts);
• want the closed-economy (Atkeson and Lucas (1992)) extension or the Zhang (2009) / Williams (2011) comparison (Section 6);
• are studying the general DARA + Markov setting beyond the CARA/d=2 numerical illustrations.

Core locators: Theorem 1 (immiseration, p. 834), Theorem 2 (backloaded incentives, p. 836), Theorem 3 / Appendix B (recursive domain, p. 853), Figure 3 (speed of immiseration, p. 842), Figure 5 (short-run wedges, p. 845).

Attribution and rights

Source: peer-reviewed, Econometrica 93(3), May 2025. This distillation was extracted by an LLM on 2026-06-26 and is not human-verified or independently reproduced. The article is paywalled; no CC licence was found in Crossref metadata.

Bloedel, Alexander W., R. Vijay Krishna, and Oksana Leukhina. “Insurance and Inequality With Persistent Private Information.” Econometrica 93, no. 3 (May 2025): 821-857. DOI: 10.3982/ECTA20404. © 2025 The Econometric Society. All rights reserved. This page extracts core results only and does not reproduce the full text.